Density-Based Model of Bending Strength for AGR Graphites Irradiated in Oxidising Environments Ernie...
24
Density-Based Model of Bending Strength for AGR Graphites Irradiated in Oxidising Environments Ernie D. Eason Modeling & Computing Services Boulder, Colorado, USA [email protected] Graham Hall Barry J. Marsden Nuclear Graphite Research Group, School of Mechanical, Aerospace & Civil Engineering, University of Manchester, UK [email protected] [email protected] Presented at INGSM-14 Seattle, Washington, USA September 18, 2013
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Transcript of Density-Based Model of Bending Strength for AGR Graphites Irradiated in Oxidising Environments Ernie...
Density-Based Model of Bending Strength for AGR Graphites Irradiated in Oxidising
Environments
Ernie D. EasonModeling & Computing Services
Boulder, Colorado, [email protected]
Graham Hall Barry J. Marsden
Nuclear Graphite Research Group, School of Mechanical, Aerospace & Civil Engineering, University of Manchester, UK
[email protected]@manchester.ac.uk
Presented at INGSM-14 Seattle, Washington, USA
September 18, 2013
2
Data Used for Density-Based Strength Models
• Trepanned data 1998 - 2010 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/S0
3
Trepanned Bending Strength Irradiated Density Model
• Model form
• The coefficients Ci and exponents Ni vary by reactor
• The Tirr term is a small correction (+3%, -1%)
• Average Tirr = 402.75C over all trepanned data
158.0
75.402
irr
Nirri T
CS i
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. 4.09 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
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
Corrected Weight Loss, %
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
HPB R3 2000 Model
HPB R3 2000 Data
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
Corrected Weight Loss, %
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
HPB R3 2003 Model
HPB R3 2003 Data
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40
Corrected Weight Loss, %
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
HPB R3 2006 Model
HPB R3 2006 Data
24 MPa 2000
26 MPa 2003
29 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
0
10
20
30
40
50
60
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm33
Po
int
Ben
din
g S
tren
gth
S,
MP
a Average BTC Data 2000 & Before
Average WNL Data 1998 - 2004
Average WNL Data 2005 & Later
0
10
20
30
40
50
60
0 5 10 15 20 25 30 35 40
Corrected Weight Loss, %
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
Average BTC Data 2000 & Before
Average WNL Data 1998 - 2004
Average WNL Data 2005 & Later
Preliminary Weight Loss Model Preliminary 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
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G1 DNB Model
G1 DNB Calibration Data
G1 DNB Chauvenet Outlier
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G2 HPB R3 Model
G2 HPB R4 ModelG2 HPB R3 Calibration DataG2 HPB R4 Calibration DataChauvenet Outliers
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G3 HNB R3 ModelG3 HNB R4 ModelG3 HNB R3 Calibration DataG3 HNB R4 Calibration DataChauvenet Outliers
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G5 HYA R1 Model
G5 HYA R2 Model
G5 HYA R1 Calibration Data
G5 HYA R2 Calibration Data
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
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G8 HRA R1 Model
G8 HRA R2 Model
G8 HRA R1 Calibration Data
G8 HRA R2 Calibration Data
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G9 TOR R1 ModelG9 TOR R2 ModelG9 TOR R1 Calibration DataG9 TOR R2 Calibration DataG9 TOR R2 Chauvenet Outlier
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G9 HYB R7 Model
G9 HYB R8 Model
G9 HYB R7 Calibration DataG9 HYB R8 Calibration Data
0.90
0.95
1.00
1.05
1.10
300 350 400 450
Irradiation Temperature, T irr, °C
Tem
per
atu
re T
erm
Trepanned Data Average = 402.75°C
+3%, -1% Over the Range of Data
11
Density-Based Strength Model Quality of Fit
• Standard Error Se = 0.05026 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
0.8
1.0
1.2
1.4
1.6
1.8
0.8 1.0 1.2 1.4 1.6 1.8
Model Value of log(S), S in MPa
Act
ual
Val
ue
of
log
(S),
S i
n M
Pa
5th percentile95th percentile90% Calibration Set, no outliers
0.8
1.0
1.2
1.4
1.6
1.8
0.8 1.0 1.2 1.4 1.6 1.8
Model Value of log(S), S in MPa
Act
ual
Val
ue
of
log
(S),
S i
n M
Pa
5th percentile95th percentile10% Validation set
Calibration Data Validation Data
The Validation Data Fit The Model as Well as the Calibration Data
13
Flat Residual Plots
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
300 350 400 450
Irradiation Temperature, Tirr, °C
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
Graphite Group/Reactor
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data G9 HYB Data
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, irr, g/cm3
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 20 40 60 80 100 120 140 160
Dose, 1020 n/cm2 EDND
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
14
Flat Residual Plots, continued
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
1.78 1.80 1.82 1.84
Virgin Density, 0, g/cm3
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB DataG3 HNB Data G5 HYA DataG8 HRA Data G9 TOR DataG9 HYB Data Residual Trend (N.S.)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10
Height in Pile, m
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB DataG3 HNB Data G5 HYA DataG8 HRA Data G9 TOR DataG9 HYB Data Residual Trend (N.S.)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
1996 2000 2004 2008 2012
Trepanning Year
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 5 10 15 20 25 30 35 40
Corrected Weight Loss, %
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB DataG3 HNB Data G5 HYA DataG8 HRA Data G9 TOR DataG9 HYB Data Residual Trend (N.S.)
15
The “Year Effect” for Weight Loss versus Density-based Models – Residual Plots
-20
-10
0
10
20
1996 2000 2004 2008 2012
Trepanning Year
Mo
del
S -
Actu
al
S,
MP
a
G1 DNB Data G2 HPB DataG3 HNB Data G5 HYA DataG8 HRA Data G9 TOR DataG9 HYB Data Residual Trend (Significant)
-20
-10
0
10
20
1996 2000 2004 2008 2012
Trepanning Year M
od
el
S -
Actu
al
S,
MP
a
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
Recommended Density-Based ModelPreliminary Weight Loss Model
-20
-10
0
10
20
1996 2000 2004 2008 2012
Trepanning Year
Mo
del
S -
Actu
al
S,
MP
a
G1 DNB Data G2 HPB DataG3 HNB Data G5 HYA DataG8 HRA Data G9 TOR DataG9 HYB Data Residual Trend (Significant)
-20
-10
0
10
20
1996 2000 2004 2008 2012
Trepanning Year M
od
el
S -
Actu
al
S,
MP
a
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
Recommended Density-Based ModelPreliminary Weight Loss Model
Significant Residual Trend No Significant Residual Trend
16
Flat Residual Plot for Inert-Irradiated Young’s Modulus Ratio
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
1 2 3 4
Inert Young's Modulus Ratio, E/E0, Dimensionless
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
)
G1 DNB Data G2 HPB Data
G3 HNB Data G5 HYA Data
G8 HRA Data G9 TOR Data
G9 HYB Data Residual Trend (N.S.)
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
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0 20 40 60 80 100 120 140 160
Dose, 1020 n/cm2 EDND
Mo
del
lo
g(S
) -
Act
ual
lo
g(S
) G2 HPB Data
G3 HNB Data
Residual Trend (Significant) Unconservative (actual S < model S)
Conservative (actual S > model S)
18
Newer Data, Received After CalibrationDensity-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 x10-11
(~0)-9.4 x10-4
(~0)
Residual Sd as log(S)
0.05003 0.04722
Residual Sd as S (MPa)
3.961 3.826
Number of Points
1831 378
Differences Not Significant
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60 70
Model Value of S, MPa
Act
ual
Val
ue
of
S,
MP
a
5th Percentile Calibration Data95th Percentile Calibration Data5 Reactors 2011 & 2012 Data
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 SA
22
Density-Based Strength Model Compared with Installed Sample Data Not Used for
Fitting (solid black symbols)
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G3 HNB R3 ModelG3 HNB R4 ModelG3 HNB R3 Calibration DataG3 HNB R4 Calibration DataChauvenet OutliersG3 Installed Samples
HNB
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G8 HRA R1 ModelG8 HRA R2 ModelG8 HRA R1 Calibration DataG8 HRA R2 Calibration DataG8 Installed Samples
HRA
23
Density-Based Strength Model and BTC Data Not Used for Fitting
The BTC data average 1 MPa below the WNL data and model
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G1 DNB Model
G1 DNB BTC Data
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G2 HPB R3 Model
G2 HPB R4 Model
G2 HPB R3 BTC Data
G2 HPB R4 BTC Data
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa G5 HYA R1 Model
G5 HYA R1 BTC Data
0
10
20
30
40
50
60
70
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Irradiated Density, g/cm3
3 P
oin
t B
end
ing
Str
eng
th S
, M
Pa
G8 HRA R2 Model
G8 HRA R2 BTC Data
24
Density-Based Strength Model & BFB and DIDO Annealed Data Not Used for Fitting
(Coefficient recalibrated for annealed tensile strength)
0
5
10
15
20
25
30
1.0 1.2 1.4 1.6 1.8 2.0
Irradiated Density, g/cm3
Ten
sile
Str
eng
th S
, M
Pa
AGL (HNB) ModelDIDO AGL DataBFB AGL DataBFB Chauvenet Outliers
0
5
10
15
20
25
30
1.0 1.2 1.4 1.6 1.8 2.0
Irradiated Density, g/cm3
Ten
sile
Str
eng
th S
, M
Pa
BAEL (HRA) Model
DIDO BAEL Data
BFB BAEL Data
0
5
10
15
20
25
30
1.0 1.2 1.4 1.6 1.8 2.0
Irradiated Density, g/cm3
Ten
sil
e S
tren
gth
S,
MP
a
UCAR (TOR) Model
BFB UCAR Data
BFB Chauvenet Outliers
