Non-Destructive Testing of the Graphite Core within an Advanced Gas-Cooled Reactor

283 Pages·2014·23.98 MB·English

Non-Destructive Testing of the
Graphite Core within an
Advanced Gas-Cooled Reactor
A thesis submitted to The University of Manchester for the degree of
Doctor of Engineering
in the Faculty of Engineering and Physical Sciences
2014
Adam David Fletcher
School of Electrical & Electronic Engineering
Contents
Contents 2
List of Figures 5
List of Tables 9
List of Abbreviations 10
Abstract 13
Declaration 14
Copyright Statement 15
Acknowledgements 16
1 Introduction 17
1.1 The Engineering Doctorate . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Industrial Background . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 The UK Energy Markets . . . . . . . . . . . . . . . . . . . 19
1.2.2 EDF Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3 Aims and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Economic . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 Regulatory . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.4 Next Generation Nuclear Plant . . . . . . . . . . . . . . . 32
1.5 Achievements and Publications . . . . . . . . . . . . . . . . . . . 33
1.6 Organisation of the Thesis . . . . . . . . . . . . . . . . . . . . . . 35
2 Nuclear Graphite and the AGR 37
2.1 AGR Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2
CONTENTS CONTENTS
2.1.1 Reactor Core . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Nuclear Graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2.1 Structure and Properties . . . . . . . . . . . . . . . . . . . 40
2.2.2 Production . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.3 Gilsocarbon Graphite . . . . . . . . . . . . . . . . . . . . . 43
2.3 Mechanisms of Graphite Degradation . . . . . . . . . . . . . . . . 43
2.3.1 Neutron Bombardment . . . . . . . . . . . . . . . . . . . . 44
2.3.2 Radiolytic Oxidation . . . . . . . . . . . . . . . . . . . . . 46
2.4 AGR Brick Failure . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Core Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 Channel Bore Measuring/Inspection Unit . . . . . . . . . . 52
2.5.2 Trepanning Tool Unit . . . . . . . . . . . . . . . . . . . . . 54
2.5.3 NICIE Mark I and II . . . . . . . . . . . . . . . . . . . . . 55
2.5.4 PoPECT and PECIT . . . . . . . . . . . . . . . . . . . . . 55
3 Review of MIT and Inductance Spectroscopy 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Non-Locality . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Non-Linearity . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.3 Ill-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.3 Forward Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Definition of the Forward Problem . . . . . . . . . . . . . 68
3.3.2 Solution of the Forward Problem . . . . . . . . . . . . . . 74
3.4 Calibration of the Forward Model . . . . . . . . . . . . . . . . . . 75
3.5 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5.1 Calculating the Jacobian . . . . . . . . . . . . . . . . . . . 81
3.5.2 Formulations of the Jacobian . . . . . . . . . . . . . . . . 88
4 Inpsection of AGR Brick Cracking 99
4.1 Historic Development . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . 115
4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5 Crack Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Inverse Problems and Regularisation Methods 140
5.1 Inversion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Gradient Descent and Newton’s Method . . . . . . . . . . 146
5.1.2 Gauss-Newton Method . . . . . . . . . . . . . . . . . . . . 149
5.2 Review of Regularisation . . . . . . . . . . . . . . . . . . . . . . . 154
3
CONTENTS
5.2.1 Theory of Regularisation . . . . . . . . . . . . . . . . . . . 156
5.2.2 Tikhonov Regularisation . . . . . . . . . . . . . . . . . . . 161
5.2.3 Parameter Choice and the L-Curve Method . . . . . . . . 164
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 Conductivity Profiling using Impedance Spectroscopy 174
6.1 Flat Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 Radial Hole Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.2.1 Cylindrical Test Cases from Simulated Data . . . . . . . . 196
6.2.2 Lattice Hole Drilling as an Analogue for Weight-Loss . . . 200
6.2.3 Inversion of Drilled Brick Test Problem . . . . . . . . . . . 205
6.3 Experimental L-Curve . . . . . . . . . . . . . . . . . . . . . . . . 221
6.4 AGR Core Measurements . . . . . . . . . . . . . . . . . . . . . . 226
6.4.1 Model without Methane Holes . . . . . . . . . . . . . . . . 228
6.4.2 Model with Methane Holes . . . . . . . . . . . . . . . . . . 233
7 Conclusions and Future Work 238
7.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8 References 250
A Analytical Solution to the Axi-Symmetric Problem 260
A.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 260
A.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 262
A.2.1 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . 264
A.3 Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
A.4 Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 267
A.4.1 Fields due to coils of finite cross section . . . . . . . . . . . 269
A.4.2 Induced Voltage . . . . . . . . . . . . . . . . . . . . . . . . 270
A.4.3 Coil Impedance . . . . . . . . . . . . . . . . . . . . . . . . 272
B Description of the Finite Element Model 273
B.1 Geometry Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
B.2 Physics Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
B.3 Meshing Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
B.4 Study Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
C Regularised Gauss-Newton Method 281
4
List of Figures
1.1 An overview of the AGR core safety assessment methodology. . . 27
2.1 The graphite core of two Advanced Gas-Cooled Reactor (AGR)
designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Brick geometry used in this work of HRA/HYA type. . . . . . . . 40
2.3 Graphite dimensional change with dose. . . . . . . . . . . . . . . 50
2.4 General arrangement of the inspection problem. . . . . . . . . . . 53
3.1 Comsol geometry used in the forward problem. . . . . . . . . . . . 69
3.2 Comsol mesh used in the forward problem. . . . . . . . . . . . . . 70
3.3 Error norm for each coil radius pair (coarse calibration). . . . . . 77
3.4 Error norm for each coil radius pair (fine calibration). . . . . . . . 78
3.5 Impedance components for the best fit excite-receive radius pairs.
The two “Best fit” curves represent the minimal error curves for
the real and imaginary impedance components (“Best fit 1” and
“Best fit 2” respectively. . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Voltage change with frequency for a range of bulk conductivity
changes (method 1). . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.7 Voltage change with ∆σ at each frequency (method 1). . . . . . . 92
3.8 Voltage change with frequency for a range of bulk conductivity
changes (method 2). . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.9 Voltage change with ∆σ at each frequency (method 2). . . . . . . 94
3.10 Voltage change with frequency for a range of bulk conductivity
changes (method 3). . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.11 Voltage change with ∆σ at each frequency (method 3). . . . . . . 96
4.1 Crack dimensioning terminology. . . . . . . . . . . . . . . . . . . 100
4.2 The geometry used in the finite element modelling. . . . . . . . . 108
4.3 Plan view of an AGR fuel channel brick (Hartlepool/Heysham 1
type). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Simulated phase change for the 120 mm sensor with respect to coil
position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5
LIST OF FIGURES LIST OF FIGURES
4.5 Simulated amplitude change for the 120 mm sensor with respect
to coil position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.6 Simulated phase and amplitude changes for three subsurface slots. 114
4.7 120 mm sensor specification . . . . . . . . . . . . . . . . . . . . . 117
4.8 120 mm sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.9 Prototype Eddy Current Inspection tool (PECIT) sensor specifi-
cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.10 PECIT sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.11 Jig used to position the sensors. . . . . . . . . . . . . . . . . . . . 121
4.12 Phase change measured for the 120 mm sensor with respect to coil
position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.13 Phase change measured for the PECIT sensor with respect to coil
position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.14 Amplitude change measured for the 120 mm sensor with respect
to coil position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.15 Amplitude change measured for the PECIT sensor with respect to
coil position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.16 Maximum phase changes measured for the two sensors used with
respect to the measured slot size. . . . . . . . . . . . . . . . . . . 129
4.17 Intersection of the slot and methane hole. . . . . . . . . . . . . . . 130
4.18 The phase change for a slot of undetermined (30 mm), 120 mm
sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.19 The phase change for a slot of undetermined size (30 mm), PECIT
sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.20 The phase change for a slot of undetermined size (50 mm), 120 mm
sesnor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.21 The phase change for a slot of undetermined size (50 mm), PECIT
sensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.1 A typical L-curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1 Method used to validate the inversion methodology. . . . . . . . . 175
6.2 Inverted profiles for profile 1. . . . . . . . . . . . . . . . . . . . . . 181
6.3 Convergence parameters for profile 1. . . . . . . . . . . . . . . . . 182
6.4 Inverted profiles for profile 2. . . . . . . . . . . . . . . . . . . . . . 183
6.5 Convergence parameters for profile 2. . . . . . . . . . . . . . . . . 184
6.6 Inverted profiles for profile 3. . . . . . . . . . . . . . . . . . . . . . 185
6.7 Convergence parameters for profile 3. . . . . . . . . . . . . . . . . 186
6.8 Inverted profiles for profile 4. . . . . . . . . . . . . . . . . . . . . . 187
6.9 Convergence parameters for profile 4. . . . . . . . . . . . . . . . . 188
6.10 Inverted profiles for profile 5. . . . . . . . . . . . . . . . . . . . . . 189
6
LIST OF FIGURES LIST OF FIGURES
6.11 Convergence parameters for profile 5. . . . . . . . . . . . . . . . . 190
6.12 Cylindrical brick with lattice of radially drilled holes. . . . . . . . 196
6.13 Inversion results for test case 1 (using simulated forward data). . . 198
6.14 Inversion results for test case 2 (using simulated forward data). . . 199
6.15 Finite Element (FE) models used to test the suitability of hole
drilling as an analogue for weight-loss. . . . . . . . . . . . . . . . 202
6.16 Calculatedimpedancefortheinnerholelattice(solidline)together
withtheimpedancesobtainedfortherangeofvariableconductivity
models (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . 203
6.17 Calculatedimpedancefortheouterholelattice(solidline)together
withtheimpedancesobtainedfortherangeofvariableconductivity
models (dashed lines). . . . . . . . . . . . . . . . . . . . . . . . . 204
6.18 Error norm for drilled lattice calibration. . . . . . . . . . . . . . . 204
6.19 Inversion results for a graphite brick with a uniform conductivity
profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.20 Inversion results for a graphite brick with a uniform conductivity
profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.21 Inversion results for a graphite brick with a uniform conductivity
profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.22 Inversion results for a graphite brick with a uniform conductivity
profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.23 Inversion results for a graphite brick with a lattice of radial holes. 212
6.24 Inversion results for a graphite brick with a lattice of radial holes. 213
6.25 Inversion results for a graphite brick with a lattice of radial holes 214
6.26 Inversion results for a graphite brick with a lattice of radial holes. 215
6.27 Inversion results for a graphite brick with a lattice of radial holes. 216
6.28 Comparison of the converged profiles for two reference conductiv-
ities applied to the uniform brick. . . . . . . . . . . . . . . . . . . 219
6.29 Comparison of the converged profiles for two reference conductiv-
ities and different λ applied to the radially drilled brick. . . . . . 221
0
6.30 An approximation of the L-curve generated from experimental data.222
6.31 Experimental L-curve analysis. . . . . . . . . . . . . . . . . . . . . 223
6.32 Experimental L-curve error data. . . . . . . . . . . . . . . . . . . 224
6.33 Heysham 1 channel plan with inspected channels . . . . . . . . . 228
6.34 Inversion results for location H13-L5-N22 using two calibrated coil
sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.35 Simulated impedance for a uniform reference profile, using two coil
calibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.36 Inversion results for Heysham 1 outage, August 2013, using a
model without methane holes. . . . . . . . . . . . . . . . . . . . . 232
6.37 CoilpositionsexaminedduringtheHeysham12013statutaryoutage.234
7
LIST OF FIGURES
6.38 Inversion results for Heysham 1 outage, August 2013, using a
model with methane holes. . . . . . . . . . . . . . . . . . . . . . . 235
6.39 Comparison of inverted profiles using the models with and without
methane holes with measurements made from trepanned samples. 237
A.1 Generic problem geometry for the 2-dimensional axi-symmetric case.261
A.2 Estimates of the A field based on the analytical formula and the
2D FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
B.1 The FE model viewed within Comsol. . . . . . . . . . . . . . . . . 274
B.2 Geometry sub-sequence. . . . . . . . . . . . . . . . . . . . . . . . 275
B.3 Physics nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
B.4 Meshing sub-sequence. . . . . . . . . . . . . . . . . . . . . . . . . 278
B.5 Solver sub-sequence. . . . . . . . . . . . . . . . . . . . . . . . . . 279
8
List of Tables
1.1 The UK electricity market as of 2014. . . . . . . . . . . . . . . . . 20
4.1 The meshing parameters used in the FE work. . . . . . . . . . . . 107
4.2 Sensor geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3 Estimated through-wall extent of two slots from experimental data.138
6.1 Summary of inversion parameters. . . . . . . . . . . . . . . . . . . 176
6.2 Approximate electrical conductivity of three graphite grades. . . . 179
6.3 Electrical conductivity of two graphite bricks. . . . . . . . . . . . 205
A.1 Boundary condition coefficients. . . . . . . . . . . . . . . . . . . . 264
A.2 Interior and exterior boundary coefficients. . . . . . . . . . . . . . 266
9
List of Abbreviations
ABWR Advanced Boiling Water Reactor
ACPD Alternating Current Potential Drop
AGR Advanced Gas-Cooled Reactor
ALARP as low as reasonably practicable
BFGSM Broyden-Fletcher-Goldfarb-Shanno Method
CAD Computer Aided Design
CBIU Channel Bore Inspection Unit
CBMU Channel Bore Measurement Unit
CCCA core component condition assessment
CRT Cathode Ray Tube
CT (X-ray) Computed Tomography
CTA Collaborative Training Account
CTE Coefficient of Thermal Expansion
DCPD Direct Current Potential Drop
DFPM Davidson-Fletcher-Powell Method
DGNM Damped Gauss-Newton Method
DTA damage tolerance assessment
EC Eddy Current
ECT Electrical Capacitance Tomography
10