Optimization of Hydride Fueled
Pressurized Water Reactor Cores
by
Carter Alexander Shuffler
B.S Mechanical Engineering
UNIVERSITY OF VIRGINIA, 2002
SUBMITTED TO THE DEPARTMENT OF NUCLEAR ENGINEERING IN PARTIAL
FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN NUCLEAR ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SEPTEMBER 2004
(©2004 Massachusetts Institute of Technology. All rights reserved.
Signature of Author:
!
!
Department of Nuclear Engineering
August 25, 2004
Certified
By:_
II.
, _
X
/7
.-
/I
-
Neil E. Todreas
KEPCO Professor of Nuclear Engineering, Professor of Mechanical Engineering
Read By:.
A
-
Thesis Supervisor
I
2
j
Pavel Hejzlar
Principal Research Scientist
l
Thesis Reader
?
Accepted By:
MASSACHUSE.ITS
INSIIT1"
OF TECHNOLOGY
IOCT 112005
LIBRARIES
Jeffrey A. Coderre
Chairman, Department Committee on Graduate Students
ARCHIVES
2
Optimization of Hydride Fueled Pressurized Water Reactor Cores
By
Carter Alexander Shuffler
Submitted to the Department of Nuclear Engineering on August 25, 2004 in partial fulfillment of
the requirements for the degree of Master of Science in Nuclear Engineering
Abstract
This thesis contributes to the Hydride Fuels Project, a collaborative effort between UC Berkeley
and MIT aimed at investigating the potential benefits of hydride fuel use in light water reactors (LWRs).
This pursuit involves implementing an appropriate methodology for design and optimization of hydride and
oxide fueled cores. Core design is accomplished for a range of geometries via steady-state and transient
thermal hydraulic analyses, which yield the maximum power, and fuel performance and neutronics studies,
which provide the achievable discharge burnup. The final optimization integrates the outputs from these
separate studies into an economics model to identify geometries offering the lowest cost of electricity, and
provide a fair basis for comparing the performance of hydride and oxide fuels.
Considerable work has already been accomplished on the project; this thesis builds on this
previous work. More specifically, it focuses on the steady-state thermal hydraulic and economic analyses
for pressurized water reactor (PWR) cores utilizing UZrH1.6 and UO2. A previous MIT study established
the steady-state thermal hydraulic design methodology for determining maximum power from square array
PWR core designs. The analysis was not performed for hexagonal arrays under the assumption that the
maximum achievable powers for both configurations are the same for matching rod diameters and H/HM
ratios. This assumption is examined and verified in this work by comparing the thermal hydraulic
performance of a single hexagonal core with its equivalent square counterpart.
In lieu of a detailed vibrations analysis, the steady-state thermal hydraulic analysis imposed a
single design limit on the axial flow velocity. The wide range of core geometries considered and the large
power increases reported by the study makes it prudent to refine this single limit approach. This work
accomplishes this by developing and incorporating additional design limits into the thermal hydraulic
analysis to prevent excessive rod vibration and wear. The vibrations and wear mechanisms considered are:
vortex-induced vibration, fluid-elastic instability, turbulence-induced vibration, fretting wear, and sliding
wear.
Concomitantly with this work, students at UC Berkeley and MIT have undertaken the neutronics,
fuel performance, and transient thermal hydraulic studies. With these results, and the output from the
steady-state thermal hydraulic analysis with vibrations and wear imposed design limits, an economics
model is employed to determine the optimal geometries for incorporation into existing PWRs. The model
also provides a basis for comparing the performance of UZrH,1 6 to UO2 for a range of core geometries.
Though this analysis focuses only on these fuels, the methodology can easily be extended to additional
hydride and oxide fuel types, and will be in the future. Results presented herein do not show significant
cost savings for UZrH1. 6, primarily because the power and energy generation per core loading for both fuels
are similar. Furthermore, the most economic geometries typically do not occur where power increases are
reported by the thermal hydraulics. As a final note, the economic results in this report require revision to
account for recent changes in the fuel performance analysis methodology. The changes, however, are not
expected to influence the overall conclusion that UZrH1.6 does not outperform UO2 economically.
Thesis Supervisor:
Neil E. Todreas
Title:
KEPCO Professor of Nuclear Engineering, Professor of Mechanical Engineering
Thesis Reader:
Title:
Pavel Hejzlar
Principal Research Scientist
3
Acknowledgements
I'd like to thank Professor Todreas for his guidance and support throughout this
thesis. It has been a pleasure working with him this year. The degree of interest he
expresses in his students on both a personal and academic level is unique among college
faculty. I'd also like to thank Professor Greenspan for his countless suggestions for
improving my analysis, and his thoughtful patience in review of my work.
Several students provided great assistance at various phases of the project.
Without their help I can confidently say this work would not have been possible. These
students are Jon Malen, Stu Blair, and Antonino Romano.
Finally, I'd like to thank those closest to me: my parents, Jason and Liz, and of
course Daniela. Your love and support is tremendous, and I'm looking forward to
spending more time with all of you soon.
4
Table of Contents
Abstract .
.........................................................................................................
3
Acknowledgements........................................................................................
4
Table of Figures............................................................................................. 8
List of Tables ...............................................................................................
1. Introduction ..........................................................................................
1.1
1.2
1.3
12
13
Overview of the Hydride Fuels Project ............................................................ 13
Design Optimization Methodology................................................................... 14
Scope of Work and Contribution to the Hydride Fuels Project ........................ 18
2. Steady-State Thermal Hydraulic Design of Square and Hexagonal
Array PW R Cores .......................................................
20
2.1
Parametric Study Overview ................................................................
20
2.1.1
Reference Core Parameters
.
.
..............................................................
21
2.1.2
Design Limits
2.1.2.1
2.1.2.2
2.1.2.3
2.1.2.4
2.1.3
................................................................
22
MDNBR ...........................................................
23
Pressure Drop
................................................................
24
Fuel Temperature
........
24
Axial Flow Velocity
................................................................25
MATLAB/V IPRE Interface .....................................................................
26
2.2
Results of the Steady-State Maximum Power Analysis ................................... 27
2.3
The Equivalence of Square and Hexagonal Array Geometries ........................ 33
2.3.1
Geometric Relationships for Square and Hexagonal Arrays .................... 33
2.3.1.1 Conversion Between Hydrogen/Heavy Metal and Pitch/Diameter
Ratio ........................... .................................................................
33
2.3.1.2 Relationship Between Square and Hexagonal Array Sub-Channels .... 38
2.3.2
Full Core Hexagonal Array VIPRE Model............................................... 40
2.3.2.1 Rod and Assembly Layout Within the Core ......................................... 41
2.3.3
Comparison of Maximum Power Predictions from Square and Hexagonal
Arrays........................................................................................................ 44
3. Vibrations Analysis for Hydride and Oxide Fueled PWRs ............. 47
3.1
Introduction ...............................................................
47
3.2
W ork Scope....................................................................................................... 47
3.2.1
Goals of the Vibrations Analysis.
.............................................................
47
3.2.2
Flow-Induced Vibration Mechanisms - Overview ................................... 48
3.2.3
Methodology
.
................. .............................................
3.3
Assumptions ......................................................................................................
50
50
3.3.1
Vibrations Analysis in the Nuclear Environment ..................................... 50
3.3.2
Key Assumptions ...................................................................................... 51
3.4
Vibration and W ear Mechanisms...............................................................
53
3.4.1
Dynamic Parameters ................................................................................. 53
3.4.1.1
Nomenclature ........................................................................................
53
3.4.1.2
3.4.1.3
Linear Mass...............................................................
53
Moment of Inertia ........................................
.........................................
54
S
3.4.1.4
3.4.1.5
3.4.1.6
3.4.2
Natural Frequency and Mode Shape
.. ............ .......................................
54
Damping Ratio .......................................................
54
Generation and Distribution of Axial and Cross-flow Velocities......... 55
Vortex-Induced Vibration ........................................
............................ 59
3.4.2.1 Vortex-Shedding Margin and Design Limit ......................................... 59
3.4.3
Fluid-Elastic Instability............................................................................. 62
3.4.3.1 Fluid-Elastic Instability Margin and Design Limits ............................. 62
3.4.4
Turbulence-Induced Vibration.......................................................
63
3.4.4.1 Turbulence-Induced Vibration in Cross-flow ....................................... 64
3.4.4.2 Turbulence-Induced Vibration in Axial Flow....................................... 66
3.4.4.3 Turbulence-Induced Vibration Response ............................................. 66
3.4.5
Wear Due to Flow-Induced Vibration ...................................................... 67
3.4.5.1 Fretting Wear Performance and Design Limits .................................... 68
3.4.5.2 Sliding Wear Performance and Design Limits .................................... 70
3.5
Summary of Constrained Parameters and Design Limits
........................ 71
3.6
Results of the Maximum Achievable Power Analysis With Vibrations-Imposed
Design Limits for Square Arrays .........................................
72
3.6.1
Results at 60 psia for UZrH.6................................................................... 72
3.6.2
3.6.3
Results at 60 psia for U0 2.........................................................................
Comparison Between UZrH1. 6 and UO2 at 60 psia ...................................
3.6.4
Results at 29 psia for UZrH.6 ................................................................... 82
3.6.5
3.6.6
Results at 29 psia for U0 2.........................................................................
Comparison Between UZrH 1. 6 and UO2 at 29 psia ...................................
3.6.7
Summary of Maximum Power Results With Vibrations and Wear Imposed
Design Limits............................................................................................ 91
Extension of Vibrations Results to Hexagonal Arrays ............................. 93
3.6.8
76
80
86
90
4. Economics Analysis of Hydride and Oxide Fueled PWRs ............... 96
4.1
Introduction .......................................................................................................
96
4.2
Work Scope....................................................................................................... 96
4.2.1
Goals of the Economics Analysis ............................................................. 96
4.2.2
Methodology ...........................................
.........................
97
4.2.2.1 Lifetime Levelized Cost Method .......................................................... 98
4.3
4.4
Assumptions .......................................................
Inputs .......................................................
100
102
4.4.1
Maximum Power.......................................................
102
4.4.2
Fuel Burnup .......................................................
105
4.4.2.1 Neutronics ........................................................................................... 106
4.4.2.2 Fuel Performance ................................................................................ 108
4.4.3
Economic Parameters.......................................................
114
4.4.3.1 Fuel Cycle Unit Costs ......................................
................. 114
4.4.3.2 Operations and Maintenance Unit Costs
.............................. 117
4.4.3.3 Capital Costs ....................................................................................... 118
4.4.3.4 Plant Operating Parameters .......................................................
120
4.5
Economics Analysis ........................................................................................
125
4.5.1
Fuel Cycle Costs ..................................................................................... 125
4.5.1.1 The Nuclear Fuel Cycle ........................................
125
6
4.5.1.2
4.5.1.3
4.5.1.4
4.5.2
Recurring Cash Flows and the Fuel Cycle ..........................................
Cash Flows for the First Operating Cycle ...........................................
126
127
Lifetime Levelized Unit Fuel Cycle Cost ........................................... 134
Operations and Maintenance Costs ........................................................
134
4.5.2.1 Annualizing O & M Cash Flows ........................................................ 135
4.5.2.2 Lifetime Levelized Unit 0 & M Cost................................................. 137
4.5.3
Capital Costs ................
.
.......................................
137
4.5.3.1 Predicting Capital Costs for PWR Backfit .......................................... 138
4.5.3.2 Lifetime Levelized Unit Capital Cost ................................................. 139
4.6
Lifetime Levelized Unit COE .............................................................. ...... 140
5.
4.6.1
Results for Major Backfit: UZrHI. 6 and UO2 at 60 psia ........................
4.6.1.1
COE Breakdown for 12.5% UZrH .1 6 and 5% UO 2 at 60 psia .............
4.6.3
Results for Major Backfit: UZrH 1.6 and UO2 at 29 psia ........................
4.6.4
Results for Minor Backfit: UZrHI . 6 and UO 2 at 29 psia ........................
140
144
154
158
Conclusions and Future Work .................................................
161
5.1
Conclusions .....................................................................................................
161
5.1.1
Steady-State Thermal Hydraulics for Square andHexagonal Array PWR
Geom etries ......................................................
162
5.1.2
Vibrations Analysis for Hydride and Oxide Fueled PWRs .................... 163
5.1.3
Economic Optimization ......................................................
165
5.2
Future Work ......................................................
168
5.2.1
Methodology ......................................................
168
5.2.1.1l Steady-State Thermal Hydraulic Analysis ........................................
5.2.1.2 Vibrations and Wear Analysis .
.......................................
5.2.1.3 Transient Thermal Hydraulic Analysis...............................................
5.2.1.4 Economics Analysis ........................................
5.2.1.5 Fuel Performance and Application of FRAPCON to Hydride Fuels..
5.2.2
New Approaches ........................................
5.2.2.1 New Fuels ........................................
5.2.2.2
Hexagonal Arrays and Wire Wrap .....................................
5.2.2.3
Reference Core.....................................
...
168
169
170
170
170
171
171
172
172
Bibliography ..............................................................................................
173
AppendixA ......
AppendixB ................
174.........
.........................
...............
Appendix C ......................................
Appendix D ........................................
Appendix E .......................................
180........
186
199
224
7
Table of Figures
Figure 1.1 Design Optimization Flow Chart ................................................................
17
Figure 2.1 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of UZrH1 .6
and UO2 at 60 psia ............................................................
29
Figure 2.2 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for Square
Arrays of UZrH .1 6 and UO2 at 60 psia ................................................................
29
Figure 2.3 Constraining Parameters vs. P/D and Drod for Square Arrays of UZrH1 .6 and
UO2 at 60 psia ................................................................
30
Figure 2.4 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UZrH1 .6
and UO2 at 29 psia ............................................................
31
Figure 2.5 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drodfor Square
Arrays of UZrH 1. 6 and UO2 at 29 psia ................................................................
31
Figure 2.6 Constraining Parameters vs. P/D and Drodfor Square Arrays of UZrHl.6 and
UO2 at 29 psia ...................................................................
32
Figure 2.7 H/HM Ratio vs. P/D and Drod for Square and Hexagonal Arrays of UZrH 1.6
and U 0 2................................................................
37
Figure 2.8 Square and Hexagonal Array Unit Cells ........................................................ 39
Figure 2.9 9 Ring Hexagonal Fuel Assembly................................................................ 41
Figure 2.10 1/6th Hexagonal Core Section................................................................
42
Figure 2.11 Channel Lumping in the 1/6th Hexagonal Core ....................................
43
Figure 2.12 1/ 6 th Section of the Hot Assembly................................................................ 44
Figure 3.1 Cross-flow Velocity Profile as a Function of Core Power ............................. 57
Figure 3.2 Cross-flow and Fundamental Mode Shape Profiles .......................................
58
Figure 3.3 Vortex-Shedding Margin vs. Hypothetical Cross-flow Velocity at the
Reference Core Geometry................................................................
61
Figure 3.4 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UZrH1.6
at 60 psia with Vibrations and Wear Imposed Design Limits .................................. 73
Figure 3.5 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for Square
Arrays of UZrHl.6 at 60 psia with Vibrations and Wear Imposed Design Limits .... 74
Figure 3.6 Power(no vibrations)
-
Power(vibrations) VS. P/D
and Drod for Square Arrays of
UZrH1, 6 at 60 psia ................................................................
74
Figure 3.7 Thermal Hydraulic Constraining Parameters vs. P/D and Drod for Square
Arrays of UZrH1. 6 at 60 psia with Vibrations and Wear Imposed Design Limits .... 75
Figure 3.8 Vibrations and Wear Constraining Parameters vs. P/D and Drod for Square
Arrays of UZrH 1.6 at 60 psia ........................................
........................
75
Figure 3.9 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drod for
Square Arrays of UZrH1.6 at 60 psia ............................................................
76
Figure 3.10 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of U0 2 at
60 psia with Vibrations and Wear Imposed Design Limits ...................................... 77
Figure 3.11 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for
Square Arrays of U0 2 at 60 psia with Vibrations and Wear Imposed Design Limits
Figure 3.12 Power(no vibrations)- Power(vibrations)
vs. P/D and Drod for Square Arrays of U0 2
at 60 psia ............................................................................
78
8
Figure 3.13 Thermal Hydraulic Constraining Parameters vs. P/D and Drod for Square
Arrays of UO 2 at 60 psia with Vibrations and Wear Imposed Design Limits.......... 79
Figure 3.14 Vibrations and Wear Constraining Parameters vs. P/D and Drod for Square
Arrays of U0 2 at 60 psia .................................................................
79
Figure 3.15 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drod
for Square Arrays of UO2 at 60 psia ................................................................
80
Figure 3.16 (Poweruo2 - PoweruzrH .6) vs. P/D and Drodfor Square Arrays at 60 psia with
Vibrations and Wear Imposed Design Limits........................................................... 81
Figure 3.17 Percentage Difference in Wear Rate Limits vs. P/D and Drod for Square
Arrays at 60 psia ................................................................
81
Figure 3.18 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UZrH1. 6
at 29 psia with Vibrations and Wear Imposed Design Limits .................................. 83
Figure 3.19 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for
Square Arrays of UZrH 1. 6 at 29 psia with Vibrations and Wear Imposed Design
Limits ................................................................
83
Figure 3.20 Power(novibrations)- Power(vibrations)
vs. P/D and Drod for Square Arrays of
UZrH ,.6 at 29 psia ................................................................
84
Figure 3.21 Thermal Hydraulic Constraining Parameters vs. P/D and Drod for Square
Arrays of UZrHI. 6 at 29 psia with Vibrations and Wear Imposed Design Limits .... 84
Figure 3.22 Vibrations and Wear Constraining Parameters vs. P/D and Drod for Square
Arrays of UZrH 1. 6 at 29 psia ................................................................
85
Figure 3.23 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drod
for Square Arrays of UZrHl. 6 at 29 psia ................................................................
85
Figure 3.24 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UO2 at
29 psia with Vibrations and Wear Imposed Design Limits ...................................... 87
Figure 3.25 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for
Square Arrays of U0 2 at 29 psia with Vibrations and Wear Imposed Design Limits
...................................................................................................................................
87
Figure 3.26 Power(novibrations)- Power(vibrations)
vs. P/D and Drod for Square Arrays of U0 2
at 29 psia ...............................................................
88
Figure 3.27 Thermal Hydraulic Constraining Parameters vs. P/D and Drod for Square
Arrays of U0 2 at 29 psia with Vibrations and Wear Imposed Design Limits.......... 88
Figure 3.28 Vibrations and Wear Constraining Parameters vs. P/D and Drodfor Square
Arrays of UO 2 at 29 psia ...............................................................
89
Figure 3.29 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drod
for Square Arrays of U02 at 29 psia ...............................................................
89
Figure 3.30 (Poweruo2- PoweruzrHl.6)vs. P/D and Drodfor Square Arrays at 29 psia with
Vibrations and Wear Imposed Design Limits........................................................... 90
Figure 3.31 Percentage Difference in Wear Rate Limits vs. P/D and Drod for Square
Arrays at 29 psia ...............................................................
91
Figure 4.1 Maximum Achievable Power and Transient Limited Regions vs P/D and Drod
for Square Arrays of UZrHl.6 and UO2 Incorporating Steady-State and Transient
Design Limits at 60 psia ...........................................................
104
Figure 4.2 Maximum Achievable Power and Transient Limited Regions vs P/D and Drod
for Square Arrays of UZrH,.6 and UO 2 Incorporating Steady-State and Transient
Design Limits at 29 psia ...........................................................
105
9
Figure 4.3 Neutronically Achievable Burnup vs. P/D and Dro for Square Arrays of
UZrH 1.6 Fuel ...............................................................
107
Figure 4.4 Neutronically Achievable Burnup vs. P/D and Drod for Square Arrays UO2
Fuel ...............................................................
107
Figure 4.5 Fuel Performance Limited Bumup vs. P/D and Drodfor Square Arrays of
UZrH 1. 6 and UO2 at 60 psia ...............................................................
111
Figure 4.6 Fuel Performance Limited Burnup vs. P/D and Drod for Square Arrays of
UZrH 1. 6 and UO2 at 29 psia ...............................................................
111
Figure 4.7 Maximum Achievable Burnup vs. P/D and Drod for Square Arrays of UZrH1.6
at 60 psia ...........................................................
112
Figure 4.8 Maximum Achievable Burnup vs. P/D and Drod for Square Arrays of UZrH1.6
at 29 psia ...........................................................
113
Figure 4.9 Maximum Achievable Burnup vs. P/D and Drodfor Square Arrays of U0 2 at
60 psia ..................................................................................................................... 113
Figure 4.10 Maximum Achievable Burnup vs. P/D and Drod for Square Arrays of U0 2 at
29 psia ..............................................
114
Figure 4.11 Unit Fabrication Cost and Rod Number vs. P/D and Drod for Square Arrays
of UO 2 .............................................................................................
Figure 4.12 Unit Fabrication Cost and Rod Number vs. P/D and
116
Drod
for Square Arrays
of UZrH 1 .6............................................................................
Figure 4.13 Plant Capacity vs. Operating Length ..........................................................
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17
117
123
Refueling Outage Activities ........................................
....................... 124
Cash Flows for Successive Operating Cycles ............................................ 127
Mass Flows for Front End Fuel Cycle Processes....................................... 128
Cash Flows for Successive Annual O & M Expenditures .......................... 136
Figure 4.18 Lifetime Levelized Unit COE vs. P/D and Drod for Square Arrays of UZrH 1.6
at 60 psia ...........................................................
142
Figure 4.19 Minimum COE and its Components vs. P/D for Square Arrays of UZrH 1. 6 at
60 psia ...........................................................
142
Figure 4.20 L,ifetime Levelized Unit COE vs. P/D and Drod for Square Arrays of UO2 at
60 psia ...........................................................
143
Figure 4.21 Minimum COE and its Components vs. P/D for Square Arrays of U02 at 60
psia ...........................................................
143
Figure 4.22 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of UZrH,.6
and UO2 at 60 psia With Vibrations and Transient Limits Applied ....................... 144
Figure 4.23 COE Breakdown for Square Arrays of 12.5% UZrH 1. 6 at 60 psia ............. 148
Figure 4.24 Plant Operating Conditions for Square Arrays of 12.5% UZrH1.6 at 60 psia
..................................................................................................................................
14 9
Figure 4.25 Plant Operating Conditions for Square Arrays of 12.5% UZrH 1. 6 at 60 psia
............ .....................................................................................................................
Figure 4.26
Figure 4.27
Figure 4.28
Figure 4.29
14 9
COE Breakdown for Square Arrays of 5% UO 2 at 60 psia ........................ 150
Plant Operating Conditions for Square Arrays of U0 2 at 60 psia.............. 150
Plant Operating Conditions for Square Arrays of U0 2 at 60 psia.............. 151
COE Difference and Fuel Enrichment for Major Backfit With UZrH 1. 6 and
UO 2 vs. P/D and Drod at 60 psia ...............................................................
152
Figure 4.30 Minor Backfit COE and its Components vs. P/D for UZrH 1. 6 at 60 psia... 153
10
Figure 4.31 Minor Backfit COE and its Components vs. P/D for U0 2 at 60 psia ......... 154
Figure 4.32 Lifetime Levelized Unit COE vs. P/D and Drod for Square Arrays of UZrH 1.6
at 29 psia ...........................................................
155
Figure 4.33 Minimum COE and its Components vs. P/D for Square Arrays of UZrH 1.6 at
29 psia ...........................................................
156
Figure 4.34 Lifetime Levelized Unit COE vs. P/D and Drod for Square Arrays of UO2 at
29 psia ...........................................................
156
Figure 4.35 Minimum COE and its Components vs. P/D for Square Arrays of UO2 at 29
psia ...........................................................
157
Figure 4.36 COE Difference and Fuel Enrichment for Major Backfit With UZrH 1. 6 and
U0 2 vs. P/D and Drod at 29 psia ..............................................................
158
Figure 4.37 Minor Backfit COE and its Components vs. P/D for UZrH .1 6 at 29 psia... 159
Figure 4.38 Minor Backfit COE and its Components vs. P/D for UO2 at 29 psia ......... 159
Figure 5.1 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UZrH 1.6
and UO2 at 60 psia and 29 psia ..............................................................
162
Figure 5.2 Maximum Achievable Power and Power Reductions vs. P/D and Drodfor
Square Arrays of UZrH 1.6 at 60 and 29 psia with Vibrations and Wear Imposed
Design Limits..............................................................
164
Figure 5.3 Maximum Achievable Power and Power Reductions vs. P/D and Drod for
Square Arrays of U0 2 at 60 and 29 psia with Vibrations and Wear Imposed Design
Limits ...........................................................
164
Figure 5.4 Major and Minor Backfit COE vs. P/D for UZrH .1 6 and UO 2 at 29 and 60 psia
.................................................................................................................................
167.
Figure 5.5 Major Backfit COE Difference vs. P/D and Drod for UZrH .1 6 and UO 2 at 60
and 29 psia ...........................................................
167
11
List of Tables
Table 2.1 Fixed Parameters for the Thermal Hydraulic Analysis .................................. 22
Table 2.2 Summary of Steady-State Thermal Hydraulic Design Limits for UZrH1. 6 and
U O 2 ...........................................................
22
Table 2.3 General Nomenclature for Geometric Relationships for Square and Hexagonal
Arrays ...............................................................
Table 2.4
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
34
Power Predictions by Full Core Square and Hexagonal VIPRE Models ........ 44
Flow-Induced Vibration Mechanisms ............................................................. 50
General Nomenclature for the Vibrations Analysis. ............................... 53
ASME Recommended Damping Ratios ......................................................... 54
Peak Cross-flow Velocities vs. Power for the Reference Geometry ............... 61
Summary of Steady-State Thermal Hydraulic Design Constraints ................. 71
Summary of Steady-State Thermal Hydraulic Results with Vibrations and
W ear Lim its ...............................................................
Table 4.1 Transient Analysis Summary ...............................................................
93
103
Table 4.2 Fuel Performance Limits for Maximum Burnup .......................................... 109
Table 4.3 PWR Fuel Cycle Unit Costs for U02 ............................................................. 115
Table 4.4 PWR Operations and Maintenance Unit Costs ..............................................
117
Table 4.5 Cost Estimates for Installed Nuclear Components ........................................ 120
Table 4.6 Fixed Plant Operating Parameters ............................................................... 121
Table 4.7 Mass Loss Fractions for Front End Fuel Cycle Processes ............................. 128
Table 4.8 Inlet and Outlet Mass Flow Stream Enrichments at the Enrichment Plant ... 129
Table 4.9 SWU Requirements for Different Enrichments of U0 2 and UZrH1. 6............ 130
Table 4.10 Mass Flows Through the Front End ............................................................ 132
Table 4.11 Fuel Cycle Unit Costs With Respect to the Heavy Metal Loading in the Core
..................................................................................................................................
133
Table 4.12 OECD/NEA Recommended Lead and Lag Times for Front and Back End
Processes ...............................................................
133
Table 4.13 Fixed and Variable Unit 0 & M Costs ........................................................ 135
12
1.
Introduction
1.1
Overview of the Hydride Fuels Project
The concentration of hydrogen in hydride fuels is comparable to that of hydrogen
in light water reactor (LWR) coolant. The additional moderation provided in the fuel
provides an optimal neutron spectrum with a smaller water volume fraction to cool and
moderate the core. The result can be either smaller core volume for new designs or
higher power output for existing LWRs that convert to hydride fuels. Candidate fuels
include but are not limited to: UZrH.1 6, PuZrH 1. 6, PuH 2-ThH 2, UH 2-ThH 2, UZrH 1 6-ThH 2,
and PuZrH 1. 6 -ThH2. Note that fuels with thorium hydride have a higher heavy metal
concentration than traditional oxide fuel. Combined with the higher power density
offered by hydride fuels, they will allow achievement of higher energy generation per
core loading. The expected outcome includes improvements in economics, resource
utilization, proliferation resistance, safety, and waste reduction.
The investigation of hydride fuels may lead to the development of LWR core
designs that may have one or more of the following advantages over contemporary
LWRs:
·
Increased power output, power density, discharge bum-up, and core lifetime;
* Reduced fuel cycle, operations and maintenance, and capital costs and therefore
reduced cost of electricity;
·
Reduced waste volume due to higher discharge burnup and partial utilization of
thoriurnm;
·
Increased utilization of commercial and surplus weapons grade plutonium due to
higher power and discharge burnup; additionally, the use of PuH2 and PuZr fuel
types eliminate fertile 238U;
* Improved proliferation resistance due to enhanced plutonium destruction and use
of thorium;
·
Increased capability of LWRs to utilize thorium fuel resources;
13
* Improved LWR safety due to the large negative temperature coefficient of
reactivity of hydride fuels;
* Reduced heterogeneity and negative void coefficient of reactivity in BWR cores
which simplifies control systems and fuel assembly design, as well as improves
stability against power oscillations.
1.2
Design Optimization Methodology
The Hydride Fuels Project aims to quantify the advantages of hydride fuel use by
developing and implementing an appropriate methodology for optimizing LWR core
designs. Two cases are considered for optimization:
* Minor Backfit: Minor backfit of existing LWRs seeks to limit the plant
modifications required for conversion to hydride fuel use by maintaining the
existing fuel assembly and control rod configurations within the pressure vessel
(i.e., maintaining the same pitch and rod number in the core). In this case,
replacement of the steam generators and upgrades to the high pressure turbine will
be required to accommodate higher powers;
* Major Backfit: Major backfit of existing LWRs does not limit the design space.
The layout of hydride fuel in the core can therefore assume any combination of
lattice pitch, rod diameter, and channel shape, further referred to throughout this
report as a design or geometry. Note that in addition to upgrades and
modifications on the steam side of the plant, replacement of the reactor vessel
head and core internals will also be necessary.
The optimization is carried out by integrating the results from separate evaluations for
thermal hydraulics, neutronics, and fuel performance into an economics model, which
identifies designs for each backfit case offering the lowest cost of electricity. To provide
a fair and consistent basis for the optimization, all analyses are carried out for both
hydride and oxide fuels.
14
The participants in the study and their respective responsibilities include: (1)
University of California, Berkeley, responsible for neutronics and materials
compatibility; (2) MIT, responsible for thermal hydraulics, fuel performance, and
economic optimization; and (3) Westinghouse, responsible for practical fuel and core
design considerations. The role played by each analysis in the optimization study is
briefly summarized below:
* Neutronics: The neutronics analysis seeks the maximum fuel burnup that
maintains the critical nuclear reaction in the core, accounting for fuel depletion. It
depends on fuel type, enrichment, and geometry. It also provides the range of
geometries with acceptable fuel and moderator temperature coefficients.
* Fuel Performance: The fuel performance analysis seeks the maximum fuel
burnup subject to design constraints that protect the integrity of the fuel pin
during irradiation. Constraints are placed on: fission gas release and internal fuel
pin pressure, clad strain, and clad oxidation. The fuel performance limited burnup
depends on the fuel type, and the power in the core.
*
Thermal Hydraulics: The thermal hydraulics analysis seeks the maximum power
that can be safely sustained in the core subject to steady-state and transient design
limits. Steady-state limits are placed on the minimum departure from nucleate
boiling ratio (MDNBR), fuel bundle pressure drop, fuel temperature, and rod
vibration and wear. Transient limits are derived by consideration of the loss of
flow and loss of coolant accidents, and an overpower transient. The maximum
power in the core depends on fuel type and geometry.
* Materials Compatibility: The materials compatibility study supports the
neutronics, fuel performance, and thermal hydraulic analyses by specifying
materials that are compatible for use with hydride fuels (i.e., cladding material,
gap fill).
15
Economic Optimization: An economics model uses the maximum bumup and
power defined by the neutronics, fuel performance, and thermal hydraulic
analyses to determine the cost of electricity (COE) as a function of fuel type,
enrichment, and geometry. Both hydride and oxide fuels are considered, and
designs are optimized for the cases of minor and major backfit by identifying the
geometries where costs are minimized, and where hydride fuel use offers potential
cost savings over oxide.
A simplified flow chart showing the interaction of these analyses with their
respective inputs and outputs is shown in Figure 1.1. Note the feedback loop connecting
maximum power and burnup to the steady-state thermal hydraulic analysis. The
vibrations design limits (see Chapter 3) for the steady-state thermal hydraulic analysis
depend on the operating cycle length, which in turn is a function of burnup and power.
The maximum power and burnup must therefore be determined in iterative fashion.
16
Figure 1.1 Design Optimization Flow Chart
Fuel Type,
Enrichment,
Geometry
Fuel Type,
Fuel Type,
Geometry
Geometry
R /
I
Fuel Type,
Geometry
RBIB
!
\
COB
17
1.3 Scope of Work and Contribution to the Hydride Fuels
Project
To date, considerable progress has been made on the project, and this report
builds on this previous work. More specifically, it focuses on the steady-state thermal
hydraulic and economic analyses for pressurized water reactor (PWR) cores. The basic
methodology for evaluating the steady-state thermal hydraulic performance of square
array PWR core designs was laid out by Jon Malen, a recent MIT graduate student [1].
As previously discussed, this analysis seeks the maximum achievable power for core
geometries subject to design limits on MDNBR, fuel bundle pressure drop, fuel
temperature, and rod vibration. Chapters 2 and 3 of this report will supplement and
complete this steady-state analysis by considering hexagonal array designs, and more
stringent limits on rod vibrations and wear. Concomitantly with this work, students at
UC Berkeley and MIT have undertaken the neutronics, materials, fuel performance, and
transient thermal hydraulic studies. Chapter 4 of this report develops an economics
model to determine the optimal PWR core designs given inputs from these analyses and
the completed steady-state results from Chapters 2 and 3. Results are reported for
UZrH1.6 and U0 2, but the methodology can be easily extended to additional hydride and
oxide fuel types.
18
19
2.
Steady-State Thermal Hydraulic Design of
Square and Hexagonal Array PWR Cores
2.1
Parametric Study Overview
A recent MIT graduate, Jon Malen (2003), performed a parametric study to
determine the steady-state maximum achievable power for square array PWR geometries
with design limits placed on MDNBR, fuel bundle pressure drop, fuel temperature, and
rod vibration [1]. The performance of new designs with respect to the limits was
determined by the VIPRE sub-channel analysis code, which was cleverly linked to a
series of student written MATLAB scripts to iteratively determine the maximum power
for a user defined range of geometries. For each channel shape, the rod diameter and
lattice pitch are the independent variables chosen to describe a specific geometry, and so
the design space for the thermal hydraulic analysis is given with respect to rod diameter
and the pitch to diameter (P/D) ratio. For square arrays, the design space is:
1.08 <p-
(2.1)
<1.55
6.5 < L)rod < 12.5
mm
(2.2)
This range is considered bounding for geometries that can be incorporated into existing
PWRs.
J. Malen performed the steady-state thermal hydraulic analysis with a single
channel VIPRE model for square and hexagonal arrays, and a full core VIPRE model for
square arrays. The single channel performance for each array type was identical for
matching combinations of rod diameter and hydrogen to heavy metal (H/HM) ratio. For
this condition, square and hexagonal sub-channels have the same coolant flow areas and
heated and wetted perimeters, which determine thermal hydraulic performance in
confined flow channels (See Section 2.3.1 for the derivation). The maximum power is
therefore expected to be the same, as confirmed by the single channel results.
20
Based on the single channel analysis, J. Malen assumed that the full core
hexagonal power could also be inferred from the full core square results. A full core
hexagonal VJIPREmodel was therefore not undertaken. Modeling a core is different than
a single channel though, because turbulent interchange of mass, momentum, and energy
occur across sub-channel boundaries. Furthermore, these interchanges are slightly
different for square and hexagonal arrays [2]. It is therefore important to verify that
differences in sub-channel communication do not significantly impact the thermal
hydraulic performance equivalence of hexagonal and square designs for matching rod
diameters and H/HM ratios. If the discrepancy is large, then the parametric study will
need to be performed separately for hexagonal array cores.
The purpose of this chapter is twofold. First, several fixed parameters in the
thermal hydraulic analysis have recently changed and require regeneration of the
maximum power results for the full core square model, using J. Malen's approach. For
continuity, the steady-state thermal hydraulic design methodology is briefly re-introduced
and summarized. Where more information is desired, the reader is referred to [1].
Second, the thermal hydraulic equivalence of square and hexagonal array designs at
matching rod diameters and H/HM ratios is proven by: deriving the geometric
relationships for flow area and heated and wetted perimeters for square and hexagonal
sub-channels; and constructing a full core hexagonal VIPRE model for a single geometry
for comparison with the square model results.
2.1.1 Reference Core Parameters
The reference core for the thermal-hydraulic analysis is the South Texas Plant, a 4
loop PWR designed by Westinghouse. It provides a set of fixed hardware dimensions
and operating conditions that define boundaries for the thermal hydraulic analysis to
ensure the feasibility of new designs (i.e., ensure that new designs can be integrated into
existing PWRs). The dimensions and operating conditions that are fixed in this analysis
are listed in Table 1.1. Parameters that have recently changed are denoted by an asterisk
*. Note that these changes are not a result of changes in the operating conditions at
21
South Texas, but rather misinterpretation in their evaluation during the initial stages of
the project.
Table 2.1 Fixed Parameters for the Thermal Hydraulic Analysis 111
Parameter
Symbol
Value
Core Radius
Rcore
-1.83 m (72")
Active Fuel Length
Lh
4.26 m (168")
Core Enthalpy Rise' *
Ah
204 kJ/kg
Inlet Temperature
Tinlet
294 C (561.2 F)
System Pressure
Psat
2250 psia
Radial Peak to Average Power
Fq
1.65
Axial Peak to Average Power *
Fq'axial
1.55
The reference core geometry and power are:
P
Dre = 9.Smm
= 1.326
(2.3)
ef = 3800MWt
2.1.2 Design Limits
The steady-state thermal hydraulic design limits, which constrain the maximum
power achievable for each geometry, are briefly discussed in this section. They were
developed with guidance from industry and MIT faculty by J. Malen to provide safety
margin and ensure technical feasibility for new designs using both oxide and hydride
fuels. It is important to note that because the limits are placed on thermal hydraulic
performance, they are equally applicable to both fuel types. The single exception is the
fuel temperature limit, which will be further discussed in Section 2.1.2.3. The limits are
summarized in Table 2.2. The changes to the core enthalpy rise and axial power peaking
factor noted in Table 2.1 have modified the MDNBR and lower pressure drop numerical
limits slightly from those implemented in J. Malen's study.
Table 2.2 Summary of Steady-State Thermal Hydraulic Design Limits for UZrH1.6 and UO2
MDNBR
2.173
Pressure Drop
29 psia, 60 psia
Fuel Temperature
750 C UZrHi.6, 1400 C UO2
Axial Velocity
8 m/s
The core enthalpy rise was fixed to protect the steam generators, which mandates that the hot leg
temperature, after core outlet and bypass flow mixing, remain below 326.7 C (620 F).
22
2.1.2.1 MDNBR
Departure from nucleate boiling (DNB) is the most limiting constraint on power
for commercial PWRs. DNB occurs at the critical heat flux, which is a function of the
geometry and operating conditions in the core. It is characterized by a sharp decline in
the heat transfer coefficient at the coolant/cladding interface, as vapor blankets the fuel
rod preventing fluid from reaching its outer surface. The result is an abrupt rise in the
temperature of both the fuel and cladding, which can damage the fuel and/or cause a
cladding breach.
The performance metric for DNB is the MDNBR, which is the minimum ratio of
the critical to actual heat flux found in the core. In commercial design, significant margin
exists in the MDNBR limit to account for transients, core anomalies (i.e, rod bow),
process uncertainty (i.e, instrument error), and correlation uncertainty. While it is
difficult to quantify the magnitude of each, a reasonable MDNBR limit can be obtained
by executing VIPRE at the reference core geometry and operating conditions. The
reference core's MDNBR limit already accounts for the needed margin. The use of the
MDNBR given by VIPRE as the MDNBR limit for the steady-state thermal hydraulic
analysis therefore ensures that all new designs will demonstrate the same level of DNB
margin as the reference core. This limit is - 2.17.
One final note on the MDNBR limit is warranted. Using the limit, as described
above, assumes that the margins built into the reference core's MDNBR limit are
sufficient for all geometries considered in this study. This may not be true, however, for
the transient contribution. Consider, for example, the loss of flow accident (LOFA).
Designs that are tighter than the reference core will coast down more quickly, and
therefore require additional margin for transients in the overall steady-state MDNBR
limit. To account for this, a fellow graduate student at MIT, Jarrod Trant, is evaluating
the maximum power with respect to specific transient design limits [3]. Ultimately, the
final maximum achievable power will be the minimum power given by either the steadystate or transient analyses, at each geometry. The transient and steady-state results are
combined in Chapter 4 of this report for the final economic optimization.
23
Also note that because the equivalence of full core square and hexagonal thermal
hydraulic models is undertaken in this chapter, the limits listed in Table 2.2 are assumed
to apply equally to both configurations. The correlation used in VIPRE to determine the
CHF is the W-3L, which can be used for both arrays if grid spacers are used. If wirewrapped designs are considered in the future, new CHF correlations will apply and the
MDNBR limit will need to be modified. This is not undertaken in this study.
2.1.2.2 Pressure Drop
The maximum pressure drop sustainable through the primary system is
determined by the capacity of the coolant pumps. Two separate pressure drop limits are
used in the steady-state thermal hydraulic analysis to reflect the current and 5 year
expected enhanced states of pumping technology. While losses occur throughout the
entire primary coolant loop, the limit is based on the pressure drop across the fuel bundle,
because it will vary most among the redesigned cores.
The lower pressure drop limit indicative of current PWR pumping capacity is
determined in the same manner as the steady-state MDNBR limit: finding the pressure
drop across the fuel bundle given by VIPRE for the reference core geometry and
operating conditions. This pressure drop is approximately 29 psia. The enhanced
pressure drop limit is based on examination of pumping capacities for the Westinghouse
AP600 and AP1000 PWR designs, and a survey of industry experts. The pressure drop in
the core nearly doubled between the design of the AP600 and AP1000, and so it is
reasonable to believe that in the next 5 years before a hydride fueled core could be
constructed, the capacity could again double. The enhanced fuel bundle pressure drop
limit is therefore 60 psia. The maximum power is presented for both pressure drop
limits in Section 2.2.
2.1.2.3 Fuel Temperature
Based on experience with hydride fuels used in TRIGA reactors, a steady-state
fuel centerline temperature limit of 750 C was adopted for this analysis [1]. The limit is
based on prevention of hydrogen release from the fuel during steady-state irradiation.
24
Excessive hydrogen release can embrittle the clad, pressurize the fuel pin, and introduce
an explosive hazard into the core. At temperatures below 750 C, the partial pressure of
hydrogen gas is low, and the gas remains evenly distributed in the fuel matrix.
Unlike hydride fuels, oxide fuels release non-negligible amounts of fission gas,
which, if not limited, can pressurize and even burst the fuel pin. Based on fuel
performance data for oxide fuels, the fission gas release fraction can be kept below 5% by
limiting the average fuel temperature to 1400 C [4]. This is the temperature limit adopted
for oxide fuel. Note that this is more limiting than imposing a peak centerline
temperature limit of 2800 C, which is the melting point of UO2. Note that this
temperature limit only applies to steady-state operation; J. Trant's thesis [3] will
determine if core designs from the steady-state analysis exceed temperature limits during
transients.
2.1.2.4 Axial Flow Velocity
Because the enthalpy rise across the core is fixed, a specific geometry can only
achieve higher powers by increasing the coolant flow through the core. As the bulk flow
gets larger, the turbulent axial and cross-flow velocities increase, making a vibrationrelated rod failure more probable. It is therefore desirable to provide design limits to
restrict rod vibrations, and resultant wear at the cladding/rod support interface.
In lieu of a detailed vibrations and wear analysis for each core design, J. Malen
imposed a single limit on the hot channel axially averaged velocity. The limit was based
on a judgment during the initial phase of the Hydride Fuels Project that vibration
problems could be avoided in PWRs by limiting the axial velocity of coolant to 7 m/s.
The limit adopted was 8 m/s, under the assumption that additional grid spacers would be
added if deemed necessary by a separate fluid-elastic instability analysis of select,
optimum geometries.
The wide range of geometries considered and the large power increases reported
by the thermal hydraulic analysis makes it prudent to refine this single limit approach. A
25
more thorough analysis of relevant vibration and wear mechanisms is required. This
analysis is performed in Chapter 3 of this report. Because the purpose of Chapter 2 is to
update J. Malen's results for steady-state maximum power, and to prove the equivalence
of hexagonal and square array designs, the single velocity limit is maintained. The final
thermal-hydraulic results used in the economic optimization study in Chapter 4, however,
will be based on the updated vibrations and wear design limits presented in Chapter 3.
2.1.3 MATLAB/VIPRE Interface
The VIPRE sub-channel analysis code was developed by the Electric Power
Research Institute (EPRI) to aid in thermal hydraulic modeling and design of LWR cores.
More specifically, it predicts the velocity, pressure, and thermal energy fields, and the
fuel rod temperatures for interconnected flow channels. VIPRE also predicts DNB
performance. To execute VIPRE, a detailed input deck must be constructed in which
every facet of the core geometry and the operating conditions must be prescribed. The
output deck is an exhaustive text document that must be manually read by the analyst to
determine if a given core is operating within pre-defined design limits. Both input deck
generation and output interpretation are prone to human error by nature of the complexity
of the formatting and the sheer magnitude of the information.
The capabilities of VIPRE have been greatly expanded through the efforts of J.
Malen and another former MIT student Stu Blair [5] by the development of student
written MATLAB scripts. S. Blair initiated the MATLABN/IPRE interface by
developing scripts that automatically generate VIPRE input decks and extract relevant
VIPRE output with minimal input required from the user. The output is saved into
MATLAB space, where it can be easily interpreted and manipulated by the user. Using
this basic construct, J. Malen coded additional scripts that iteratively determine the
maximum power for a range of geometries subject to user defined design limits. The
programs execute according the following sequence:
26
* The design space is specified by the user, which includes the desired range of rod
diameters and P/D ratios, as well as the discretization within each range (i.e, 20
equally space rod diameters between 6.5 mm and 12.5 mm);
·
MATLAB programs generate a VIPRE input deck for the first geometry in the
parametric study, and supply an initial guess for power;
* VIPRE is executed, and the constrained parameters (i.e, MDNBR, pressure drop,
flow velocity, fuel temperature) extracted from the VIPRE output file are
compared with the design limits within MATLAB;
* If no design limits are exceeded, the power estimate is increased, and VIPRE is
executed with an updated input deck;
*
If one or more design limits are exceeded, the power estimate is decreased, and
VIPRE is executed with an updated input deck;
* Each VIPRE execution is followed by a comparison of the constrained parameters
with their respective design limits;
*
The maximum power occurs when one constrained parameter meets its design
limit, and the remaining parameters stay below their design limits;
This procedure continues for every geometry in the parametric analysis. Note that
the programs can only be used to evaluate maximum power for square array designs.
Considerable modifications to the scripts would be necessary to consider hexagonal
arrays. The reader is referred to [1] for a thorough description of the MATLAB scripts.
2.2 Results of the Steady-State Maximum Power Analysis
In preparation for the thermal hydraulic comparison of square and hexagonal
array PWR geometries, the updated maximum power results for square arrays are
presented. Figures 2.1 and 2.4 show the maximum achievable power for square arrays of
UZrH1. 6 and U0 2 as a function of P/D ratio and rod diameter for both pressure drop
limits. Color maps power, where "hotter" colors indicate higher powers. The color scale
is shown at the right of each plot. The maximum power for the enhanced pressure drop
case occurs at: P/D - 1.42, Drod- 6.8 mm. The power is - 5458 MWth, almost a 45%
27
increase over the reference core power. A less substantial, but still sizeable increase is
realized for the lower pressure drop case, where the maximum power is -4245 MWth.
The geometry is shifted slightly from the high pressure drop case: P/D - 1.48, Drod= 6.5
mm.
It is helpful to understand how the power gains are realized over the design space.
For a fixed core length, the maximum power, L~, depends on the average linear heat rate,
q', and the number of rods in the core, N.
-- '- N L
(2.4)
As long as a design limit is not exceeded, the power can be raised by increasing the
average linear heat rate and/or the rod number. Figures 2.2 and 2.5 show the ratios of the
rod number, average linear heat rate, and power for new geometries to the reference core
rod number, average linear heat rate, and power. Note that the product of Figures
2.2A/2.5A and 2.2B/2.5B yield Figure 2.2C/2.5C. A black line is used to denote the
contour where each ratio is unity.
Figures 2.3 and 2.6 plot the constrained parameters that determine the maximum
achievable power for each geometry. Regions where constraints are limiting are shown
as dark red. Note that because the fuel temperature limits for UO 2 and UZrH .1 6 are never
reached, their maximum achievable powers are identical.
28
Figure 2.1 Maximum Achievable Power vs. P/D and Drod for Square Arrays of tUZrH
60 psia
Core Power (x10 6 kWlkt;
6
and U0
2
at
a
U
1'2
5.5
5
1 1
45
4
10
3.5
"o,
:
Q
3
2.5
g
2
1.5
1.1
1 15
1.2
1.25
1.3
'
1.35
1.4
1
1.45
1.5
P/I
Figure 2.2 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drod for Square Arrays of
UZrH. 6 and UO2 at 60 psia
B: q/qre
A: N/N ref
12
12
11
11
10
2
o
1.5
IC
'
1
.0
1
I
7
1.1
1.2
1.3
1.4
0.5
7
I
.7
0
1.5
P/D
C PP
1.1
1.2
1.3
P/D
1.4
1.5
12
1.5
11
1
0.5
7
f:9
1.1
1.2
1.3
1.4
1.5
I
P/D
29
Figure 2.3 Constraining Parameters vs. P/D and D,,d for Square Arrays of UZrHL.6 and ITO2 at 60
psia
A: Pressure Drop (psia)
ill
"a
B: W3-L MDNBR
--
60
12
-2.5
11
11
-3
1(
1-1
I
20 O
-3.5
8
7
-4
rl
1.1
1.2
1.
14
1.1
1.5
1.2
;
1.4
1.5
D: Fuel CL Temp. UZrH. , (C)
8
12
12
6
11
700
-11
600
:10
9
4
019
500
8
8
7
2
7
1.1
1.2
13
1.4
1.1
1.5
1.2
1.3
1.4
1.5
400
ETemp
Fuel
UO(C)PD
Ave.
A
_
^.
' 14UU
12
1200
1000
10
9
c
800
8
600
7
Ann
-tuu
1.1
1.2
1.3
1.4
1.5
P/D
30
Figure 2.4 Maximum Achievable Power vs. P/D and D,.,d for Square Arrays of UZrHI.6 and UO2 at
29 psia
Core Power (xl 06kWth)
6
12
5.5
5
11
K
45
4
1'
3.5
11
3
2.5
2
1.5
7
1
'1
1.15
1
125
1.2
1.3
1.3
1.4
1.45
1.5
P/D
Figure 2.5 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drodfor Square Arrays of
UZrH *1 6 and UO02at 29 psia
A: N/N
B: q/%f
ref
1
3
'1
10
'O
.. ) 9
2
1
1
1.5
1
1
0.5
8
1.1
1.2
1.3
P/D
1.4
1.5
I
n
v
1.1
C P/P
1.2
1.3
P/F
1.4
15
I
2
1.5
1
I:::
0.5
0
9
0.5
1.1
1.2
1.3 1.4
PAD
1.5
I
31
Figure 2.6 Constraining Parameters vs. P/D and D,.odfor Square Arrays of UIZrHI.6and UtO2 at 29
psia
B: W3-L IvDNBR
A: Pressure Drop (psia)
12
I
1 '1
c1c
-2.5
11
20
1l0
15
9
Q)
12
25
-3
0t 1yA
10
8
8
l
5
I
1.1
1.2
1.3
P/D
1.4
7
1n
U
1.5
1.1
12
.D
1.4
1.5
-3.5
A
-~t
D: Fuel CL Temp. UZrH. (C)F
C: Axial
rd
12
I 1U
11
I
12
18
11
-
I
700
600
210
4
O9
500
8
8
2
7
f:
1.1
1.2
J
1.4
1.1
1.5
1.2
1.3
1.4
1.5
I
Ann
Y.UU
P/D
E: Fuel Ave. Temp UQ(C)
14UU00
12
,
I
11
1200
10c
1000
0
800
Q 8
600
7
I
1.1
1.2
1.3
1.4
Ann
'-tU
1.5
P/D
32
2.3
The Equivalence of Square and Hexagonal Array
Geometries
Confined square and hexagonal array sub-channels at the same H/HM ratio and
rod diameter have the same coolant flow areas and heated and wetted perimeters, which
determine thermal hydraulic performance. It is this fact which motivates the extension of
the square array results presented in Section 2.2 to hexagonal designs. It is desirable,
however, to prove this equivalency between full core VIPRE models, which, unlike the
single channel analysis, accounts for the turbulent interchange of mass, momentum, and
energy among sub-channels.
2.3.1 Geometric Relationships for Square and Hexagonal Arrays
Before discussing the construction of the full core hexagonal VIPRE model, the
relationships between the P/D and H/HM ratios for square and hexagonal arrays are
developed.
nce defined, the geometric equivalence of square and hexagonal arrays at
the same H/HM ratio and rod diameter is shown. Note that this information was first
developed by J. Malen and is presented in [1]. Recent changes to the fuel rod cladding
and gap thickness correlations, however, invalidate portions of the original derivation.
The relationships are therefore re-developed in this report incorporating all updated
information.
2.3.1.1 Conversion Between Hydrogen/Heavy Metal and Pitch/Diameter
Ratio
The general notations for variables used in this derivation are defined in Table
2.3.
33
Table 2.3 General Nomenclature for Geometric Relationships for Square and Hexagonal Arrays
-
Name
Avogadro's Constant
Symbol
Cladding Thickness
tcl
Units
Value UZrH,. 6 '----
mm
Value U0 2
6.02x1023
6.02xl 023
(2.14) & (2.16)
(2.14) & (2.16)
NA
3
-6
10.43x10 - 6
Densityof the Fuel
Pfuel
kg/mm
Density of Water (at 700 F)
PH20
kg/mm 3
6.67x10- 7
6.67x10- '
Diameterof Fuel Pellet
Diameterof Fuel Rod
Dpellet
mm
(2.18)
(2.18)
D
mm
tg
mm
(2.15) & (2.17)
(2.15) & (2.17)
MHM
kg/kmol
237.85
237.85
MMatrix
kg/kmol
93.2
MH20
kg/kmol
18
18
I
I
Gap Thickness
Molecular Weight of Heavy Metal
MolecularWeightof Fuel Matrix (ZrH,.6 )
Molecular Weight of Water
Number of Heavy Metal Atoms Per Unit of
the Fuel Matrix Element
8.256x10
Y
HM
Number of Heavy Metal Atoms
Number of Hydrogen Atoms Per Unit of the
Fuel Matrix Element
X
Number of Hydrogen Atoms
H
Pitch
P
Volume
V
Weight Percent Heavy Metal
w
1.6
mm
mm 3
.45
.8813
The number of hydrogen atoms in a prescribed volume of water and fuel (i.e. in a
sub-channel/unit cell) is given by:
(2.5)
H = HHI + Hfuel
where,
HH
Hfu
0
-=2'
NA PHO
.
VHO
MHO
Hel=X. NA Pfuel' Vfuel ( - w)
=: X e(MMatr
(2.6)
(2. 7)
Matrix
where,
X:
number of hydrogenatomsper unit of thefuel matrix
w:
weight percent heavy metal in thefuel
Note that Hfuelis 0 for UO2. The number of heavy metal atoms in a prescribed volume of
fuel is given by:
34
HM = YNA
Pfiel Vie W
(2.8)
MHM
where,
Y:
number of heavy metal atoms per unit of the fuel matrix
Taking the ratio of equations (2.5) to (2.8) gives the H/HM ratio:
H_
HM
(.2).(1).MM
Y
w
MHO )
.+(.M.
Pfuel )
fuel
Y
mMatrix
W
W)
)
(2.9)HYMNM
Square Array
For square arrays,
VH 2 0 : = Aflow-sq
L
(2.10)
Vfuel = Afuel* L
Aflow-square
= pq ,r
(2.11)
4
2
pellet
4
Afue
Afuel
=
(2.12)
(2.13)
To determine the diameter of the fuel pellet, the radial gap and cladding
thicknesses must be specified. The original correlations for gap and cladding thickness
used in [ 1] scaled linearly with rod diameter. It is believed by industry experts, however,
that this leaves the gap and cladding too thin at small rod diameters. New correlations
were therefore adopted that impose a minimum cladding and gap thickness.
ifDrod < 7.747 mm
to= .508 mm
tg
= .0635 mm
(2.14)
(2.15)
ifDrod > 7.747 mm
tc
1 = .508 +.0362 (D - 7.747) mm
(2.16)
tg =.0635+.0108. (D- 7.747) mm
(2.17)
Dpellet:=D - 2 -t -2 t
(2.18)
35
Substituting constants from Table 2.3 and equations (2.12) and (2.13) into equation
(2.9)
gives the H/HM ratios for square arrays of UZrH1.6 and U0 :
2
HM
=4.745.
+ 4.991
UZrH1.6
= 1.918
(2.19)
(2.20)
(HM UO2
Rearranging and solving for P/D gives the desired relationship between the P/D
and
H/HM ratios:
r
(P.
=j. 166 l
D
PDsq,UZrH1.6
(D sq,Uo2
1
(41Dpellet)
991 + 0.785
)HM
H
+ 0.785
(2.21)
(2.22)
Hexagonal Array
Equation (2.9) can also be applied to hexagonal arrays.
L
VH20 = Aflow-hex
Aflow-hx
=:
J.p2*
4
(2.23)
-
4
D2
8
(2.24)
Substituting equations (2.23) and (2.24) and the constants in Table 2.3 into equation
(2.9)
gives the H/HM ratios for hexagonal arrays of UZrH 1.6 and UO :
2
2
HM
)UZrH .6
Z=
47451
TD +4.991
4.-
Ph2x D2 _
Dpellet
(2.25)
36
H
HAL
(2.26)
=1.918
=
U~o 2
Rearranging and solving for the P/D ratios gives:
H
pellet
HM
eD
hx,'H11 16
(PI =
h
(227)
+0.
0.191.~~~~~~~~~~~~
ts=
0.473
pelet
j(
(2.28)
+0.907
*
Thus it is shown that the P/D ratio depends on both the HIHM ratio and the rod diameter.
The H/HM ratio is shown graphically in Figure 2.7 as a function of rod diameter and P/D
ratio for square and hexagonal arrays of UZrH.
6
and U0 2. Note that the rod diameter has
very little influence on the H/HM ratio.
Figure 2.7 H/HM Ratio vs. P/D and Drod for Square and Hexagonal Arrays of UZrH. 6 and UO2
H/HM: Hex. Arrays of UZrH 16
H/HM: Sq. Arrays of UZrH 1.6
201
--
18
20 1'.
116
:
16
14
12
.,:
0
rod (mm) 8
H/HM: Sq. Arrays of UO 2
8--
cL
_
8l.
42-,
012
od 8
Do(mm)
1.1
130
' 13
PD
12 PD
HIHM: Hex. Arrays of UO 2
-l
1W-
18
10;
10
i
12:
1W
10
5
4
4:
2-
3
01
1.512
0P/
. .
2
7
.
641~
r
2
12
D, (mm)8(mm)
1.4
.
1.2 P/D
37
2.3.1.2 Relationship Between Square and Hexagonal Array Sub-Channels
To prove the geometric equivalence of square and hexagonal array sub-channels
at the same rod diameter and H/HM ratio, a relationship between the square and
hexagonal pitch is determined. This proof is only carried out for UZrH 1.6, but could
easily be performed for UO2 given the equations in Section 2.3.1.1.
For equivalent rod diameters and H/HM ratios, a constant C is defined such that,
Dpellet 2
(H
(2.29)
Substituting equation (2.29) into equations (2.21) and (2.27) gives the square and
hexagonal P/D ratios for UZrH1 .6 with respect to C.
=O.166 C+0.785
PI
D
= Jo ll
(P)
D
(2.30)
sq,UZrHI.6
c+0.907
(2.31)
hex,UZrH1.6
Solving for Psqand Phe:
PSq= D
.166C + 0.785
(2.32)
Phe,= D
0.191.C + 0.907
(2.33)
1.0746. Pq
= Phe
(2.34)
For equivalent combinations of rod diameter and H/HM ratio, equation (2.34) can be
used to relate the rod pitch between square and hexagonal lattices. This relationship also
holds for UO2, though the derivation is not provided.
The flow areas and heated and wetted perimeters are now presented for square
and hexagonal arrays. Square and hexagonal unit cells are shown in Figure 2.8.
38
Figure 2.8 Square and Hexagonal Array UJnitCells
jr-f6>(~N
TI
\ '~I
'(
{ A )
Phex/
Psq \
f2
Drod
)
Drod
For the square array, the geometric relationships are:
Aflo-q ..
-
P,sq = Psq =
(2.35)
4
(2.36)
D
The geometric relationships for the hexagonal sub-channel, with Phexwritten with respect
to Psy according to equation (2.34), are:
/3 ·(1.0746.P
A
i-hex
P,,,.hex = fh,he.- =
4
q
.
Dr.
(2.37)
-. = 0.5 A2orq
8
= 0. 5
2
Z
Psq
(2.38)
= 0.5. Ph,,q
The flow area and heated and wetted perimeters for the hexagonal sub-channel are
exactly one half the corresponding values for the square sub-channel.
They are identical,
however, on a unit rod basis (hexagonal sub-channels have 0.5 rods and square subchannels have I rod). Because the thermal hydraulic performance of new core designs
depends on the total flow area and total heated and wetted perimeters, hexagonal and
square designs with the same rod number, H/HM ratio, and rod diameter will have the
same total flow area and heated and wetted perimeters. Thus the geometric equivalence
between square and hexagonal arrays is shown, and it is believed that the square array
steady-state thermal hydraulic results presented in Section 2.2 can be extended to
hexagonal arrays by modifying the P/D ratio according to equation (2.34):
)
D
= 1.0746
hex
()
(2.39)
D q
39
This means that the power at Drod = 6.5mm and (P/D)sq = 1.29 should be the same as that
for Drod = 6.5mm and (P/D)hex
=
1.387, if differences in lateral mixing are negligible.
This is proven in the next section by constructing a full core hexagonal VIPRE model.
2.3.2 Full Core Hexagonal Array VIPRE Model
As discussed in Section 2.1, the mixing characteristics that determine the
interchange of mass, momentum, and energy among sub-channels are slightly different
for square and hexagonal array designs. To verify that this discrepancy does not translate
into more significant differences in core power, a full core hexagonal VIPRE model was
constructed for a single geometry for comparison with the square results. Like the
analysis for square arrays, the maximum power was determined subject to design limits
on MDNBR, pressure drop, fuel centerline temperature, and axial velocity.
It is assumed that the reader has some familiarity with the VIPRE code; it is
therefore not the intent of this section to explain in exhaustive detail the construction of a
VIPRE input deck and the interpretation of the code's output. Rather an overview of the
full core model is provided, followed by the presentation of results. The fixed operating
conditions and dimensions adopted from the reference core for the square analysis are
also applied to the hexagonal core. For readers interested in these details, please refer to
Appendices B and C, where the thermal hydraulic assumptions and the VIPRE input deck
are provided.
The square and hexagonal geometries chosen for this single model verification
are:
Dsq = Dhex = 6.5mm
(2.40)
(P
(2.41)
=1.29
(De=
H sq
HM
1.0746 = 1.386
-
q
(H
M
e
= 1281
(2.42)
(2.43)
40
2.3.2.1 Rod and Assembly Layout Within the Core
Due to symmetry, a full hexagonal core can be adequately represented by a 1/6t"
section. Because the pressure vessel radius, rod diameter, and pitch are fixed, the number
of hexagonal fuel assemblies that can fit in the core depends on the number of rings in
each assembly. A 9 ring hexagonal fuel assembly was adopted because of its common
use in liquid metal cooled fast breeder reactor designs.
Figure 2.9 9 Ring Hexagonal Fuel Assembly
L
DI
3
Dft
The dimensions of the fuel assembly are defined by the length of one side of the hexagon,
D,, and the distance across the flats of the hexagon, Dft.
Dj1 =
j2 .Nrings
Phex
+g
(2.44)
D). ()5
(2.45)
+
where,
g
= (Phx
l,=(-p¢1)-"he.
+3
.2+g
3
22)
(2.46)
where,
Nps:
number of rods per side
The rod diameter and pitch are given by equations (2.40) and (2.42), and the reference
core pressure vessel radius is given in Table 2.1. With this information, the number of 9
41
ring fuel assemblies that fit radially in the core starting at its center (with the center rod of
the first assembly at the center of the core) is 11.5. The 1/6th core section is shown in
Figure 2.10.
Figure 2.10
1 /6th Hexagonal
Core Section
11
VIPRE requires that every sub-channel in the core model be specified in the input
deck, which can be a tedious task for the user. For the design range considered in the
parametric study, the number of channels in the core can easily approach 100,000. To
ease this burden, and lessen the computational strain for program execution, VIPRE
allows channel lumping. A lumped channel is simply a large grouping of individual subchannels; it is specified by the total heated and wetted perimeters and flow areas of the
individual channels that it comprises. Communication of mass, momentum, and energy
are still allowed, although fluid properties and conditions are uniform within each lumped
channel. The lumped channels for the 1/6th core section (# 82 - 91) are shown in Figure
2.11.
The radial power distribution is peaked near the center of the core, so the hot subchannel in the core always occurs within the central assembly2 . A fine mesh of
individual sub-channels is maintained throughout this central assembly, and outer
2 The
radial power profile is provided in Appendix B. 1.9
42
assemblies are lumped together in increasingly larger areas. This lumping effect is
shown in Figure 2.1 1, where the hot assembly comprises the first 82 flow channels, and is
represented by the 1/6th assembly section at the top.
Figure 2.11 Channel Lumping in the 1/6h Hexagonal Core
Figure 2. 12 shows a close up of the hot assembly section, and demonstrates the
correct numbering technique for fuel rods and sub-channels in VIPRE. Note that control
rods are not numbered. The position of control rods in the assembly was randomly
selected, but the number is chosen so that the number of fuel rods per control rod in the
reference core is maintained in the hexagonal core.
43
Figure 2.12 1/6thSection of the Hot Assembly
2.3.3 Comparison of Maximum Power Predictions from Square
and Hexagonal Arrays
The powers predicted by the square and hexagonal full core models are shown in
the Table 2.4.
Table 2.4 Power Predictions by Full Core Square and Hexagonal VIPRE Models
SquareArray
HexagonalArray
Drod
6.5 mm
6.5 mm
P/D
1.29
1.387
H/HM
12.21
12.21
Heated Rods
q'
Limiting Constraint
111,760
108,433
2.49 kW/ft
2.46 kW/ft
3933 MWth
3740 MWth
Pressure Drop
Pressure Drop
44
The difference in predicted power is - 5%. The majority of this discrepancy has
to do with how efficiently the core cross section is utilized by the fuel assemblies. Both
models demonstrate the same heated and wetted perimeters and flow areas per unit rod,
because the rod diameter and H/HM ratios are the same. Notice, however, that the
number of heated rods in the core is larger for the square model. This is because the
pressure vessel dimensions are fixed, and the hexagonal and square assemblies do not
utilize the space in the core with the same degree of efficiency (one model leaves more
wasted, unheated flow area at the core's periphery). Vessel space could be optimized for
the hexagonal model by changing the number of rings in each assembly, but this is not
considered in this analysis. The core power is simply the product of the number of
heated rods and the average linear heat rate, which is independent of the core cross
section. The linear heat rate is therefore a better indicator of thermal hydraulic
performance than the power.
The linear heat rates for the square and hexagonal array models are very close,
differing by - 1.2%. This discrepancy could be caused by several reasons: the precision
of convergence to thermal hydraulic design limits; differences in the sub-channel friction
factor; and differences in the turbulent interchange of mass, momentum and energy.
Because both models are limited by pressure drop, the friction factor difference is most
likely the culprit. The difference, however, is so minute that it is safe to conclude that the
steady-state thermal hydraulic analysis for full core square arrays can be safely extended
to hexagonal arrays for matching rod diameters and H/HM ratios.
45
46
3.
Vibrations Analysis for Hydride and Oxide
Fueled PWRs
3.1
Introduction
Dynamic forces generated by the turbulent flow of coolant in PWR cores cause
fuel rods to vibrate. Flow-induced rod vibrations can generally be broken into two
groups: large amplitude "resonance type" vibrations, which can cause immediate rod
failure or severe damage to the rod and its support structure, and smaller amplitude
vibrations, responsible for more gradual wear and fatigue at the contact surface between
the fuel cladding and rod support. While the former group is typically prevented by
adequate structural design of the fuel assembly, the latter is unavoidable. Sufficient wear
resistance must therefore exist in the fuel assembly components to preclude excessive
damage. Ultimately, both vibration types can result in a cladding breach, and therefore
must be accounted for in the thermal hydraulic design of hydride and oxide fueled PWRs.
The thermal hydraulic analysis to determine the maximum achievable power for
hydride fueled cores did not account for specific vibration mechanisms; instead, a single
limit on the axial flow velocity was imposed [1]. This limit was based on a judgment
during the initial phase of the Hydride Fuels Project that vibration problems could be
avoided in PWRs by limiting the coolant axial velocity to 7 m/s in the core. The wide
range of core geometries considered and the large power increases reported by the
thermal hydraulic study makes it prudent to refine this single limit approach. A more
thorough analysis of relevant vibration and wear mechanisms is needed, with appropriate
design limits imposed for each mechanism.
3.2
Work Scope
3.2.1 Goals of the Vibrations Analysis
The thermal hydraulic analysis for hydride fueled PWRs linked a series of student
developed Matlab programs and the VIPRE sub-channel analysis code to iteratively
determine the maximum achievable power for a range of core geometries, subject to user
47
defined design constraints. Constraints included minimum departure from nucleate
boiling ratio (MDNBR), fuel centerline temperature, bundle pressure drop, and axial flow
velocity. The maximum power reported by the study for a given geometry was the
highest power for which no constrained parameter exceeded its design limit.
The goal of the vibrations analysis is to develop and incorporate new design limits
for flow-induced vibration and wear mechanisms into the existing thermal hydraulic
programs, replacing the single limit on axial velocity. The results will include new maps
of steady-state power for PWR geometries utilizing hydride and oxide fuels. Combined
with the transient analysis performed by J. Trant [3], the thermal hydraulic analysis for
maximum power will be complete.
3.2.2 Flow-Induced Vibration Mechanisms - Overview
Three primary types of flow-induced vibration are observed for cylindrical fuel
elements subject to cross and axial flow:
*
Vortex-Induced Vibration: Vortex-induced vibration can occur by two means:
vortex shedding lock-in and vortex-induced acoustic resonance. In vortex
shedding lock-in, the frequency of the vortices shed by cross-flow over the
fuel rod "lock in" to the rod's structural frequencies, causing resonant
vibration. In vortex-induced acoustic resonance, the shedding frequency
excites standing acoustic waves created by the operation of fans, pumps,
valves, etc. in the coolant loop3 . Because the rules to avoid lock-in are more
conservative than the rules to avoid acoustic resonance, only vortex shedding
lock-in is considered in this analysis.
* Fluid-Elastic Instability: Fluid-elastic instability of a rod bundle occurs when
the cross-flow velocity exceeds the critical velocity for the bundle
3 Standing waves are required for the acoustic resonance condition. They are formed when acoustic waves
traveling in opposite directions (as when an acoustic wave deflects off of fuel rods) superimpose onto one
another.
48
configuration, at which point the rod response increases uncontrollably and
without bound.
Turbulence-Induced Vibration in Cross and Axial Flow: The fluctuating
pressure fields generated by cross and axial flow turbulence in the core exert
random forces on the fuel rods, causing vibration.
The vibration amplitudes associated with vortex shedding lock-in and fluid-elastic
instability are generally very large, and can quickly cause severe damage to the fuel rod
and its support structure. If the pitch is tight enough, rod failure by tube-to-tube
impaction is also possible. Fortunately, these devastating mechanisms can be prevented
by adequate design of the fuel assembly structure for the flow conditions in the core (i.e.
using an appropriate number of grid supports and providing adequate stiffness to the fuel
rod).
Unlike vortex shedding lock-in and fluid-elastic instability, turbulence-induced
vibration is generally of small amplitude and cannot be avoided. The principle design
concern is therefore not the prevention of the vibration mechanism, but the limitation of
resultant wear at the cladding/rod support interface. Wear is a concern for two reasons.
First, excessive wear can directly breach the clad or increase the likelihood of a breach
from other rod damage mechanisms (i.e. impact stress and fatigue). Second, wear at the
cladding/rod support interface lowers the structural frequencies of the rod, making it
more susceptible to vortex-induced vibration and fluid-elastic instability.
The most common wear mechanism, and historically the most costly flowinduced vibration problem in the nuclear industry, is fretting wear. Fretting results from
combined rubbing and impaction between the fuel rod support and the cladding surface.
Sliding, or adhesive, wear also occurs where the grid support springs and rod rub against
one another. Both wear types are considered in this study.
49
The mechanisms considered and their respective design concerns are summarized
in Table 3.1.
Table 3.1 Flow-Induced Vibration Mechanisms
Flow-Induced Mechanism
Vortex-Induced Vibration
*
Fluid-Elastic Instability
*
Large amplitude vibrations occur when cross-flows exceed
the critical velocity for the rod bundle configuration
Turbulence-Induced Vibration in
Cross and Axial Flow
*
Small amplitude rod vibrations from turbulence generated
pressure fields cause excessive fretting and sliding wear at
the cladding/rod support interface
Design Concern
Large amplitude vibrations occur when vortex shedding
frequencies lock-in to the structural frequency of the rod
3.2.3 Methodology
The vibrations analysis is performed using the MATLAB/VIPRE interface
described in Section 2.1.3. In order to consider the vibrations and wear mechanisms
within this construct, constrained parameters with appropriate design limits need to be
developed, which is undertaken in Section 3.4. Unlike the steady-state analysis described
in Chapter 2, the constrained parameters for this vibrations and wear analysis cannot
simply be "measured" during core operation. Rather they are dependent on multiple
parameters including the structural properties of the fuel rod, sub-channel geometry, and
the cross and axial flow distributions within the core. With additional MATLAB scripts,
the new limits are incorporated into the existing thermal-hydraulic program structure,
where the iterative approach to maximum power can once again be employed.
3.3
Assumptions
3.3.1 Vibrations Analysis in the Nuclear Environment
Vibrations analysis for nuclear fuel rods is extremely complex given the random
nature of turbulent flow in the core. Adding to the complexities, severe thermal,
mechanical, and radiation loads, as well as wear accumulation, continuously change the
structural mechanics of fuel assembly components. Accurately predicting the vibration
50
response of fuel rods over time is therefore an arduous task at best, and even impossible
without the aid of state-of-the-art finite element analysis (FEA) codes.
3.3.2 Key Assumptions
The following simplifying assumptions are made so that the vibrations analysis
can be performed without the aid of advanced computational tools and with best-practice
guidance in the academic literature:
The fuel rod is modeled as a linear structure: This assumption is based on
treating the grid supports as single pin supports. In reality, the gapped support
condition between the grid spacer and the fuel rod allows relative movement
between the two components. With this movement, non-linear FEA codes are
needed to quantitatively model the rod response, which is beyond the scope of this
work. A linear rod model and experimental correlations for rod response are used
as a substitute.
* Changesto thefuel assemblystructureover time are not considered: Core
operating conditions play a significant role in the structural mechanics of fuel
assembly components during fuel irradiation. For example, creep-down of the
cladding due to pressure forces, support spring relaxation due to irradiation, and
wear accumulation combine to slowly open the gap between the fuel rod and its
support. Oxidation from temperature extremes and irradiation change the
material properties of all structural components in the core. Rod bow may also
occur, changing the rod/support structure interaction and the flow distribution of
coolant in the core. Because of the difficulty associated with modeling these
effects, and the lack of guidance outside of proprietary vendor computer codes,
structural changes to the rod and support structure are neglected.
* Only the cladding structure is considered in thefuel rod model: A gap exists for
fresh fuel rods between the fuel pellets and the cladding. Over time, the fuel
swells closing this gap, and it contacts the cladding surface. In addition, gases
51
generated by the fission process and any burnable absorbers present pressurize the
pin. For conservatism, the additional rigidity provided by fuel swelling and rod
pressurization is not considered.
*
Only the first vibration mode is considered: The first vibration mode
(fundamental mode) typically has the largest impact on rod vibration. With
regard to vortex shedding lock-in and fluid-elastic instability, the use of the first
mode typically yields the most conservative design margin. Furthermore, several
correlations used in place of FEA codes for modeling turbulence-induced
vibration response are only applicable for the fundamental mode.
* Corepower is the only operatingparameteraffecting the vibrationsperformance
of new designs: Implied in the iterative approach to maximum power with
vibrations imposed design limits is the sole dependence of vibrations performance
on core power. A design that fails a vibrations design limit can be made
acceptable, however, by changing other parameters. One example is modifying
the fuel rod and assembly hardware to affect the vibration frequency and
amplitude of the rod. This can be accomplished by using additional grid spacers,
thicker cladding, or a combination of the two. Both, however, have trade-offs.
Thicker cladding implies less fuel per rod, and more grid spacers, while making a
design more vibration resistant, will increase the pressure drop across the core.
For this analysis, the cladding thickness is not considered variable for vibrations
design purposes; additional grid spacers will only be considered if a geometry
fails to meet all vibrations design limits with modest power reductions.
These are the key assumptions used in the vibrations analysis. Additional
simplifications for specific vibration and wear mechanisms, where needed, are presented
in subsequent sections.
52
3.4 Vibration and Wear Mechanisms
3.4.1 Dynamic Parameters
The dynamic parameters and material properties used in the vibrations analysis
are defined in this section. They are derived from [6].
3.4.1.1 Nomenclature
Table 3.2 defines the general notation for properties and parameters used in the
derivations throughout this section.
Table 3.2 General Nomenclature for the Vibrations Analysis
Name
Added Mass Coefficient
Average Length Between Spacers
Cladding Density
Cladding Inner Diameter
Cladding Linear Mass
Cladding Moment of Inertia
Cladding Outer Diameter
Symbol
Units
Cm
Ls
m
Pet
Dcl,
mc,
Ici
Dco or D
kg/m 3
m
kg/m
m
tcl
m
E
Pfl
mfl
C
N/m 2
kg/m 3
kg/m
Cladding Thickness 4
Cladding Young's Modulus
Coolant Density
Coolant Linear Mass
Damping Ratio
Value
6550
m
9.9063 x 101°
700
Mode Shape Function
NaturalFrequency
fn
Number of Grids
Number of Spans
ng
ns
Pitch
P
m
Total Fuel Rod Length
L,
m,
m
kg/m
Total Linear Mass
s
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
....
3.4.1.2 Linear Mass
Cladding Linear Mass:
mcI = Pcl
. (Dc - D )
(3.1)
4 Cladding thickness depends on the fuel rod diameter. See equations (2.14) - (2.17).
5 Coolant density is assumed to be constant, and equal to the average coolant density in the reference core.
53
Coolant Linear Mass:
Added Mass Coefficient:
where,
Total Linear Mass:
Mfl=Cm
"Il~e4
.(f)pfl D2,
(D *2 +1)
C = (D2
(3.3)
1)
}(P
D*(1+
(3.2)
(3.4)
~D
(3.5)
mt = mcl +mf
3.4.1.3 Moment of Inertia
Cladding Moment of Inertia:
(
ICI = 6 4
i)
o-
(3.6)
3.4.1.4 Natural Frequency and Mode Shape
1 st
Natural Frequency:
f(=2
.s
(3.7)
in
1 stMode
Shape Function:
where,
/l (x)=
jns
Lx
n, =ng -1
(3.8)
(3.9)
3.4.1.5 Damping Ratio
The following table shows ASME recommended fluid damping ratios for
different rod support conditions.
Table 3.3 ASME Recommended Damping Ratios [61
Mean Values
Conservative
Values
Cn,water or wet steam,
tightly supported tube
0. 015
C,nloosely supported tube
0. 01
0.03
0.05
Because a smaller damping ratio leads to larger vibration amplitudes and less margin
with regard to fluid-elastic instability, the most conservative value is chosen: = 0.01.
54
3.4.1.6 Generation and Distribution of Axial and Cross-flow Velocities
The vibrations performance of fuel rods depends on both the structural properties
of the fuel assembly and the distribution of axial and lateral flows in the core. Once the
geometry and power level are specified, VIPRE computes the coolant axial and lateral
(cross-flow) velocity profiles for each channel and gap specified in the full core model.
The distribution of flow around individual rods is accurately captured for the hot
assembly, which is at the center of the core. In the outer channels where lumping is used,
VIPRE computes average cross-flow and axial velocities. Because the flows are greatest
in the hot assembly, and the individual flow distributions around each rod are computed,
the vibrations analysis will only consider the coolant velocities in the hot assembly.
The vibrations and wear performance of new designs will depend on the
magnitude of the cross-flow and axial velocities. It is therefore important to establish
their relationship with core power, given the need for each constrained parameter in the
thermal hydraulic analysis to scale with power. Though this relationship may seem
intuitive, simplified proofs and examples are presented below.
3.4.1.6a Axial Velocity
Because the enthalpy rise, Ah, in the parametric study is fixed to the enthalpy rise
in the reference core, the power l for a specific geometry can be changed by either
increasing or decreasing the mass flow rate
.l=cor e Ah
lScore through
the core.
(3.10)
The mass flow rate is simply the mass flux G multiplied by the total core flow area.
nore
=
G Alow
(3.11)
For the conditions of low void fraction that exist in a PWR core, an average mixture
density Pmcan be used to relate the axially averaged coolant velocity to the mass flux.
G
Vaxial-
--
(3.12)
Pm
Making appropriate substitutions yields the desired relationship between average axial
velocity and core power.
55
Vaia
-=
fl
Ah =c
Pm
Ajow' Ah
(3.13)
It is therefore seen that the axial velocity for a specific geometry is directly proportional
to core power.
3.4.1.6b Cross-flow Velocity
For bare rod bundles, cross-flows between adjacent channels are generated by two
mechanisms: turbulent fluctuations in the axial flow that drive turbulent mass
interchange, and transverse pressure gradients that drive diversion cross-flow [2].
Turbulent interchange can be thought of as eddies of equal volume that cross the subchannel boundary. If these eddies are also of equal density, as in the case of single phase
flow, no net mass is exchanged. In two phase flow, however, mass is transferred from
channels of higher density to channels of lower density. Because the density difference
between adjacent channels in a PWR is relatively small, the cross-flow velocity
contribution from turbulent interchange is minimized. Diversion cross-flow is therefore
the dominant factor driving cross-flows.
The transverse pressure gradients that drive diversion cross-flow arise for two
reasons: geometric anomalies in the core (i.e., rod bow), and non-uniform changes in
coolant density. Geometric anomalies are not considered in this analysis. Density
changes between adjacent channels are due to radial heat flux variations, and larger local
differences accompanying the onset of boiling. The radial power profile peaks near the
core's centerline, and tapers off toward the outer assemblies. Moving through the core,
the enthalpy rise is greater for coolant in the centrally located channels. Adjacent
channels therefore undergo phase and property changes to a different extent at the same
axial location. This gives rise to pressure gradients across channels due to the
dependence of pressure drop on local quality and mixture density. These gradients drive
diversion cross-flow. As the power increases, the amount of coolant flowing into and out
of the hottest channels will also increase. Cross-flow velocities are therefore expected to
scale with core power.
56
This effect is illustrated in Figure 3.1, which plots the cross-flow velocity profile
given by VIPRE for a randomly selected gap in the hot assembly at three different
powers. There are two important things to notice from this figure. First, the cross-flows
are greatest for the highest power, and smallest for the lowest power. Hence, the crossflows do scale with core power. Second, the effect of grid spacers on the local cross-flow
velocity is to sharply increase its magnitude at each grid location. The first two grids are
non-mixing, and therefore the deviations from the normal velocity profile are small. The
next eight, however, have mixing vanes, and the magnitudes of the deviations are
significantly larger. The negative values indicate that coolant is flowing into the
channel.
Figure 3.1 Cross-flow Velocity Profile as a Function of Core Power
U.1
0.i
0.(
2O.
t
> -0.1
O-0
-0.1
0
20
40
60
80
100
AxialPositionon)
120
140
160
180
3.4.1.6c Cross-flow and Axial Velocity Inputs to the Vibrations Analysis
For conservatism, the local peak cross-flow and axial velocities are used in
performing the vibrations and wear analyses, with two exceptions. First, the large crossflows occurring at grid spacer locations are ignored because they are not representative of
the remaining cross-flows in the gap. Because they only occur at the grid locations where
the rod motion is constrained, their contribution to the overall vibration response of the
rods will be negligible anyway. A second exception applies specifically to the case of
fluid-elastic instability, where Au-Yang [6] recommends the use of an "effective" cross-
57
flow velocity in evaluating the fluid-elastic instability margin (See Section 3.4.3). The
effective cross-flow velocity for the nht mode is obtained by weighting the cross-flow
velocity profile in each gap by the vibration mode shape of the rod. Cross-flows
occurring at the mid-point between spacers, where the vibration amplitude is largest, will
therefore be given more weight than cross-flows occurring at the supports, where the
amplitudes are approximately zero. Neglecting changes in the coolant density and linear
mass axially along the gap, the effective cross-flow velocity for the first mode is given
by:
rz2 _ 0
reJj -
(3.14)
L,
Figure 3.2 is a plot of the first vibration mode shape function for a rod simply
supported by 10 grid spacers. Also shown is a representative cross-flow velocity profile
from VIPRE, with the cross-flows at the grids removed. The maximum, effective, and
average cross-flow velocities for the profile are reported on the figure. The effective
cross-flow velocity typically lies between the channel maximum and average values.
Figure 3.2 Cross-flow and Fundamental Mode Shape Profiles
0. 08
0.06
0.04
.5
0
U,
0.02
0
C,
(I)
0
2
U
-0.02
()rAt
-U.Ul
0
20
40
60
80
100
120
140
160
180
Axial Position (in)
58
3.4.2 Vortex-Induced Vibration
Cross-flow over cylindrical elements generates vortices that are shed alternately
from one side of the structure to the other. This shedding produces an uneven pressure
distribution around the rod, and resultant forces act on the structure in both the lift
(perpendicular to the flow) and drag (parallel to the flow) directions. The force
component in the lift direction has a frequency equal to the vortex shedding frequency,f,,
while the drag component has a frequency equal to twice the vortex shedding frequency.
When the vortex shedding frequencies are well separated from the natural
frequency of the structure, only mild vibrations occur. When the shedding frequency in
either the lift or drag direction approaches any of the rod's natural frequencies, a
phenomenon called lock-in occurs. In this event, the shedding frequency assumes the
natural frequency of the vibrating rod, causing larger resonance-type vibrations. This
effect may cause immediate failure of the rod, or lead to premature wear. Because the
shedding frequency may actually shift to the natural frequency of the rod, sufficient
separation must exist between the two frequencies to preclude this resonant behavior.
3.4.2.1 Vortex-Shedding Margin and Design Limit
The vortex shedding frequency is given by:
f=s=SV
D
(3.15)
where the Strouhal number, S, was found by Weaver and Fitzpatrick [6] to depend on the
P/D ratio and channel shape. For square arrays,
S = 2(P/D
/ - 1)
)
(3.16)
and for hexagonal arrays,
S-i=
1
DII~
1
1.73(P/D-1)
(3.17)
(3.17)
To assess the susceptibility of new core designs to vortex-shedding lock-in, a
vortex shedding margin (VSM) is defined in both the lift and drag directions. This
margin indicates the separation distance between the rod and shedding frequencies. Au-
59
Yang [6] recommends that lock-in will be avoided as long as this margin is greater than
or equal to 30%. The vortex shedding margins in the lift and drag directions are defined
as:
VSMif
=ffI
VSM dragIf,
(3.18)
-2
(3.19)
2 fs
where,
f:
fundamentalfrequencyof the rod
And the design limit to avoid vortex-shedding lock-in is:
VSMliQ & VSM drag 0.30
(3.20)
Lock-in can occur for any of the fuel rod's natural frequencies. For the range of
geometries considered in this study and the corresponding magnitude of cross-flow
velocities reported by VIPRE, the shedding frequencies are closest to the fundamental
rod frequency. The use of the first natural frequency in the VSM is therefore the most
conservative approach; if a core design avoids lock-in with the fundamental mode, then
all higher order frequencies will avoid lock-in too.
The vortex-shedding margins in both the lift and drag directions for the reference
core geometry are plotted as a function of a hypothetical cross-flow velocity in Figure
3.3. It is evident that, depending on the cross-flow velocity, vortex-shedding margins can
be increased by either raising or lowering the cross-flow velocity. Because the crossflow velocities scale with power, this implies that designs failing a vortex-shedding
design limit can be made acceptable by either increasing or decreasing the power in the
core. This could prove problematic when incorporating the vortex-shedding design limits
into the thermal-hydraulic codes, which are set up to reduce, not increase, power if a
design limit is exceeded. To resolve this issue, it is helpful to examine the magnitudes of
the peak cross-flows at the reference core geometry for several different powers. These
are shown in Table 3.4 below. Notice that even for very large powers, the cross-flow
velocities stay well within the desirable cross-flow range where reducing power will
60
increase the vortex-shedding margin. It is therefore concluded that lowering the power is
a viable approach for improving the vortex-shedding performance of new designs.
Figure 3.3 Vortex-Shedding Margin vs. Hypothetical Cross-flow Velocity at the Reference Core
Geometry
A C'..
I
I
I
:
40)
--
-|
|
:
|
S
VSM LiftDirection 1
Drag Direction ]-
-iVSM
_
I
2) 300
>
I
I
I
I
I
I
I
_
I
20()
n0
I
I
25()
'
~
I
350
Ž
I
I-
-
I
L
r -
-r
-
-
L
.
I
I_
L
. _
~
~
I
I
-----
I
I
.
I
I
~
{
_
~~
~~~I
I
I
I
I
I
I
I
I
_
I
-.
...-.........
I
--
-I
-_
I
_
.
-~r
I
_
a)o
10()
_ .
_
I
I\
-
50 0
I \
I(
-
I
-X-I
--
\
-I
I
------
I
I
-I
0.1
.
I
I
.
I
--
0.05
-
0.15
0.2
I
I
T
0.25
0.3
-
.
I
I
I
-
__
0.35
0.4
Cross-flow Velocity (m/s)
Table 3.4 Peak Cross-flow Velocities vs. Power for the Reference Geometry
Power (MWth)
Cross-flow Velocity (m/s)
2000
0.0313
3000
0.0473
4000
0.0696
5000
0.0985
A final note on vortex-induced vibration: even if lock-in is avoided, the shedding
of vortices contributes to small amplitude rod vibrations, and therefore rod wear and
fatigue with time. The magnitude of this vibration, however, is usually small and
bounded by turbulence-induced
vibration, which is considered in Section 3.4.4. The off
lock-in vortex-shedding contribution to fretting and sliding wear is therefore not
considered.
61
3.4.3 Fluid-Elastic Instability
Fluid-elastic instability of a tube bundle occurs when the cross-flow velocity
reaches the critical velocity, at which point the vibration response of the tubes suddenly
increases uncontrollably and without bound. Unlike other vibration mechanisms, the
instability of the tube, and therefore its vibration amplitude, continues to increase as the
cross-flow velocity rises above the critical value. A tube that is elastically unstable can
experience rapid structural damage by any combination of the following damage
mechanisms:
*
Tube-to-tube impaction: this causes impact wear at the midpoint of a span
length. Within a few days, or even hours in the most extreme cases, enough
material can be removed from the cladding surface to burst the fuel rod.
* Fatigue failure: this occurs when stress in the fuel rod exceeds the endurance
limit for the cladding material. The fuel rod will sever at the location where
the stress is highest, often at the grid supports.
* Acceleratedfretting wear: accelerated wear rates accompanying the large
vibration amplitudes can quickly wear through the cladding surface at contact
points.
3.4.3.1 Fluid-Elastic Instability Margin and Design Limits
A fluid-elastic instability margin (FIM) is defined to quantify a tube bundle's
performance with regard this mechanism. It is the ratio of the maximum effective crossflow velocity in the hot assembly, Veff,to the critical velocity for the bundle geometry
Vcritical:
rFIM-
ef
(3.21)
Vcritical
62
The FIM' can be thought of as a safety margin: as long as the effective cross-flow
velocity remains below the critical velocity (and the FIM remains below 1), fluid-elastic
instability will not occur. The design limit is therefore:
FIM <1
(3.22)
The most widely accepted correlation for estimating the critical velocity for a tube
bundle is Connor's equation [6]:
Vcritical
=Pnf
'f
Pfl
(3.23)
where Pettigrew [7] suggested a P/D effect on Connors' constant:
/ = 4.76. P-1)+0.76
(3.24)
The critical velocity is constant for a fixed geometry and, with the exception of
small changes in coolant density, does not depend on the power and flow conditions in
the core. Evident in Connor's equation is the conservatism accompanying the use of the
first natural frequency, which yields the lowest critical velocity and the largest FIM.
Because of the relationship between effective cross-flow velocity and power, the FIM
will scale with power and can therefore be incorporated as a design constraint into the
existing thermal hydraulic analysis.
3.4.4 Turbulence-Induced Vibration
Turbulence from cross and axial flows generate random pressure fluctuations
around fuel rods, causing them to vibrate. The energy associated with the pressure
fluctuations is distributed over a wide range of frequencies. Vibration occurs when the
rod selects the portion of this energy that is closest to its natural frequency. Unlike fluidelastic instability or vortex shedding lock-in, whose respective impacts can be minimized
and even eliminated in the design phase of the fuel assembly structure, turbulenceinduced vibrations cannot be avoided. Furthermore, no specific design limits are
applicable because the vibration amplitudes accompanying turbulence are small. The
63
principle design concern is therefore not the prevention of the mechanism but the
limitation of resultant wear at the cladding/rod support interface over time.
Assessing the wear performance of new core designs requires knowledge of the
rod response, or its displacement amplitude. Turbulent flows and the associated pressure
distributions causing rod vibration, however, are random; it is therefore impossible to
determine a detailed time history of the rod response. Instead, probabilistic methods are
used to estimate the root mean square (RMS) response from both cross and axial flow
turbulence. The total rod response is simply the summation of the cross and axial flow
contributions, and becomes the primary input to the wear performance portion of the
vibrations analysis (See Section 3.4.5).
3.4.4.1 Turbulence-Induced
Vibration in Cross-flow
Despite the latest advancements in computational fluid dynamics, the turbulent
rod response remains best characterized by a combination of experimental and analytical
J
techniques and correlations. Assuming the vibration modes of the rod are well separated,
Au-Yang [6] recommends the following relationship for estimating the RMS rod
response from cross-flow turbulence:
(2
.)
- 64*
where,
GF:
Jnn.'
. mt
(Xross
n
n
(3.25)
randomforce power spectral density
joint acceptance
The turbulent forcing function that drives the rod response is characterized by the
random force power spectral density (PSD), GF, and the joint acceptance Jn. The joint
acceptance is the probability that a structure originally vibrating in the nth mode will
remain in the nth mode under the excitation of the forcing function. Assuming that the
correlation length for the forcing function is equal to the length between grid spacers, and
remembering that only the first vibration mode is considered, Au-Yang recommends a
value of 0.64 for the joint acceptance of a simply supported tube.
JI = 0.64
(3.26)
64
Based on experimental testing, Pettigrew and Gorman [6] suggested the following
empirical equation for the random force PSD for single-phase turbulent cross-flow over
tube bundles:
GF = CR D2
Pl
(3.27)
ross
The random lift coefficient, CR,depends on the normalized random pressure PSD,
GP, which has been correlated by Au-Yang to the dimensionless frequency, F. The
relationships are provided here:
F=
CR
(3.28)
VI.S
=
J7
GP
(3.29)
where,
GP =
0.01
for
F <0.1
0.2
for
0.1 <F <0.4
5.3e4F 7 /2
for
F > 0.4
Referring back to equation (3.25), the mode shape function, ,if (x), is assumed
constant and equal to its maximum value. This adds conservatism to the determination of
the RMS response. The maximum of the mode shape function is easily obtainable from
equation (3.8):
Y/l,max=
(3.30)
L
The final form of equation (3.25) for the upper bound RMS rod response from
turbulent cross-flow is given by:
Yrms-cross 64
3
m
3
(3.31)
65
Vibration in Axial Flow
3.4.4.2 Turbulence-Induced
For comparable velocities, axial-flow induced vibrations are generally of much
less concern than cross-flow-induced vibrations. In PWR cores, however, the axial
velocity is often two or more orders of magnitude larger than the peak cross-flow. With
this in mind, the rod response contribution from turbulent axial flow is often more
significant than the cross-flow contribution.
The method for determining the RMS rod response from cross-flow turbulence
can be applied to axial flows, with the use of appropriate PSD forcing functions. A
simpler approach, however, is recommended by Paidoussis [6], who developed the
following correlation from experimental data to estimate the upper bound maximum rod
response from axial-flow turbulence:
1.6
=5e-5
Ymax-axial
1.8
4
.25
(
- 4
l+u2
'
1+4
.)
(3.32)
where,
K:
a:
5. Ofor turbulentflow
7rfor simply supported rods
V:
Vaxialvs
p f A
it
V,,ial: peak axial velocity
A,:
outer rod cross sectionalarea
e:
Re:
LID
Reynold's number
hydraulic diameter
Dh:
:
pfl A,/m,
Pfl:
coolant density
m,:
total linear mass
Assuming the rod response follows a normal distribution, the RMS axial response
is approximately one third of the peak response:
YYrms-axial
- Y m a x -axial
(3.33)
3
3.4.4.3 Turbulence-Induced Vibration Response
The total RMS rod response is equal to the sum of the cross and axial flow
contributions:
Yrms = Yrms-axial + Yrms-cross
(3.34)
66
This is the primary input for the wear portion of the analysis.
3.4.5 Wear Due to Flow-Induced Vibration
There are three primary types of wear mechanisms caused by flow-induced
vibration: impact, fretting, and sliding wear. Impact wear, or tube-to-tube impaction,
occurs when fuel rods experience very large amplitude vibrations. Because this behavior
is precluded by design limits for fluid-elastic instability and vortex-induced vibration,
impact wear is not considered. Fretting wear is the most common wear mechanism, and
historically the most costly flow-induced vibration problem in the nuclear industry.
Fretting results from combined rubbing and impaction between the fuel rod support and
the cladding surface. Sliding, or adhesive, wear occurs by the rubbing motion between
the grid support springs and rod. It is often difficult to distinguish between these latter
two wear types. Fretting wear is typically associated with smaller vibration amplitudes
for gapped supports, where both wear and impact stress make contributions to material
degradation. Sliding wear is typically associated with slightly larger vibration
amplitudes, and results from the relative motion of two surfaces in continuous contact
with one another.
Modeling the rate and accumulation of wear for nuclear fuel rods is extremely
complex. Because specific design limits for wear are vague, cumulative rod wear for
new designs will be limited by the end of life cumulative wear calculated for reference
core fuel pins. To estimate wear, the wear rate must be calculated, which requires
knowledge of the relative motion between the rod and its support. Because the linear rod
model assumed in this analysis treats the grid spacers as pinned supports, no relative
motion exists. Once again, non-linear structural dynamics and FEA codes are needed to
solve the problem quantitatively. The academic literature, however, has simplified
models for qualitatively assessing the fretting and sliding wear performance of new
designs using a linear rod model under known flow conditions. This is the approach
adopted for the wear analysis.
67
3.4.5.1 Fretting Wear Performance and Design Limits
Determining the cumulative fretting wear requires knowledge of both the fretting
wear rate,
fretting and the wear
coefficient, Kr d . Because most of the damping for a
vibrating fuel rod comes from the interaction between the rod and its support, Yetisir et al
[6] suggested that the fretting wear rate can be approximated as the power dissipated by
the vibrating rod. The power dissipation depends on the structural properties of the rod
and the RMS response from flow-induced vibration. The power dissipation, or fretting
wear rate, for the first vibration mode is given by:
'fretting
=32,r 3 . r ' f 13 L mt y2
rs
(3.35)
Notice that this equation implies that the fretting wear rate is independent of time;
in reality this is not true. Reasons for this include: spring relaxation, cladding and grid
material degradation, and wear accumulation. These factors combine to slowly open the
gap between the fuel rod and its support, changing both the rod's vibration frequency and
amplitude. To accurately determine the cumulative wear as a function of time, an
iterative approach incorporating non-linear FEA codes is required to continuously update
the wear rate given by equation (3.35) as the dynamic properties of the fuel rod change.
This is beyond the scope of this work. It is generally accepted by industry, however, that
after a brief "break-in" period, the wear rate does not change significantly, unless the rods
are allowed to remain in the core until failure. Because the reference core design is
sufficiently robust to prevent wear-related rod failures, and the cumulative wear of new
designs is limited to the cumulative wear of the reference core, a constant wear rate can
safely be assumed for this analysis.
The cumulative volume of material removed by fretting wear as a function of
time, T is approximated by Archard's equation [6]:
Qfretting=
fretting
Krod T
(3.36)
The wear coefficient, Krod, is material dependent and must be determined
experimentally. Because the cladding and grid materials are identical for both new and
68
the reference designs, the wear coefficients will be the same. By taking the ratio of the
cumulative fretting wear for new designs to the cumulative fretting wear for the reference
core, the wear coefficients cancel.
Qfretting,new
-f
retting,new Tnew
Teref
Qfretting,ref
mt
(3
Yrmsnew
337
Tnew
ref ms Tref
Recognizing that the lifetime of the fuel in the core is equal to the product of the
number of batches, n, and the cycle length, T,, and that the cumulative fretting wear ratio
for the end of life fuel must remain below 1, equation (3.37) becomes:
Qfreting,new (
mt YrMs
2 new
Qfretting,ref
mt Yrms f
,e
(3.38)
(3.38)
n Tc,ref
Rearranging and canceling terms yields:
Afretting,new =
t Yms
fretting,ref
m, ymsref
Inew
<cre
f
c,new
(3.39)
This is the desired relationship for the fretting wear analysis: the wear rate ratio is
the constrained parameter, and the ratio of the cycle lengths is the design limit. If a new
design has a shorter cycle length than the reference core, then it can safely accommodate
a higher rate of wear. Note that the wear rate limit, due to its dependence on cycle
length, will depend on both the power and the fuel bumup. The power, however,
depends on the wear rate limit, and the burnup, when limited by fuel performance
constraints, depends on the power. The relationships among the individual analyses in
the optimization study were illustrated in Figure 1.1. Several iterations are therefore
required to determine the core power subject to the design limit for fretting wear.
Another comment on the wear rate limit is also warranted. The Hydride Fuels
Project is examining different enrichments of UZrH1 .6 for potential use in PWRs; each
enrichment, via the neutronics analysis, has a different achievable burnup for the same
geometry. Burnup increases with enrichment, and so does the cycle length. The
economics analysis presented in Chapter 4 will show that costs are minimized at higher
69
enrichments for UZrH1. 6 and lower enrichments for U0 2. The wear rate limits are
therefore determined for the highest enrichment (12.5%) hydride fuel and the lowest
enrichment (5%) oxide fuel. Note that a consequence of this is imposing a more stringent
wear rate limit on hydride fuel than oxide fuel. The benefit, however, is that the
maximum power input to the economics analysis will be specifically adapted to the
lowest cost enrichments for each fuel type. Ideally, the vibrations analysis would be
performed separately for all enrichments of hydride and oxide fuels, but this is not
undertaken in this work.
3.4.5.2 Sliding Wear Performance and Design Limits
Connors [6] suggested that the sliding wear rate is equal to the product of the
normal contact force between the rod and support spring, F,, and the total sliding
distance, Sd.
siding =
(3.40)
Fn ' Sd
The normal contact force is given by:
- D y,,(3.41)
- s
F =
L,
0D · L
--+
LA,j E+ 4E,, I,,
s
where,
u:
coefficient offriction
Acl:
claddingcross-sectionalarea
The sliding distance is given by:
Sd
= Tr.f, g. T
(3.42)
where,
g:
diametralgap betweenthe tubeand support
Assuming a constant wear rate, the cumulative sliding wear at time t is:
Qsliding = T' Krod Fn ' f'g
'gT
(3.43)
As in the case of fretting wear, the cumulative sliding wear in new designs is
limited to the cumulative sliding wear in the reference core. Taking the ratio of the wear
70
rates and reducing like terms yields the desired constrained parameter and design limit
for the sliding wear analysis:
(*
sidingref
1 D
__I+ --
rs
*w
l
(D. Yrms fAl)re--
2
A,, 41
2
s~1idingref
Ac+
4n
l
A,
IC
(3.44)
Tc,new
e
e
The wear rate limit is once again equal to the ratio of cycle lengths.
3.5
Summary of Constrained Parameters and Design Limits
Table 3.5 summarizes the final list of constrained parameters and design limits
used in the steady-state maximum power analysis for hydride and oxide fuels. The first
four are the new relationships developed in this chapter to limit flow-induced vibrations
and wear to acceptable levels. The remaining limits were carried over from the original
analysis presented in Chapter 2.
Table 3.5 Summary of Steady-State Thermal Hydraulic Design Constraints
Design Constraints For:
Constrained Parameters
Vortex-Shedding Lock-in
VSMif,, VSMdrag
Fluid-Elastic Instability
FIM
Fretting Wear
frettng,n
Design Limit
Equatosce
Equations
> 0.3
(3.18), (3.19)
<1
(3.21)
(3.39)
<
cnew
fretting,ref
SlidingWear
siding,new
DNBR
MDNBR
Pressure Drop
AProdbundle
Fuel Temperature
(3.44)
T,r
sliding,ref
Tcenterline- UZrHl. 6
Taverage
- U0 2
cnew
> 2.17
< 29 psia, 60 psia
< 750 C
< 1400 C
71
3.6
Results of the Maximum Achievable Power Analysis With
Vibrations-Imposed Design Limits for Square Arrays
3.6.1 Results at 60 psia for UZrH1 . 6
The maximum achievable power for UZrH 1.6 at 60 psia is shown in Figure 3.4
with vibrations and wear imposed design limits. The peak power geometry, occurring at
P/D - 1.37 and
Drod
= 6.5 mm, is - 5017 MWth. This is -450 MWth lower than the peak
power determined by the thermal hydraulic analysis without the vibrations and wear
limits (See Figure 2.1). Figure 3.5 shows the ratios of the rod number, average linear
heat rate, and power for the new geometries to the reference core rod number, average
linear heat rate, and power. A black line is used to denote the contour where each ratio is
unity. The regions where the hydride core has a higher power than the reference core
have been reduced significantly; no regions exceed the reference linear heat rate
(compare with Figures 2.2B and 2.2C). The reduction is more clearly seen in Figure 3.6,
which plots the difference in power achievable with and without the vibrations and wear
limits imposed. The peak power reduction is close to 2200 MWth, and occurs at a P/D 1.16 and Drod
11.8 mm.
Figures 3.7 and 3.8 plot the constrained parameters, where dark red shading
indicates that a design limit has been reached (and therefore constrains power). While
MDNBR and pressure drop continue to limit large regions of the design space, the area
originally occupied by the axial velocity limit has been replaced and enlarged by limits
on fretting and sliding wear. Recall that one of the primary inputs to the wear rate
equations is the RMS rod response, which is composed of axial and cross-flow
components (equation (3.34)). Because the axial velocities in the core are orders of
magnitude larger than the cross-flows, the axial flow term dominates the RMS rod
response. It is therefore expected that the wear limits will be reached where the axial
velocities are greatest, which is the observed behavior.
The fluid-elastic instability margin, as shown in Figure 3.8C, never approaches its
limit of 1; its maximum value over the entire design range is 0.34. The vortex shedding
72
margins are plotted separately in Figures 3.9A and 3.9B in both the lift and drag
directions. Unlike the wear parameters, the vortex-shedding margins are very sensitive to
changes in the cross-flow velocity. As shown in Figure 3.9C, VIPRE experienced
difficulty in converging to cross-flow solutions for P/D ratios less than - 1.2. The code's
output for several geometries, apparent by large velocity spikes, do not appear likely and
made it difficult to maintain the vortex-shedding design limits in the thermal hydraulic
code in this region. The limits were therefore only imposed for geometries with P/D
ratios greater than 1.2. As a result, several of the vortex shedding margins at the tighter
geometries are less than 30%, as evident in the figure. These need to be re-investigated,
particularly if the economically optimum geometries fall into this region. It is hoped that
an updated version of the VIPRE code will soon be available at MIT, and that the
convergence problems will be resolved.
Figure 3.4 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of UZrH1.6 at 60 psia
with Vibrations and Wear Imposed Design Limits
Core Power (xl 06 kWh )
5
12
4.5
4
11
135
'10
B
9
2.5
1.5
7
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/
73
Figure 3.5 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and D,.odfor Square Arrays
of UZrH1. 6 at 60 psia with Vibrations and Wear Imposed Design Limits
B: q/q.,
A NI/Nf
12
13
12
1.5
11
<11
' 10
2
110
1g
8
1
Q
7
I0
1.1
1.2
1.3
P/D
1.4
1
88
0.5
I
7
1.1
1.5
1.2
O-Dlt
1.3
P/D
1.4
1.5
12
1.5
11
1
c
108
7
1.1
1.2
1.3
1.4
1.5
I
P/D
Figure 3.6 Power(,, vibrations) - Power(vibrations)VS.P/D and Drodfor Square Arrays of UZrH1.6 at 60
psia
Core Power Difference (kWh)
x106
I 10
. ...
IZ.D
1:
2
11
1
5
IU
Q
c
8
5
7
1.1
1.15
1.2
1.25
1.3
P/D
1.35
1.4
1.45
1.5
74
Figure 3.7 Thermal Hydraulic Constraining Parameters vs. P/D and D,.,,dfor Square Arrays of
UZrH1.
6
at 60 psia with Vibrations and Wear Imposed Design Limits
B: W-3L MDNBR
A: Pressure Drop (psi)
60
12
12
I
11
10
20
1.1
1.2
1.3
1.4
1.5
I0
P/D
C: Axial Velocity (m/s)
1.2
1.3
P/D
14
1.5
D: Fuel CL Temperature (C)
12
11
l 700
10
600
'a9
4
o
I -4
1.1
la
11
-3 5
7
8
12
-3
7
8
8
7
I -2.5
11
11
10
40
500
8
I
7
1.1
1.2
1.3
R/D
1.4
2
I
1.5
1.1
1.2
1.3
1.4
Adln
-J
1.5
P/D
Figure 3.8 Vibrations and Wear Constraining Parameters vs. P/D and Drodfor Square Arrays of
UZrH1. 6 at 60 psia
A: Fretting Wear Limit
B: Sliding Wear Limit
12
12
I 0.8
11
I 10
0.4
0.4
9
C-)
8
0.2
I0
7
11
1.2
1.3
1.4
1.5
I 0.8
10
006
'O9
0.4
8
0.2
I0
7
1.1
PID
C: FIM
I
11
1.2
1.3
PJD
1.4
1.5
1
12
I 0.8
1 '1
0.6
O
0.4
0.2
7
1.1
1.2
1.3
1.4
1.5
I0
PID
75
Figure 3.9 Vortex-Shedding
Margins and Peak Cross-flow Velocity vs. P/D and D,.,t for Square
Arrays of UZrH1. 6 at 60 psia
A: VSM Lift (%)
B: VSM Drag (%)
12
12
-100
11
10
I
7
1.2
1.3
1.4
-200
o9
-400
1.1
03
-200
-300
8
I -100
11
1.5
-300
8
7
1.1
P/D
C: Peak Cross-flow Velocity (/s )
1.2
1.3
1.4
1.5
I -400
PI/
0.2 .......
0.1
12
:ro
d
. 8 -
(mm)
-
_
P/D
_1.4
3.6.2 Results at 60 psia for UO2
The maximum achievable power for UO2 at 60 psia is shown in Figure 3.10. The
peak power geometry and peak power are almost identical to hydride: 5045 MWth at P/D
- 1.37,
Drod =
6.5mm. Figure 3.11 shows the ratios of the rod number, average linear
heat rate, and power for the new geometries to the reference core rod number, average
linear heat rate, and power. The regions where the oxide core has a higher power and
linear heat rate than the reference core have been reduced, though not to the extent
experienced by the hydride fuel (compare with Figures 2.2B and 2.2C). The reduction is
more clearly seen in Figure 3.12, which plots the difference in power achievable with and
without the vibrations and wear limits imposed. The peak power reduction is 1435
MWtl,, and occurs at a P/D - 1.2 and Drod
=
12.5 mm.
Figures 3.13 and 3.14 plot the constrained parameters, which demonstrate similar
behavior to the case of hydride fuel. MDNBR and pressure drop continue to limit large
regions of the design space, and the axial velocity limit is replaced by fretting and sliding
76
wear, once again because of the dominance of the axial flow term in the rod response
equation. The fluid-elastic instability margin remains well below 1. The vortex shedding
margins and the peak cross-flow velocities are shown in Figure 3.15. The cross-flow
convergence problems are again evident, and so the vortex shedding design limits were
not maintained below a P/D of 1.2.
Figure 3.10 Maximum Achievable Power vs. P/D and
Drod
for Square Arrays of UO2 at 60 psia
with Vibrations and Wear Imposed Design Limits
Core Power (xl 0 6 kWth)
12
5.5
5
11
4.5
'-
10
.5
~o
.5
8
.5
7
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/D
77
Figure 3.11 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and D,.,,dfor Square Arrays
of UJO2 at 60 psia with Vibrations and Wear Imposed Design Limits
A: N[Neref
B q/q,%f
12
12
3
1.5
11
11
a::
7
2
10
1
8
S
I
1.1
1.2
1.3
PfD
C
1.4
05
7
1.5
1.1
1.2
DD
1.3
P/D
1.4
15
I
12
1.5
11
0O5
8
I
7
1.1
1.2
1.3
1.4
1.5
P/D
Figure 3.12 Power(novibrations)- Power(,brations)
VS.
P/D and Drodfor Square Arrays of U0
Core Power Difference (kWtU)
2
at 60 psia
XI196
Z.D
12
L
11
1.5
- 10
1
8
05
7
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Q
P/D
78
Figure 3.13 Thermal Hydraulic Constraining Parameters vs. P/D and Drodfor Square Arrays of
U0 2 at 60 psia with Vibrations and Wear Imposed Design L,imits
B: W-3L MDNBR
A: Pressure Drop (psi)
60
12
I 40
11
12
I -25
11
11
-3
20
8
I En
1.1
1.2
1.3
1.4
-3-5
8
7
I -4
1.1
1.5
P/D
1.2
V
C: Axial Velocity (m/s)
0
12
I
11
6
1c
1.3
P/D
1.4
15
D. Fuel Ave Temoerature (C)
12
lI
11
I 10
4
I
1.1
1.2
1.3
1.4
1400
1000
*9
8
7
2
1200
1.5
I
1.1
1.2
1.3
1.4
800
1.5
P/D
P/D
Figure 3.14 Vibrations and Wear Constraining Parameters vs. P/D and Drodfor Square Arrays of
UO2 at 60 psia
A: Fretting Wear Limit
12
B: Sliding Wear Limit
I
1
'O
71
7
1.1
1.2
1.3
1.4
1.5
12
0.8
I 0.8
3611
0.6I1
0.6
0.4
9
0.2
8
7
I0
P/D
0.4
0.2
I0
1.1
1.2
1.3
14
1.5
P/D
1
12
0.8
11
0.6
O(
0.4
0.2
0
1.1
1.2
1.3
1.4
1.5
P~
79
Figure 3.15 Vortex-Shedding
Margins and Peak Cross-flow Velocity vs. P/D and D.od for Square
Arrays of UO2at 60 psia
A: VSM: Lift (%)
B: VSM: Drag (%)
'12
I -100
11
10
2I
8
1.1
1.2
1.3
1.4
1.5
I
PfD
C: Peak Cross-flow Velocity (s)
I -100
11
-200 v10
-200
-300
9
-300
-400
8
7
-400
-jnn
--- v
I
1.1
1.2
1.3
1,4
Inn
-J UU
1.5
P'D
03
U.2
0.
12
8'- (m)
Drod (M)
1.2
P/D
3.6.3 Comparison Between UZrH1.6 and UO2 at 60 psia
Figure 3.16 shows the difference in maximum achievable power for UO2 and
UZrH .6 at 60 psia with vibrations and wear imposed design limits. Note that oxide fuel
has a power advantage in the sliding and fretting wear limited regions because of the
difference in wear rate limits applied to each fuel. Recall Section 3.4.5.1, which
explained the rationale for using a more stringent wear rate limit for UZrH1.6. The limit
depends on cycle length, which will vary with enrichment for each fuel type. The idea is
that the wear rate limit for each fuel is determined at the enrichment that is expected to
yield the most desirable economics. As will be revealed in Chapter 4, costs are
minimized by using 12.5% UZrH 1 6 and 5% UO2. This approach ensures that the
maximum power input to the economics analysis will be specifically adapted to the
lowest cost fuels. Figure 3.17 shows the difference in the wear rate limits between the
two fuels that causes the power discrepancy in the wear limited regions. The difference
is quantified as the percentage by which the oxide limit is larger than the hydride limit.
In the wear limited design regions, the percentage difference ranges from - 10% to 65%.
80
Figure 3.16 (Powertuo2 - Powertz,.Ii. 6) vs. P/D and Drod for Square Arrays at 60 psia with
Vibrations and Wear Imposed Design Limits
Core Power Difference (U0 2 - UZrH1 .6 ) (x 106 kWth)
2
1. 8
11
6
1.
1.4
ic
z
-
12
E
E
v
iD
-
3
4
E
2
7
1.1
1.15
1.2
1.25
1.3
P/D
1.35
1.4
1.45
1.5
Figure 3.17 Percentage Difference in Wear Rate Limits vs. P/D and Drod for Square Arrays at 60
psia
Wear Rate Difference [(UO2 - UZrH1 .6 )/U 2] (%)
12
11
.,
1C
50
E
2
D~ 9
E
I
1.1
1.15
1.2
1.25
1.3
P/D
1.35
1.4
1.45
1.5
81
3.6.4 Results at 29 psia for UZrH1. 6
The maximum achievable power for UZrH1 .6at 29 psia is shown in Figure 3.18
with vibrations and wear imposed design limits. The peak power geometry, occurring at
P/D - 1.49 and Drod= 6.5 mm, is - 4245 MWth. This is unchanged from the peak power
determined by the thermal hydraulic analysis without the vibrations and wear limits (See
Figure 2.4). The ratios of the rod number, average linear heat rate, and power for the new
geometries to the reference core rod number, average linear heat rate, and power are
shown in Figure 3.19. Comparing with Figures 2.5B and 2.5C, it is evident that the
regions where the hydride core has a higher power than the reference core have been
reduced, though not as significantly as for the 60 psia pressure drop case. Note that the
reference core's linear heat rate is not exceeded at any geometry. The difference in
achievable power with and without the vibrations and wear limits imposed is plotted in
Figure 3.20. The peak power reduction is - 1210 MWth, and occurs at a P/D - 1.23 and
Drod= 12.5 mm.
Figures 3.21 - 3.23 plot the familiar constraining parameters for the thermal
hydraulic and vibrations analysis. MDNBR and pressure drop continue to constrain
power for large regions of the design space. Because the axial velocities are lower at 29
psia, however, the wear limits are not as constraining on power as for the higher pressure
drop case. This is most evident when comparing Figures 3.22A and 3.22B with Figures
3.8A and 3.8B. In fact, the fretting wear limit is never reached at 29 psia, and the region
where the sliding wear constrains power is reduced.
As expected, Figure 3.22C shows that no geometries have problems with fluidelastic instability. The vortex shedding margins are again plotted separately in Figure
3.23, along with the peak cross-flow velocity. As for 60 psia, VIPRE experienced
difficulty converging to cross-flow solutions for P/D ratios less than - 1.2, and so the
vortex shedding design limits were not imposed in this region.
82
Figure 3.18 Maximum Achievable Power vs. P/D and Drod for Square Arrays of UZrH .1 6 at 29 psia
with Vibrations and Wear Imposed Design Limits
Core Power (xl 06 kWth)
4
12
.35
11
I
.9I
3
10
2.5
Q.
9
2
8
1.5
1
7
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/D
Figure 3.19 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drodfor Square Arrays
of IIZrH
1 6 at 29 psia with Vibrations and Wear Imposed Design Limits
A: N/Nre f
B: q/qf
12
12
1.5
11
2
1
8
1
8
.7
I0
7
a,
1.1
1.2
1.3
1.4
P/D
C·
1.5
MD
10
1
0.5
1.1
1.2
1.3
P/D
1.4
1.5
I
12
1.5
11
10
0.5
8
I
7
1.1
1.2
1.3
1.4
1.5
P/D
83
Figure 3.20 Power(,{, vibrations)
- Power(,ibrations)vs. P/D and D,od for Square Arrays of UZrH. 6 at 29
psia
Core Power Difference (kWt)
-I
1 5
13
11
12
11
q 10
Qa
5
8
7
J
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/D
Figure 3.21 Thermal Hydraulic Constraining Parameters vs. P/D and Drod for Square Arrays of
UZrH1.6 at 29 psia with Vibrations and Wear Imposed Design Limits
B: W-3L MDNBR
A: Pressure Drop (si)
12
25
I
11
12
-2.5
I
20
15
-3
10
10
8
5
7
I
1.1
1.2
1.3
P/D
1.4
n
-3.5
8
7
-4
-u
1.'5
1.1
C: Axial Velocity (m/s)
1.2
1.3
P/D
1.4
1.5
D: Fuel CL Temoerature (C}
o
0
12
12
I
11
700
11
600
C
8
7
1.1
1.2
1.3
P/D
1.4
1.5
I
2
500
8
7
400
1.1
1.2
1.3
P/D
1.4
1.5
84
Figure 3.22 Vibrations and Wear Constraining Parameters vs. P/D and D,.odfor Square Arrays of
UZrH1
6
at 29 psia
A: Fretting Wear Limit
B: Sliding Wear Limit
12
12
I
11
0.8
8
7
I 0.8
11
0.6
10
0.4
' 9
0.2
8
7
I In
I
1.2
1.3
P/D
1.4
I
1.1
1.5
C: FIM
0.4
0.2
v
1.1
0.6
1.2
1.3
P/D
1.4
I0
15
1
12
I
11
0.8
10
0.6
9
0.4
.9
8
0.2
In
7
v
'1.1
1.2
1.3
1.4
1.5
P/D
Figure 3.23 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drodfor Square
Arrays of UZrH1. 6 at 29 psia
D: VSM Drag (%)
D: VSM Lift (%)
12
12
11
I
I -100
-100
-200 1C
ic
-300
7
-400
1.1
1.2
1.3
1.4
-200
-300
8
7
1.5
P/D
Peak Cross-flow Velocity (m/s)
1.4 1.5
I -400
P/D
0.3
0.2
0.1
10
Dro d (mm)
P/D
85
3.6.5 Results at 29 psia for U0 2
The maximum achievable power for UO2 at 29 psia is shown in Figure 3.24 with
vibrations and wear imposed design limits. The peak power geometry, occurring at P/D
- 1.49 and
Drod
= 6.5 mm, is - 4245 MWth. This is unchanged from the peak power
determined by the thermal hydraulic analysis without the vibrations and wear limits (See
Figure 2.4), and is identical to UZrH1. 6 (see Figure 3.18). The ratios of the rod number,
average linear heat rate, and power for the new geometries to the reference core rod
number, average linear heat rate, and power are shown in Figure 3.25. Comparing with
Figures 2.5B and 2.5C, it is evident that the regions where the oxide core has a higher
power and linear heat rate than the reference core have been reduced. The difference in
achievable power with and without the vibrations and wear limits imposed is plotted in
Figure 3.26. The peak power reduction is - 589 MWth, and occurs at a P/D - 1.19 and
Drod =
12.5 mm.
The constraining parameters for the thermal hydraulic and vibrations analysis are
plotted in Figures 3.27 - 3.29. MDNBR and pressure drop are the primary constraints.
Like UZrH 1. 6, the fretting wear limit is never reached and the region where sliding wear
constrains power is reduced because of the lower axial velocities. Figure 3.28C shows
that fluid-elastic instability is not constraining for any geometries, and Figure 3.29 shows
the vortex shedding results, where design limits were not incorporated for P/D ratios <
1.2 because of the cross-flow convergence problem.
86
Figure 3.24 Maximum Achievable Power vs. P/D and D,.odfor Square Arrays of lOz2 at 29 psia
with Vibrations and Wear Imposed Design Limits
Core Power (xl 06 kWth)
6
5.5
5
11
4.5
,p 1C
4
A
35
8
c
3
2.5
8
2
1.5
7
1
11
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/D
Figure 3.25 Rod Number, Linear Heat Rate, and Power Ratios vs. P/D and Drodfor Square Arrays
of lJO2 at 29 psia with Vibrations and Wear Imposed Design Limits
B: qqref
A: N/N e f
12
3
11
'
r
12
1.5
11
10
2
I10
1
Q
9
8
7
1.1
1.2
1.3
P/D
r P/
1.4
1.5
I
0.5
I
7
0
1.1
1.2
1.3
P/f
1.4
1.5
12
1.5
11
10
1
0.5
:3
8
*/
1.1
1.2
1.3
1.4
1.5
I
P/D
87
Figure 3.26 Power(n,,
ibraltions) -
Power(vibrations)vs. P/D and Drod for Square Arrays of U0
Core Power Difference (kWt)
2
at 29 psia
x10
I1
11
10
-' 10
9
5
8
7
0
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
P/D
Figure 3.27 Thermal Hydraulic Constraining Parameters vs. P/D and Drodfor Square Arrays of
UO2 at 29 psia with Vibrations and Wear Imposed Design Limits
A: Pressure Drop (psi)
B: W-3LMDNBR
12
25
I
11
1.,0
15
9
I
20
12
I
11
10
-3
10
8
7
I
1.1
1.2
1.3
1.4
-3.5
8
5
7
0
1.1
1.5
1.2
P/D
C: Axial Velocity-r (m/s)
-
1.3
P/D
1.4
1.5
) F l A1e TPmnPrnter.h C)f
8
12
12
I6
, 10
-2.5
I -4
1400
I 1200
11
1000
"1;
4
8
7
1
1.1
1.2
1.3
P/D
1.4
1.5
i
2
800
*9
8
600
7
I 400
1.1
1.2
1.3
1.4
1.5
P/D
88
Figure 3.28 Vibrations and Wear Constraining Parameters vs. P/D and D,.,rd
for Square Arrays of
U0
2
at 29 psia
A: Fretting Wear Limit
I
B: Sliding Wear Limit
12
12
I
'11
0.8
11
0.4
0.2
I0
7
1.1
1.2
1.3
PD
1.4
8
7
1.1
1.5
1.2
1.3
1.4
I
0.8
0.6
0.4
j I
0.2
0
1.5
P/D
C: FIM
1
'11
I 0.8
11
0.6
1
I9.
2
0.4
0.2
I
I0
1.1
1.2
1.3
1.4
1.5
PD
Figure 3.29 Vortex-Shedding Margins and Peak Cross-flow Velocity vs. P/D and Drodfor Square
Arrays of UO2 at 29 psia
A: VSM: Lift (%)
B: VSM: Drag (%)
12
I
11
-100
-200
I
-200
10
-300
7
1.1
1.2
1.3
1.4
8
I -500
7
:
P/D
D: Peak Cross-flow Velocity (m/s)
.. .
....
0:3
02]...
od (mm)
rod
-400
1.1
1.2
1.3
1.4
I -500
1.5
P/D
. . I. . ... .
0.1
01
12
-300
-400
1.5
-100
I
-v
I
-
-1
1.2
P/D
89
3.6.6 Comparison Between UZrH 1. 6 and UO2 at 29 psia
Figure 3.30 shows the difference in maximum achievable power for UO0
2 and
UZrH. 6 at 29 psia with vibrations and wear imposed design limits. As discussed in
Section 3.6.3 the oxide fuel has a power advantage in the wear limited regions because of
the more stringent wear rate limits applied to UZrH 1 6. Figure 3.31 shows the
corresponding difference in the wear rate limits, quantified as the percentage by which
the oxide limit is larger than the hydride limit, that causes the power discrepancy.
In the
wear limited design regions, the percentage difference ranges from - 20% to 50%.
Figure 3.30 (Powero02 - PowertzrHl.6) vs. P/D and Drodfor Square Arrays at 29 psia with
Vibrations and Wear Imposed Design Limits
Core Power Difference (UO2 - UZrH1
)
6
(x 108 kWth)
1.b
1
11
E
E
0.5
(q
O
1.1
1.15
1.2
1.25
1.3
P/D
1.35
1.4
1.45
1.5
90
Figure 3.31 Percentage Difference in Wear Rate Limits vs. P/D and D,0 d fior Square Arrays at 29
psia
Wear Rate Difference [(UO2 - UZrH1 6 )/UO 2 ] (%)
80
12
70
11
Rn
50
103
4U
30
3
20
10
7
K.
1.1
1.15
1.2
1.25
1.3
P/D
1.35
1.4
1.45
1.5
3.6.7 Summary of Maximum Power Results With Vibrations and
Wear Imposed Design Limits
Table 3.6 summarizes the steady-state thermal hydraulic results for UZrH 1 6 and
UO2 at 60 psia and 29 psia with vibrations and wear limits imposed. Three geometries
are considered: the peak power geometry, the reference core geometry, and the geometry
at the reference pitch offering the maximum power. This last one is chosen because of its
relevance to the minor backfit scenario considered later in the economics analysis (Refer
again to Section 1.2 for a definition of minor backfit).
Note in examining the results in Table 3.6 that the predicted power for oxide at
the reference core geometry is - 3800 MWth at 60 psia, and 3420 MWth at 29 psia.
As
discussed in Chapter 2, the reference core geometry is MDNBR limited, and so the
predicted powers should be independent of the pressure drop limit. The vibrations
analysis is not expected to impact it either because the wear limits are based on
maintaining the same margin for cumulative fretting and sliding wear as the reference
91
core. The power is correctly predicted, however, only for the higher pressure drop limit.
There are three primary reasons for this. First, the reference core geometry is not directly
included in the parametric analysis, and so all data for this configuration must be
interpreted from output at nearby geometries6 . Second, there is a convergence limit of
1% for all thermal hydraulic design limits, which means that the peak power is reached
when the constraining parameter is within 1% of its respective limit (i.e., when the
MDNBR is less than or equal to 1.01 x 2.173 = 2.194). This causes the power to be
slightly lower than if the limit were perfectly met. A convergence limit is necessary,
however, to expedite execution of the programs. Finally, the interpolation technique
employed introduces error (i.e, linear, cubic, spline, etc.), which combined with the
convergence limit, causes the predicted power for the reference core geometry to deviate
from its true value. The exact reason the discrepancy is only apparent at 29 psia is
unknown, although it most likely has to do with interpolation error with the lower powers
at 29 psia. Another reason may be that the reference core geometry at 29 psia is limited
by two thermal hydraulic design limits (MDNBR and pressure drop). If the code
converges to the pressure drop limit first, than the convergence limit's impact on power
will be more significant than for the higher pressure drop case (where only MDNBR
constrains the reference core power).
6 The
parametric study considers 20 equally spaced rod diameters between 6.5 and 12.5 mm, and 20
equally spaced P/D ratios between 1.08 and 1.55. The reference core geometry is not part of this design
range.
92
Table 3.6 Summary of Steady-State Thermal Hydraulic Results with Vibrations and Wear Limits
60 psia
Power (MW,)
Q/Qref
q/qref
UZrH 1. 6
5017
1.32
0.66
1.37
6.5
Pea2
5045
1.33
0.66
1.37
6.5
UZrH.6
3471
0.91
0.91
1.326
9.5
UO 2
Ref. Geom
3821
1
1
1.326
9.5
Ptch
UZRef.
Ref. Pitch
3503
0.92
0.92
1.3
9.66
Ref.Pitch
4066
1.07
1.07
1.264
9.97
Peak Power
Peak Power
Ref. Geom
Ref. Pitch
P
D (mm)
29 psia
Power(MW,)
Q/Qref
q'/qef
P/I
D (mm)
4245
1.12
0.66
1.49
6.5
4245
1.12
0.66
1.49
6.5
.6
UZrH.1e3332
0.875
0.875
1.326
9.5
UO
2 Ge3420
Uf2
0.9
0.9
1.326
9.5
UZrH 1 .6
kUZrH.6
Peak Power
UO 2
Peak Power
UZrH
Ref. Geom
Ref. Geom
UZrH 1. 6
UZrH.6
3314
0.875
0.875
1.3
9.66
U0 2
Ref. Pitch
Ref. Pitch
3511
0.924
0.924
1.3
9.66
Ref. Pitch
3.6.8 Extension of Vibrations Results to Hexagonal Arrays
The vibrations and wear analysis was only performed for square array designs. It
is believed, however, that the results can confidently be extended to hexagonal
geometries supported by grid spacers. Recall from Chapter 2 that the maximum power
obtained for square and hexagonal array geometries for matching rod diameters and
H/HM ratios was found to be the same. This is because the flow areas are identical per
unit rod, and minor differences in the turbulent interchange of mass, momentum, and
93
energy and the friction factor between the two configurations do not significantly impact
the overall power. For equivalent geometries, square and hexagonal arrays will therefore
have the same mass flux, flow area, and average axial velocity. Because the vibrations
limits that constrained power for square arrays involved sliding and fretting wear, and
wear is dominated by the axial flow, it is expected that the maximum achievable power
for hexagonal arrays subject to vibrations and wear limits will be identical. If in the
future wire wrapping is used in hexagonal designs (i.e, for tighter geometries where a
pressure drop benefit can be realized), then a separate analysis will be required to verify
that flow-induced vibration limits are not exceeded. J. Trant has provided some
preliminary information on this in his thesis [3].
94
95
4.
Economics Analysis of Hydride and Oxide
Fueled PWRs
4.1
Introduction
For energy production technologies, the cost of electricity (COE) is used to make
comparisons and guide investment decisions among competing options. To this end, an
economic analysis is performed to identify the optimum PWR geometries using UZrH1.6
and UO2 fuels, and provide a fair basis for their comparison over the design range of the
parametric study. In this chapter, an economic model is developed and applied to both
fuel types with inputs from recent work in thermal hydraulics and fuel design. While
results are presented for UZrH 1. 6 and UO2, the model can be easily adapted to other
hydride fuels. Specific assumptions are made regarding plant and fuel costs and the
economic parameters that influence them over time. Due to these assumptions and the
numerous models available for both collecting and interpreting economic information for
nuclear facilities, the cost of electricity (COE) results may not necessarily appear
consistent with the financial accounts of utilities. Because the model is equally applied to
UZrH .1 6 and UO 2 fuels, however, cost comparisons can be made with less regard for the
specifics of the modeling techniques and any discrepancies with commercial records.
4.2
Work Scope
4.2.1 Goals of the Economics Analysis
The COE is broken into three cost components: fuel cycle, operations and
maintenance (O & M), and capital costs. The economics analysis seeks the COE via
these components for square and hexagonal array PWR designs using UZrH 1. 6 and UO2
fuels. Results are presented for both fuels at enrichments of 5%, 7.5%, and 10%. Results
are also presented for 12.5% enriched UZrH.1 6.
The Hydride Fuels Project aims to quantify the benefits of hydride fuel use in
existing LWRs for two cases:
96
* Minor backfit, where the plant modifications required for fuel conversion are
minimized by maintaining the existing fuel assembly and control rod
configurations within the pressure vessel (i.e., maintaining the same pitch and rod
number in the core). In this case, replacement of the steam generators and
modifications to the high pressure turbine are required to accommodate designs
offering higher powers than the reference core;
* Major backfit, where the layout of fuel in the core can assume any geometry.
Note that in addition to upgrades of the steam generators and high pressure
turbine, this option will require replacement of the vessel head and the core
internals to accommodate the new layout of fuel assemblies and control rods at
increased power.
The final COE results are used to identify optimum geometries where the overall costs
are minimized, and where the use of UZrH1.6 fuel offers cost savings over U0 2.
In addition to presenting cost estimates, the economics analysis provides
important information for utilities considering the fuel switch from UZrH .1 6 to U0 2. For
example, the operating cycle length, plant capacity factor, annual energy production and
outage length are all byproducts of the COE calculations.
4.2.2 Methodology
The structure for the economics analysis was laid out by a former MIT student,
Jacopo Saccheri [8], and is consistent with OECD/NEA reports on the economic
evaluation of nuclear fuel cycles [9]. The primary inputs include the maximum power
from the steady-state and transient thermal-hydraulic analyses, and the maximum burnup
from neutronics and fuel performance studies. The fuel cycle, O & M, and capital costs
are then determined for each geometry and fuel type.
97
4.2.2.1 Lifetime Levelized Cost Method
The methodology employed is the lifetime levelized cost method, which
determines the levelized cost of electricity per unit of energy produced over the plant's
lifetime. It is also called the levelized busbar unit cost of electricity, to reflect that all
expenses up to the plant/transmission line interface are included. The term "levelized"
simply means that the costs, originally incurred at discrete times, are transformed into
one equivalent, continuous stream of expenditures. This method therefore allows direct
comparison of alternatives having vastly different cash flow histories. The basic
equations relating discrete cash flows and levelized costs are presented in the following
paragraphs.
In order to charge the correct price for service, a utility must first decide on the
rate of return on investment, r, that is desired. The rate of return is also commonly called
the discount rate, or nominal interest rate. Discrete expenditures for fuel cycle, O & M,
and capital costs are incurred at different times during the plant's life. To get the
levelized cost, these expenditures are discounted back to a reference date, which is
chosen to be the start of irradiation for the first fuel cycle (i.e., t = 0). Discounting all
expenditures to this date with continuous compounding of interest yields the present
value of all costs, PVos,
PVcoss=
and a relationship with the lifetime levelized cost,
Clev.
.'Ce
e-rtdt
'CN e -rN = P
(4.1)
N
where,
CN:
TN:
N h discretecashflow
time relative to the ref date for the Nh discrete cash flow
Tpant,:
plant
life
In addition to the discount rate, an escalation rate, g, can be included for recurring
cash flows to account for non-inflationary price increases with time. Rewriting equation
(4.1) to include this cost escalation effect yields:
PVCot=
E CN
e - rTN =E
N
CT
ePlant
.e-rtdt
(4.2)
N
where,
Co:
jSt
discretecashflow at referencedate
98
Integrating equation (4.2) with respect to time and solving for the lifetime levelized cost
yields:
Clev = P
coss [1- r
(4.3)
where:
capitalrecoveryfactor
[1
(4.4)
eplnl
The capital recovery factor, or carrying charge rate, relates the lifetime levelized
cost of electricity,
Clev,
to the present value of all expenditures. It correctly accounts for
the time value of money at the desired rate of return on investment. The lifetime
levelized unit cost of electricity, CIev,is obtained by normalizing Ce by the energy
production from the plant. If energy production is assumed to occur at a uniform rate
over time, the lifetime levelized unit cost is simply the levelized cost ($/yr) divided by the
annual energy production (kW-hre/yr).
The annual energy production, Eannual,from the
plant is:
(4.5)
Eannua,,
= th .-. L.8766
where:
Q,h:
core thermalpower
a:
plant thermal efficiency
L:
plant capacityfactor
The final relationship for the lifetime levelized unit cost of electricity is given by
the quotient of equations (4.3) and (4.5):
_
Clev
Eannual
L 8.766
(4.6)
rTp
PVcosts
th *77
1-e
.
If costs and the power are recorded in $ and kWth, and interest rates are
annualized (yr-l), then equation (4.6) reports the levelized unit COE in mills/kW-hre, the
desired units for this analysis. To get the individual levelized unit costs for the fuel cycle,
O & M, and capital components, equation (4.6) is applied with the relevant cash flow
99
histories incorporated into the PVc,,, term. The levelized unit COE is simply the sum of
the cost contributions from these individual components.
Clev = Clev-fcc + Clev-O&M + Clev-cap
(4 7)
The details of the cash flow histories for fuel cycle, O & M, and capital costs are
presented in Section 4.5.
4.3
Assumptions
The lifetime levelized cost method relies on accurate predictions for nuclear costs
and the terms under which they are financed (i.e. rate of return) for the length of plant
operation. Forecasting these costs and conditions, however, is a difficult task. For
example, economists are hesitant to project long term nuclear fuel prices, plant O & M
costs, and the rate of return on capital demanded by nuclear investors because of
uncertainties in the future accessibility to uranium at world prices, the quality and
quantity of tomorrow's nuclear work force, and the future of energy market competition.
With the hiatus on nuclear construction orders in the US approaching 25 years,
significant debate also exists over many of the current cost estimates for constructing new
or upgrading existing plants. The use of hydride fuel, which is new to commercial
reactors, will undoubtedly involve costs that are not currently accounted for in the three
component COE model. Because of the uncertainty and the speculative nature of most
long term cost projections for nuclear power, several assumptions are necessary to keep
this analysis simple and ultimately focused on the comparison of UZrHI.6 and UO2
performance in PWRs.
Only direct costs associatedwith thefuel cycle, 0 & M, and capital are
considered: The use of hydride fuels commercially will involve additional firstof-a-kind (FOAK) costs that accompany the introduction of new technologies
within the nuclear realm. Examples include fees associated with NRC licensing
and certification, research and development, and plant design. Because FOAK
100
costs are only incurred once, this economics analysis will focus on the Nthhydride
plant, where the 3 component COE model sufficiently covers all costs.
* Discountand escalationrates are constantover theplant's life: The discountrate
can be thought of as the sum of three components: a basic market growth rate, an
inflation rate, and an allowance for risk. The risk allowance portion scales with
investment risk which, for nuclear power, tends to be higher than for other energy
technologies because of the greater probability for investment loss. The market
growth and inflation rate portions are less dependent on the specific technology
and more heavily tied to the state of the overall economy. All three components
fluctuate in time, and it is beyond the scope of this report to either analyze or
attempt to predict this variability. As a result, a fixed discount rate of 10%/yr is
chosen to account for all three components. Similar arguments can also be made
for the escalation rates used to project non-inflationary price increases for fuel
cycle and O & M costs. Their contribution will also be assumed constant at
1%/yr. Because considering discount and escalation rate variability would have
the same impact on both fuels, this assumption is expected to have little impact on
the final economic comparison.
* Interest is compounded continuously: Because the lifetime levelized COE
represents a continuous stream of expenditures, the escalation and discount rates
are compounded continuously through time for all cost calculations.
*
The unit cost information available for commercial PWRs is cautiously extended
to new hydride and oxidefueled designs: The unit costs for the fuel cycle, O &
M, and capital portions of the COE analysis are widely available for existing
PWRs using UO2 fuel. Several of the unit costs, however, require modification to
accommodate: UZrH1. 6 fuel, for which no economic data is currently available,
and the large range of geometries considered in this study that deviate
significantly from standard PWR designs. The details of the assumptions are left
for Section 4.4.3.
101
4.4
Inputs
The primary inputs to the economics analysis are presented in this section. They
include the maximum power from the steady-state and transient thermal-hydraulic
analyses, the maximum burnup from the neutronics and fuel performance studies, and
specific unit cost estimates for front and back end components of the fuel cycle, O & M
activities, and plant capital.
4.4.1 Maximum Power
The vibrations analysis in Chapter 3 completed the steady-state thermal-hydraulic
analysis for maximum power in square and hexagonal array PWR cores using UZrH.6
1
and UO2 fuels. Parallel to this work, another MIT student J Trant [3] performed a
transient analysis for the same core designs, in which additional limits were imposed on
the maximum achievable power. The transients considered were the loss of coolant, loss
of flow, and overpower events. Each is briefly discussed below.
* Loss of Coolant Accident (LOCA): The LOCA postulates a break in the primary
coolant loop, at which point the pressurized fluid flashes to steam and blows
down into the containment structure. Though the control rods are immediately
driven in to stop the nuclear reaction, decay of fission products provides a
continued heat source that must be removed to protect the integrity of the clad and
fuel. LOCA performance is evaluated by considering the time history of the peak
clad temperature following the steam line break. As long as the peak clad
temperature for new designs remains below the peak clad temperatures for the
reference core at each time step, the LOCA is not limiting. Because the clad
temperature is driven by the decay heat in the fuel, which depends on the steadystate power prior to the accident, a design that exceeds the LOCA limit can be
made acceptable by reducing the steady-state power in the core.
* Overpower Transient: The overpower transient postulates a rod bank withdrawal
during steady-state operation, which causes a sudden increase in power (assumed
to be 18% of the steady-state power), which may lead to DNB. Like the steady102
state thermal hydraulic analysis for maximum power, DNB performance during
transients is measured by the MDNBR. The overpower design limit is therefore
the MDNBR of the reference core during the overpower event. If the limit is
exceeded, the steady-state power is reduced until the design is acceptable.
Loss of Flow Accident (LOFA): The LOFA postulates a loss of flow in the
primary coolant loop, which, depending on the power and flow conditions in the
core during coast down, may also lead to DNB. The LOFA design limit is
therefore the MDNBR for the reference core during coast down. If the limit is
exceeded, the steady-state power is reduced until the design is acceptable.
The transients and their respective design limits are summarized in Table 4.1.
Table 4.1 Transient Analysis Summary
Transient
Constrained Parameter
LOCA
Peak cladding temperature
LOFA
MDNBR
Overpower
MDNBR
Design Limit
*
The time history of the peak
clad temperature following a
LOCA for the reference core
*
The MDNBR during coast
down for the reference core
*
The MDNBR during an 18%
overpower transient for the
reference core
The transient analysis yields new maps for maximum power. The thermalhydraulic input to the economics study is the most limiting power reported by either the
transient or steady-state analyses, for each geometry. Because of differences in the
vibrations and transient performance, discrepancies exist between the maximum
achievable powers for UZrHI.6 and U0 2 fuels. The final power maps incorporating the
steady state and transient design limits are presented below for square arrays. Recall
from Chapter 2 that the thermal hydraulic performance for square and hexagonal arrays is
identical for equivalent rod diameters and H/HM ratios. The hexagonal array results can
103
therefore be inferred from the figures by adjusting the P/D ratio according the following
relationship: (/D)hex
= 1.0746 (P/D),q.
Figures 4.1 and 4.2 show the maximum achievable powers at the 60 and 29 psia
pressure drop limits for square arrays of UZrH .1 6 and UO2. The peak powers, as shown in
the plots, are the same for both fuels and occur at: P/D - 1.39, Drod = 6.5 mm for 60 psia,
and P/D - 1.49,
Drod
= 6.5 mm for 29 psia. The peak powers are very close to the peak
powers reported by the steady-state thermal hydraulic analysis (See Figures 3.4, 3.10,
3.18, and 3.24). As discussed in Chapter 3, more stringent wear rate limits lead to
additional power reductions in the wear rate limited regions for UZrHi.6; this is
particularly evident when comparing the crest of the maximum power regions below.
Figures 4.1 and 4.2 also show the regions (shaded in white), where the steady-state power
has been further reduced by transient limits.
Figure 4.1 Maximum Achievable Power and Transient Limited Regions vs P/D and D,.odfor
Square Arrays of UZrH1.6 and U0 2 Incorporating Steady-State and Transient Design Limits at 60
psia
Power TJZrH (xl o6kW
Power UTO (xl 06 kW.)
_
_D
a
C0
1
11
I
iQ
1.1
1.2
1.3
1.4
5
4
14
P/D
Transient Limited Region: UZrH
4
'1
3
3
2
2
I
1.5
5
1
1
1.1
1.2
1.3
15
1.4
I
i
P/D
Transient Limited Region: UO,
6
12
11
o10
.0
1.0
Q
Q
1.1
1.2
1.3
P/D
1.4
15
1.1
1.2
1.3
1.4
1.5
P/D
104
Figure 4.2 Maximum Achievable Power and Transient Limited Regions vs P/D and Drodfor
Square Arrays of UZrH1.6 and U0 2 Incorporating Steady-State and Transient Design Limits at 29
psia
Power UZrH . 6 (xl 06 kWth)
Power UO2 (xl 06 kWth )
_Q
6
1
12
12
5
-11
5
11
4
10
3
8
I
7
1.1
1.2
1.3
1.4
4
3
E~
2
2
P/D
-
[
12
11
11
. 10
10
rrV__:
;
9
8
8
7
7
13
P/D
1.4
1.5
1.
Transient Limited Region: UO,
12
1.2
14
1.1
P/D
Transient Limited Region: UZrH, ,O.
11
I1
1I
1.5
9.0 -. 0
1.1
1.2
1.3
1.4
i.
1.5
P/D
4.4.2 Fuel Burnup
Similar to the transient and steady-state approaches to maximum power, the
achievable burnup for UZrH 1 6 and UO2 fuels depends on the results from two separate
and independent analyses. Performed at UC Berkeley and MIT, the analyses seek the
maximum discharge bumup subject to neutronics and fuel performance constraints,
respectively.
Results from both are presented in this section. As in the case of maximum
power, the input to the economics analysis is the minimum discharge burnup reported by
either the neutronics or fuel performance studies for each geometry. Note that the bumup
is determined independently of cycle length, which is considered variable in this
economics analysis.
105
4.4.2.1 Neutronics
The neutronics analysis serves two roles. First, it provides the maximum burnups
for hydride and oxide fuels that can maintain the critical nuclear reaction in the core.
Second, it provides the range of geometries with acceptable (negative) fuel and
moderator temperature coefficients.
UC Berkeley employed SAS2H, which is part of the SCALE module, to
determine the unit cell 3-batch burnup for UZrH1. 6 and U0 2 fuels as a function of pin
geometry and fuel enrichment. The maximum burnup is reached when the core average
K, drops below 1.05. The margin of 0.05 is provided to account for leakage. Results for
square arrays are presented in Figures 4.3 and 4.4. As expected, higher burnups are
obtained for higher enrichment fuels. Note the difference in the curve shapes for hydride
and oxide fuels. The addition of hydrogen to the fuel matrix shifts the optimum
geometries to tighter configurations, because less coolant is needed for neutron
moderation. The neutronics performance depends primarily on the H/HM ratio, and so
the neutronically constrained burnups for hexagonal arrays can be inferred from the
square results for matching rod diameters and H/HM ratios.
106
Figure 4.3 Neutronically Achievable Burnup vs. P/D and Di.d for Square Arrays of UZrH.6 Fuel
5% (MWD/kgHM)
40
7 5% (MWDkgHM)
i
35
10
0
I
1.5
ROO(mmM8
10% (MW/kg
H,M)
110 '
48
80
78
46
75
76
44
70
74
42
65
12
72
40
10
DROD
(mm) 8
I
106
105
150
.
104
130
120
95
102
110
I
130
DRo (mm) 8
60--:
40-
40
.
0
12
,
.
.
.
---: 2...
IV'"
DROD
(mm)8
1.1
.
.
20
.
>-
"
30
.
.
for Square Arrays UO2 Fuel
Drod
.
100
80
,
I 120
1.3
1
7.5% (MWD/kgHM)
60
50
20
125
.
Figure 4.4 Neutronically Achievable Burnup vs. P/D and
5% (MWD/kg HM)
135
.
1"
100
70
140
.
.
100
13
I
*. :.
.;.-;'
140
DRCD(mm)8
P
12.5% (MWD/kgHM)
108
-
1.5
1
1.3
1..5
P/D
I 10
. M"
.
80
.
-
; -:
60
40
60
20
1"
"~
.
1.3
1.1 P/D
NvL~
DROD
(mm) 8
.5
40
A
I
10% (MWD/kg HM)
.'.-:
'.
110
120- -
100
nn '
-1
IUVVU
90
80
80
60
70
12
'
*
·
.
OD (mm)D
'8
1.1
..
P/D
I
PtD6
.5
I
60
107
The range of geometries with acceptable temperature coefficients for UO2 is:
1.0738<(P)
D sq
()
,
<1.4
1.5
6
.5<Drod <12.5mm
Drod
< 8.5mm
sq
All other geometries in the parametric study are unacceptable.
Unfortunately, the outlook for UZrH,.6 is not as optimistic. No geometries
considered in this parametric study had negative moderator temperature coefficients
(MTC). The cause is the large amount of soluble boron needed in the coolant to combat
the excess reactivity in UZrH 1.6. UC Berkeley is currently investigating ways to resolve
this issue. Possible solutions include introducing thorium into the fuel matrix, and using
different burnable absorbers to reduce the amount of soluble boron required. Only the
former has been considered in detail, and preliminary results indicate that utilizing
UZrHi.6-ThH2 provides a large design range with acceptable temperature coefficients. In
light of this information, the economic optimization is continued in this chapter for
UZrH1. 6 because it is believed that the poor MTC performance can be improved. It is
also important to firmly establish the optimization methodology because it applies to all
hydride fuels.
4.4.2.2 Fuel Performance
In addition to reactivity considerations, the discharge burnup is also limited to
protect the integrity of the fuel pin during irradiation. In the fuel performance analysis,
design constraints are placed on: internal fuel pin pressure and fission gas release, clad
strain, and clad oxidation. Each is briefly discussed below.
Internal Pressure: Gases generated during the fission process are originally
trapped in the fuel matrix. As the fuel heats up, this gas is released into the
plenum space, where it exerts an outward force on the pin. This may cause
outward creep and ballooning of the clad, and a rise in the conductance across the
pellet/clad gap. With more resistance to heat transfer, the fuel temperature rises
108
and more fission gas is released into the plenum. The potential therefore exists
for a positive feedback loop that can severely damage or even burst the pin.
Gases are also generated when burnable poisons are used to compensate the
negative reactivity effects of fuel depletion. The fission gas and burnable poison
contributions to overall fuel pin pressure need to be taken into account.
* Clad Strain: Clad strain is predominantly the result of three mechanisms:
external coolant pressure, differential thermal expansion between the fuel and
clad, and fuel swelling due to irradiation and the accumulation of fission gas in
the fuel matrix. Excessive strain can induce failure of the pin.
* Clad Oxidation: The outer surface of the clad is continuously oxidized by
primary coolant during irradiation, and can lead to both material and thermal
degradation of the cladding material. For example, a thickening oxide layer
reduces the wall thickness, possibly rendering the clad unfit to withstand the
stresses and strains imposed on it. Oxide buildup increases the thermal
conductivity of the clad; this raises its temperature and accelerates the rate of
oxidation. For Zircaloy materials, the oxidation process also generates hydrogen,
which migrates into the clad to form platelets of zirconium hydride. These
platelets embrittle the material, deteriorating its mechanical properties.
With guidance from industry, design limits were quantified for each fuel
performance constraint; these are listed in Table 4.2.
Table 4.2 Fuel Performance Limits for Maximum Burnup 11]
Fuel InternalPressure (psia)
Clad CorrosionThickness
CladStrain
(mm)
(%)
U02
.01
1.0 (tension)
2500
UZrH.6
Note that the limits on clad corrosion and strain apply equally to UO2 and UZrH1.6 fuel
rods. The internal pressure limit, however, is neglected for hydride fuel pins because of
low fission gas generation.
109
The maximum burnup with fuel performance imposed design limits is evaluated
using the FRAPCON code, which simulates the thermal-physical properties of U0 2 fuel
pins under steady-state conditions. The input requirements for FRAPCON include core
geometry, peak linear heat rate, and mass flow rate. All are provided by the steady-state
thermal hydraulic analysis7. For conservatism, the fuel is assumed to operate at the peak
linear heat rate in the core, and the code progressively outputs clad strain, oxide layer
thickness, and pin pressure as a function of burnup. The maximum burnup is achieved
when one constraint reaches its limit, and the others remain below their respective limits.
Because FRAPCON can only be used to simulate burnup in UO2 fuel, additional
assumptions were required to extend its use to UZrH1 .6.
* The internal pressure constraint was neglected due to the low fission gas
release.
*
The thermal expansion properties of the fuels, which affect clad deformation
and the limit on clad strain, were assumed to be identical.
T The fuel performance limit that dominates over most of the design range is
clad corrosion, which depends on the amount of time the fuel remains in the
core. Because the heavy metal loading in U0 2 is 2.5 times greater than
UZrH 1.6 at the same geometry, the residence time for UZrH .1 6 will be 40% of
the residence time for UO2 if the same burnup limit is applied. It is therefore
assumed that an equivalent FRAPCON burnup for UZrH 1.6 can be obtained by
multiplying the output from the code without the internal pressure constraint
by, 2.5. This ensures that the residence time for both fuels in the core at
discharge will be the same.
Of the three, the second assumption is the most difficult to justify, but its impact is
minimized because clad oxidation limits most geometries.
Because the FRAPCON analysis was performed before the transient limits were applied to the maximum
power, they are not incorporated into the maximum burnup results. This adds conservatism, however,
because the discharge burnup would increase if transient limits on core power were imposed.
7
110
The FRAPCON analysis was undertaken by a recent MIT post-doc, Antonino
Romano. The following figures show the maximum burnups for square arrays of UZrH 1.6
and UO2 at both pressure drop limits, subject to fuel performance design constraints.
Because the FRAPCON burnup depends on the output from the thermal hydraulic
analysis, the hexagonal results can once again be inferred by adjusting the P/D ratios.
The fuel performance limited burnup, unlike the neutronics analysis, is independent of
enrichment.
Note the large indentations in the oxide curves at both pressure drop limits
that occur in the regions of higher power. The same effect is not observed in the hydride
figures because the limiting constraint is internal pressure.
Figure 4.5 Fuel Performance Limited Burnup vs. P/D and Drod for Square Arrays of UZrH. 6 and
UO 2 at 60 psia
UZrH, ; (MWD/kg9M)
UO2 (MWD/kgHS
'.
"-
:
'
200
- 180
200,, :
.' ..'
..
60
150.,
.
140
14
120
50
100.
50
40
'. 100
12
/4
15 *80
13
12
"""
rod ......
-
PID
5
30
cn
U
11
11
.:,:':,
Figure 4.6 Fuel Performance Limited Burnup vs. P/D and Drodfor Square Arrays of UZrH. 6 and
UO 2 at 29 psia
IJZrH 6 (MWD/kg HM
)
UJO (MWDkHM)
200
2
.
200 -:-
.
:
180
,
160
150
140
100 ,
120
50\
15
,
12
100
n
rod .......
3
8
-. _
12
1.1
PD
PD
14
80
o80
..
80.
80.:- ,:i .
60.
75
:
,
70
,
65
60
55
''
40,
50
45
20.
12
-
Dro
d (mm)
8 r- h1
11
14
0.I
r1 ;i1 114
2
P/D
15
5
40
35
30
111
The maximum achievable bumup for UZrHi.6 and U0 2 fuels is the minimum
bumup reported by the neutronics or fuel performance studies. Figures 4.7 - 4.10 present
the combined results as a function of enrichment, pressure drop, and geometry for UO2
and UZrHi. 6. With the exception of very small rod diameters and P/D ratios, the burnup
for tJZrH 1 .6 is limited by the neutronics. UO 2 is limited primarily by neutronics for
smaller P/D ratios, and fuel performance for larger P/D ratios, though as the enrichment
increases fuel performance takes on a more dominant role.
Figure 4.7 Maximum Achievable Burnup vs. P/D and D,,d for Square Arrays of UZrHi. 6 at 60 psia
A: 5% UZrH
6
B: 7.5% UZrH .6 (MWD/kgH
,,,)
(MWD/kgHM)
VIM'
50
48
80
45
46
70 - .
44
60
42
50
40
12
Dd (mm) 8
35 12
10
8
Dd (mm)
1.4
1 2 P/S
I
75
70
65
;
60
D: 12.5% UZrH
C: 10% UZrH 1 6 (MWD/kgHM)
I 55
1.2 P/D
(MWD/kg,,
,
,)
140
.
100
.
120
I .
90
80
70
I 60
.
..
.,
100 .
50
100
80
;
12
(mm]Q
'rod """'
.
I
1
60
P/D
112
Figure 4.8 Maximum Achievable Burnup vs. P/D and D.od for Square Arrays of UZrHI. 6 at 29 psia
A: 5% UZrH1 6 (MWD/kgHM)
B: 7.5% UZrH 16
16 (MWD/kgHM)
45
40.
-
I
35
12
Drd (mm) 8
48
78
46
76
74
44
72
42
70
40
I 68
12 P/D
D: 12.5% UZrH , (MWD/kg,,,)
C: 10% UZrH 1 6 (MWD/kgHM)
140
100
110 . -
90
--
80
:
:
100
.
80 50:;
70 -
)8
12
150
120
100
90
12
rod(mm)
12 PD
I 70
100
.o
,
10
n
/mm \
umd
"
_
14
I .Z P/D
Irrrrio
I
80
Figure 4.9 Maximum Achievable Burnup vs. P/D and Drod for Square Arrays of U0 2 at 60 psia
A: 5% UO 2 (MWD/kgHM)
60--
:
:
B: 7.5% UO 2 (MWD/kgHM)
-.
I
,
40
40
20
30
40
20
20
1.
1.4
.2 P/D
D_,
,.
C: 10% UO 2 (MWD/kgHM)
80 1-' i-
I
60
0U
50
40
Dod(mm) 8
1.2 P/D
I 30
80
70
:
60
40-
50
40
20- ;
Drod (mm) 8
70
.I
IA
u
60-
12.
- ---,
' 1.
30
1.2 P/D
113
Figure 4.1) Maximum Achievable Burnup vs. P/D and D,.,d for Square Arrays of UO2 at 29 psia
A: 5% UO 2 (MWD/kgHM)
60--'"'
'-, .
B: 7.5% UO
.
.
50
~
A| -
awn
401
-v
20
- .
1. 4
.
12
C: 10% U0 2 (MWD/kgHM)
80 -
.
.
'
-iH
i ,
.
80 -
~
cn,
I
OU -
30
40'4
20
12
I 10
1.2 P/
Dd (mm) 8
(MWD/ka,..
'- -
.
'.
.
11
'dk...-
ou
70
.
60
uu-~~~~~~~~~~~'
50
40
120
II
~
",
-
14
'. PD
1
I 30
dU
70
60
40
12
D,,d(mm) 8
j.t .
'
50
40
-.
'
1.2p/D
I 30
4.4.3 Economic Parameters
This section presents the unit cost information and plant operating parameters
necessary to determine the cash flow histories for the fuel cycle, O & M, and capital cost
portions of the economics analysis.
4.4.3.1 Fuel Cycle Unit Costs
The nuclear fuel cycle has costs associated with procurement and fabrication of
fuel assemblies, and storage and disposal of spent fuel. For this analysis, this data is
derived from two sources: the June, 2004 US spot market for uranium services, and
OECD/NEA recommendations [9].
114
Table 4.3 PWR Fuel Cycle Unit Costs for U0 2
Cost Component
Symbol
Unit Price
Mining/Ore
Core
$4 l/kgHM
Conversion
Cconv
$8/kgHM
Cenr,SWU
$108/kgswu*
Enrichment
Cfab
$
Temporary Spent Fuel Storage
Cstor
$
Waste Disposal
Cdisp
275
Fabrication
25
/kgHM
0/kgHM
1 mill/kW-hre
* Separative Work Units (SWU) are a measure of the amount of work necessary to separate the
enrichment plant feed into the desired enriched product and a depleted waste stream. Determining
the mass of SWU required to achieve a desired enrichment is discussed in Section 4.5.1.3a.
4.4.3.1a Fabrication Costs for UO2
The unit fabrication cost is given with respect to the unit mass of heavy metal and
is based on the cost to manufacture fuel assemblies for commercial PWRs. It has been
suggested by industry experts, however, that as much as 50% of the total fabrication cost
depends on the number of fuel rods in the core. Therefore, to account for geometries
offering larger and smaller numbers of fuel rods than the reference core, an adjustment is
made to the unit fabrication cost via a scaling factor. This scaling factor depends on both
the portion of fabrication costs that scale with the rod number, x, and the difference
between the number of rods in a new geometry, N, and the reference core, Nref. The new
unit fabrication cost for oxide fueled cores is given by:
Cfab,oxide
Cfab (-X) +(Cfb 'X) 1+
Nref
LKrefrel11
(4.8)
(4.8)
Simplifying the expression yields:
Cfab,oxide = Cfab + (Cfb
x).
N-N
ref
Nre
Nref
1
(4.9)
Assuming that x = 50%, the unit fabrication cost and rod number are shown in Figure
4.1 1 for square arrays of UO2. The unit fabrication cost and rod number for the reference
core are denoted by black lines.
115
Figure 4.11 UTnitFabrication Cost and Rod Number vs. P/D and D,odfor Square Arrays of tO 2
UnitFabrication
Cost($/kgrM)
-
-I-
,
I·
.
RodNumber(x 14)
* .
*15
114
~Enn
Mr
600
450
15
12
400
10
10
- .
350
5
'-8
5
300
2250
1
I!
500
400
_
300
8
200
1.5
1.3
I1-2-Of
P/D
1.1
-
9
1-00o
12.5
I
h
~1C"0200
PD
1.1
12.5
65
4
4.4.3.1b Fabrication and Storage Costs for UZrH1 .6
The mining, conversion, and enrichment processes are independent of the fuel
type, and so their associated unit costs are identical for UZrH. 6 and UO2 fuels. The
fabrication cost, however, is based on the specific chemical processes necessary to
manufacture 1JO2fuel pellets from enriched UF6 gas, and so does not apply to UZrH
6.
With no cost information available from industry, the fabrication cost must be
approximated.
An assumption is therefore made that the fabrication costs for UZrHt. 6
and UO2 are identical for equivalent volumes of fuel. The unit fabrication cost for
UZrHI 6 is therefore given by:
(4.
l.oxide 'Poxide
C(W'HA
(fib.hdride =
(4.10)
fihaoxide
M'[Li hydride Phyvdride
where,
wMHt: weightpercent heavy metal in the fuel matrix
p:
fuel density
The unit fabrication cost for UZrH.
6
is therefore larger than oxide because of its
lower heavy metal content. Assuming again that x = 50%, the unit fabrication cost and
rod number are shown in Figure 4.12 for square arrays of UZrH 1 .6, with the values at the
reference geometry denoted by black lines.
116
Figure 4.12 Unit Fabrication Cost and Rod Number vs. P/D and D,.,dfor Square Arrays of
UZrHi.6
UnitFabrication
Cost($/kIk
Rod Number(x 1i)
16
1400
- . ~~~~~681
:"
.
I DUU
I ,
.
.
. . .-:
50952 .
14
1200
1000
1000
15
12
10
10
5
8
800
500
-
1
65
8
1;.5,@h
PID 1 2
' :",
600
0
1
65
6
4
11
1F1 12 5
Like the fabrication costs, the unit cost for UO2 spent fuel storage is based on the
heavy metal content in the fuel. Assuming that storage costs are dominated by the
availability of space in the spent fuel pools and dry cask storage, storage costs for UO2
and UZrH 1. 6 should be identical for equivalent fuel volumes. The same relationship used
to relate the unit fabrication costs is thus applicable to the unit storage costs:
'H
WH.oxide 'Poxide
L stor.hlldride-=- stor.oxide'
/
I II
(I . I
'HAhvdride ' Phydride
4.4.3.2 Operations and Maintenance Unit Costs
O & M includes all costs associated with energy production that are not directly
related to the fuel cycle. Examples include material and manpower costs for outages,
replacement energy costs, and the salaries of year round personnel. They are categorized
as fixed or variable. Fixed costs do not depend on the operating conditions of the plant,
and variable costs do. The O & M unit costs are derived from expert recommendations
acquired by Jacopo Saccheri at MIT [8].
Table 4.4 PWR Operations and Maintenance Unit Costs
Cost Category __
Variable
Fixed
Cost Component
Symbol
Unit Price
Refueling Outage
CRO
$800,000/day
Forced Outage
CFO
$100,000/day
Replacement Energy
('repP
30 mills/kW-hre
Personnel
Cpers
$150,000/person-yr
Number of Personnel
Npers
600
117
All costs presented in Table 4.4 are considered in the O & M analysis except
replacement energy. Note that equation (4.6) shows that the levelized unit cost is
obtained by normalizing the total cost by the energy produced from the plant; this does
not include replacement energy purchased from other utilities during outages. Including
the replacement energy costs unjustly increases the COE, because the utility receives
revenue from its sale that is unaccounted for in the economics analysis. It is therefore
assumed that the costs and revenues associated with replacement energy are perfectly
balanced, and do not affect the overall COE.
4.4.3.3 Capital Costs
With the lack of new plant construction in the US over the last two decades,
vendor estimates for capital costs are regarded with extreme caution by potential
investors. Historically, the difference between initial estimates and final realized costs
for nuclear construction was large, often by a factor of 3 or more. Reasons for this
include but are not limited to the lack of owner control of the construction process,
"regulatory ratcheting" by the NRC in the wake of the accident at Three Mile Island, and
growing public opposition to nuclear power. While it is not the intent of this report to
examine the current state of these factors and their ability to contribute further price
increases to the industry, it is important to note that any cost projections for nuclear
power will be regarded as highly speculative until demonstrated by construction of a new
plant in the US.
Uncertainties aside, there are two capital cost estimates addressed in this
economics analysis, both of which involve backfitting existing PWRs for use with
hydride fuels. The first scenario, minor backfit, is geared toward integrating hydride
fuels into existing plants without major modifications to the pressure vessel. This is
accomplished by maintaining the fuel assembly and control rod configurations; the
design space is therefore limited to geometries at the existing core pitch. The conversion
costs from the point of view of capital include the costs to replace the steam generators
118
and upgrade the turbine units , if the new geometries offer increased power. Coolant
pumps will also require upgrades, but their contribution to the capital investment is small
compared to the turbine and steam generators, and so will not be considered. The major
backfit scenario considers retrofitting a much wider range of geometries into existing
PWR cores (i.e, all remaining geometries in the parametric study). The potential power
upgrades are larger, and in addition to steam generator replacement and turbine upgrades,
the vessel head and core internals will require replacement.
An additional cost should also be considered. Existing units that decide to switch
to hydride fuel will have to discard partially irradiated UO2 fuel assemblies. Since the
utility has already incurred the cost of this fuel, a simple model for fuel depreciation is
needed so that the remaining "worth" of the fuel can be estimated and added to the
overall capital cost of the conversion process. In this analysis, it is assumed that the
value of the fuel depreciates linearly with time, and so the total value of the fuel in the
core at the end of an operating cycle is approximately 1/3 of the total cost of the fuel
assemblies when new9.
The costs for new steam generators, vessel heads, and core internals are available
from current industry experience. It is assumed that the vessel head and core internal
replacement costs are fixed, and not dependent on the power. The steam generator and
turbine upgrade costs, however, are expected to demonstrate an economy of scale. This
is based on historical economic information, that shows that the unit cost (i.e, $/kW) for
these components is lowered as the power output from the plant increases. A scaling
argument is therefore used to predict the capital cost of major nuclear plant components
as a function of the reference cost, Costref, power,
, and a scaling factor, m [10]. The
cost for a new component is given by:
8 It
is assumed that the turbine has untapped capacity that can be exploited with appropriate modifications
and upgrades. New turbines are therefore not required.
9 Assuming a linear depreciation model, at the end of an operating cycle 1/3 of the fuel is worth 0, 1/3 of
the fuel has 1/3 of its value, and 1/3 of the fuel has 2/3 of its value. The average worth is therefore - (1/3)
* (O + 1/3 + 2/3) - 1/3 of the total cost of the fuel for a new core.
119
(t~
Lmcompo.nr
(4. 12)
:= Costref kne
Costnew
The upgrade cost is approximated as the difference in cost between the component sized
to operate at a new power, and the component sized to operate at the reference core
power.
Costupgrade= CoStnew - Costref = Costref
e
f
J
-.
The costs of components for the major and minor backfit scenarios are summarized in
Table 4.5. It is assumed that these estimates are for installed components, and so the cost
of labor is not considered separately. The reference turbine cost and the scaling factors
are taken from historical data available in the Department of Energy's "Nuclear Energy
Cost Database" [10], while the cost estimates for the reference steam generators, vessel
head, and core internals are obtained from Professor R. Ballinger of MIT, an observer
familiar with industry cost experience. The value for existing UO2 fuel that is lost when
the backfit is performed is obtained from equation (4.40), presented later in this Chapter.
Table 4.5 Cost Estimates for Installed Nuclear Components
Cost Component
Steam Generators (Ref Q)
Vessel Head
Core Internals
Turbine Generator (Ref Q)
Existing Fuel Value
Symbol
Price
Scaling Factor, m
CsGref
$100,000,000
0.6
Chead
$25,000,000
Cint
$25,000,000
Cturb,rf
$338,000,000
Cfuel
$66,586,000
0.8
4.4.3.4 Plant Operating Parameters
The following operating parameters are fixed for the economics analysis.
120
Table 4.6 Fixed Plant Operating Parameters
Parameter
Refueling Outage Length (ref. core)
Symbol
TROref
Value
20 days/cycle
FOR
0.01
L'
0.99
Thermal Efficiency
1
0.33
Batches
n
3
ForcedOutageRate
Availability
Plant Life
Tplant
20 yrs
The 20 year plant life is based on current NRC license extensions for existing LWRs. It
is expected that an existing unit that converts to hydride fuel can expect to operate for this
period of time. The refueling outage length is based on the most efficient units operating
commercially today.
4.4.3.4a The Operating Cycle
Two significant indicators of plant performance are missing from Table 4.6: the
capacity factor, L, and the cycle length, Tc. The cycle length, which is the time between
successive startups, is made up of three components: the duration of fuel irradiation and
the downtime for forced and refueling outages. The effective full power years (EFPY) at
discharge is used to denote the total irradiation time for the fuel. It depends on the
thermal power, Qth,and the amount of energy available from the fuel through the burnup,
Bu, and heavy metal loading in the core, MHM. Because the fuel is handled n times during
its lifetime before being discharged from the core, the irradiation time for an individual
cycle is EFPYC,which is equal to the quotient of EFPY and n. The cycle length is
therefore given by:
Tc = EFPY +
TFO + TR °
365.25
(yrs)
(4.14)
where EFPYC is in years, and the forced outage length per cycle, TFO,and the refueling
outage length per cycle, TRO,are in days. EFPY and EFPYC can be determined by:
EFPY = EFPYcn
BuEP
-M
MHM
A1h
1000
10(4.1
365.25j
5)
121
where the units are:
t
MWD(kgM =1000
j(Birth)
365.25(dSj
J
=(Yr)(4.16)
__-~~~~
\M}-
(yr)
The availability is a measure of the plant's operating capacity, accounting for outages that
are not planned and require the plant to be either de-rated in power or shutdown. Both
EFPY (yrs) and EFPYC (yrs) are related to the forced outage length, TFO(days), through
the availability. Note that the sum of EFPYCand the forced outage length is the
available cycle length, TAV(yrs).
EFPY =(EFPY+
TFO .L'=(EFPY
n
+ 365
L'=.25
. n TA L' (yrs)
(4.17)
Solving for the forced outage length yields:
TFO
= EFPYC
-(
-1) 365.25 (days)
(4.18)
With the refueling outage length given in Table 4.6, all unknowns in equation (4.14) for
the cycle length have been defined. The cycle length is then used to solve for the
capacity factor, which is a measure of the plant's operating capacity accounting for both
planned and unplanned shutdowns:
L EFPYC
-
(4.19)
r
Substituting for both Tc and EFPYC, the capacity factor can also be written as:
L
Bu
B//L, + 0.001 T
(4.20)
Q
.sn
where the specific power, Qp, is:
aOA
Mp=
kth
MH
kgHM
,
(4.21)
122
Thus it is seen that the plant capacity factor depends on the burnup in the fuel, the mass
of heavy metal loaded in the core, and the power.
A helpful way to visualize the relationship among these terms is by constructing a
simple plot of plant capacity versus time, for three different operating scenarios. This
plot is shown in Figure 4.13. The plant capacity is the fraction of rated power at which
the plant operates. When the plant operates at 100% capacity, the operating cycle length
(the length between shutdowns) is EFPYC. When the plant operates at its availability,
which accounts for time down due to forced outages, the length of operation is the
available cycle length, TAV. And finally, when the plant is operating according to its true
capacity, which accounts for time down due to both forced and refueling outages, the
operating length is equal to the cycle length, Tc. Because the total energy available from
the core is the same for all three operating scenarios, the areas represented by each
combination of plant capacity and operating length are equivalent.
Figure 4.13 Plant Capacity vs. Operating Length
Plant
Capacity
TFO
TRO
::~~~1 h1-p
1
L
L
· ·---
· r----
::
::
::·:··::
·
:::·
:':'"·'::'::
·
10
EFPYc
TAV
Tc
Time
There is one final note regarding the refueling outage length. The value listed in
Table 4.6 is exemplary of the most efficient PWR units, which typically operate on an 18
month cycle. During the refueling outage two types of maintenance activities occur:
those associated with refueling evolutions, and those associated with maintenance and
123
inspections of the plant equipment (i.e., reactor vessel, steam generators, turbine, etc.).
With advice from plant engineers in industry, the average time devoted to each outage
activity was approximated for a typical refueling outage.
Figure 4.14 Refueling Outage Activities
TRO
_
_
L
._
3 daysfor reactor shutdownand startup
10 daysfor refuelingevolutions
2:
3+ 4: 20 daysfor maintenanceand inspections
1:
Because of the large variation in cycle length for the designs in the parametric
study, it is believed that the critical path maintenance devoted to non-refueling activities
(i.e, 4 in Figure 4.14) should scale with the time between successive shutdowns. For
example, a PWR operating on a 9 month cycle still requires a total of 13 days for
refueling evolutions, but the remaining critical path maintenance can most likely be
performed in fewer than the originally allotted 7 days. An assumption is therefore made
that the critical path maintenance devoted to non-refueling activities scales linearly with
the cycle length. A PWR operating on a 9 month cycle would therefore need a total of
16.5 days for its refueling outage (13 days for refueling, and 3.5 days for remaining
critical path maintenance).
TR =13T
ref
(days)
(4.22)
Equation (4.14) shows that the cycle length depends on the refueling outage
length. Multiple iterations are therefore required for the refueling outage and cycle
lengths to converge to constant values. The procedure is as follows: an initial guess for
the cycle length is made using the reference refueling outage length; the new refueling
outage length is determined using equation (4.22); a new cycle length is then determined.
124
The process repeats itself until neither the refueling nor cycle lengths change significantly
between iterations.
4.5
Economics Analysis
This section develops and discusses the details necessary to determine the cash
flows for the fuel cycle, O & M, and capital cost components of the COE, subject to the
range of operating conditions and core geometries considered in the parametric study.
The results for the lifetime levelized unit cost components are presented, but the
combined COE analysis is delayed until Section 4.6.
4.5.1 Fuel Cycle Costs
4.5.1.1 The Nuclear Fuel Cycle
The nuclear fuel cycle includes all activities related to the procurement, use,
storage, and disposal of nuclear fuel. It can be open or closed. In the open, or oncethrough cycle, uranium is mined from ore deposits, enriched and fabricated into fuel
assemblies, burned in the reactor, and disposed of directly in a geologic repository after a
period of on-site storage. In the closed cycle reprocessing of the discharged fuel is
employed to chemically separate fissile material from the fission products for later
incorporation into fresh fuel assemblies. There are significant safety concerns
accompanying spent fuel reprocessing and the potential use of the technology as the basis
for an advanced nuclear weapons program. Because of these concerns, and the stable low
cost of uranium ore and nuclear fuel services, the US has adopted the open cycle for
today's operating reactor fleet. The fuel cycle cost analysis therefore only considers the
once-through direct fuel cycle option. In future analyses of hydride fuels incorporating
plutonium, reprocessing cost estimates will need to be included.
The once-through fuel cycle is typically broken into two stages:
125
*
Thefront end: The front end encompasses all activities from the initial mining of
uranium ore to the fabrication of new fuel assemblies. More specifically, it
includes all the costs associated with: extraction/mining of uranium ore,
conversion of ore to gaseous UF6,
235U
enrichment, and fabrication of fuel pellets
and fuel assemblies.
*
The back end: The back end includes all costs associated with the temporary
storage of spent fuel at the reactor site and permanent disposal in a geologic
repository.
4.5.1.2 Recurring Cash Flows and the Fuel Cycle
The lifetime levelized unit fuel cycle cost is given by:
Clev,fcc
Clevc
-
CEannual
PVfcc
th
L 8.766
r
'1
(4.23)
r
e-rTplanI
All terms in this equation have been previously defined except for the total present value
of fuel cycle costs, PVfc, which depends on the cash flow histories for all operating
cycles over the lifetime of the plant. To predict the cost for the nth successive cycle, the
present value for the first operating cycle, PVfcc,,,,is found, and projected forward using
the escalation rate and cycle length.
PVfcc, = PVfcco en g T C
'
(4.24)
In this way, the fuel cycle costs are modeled as discrete cash flows through time, as
shown in Figure 4.15.
126
Figure 4.15 Cash Flows for Successive Operating Cycles
Time
PVfc,o
PVfP,
4
PVfcc,2
PVfcc,3
PVfcc,4
1"V fcc,5
PVfcc, 6
The total present value can then be found by discounting each cash flow expenditure back
to the reference date by the discount rate, r. The reference date is the start of irradiation
for the first operating cycle.
PVfc =· PVfcc,o+ EPVfcc,n .e - nr' T
(4.25)
n=l
This is the desired cost term in equation (4.23). The only unknown is the present value of
costs for the first operating cycle, PVfc,,,, which is discussed in the next section.
4.5.1.3 Cash Flows for the First Operating Cycle
To obtain the desired cash flows for the front and back end components of the
first operating cycle, four things are needed: a mass balance of material flows through the
front end processes, the unit cost for each component based on the heavy metal loading in
the core ($/kgHM),the time relative to the start of irradiation when each cost is incurred,
and the mass of heavy metal loaded into the core.
4.5.1.3a Front End Mass Balances
The front end unit costs listed in Table 4.3 are based on the mass of heavy metal
entering the individual processes. For example, the unit conversion cost is $8 per
kilogram of heavy metal entering the conversion plant. Due to material losses, however,
the mass of heavy metal flowing through each process is not the same as the mass of
heavy metal that ends up in the core. Thus, the production of 1 kg of heavy metal for
127
core loading requires more than 1 kg of heavy metal mined from uranium deposits. Mass
flow balances of the front end are therefore used to determine the equivalent front end
costs with respect to the unit mass of heavy metal loaded into the core.
The mass flow balance is performed using mass loss fractions, 1, as recommended
by NEA/OECD [9]. The mass loss fractions, given in Table 4.7, relate the mass flows,
M, into and out of each process.
Mm" ( 1
Mout
(4.26)
-
Table 4.7 Mass Loss Fractions for Front End Fuel Cycle Processes
Mass Loss Fraction
Symbol
Mining/Ore
Value
lore
0
Conversion
Iconv
.005
Enrichment
lenr
equation(4.32)
Fabrication
lfab
.01
Figure 4.16 traces the flow of heavy metal through the front end of the fuel cycle
required to produce 1 kg of heavy metal at the reactor. The individual mass flow terms,
Mx, are developed below.
Figure 4.16 Mass Flows for Front End Fuel Cycle Processes
Mtails
Power Plant
The unit mass of heavy metal loaded into the power plant is 1 kg.
Mpp= 1
kgHM
(4.27)
128
Fuel Fabrication
Mass flows into the fabrication plant as enriched UF6 , and flows out as the mass
of heavy metal in either U0 2 or UZrH,.6.
Mo~fab
(ll
0IfabJ
1|(4.28)
M pp
-lfab
Fuel Enrichment
The enrichment loss, lenr, depends on the fuel enrichment, and so it is not assigned
a specific value in Table 4.7. For this study the fuel enrichment varies between 5% and
12.5% for UZrHI.6and between 5% and 10% for UO2. Unlike the other front end
processes that have a single output mass flow stream, the enrichment plant has two: the
tails (or waste) stream, Mt,,is,with the depleted uranium by-product, and the enriched
product stream, Mfab,headed to the fuel fabrication plant. Both depend on the product
enrichment. I:[nthis study, the feed and tails enrichments are fixed, and the product
enrichment is variable, as shown in Table 4.8.
Table 4.8 Inlet and Outlet Mass Flow Stream Enrichments at the Enrichment Plant
Flow Stream
Symbol
Mass Fraction Enrichment
Feed
f
0.00711
Tails
t
0.003
Product
p
0.05, 0.075, 0.10, .125
Simple mass flow balances on 235 U and the total mass of heavy metal are applied to
obtain the desired ratio between the feed and product mass flow streams.
Menr, f =Mtas t+Mfab p
(4.29)
Mtails = Menr - Mfab
(4.30)
and,
Substituting equation (4.30) into equation (4.29) and rearranging terms yields the desired
mass flow ratio.
Menr
( -t)
Mfab f
-t)
~~Mnrloa~~~
~(4.31)
(p~-)
129
where,
(IlenrJ)(
(4.32)
There is an additional step, however, to finalize the unit cost for the enrichment
process. Table 4.3 lists the unit cost for enrichment with respect to kg of SWU, or
separative work units. Like the mass flow streams, the unit enrichment cost also depends
on the final enrichment of the fuel. This is accounted for by determining how many units
of "separative work" are required to separate the feed into output streams with desired
enrichments. SWU depends on a potential function, V,which is applied to the input and
output streams at their respective enrichments, e.
V(E) =:(2E -1). ln
(4.33)
1--E
The enrichment dependent SWU is defined using this potential function and the mass
flows for each stream connected to the enrichment plant:
Mtais V(t) - Menr,,V(f)
(4.34)
(kgswu
kgHM
M fab
Substituting for the mass flows from equations (4.29) and (4.30), and for the potential
function from equation (4.34), it can be shown that:
SWU =[V(p)- V(t)]- (P)
[V(f
-
V(t)] (kg
u)
(4.35)
The unit SWU requirements for the 4 enrichments considered in this study are listed in
Table 4.9.
Table 4.9 SWU Requirements for Different Enrichments of UO2 and UZrH1.6
SWU (kgswU/kgHM)
5%
7.5%
10%
12.5%
UO 2
7.19
12.17
17.28
-
UZrHI. 6
7.19
12.17
17.28
22.48
130
Note that the product of SWU and the unit enrichment cost listed in Table 4.3
($108/kgswu), yields the desired enrichment cost per kg of heavy metal loaded into the
core (Table 4.9 lists the kg of SWU per kg of heavy metal loaded into the core).
Fuel Conversion
The conversion process transforms uranium ore, U3 08 , into gaseous UF6,
according to the following mass balance:
Menr
(4.36)
Mc
=
1 |
1conv )
Uranium Mining/Ore
There are no losses in the mining and milling operations, so,
More (
1
Mc
=1
(4.37)
'onv
re
Combining equations (4.27), (4.28), (4.31), (4.36), and (4.37), the mass of natural
uranium required to produce 1 kg of heavy metal at the reactor is:
More
--
l
(P-t
fco
-t
.M
1-lfab
kg(I
(4.38)
These equations can also be used to get the mass flows through each front end process
necessary to yield 1 kg of heavy metal at the reactor. This information is given in Table
4.10 for different enrichments of U0 2 and UZrH1 .6. Note that the units are kg of heavy
metal through the process per kg of heavy metal loaded into the core.
131
Table 4.10 Mass Flows Through the Front End
Front End Unit Mass Flows (kP HM throuh nrocess/k HM in the core)
UO
UZrH. 6
5%
7.5%
10%
5%
7.5%
10%
12.5%
1
1
1
1
1
1
1
Fabrication
1.01
1.01
1.01
1.01
1.01
1.01
1.01
Enrichment
11.55
17.7
23.84
11.55
17.7
23.84
29.98
Conversion
11.61
17.79
23.96
11.61
17.79
23.96
30.13
Mining/Ore
11.61
17.79
23.96
11.61
17.79
23.96
30.13
Reactor
4.5.1.3b Back End Mass Balances
The back end unit costs for spent fuel storage and disposal are listed in Table 4.3.
The disposal fee is already levelized with respect to the energy production from the plant,
and need only be added to the lifetime levelized unit fuel cycle cost as determined by
equation (4.23) for the remaining front and back end processes. The unit storage cost
requires no mass flow balance because it is already based on the heavy metal loading in
the core. Even though the burnup in the fuel changes this mass between initial loading
and final discharge, this analysis assumes that the storage costs, like the front end costs,
are based on the original heavy metal loading in the core.
4.5.1.3c Fuel Cycle Cash Flows
With the information in Tables 4.3, 4.9, and 4.10, the unit costs for the front and
back end components can be determined with respect to the initial heavy metal loading in
the core.
132
Table 4.11 Fuel Cycle Unit Costs With Respect to the Heavy Metal Loading in the Core
Unit Cost (/kg H
Sym.
5%
UO
7.5%
10%
5%
UZrHj.6
7.5%
10%
12.5%
Mining/Ore
UCore
476.01
729.39
982.36
476.01
729.39
982.36
1235.33
Conversion
UCo,,n
92.81
142.32
191.68
92.81
142.32
191.68
241.04
Enrichment
UCenr
776.52
1314.36
1866.24
776.52
1314.36
1866.24
2427.84
Fabrication*
UCfab
277.75
277.75
277.75
687.81
687.81
687.71
687.71
Storage
Fuelmp.
Fuel
Storage
UCtor
250
250
250
618.54
618.54
618.54
618.54
* This is the unit fabrication cost for the reference core geometry. For geometries with larger and smaller
rod numbers, the unit fabrication costs are still given by equations (4.9) and (4.10), with Cfabreplaced by
UCfab.
The front and back end costs occur at different times relative to the start of fuel
irradiation and therefore must be referred to this date using the discount rate by
appropriate lead and lag times to obtain their true present value. OECD/NEA
recommended lead and lag times for the front and back end processes are given in Table
4.12 [9].
Table 4.12 OECD/NEA Recommended Lead and Lag Times for Front and Back End Processes
Symbol
Value
Fuel Fabrication
Tfab
1 yr
Uranium Enrichment
Tenr
1.5 yr
Uranium Conversion
Tco,,v
1.5 yr
Transaction
2 yr
To
Uranium Ore Purchase
- Tc*
Tstor
Temporary Spent Fuel Storage
* A negative sign denotes that the storage costs are incurred after the start of fuel irradiation and
so the cost must be referred back in time to the reference date
Keeping in mind that the disposal fee is added at the end, the present value of unit
costs for the first operating cycle is given by:
PV(UCfC,G)
TJCoree
+
con
r er
+ UCenre
+ C fabeUCbstor
e - r Tc
(4.39)
And the total present value for the first operating cycle is given by:
PVc,
o
= PV(UCfcC,O) MHM
($)
(4.40)
133
The mass of heavy metal in the core, MHM,is:
MHM = N (4)
(D-2tcI
-2 tg )
Lh
W
(kgH)
(4.41)
where,
N:
number offuel rods in the core
D:
clad outerdiameter
tcl:
clad thickness
tg:
Lh:
p:
w:
radialgap thickness
fuel rod heatedlength
densityof thefuel
weightpercentheavy metal of thefuel
4.5.1.4 Lifetime Levelized Unit Fuel Cycle Cost
Using the present value for the first operating cycle given by equation (4.40), the
total present value for all fuel cycle costs incurred over the life of the plant can be
determined:
PVfcc=PVfcc,o+PVfcc,n .e-'e
(4.42)
n=l
where,
PVfcc,, = PVfcc,o e g
(4.43)
And the final lifetime levelized unit fuel cycle cost is given by:
Ecc
annual
L,it7 *L
8.766 1-e rTplat
where Cdispis given in Table 4.3.
4.5.2 Operations and Maintenance Costs
Operations and maintenance costs include all production costs associated with
energy generation from the plant not included with the fuel cycle, and can generally be
divided into two categories: fixed and variable. Fixed costs are steadily incurred over the
life of the plant and do not depend on operating parameters (i.e, operating cycle length,
capacity factor, energy production per cycle, etc.). Variable costs take these factors into
consideration. The fixed and variable costs considered in this analysis and their
respective unit costs are given in Table 4.13.
134
Table 4.13 Fixed and Variable Unit O & M Costs
Cost Category
Symbol
Unit Cost
Fixed Costs
Plant personnel
Cpers
$150,000/pers-yr
Refueling Outage
CRo
$800,000/day
Forced Outage
CFO
$150,000/day
Variable Costs
As discussed in Section 4.4.3.2, the expenditures and revenues associated with the
purchase and sale of replacement energy during plant outages are assumed equal, and so
are not included.
4.5.2.1 Annualizing O & M Cash Flows
The lifetime levelized unit O & M cost is given by equation (4.6):
levO&M
&M
Clev"O&M -- ,,Eannual
Eannuali
*ith
PVO&M
1 L 8.766
r
j
L1eraa,
-r
(445)
where all temls have been previously defined except the total present value of O & M
expenditures over the plant's life, PVo&M.Like the fuel cycle cost analysis, it is helpful
to redefine the O & M unit costs with respect to consistent units. For example, in Table
4.13 the unit personnel costs are in $/year, while the outage costs are in $/day. For ease
of calculations, all O & M unit costs will be annualized so that their sum constitutes the
total O & M expenditures for the first year of plant operation.
With the refueling outage length, TRO,defined in equation (4.22), the annual cash
flow for refueling costs is:
CFRo =: ROTRO
TC
($
(4.46)
yr
The units for TRo and Tc are (days/cycle) and (years/cycle).
Similarly, the forced outage length defined in equation (4.18) can be used to determine
the annual cash flow for forced outage costs:
135
CFFO = CFO TFO
$
(4. 47)
TC
where TFOis in (days/cycle).
The product of the number of plant personnel and their annual salaries yields the total
annual personnel costs:
(4.48)
CFpers = Npers Cpers
The total annual 0 & M expense for the first year of plant operation is therefore:
CF&M, = CFRo + CFFo + CFpers
(4.49)
)
Recall that a constant escalation rate is used to account for non-inflationary
price
increases in the cost of labor and materials over time. The nth successive annual O & M
expenditure is therefore given by:
CFO&M
, = CFo&M,o eg n
'
(4.50)
'
where the initial cost, CFo&M,o,is assumed to occur at the end of the first year of plant
operation. This is shown in Figure 4.17.
Figure 4.17 Cash Flows for Successive Annual O & M Expenditures
Time
CFo&NI,O
C&
CFo&M,
CFo&M,2
CFo&M,oU
3
CFO&M,4
CFo&M,S
136
4.5.2.2 Lifetime Levelized Unit O & M Cost
With discrete O & M expenditures projected over the plant's life, the total present
value is determined by discounting each cost term back to the present by the discount
rate.
PVO&M =
(4.51)
CFO&M,n e-r(n+l)
n=O
The lifetime levelized unit O & M cost can then be determined according to equation
(4.45), which is repeated below:
EvO&M&Eannual
th
PV&M
-r -L8.766
r
1-e-rTp](.2
(4.52)
4.5.3 Capital Costs
Capital is typically the largest component of the COE, and includes costs
associated with siting, design, construction, refurbishment, and upgrades of nuclear
facilities. They are unique among the COE components because they are not
continuously incurred over the life of the plant. Instead, these sunk costs occur during
the plant's construction phase, when aging equipment needs replacement, and when
equipment modifications/replacements are made in preparation for power upgrades.
Because the Hydride Fuels Project seeks to quantify the benefits of hydride fuel use in
existing LWRs, only capital costs associated with the backfit are considered for the final
optimization. These expenditures are only a fraction of the design and construction costs
for a new plant, and so their contribution to the overall COE will not differ significantly
from the fuel cycle and O & M components.
In the minor backfit scenario, the existing fuel assembly and control rod
configurations are maintained and capital costs are incurred to upgrade the steam side of
the plant (the steam generators and turbine) for operation at higher powers. Because the
coolant temperatures at the inlet and outlet of the core are fixed, it is assumed that no
further modifications are required to the pressure vessel. For the major backfit scenario,
the layout of fuel in the core can assume any geometry within the bounds of the
137
parametric study, and so in addition to steam generator and turbine upgrades, replacement
of the vessel head and core internals is also required. Note that designs offering lower
powers than the reference core for both backfit scenarios will not incur capital costs for
the turbine and steam generator units. The major backfit case will still require
replacement of the vessel head and core internals, however, to accommodate the new
geometry of the fuel assemblies. Both the major and minor backfit cases will also incur
the cost of discarded UO2 fuel assemblies.
4.5.3.1 Predicting Capital Costs for PWR Backfit
Section 4.4.3.3 discussed the capital expenditures assumed for the different
backfit scenarios. They are summarized below:
Major Backfit Capital Costs
·
Steam Generators - Replace ifQnew
> Qref
* VesselHead- Replacefor all Qnew,,
* Core Internals- Replacefor all Qnew
* TurbineUnit - UpgradeifQnew > Qref
·
ExistingFuel - Add remaining value of discardedfuelfor all Qnew
Minor Backfit Capital Costs
·
Steam Generators - Replace ifQnew >
·
Turbine Unit - Upgrade if Qnew> Qref
·
ExistingFuel - Add remainingvalue of discardedfuel for all Qnew
Qref
Using the capital costs defined in Table 4.5, and the scaling arguments provided in
equations (4.12) and (4.13), the backfit expenditures for each scenario can be determined:
If Qnew> Qref
jChead
+
+
Cbacfit,major= head + Cint +
c
fuel
+c
.
+ CSG,ref
C
+C
new
)ref
J
ewre
(4-53)
138
Cbactfiminor = Cfuel +CSG,ref r
+ Cturbref
e
y ef
J
[L
) mL
(4-54)
ref)
If Qnew< Qref
Cbacft,major = Chead + Cint + Cfitel
(4-55)
Cbackft,minor= Cfuel
(4-56)
As in the case of O & M, it is assumed that the replacement energy costs incurred during
the fuel conversion outage are directly passed onto the consumer, and so their costs and
associated revenues balance.
4.5.3.2 Lifetime Levelized Unit Capital Cost
The general form of the equation for the lifetime levelized unit capital cost is:
lev'cap
lev,cap
Eannual
where PVcapis equal to
(4.52)
r
77.-L8.766 l-eT
-,-
COStbackfitminor
l
ant
orCStbackfit,major
Unlike fuel cycle and O & M costs, the capital costs for hydride fuel conversion
are only incurred once during the plant's lifetime. The selection of Tplanttherefore has a
significant impact on Cevcap because it determines the total amount of energy produced
that can be sold to recover the initial capital investment. If Tplantdecreases, the energy
production decreases, and Cle,,capincreases (meaning the utility must charge more for the
electricity consumed to recover the capital investment). The impact on the fuel cycle and
O & M costs is much less significant because they are recurring in time, and the energy
production to recover their investment is fixed by the spacing between recurring cash
flows (i.e., 1 yr. for O & M, or Tc for fuel cycle costs). The plant life chosen for this
analysis as listed in Table 4.6 is 20 years. It is based on the life of NRC license
extensions granted for approved LWRs.
139
4.6
Lifetime Levelized Unit COE
The results from the economics analysis are presented in this section for the major
and minor backfit scenarios at both pressure drop limits. Throughout this report, the
equivalence of square and hexagonal arrays has been suggested for identical
combinations of rod diameter and H/HM ratio. It is believed that this relationship holds
for the thermal hydraulic, neutronics, and fuel performance studies, which comprise the
primary inputs to the economics analysis. The relationship should therefore hold for the
cost analysis. In this section results are only presented for square arrays, but the
hexagonal costs can be inferred for equivalent geometries.
4.6.1 Results for Major Backfit: UZrH1.6 and UO2 at 60 psia
The lifetime levelized unit cost of electricity for the major backfit scenario at 60
psia is shown in Figures 4.18 and 4.20 for different enrichments of UZrH 1.6 and U0.2
Also shown on each plot, as a black line, is the minimum COE at each P/D ratio. These
lines are re-plotted in Figures 4. 19A and 4.21A independent of rod diameter to provide a
clearer comparison of the COE among different enrichments and fuel types. Also shown
in Figures 4.1.9 and 4.21 are the fuel cycle, O & M, and capital costs that comprise the
minimum COE (i.e., adding Figures 4.19(B-D) gives 4.19A). The minimum COE for
UZrH1 .6 is 18.4 mills/kW-hre for 12.5% enriched fuel at: P/D = 1.32, Drod= 9 mm. Note
that this is very close to the reference core configuration, but that there is a large range of
P/D ratios with costs that are within a small fraction of this minimum value (i.e, 1.22 <
P/D < 1.42). The minimum COE for U02 is slightly lower at 17.9 mills/kW-hre for 5%
enriched fuel at: P/D = 1.39, Drod = 6.5 mm. The range of P/D ratios with costs close to
this minimum value, however, is narrower than for UZrH1.6 . UZrH1. 6 appears to offer the
potential for cost savings over UO 2 in the P/D range 1.2 to - 1.35. A more detailed
comparison of the results is presented in Section 4.6.1.2.
There is a lot of useful information in these figures. First, examine Figure 4.22,
which plots the familiar maximum powers for UZrH1.6 and UO2 . Lines showing the
maximum power as a function of P/D ratio have been added. Note the similarities
140
between these maximum power lines and the minimum COE lines plotted in Figures 4.18
and 4.20. They do not perfectly overlay one another, but their shapes and placements on
the parametric map are similar, thus establishing a strong correlation between minimum
cost and high power. There are exceptions to this rule. For example, Figure 4.18 shows
of P/D ratio for each enrichment of UZrH 1.6
that the minimum COE as a fimunction
plateaus at a rod diameter of - 9 mm, before making a sudden jump to lower rod
diameters where the achievable power is higher.
Next compare the minimum COE curves shown in Figures 4.19A and 4.21A. The
costs for UZrH 1.6 are minimized at its highest enrichment; the opposite is true of oxide
for most of the geometry range. Also note the difference in the curve shapes. As P/D
increases, UZrH .1 6 costs rapidly approach the most economical geometries, and then
begin a gradual trend of increased cost. The reason for this, revealed in Figures 4.19B
and 4.19C, is rising fuel cycle and O & M costs. The UO 2 costs, however, experience a
more gradual descent and are not redirected upward like UZrH1.6. Figure 4.21B shows
that the fuel cycle costs are consistently decreasing (with the exception of a brief increase
around P/D = 1.4 for 7.5% and 10% enriched fuels), and therefore keep the overall COE
from rising. This reason for this is explained in detail in Section 4.6. 1.1.
To provide increased understanding of the behavior of the COE curves, the next
section examines in detail the COE and its individual components for 12.5% UZrH 1. 6 and
5% UO2. These enrichments are chosen because they provide the lowest overall COE.
Following this examination is the final comparison of costs for the major backfit scenario
employing UZrH 1. 6 and UO2, with specific attention given to geometries where cost
savings can be realized for hydride fuel.
141
Figure 4.18 Lifetime Levelized Unit COE vs. P/D and Drodfor Square Arrays of UZrH.6 at 60 psia
,4: 5% UZrH 1.6 (mills/kW-hre)
B: 7.5% UZrH 1.6 (mills/kW-hre)
60
12
11
12
E 10
50
11
50
E
40
40
CE
8
30
8
30
7
7
1.3 1.4 1.5
Pf/D
0
C' 1no, I 7rH
(milIqk ALhrp)
1.1
1.2
1.3 1.4 1.5
PfD
19 r°F, I 17rH
{milIQkALhrp.
1.1
1.2
fl
60
12
11
12
60
11
50
10
50
10
40
:c 9
8
30
7
n
1.2
1.3
1.4
30
8
7
20
1.1
40
9
20
1.5
1.2
1.1
1.3
PID
P/D
1.4
1.5
Figure 4.19 Minimum COE and its Components vs. P/D for Square Arrays of UZrHI.6at 60 psia
.
8 MinimumFuel CycleCost vs PD
A: MinimumCOE vs P.O
35 -
11'
12
30A1
. ......
. ........- .....
----------
_:
0
A:m
8F:
;
V.
LLI
i '~-m
: o ~:~
CS
:
°
:
1--
:El
i
oo:
9 ,7 ......
(
LL
120
:
,
11
---.--.
,
......... -. I o
12
13
14
-- --- ----- --
.
A,,
7
,
6
5% Enrichment
7 5% Enrichment
11)%Ernllchent
12.5% Enrichment
12
.
1:3
.
14
15
PiO
O MinimumCapital Costvs PDO
"-
15
C10
MinimumP/O
&M ost vs P.
0
..oc
AL
.-I-----
183
11
15
00
- 3.5 ---:-
'Z
i=
-
. ,
.
.
:
t
3.
3-.
E
ci
~
9000
a
C
! 7 '7t
: , '
b
:
:
,
2
a
I ---t9
12
13
P/D
14
15
.... ....... .. ........ -8--.
.. . . . .1-- ---------
;IZ
- 15-
:a~
:;
11
2 5 . .....
I
(3j
()
1
: -- - - - - : :
12
66,14
13
14
,~4
t" I
15
P/D
142
Figure 4.20 Lifetime Levelized Unit COE vs. P/D and D,.odfor Square Arrays of UO 2 at 60 psia
B 7.5% UO
A: 5% UO2 (mills/kW-hre)
(mills/kW-hre)
12
1:
80
11
'~- 1C
-
2
60
I
11
50
60
C
oI
IliD
40
40
0
30
.
7
7I
20
20
1.1
1.2
1.3 1.4 1.5
P/D
C: 10% UO. (mills/kW-hre)
1.1
1.2
1.3
P/D
1.4
1.5
70
1
1
60
E1 1
E
50
,,
40
Q
30
.
.
.
1.2
1.1
.
.
.
1.3
P/D
20
_
,
,
1.4
1.5
Figure 4.21 Minimum COE and its Components vs. P/D for Square Arrays of U0 2 at 60 psia
A:MinimumCOEvs.PD
/I1~yJ
40
---
-
--;------
---------
B: MinimumFuelCycleCostvs.P/D
1S
_
r
-----
12.5
.I
:35
30
-----
II
t
-
A.
:
-
---
- ------
--
15
--- ---
I
o
E
:
10
:
:*
c 25
I 1C
3
--------
6A
4
:
20
;
;
[
L___.:
15
---------------
11
12
:
..
1.3
PiD
--------
oCLi-)
--
+~-
+n*
AD
A a
.
:.:.,.
14
,
75
'
::
2
nA
nA4A
;
1
15
A
i
**
11
5%Enrichment
7.5%Enrichment
10%
10 Enrichment
Enchment
1.2
1.3
P/D
15
I..I... ..
3 ---2
.
14
C: Minimum
CapitalCostvs PD
4
20
.
,
2
C Minimum O & M Cost vs. PID
14
. -.X
.:
- .------ E
j -. -----
1.
aa
:
'
-----I - -- r
..
8
'"-_
.....
..... .......
' ''5
a 6: ::
'.
11
12
13
PD
0*.:*
-
...
-
:
I
:~~~~~~~~~~~~~~~~~~~~~~~~~~~~
:
.
'
14
15
11
12
13
PiD
14
15
143
Figure 4.22 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of UZrH.6
1 and UO2
at 60 psia With Vibrations and Transient Limits Applied
,Avi-n a D,
TT7rH
-,a
ivlt
6
lrl
i,.m m D,.rr TTf /-1
'
6
1r,; \
5
5
11
45
4
45
2e 1
5
'9
3
2.5
25
2
15
1
15
1
IL2
1.
1
13
P/D
4.6.1.1 COE Breakdown for 12.5% UZrH 1. 6 and 5% U0 2 at 60 psia
Figures 4.23 and 4.26 plot the COE and its individual cost components for the
major backfit scenario considering 12.5% UZrH. 6 and 5% UO2. Figures 4.24, 4.25,
4.27, and 4.28 plot the corresponding power, specific power, burnup, annual energy
production, capacity factor, cycle length, annual outage length, and planned outage
length.
Following is a discussion of the cost components of the COE for each fuel as
they relate to the operating parameters in Figures 4.24, 4.25, 4.27, and 4.28.
Fuel Cycle Cost
The fuel cycle cost analysis was presented in Section 4.5.1, where it was shown
that the total cost in $ for a single operating cycle depends on the mass of heavy metal in
the core (See equation (4.40)). The lifetime levelized unit fuel cycle cost, cevfcc, is equal
to the refueling cost divided by the energy produced during the operating cycle. The
energy production is the product of the fuel burnup and the mass of heavy metal in the
core. Thus for a fixed geometry,
evfcccan be increased or decreased by varying the
burnup in the fuel. It is therefore not a surprise that the plots of Ce,ftc for 12.5% UZrH .1 6
and 5% UO2 in Figures 4.23B and 4.26B show a striking resemblance to the fuel bumups
in Figures 4.24C and 4.27C. The figures are practically inverted, with the minimum fuel
cycle costs corresponding to regions of maximum burnup.
144
The core power also plays a role in the determining the fuel cycle cost through the
specific power, though its contribution is less significant than burnupl°. For a fixed
burnup, ,,evf,,c
will decrease as the specific power increases, because the operating cycle
length, and therefore the period of time that the plant operates to recover the fuel cycle
expenditure, is reduced. The effect of specific power on the fuel cycle cost is therefore a
simple matter of the interest accrued between successive refuelings. Because this effect
is marginal, the minimum fuel cycle costs will favor regions of higher burnup. Figures
4.24A, 4.24B, 4.27A, and 4.27B plot the core power and specific power. Note their
similarity. Note that regions where the power and specific power are largest do not
coincide with areas where the fuel cycle costs are minimized. This reinforces the primary
dependence of fuel cycle costs on bumup.
Figures 4.1 9A and 4.1 9B revealed that one of the reasons the minimum COE for
UZrH,.6 increases at larger P/D ratios is rising fuel cycle costs. Tracing the path of the
minimum COE line in Figure 4.23A through the burnup curve in Figure 4.24C, it is
evident that the burnup at the most economic geometries decreases as the P/D ratio
increases. This is the reason for the rise in fuel cycle costs. The fuel cycle costs shown
in Figure 4.21 B for 5% U0 2, however, do not experience an increase for larger P/D
ratios because the burnup continues to increase, as shown in Figure 4.27C.
Another feature in Figure 4.19 warrants explanation in this section. Unlike the
UO2 costs, the overall COE and fuel cycle costs for UZrH1. 6 are minimized for higher
enrichments. This is contradictory with industry experience and the COE results
presented for UO2. From a cost perspective, increasing the enrichment does two things.
First, it increases the front end enrichment costs (Refer back to Table 4.11). Second, it
allows neutronically the attainment of higher bumups which, if not limited by fuel
performance constraints, lowers the overall fuel cycle costs. The two effects therefore
compete to drive changes in the fuel cycle cost. For UZrH1.6, the cost savings attributed
to larger burnups at the higher enrichments outweigh the cost penalty at the enrichment
plant. The same effect is not realized for UO 2 fuel, because the marginal gain in bumup
10 Recall, however,
where limited by fuel performance constraints, the burnup does depend on power.
145
at higher enrichments is smaller due to more stringent fuel performance constraints (i.e,
from considering the internal pressure limit). The added enrichment costs therefore
outweigh the small savings attributed to increased energy production. Refer again to
Figures 4.7 and 4.9 for proof of this.
O & M Cost
The C)& M costs, as detailed in Section 4.5.2, are comprised of both fixed and
variable expenditures that are recurring in time. The fixed costs involve the salaries of
plant personnel, which are constant each year and independent of the plant's operating
performance. Their contribution to the lifetime levelized unit O & M cost, CevO&M,
is
therefore determined by the annual energy production from the plant, which is shown in
Figure 4.24D to depend predominantly on power (This relationship was also established
by equation (4.5)). The fixed O & M component and power are therefore inversely
proportional. Annually, the variable O & M component involves costs associated with
both forced and refueling outages. Unlike the fixed component, they depend on the
plant's performance (i.e, the parameters that determine total annual outage length), as
well as the annual energy production from the plant. For most reasonable cycle lengths
(i.e., > 6 months), the fixed component comprises the majority of the annual O & M
cost . It is therefore expected that the behavior of cZev,O&M
will scale with the fixed
component, which as just discussed is inversely proportional to power. The lifetime
levelized unit O & M cost should therefore be lowest where the power is maximum,
which is evident when comparing Figures 4.23C to 4.24A and Figures 4.26C to 4.27A.
The annual outage lengths are plotted in Figures 4.25C and 4.28C. Note that
ClevO&M
is lowest where the outage lengths are longest, which would not be observed if
the variable component had greater influence on the overall O & M costs. This fact
therefore reinforces the primary dependence of l,,,O&M
on its fixed component.
" From Table 4.4, the annual fixed cost is 150,000/pers-yr x 600 pers = $90 million/year. The variable
costs, assuming a 1 year cycle length, are approximately
$800,000/day x 20days/cycle x 1 cycle/year +
$150,000/day x .01 forced outage days/day x 365 days/year = $16.5 million/year.
146
Like the fuel cycle costs, Figure 4.19C shows that O & M costs are minimized for
higher enrichments of UZrHl.6. This is once again a result of increased burnup in the
fuel. As the burnup increases the capacity factor increases, and with it the annual energy
production from the plant. This brings Zev,O&Mdown. This effect is not observed,
however, for UO2 because the marginal gain in burnup for higher enrichments is much
smaller. The O & M costs for U0 2 in Figure 4.21 C therefore appear independent of
enrichment.
Capital Costs
The capital costs are unique among the COE components because they occur only
once, at the time of the expenditure. The lifetime levelized unit capital cost, lev,cap,
depends on the amount of this expenditure and the energy produced to recover it over the
life of the plant. For the major backfit scenario, the capital costs depend on two
conditions: (1) whether a geometry offers increased power relative to the reference core;
and (2) if an increase is reported, its magnitude.
lev,ca,
is plotted in Figures 4.23D and
4.26D. Each cost figure is clearly divided into two sections per the conditions described
above. In the region where the power is below the reference core power, the capital
expenditure is fixed (replacement of core internals, vessel head, remaining value of lost
fuel), and so
lev cap
depends solely on the power/energy production from the plant. This
is evident when comparing Figure 4.23D to Figures 4.24A and 4.24D and Figure 4.26D
to Figures 4.27A and 4.27D. For regions where the maximum power is greater than the
reference core power, additional costs are incurred for replacement of the steam
generators and upgrades to the turbine. lev,captherefore increases in this region where
power increases are reported. The magnitude of the increase, however, is limited because
the power and energy production in this region also increases, which in turn drives
reductions in the O & M costs. If the power increase is large enough, the O & M
reductions more than compensate for the added capital component and overall costs are
lowered. This is the reason the minimum COE lines plotted in Figures 4.23A and 4.26A
pass through the maximum power region, despite the increased capital costs.
147
In summary, the behaviors of each COE component can be explained by
examining the operating parameters for the plant and the individual cost assumptions. It
is believed that the tools needed to understand the COE curves have now been provided
to the reader. Subsequent cost results will therefore not be examined in such detail as
provided in this section, although sufficient information graphically will be provided for
such an interpretation.
Figure 4.23 COE Breakdown for Square Arrays of 12.5% UZrH1. 6 at 60 psia
A: COE (mills/kW-hre)
B: FCC (mills/kW-hre)
12
60
1
50
1
11
,,
-ff
E
E
40
-
30
El
10
-o
9
cz
8
20
7
1.3 1.4 1.5
P/D
C: O & M (mills/kW-hre)
1.1
1.2
1.3 1.4 1.5
P/D
D: Capital (mills/kW-hre)
1.1
40
1
1.2
12
5
11
1
30
E
20
(2
4
E10
E
9
3
8
2
10
1.1
1.2
1.3
P/D
1.4
1.5
7
1.1
1.2
1.3
P/D
1.4
1.5
148
Figure 4.24 Plant Operating Conditions for Square Arrays of 12.5% UZrH 1.
A: Core Power (x 10 6 kWth)
6
B: Specific Power (kWth/kgHM)
140
12
12
4
11
E
10
E
120
11
100
10
3
80
g
60
8
7
at 60 psia
2
k,.J
1.3 1.4
1.5
P/D
C: Burnup (MWD/kg u,,)
1.1
1.2
12
11
E 10
lO
E
8
40
7
1
20
1.3 1.4 1.5
P/D
D: Annual Energy Prod. (x 1010 kW-hre)
140
12
1.2
11
120
1
1.1
1.2
100
8
80
7
An
vv
1.1
1.2
1.3
P/D
1.4
0.8
0
0.6
8
0.4
7
1.1
1.5
1.2
1.3
P/D
1.4
1.5
Figure 4.25 Plant Operating Conditions for Square Arrays of 12.5% tJZrH1. 6 at 60 psia
A: Capacity Factor
B: Cycle Length (yrs)
12
1
0.96
11
E
E 1C
5
1
4
n1
0.95 -
-O 9
7
3
2
0.94
J
1
1.3 1.4 1.5
P/D
D: Planned Outage Length (days/cycle)
-
1.3 1.4 1.5
P/D
C: A,nnual Outage Length (days/yr)
1.1
1.2
1.1
20
12
11
18
11
E. 1C
16
E
1.2
40
35
E10
14
n
9
12 10
1.1
1.2
1.3
P/D
1.4
1.5
30
8
25
7
20
1.1
1.2
1.3
P/D
1.4
1.5
149
Figure 4.26 COE Breakdown for Square Arrays of 5% UO2at 60 psia
B: FCC (mills/kW-hre)
A: COE (mills/kW-hre)
12
1
80
1
30
10
60
E
40
11
~...
O g
40
20
8
10
7
20
1.1
1.2
P/D
1.4
1.5
40
1
5
1
1
EE
1.3
P/D
1
QI
30
E 1(
20
n
10
4
3
o
l
]
1.1
1.2
P/D
1.3
PfD
1.4
1.5
2
Figure 4.27 Plant Operating Conditions for Square Arrays of UO2at 60 psia
A: Core Power (x 106 kWth)
B: Specific Power (kWth/kgHM)
12
12
50
11
4
11
ic
3
10
30
8
2
Q 8
20
7
10
3
1
11
12
13
14
15
40
11
12
P/D
1.3 14
P/D
C:
15
.
D: Anr
V-hre)
12
1
50
11
40
-p10
30
9
c
0El
20
7
10
1.1
1.2
1.3
P/D
1.4
1.5
1.2
11
0.8
o8
0.6
0.4
7
1.1
1.2
1.3
P/D
1.4
1.5
150
Figure 4.28 Plant Operating Conditions for Square Arrays of UO2at 60 psia
A: Capacity Factor
B: Cycle Length (rs)
12
0.96
11
3
12
2.5
0.955
E 10
2
- 9
0.945
8
8
1.5
0.94
7
1
11
1.2
1.3
Pin
|
1.4
1.5
1.1
1.2
1.3
1.4
15
Pin
t
Vcle)
D:
C::i
20
12
11
18
10
16
, 9
14
8
12
30
28
E
26
24
22
7
20
10
1.1
1.2
1.3
P/D
1.4
1.5
1.1
1.2
1.3
P/D
1.4
1.5
4.6.1.2 Final Comparison for Major Backfit: UZrH1. 6 and U0 2 at 60 psia
The bottom line for the major backfit scenario is the COE, and how the cost
performance of UZrH1. 6 compares with UO2. As discussed in Section 4.6.1, the honor of
the lowest COE for the major backfit scenario is reserved for U0 2 at the geometry: P/D
= 1.39, Drod =: 6.5 mm. The cost is 17.9 mills/kW-hre. The minimum cost for UZrHi.6 is
slightly higher at 18.4 mills/kW-hre at a tighter configuration: P/D = 1.32, Drod = 9 mm.
A large range of geometries remain, however, where cost savings may be realized
for UZrH 1. 6 fuel. Figure 4.29A illustrates this as a plot of the difference in the minimum
COE for UZrHI
. 1 6 and UO2 as a function of P/D ratio and rod diameter. Where the
difference is positive, U0 2 offers the lowest COE. A black contour is provided to
indicate where the cost difference is zero, and divides the plot into regions where each
fuel provides the minimum COE.
Figure 4.29B plots the enrichments that correspond to
fuel providing the minimum cost at each geometry (i.e., the majority of the region where
UO2 costs are lowest corresponds to 5% enrichment, although 7.5% UO2 is most
economical in the top left portion of the figure). As expected, higher enrichments prevail
151
in the regions where UZrHi 6 is optimum, and in regions where UO2 is optimum, the
enrichments are lower.
Overall, Figure 4.29 shows that the costs are fairly comparable for the two fuels.
The cost savings in the regions where UZrH 1.6 appears beneficial are not significant (< 2 mills/kW-hre), and the corresponding powers are typically lower than the reference
core power. With this in mind, and recalling the uncertainties in the cost assumptions for
UZrH 6 (i.e, fabrication cost), Figure 4.29 can hardly be used as a strong argument for a
switch to hydride fuel.
Figure 4.29 COE Difference and Fuel Enrichment for Major Backfit With UZrH1. 6 and UO2vs.
P/D and Drodat 60 psia
B: Enrichment
10
12
8
6
11
4
10
2
E
E
9
0
E
1:g
-2
8
-4
7
-6
6
-8
-10
5
"-
P/D
"-
P/D
4.6.2 Results for Minor Backfit: UZrH1.6 and U0 2 at 60 psia
For the minor backfit scenario, the design range is restricted to the reference core
pitch. The fuel cycle and O & M costs are unchanged from the major backfit scenario,
but the capital expenditure is reduced to reflect the use of the existing unit's vessel head
and internals. Figures 4.30 and 4.31 plot the COE and its cost components as a function
of P/D ratio for the minor backfit scenario employing UZrH 1. 6 and U0 2. Immediately
obvious is that the minimum COE for each fuel occurs very close to the reference core
geometry. For UZrH1.6, the minimum COE is 18.5 mills/kW-hre for 12.5% enriched fuel,
and occurs at P/D - 1.30,
Drod
= 9.66 mm. This is almost identical to the major backfit
minimum COE, which is marginally lower at 18.4 mills/kW-hre. For UO2, the minimum
152
COE is 19.7 mills/kW-hre for 5% enriched fuel, and occurs at P/D - 1.35, Drod
=
9.33
mm. This is larger than the major backfit minimum COE, which is 17.9 mills/kW-hre.
For utilities considering minor backfit, UZrH1. 6 may therefore offer an economic
advantage at 60 psia. Because the minimum cost occurs very close to the reference core
configuration, changing the existing core geometry may not be necessary. This would
allow the incorporation of hydride assemblies into the existing core via the regular
refueling outage schedule (i.e., replacing discarded U0
2
fuel assemblies with UZrHI.6
fuel assemblies for 3 successive outages until the core is operating entirely on hydride
fuel). This would eliminate the capital cost penalty associated with the discarded UO2
fuel and improve the overall economics. As for major backfit, however, the magnitude of
the cost advantage for UZrHt 6 is not large given the uncertainties in the overall
economics analysis.
Figure 4.30 Minor Backfit COE and its Components vs. P/D for UZrH1.6 at 60 psia
A. COEvs P/D
0MCostvs P
:C
E7J;-1
-'
B FCCvs Pr
2,~-.....:'Ets
~'
4
---
T
,,
!
1X
12
13
14
2
10
Enrichment
:
?--~-i-,:
o
CapaCostvs
- P: -
-............
.
;
....
.......................
~s,-.....~..........c
nictinent
1
51
no 7 5°%Q
Ervtil en)
='%Eichmen
...........
..........
,t4-2r
------_
-:
:
-
----
---
.a...-:
014
...
..........
1i.. ......
2 - - ii
- .-......
.-.-
.
1- - --..
--.~ ......... ..................
.1.5
.i..1 3 ___-_-_
10
L------14
1.1
12
P.D
..
11
..........
._._._
12
....
........
.. ..
13
P/D
14
1.5
153
Figure 4.31 Minor Backfit COE and its Components vs. P/D for UO2 at 60 psia
A cOE .i PD
*"
8 FCCvs PO
2(0
:i
3
-u
J
------- -*i
r
-..,...........----I ::...........
-........
-
L
E
12
c
b
Eu
'
(2
-----,.-
:. . . .
..
. . .
:/
',y
.__
__
1
12
13
_
14
:
1 5I
12
+
C 0 & M ¢o;tvs
O0Capltal cost; V( PO
2
~
~
....................
- ...............
,-.
i,~
i --- ---------------i-------
1
:....................
<T>
I
I
15
Erllllcrenrl[t
-
Co:
,
14
PD
; PO
i-
.
l",
:
1:3
5 % Erln: lrler
el
!o)5 Erlri: lrnerlt
PiD
IT
~
if
:
' , ............
...............
.
......... -------1
P9D
1-6
-
. ...
.. .. . .. . .. =
11'
..... .
.................
1
~ ...
i3 1
-
------
.
i
12.
----
1
_:
__-:
PO
4.6.3 Results for Major Backfit: UZrHI.6 and UO2 at 29 psia
The lifetime levelized unit cost of electricity for the major backfit scenario at 29
psia is shown in Figures 4.32 and 4.34 for different enrichments of UZrH1. 6 and U0 2.
The minimum COE at each P/D ratio is fitted to a black contour on each plot. Figures
4.33A and 4.35A plot these lines independent of rod diameter to provide a clearer
comparison of the COE among different enrichments and fuel types. Also shown in
Figures 4.33 and 4.35 are the fuel cycle, O & M, and capital costs that comprise the
minimum COE. The minimum COE for UZrHI 6 is 19 mills/kW-hre for 12.5% enriched
fuel at: P/D = 1.37, Drod= 8.4 mm. The minimum COE for UO2 is lower at 18 mills/kWhre for 5% enriched fuel at: P/D = 1.47, Drod= 7.13 mm. The cost gap between the most
economic UZrHI 6 and U0 2 geometries has therefore widened at 29 psia (1 mill/kW-hre
at 29 psia vs. 0.5 mills/kW-hre at 60 psia). Like the 60 psia results, the minimum COE
lines tend to follow the crest of the maximum power plots for both UZrH,. 6 and UO2
(compare with Figure 4.2). The costs again are minimized for UZrH. 6 at higher
enrichments and for U0 2 at lower enrichments.
The analysis provided in Section 4.6.1.1
154
to explain the behavior of the COE curves at 60 psia is not repeated here. Because the
logic is similar, the figures for the COE breakdown and corresponding plant operating
conditions for 12.5% UZrHi.6 and 5% U0 2 are presented in Appendix E.
Figure 4.32 Lifetime Levelized Unit COE vs. P/D and Drodfor Square Arrays of UZrH.6 at 29 psia
B. 7.5% UZrH 1 6 (mills/kW-hre)
A: 5% UZrH 1 6 (mills/kW-hre)
12
70
E
E10
II 60
50
50
9
U
II 70
12
I 60
11
40
8
7
11
12
13
14
15
I
010
I
40
30
30
11
P/D
C: 10% UZrH · - (mills/kW-hre)
J
19 ,
1.2 1.3 14
15
P/D
I 17rH (m illkLAN-hr'·
^^
1bU
12
70
11
I 60
c~
I 60
EjC
10
E
1
50
9
40
8
E
E
40
30
7
I 20
1.1
1.2
1.3
P/D
1.4
1.5
7
20
_1.1
1.2
1.3
P/D
1.4
1.5
155
Figure 4.33 Minimum COE and its Components vs. P/D for Square Arrays of IZrH 6 at 29 psia
A: Minimum COE vs F'D
B: Minimum Fuel Cycle Costvs. P.D
13
9|5i
5
....
...........
...
. ..
40n
............
12
-
.. .....
11
El!
..
..-.
'!
.........
....
.,
10
:. ..
..
..
.
0a c
1i
:
23
at·?S
Si
00Lj
08
a
-
7"ms
J
LL
=·
,.
-
12
13
P:D
14
1.1
15
C Mlnimtum i & M Co:lstvs
o
+
P-
PD
J0)
12
5% Ennicrment
7 5% Enrichment
10% Enrichment
12 5% Enrichmenl ,
A,
3
L.. . . . . .. . . ...
i......
1
4
.. ......
.. ;
4
..
:
O
,
15
11
Figure 4.34 Lifetime Levelized Unit COE vs. P/D and
,
oi
13
P'D
12
Drod
.
*
.
:
.
..
,
',
14
15
.
for Square Arrays of U0
2 at
29 psia
B: 7.5% UO 2 (mills/kW-hre)
A: 5% U0 2 (mills/kW-hre)
I 80
-1
I
11
60
9
40
8
n
7
80
I
12
100
11
...
*i
2
P:D
E
El(
E10
15
16
-15
1
14
.
25
1
....
. .. ... .. . . . - . . . ... . . ---- ; .. . .
i ............
~o
o
13
P')
.......
, _
-35
oi
-------
C Minimum Capital Cost vs P)
fit
_
...............
...
6
11
C
-:
o
.
60
40
Q
7i
1.3 14
1.5
PID
C: 10% UO, (mills/kW-hre)
1.1
1.2
I
,
12
80
11
I 70
E10
60
7
20
1.1
1.2
1.3
P/D
1.4
1.5
50
40
8
30
7
1.1
1.2
1.3
P/D
1.4
1.5
i
156
Figure 4.35 Minimum COE and its Components vs. P/D for Square Arrays of U0 2 at 29 psia
A: MinimumCOE vs. P:D
, ,
,
8 MinimumFuel Cycle Costvs PD
Zz
I
L.
45
. 40
.
.
* 35
7
.
+
E 30
0
(
,
---
L] i
25
.+
.
+ *
---
-
20
. ,e .-1
2
c 1
10
13
P."D
14
15
a
,
:
11
_
,:-an9L22 _I8
-
12
_:
-
5%Enrichment
7 5% Enrichment
0% Ennchment
1
PiD
14
~
15
C Minimum
CapitalCostvs PD
C MinimumO &M Cost vs P. D
:
.........
+
A uAh
+
. .....
0 000
:
E
81
:
a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
..............
cd
o 10
........
i.t= ~.2i... ..... ..........
11
12
1.3
14
15
PD
11
12
:
:
13
P.D
14
:~~,
15
4.6.3.1 Final Comparison for Major Backfit: UZrH. 6 and UO2 at 29 psia
The lowest COE for the major backfit scenario at 29 psia is achieved with oxide
fuel, though as observed for major backfit at 60 psia, the costs are comparable for both
fuels over most of the design range. Figure 4.36A plots the difference in the minimum
COE for UZrHI
6
and UO2 as a function of P/D ratio and rod diameter. A black contour
is provided to indicate where the cost difference is zero, and divides the plot into regions
where overall costs are minimized with each fuel. Figure 4.36B plots the enrichments
that correspond to the minimum cost fuel at each geometry. As expected, higher
enrichments prevail in the regions where UZrHI.6 is optimum; the opposite is true in
regions where costs are minimized with UO2. Though Figure 4.36A shows a large region
where savings may be realized with UZrHI.6 (P/D < 1.35), the magnitude of the savings
are not large enough to provide overwhelming approval for hydride fuel, particularly in
light of the uncertainties in the economics analysis.
157
Figure 4.36 COE Difference and Fuel Enrichment for Major Backfit With UZrHI1 6 and U O2 vs.
P/D and Drodat 29 psia
A: UZrH. COE- UO, COE
FB
Enrichmant
10
12
12
12
8
11
6
11
11
4
2
E- 10
I
10
E- 10
0
C 9
-2
-4
F
9
9
8
8
7
-6
6
-8
5
-10
-
P/D
P/D
4.6.4 Results for Minor Backfit: UZrH1.6 and UO2 at 29 psia
The minor backfit results are presented in Figures 4.37 and 4.38 for UZrHt 6 and
UO2 at 29 psia. Like the higher pressure drop case, the minimum COE for each fuel
occurs very close to the reference core geometry. For IJZrHI 6 the minimum COE is 19.2
mills/kW-hre at P/D - 1.3, Drod= 9.66 mm. For U0 2 the minimum COE occurs at the
same geometry but its value is higher at 20 mills/kW-hre.
Once again, UZrH.
6
appears
to provide the best option for the minor backfit scenario, but not by a significant margin.
158
Figure 4.37 Minor Backfit COE and its Components vs. P/D for UZrH1. 6 at 29 psia
A: COE vs. PtD
B: FCCvs PD
40
11
A
105
10
:A
20
gA
Z
a
2i
95
A.
9
A
~~~~~~~~
8rq
LLIj
20
- - - -- -- --
j..,
:
~
. ....
:._~..........
·---
'9
.....-.......
.. --3
[+;A
FJ []
[]
_ _ __ _ i _ _ _
12
14
13
Is
-. . . . . C-"
. .
7
_ 1 . . .7
15
11
5% Ennchment
7 5% Ennrichment
10% Enrichment
A
I~
I
12.1b%Enchmer ,t
2
PD
c
.
P o
13
12
:
;
',,
,J~~~~~7
15 i
11
A
<
o
o
+, C
A
_ __ __
14
15
P"D
a
u
C
& M CCostvs
)
I
D CaltalCost vs FD
18
.;:
i&20
iA
t
I
y
A~~~
~d 15
O
14
= 1
I1
AI
40
fE :1
I
'~ 1.2
c)
1
A
R
11
..........
. .......
----------. -----.---- ---- -
'~ 10
!-~
.---
12
1.4
08
t
15
1.1
1.2
.~~9
1.3
14
15
PT)
PT:
Figure 4.38 Minor Backfit COE and its Components vs. P/D for UOz at 29 psia
8. FCCvs PD
A.COE vs. PD
20
45
40
a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
18.---------
~ ------------~ --- ·- ·
------- --A::
.. ...........
'
,-----------......
:..
------- ....................
i
35
~i
14 ......
-E 30
L'i
0
(
.a.
25
12
16
10 10
I-
+
......... ............ ...... --..............
--"'---.
-
,-r.. -...--.... ~--~
i--.
20
11
12
1.
14
15
P"D
A
A
C 0 & M Costs vs P/D
20,
~
a
8
11
tr-----2
12
1.3
~~
1
14
A
.. 1
15
P"D
5% Enrichment
7 5% Ennchment
10% Enrichment
D CapitalCostvsPD
16
!
A
1
..
43
1I
a
S?
15t·i--'
~
·
a;
............ ............
' ...
I
....',
,a,
!R
a_
1
--
lt
lrl
11
12
13
P"D
-.
: :a
14
15
0.8
11
12
13
PD
14
15
159
160
5.
Conclusions and Future Work
This section summarizes the major findings from this report and discusses future
work for subsequent researchers on the Hydride Fuels Project. The reader is encouraged
to study both sections carefully, as several analyses that support the overall design
optimization require modification and improvements in the near futurel2. These changes
may affect the final results and conclusions drawn in this report, but their magnitude is
unknown at this time. With this said, the framework for the overall optimization
methodology laid out is correct, and will continue to be used for quantifying the
performance of hydride fuels in PWRs.
5.1
Conclusions
This report had three distinct goals:
* Updating the steady-state thermal hydraulic analysis performed by J. Malen [1]
incorporating recent changes to fixed operating parameters, and verifying the
thermal hydraulic equivalence of square and hexagonal array PWR geometries at
the same rod diameter and H/HM ratio;
* Replacing the single thermal hydraulic design limit on axial velocity with more
specific vibrations and wear design limits;
* Combining the results from the steady-state and transient thermal hydraulic,
neutronic, and fuel performance analyses into an economics model to optimize
UZrH1. 6 and U0 2 geometries for incorporation into existing PWRs.
12
The key study that needs to be changed is the fuel performance analysis, which determines the maximum
discharge burnup for hydride and oxide fuels subject to constraints on fission gas release and internal
pressure, clad oxidation, and clad strain - See Future Work Section 5.2.1.5 for a list of the required
modifications.
161
5.1.1 Steady-State Thermal Hydraulics for Square and
Hexagonal Array PWR Geometries
The methodology for performing the steady-state thermal hydraulic
analysis was established by a recent MIT graduate student, J. Malen, who developed an
interface between MATLAB and VIPRE to iteratively determine the maximum
achievable power for a range of PWR geometries subject to user defined design limits.
Design limits were placed on MDNBR, fuel bundle pressure drop, axial flow velocity,
and fuel temperature.
Recent project decisions to modify the axial power profile and core
enthalpy rise used in this analysis required regeneration of J. Malen's maximum power
results. These are presented for two pressure drop limits in Figure 5.1 for UO2 and
UZrH1 6. Fuel temperature is the only limit that depends on the fuel type. Because it
never constrains power, the maximum power plots are identical for UZrH1.6 and UO 2.
Figure 5.1 Maximum Achievable Power vs. P/D and Drodfor Square Arrays of UZrH1.6 and UO2at
60 psia and 29 psia
Maximum Achievable Power (x 106 kWV.: AP = 29 osia
Maximum Achievable Power (x 106 kW. ): AP = 60 osia
___
- -
-- -
-t
-
6
5.5
1
5
45
E
45
4
35
35
3
2*5
25
2*1
2
15
15
1
1 '1
1
3
a
I 1
1z
13
14
1
I
POD
Power increases over the reference design are reported at both pressure drop limits. The
maximum power at 60 psia is 5458 MWth at the geometry: P/D = 1.42, Drod = 6.8 mm.
The maximum power at 29 psia is 4245 MWth at the geometry: P/D = 1.48,
Drod
= 6.5
mm.
The steady-state thermal hydraulic analysis only considered square arrays, under
the assumption that the results could be extended to hexagonal arrays for equivalent rod
162
diameters and H/HM ratios13 . For this condition, square and hexagonal arrays have the
same flow area and heated and wetted perimeters and so the thermal hydraulic
performance should be similar. A full core hexagonal VIPRE model was constructed at a
single geometry for comparison with the square results and the predicted powers were
within 1% of each other. The maximum power for hexagonal array geometries can
therefore be inferred from the plots for square arrays at the same rod diameter and H/HM
ratio. The H/HM ratios are identical when: (P/D)h =1.0746. (PD)s
5.1.2 Vibrations Analysis for Hydride and Oxide Fueled PWRs
The steady-state thermal hydraulic analysis did not account for specific vibration
and wear mechanisms; instead, a single limit on the axial flow velocity was imposed.
The wide range of core geometries considered and the large power increases reported by
the thermal hydraulic study makes it prudent to refine this single limit approach. Design
limits were therefore developed for key flow-induced vibration and wear mechanisms,
and incorporated into the existing thermal hydraulic codes for determining the maximum
power. The mechanisms considered were: vortex-induced vibration, fluid-elastic
instability, turbulence-induced vibration in cross and axial flow, fretting wear, and sliding
(or adhesive) wear.
The maximum achievable powers for square arrays of UZrH 1.6 at 60 and 29 psia
with vibrations and wear imposed design limits are shown in Figures 5.2A and 5.2C.
Figures 5.2B and 5.2D plot the power reductions accompanying the consideration of the
additional vibrations and wear limits (i.e, Figure 5.2B is the difference between Figure
5.1A and 5.2A).
13
The same information is plotted in Figure 5.3 for UO 2.
This refers to hexagonal geometries using grid spacers. Wire-wrapping
was not considered in this
report- see Section 5.2.2.2
163
Figure 5.2 Maximum Achievable Power and Power Reductions vs. P/D and Dr,d for Square
Arrays of UZrH1 6at 60 and 29 psia with Vibrations and Wear Imposed Design Limits
A: Max.Power(kWh,):AP = 60psia
B: PowerDifference(kW,): AP= 60psia
x1 P
_
1-
12
5
11
2
1
4
15
!I
,13
3
7
I
I
.5
1Ii 4
1
I
I
1
I1
PfD
Io
PID
C: Max.Power(kWh):AP = 29psia
14
19i
I
05
0
v
D: PowverDifference(kWt,): P = 29 psia
Y
1 1)
r]
5
2
4
g1
II
3
11
12
1
PlD
14
1s
u-0
05
2
I
1
11
12
13
PID
14
"
I
n
-J
Figure 5.3 Maximum Achievable Power and Power Reductions vs. P/D and Drodfor Square
Arrays of U0 2 at 60 and 29 psia with Vibrations and Wear Imposed Design Limits
A: Max.Power (kW,,):AP= 60 psia
B: PowerDifference(kW): AP= 60 psia
x 10
r
-i6
x5
ArV
1
5
lU
4
E
3
5
2
I
I-I
D:
Power
Dine
(kW): P=29psia
C: Max.Power(kW,,):AP= 29psia
D: PowerDifference(kWh):AP= 29 psia
xjLB
-- l
x 10
"- 1'1
12
12
11
11
I
10
iA
12
3
10
?a9
5
2
I
,I
z
la
PID
i*
I ,
I I
I
I
P/D
164
The fluid-elastic instability and vortex-induced vibration design limits were not
constraining for any geometries 4 . Sliding and fretting wear, however, limited large
regions where the axial velocities were greatest, which correspond to the highest power
designs. This is why the power reductions shown in Figures 5.2B and 5.2D, and 5.3B
and 5.3D follow the crest of the maximum power plots. The peak power geometries at 60
psia for both UZrH. 6 and U0 2 were reduced by - 450 MWth (Refer to Figure 5.1). The
peak power geometries at 29 psia, however, were unaffected because the flows in the
core are lower. Note that the power reductions plotted in Figures 5.2 and 5.3 are larger
for UZrHI.6 than for UO2, because the wear limits applied to UZrHi.6 were more
stringent. The wear limits depend on cycle length, which varies by enrichment and fuel
type. As revealed in Chapter 4, costs are minimized for 12.5% UZrH 1. 6 and 5% U0 2, and
so the cycle lengths from these fuels were used to determine the wear limits. Because
longer cycle lengths are achieved with 12.5% UZrH 1.6, the wear limits were stricter and
resulted in larger power reductions than for U0 2. This approach ensured that the
maximum power input to the economics analysis was specifically adapted to the lowest
cost fuels, although ideally a separate vibrations analysis would be performed for each
enrichment and fuel type (See Future Work).
5.1.3 Economic Optimization
The Hydride Fuels Project aims to quantify the benefits of hydride fuel use in
existing PWRs for two cases: major backfit, where the layout of fuel in the core can
assume any combination of lattice pitch and rod diameter; and minor backfit, where new
designs must maintain the existing core pitch. To this end, an economics analysis was
performed to estimate the cost of electricity for retrofit and operation of existing PWRs
using UZrH 1.6 and UO2 fuels. Cost estimates were provided for each backfit scenario
over a range of core geometries. Fuel cycle, operations and maintenance, and capital
costs were considered. The inputs to the economics model included the maximum power
from the steady-state and transient thermal hydraulic analyses, and the maximum burnup
from the neutronics and fuel performance studies.
Recall from Section 3.6.1 that the vortex-shedding design limits were not maintained for P/D ratios < 1.2
because VIPRE experienced problems converging to cross-flow solutions.
14
165
A comparison of the minimum COE for the major and minor backfit scenarios as
a function of P/D ratio is shown in Figure 5.4. Subplots A and B provide the major
backfit COE for 12.5% UZrH .1 6 and 5% UO2 at 60 and 29 psia.
Subplots C and D
provide the minor backfit COE for 12.5% UZrH 1. 6 and 5% UO02 at 60 and 29 psia. The
two enrichments are chosen because they provide the lowest minimum cost for each
scenario5. The lowest COE is obtained for major backfit with UO2 and minor backfit
with UZrH1 .6,though the respective cost advantage for each fuel is not greater than - 1
mill/kW-hre.
Figure 5.5 shows the difference in the major backfit COE for UZrH1. 6 and U0 2 at
both pressure drops over the whole design range. Each region is labeled with the fuel
that provides the lowest COE; the magnitude of the cost advantage can be read from the
colorbar. Note that the enrichment (not shown) varies across each plot (i.e, even though
the lowest cost geometries for each backfit scenario are obtained with 12.5% UZrH1. 6 and
5% UO2, costs are minimized at different enrichments for non-optimum geometries - See
Figures 4.29 and 4.36). Overall, costs are fairly comparable for both fuels and neither
provides a significant economic benefit for most of the design region (cost difference is
typically < - 2 mills/kW-hre). The region where UZrH1. 6 does provide the lowest costs,
however, corresponds to geometries with lower maximum powers than the reference
core. In light of this, and recalling the uncertainties in the cost assumptions for UZrH 1.6
(i.e, fabrication cost), a strong argument cannot be economically for its use in either the
major or minor backfit scenarios over UO.2
15 Note that
this does not mean that costs are minimized over the entire design range with 12.5% UZrH 1.6
and 5% UO2. It means that the lowest cost (i.e., at the single optimum geometry) for each backfit scenario
is obtained with 12.5% UZrH1. 6 and 5% U0 2.
166
Figure 5.4 Major and Minor Backfit COE vs. P/D for 1TZrH
.6
1 and
45
------------------40
-i·
A: MajorBackfitCOEvs. P/D; AP = 60 psia
B: MajorBackfitCOE vs.P/D;AP = 29 psia
4
40
"::-........
OM-17
------.....................
:
~-~·----
35
;UO 0.0~~~~
Min= 179
-
.
18.
.E
_)
i
.l0
30
uJ
at 29 and 60 psia
U02
25
..
,
.
.
....L
D
..... '...... ......-...............
O2=-- ,'-
UZrH6 Mm
;.
19
w
20
20
15
40
1.1
12
13
P:D
1.4
1.5
C: MinorBackfitCOE vs. P/D; AP = 60 psia
""
_
.
13
P;D
12
O 12 5%UZrH16
5% U02
........
30
,--
D: MinorBackfitCOE vs.P/D; AP = 29 psia
- -. -
-
............--
.
- -
. ...... .
- 40
..
.
.
o
2
MIln - Il
35
185
E- 30
12
11
13
14
-,-O
----
UZrH16Mn. 19.2
o 25
- 8 6 \../.
20
4\..·..
15
15
.......
.
.
.oL .
o
......- ":...
b - *~"
\ ......- ...............
::.0..
. --·--% .... .:.?
20 ....
UO Min.=20
/
.------
.-----------.
UZrH 6 Min
E
15
15
45
35
Uj
J
ii
14
11
12
- --------
............
~.---...
13
14
1'5
PiD
PiD
Figure 5.5 Major Backfit COE Difference vs. P/D and Drod for UZrH1. 6 and UO2 at 60 and 29 psia
lJZrH.,
6 COE- UO2 COE: P 60 psia
UZrH1.COE- UO2 COE:AP = 29 psia
10
10
8
8
6
6
4
4
2
2
E
0
0
-2
C)
-2
-4
-4
-6
-6
-8
-8
-10
I
I1
IZ
P/D
14
I I
1
IJ.
PiD
Iq
1
-10
Note that all cost figures provided in this section are for square array geometries, but the
costs for hexagonal designs can be inferred for the same rod diameters and H/HM ratios.
167
A final note is warranted on the economics results. As mentioned in the
beginning of Section 5, the fuel performance limited burnups may not be accurate
because of mistakes in the existing analysis. These mistakes will be corrected shortly,
but not soon enough for incorporation in this report. The economic impact will most
likely be limited, however, to UO2 because the achievable burnups for UZrH 1. 6 as
presented in Section 4.4.2 are constrained primarily by neutronics limitations.
5.2
Future Work
The issues that arose during this thesis requiring attention by subsequent
researchers on the Hydride Fuels Project are discussed in this section. The future work is
divided into two categories: methodology and new approaches. The methodology
section provides suggestions for improving the existing analysis for the design and
optimization of hydride fueled PWRs (i.e, as discussed in this report and shown in Figure
1.1). New approaches apply to changes in the underlying assumptions and structure for
the analysis that may improve the project's ability to assess the feasibility of the use of
hydride fuels in LWRs.
5.2.1 Methodology
5.2.1.1 Steady-State Thermal Hydraulic Analysis
As discussed in Section 2.1.2.1, the MDNBR limit for the steady-state thermal
hydraulic analysis was obtained by executing VIPRE for the reference core geometry and
operating conditions. The W-3L correlation was adopted for predicting the critical heat
flux (CHF). It is thought that this correlation may be limited for the large range of
geometries considered in this study. More specifically, the applicability of the W-3L
correlation for smaller rod diameter geometries has been questioned. Among other
things, it is believed that the CHF depends on the spacing between mixing grids, L, and
the equivalent diameter of the sub-channel, De. Quantified as an L/De factor, the effect
reduces the critical heat flux for fixed operating conditions as the rod diameter decreases.
There is no L/De factor in the W-3L correlation, and the large power estimates from the
168
steady-state thermal hydraulic analysis for small rod diameters may be too optimistic as a
result. While the majority of CHF correlations are proprietary and cannot be released for
academic pursuits, further discussion with industry may provide a reasonable
approximation of the scaling law for an L/De factor in the W-3L correlation.
5.2.1.2 Vibrations and Wear Analysis
1) The dependence of the wear limits on the cycle length tie the vibrations
analysis to the fuel burnup. As discussed in Section 3.4.5.1 and shown in Figure 1.1, the
maximum achievable burnup subject to fuel performance constraints depends on the
maximum power in the core. Due to this relationship, several iterations are required until
both the fuel performance limited burnup and the steady-state power with vibrations and
wear limits converge to constant values. Due to time constraints, this iteration was only
performed once. Recent work, particularly the development of scripts that automate the
generation and extraction of burnup from FRAPCON, will make this task more efficient
for future users.
2) Based on results from the economic optimization, the wear limits for UZrH1.6
and U0
2
were determined for single enrichments (i.e, 12.5% UZrH 1. 6 and 5% U0 2). The
optimization can be improved by applying enrichment specific wear limits to all fuels in
the vibrations and wear analysis. The maximum power will therefore depend on fuel
type and enrichment, instead of just fuel type as presented in this report.
3) As discussed in Section 3.6.1, VIPRE experienced difficulty converging to
cross-flow solutions for tighter geometries, which precluded the inclusion of the vortex
shedding design limits for P/D ratios less than 1.2 in the thermal-hydraulic analysis.
Tighter cores are more susceptible to vortex shedding lock-in, and so this problem should
be investigated more thoroughly, particularly if future recommendations for new hydride
fueled designs fall into this range. It is hoped that an updated version of the VIPRE code
will soon be available at MIT, and that the convergence problems will resolved.
169
5.2.1.3 Transient Thermal Hydraulic Analysis
The results from the transient thermal hydraulic analysis are important to the
overall optimization of hydride and oxide fueled PWRs. The reader is referred to J.
Trant's report on transient design [3] for future work in this area.
5.2.1.4 Economics Analysis
It is believed that the most reasonable estimates available for the fuel cycle, O &
M, and capital cost components of the COE were provided in this analysis, but there is
always room for improvement. Examples include: improving the estimate for UZrH1. 6
fabrication costs (or any hydride fuel being examined, for which little to no cost data is
available); and solidifying the major plant components requiring modification or
replacement for conversion to hydride fuel use and for operation at higher powers. This
analysis assumed that the vessel head, core internals, and steam generators required
replacement, but additional capacity could be extracted from the turbine with appropriate
modifications.
5.2.1.5 Fuel Performance and Application of FRAPCON to Hydride Fuels
1) The FRAPCON code is used to simulate and model the thermal-physical
properties of UO2 fuel pins during irradiation. As discussed in Section 4.4.2.2, several
assumptions were made to extend its use to hydride fuels. For example, the fuels were
assumed to behave identically during irradiation with respect to thermal expansion and
fuel swelling. Additionally, the effect of fission gas release and the generation of helium
from the use of burnable absorbers were not considered for the hydride case. For a more
accurate picture of the achievable burnup in hydride fuel, these effects require
consideration. This can be accomplished by incorporating thermal properties and
correlations for fission gas release and swelling in hydride fuels directly into the
FRAPCON code.
2) In the near future, several changes will be made to the FRAPCON analysis to
correct existing mistakes, which will improve the estimates for the fuel performance
limited burnups for both fuels. These changes are summarized below:
170
* The axial power profile used in the FRAPCON analysis was incorrect, and needs
to be changed to the profile used in the thermal hydraulic analysis (chopped
cosine) with the same peaking factor.
* For conservatism, the fuel performance analysis assumed that all fuel assemblies
in the core were subjected to the operating conditions for the peak pin over the
entire operating cycle' 6. The burnup used in the economic optimization should
therefore be the average burnup given by FRAPCON for this peak pin. The
bumup in this report, however, was mistakenly taken from the peak fuel pellet in
the peak pin, which is larger than the average bumup for the rod. Making this
correction will have a greater impact on the U02 results because most of
achievable bumups for the parametric design range were limited by fuel
performance limits.
5.2.2 New Approaches
5.2.2.1 New Fuels
While this report focused on establishing a methodology for the design and
optimization of hydride fueled PWR cores, results were only presented for UZrHI.6 and
UO2 . As discussed in Section 4.4.2.1 none of the geometries in this study demonstrated
acceptable moderator temperature coefficients, which will deter industry from its use
unless resolved. Preliminary neutronics work indicate that other hydride fuels (i.e.,
UZrH,.6-ThH2) do not have this problem. The design and optimization should therefore
be carried out for new hydride fuels. A list of candidate fuels being considered now by
UC Berkeley include: UZrHI. 6, PuZrH. 6, PuH 2-ThH 2, UH 2 -ThH 2, UZrH. 6-ThH 2, and
PuZrH1 .6 -ThH2. Note that the fuel performance, steady-state thermal hydraulics, and
vibrations work can be performed for the new fuel types using the construct provided in
this report. Minor modifications will be required to the economics analysis to estimate
16 The linear heat rate input to FRAPCON was the average linear heat rate given by the thermal hydraulic
analysis multiplied by both the radial and axial peaking factors (i.e, x 1.65 x 1.55)
171
costs associated with: fuel fabrication, the use of thorium, and reprocessing to extract
plutonium.
5.2.2.2 Hexagonal Arrays and Wire Wrap
The square and hexagonal array PWR designs considered in this report employ
grid spacers. The use of wire wrapping in hexagonal arrays, however, may improve the
thermal hydraulic performance in the pressure drop limited regions (i.e, tighter
geometries) because the pressure drop across the fuel bundle will be reduced. To
quantify this benefit, the vibrations performance of wire wrapped fuel assemblies will
need to be assessed, and new CHF correlations applicable to wire wrapped fuel rods will
need to be adopted. J. Trant performed some preliminary work regarding the
performance of wire-wrapped fuel assemblies; the reader is referred to [3] for this
information.
5.2.2.3 Reference Core
The reference core adopted for this analysis is the South Texas Plant, which is
unique among current US LWRs because of its longer core length. The standard LWR
core is 12', but South Texas is 14'. To better quantify the performance and benefits of
hydride fuel use for the remaining US fleet, the project may consider changing the
reference core to a more representative LWR.
172
Bibliography
[1] J.A. MALEN, N.E. TODREAS, A. ROMANO, "Thermal Hydraulic Design of
Hydride Fueled PWR Cores," MIT-NFC-TR-062, MIT, Department of Nuclear
Engineering, March 2004.
[2] M.S. KAZIMI, N.E. TODREAS, Nuclear Systems II, Elements ofThermal Hydraulic
Design, Taylor & Francis, 1993.
[3] J.M. TRANT,"TransientAnalysis of HydrideFueledPressurized WaterReactor
Cores, " S.M Thesis, MIT, Department of Nuclear Engineering, August 2004.
[4] M.S. KAZIMI, N.E. TODREAS, Nuclear Systems I, Thermal Hydraulic
Fundamentals, Taylor & Francis, 1993.
[5] S. BLAIR,"Thermal HydraulicPerformanceAnalysisof a SmallIntegralPWR
Core," Engineers Thesis, MIT, Department of Nuclear Engineering, Sept. 2003.
[6] M.K. AU-YANG,Flow-InducedVibrationof Power and Process Plant Components,
pgs. 62, 141, 149, 166-167, 181, 259, 265-267, 308, 359, ASME Press, New York, 2001.
[7] M.S. KAZIMI, "High Performance Fuel Design for Next Generation PWRs: Annual
Report, " pg. 84, MIT-NFC-PR-048, MIT, Center for Advanced Nuclear Energy Systems,
August 2002.
[8] J.G.B. SACCHERI, "A Tight Lattice, Epithermal Core Design for the Integral
PWR, " Ch. 4, PhD Dissertation, MIT, Department of Nuclear Engineering, August 2003.
[9] NEA/OECD, The Economics of the Nuclear Fuel Cycle, OECD Publications, Paris,
1994.
[10] DOE, NuclearEnergy CostData Base: A ReferenceData Basefor Nuclearand
Coal Fired Powerplant Generation Cost Analysis, DOE/NE-0095, Washington, DC,
Sept. 1988.
173
APPENDICES TABLE OF CONTENTS
A NOMENCLATURE
A. 1
A.2
....................................................................
GENERALNOTATION.........................
............................................
SUBSCRIPTS.....................................................................
B LISTED ASSUMPTIONS OF VIPRE THERMAL HYDRAULIC ANALYSIS
B. 1
B 1.1
B. 1.2
B. 1.3
B. 1.4
B. 1.5
B. 1.6
B. 1. 7
B. 1.8
B. 1.9
175
175
178
...............
ASSUMPTIONS COMMON TO FULL CORE SQUARE AND HEXAGONAL ARRAY ANALYSIS ...........
onstraints..........................................................................................................................
Channel Geometry..............................................................................................................
Fuel Rod Geometry.............................................................................................................
Operating Conditions ........................................
Grid Spacers ..............................
Pressure Loss Correlations ........................................
NB, Heat Transfer Correlations.............................
Material Properties .............................................................................................................
Radial Power Profile ........................................
B.1.10 LateralDrag....................................
180
180
180
180
180
180
181
1.........................................
181
182
184
184
185
C VIPRE INPUT DECK .................................................................................................................... 186
D MATLAB TOOLS ....................................................................................................................... 199
D.1
D.2
D .3
'SQ CORE MAX VIBRATIONS"....................................
199
"VIBRATIONS".........................................................................................................................
"E ONOMICS .........................................................................................................................
205
212
E COE COMPONENTS AND PLANT OPERATING CONDITIONS FOR 12.5% UZRH1.6 AND
224
5% U0 2 AT 29 PSIA FOR MAJOR BACKFIT....................................
174
A NOMENCLATURE
A.1 GENERAL NOTATION
A:
Ac,:
Aflo,:
ChannelArea
CladdingArea
ChannelFlow Area
Afuet.' FuelArea
Agrid. Wetted Area of Grid
Ao.
Arod.'
Rod Area
BOP.
Wetted'Area of Rod at GridSpacer
Balanceof Plant
Bu.
C:
Burnup
Unit Cost
CB,.'
CB,2:
CF.
Non Mixing Vane Drag Coefficient
Mixing Vane Drag Coefficient
Cash Flow
Cf
FrictionalDrag Coefficient
C'ev":
Lifetime Levelized Unit Cost
Clev:'
Lifetime Levelized Cost
Cm.'
CN
C:.'
Added Mass Coefficient
Nth Cash Flow
Initial Cash Flow
cp.
constantpressurespecific heat
CR.'
Random Lift Coefficient
D.'
Rod Diameter
DJ.:
Length Across Hexagon Flats
Dh:.'
HydraulicDiameter
DI.
Length of Hexagon Side
Eannu,,a.Annual Energy Production
EFPY.' Effective Full Power Years
EFPYc. Effective Full Power Yearsper Cycle
E.'
Young's Modulus
f
'
FrictionFactor
Feed Enrichment
NaturalFrequency
F.'
Dimensionless Frequency
FIM.'
Fluid-ElasticInstabilityMargin
FN.'
FOR.'
Normal Contact Force
Forced Outage Rate
Fq.:
RadialPeakingFactor
Fq a.ial. Axial Peaking Factor
f:
VortexSheddingFrequency
g:
Radial Gap Thickness
EscalationRate
175
Rod to Duct Spacingfor HexagonalAssembly
G:
GF:.
Mass Flux
Random Force Power Spectral Density
Gp.
PressurePowerSpectralDensity
h:
Enthalpy
Heat Transfer Coefficient
Enthalpy Rise of Coolant
# of Hydrogen Atoms
Ah:
H:
GridSpacer Height
HM:
# of Heavy Metal Atoms
H/HM. Hydrogen to Heavy Metal Ratio
I:
J:
Momentof Inertia
JointAcceptance
Krod.'
Material Wear Coefficient
L.'
Length
Ls:
CapacityFactor
SpacingLengthfor GridSpacers
L,:
Total Fuel Rod Length
L':
l:
Availability
Loss Coefficientfor Front EndFuel CycleProcesses
m:
Capital Cost Component Scaling Factor
LinearMass
n:
M.'
Mass Flow Rate
Mass
Molecular Weight
n:
Mass Flowsfor Front and Back End of Fuel Cycle
# of batches
N:
Vibration Mode #
# of Rods
NA:
ng:
n,:
Npers:
Nps:
Avagadro's #
# of GridSpacers
# of RodSpans
# of Plant Personnel
# of Rods Per Side in HexagonalFuelAssemblies
Nrigs:
# of Rings in Hexagonal Fuel Assemblies
p:
ProductEnrichment
P.'
Lattice Pitch
System Pressure
AP:
PressureDrop
Pitch to DiameterRatio
P/D:
Heated Perimeter
Wetted Perimeter
Pw.'
PVot,,,. Present Value of Costs
Ph:
q' .'
CoreAverage LinearHeat Rate
q"
Heat Flux
0:
CorePower
176
Q:
,.*
R:
r.·
Re.'
s:
Volumetric Wear
SpecificPower
Radius
Radial Location
Nominal Interest Rate/Discount Rate
Reynolds #
Gap Width
S.: Strouhal
#
Sd:
t:
Slip Ratio
Sliding Distance
Thickness
Tails Enrichment
T.
Time
TAV:
Tc:
Temperature
Available Cycle Length
Cycle Length
TN:
Timefor Nth Cash Flow
Tant,:
Plant Lifetime
UC.
Unit Cash Flow per Kg of HM in the Core
v:
Velocity
V..
Velocity
Volume
PotentialFunctionfor SWU Calculations
Vcri,icaI:Critical Velocity
VSM:
w.
Vortex Shedding Margin
Weight Percent Heavy Metal
W.'
wi:
x:
X:
Wear Rate
Mass Flow Rate Between Adjacent Channels i andj
Quality
# of Hydrogen Atoms Per Unit of the Fuel Element
xe.
Equilibrium Quality
Y:
yrms.
Ymax:
z:
# of Heavy Metal Atoms Per Unit of the Fuel Element
Root Mean Square Rod Response
Maximum Rod Response
Axial Position
ZD:
Axial Position for Onset of Significant Void Fraction
a:
VoidFraction
f:
Connors' Constant
Mixing Coefficient
Relative Plugging Ratio
c:
L/D
Enrichment
DampingRatio
ll:
p:
,u:
Thermal Efficiency
Density
Coefficient of Friction
177
{o:
Viscosity
Axial Flux Distribution
Pp%: twophasefriction multiplier
a1:
Mode Shape
RadialPowerDistribution
A.2 SUBSCRIPTS
axial:
b:
cap.
CB:
Axial Direction
bulk Property
Capital
Condie Bengston
cl.'
Clad
CL.
Centerline
cli:
CladInner
clo:
conv:
core:
Clad Outer
Conversion
Core
cross: Cross/LaterialDirection
DB.'
disp:
Dittus Boelter
Waste .Disposal
drag:
DragDirection
eff.'
Effective
enr:
Enrichment
fab:
SaturatedLiquid
Filmfor GroenveldCorrelation
Fabrication
fcc.
fg.'
Fuel Cycle Cost
Latent Heat of Vaporization
fl.
Fluid/Coolant
FO.'
ForcedOutage
form.'
Form Loss
fretting: Fretting Wear
fric.'
FrictionLoss
g:
Saturated Vapor
Gap
grid:
Grid
h:
Heated
hex:
HexagonalArray
H,O0.
Water
HM.
Heavy Metal
i:
IncipientBoiling
inlet:
Core Inlet
1:
LiquidPhase
L:
Laminar
178
lo:
lift:
Liquid Only
Lift Direction
MixtureProperty
matrix: Fractionof Fuel That is Not HeavyMetal
m:
mv.
Mixing Vane
n.
ModeShape #
new.
New Design/Geometry
O&M. Opeationsand Maintenance
ore.
Mining/Ore
pellet: Fuel Pellet
pers: PlantPersonnel
plant: Plant
PowerPlant
PP'
ref:
repl.
RO.
rod:
Reference Core
Replacement Energy
Refueling Outage
sat.
Fuel Rod
Saturation
SG:
Steam Generator
sliding: Sliding Wear
sq:
SquareArray
stor: Spent Fuel Storage
t:
T.
Total
Turbulent
Thermal
Turbine
th:
turb.
vessel: Pressure Vesseland Head
w.
Wall
179
B LISTED ASSUMPTIONSOF VIPRE THERMAL
HYDRAULIC ANALYSIS
The thermal hydraulic assumptions are taken from [1], with appropriate updates for the
changes discussed in this report.
B.1 ASSUMPTIONS COMMON TO FULL CORE SQUARE AND
HEXAGONAL ARRAY ANALYSIS
B.1.1 CONSTRAINTS
1. UZrH 1. 6 fuel CL temperature limited to 750 0 C (1292°F)
2. UO2 fuel average temperature limited to 14000 C (2552°F)
3. MDNBR limit: 2.173 using W3-L DNB Correlation
4. Core average enthalpy rise: 204 kJ/kg
B.1.2
1.
2.
3.
4.
CHANNEL GEOMETRY
Square and hexagonal
Core height: 4.59m (181.1")
Active fuel length: 4.267m (168")
Axial nodes: 100
B.1.3 FUEL ROD GEOMETRY
i. Clad/gap thickness
ifDrod < 7.747 mm
ifDr
t,,= .508mm
(B. 1)
tg =.0635 mm
(B. 2)
:>7.747 mm
t,l= .508 +.0362(D- 7.747) mm
(B. 3)
tg =.0635+.0108.(D-7.747) mm
(B. 4)
2. Radial nodes: 6
3. Gap geat transfer coefficient (LM only): hg = kg/(Rfoln(Rci/Rfo))
4. LM material properties are listed below
B.1.4
OPERATING CONDITIONS
1. Axial power profile: chopped cosine curve
2. Radial Power Profile Peaking Factor: Fq' = 1.65 Peak/Average
3. Inlet Temperature: 294°C (561.2 °F)
4. Operating Pressure: 15.513 MPa (2250 psia)
180
5. Heat generated directly in the coolant scaled as Hc/(Hc+HF) where Hc is the
amount of H in the coolant and HF is the amount of H in the hydride fuel. The
reference case is 2.6% direct deposition
B.1.5 GRID SPACERS
1. Square: 10 grids spaced axially at the following positions, subscript corresponds
to loss coefficient used: 0.001, 0.0601, 0.1482, 0.5292, 1.0512, 1.5732, 2.3562,
2.8782, 3.1392, 3.6612, 4.1832 m (0.00001, 5.84001, 20.84002, 41.39002, 61.94002,
82.49002, 103.04002, 123.59002, 144.14002, 164.69002 inches)
2. CB,I for non mixing vane grids is evaluated from In Et Al.
Arod
1
c
(1 )2
fricrd A
Agrid
) + Cfric,gridAiow (1 -)2
B,I = Cd,form (I
Agridfrolfal((B-6)
Ag=
dfro
Arod = Pw,rodH
Agrid= HPw,grid
(B-5)
(B-6)
flow
(B- 7)
H = 38mm
tsacer =.5mm
(B-8)
Cd,form = 2.750- .27 log 10 ReD
H
Cfric,grid= CL,grid +CTgrid
H
LI
Re,
=,EG
30000,u
1.328
CL,gri
d
G
Cfric,rod
A
Aflow
CT,gid
ReL,
(B-9)
H
= f H =.184Re;2
DH
.523
[1n(.06ReHL
.2
)]2
(B-IO)
(B-Hl)
DH
3. CB,2for mixing vane grids included an additional term due to the additional flow
constriction of mixing vanes.
(B-12)
CB,2 = CB,I + CB,mv
V
(CBmvCd
mv
6
Cd,mv = 0.72
A
(B-13)
B.1.6 PRESSURE LOSS CORRELATIONS
The Cheng-Todreas for friction in square and triangular rod bundles was chosen. The
coefficients are listed in
Table B-l: Coefficients for Cheng Todreas Friction Loss Correlation
fL
=
Re
Cf
=> CfL = a + b (P/D -1)+ b2(P/D - 1)2
(B-14)
= a+b(P/D -1)+
Re.'"~~~+b
-) (P/D
(B-15)
frT = Re, 8 =C
b2
1)
181
Table B-1: Coefficients for Cheng Todreas Friction Loss Correlation
1.1 < P/D < 1.5
1.0 < P/D < 1.1
b2
bl
-493.9
374.2
a
bl
b2
35.55
263.7
-190.2
-.09926
Square Laminar
a
26.37
Square Turbulent
.09423
.5806
-1.239
.1339
.09059
Triangular Laminar
26.00
888.2
-3334
62.97
216.9
-190.2
.03632
-.03333
Triangular Turbulent
.1458
-8.664
1.398
.09378
B.1.7 DNB, HEAT TRANSFERCORRELATIONS
1. Flow Correlations
a. Subcooled void: Levy
X(Z)
=
Xe (Z)-
x
Xe (ZD ) exp
X
e
('
qt
:>ATd =
hfg
ph,
zO)
YB<O
PhhDB
Xe(ZD) = -
(B-16)
( (z) -1
-5QPr
O<Y_<5
YB
(B-1 7)
p.'-~nn-5Q Pr+1n[+Pr(YB5
-1)11
phhDB-5QPr+n[1+5Pr]+.5n(YB30)}
C gcDe
YB
0.015
#.f
Vf
2
Q= q'lP
Re
hDB = .023
'8
Pr .4
f = CD Re - -' 8 (B-18)
Gcp(.125f)Y2
b. Bulk void/quality: homogeneous
a(z) =
=>S =
1
+I-x(Z)p,
1+
-SS
x(z) p,
(1
-19)
c. Two phase friction multiplier: homogeneous
?-20)
Pm (Z) [(
/lr
:
)P
Pm(Z)= a(z)Pv +(1- (z))p]
( ?-21)
2. Heat Transfer Correlations
a. Single phase forced convection: Dittus-Boelter for turbulent flow
8
hDB =.023 Re- Pr'
4
( B-22)
b. Subcooled nucleate boiling: Thom w/ Dittus- Boelter for single phase
c. Saturated nucleate boiling: Thom w/ Dittus - Boelter for single phase
182
hThom= [(T w - Tsa,)exp(p/1260)12
.0722 (T - Tb)
T,,t = Fluid Saturation Temp (F)
Tb = Bulk Fluid Temp (F)
T = Wall Temperature (F)
(B-23)
p = system pressure (psia)
d. CHF Correlation: W3-L (L-grid factor = .042, grid spacing factor =.066,
L-grid coefficient = 1.0)
q 'cri,
=[(
-b2p)
+
(b3
b4 p)e(b 5 b6P)
e
][(b 7 - b8e
+ bgXe IXe)G +
0
]
74)
-b l
[b11- b52Xe][b 3 + bl4e De][b16 + b7 (hf - hinet)] :>f (p,G,e,De,hf ,hinet )
/ = 2.022
b7 =.1484
b13 =.2664
b2 =.0004302
4 = 1.596
44 =.8357
=
b5
bg .1729
b3 =.1722
=
bo 1.037
b4 =.0000984
b5= 18.177
=
3.151
25)
=
b 16 .8258
b,7 = .000794
1 = 1.157
b12 =.869
b6 =.004129
e. Transition Boiling: Condie-Bengston
q"cB = C 1 (T - Tsa ) exp
C,= exp[In (q"chf q"b ) + 0.5(Twchf Tsat)I/2 -In (Tw-chfq"chf =
q"j
Tsat)]
(B-27)
criticalheatflux
= film boiling heatflux
q "chf = h (T-chf - Tf )
f.
(B-26)
2 ,,at)/2
(T
(B-28)
**.qcB = qchf - q jb
Film boiling: Groeneveld 5.7
q "b = hfb (Tw - Ta
k
hfb = 0.052k
Re.
DH
68 8
(B-29)
)
Pr'
26
Ho
f
y,.06
(B-30)
1
(B-31)
Y = 1.0-0.1 (I-x) (P1
SCp-vv )
ReHom =GxD
ga
(B-32)
(B-33)
3. DNB Correlation: W3-L, see Heat Transfer Correlations, 2.d
183
B.1.8
MATERIAL PROPERTIES
The material properties are organized in Table B-2.
Table B-2: Material Properties used for the Parametric Study
P
Cp
k
UZrHi 6
17.6 W/m-K
(10.169 Btu/hr-ft-F)
UThH2
Same as UZrHi. 6
Same as UZrH6
-3
+3.3557E-6T
1.2058E-3
2
VIPRE lookup table
Zirc~
2 3218E-(T
Cp(u) (1.305E-4)T+O. 1094 J/g-C
J/g-C
(.06976T+33.706)/92.83
Cp(zrH1.6)
Cp(UZrHI.6)= .45 Cp¢(L)+ .55CP(ZrHI.6)
+1.2315E-12*T
3/ft3
(515.4 ibm/f)
10.852 g/cm 3
(677.46 bm/f 33))
(677.46 lbm/ft
6.55 g/cm 3 3
(409.7
Ibm/ft
)
(409.7 lb
7.968 g/cm 3
Btu/s-ft-F
9.01748+1.62997E-2T-
SS
8.256 g/cm 3
(515.4
3
18422E-9T
4.80329E-6T2+2.
(497.4 bm/ft3)
Not Provided
W/m-k
9.049 g/cm 3
173.4 J/kg-K
35 W/m-K
(564.9 Ibm/ft3 )
(0.0414 Btu/lbm-F)
(20.222 Btu/hr-ft-F)
*indicates that VIPRE defaults were used instead of values from Reference Database
LM
B.1.9 RADIAL POWER PROFILE
The radial power profile, shown in Figure B-l has the following properties:
core
core = j2
0J
r = 3" >
=O
dr
(B-34)
(r)rdrdO
R
V(r)=Fq,
(B-35)
= 1.65
Figure B-l: Radial Peaking Factor vs. Core Radius
1
T-
T-
I
I
16
I
-
---
--
--
T
I
..
I
I
-
- --
T
I
I
I
I
I
15
14
~~
I~
-
----
I
I
!
13
I-l--
I
.
-1
C-I-
~ ~~~~~
11
I
I
I
1
I
I
1
09
08
3
Fql= 8 8628e-O36*r-0 00081337'r +0 0657179r+1 Q402
A
.
S
0
10
A
I
20
L
1
30
I
I
40
radius r (rches)
50
-L
I
I
60
70
B.1.9.1 Mixing Coefficient for Internal Subchannel: Rogers-Tahir
/3 = frod
184
+ fSgrid
(B-36)
IProds W
(B-37)
s=P-D
(B-38)
Gs
square arrays: w
hex. arrays:
9
= .005 uRe
1
wiu= .0018Re9
(B-39)
S
0.(B-40)
a conservative estimate for the total mixing coefficient for the reference core is .038, and
the grid component is found using equation (B-36)
/Jgrid = .038- rods-ref = .0 33 5
(B-41)
the mixing coefficient between lumped channels equals the mixing coefficient between
individual subchannels divided by the centroid to centroid distance of the lumped
channels
(B-42)
Flumped
i-lumped
B.1.10
LATERAL DRAG
VIPRE default of uniform lateral resistance factor multiplied by centroid to centroid
distance divided by the pitch
CD-_ateral =.5
P
(B-43)
185
C VIPRE INPUTDECK
* Input for Hydride Core With Hexagonal Arrays
* Drod = 6.5 mm, P/D = 1.3866
* VIPRE CASE CONTROL
* VIPRE.1: KASE, IRSTRT, IRSTEP
1,0,0
Hydride Core Hexagonal Subchannel
* CHANNEL GEOMETRY
*
____
=
_____
=-------
____
* GEOM.1
geom, ?* INFLAG
91, ?* NCHANL, # of channels*****************
0, ?* NCARD, 0 for compressed geometry option
48, ?* NDX, # of axial nodes
0, ?* NAZONE, 0 for uniform axial node length
12, ?* NCTYP, # of channel types
0 * MBWR, 0 for non-BWR geometry
* GEOM.2
181.10, ?* ZZ, bundle length (inches)
0.0, ?* THETA, bundle orientation
0.0 * SL, gap/length parameter
* GEOM.5
57, ?* MCHN, # of channels of this type
0.028804, ?* CAREA, channel flow area (inA2)
0.401975, ?* CPW, channel wetted perimeter (in)
0.401975 * CPH, channel heated perimeter (in)
*
* GEOM.6
1,2,3,4,5,6,10,11,12,16,20,21,22,23,24,25
26,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44
48,49,50,51,55,56,57,58,62,63,64,65,66,67,71,72
73,74,75,76,77,78,79,80,81
24,0.021796,0.448625,0.267984
7,8,9,13,14,15,17,18,19,27,28,29,45,46,47,52
53,54,59,60,61,68,69,70
186
1,0.264293,3.751771,3.751771 * 1st lumped subchannel - outer edge of 1/6th
82
1,7.287642,112.29415,99.28794 * 1/2 of assembly
83
1,29.15058,449.1766,397.1518 * 2 assemblies
84
1,48.5843,748.62767,661.91967 * 3.33 assemblies
85
1,68.018,1048.0787,926.6875 * 4.67 assemblies
86
1,87.45174,1347.5298,1191.4554 * 6 assemblies
87
1,106.885,1646.979,1456.223 * 7.33 assemblies
88
1,126.3192,1946.4319,1720.9911 * 8.67 assemblies
89
1,145.7529,2245.883,1985.759 * 10 assemblies
90
1,444.546,6849.94,6056.5639 * 30.5 assemblies
91
*GEOM.7
126, ? * NK, number of gaps
0, ? * NGTYP, gaps are input individually
1, ? * NGAP, gaps are input individually
0, ? * NROW, gaps are input individually
0, ? * ISYM, gaps are input individually
0
* NSROW, gapas are input individually
*GEOM.
11
1, ? * K, gap indentification number
1, ? * I, lower channel connected
3, ? * J, upper channel connected
0.09894, ? * WIDTH, width of gap
0.20487 * CENT, centroid distance for gap
*
187
2,2,3,0.09894,0.20487
3,2,6,0.09894,0.20487
4,3,4,0.09894,0.20487
5,4,8,0.09894,0.20487
6,5,6,0.09894,0.20487
7,5,11,0.09894,0.20487
8,6,7,0.09894,0.20487
9,7,8,0.0544,0.20487
10,7,13,0.0544,0.20487
11,8,9,0.0544,0.20487
12,9,15,0.0544,0.20487
13,10,11,0.09894,0.20487
14,10,18,0.09894,0.20487
15,11,12,0.09894,0.20487
16,12,13,0.09894,0.20487
17,12,20,0.09894,0.20487
18,13,14,0.0544,0.20487
19,14,15,0.0544,0.20487
20,14,22,0.09894,0.20487
21,15,16,0.09894,0.20487
22,16,24,0.09894,0.20487
23,17,18,0.0544,0.20487
24,17,27,0.0544,0.20487
25,18,19,0.0544,0.20487
26,19,20,0.09894,0.20487
27,19,29,0.0544,0.20487
28,20,21,0.09894,0.20487
29,21,22,0.09894,0.20487
30,21,31,0.09894,0.20487
31,22,23,0.09894,0.20487
32,23,24,0.09894,0.20487
33,23,33,0.09894,0.20487
34,24,25,0.09894,0.20487
35,25,35,0.09894,0.20487
36,26,27,0.09894,0.20487
37,26,38,0.09894,0.20487
38,27,28,0.0544,0.20487
39,28,29,0.0544,0.20487
40,28,40,0.09894,0.20487
41,29,30,0.09894,0.20487
42,30,31,0.09894,0.20487
43,30,42,0.09894,0.20487
44,31,32,0.09894,0.20487
45,32,33,0.09894,0.20487
46,32,44,0.09894,0.20487
47,33,34,0.09894,0.20487
188
48,34,35,0.09894,0.20487
49,34,46,0.09894,0.20487
50,35,36,0.09894,0.20487
51,36,48,0.09894,0.20487
52,37,38,0.09894,0.20487
53,37,51,0.09894,0.20487
54,38,39,0.09894,0.20487
55,39,40,0.09894,0.20487
56,39,53,0.09894,0.20487
57,40,41,0.09894,0.20487
58,41,42,0.09894,0.20487
59,41,55,0.09894,0.20487
60,42,43,0.09894,0.20487
61,43,44,0.09894,0.20487
62,43,57,0.09894,0.20487
63,44,45,0.09894,0.20487
64,45,46,0.0544,0.20487
65,45,59,0.0544,0.20487
66,46,47,0.0544,0.20487
67,47,48,0.09894,0.20487
68,47,61,0.0544,0.20487
69,48,49,0.09894,0.20487
70,49,63,0.09894,0.20487
71,50,51,0.09894,0.20487
72,50,66,0.09894,0.20487
73,51,52,0.09894,0.20487
74,52,53,0.0544,0.20487
75,52,68,0.0544,0.20487
76,53,54,0.0544,0.20487
77,54,55,0.09894,0.20487
78,54,70,0.0544,0.20487
79,55,56,0.09894,0.20487
80,56,57,0.09894,0.20487
81,56,72,0.09894,0.20487
82,57,58,0.09894,0.20487
83,58,59,0.09894,0.20487
84,58,74,0.09894,0.20487
85,59,60,0.0544,0.20487
86,60,61,0.0544,0.20487
87,60,76,0.09894,0.20487
88,61,62,0.09894,0.20487
89,62,63,0.09894,0.20487
90,62,78,0.09894,0.20487
91,63,64,0.09894,0.20487
92,64,80,0.09894,0.20487
93,65,66,0.09894,0.20487
189
94,65,82,0.09894,1.4307
95,66,67,0.09894,0.20487
96,67,68,0.09894,0.20487
97,67,82,0.09894,1.0793
98,68,69,0.0544,0.20487
99,69,70,0.0544,0.20487
100,69,82,0.09894,0.73215
101,70,71,0.09894,0.20487
102,71,72,0.09894,0.20487
103,71,82,0.09894,0.39786
104,72,73,0.09894,0.20487
105,73,74,0.09894,0.20487
106,73,82,0.09894,0.17994
107,74,75,0.09894,0.20487
108,75,76,0.09894,0.20487
109,75,82,0.09894,0.39786
110,76,77,0.09894,0.20487
111,77,78,0.09894,0.20487
112,77,82,0.09894,0.73215
113,78,79,0.09894,0.20487
114,79,80,0.09894,0.20487
115,79,82,0.09894,1.0793
116,80,81,0.09894,0.20487
117,81,82,0.09894,1.4307
118,82,83,3.39,1.7076
119,83,84,6.79,4.567
120,84,85,13.56,5.743
121,85,86,20.34,5.817
122,86,87,27.12,5.842
123,87,88,33.9,5.8535
124,88,89,40.68,5.8593
125,89,90,47.46,5.863
126,90,91,54.24,9.74
* FLUID PROPERTIES
* PROP.1
prop, ? * INFLAG
0, ? * NPROP,# of entries in fluid properties table
0, ? * ISTEAM, 0 for no superheated steam properties
2, ? * NFPROP, 2 for EPRI water properties
1 * IPVAR, 1 for property evaluation at local pressure
*
190
* ROD INPUT
* RODS.1
rods, ? * INFLAG
1, ? * NAXP, # of axial power profiles
60, ? * NROD, # of rods
1, ? * NC, 1 to use conduction model
2, ? * NFUELT, # of rod geometry types
1, ? * NMAT, # of rod/wall properties to be input
0, ? * IGPFF, 0 - no gap conductance forcing fct.
0,
0,
0,
0,
0
? * NGPFF, 0 - no gap conductance forcing fct.
? * NOPT, 0 - normal rod layout
? * IPOWV, 0 - constant axial power profiles w/ time
? * ICPR, 0 - no CPR calculations
* IRFF, 0 - cst. radial power factors w/ time
* RODS.2
168, ? * ZZH, heated length (inches)
3.5, ? * ZSTRT, beginning of heated profile (inches)
0, ? * NODALS, 0 - default
0 * NODALT, 0 - default
* RODS.3
-1 * NAXN, # of points in axial power profile table, -1 for chopped cosine
* RODS.5
1.55 * PSTAR, Peak/Average axial power
* RODS.9
1, ? * I, rod ID #
1, ? * IDFUEL, rod geometry type
1.6402, ? * RADIAL, rod radial power factor
1, ? * IAXP, axial power profile table flag for rod I
1, ? * LRDUM, enter the channel number that receives heat from rod I
0.1666 * PHIDUM, fraction of rod outside perimeter facing channel LRDUM
2,1,1.6421,1,*,0.1666,3,0.1666,2,0.1666
3,1,1.6421,1,1,0.1666,3,0.1666,4,0.1666
4,1,1.6438,1,2,0.1666,6,0.1666,5,0.1666
5,1,1.6434,1,2,0.1666,3,0.1666,4,0.1666,6,0.1666,7,0.1666,8,0.1666
6,1,1.6438,1,4,0.1666,8,0.1666,9,0.1666
7,1,1.6453,1,5,0.1666,11,0.1666,10,0.1666
8,1,1.6448,1,5,0.1666,6,0.1666,7,0.1666,11,0.1666,12,0.1666,13,0.1666
191
9,1,1.6453,1,9,0.1666,15,0.1666,16,0.1666
10,1,1.6466,1,10,0.1666,18,0.1666,17,0.1666
11,1,1.6461,1,10,0.1666,11,0.1666,12,0.1666,18,0.1666,19,0.1666,20,0.1666
12,1,1.6459,1,12,0.1666,13,0.1666,14,0.1666,20,0.1666,21,0.1666,22,0.1666
13,1,1.6461,1,14,0.1666,15,0.1666,16,0.1666,22,0.1666,23,0.1666,24,0.1666
14,1,1.6466,1,16,0.1666,24,0.1666,25,0.1666
15,1,1.6477,1,17,0.1666,27,0.1666,26,0.1666
16,1,1.647,1,19,0.1666,20,0.1666,21,0.1666,29,0.1666,30,0.1666,31,0.1666
17,1,1.647,1,21,0.1666,22,0.1666,23,0.1666,31,0.1666,32,0.1666,33,0.1666
18,1,1.6473,1,23,0.1666,24,0.1666,25,0.1666,33,0.1666,34,0.1666,35,0.1666
19,1,1.6477,1,25,0.1666,35,0.1666,36,0.1666
20,1,1.6485,1,26,0.1666,38,0.1666,37,0.1666
21,1,1.6482,1,26,0.1666,27,0.1666,28,0.1666,38,0.1666,39,0.1666,40,0.1666
22,1,1.6479,1,28,0.1666,29,0.1666,30,0.1666,40,0.1666,41,0.1666,42,0.1666
23,1,1.6479,1,30,0.1666,31,0.1666,32,0.1666,42,0.1666,43,0.1666,44,0.1666
24,1,1.6479,1,32,0.1666,33,0.1666,34,0.1666,44,0.1666,45,0.1666,46,0.1666
25,1,1.6482,1,34,0.1666,35,0.1666,36,0.1666,46,0.1666,47,0.1666,48,0.1666
26,1,1.6485,1,36,0.1666,48,0.1666,49,0.1666
27,1,1.6492,1,37,0.1666,51,0.1666,50,0.1666
28,1,1.6489,1,37,0.1666,38,0.1666,39,0.1666,51,0.1666,52,0.1666,53,0.1666
29,1,1.6487,1,39,0.1666,40,0.1666,41,0.1666,53,0.1666,54,0.1666,55,0.1666
30,1,1.6486,1,41,0.1666,42,0.1666,43,0.1666,55,0.1666,56,0.1666,57,0.1666
31,1,1.6486,1,43,0.1666,44,0.1666,45,0.1666,57,0.1666,58,0.1666,59,0.1666
32,1,1.6489,1,47,0.1666,48,0.1666,49,0.1666,61,0.1666,62,0.1666,63,0.1666
33,1,1.6492,1,49,0.1666,63,0.1666,64,0.1666
34,1,1.6497,1,50,0.1666,66,0.1666,65,0.1666
35,1,1.6495,1,50,0.1666,51,0.1666,52,0.1666,66,0.1666,67,0.1666,68,0.1666
36,1,1.6492,1,54,0.1666,55,0.1666,56,0.1666,70,0.1666,71,0.1666,72,0.1666
37,1,1.6492,1,56,0.1666,57,0.1666,58,0.1666,72,0.1666,73,0.1666,74,0.1666
38,1,1.6492,1,58,0.1666,59,0.1666,60,0.1666,74,0.1666,75,0.1666,76,0.1666
39,1,1.6493,1,60,0.1666,61,0.1666,62,0.1666,76,0.1666,77,0.1666,78,0.1666
40,1,1.6495,1,62,0.1666,63,0.1666,64,0.1666,78,0.1666,79,0.1666,80,0.1666
41,1,1.6497,1,64,0.1666,80,0.1666,81,0.1666
42,1,1.6499,1,65,0.1666,82,0.33333
43,1,1.6498,1,65,0.1666,66,0.1666,67,0.1666,82,0.5
44,1,1.6497,1,67,0.1666,68,0.1666,69,0.1666,82,0.5
45,1,1.6497,1,69,0.1666,70,0.1666,71,0.1666,82,0.5
46,1,1.6496,1,71,0.1666,72,0.1666,73,0.1666,82,0.5
47,1,1.6496,1,73,0.1666,74,0.1666,75,0.1666,82,0.5
48,1,1.6497,1,75,0.1666,76,0.1666,77,0.1666,82,0.5
49,1,1.6497,1,77,0.1666,78,0.1666,79,0.1666,82,0.5
50,1,1.6498,1,79,0.1666,80,0.1666,81,0.1666,82,0.5
51,1,1.6499,1,81,0.1666,82,0.33333
52,2,1.6491,1,83,123.5
53,2,1.6271,1,84,494
54,2,1.5633,1,85,823.33
192
55,2,1.4674,1,86,1152.67
56,2,1.3495,1,87,1482
57,2,1.2208,1,88,1729
58,2,1.0917,1,89,2140.67
59,2,0.97321,1,90,2470
60,2,0.82846,1,91,7533.5 * not positive about the centroid location in this outer lumped
channel
0 * 0 terminates rod layout input
RODS.62
1, ? * I, rod geometry type #
nucl, ? * FTYPE, rod type
0.2559, ? * DROD, rod outside diameter
0.2194, ? * DFUEL, fuel pellet diameter
6, ? * NFUEL, number of radial nodes in fuel pellet
0.0, ? * DCORE, central void diameter
0.0154 * TCLAD, cladding thickness
*
RODS.63
0, ? * IRADP, 0 - uniform radial power profile in pellet
1, ? * IMATF, use fuel properties stored in table 1
0, ? * IMATC, 0 - use VIPRE zircaloy tables for cladding
*
0, ? * IGPC, 0 - uniform gap conductance
0, ? * IGFORC, 0 - gap conductance is constant in time
86184.361, ? * HGAP, constant gap conductance
0.9550, ? * FTDENS, fuel theoretical density
0.0000 * FCLAD, fraction of applied power generated in clad
RODS.68
2, ? * I, rod geometry type #
dumy, ? * FTYPE, rod type
0.2559, ? * DROD, rod outside diameter
0.0, ? * DFUEL, 0 if FTYPE not a tube
0 * NFUEL, 0 - if DUMY type rod
*
RODS.70
1, ? * N, material type #
140, ? * NNTDP, number of entries in material properties table
515.4 * RCOLD, cold state density of material (lbm/ftA3)
*
RODS.71
*32.0, ? * TPROP, temperature F
*0.05985, ? * CPFF, specific heat (Btu/lbm-F)
*10.16909, ? * THCF, thermal conductivity (Btu/hr-ft-F)
32.0,0.05985,10.16909, 77.0,0.06228,10.16909
122.0,0.06468,10.16909, 167.0,0.06707,10.16909
*
193
212.0,0.06943,10.16909, 257.0,0.07178,10.16909
302.0,0.07411,10.16909, 347.0,0.07642,10.16909
392.0,0.07872,10.16909, 437.0,0.08100,10.16909
482.0,0.08326,10.16909, 527.0,0.08552,10.16909
572.0,0.08776,10.16909, 617.0,0.09000,10.16909
662.0,0.09222,10.16909, 707.0,0.09444,10.16909
752.0,0.09665,10.16909, 797.0,0.09885,10.16909
842.0,0.10105,10.16909, 887.0,0.10324,10.16909
932.0,0.10543,10.16909, 977.0,0.10762,10.16909
1022.0,0.10981,10.16909, 1067.0,0.11200,10.16909
1112.0,0.11419,10.16909, 1157.0,0.11638,10.16909
1202.0,0.11858,10.16909, 1247.0,0.12078,10.16909
1292.0,0.12299,10.16909, 1337.0,0.12520,10.16909
1382.0,0.12742,10.16909, 1427.0,0.12965,10.16909
1472.0,0.13189,10.16909, 1517.0,0.13414,10.16909
1562.0,0.13641,10.16909, 1607.0,0.13868,10.16909
1652.0,0.14097,10.16909, 1697.0,0.14328,10.16909
1742.0,0.14560,10.16909, 1787.0,0.14794,10.16909
1832.0,0.15030,10.16909, 1877.0,0.15268,10.16909
1922.0,0.15508,10.16909, 1967.0,0.15750,10.16909
2012.0,0.15994,10.16909, 2057.0,0.16241,10.16909
2102.0,0.16490,10.16909, 2147.0,0.16742,10.16909
2192.0,0.16996,10.16909, 2237.0,0.17254,10.16909
2282.0,0.17514,10.16909, 2327.0,0.17777,10.16909
2372.0,0.18043,10.16909, 2417.0,0.18313,10.16909
2462.0,0.18586,10.16909, 2507.0,0.18863,10.16909
2552.0,0.191.43,10.16909, 2597.0,0.19426,10.16909
2642.0,0.19714,10.16909, 2687.0,0.20005,10.16909
2732.0,0.20300,10.16909, 2777.0,0.20600,10.16909
2822.0,0.20903,10.16909, 2867.0,0.21211,10.16909
2912.0,0.21524,10.16909, 2957.0,0.21840,10.16909
3002.0,0.22162,10.16909, 3047.0,0.22488,10.16909
3092.0,0.22819,10.16909, 3137.0,0.23155,10.16909
3182.0,0.23496,10.16909, 3227.0,0.23842,10.16909
3272.0,0.24194,10.16909, 3317.0,0.24551,10.16909
3362.0,0.24913,10.16909, 3407.0,0.25281,10.16909
3452.0,0.25655,10.16909, 3497.0,0.26034,10.16909
3542.0,0.26420,10.16909, 3587.0,0.26811,10.16909
3632.0,0.27209,10.16909, 3677.0,0.27612,10.16909
3722.0,0.28022,10.16909, 3767.0,0.28439,10.16909
3812.0,0.28862,10.16909, 3857.0,0.29292,10.16909
3902.0,0.29728,10.16909, 3947.0,0.30172,10.16909
3992.0,0.30622,10.16909, 4037.0,0.31079,10.16909
4082.0,0.31544,10.16909, 4127.0,0.32016,10.16909
4172.0,0.32495,10.16909, 4217.0,0.32982,10.16909
4262.0,0.33477,10.16909, 4307.0,0.33979,10.16909
194
4352.0,0.34489,10.16909, 4397.0,0.35007,10.16909
4442.0,0.35533,10.16909, 4487.0,0.36067,10.16909
4532.0,0.36609,10.16909, 4577.0,0.37160,10.16909
4622.0,0.37719,10.16909, 4667.0,0.38287,10.16909
4712.0,0.38864,10.16909, 4757.0,0.39449,10.16909
4802.0,0.40043,10.16909, 4847.0,0.40647,10.16909
4892.0,0.41259,10.16909, 4937.0,0.41881,10.16909
4982.0,0.42512,10.16909, 5027.0,0.43152,10.16909
5072.0,0.43802,10.16909, 5117.0,0.44461,10.16909
5162.0,0.45131,10.16909, 5207.0,0.45810,10.16909
5252.0,0.46499,10.16909, 5297.0,0.47198,10.16909
5342.0,0.47908,10.16909, 5387.0,0.48627,10.16909
5432.0,0.49357,10.16909, 5477.0,0.50098,10.16909
5522.0,0.50849,10.16909, 5567.0,0.51611,10.16909
5612.0,0.52384,10.16909, 5657.0,0.53167,10.16909
5702.0,0.53962,10.16909, 5747.0,0.54768,10.16909
5792.0,0.55585,10.16909, 5837.0,0.56414,10.16909
5882.0,0.57254,10.16909, 5927.0,0.58105,10.16909
5972.0,0.58968,10.16909, 6017.0,0.59843,10.16909
6062.0,0.60730,10.16909, 6107.0,0.61629,10.16909
6152.0,0.62540,10.16909, 6197.0,0.63463,10.16909
6242.0,0.64399,10.16909, 6287.0,0.65347,10.16909
* OPER.1
oper, ? * INFLAG
1, ? * IH, inlet condition specified as uniform inlet temperature
2, ? * IG, inlet flow condition specified by mass flux
0, ? * ISP, 0 - equal mass flux per channel at inlet
0, ? * NPOWR, 0 - power specified in units (kW/ft)
0, ? * NDNB, 0 - no MDNBR iteration
0, ? * IRUN, 0 - run only one case
0, ? * IFCVR, 0 - constant direct heat generation in coolant
0, ? * LUF, 0 - no forcing functions
0 * IHBAL, 0 - specify inlet enthalpy directly
OPER.2
0.0, ? * DPS, 0 - use inlet flow BC specified by IG and ISP
0.0, ? * DNBRL, 0 - 0 if NDNB = 0
1.9597, ? * FCOOL, percent of heat generated in coolant
0.001, ? * DNBRC, convergence factor for CHF iterations
0 * IHROD, 0, if NDNB = 0
*
OPER.5
2250.0, ? * PREF, operating system pressure (psia)
561.2, ? * HIN, enter average inlet temperature (F)
3.2746, ? * GIN, average inlet mass flux (Mlbm/hr-ft2)
*
195
2.4637, ? * PWRINP, core average power (kW/ft)
0 * HOUT, 0 - since exit flow don't reverse
OPER.12
0,0,0,0,0,0
*
*
CORR.1
corr, ? * INFLAG
2, ? * NCOR, # of CHF correlations to use for DNBR calculations
1, ? * NHTC, use correlations for single-phase convection and nucleate boiling only
0 * IXCHF, 0 ifNHTC = 1
*
CORR.2
levy, ? * NSCVD, subcooled void correlation
homo, ? * NBLVD, bulk/void quality correlation
homo, ? * NFRML, two-phase friction multiplier
none * NHTWL, no hot wall correction factor
*
CORR.6
ditb, ? * NFCON, single phase forced convection correlation
thsp, ? * NSUBC, subcooled nucleate boiling correlation
thsp, ? * NSATB, saturated nucleate boiling correlation
epri, ? * NCHFC, critical heat flux correlation
cond, ? * NTRNB, transition boiling correlation
g5.7 * NFLMB, film boiling correlation
*
CORR.9
w-31,? * NCHF, DNB correlation
epri * NCHF, DNB correlation
*
CORR.11
0.042, ? * TDCL, L-grid mixing factor
*
0.0660, ? * SPK, grid spacing factor
1.0 * FLGRD, L-grid factor converted to R-grid factor
*
CORR.16
0, ? * KBWR, 0 for no cold wall correction factor
0, ? * NUC, 0 for no nonuniofrm axial flux correction factor
0.85 * CGRID, grid loss coefficient for mixing grids
*
*
MIXX.1
mixx, ? * NFLAG
0, ? * NSCBC
0, ? * NBBC, 0 if 2 phase mixing computed as for single phase
1 * MIXK, 1 if a pair of coefficients entered for each gap
*
196
* MIXX.2
0.0000, 0.0000, 0.0000
* MIXX.3
0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634,
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.061634, 0.0000, 0.061634, 0.0000, 0.061634, 0.0000, 0.061634,
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.061634, 0.0000, 0.008826, 0.0000, 0.0616, 0.0000, 0.0616, 0.0000
0.0117, 0.0000, 0.0616, 0.0000, 0.0616, 0.0000, 0.0172, 0.0000
0.0616, 0.0000, 0.0616, 0.0000, 0.0317, 0.0000, 0.0616, 0.0000
0.0616, 0.0000, 0.0701, 0.0000, 0.0616, 0.0000, 0.0616, 0.0000
0.0317, 0.0000, 0.0616, 0.0000, 0.0616, 0.0000, 0.0172, 0.0000
0.0616, 0.0000, 0.0616, 0.0000, 0.0117, 0.0000, 0.0616, 0.0000
0.0088, 0.0000, 0.0074, 0.0000, 0.0028, 0.0000, 0.0022, 0.0000
0.0022, 0.0000, 0.0022, 0.0000, 0.0022, 0.0000, 0.0022, 0.0000
0.0022, 0.0000, 0.0013, 0.0000
* DRAG.1
drag, ? * INFLAG
1, ? * NCHTP, # of axial friction correlations to be specified
0, ? * NGPTP, 0 - apply constant loss coefficients to all gaps
1 * KIJOPT, lateral resistance option
* DRAG.2
0.1541, ? * ATF, coefficient in turbulent rod friction factor correlation
-0.1800, ? * BTF, constant in turbulent rod friction factor correlation
197
0.0, ? * CTF, constant in turbulent rod friction factor correlation
118.396, ? * ALF, coefficient in laminar rod friction correlation
-1.0, ? * BLF, constant in laminar rod friction correlation
0.0 * CLF, constant in laminar rod friction correlation
DRAG.5
0.5, ? * DUMKIJ, lateral resistance factor applied to all gaps
0.35484 * PPITCH, rod pitch (in)
*
GRID. 1
grid, ? * INFLAG
*
0, ? * KOPT, 0 - constant local loss coefficients
2 * NKCOR, # of correlation sets to be supplied
GRID.2
1.4, ? * CDK(1), loss coefficient for non-mixing grid
1.625 * CDK(2), loss coefficient for mixing grid
*
GRID.4
-1, ? * NCI, all NCHANL channels are in this location set
10 * NLEV, enter number of axial locations
*
GRID.6
0.0000,1,5.8400,1,20.8400,2,41.3900,2,?
61.9400,2,82.4900,2,103.0400,2,123.5900,2
144.1400,2,164.6900,2,
*
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E COE COMPONENTS AND PLANT OPERATING
CONDITIONS FOR 12.5% UZRH,.6 AND 5% UO2
AT 29 PSIA FOR MAJOR BACKFIT
Figure E-I COE Breakdown for Square Arrays of 12.5% UZrH1.6 at 29 psia
A: COE (mills/kW-hre)
80
12
-
12
11
11
60
E10
9
40
8
E
E
10
l
E
8
20
1.3 1.4 1.5
P/D
C: 0 & M (mills/kW-hre)
1.1
1.1
I
11
7
1.3 1.4 15
P/D
D: Capital (mills/kW-hre)
1.2
12
1.2
7
50
I 40
E10
I
f
8
20
7
I 10
1.1
1.2
1.3
P/D
1.4
1.5
6
5
E
30
9
224
9
4
3
1.1
1.2
1.3
PID
1.4
1.5
I
2
Figure E-2 Plant Operating Conditions for Square Arrays of 12.5°/, UiZrH.6at 29 psia
A: Core Power (x 106 kWth)
B: Specific Power (kWth/kgHM)
A
12
I
11
10
C
1'
I 100
3
E 1
80
2
02
I
12
11
I 604(
7-
1
1.3 1.4 1.5
P/D
C: Burnup (MWD/kgHM)
11
120
._P
8
7
12
11
3
I
7
i
1.2
2(
1.3 1.4 15
P/D
): Annual Enerav Prod. x 10 10 kW-hre)
1.1
140
12
I 120
11
1
0. 8
E
E C
c
100
I
1.1
1.2
1.3
PID
1.4
0. 6
70'
0.'4
80
7
1.5
0.2
1.2
1.1
1.3
P/D
1.4
1.5
Figure E-3 Plant Operating Conditions for Square Arrays of 12.5% UZrHI.6 at 29 psia
B: Cycle Length (vrs)
A: Capacity Factor
0.97
12
8
11
0.96
E
E
6
1(
0.95
CI
r"'
-B.
F
4
0.94
1.1
1.2
1.3
1.4
1.5
2
1.1
1.3 1.4 1.5
P/D
D: Planned Outage Length (days/cycle)
P/D
C: Annual Outage Length (days/yr)
1.2
20
12
61i
11
15
Et: 1
F
5'
-o
0
n2
4
9
E'
C1 C
..
D9
10
990
3' 0
1
0
21
1.1
1.2
1.3
P/D
1.4
1.5
1.1
1.2
1.3
P/D
1.4
1.5
225
Figure E-4 COE Breakdown for Square Arrays of 5% U0 2 at 29 psia
A: COE (mills/kW-hre)
B: FCC (mills/kW-hre)
12
11
E
1
100
40
11
80
30
10
60
O1(
40
7
20
8
10
20
1.3 1.4 15
P/D
D: Capital (mills/kW-hre)
1.3 1.4 1 5
PID
C: O & M (millsIkW-hre)
1.1
1.1
1.2
12
I
11
E10
E
1.2
1
50
I
1
40
4
3
20
7
1.1
1.2
1.3
P/D
1.4
1.5
I
6
5
E
30
Q8
7
I
10
2
P/D
Figure E-5 Plant Operating Conditions for Square Arrays of 5% UO2 at 29 psia
B: Specific Power (kWth/kgHM)
A: Core Power (x 106 kWth)
14
II 3
tf.
~._.
,C:
I
50
11
10
9
40
30
2
9
1 8
20
1
7
10
1.3 1.4 1.5
P/D
D: Annual Energy Prod. (x 10 10 kW-hre)
1.1
1.3 1.4 1.5
PID
C: Burnup (MWDkg,,,)
1.1
E
12
1.2
1.2
12
12
1
50
11
E
F
40
10
0.8
E10
._
E3c
30
9
0.6
8
20
8
0.4
7
1.1
226
11
I
1.2
1.3 1.4
P/ID
1.5
l
10
0.2
1.1
1.2
1.3 1.4
P/ID
1.5
Figure E-6 Plant Operating Conditions for Square Arrays of 5%/1l102 at 29 psia
A: Capacity Factor
B: Cycle Length (yrs)
12
12
0.96
11
3
11
0.955
E
Ei_ 109
2.5
E10
2
0.95
8
0.94'
7
0.94
1.5
7
1.1
1.3 1.4 1.5
PID
D: Plainned Outage Length (days/cycle)
1.3 1.4 1.5
P/D
C: Annual Outage Length (days/yr)
1.1
8
1.2
1.2
12
18
12
30
11
I 16
11
28
10
26
EE-10
14
24
C
12
8
7
I 10
1.1
1.2
1.3
P/D
1.4
1.5
28
22
20
7
1.1
1.2
1.3
1.4
1.5
P/D
227
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