CORE DESIGN ASSESSMENT AND SAFETY ANALYSIS OF A FAST SPECTRUMMOLTEN CHLORIDE SALT REACTOR
By
ALEXANDER J. MAUSOLFF
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2019
c© 2019 Alexander J. Mausolff
To my parents
ACKNOWLEDGMENTS
The amount of people I have to thank for supporting me on this journey is immense.
To begin, none of this would have been possible without the unwavering support of my
research advisor Dr. Sedat Goluoglu. With his guidance I was able to investigate many
research pathways and remain heavily involved in professional development activities.
Additionally, I am very grateful to my committee, Dr. Mark DeHart, Dr. Richard Hennig,
and Dr. Greg Stitt for feedback and support.
I was fortunate to spend several summers at several national laboratories. I first
would like to thank Dr. Leslie Kerby and Dr. Mark DeHart for bringing me out to Idaho
National Laboratory as a young graduate student. Additionally, there are many people
and friends at Argonne National Laboratory who helped me along this journey. Dr.
Emily Shemon provided me with tremendous guidance over a summer at Argonne and
trusted me to spend precious CPU hours on the MIRA supercomputer. Dr. Mike Smith
deserves a special thanks for helping me come out to Argonne on two more occasions.
The discussions and feedback provided by Dr. Mike Smith were unparalleled and I cannot
thank him enough for taking time out of his day to help me.
There are many people apart of community growing up who were instrumental in me
even getting to attend graduate school. One of my youth soccer coaches, Marcos Mercado,
taught me so much about certitude, discipline, and toughness, all of which transcend
sports and have become integral parts of my life. I cannot thank Mr. Saylor enough for
inspiring me to pursue a degree in science through his amazing High School physics class.
I would like to thank Dr. Horacio Camblong for supporting me pursue a degree in physics
at The University of San Francisco. Additionally, I have to thank Dr. Thomas Bottger
for teaching me how to approach research and for the support writing my undergraduate
dissertation. I would like to thank Dr. Seth Foreman whose excitement and scientific
approach was highly influential in honing my problem solving capabilities.
4
Of course the support of my friends and family has been nothing short of amazing.
I have made many amazing friends while in Florida, Pat, Zach, Daken, Paul, Zach
Mannes, Justin, Max, and many others, without whom would have made this journey
quite dull. I especially have to thank for the very special Tory, who I met in graduate
school, for the constant love and support. My family has always been supportive of my
education, in particular, my great Uncle, Dr. Jim Hoch and great Aunt Dr. Sallie Hoch
were extremely generous in supporting my undergraduate education and providing role
models as scientists. I must thank my Uncle, Dr. Chris Mausolff for his feedback and
advice on getting through graduate school. A special thanks is given to my cousin Jacey,
for his always insightful opinions, mutual love for the Salty Dog Saloon, and conveying
the importance of networking. My Aunt, Professor Sarah Buel, deserves thanks for her
amazing ability as a role model and support through the years. Of course I need to thank
Grammie, whose love and understanding are more than I could ever ask for. My brother
Harrison has become a close friend and I am thankful to have him as a younger brother,
as he is the real scientist in the family. I owe so much to my dad for his lack of judgment
and unflinching support of just about everything I wanted to do. My mothers kindness,
love, and support has been absolutely amazing over the years, without which I do not
believe I would be at this point in my life.
I would finally like to express my deepest gratitude to the U.S. Department of
Energy Nuclear Engineering Universities Program (NEUP) and their Integration Office
for granting me the fellowship that allowed me to pursue this Ph.D. The material in this
thesis is based on work supported under a NEUP graduate fellowship. Any opinions,
findings, conclusions, or recommendations expressed in this work are those of the author
and do not necessarily reflect the views of the Department of Energy Office of Nuclear
Energy. Lastly, I would like to thank the staff and faculty at the University of Florida for
their assistance in helping me pursuing a Ph.D.
5
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ABBREVIATIONS, MATHEMATICAL CONVENTION . . . . . . . . . . . . . . . . 13
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2 Research Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Historical Development of Molten Salt Reactors . . . . . . . . . . . . . . . 212.2 Survey of Fast Spectrum Molten Chloride Reactors in the Open Literature 222.3 Physical Phenomena in Molten Salt Reactors . . . . . . . . . . . . . . . . . 29
2.3.1 Neutron Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Delayed Neutron Precursors . . . . . . . . . . . . . . . . . . . . . . 322.3.3 Transport of Delayed Neutron Precursors . . . . . . . . . . . . . . . 332.3.4 Temperature and Fluid Flow . . . . . . . . . . . . . . . . . . . . . . 35
3 SURVEY OF SIMULATION METHODS FOR TRANSIENT ANALYSIS . . . . 37
3.1 Point Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.1 Overview of the Point Kinetics Equations for Stationary Fuel . . . . 393.1.2 Point Kinetics Modification for Molten Salt Reactor Systems . . . . 41
3.2 Quasi-Static Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Quasi-Static Methods for Molten Salt Reactors . . . . . . . . . . . . . . . 45
4 DEVELOPMENT OF A SIMPLE DYNAMICS CODE FOR MOLTEN SALTREACTOR SAFETY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Prototypical One-Dimensional Molten Salt Reactor Model . . . . . . . . . 474.2 Discontinuous Galerkin Finite Element Method . . . . . . . . . . . . . . . 48
4.2.1 Discretization of the Power Amplitude Equation . . . . . . . . . . . 514.2.2 Determination of Time Stable Modified Point Kinetics Equations . . 534.2.3 Discretization of the Precursor Equation . . . . . . . . . . . . . . . 544.2.4 Discretization of the Heat Equation . . . . . . . . . . . . . . . . . . 564.2.5 Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Coupling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6
4.4 Steady State System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 Time-Dependent System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.1 Explicit Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.5.2 Implicit Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.3 Reactivity Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Computer Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 VERIFICATION OF THE TRANSIENT SOLUTION METHOD . . . . . . . . 67
5.1 Step Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1.1 Physics Based Verification . . . . . . . . . . . . . . . . . . . . . . . 675.1.2 Step Perturbation Verification . . . . . . . . . . . . . . . . . . . . . 685.1.3 Zig-zag Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Power Stabilization at New Flow Speed . . . . . . . . . . . . . . . . . . . . 715.3 MSRE Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 MOLTEN CHLORIDE SALT REACTOR DESIGN . . . . . . . . . . . . . . . . 76
6.1 Design Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2 Design Approach and Tools Used . . . . . . . . . . . . . . . . . . . . . . . 776.3 Material Considerations and Properties . . . . . . . . . . . . . . . . . . . . 78
6.3.1 Fuel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3.2 Density of Fuel Salt . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3.3 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.4 Overview of Thermophysical Properties Selected . . . . . . . . . . . 826.3.5 Vessel and Reflector Materials . . . . . . . . . . . . . . . . . . . . . 836.3.6 Primary Loop Mass Flow Rate . . . . . . . . . . . . . . . . . . . . . 84
6.4 Simple Tank Molten Chloride Fast Reactor Model . . . . . . . . . . . . . 856.5 Refined Core Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.6 Steady State Analysis with DIF3D . . . . . . . . . . . . . . . . . . . . . . 87
6.6.1 Cross Section Processing . . . . . . . . . . . . . . . . . . . . . . . . 886.6.2 Core Coolant Paths Assessment Method . . . . . . . . . . . . . . . 886.6.3 Reflector and Shielding Cooling Assessment . . . . . . . . . . . . . . 916.6.4 Coolant Flow Path Results . . . . . . . . . . . . . . . . . . . . . . . 936.6.5 Core Component Lifetimes . . . . . . . . . . . . . . . . . . . . . . . 956.6.6 Core Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.6.7 Enrichment of Chlorine-37 . . . . . . . . . . . . . . . . . . . . . . . 1006.6.8 Enrichment of Uranium-235 . . . . . . . . . . . . . . . . . . . . . . 102
6.7 Heat Exchanger Sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.8 Generation of Point Kinetics Data with PERSENT . . . . . . . . . . . . . 1086.9 Reactivity Feedback Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 110
7 SAFETY ANALYSIS OF THE MOLTEN SALT REACTOR DESIGN . . . . . 113
7.1 Primary Fuel Pump Failure Transient Simulations . . . . . . . . . . . . . . 1137.2 Quantification of Precursor Loss . . . . . . . . . . . . . . . . . . . . . . . . 1177.3 Primary Fuel Pump Over Speed . . . . . . . . . . . . . . . . . . . . . . . . 120
7
7.4 Reduction in Heat Sink Transients . . . . . . . . . . . . . . . . . . . . . . 1237.5 Heat Sink Overcool Transients . . . . . . . . . . . . . . . . . . . . . . . . . 125
8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
APPENDIX
A STEADY STATE SOLUTION ALGORITHM . . . . . . . . . . . . . . . . . . . 133
B TRANSIENT SOLUTION ALGORITHMS . . . . . . . . . . . . . . . . . . . . 134
C CROSS SECTION DIAGRAMS FOR FUEL SALT ATOMS . . . . . . . . . . . 135
REFERENCE LIST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8
LIST OF TABLES
Table page
2-1 Summary of operating parameters for the primary MCFRs in the literature. . . 29
2-2 Delayed neutron fraction data for each precursor group for prototypical thermaland fast neutron spectrum systems. . . . . . . . . . . . . . . . . . . . . . . . . . 34
5-1 Decay constant (λ) and delayed neutron fraction (β) values per delayed family(i) for the point kinetics physics based verification problem. . . . . . . . . . . . 68
5-2 Point kinetics parameters for the step perturbations. . . . . . . . . . . . . . . . 69
5-3 Comparison of calculated amplitude with forward Euler time discretization (FETD)and backward Euler time discretization (BETD). . . . . . . . . . . . . . . . . . 69
5-4 Point kinetics parameters for the zig-zag perturbations. . . . . . . . . . . . . . . 70
5-5 Amplitude values are compared at several time steps for the zig-zag perturbation. 70
5-6 Delayed precursor parameters for flow transition simulation verification. . . . . . 72
5-7 Summary of MSRE experimental values as reported. . . . . . . . . . . . . . . . 74
5-8 Summary of kinetics data used in MSRE theoretical calculations. . . . . . . . . 74
5-9 Comparison of calculated loss in delayed neutron fraction (in units of pcm) betweenexperiment and simulated results. . . . . . . . . . . . . . . . . . . . . . . . . . 75
6-1 Reported values of the heat capacity in several MSR design studies. . . . . . . . 82
6-2 Nominal values selected and compared with typical reactor coolants. . . . . . . . 83
6-3 Summary of nominal design parameters in the revised MCFR design. . . . . . . 88
6-4 Calculated parameters for the core inlet and outlet flow paths. . . . . . . . . . . 94
6-5 Calculated parameters for the radial reflector flow paths. . . . . . . . . . . . . . 94
6-6 Calculated parameters for the inner shield radial flow paths. . . . . . . . . . . . 94
6-7 Thermophysical properties selected for FLiNaK. . . . . . . . . . . . . . . . . . . 104
6-8 Heat exchanger design parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 105
6-9 Estimated heat exchanger thermal design specifications. . . . . . . . . . . . . . 108
6-10 Point kinetics parameters generated by PERSENT. . . . . . . . . . . . . . . . . 110
6-11 Doppler and density coefficients compared to the REBUS-3700 MCFR. . . . . . 111
7-1 Decay constants and half-lives per precursor group. . . . . . . . . . . . . . . . . 118
9
LIST OF FIGURES
Figure page
2-1 Schematic of MSRE core and vessel. . . . . . . . . . . . . . . . . . . . . . . . . 22
2-2 Early molten chloride reactor concept. . . . . . . . . . . . . . . . . . . . . . . . 24
2-3 Homogeneous molten chloride fast reactor design. . . . . . . . . . . . . . . . . . 25
2-4 CHLOROPHIL reactor schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2-5 SOFT reactor concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2-6 Reactor schematic of the REBUS-3700 design. . . . . . . . . . . . . . . . . . . . 28
2-7 Illustration of the mean generation time in a nuclear system. . . . . . . . . . . 31
2-8 Prompt and delayed neutron production and their relative time scales. . . . . . 32
2-9 Simplified view of an active MSR core and the possible decay of precursors outsideof the core. The V1 and V2 indicate two different velocities, λi is a average decayconstant for a given family i , and βi is the effective loss in β for a given family. . 33
2-10 Comparison of power amplitude for reactivity insertions in fast and thermalspectrum systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2-11 Comparison of a prompt-critical reactivity insertion in a fast and thermal spectrumcore. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3-1 Representation of the time scale in a generic QS method. . . . . . . . . . . . . . 45
4-1 One-dimensional MSR model with an active core region (fission occurs here),external piping, a heat exchanger, and pump. Note, the flow circulates in thismodel with the flow out becoming the flow in. . . . . . . . . . . . . . . . . . . . 48
4-2 Lagrange interpolation functions over an element with a size of 1.0. . . . . . . . 50
4-3 Sample prescribed power profile for a case with 10 nodes and an active fuel regionof 7 nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5-1 The first 10 seconds of a simulation are shown where a step perturbation is introducedand maintained for 0.2 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5-2 Variation in reactivity for the zig-zag test problem. The reactivity as a functionof time is given on the left and the normalized power amplitude is given on theright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5-3 Reactivity inserted in the system as a function of time. . . . . . . . . . . . . . . 72
5-4 Amplitude change over time for the flow transition test problem. . . . . . . . . . 73
10
5-5 Normalized precursor distribution of each group for the steady state condition. 75
6-1 Overview of each codes role in the analysis of an MCFR. . . . . . . . . . . . . . 77
6-2 Reported density values as a function of temperature for several molar compositionsof UCl3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6-3 Cutaway view of a simple tank MCFR model. . . . . . . . . . . . . . . . . . . . 86
6-4 Axial view of the updated MCFR design. . . . . . . . . . . . . . . . . . . . . . . 87
6-5 R-Z core model used in TWODANT flux calculations. . . . . . . . . . . . . . . . 89
6-6 Iterative process for determining necessary fuel salt coolant in reflector and innershield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6-7 The lower reflector fluence is plotted as a function of time in each radial region,where the dashed line represents the structural fluence limit. . . . . . . . . . . . 96
6-8 The upper reflector fluence is plotted as a function of time in each radial region,where the dashed line represents the structural fluence limit. . . . . . . . . . . . 97
6-9 Variation in eigenvalue as a function of core width. . . . . . . . . . . . . . . . . 99
6-10 Calculated eigenvalue for different core heights. . . . . . . . . . . . . . . . . . . 99
6-11 Eigenvalue plotted as function of the 37Cl enrichment. . . . . . . . . . . . . . . . 101
6-12 Calculated eigenvalue as a function of 235U enrichment. . . . . . . . . . . . . . . 103
6-13 Spatial dependence of Doppler and fuel expansion reactivity changes. . . . . . . 112
6-14 Density comparison between NaCl-UCl3 and solid UO2 fuel. Note, in both casesall values are normalized by the starting density value evaluated at 600 K. . . . 112
7-1 Power as a function of time for the first 100 seconds of each simulated pumpcoast down. Each dashed line represents the time it took to reach the lower massflow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7-2 Different time steps employed in the calculation of the power as a function oftime for a transient where the mass flow rate is reduced in 1.6 seconds. . . . . . 116
7-3 Average temperature across the active core as a function of time for each simulatedpump coast down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7-4 Calculated delayed neutron fraction in the core at different steady state massflow rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7-5 Fractional contribution of each precursor group to the total fraction of delayedneutrons at each mass flow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
11
7-6 Power amplitude as a function of time for different transient simulations whereeach line represents the time taken to reach the new flow rate. . . . . . . . . . . 121
7-7 Comparison of the power trace with different time steps for a 10% increase inmass flow rate over 1.6 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7-8 The average temperature is plotted as a function of time. On the left the first20 seconds of the transients are shown, on the right the first 250 seconds. . . . . 122
7-9 Power profile for different amounts of heat removed from the heat exchanger. . . 124
7-10 Average core temperature over time for different temperature reductions acrossthe heat exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7-11 Power amplitude (left) and average temperature (right) as a function of timefor a 10 K reduction in the temperature across the heat exchanger. . . . . . . . 125
7-12 Power as a function of time for several heat exchanger temperature drop overcool transients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7-13 Average core temperature for several heat exchanger temperature drop over cooltransients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C-1 22Na neutron cross section as a function of energy plot from ENDF/B-VII.1. . . 135
C-2 37Cl neutron cross section as a function of energy plot from ENDF/B-VII.1. . . 136
C-3 35Cl neutron cross section as a function of energy plot from ENDF/B-VII.1. . . 137
12
ABBREVIATIONS, MATHEMATICAL CONVENTION
MSR Molten Salt Reactor
MCFR Molten Chloride Fast Reactor
LWR Light Water Reactor
SFR Sodium cooled Fast Reactor
HTGR High Temperature Gas cooled Reactors
ARE Aircraft Reactor Experiment
MSRE Molten Salt Reactor Experiment
ORNL Oak Ridge National Laboratory
BE Backward Euler
FE Forward Euler
OS Operator Split
MFNK Matrix Free Newton Krylov
STP Standard Temperature and Pressure
FEM Finite Element Method
DG-FEM Discontinuous Finite Element Method
QS Quasi-Static
IQS Improved Quasi-Static
~~A Denotes a m by n matrix A
~x Denotes vector x of length n
~f (z) Lagrange interpolation functions
13
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
CORE DESIGN ASSESSMENT AND SAFETY ANALYSIS OF A FAST SPECTRUMMOLTEN CHLORIDE SALT REACTOR
By
Alexander J. Mausolff
August 2019
Chair: Sedat GoluogluMajor: Nuclear Engineering Sciences
In recent years a resurgence of interest in flowing fuel fast spectrum molten chloride
salt reactors has been observed. However, no such reactors have been constructed and
existing safety assessment tools do not consider the movement of nuclear fuel. In
this work it is hypothesized that flow perturbations could rapidly change the delayed
neutron fraction in the core potentially causing a power excursion. This would pin the
neutronics and thermal-hydraulics equations to a similar time-scale thus requiring a
high-order time integration scheme to efficiently solve the coupled set of equations. To
test this hypothesis a modified point kinetics code encapsulating the relevant physics is
developed to assess the time response of flow perturbations. Additionally, all chloride
reactor designs proposed to date have been compared revealing a lack of consistent
thermophysical properties, spherical core geometries with recirculation zones concerns,
minimal consideration for component lifetimes, and a focus on fuel cycle optimization
rather than designs constrained by engineering limitations. To provide a meaningful
safety assessment a reactor core design is developed with constrained flow paths and
replaceable reflector. Testing the hypothesis on this new core design through simulation
of primary and secondary side flow changes resulted in the hypothesis being rejected.
It is found that changes to the delayed neutron fraction can inject reactivity but such
changes are felt on the order of 1–2 seconds due to the dominance of the fourth and fifth
precursor groups, which have half-lives on that time scale. However, the the large negative
14
density reactivity coefficient mitigates any reactivity change introduced by the precursor
distribution. In addition, the large core volume and high mass flow rate are such that the
fuel spends about 2 seconds across the core, thus further reducing the influence of the
precursor redistribution. Of greater consequence in transient conditions is the sustained
temperatures (>1400 K) observed in the active core. Furthermore, the design developed
reveals significant challenges for any molten salt system such as a very large core and
heat exchanger sizes, inner reflector lifetimes on the order of 5 years, a lack of verified
thermophysical properties, and an operational temperature range above any nuclear
qualified materials.
15
CHAPTER 1INTRODUCTION
The civilian nuclear power sector in the United States (U.S.) appears to be in
a challenging position. Since 2013 several plants have closed prematurely and more
appear on the way out. Furthermore, the pace of closures outweighs the construction
of replacement plants. The lack of new construction is in part due to economic and
regulatory challenges. Issues are compounded by a lack of direction from government
agencies and the research community as to which reactor variants might replace the
Light Water Reactor (LWR) fleet, or if doing so is necessary. The next generation of
non-LWR concepts such as Sodium Cooled Fast Reactors (SFRs), High Temperature Gas
Cooled Reactors (HTGRs), etc., have been in development for some time and some have
even operated in the U.S. and abroad [1]. The so called ‘generation IV’ nuclear reactors
are tremendously promising but still have economic, regulatory, public perception, and
technical issues to resolve. Technical issues are solvable, but it is concerning that many
current research projects appear not to take a holistic approach to reactor development.
For instance, computational tools are developed that are interesting as academic objects
but often do little to further the ability to understand the dynamics of new reactor
concepts. While research does not always require a clear use case, given the state of the
nuclear industry in the U.S., it seems there is an extra burden on the computational
nuclear engineer to at least be aware of how his or her research may benefit the future of
the industry.
The economic, regulatory, and public perception problems in nuclear power have been
compounded by difficulties constructing new plants in the U.S. [2]. However, worldwide
there are approximately 60 reactors under construction as growing countries such as China
and India incorporate nuclear reactors to reduce carbon emissions [3]. The U.S. has the
option to continue leading the development of nuclear technology or let the global industry
be dominated by others. A potential benefit to the U.S. industry may be to develop
16
reactor concepts that cost less, are simpler to build, offer possible use cases beyond power
generation, and potentially safer operation. There are many reactor concepts being
pursued privately and researched at U.S. national laboratories. One such concept that
has garnered significant attention is the Molten Salt Reactor (MSR). Several venture
capital funded nuclear start up companies are looking to develop MSR concepts because
of the possibilities MSRs open up to recycling nuclear fuel, improving safety margins, and
enhancing economic competitiveness.
The class of reactors known as MSRs differs from other designs such as commercial
LWRs and SFRs as the primary cooling and heat transfer systems utilize salt heated past
the melting temperature as the working fluid. These salts can be composed of familiar
table salt (NaCl) or lithium-fluoride salts, and can have fissile (e.g. 235U) and fissionable
(e.g. 232Th) material mixed in. It is important to note that not all MSR designs have
nuclear fuel combined with the salt. For instance, some MSR concepts have solid fuel rods
or Tristructural-isotropic (TRISO) fuel pellets, making use of molten salt (absent of fissile
material) as a coolant and heat transfer fluid only [4]. However, the work presented in
this dissertation will consider reactors where nuclear fuel is dissolved in salt and pumped
through the primary core circuit. These flowing fuel concepts can operate with a thermal
or fast neutron energy spectrum and are envisioned to have secondary sides for power
generation or industrial heat applications.
One advantage of MSRs is molten salt can be heated to high temperatures (greater
than 700 C) without boiling while remaining at near atmospheric pressure. This is
advantageous as there is no need for a vessel (where the reactor is placed) capable of
maintaining high pressure. Conversely, in LWRs a pressure vessel is required to increase
the temperature of the water without boiling it. These pressure vessels are expensive and
at the moment the U.S. does not have the facilities to produce them [5]. Additionally,
since molten salts can be heated to higher temperatures than pressurized water, this
allows for improved operating efficiency compared to LWRs. Other often touted benefits
17
of MSRs are their abilities to recycle fuel, operate on a variety of fuel cycles, and breed
fuel. While these claims may be theoretically possible, there still remain material, legal,
and regulatory challenges, particularly related to proliferation concerns. For additional
clarification, a further distinction between MSR concepts can be made between single and
two-fluid designs. The single fluid design contains both fissile material and potential fertile
material for breeding mixed together. Alternatively, in the two-fluid design salt containing
fertile nuclear material is separate from the fuel salt and flows through a second system
in close proximity to the loop containing fissile salt [6]. Distinction is made between these
various reactor concepts as often all of these are generically referred to as MSRs but there
are a variety of fundamental design differences. During the course of this work the focus
will be on the single-fluid Molten Chloride Salt Fast Reactor (MCFR).
1.1 Motivation
The authors of early work in numerical methods for nuclear reactor applications
made numerous remarks noting the methodology and assumptions stated to simulate the
physics only hold true for stationary nuclear fuel [7, 8]. It was clear these early pioneers
were aware of the fundamental challenges flowing nuclear fuel would bring about when
trying to simulate the underlying physics. Thus the development of tools to simulate
reactor concepts such as the molten salt variety, specifically those with flowing fuel,
requires careful consideration to accurately capture the physics correctly. Not surprisingly,
these considerations are of particular importance when time-dependent phenomena are
of interest. Specifically, consider that the flowing of irradiated nuclear fuel results in
fission products (including delayed neutron precursors) being produced at the fission
site which then may be carried away post-fission due to the fuel movement. In this case
the movement of the fission products is now coupled to the time scale of the fluid they
are travelling in. This is not the case in a solid fueled reactor, where fission products
are produced at the location of the fission event and do not move from there (on the
time scale of any transient scenario). The decay of delayed neutron precursors yields
18
neutrons that are produced on the order of seconds to minutes which allows the reactor
to be controlled on that time scale. Without delayed neutrons reactor control would
be tied to the prompt neutron lifetime, which is on the order of 10−7 seconds in fast
spectrum reactors. The disparate time scales between the prompt neutron lifetime and
delayed neutron production (which now varies with flow rate) motivates assessing the time
response to flow perturbations in MCFRs.
Considering MCFR designs are sparse in the literature, have fundamental design
differences with conventional reactors, and have no operational experience it seems
prudent to perform a numerical experiment to test a simple, yet crucial, hypothesis. The
hypothesis of this work is that flow perturbations introduced in a fast MSR system will
pin the fluid flow and neutronics to a similar time scale and require development of a
high-order time integration approach to solving the governing system of equations. The
hypothesis is motivated in part because the flowing fuel can lead to a significant fraction
of the delayed neutrons being produced outside of the core. Rapid changes in the delayed
neutron fraction in a fast spectrum system could potentially lead to large power excursions
due to the very small average prompt neutron lifetime relative to thermal spectrum
systems. The extent of these changes to the delayed neutron spectrum is difficult to
evaluate given the uncertainty in nearly all aspects of MCFR designs.
1.2 Research Approach
The methods to assess the time response to various perturbations in the system
ideally would be understood by solving a set of nonlinear, multi-physics, time-dependent
partial differential equations defining all of the physics. Considering this is immensely
challenging and computationally intensive at this point in time there is motivation to
reduce the problem to the simplest form to understand the dynamics under certain
conditions. This was the approach taken early on by nuclear engineers for kinetics
problems. The approach is referred to as the point kinetics model, which is a set of
19
6–8 ordinary differential equations derived from the time-dependent neutron transport
equation by assuming no spatial changes to the neutron flux occur over time.
To test the hypothesis discussed in Section 1.1 a simple code is developed that
captures the time response to changes in flow rates. This simplified approach makes use
of the point kinetics equations but modified to incorporate precursor flow, temperature
changes, and feedback mechanisms. The fluid and temperature changes account for the
nonlinear material properties and the feedback effects introduced as this system evolves in
time. In addition to the development of a safety analysis code, a core design is developed
with the use of Argonne National Laboratory’s suite of fast reactor tools [9]. These reactor
tools do not explicitly account for the movement of the precursors but do provide an
idea of the steady state core behaviour and dimensions. Calculations are performed to
ensure criticality with the given salt composition, develop core dimensions, assess radiation
damage to core components, and calculate reactivity coefficients for transient analysis. In
addition, the selection of thermophysical properties is discussed in detail and considers
recent evaluations of some properties. A tube-and-shell heat exchanger sizing study, with
a secondary heat transfer salt, is performed to estimate the overall amount of fuel salt
required in the system. The motivation in coming up with a core design is to provide a
plausible and well understood starting set of parameters to assess the time-response of an
MCFR system and investigate the feasibility of this reactor type.
20
CHAPTER 2BACKGROUND
2.1 Historical Development of Molten Salt Reactors
The notion of using fluid-fuel in the form of a molten salt is not a novel idea and has
been around since the 1950s. An early project beginning in 1952 produced several reports
discussing the problems of fluid-fueled reactors [10, 11]. In these reports it was suggested
that chloride salts would be ideal for fast spectrum reactors, while fluoride salts should
be reserved for thermal spectrum systems [10]. Several years later the first demonstration
of a fluid-fueled Molten Salt Reactor (MSR) began in 1954, with the Aircraft Reactor
Experiment (ARE) [12]. The ARE was operated for several hundred hours using molten
fluoride salt for fuel with a beryllium oxide moderator and a liquid sodium secondary
side for cooling. Operation of the ARE demonstrated several of the attractive features,
such as a strong negative temperature coefficient (safety feature), removal of fission
products during operation, and management of the power levels without control rods.
Development of MSRs continued at Oak Ridge National Laboratory (ORNL) into the late
1960s, the highlight of which was the successful operation of an 8.2 MW thermal MSR
from 1965 to 1969, which is commonly referred to as the Molten Salt Reactor Experiment
(MSRE), shown in Figure 2-1 [13]. The MSRE was a single-fluid design and operated
with a thermal neutron spectrum with several varieties of fuel salt containing 233U,
235U, and 239Pu, all moderated with graphite. The operation of the MSRE led to the
conceptual design of a thermal spectrum and thorium fuel cycle 1000 MWe Molten Salt
Breeder Reactor (MSBR) at ORNL [6]. Other MSR concepts were investigated such as
fast spectrum MSRs with chloride fuel, all primarily focusing on breeding nuclear fuel and
will be discussed in greater detail in Section 2.2 [15–20].
The MSR concept has come back into favor in nuclear research and now is an official
part of several large scale research initiatives all over the world. In addition, there is
significant support from the private sector into MSR research, with several companies
21
Figure 2-1. Schematic of MSRE core and vessel [14].
pursuing designs. Several private companies are pursuing MCFRs such as Elysium,
TerraPower, and Southern Company [21, 22]. As no MCFRs have been developed there is
less information regarding the material properties and schematics of these designs. Thus
there is a need for development of research tools to design and assess MCFRs.
2.2 Survey of Fast Spectrum Molten Chloride Reactors in the OpenLiterature
After the Aircraft Reactor Experiment (ARE), work on MSR designs was motivated
by concerns over uranium shortages and that breeding would become necessary to make
nuclear power viable. As such, much of the focus was on the fuel cycle and how to design
thermal or fast systems to breed nuclear fuel as the reactor operates. The MCFR designs
offered tremendous flexibility due to the very hard neutron spectrum and possibility of
removing fission products during routine operation. At the time of many of the early
studies neither the computational power nor the tools were available for in-depth reactor
22
physics or safety analyses. Considering our goal is to assess the time response of an
MCFR, it seems prudent to make such an assessment on a core geometry with plausible
operating parameters. Upon investigation of molten chloride concepts it appears many
of the designs in the literature are not well described and do not consider realistic design
constraints, nor experience garnered from solid fuel fast reactor research. Additionally,
it seems there are minimal comparisons between previously conceived chloride reactor
concepts or comparisons to commercially operating reactors. To illustrate these points a
review of the MCFRs in the literature will be discussed.
The first reported investigation of an MCFR type of design was described in a
series of reports from 1952 carried out by the Massachusetts Institute of Technology
[10, 11, 23]. The motivation for this investigation was to determine the suitability of
a molten salt reactor for the production of plutonium and to recommend a research
plan if such a reactor were to be developed. The report focusing on nuclear calculations
gave the first indication that chloride salts would be favorable compared to fluoride as
fluoride moderates the neutron energy spectrum more than chloride salts [11]. Much
of the analyses and conclusions drawn were quite general as the study was focused on
plutonium breeding and the kind of research program which would be required for the
development. Additionally, at that time the availability of nuclear cross section data was
minimal and detailed flux calculations and safety analyses were not possible. The nuclear
data available for the fast spectrum calculations were lacking for the inelastic and elastic
scattering of chloride so it was concluded that not enough information was present to say
whether a chloride salt fast spectrum reactor would be economically feasible for plutonium
production [11]. Nevertheless much of the information provided is quite valuable for future
chloride designers; in particular, the discussion of the chemical problems highlights many
of the materials challenges still present today [10].
In 1956, a report by ORNL investigated the possibility of developing a chloride fast
spectrum breeder reactor for civilian use as a high temperature source of industrial process
23
Figure 2-2. Early molten chloride reactor schematic [15].
heat, and possibly to circumvent problems (at the time) with solid fuel fast reactors [15].
This design used plutonium-fused chloride salts for the fuel and depleted uranium as a
fertile blanket for breeding purposes. The goal was to produce 700 MW thermal power
using a spherical core, shown schematically in Figure 2-2. Control of the nuclear chain
reaction was envisioned using a molten lead reflector to enable dynamic reactivity control
without the use of control rods. Many engineering level calculations were performed to
develop what was at the time a reasonable overview of a reactor concept [24]. Much of the
analysis was performed with one-dimensional diffusion theory using 10 energy group cross
sections and safety analysis was carried out using whole core reactivity coefficients derived
from these simple calculations.
Nearly 15 years after the initial MIT reports, the topic of MCFRs was revisited
in greater detail, this time focusing on commercial power production [16]. The fuel
salt composition selected was PuCl3-UCl3-NaCl-MgCl2. However, for the neutronic
calculations it was assumed the magnesium was actually sodium as at the time there was
24
Figure 2-3. Homogeneous molten chloride fast reactor design [16].
no reliable cross section information for magnesium. Much of the thermophysical data was
based on measurements done on fluoride based fuels. The reactor physics calculations were
more sophisticated than the work in the 1950s but there still was not a detailed account
of all core dimensions nor safety based simulations. Again the core proposed was spherical
with no details on the inlet or outlet paths to the core. A schematic of the reactor system
is shown in Figure 2-3.
Even in the late 1960s, a period of tremendous innovation and optimism for nuclear
technology, the results of the analyses were that the core would be quite large, have
material problems containing the fuel salt, complicated design, and possibly a large
plutonium inventory [16]. The favorable characteristics were that a high breeding ratio is
possible, the core exhibits a large negative reactivity coefficient due to fuel salt expansion,
and low fuel-cycle costs are anticipated. However, despite the positive aspects of these
reactors the overall conclusion was that a sizable program would be required to assess the
25
Figure 2-4. CHLOROPHIL Reactor schematic [18].
feasibility and to construct the auxiliary components necessary to operate a commercial
plant.
Beginning in the early 1970s, several studies were undertaken focusing on plutonium
and uranium MCFR designs for power production [17, 18]. The liquid fuel consisted of
PuCl3-NaCl with a 238UCl3-NaCl fertile blanket surrounding the core region. Once again
the analysis focused on the fuel cycle. Calculations of the neutron flux based on a 22
neutron energy group structure were performed and some safety analysis based on whole
core reactivity coefficients were determined. An interesting conclusion at the time was
that natural chlorine could be used and it was believed enrichment of chlorine to high
levels of 37Cl would not be needed. A schematic of the reactor concept, referred to as
CHLOROPHIL, is given in Figure 2-4. Again, based on the schematic in Figure 2-4 it
appears to use a spherical core and insufficient data is available for performing a more
detailed safety analysis.
26
In 1980, another MCFR concept was proposed focused on highlighting how a lower
fissile inventory could be achieved [19]. The thought was this would increase safety due
to a lower chance of exposing the public to a large dose [19]. The design, referred to as
the “SOFT” reactor, is described in greater detail than previously proposed designs. It is
worth pointing out a great peculiarity of this concept, which is that the pump is placed
before the heat exchanger [19]. Placing the pump before the heat exchanger would likely
decrease thermal efficiency and subject the pump to higher temperatures than necessary.
Once again the spherical tank core design is proposed as shown in Figure 2-5. The design
appears to gloss over many issues related to the flow of the fuel and appears to be useful
only as an academic paper model.
Figure 2-5. SOFT reactor concept [19].
In the early 1990s an MCFR concept was investigated in the context of the
development of a high-flux test reactor for possible replacement of the Advanced Test
Reactor [25]. The very fast neutron spectrum and minimal downtime for refueling of a
molten chloride reactor motivated the authors to investigate the feasibility of the concept.
In general the work provided a broad summary of research on MCFRs to date but did
27
Figure 2-6. Reactor schematic of the REBUS-3700 design [20].
not provide a general reactor concept and most of the discussion was not tailored to
commercial power generation.
More recently, in 2006, an MCFR known as REBUS-3700 was proposed [20]. The
design follows much of the literature in that the reactor details are not well described nor
are many important reactor physics and engineering concerns addressed. The analysis uses
more modern tools than previous works, but the design still appears to resemble a simple
tank model. Reactor safety studies come in the form of whole core reactivity coefficient
calculations. Similar conclusions are made to the previous literature without discussing
many of the challenges nor adding detailed information like possible inlet and outlet core
geometries, how the core might be shielded, or any kind of economic analysis.
The body of work in the open literature on MCFRs has largely focused on fuel
cycle issues and has spent little time on detailed reactor physics calculations, plausible
engineering constraints, or safety analysis. To illustrate much of the inconsistency in
MCFR designs, several of the main operating parameters and thermophysical properties
from the MCFRs in the literature are listed and compared in Table 2-1. Inspecting Table
2-1 highlights the many differences in assumed material properties and the calculated
operating parameters. Much of the early work in the 1950s and 1960s assumed the
chloride fuel salt has similar properties to fluoride salt, however the fluoride based salts
have significantly better heat transfer capabilities than chloride (about twice as good),
which significantly impacts how the core is designed. The lack of consistent material data,
28
minimal details on core geometries, and implausible flow paths motivates developing a new
MCFR design.
Table 2-1. Summary of operating parameters for the primary MCFRs in the literature.Reactor [15] [16] SOFT CHLOROPHIL REBUS-3700Power [MWth] 700 2500 2000 3000 3686Inlet T [K] 838 898 1240 743 923Outlet T [K] 1005 1013 1274 923 1003∆ T 167 115 34 180 80m [kg/s] 4,657 25,973 48,959 5,733 50,743Core volume [m3] 3.23 10.00 8.75 74.99 36.85Temperature Evaluation [K] 923 923 1257 1257 963Density [g/cm3] 2.5 3.0 2.3 3.3 3.6Liquidus [K] 708 798 958 >743 873Viscosity [cP] 0.10 4.2 2.17 1 2
Power Density [kW/L] 217 250 220 40 100
2.3 Physical Phenomena in Molten Salt Reactors
The physics in an MSR requires developing an ability to determine the neutron
flux, temperature distribution, fluid flow, structural strain, etc. The challenge with MSR
analysis is that the nuclear fuel is dissolved into the salt and couples nuclear calculations,
heat transfer, and fluid flow simultaneously. This motivates highlighting what differences
are of importance for safety considerations and transient simulations. In the following
sections key differences with MSRs compared to solid fueled reactors are pointed out
and discussed in regards to the impact on simulating the time-dependent behavior. The
physical phenomena discussed are not exhaustive but should provide a reasonable overview
for the purposes of this work.
2.3.1 Neutron Transport
In nuclear reactor theory the transport of neutrons is a topic of immense study.
The study of neutron transport can be understood through the Boltzmann transport
1 Value not reported.
2 Value not reported.
29
equation, which describes the rate at which neutrons are produced, lost, and moved
when interacting with fissionable and non-fissionable material [7]. In this work the
neutron transport equation is not going to be studied in detail but it is important to
recall it is governing the underlying physical processes. What is important to note is
the interpretation of cross sections in the transport equation. Neutron interactions with
a given atom are described as a probability per unit path length with what are known
as cross sections. When a neutron interacts with an atom of a given material it can be
absorbed, scatter, or cause the atom to undergo fission. The chance of any of these events
happening depends on the atom’s size and type, internal (quantum) energy state, velocity,
and the relative velocity and direction of the incoming neutron. It is vital to understand
the aggregate effect of these neutron-material interactions to have a functioning nuclear
reactor.
For a reactor to operate there must be a self-sustaining chain reaction where neutrons
produced go on to induce fission in other atoms. Neutrons are primarily produced through
the fissioning of fissile material. A small fraction, about 1% of the total neutrons in a core,
are produced from the decay of certain fission products and are referred to as delayed
neutrons. Neutrons can be lost (in the chain reaction) due to absorption within a material
or leak out of the system. The term multiplication factor (sometimes referred to as ‘keff ’
or the eigenvalue) is an important definition as it describes the mean number of fission
neutrons produced by a neutron during its life within the system [26]. It follows that keff
= 1, if the system is critical; keff < 1, if the system is subcritical; keff > 1, if the system is
supercritical. An operating reactor at steady state requires a keff = 1. Typically, the goal
of steady state analysis of a nuclear system is to make sure the system is critical.
Time-dependent phenomena in a nuclear reactor can be understood as what happens
to the system as the multiplication factor deviates from unity. The concept of reactivity,
commonly denoted as ρ, encapsulates changes in the multiplication factor and can be
30
defined as:
ρ = 1− 1
keff
. (2–1)
How quickly a reactivity change will occur in a core depends on how long it takes a
neutron produced from fission to then strike an atom and cause the atom to undergo
fission. This can be defined as the prompt neutron lifetime or mean generation time,
denoted as Λ. The mean neutron generation time concept is illustrated in Figure 2-7.
Λ [s]
Birth of neutron
Absorption of neutron, leading to fission
Figure 2-7. Illustration of the mean generation time in a nuclear system.
As mentioned, some of the neutrons in a system come from the decay of certain
fission products and are of great importance for reactor control and understanding
time-dependent phenomena. When fission events occur a large amount of energy is
released as the atom undergoing fission is split into two or more elements (fission
products), along with neutrons, gamma rays, electrons and neutrinos. These fission
products are typically neutron-rich and therefore unstable and may undergo several more
decays giving off gamma rays, electrons, and neutrons. Some of these fission products
can also be referred to as delayed neutron precursors as they decay with the emission of a
neutron with some time delay compared to the prompt production of neutrons produced
31
directly at the time of the fission event. The difference is highlighted in Figure 2-8.
Nuclear cores are designed to become critical from the contribution of delayed neutrons.
If a reactor was critical just based upon the production prompt neutrons, referred to as
‘prompt-critical’, then the reactor would need to be controlled on the time scale of the
prompt generation time (10−4 - 10−7 seconds). Fortunately, designing the system to be
critical with delayed neutrons, in a so called ‘delayed-critical’ mode, allows the reactor
control to be tied to a time scale of seconds to minutes. Delayed neutron precursors have
Energy~200 MeV
~1x10 seconds 0.2 - 55 seconds
Prompt
Delayed
-14
Figure 2-8. Prompt and delayed neutron production and their relative time scales.
important implications for reactor control and require fundamentally different treatment
in MSR systems; they will be discussed in detail in Sections 2.3.2 and 2.3.3.
2.3.2 Delayed Neutron Precursors
As nuclear fuel is irradiated there are approximately 500 different fission product
nuclides produced, about 40 of which produce a delayed neutron somewhere in their
decay chain [27]. The relative yields of the fission products is dependent on the fuel
composition. The 40 or so delayed neutron precursors decay at different rates so they give
off a neutron at different rates, which must be accounted for in time-dependent problems.
It is impractical to consider each precursor directly as the lifetime of many precursors is
not known exactly, and many of the precursors are products of one or more beta decays,
which would need to be included in the theoretical formulation of the problem [27].
32
Instead, precursors are condensed into “delayed groups” or “families”, typically 6, that
represent the superposition of the contributions from each precursor. Throughout this
work the 6 delayed group convention will be used.
2.3.3 Transport of Delayed Neutron Precursors
In a solid fueled reactor fission events occur producing delayed neutron precursors
and subsequently delayed neutrons at the location of the fission site. This simplifies the
treatment of the precursor production as there is no need to keep track of the spatial
location of the precursors. Conversely, in a flowing fuel MSR the precursors are born in
one location but are transported with the flow of the fuel. The result is the precursors can
give off delayed neutrons in a different location than where the fission event occurred as
highlighted in Figure 2-9. Technically, the delayed neutron fraction (commonly denoted
Fissioned Fuel Decay @ 10*λi
V1
V2
Recovered/lost βi
Active Core
Figure 2-9. Simplified view of an active MSR core and the possible decay of precursorsoutside of the core. The V1 and V2 indicate two different velocities, λi is aaverage decay constant for a given family i , and βi is the effective loss in β fora given family.
as β) is constant during transients but is effectively reduced in an MSR core due to the
distribution of precursors. In an MSR running at steady state the β value is constant,
but reduced compared to non-flowing fuel depending on the flow rate. Any deviations
from the steady state flow rate will adjust the β observed in the core over time. Clearly,
33
the variation in the precursor distribution as the flow speed changes in time will be an
important characteristic to understand in MSR systems.
To illustrate why changes in the precursor distribution might be more problematic in
an MCFR, several transient simulations will be performed comparing the time response
in a fast and thermal system. Primary differences (from a kinetics point of view) between
fast and thermal spectrum systems are the delayed neutron yield and mean neutron
generation time. In a fast spectrum reactor the delayed neutron fraction is lower and the
mean neutron generation time is much shorter than in a thermal spectrum system. The
result is that the effects of reactivity insertions in a fast spectrum system occur faster and
potentially reach higher power levels depending on the feedback mechanisms in the core.
This can be shown comparing several reactivity insertions in a fast and thermal
system where negative feedback mechanisms are not considered. Prototypical fast reactor
data is found from the literature and provided in Table 2-2 [28]. The prompt neutron
lifetime was set as 1× 10−4 seconds for the thermal spectrum system and 1× 10−7 seconds
for the fast spectrum. Simulations are carried out using the point kinetics equations for
solid fueled reactors. The details of point kinetics equations will be discussed in Section
3.1.1. In all simulations in Figure 2-10, the reactivity was increased linearly for 1.0 second
to a final value of (a) 10 or (b) 50 pcm. Both of these reactivity insertions are less than
the total β value. The power profiles in each simulation are given in Figure 2-10.
Table 2-2. Delayed neutron fraction data for each precursor group for prototypical thermaland fast neutron spectrum systems.
Group 1 2 3 4 5 6βi Thermal 2.7 × 10 −4 1.5 × 10−3 1.3 × 10 −3 2.8 × 10−3 9.0 × 10−4 1.8 × 10−4
βi Fast 7.9 × 10 −5 7.3× 10 −4 6.4× 10 −4 1.3 × 10−3 5.7 × 10−4 1.6 × 10−4
If instead the reactivity insertion is equivalent to β then the reactor starts to operate
in a prompt-critical regime. When a prompt-critical transient occurs the power increases
very rapidly as the power rises on the order of the mean neutron lifetime, creating a
dangerous situation. In a fast spectrum reactor, where the prompt neutron lifetime is
34
0 1 2 3 4 5Time [s]
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
Powe
r Am
plitu
de
Thermal = 10 pcmFast = 10 pcm
Thermal = 50 pcmFast = 50 pcm
Figure 2-10. Comparison of power amplitude for reactivity insertions in fast and thermalspectrum systems.
very small, the power increase occurs extremely fast compared to thermal spectrum
cores. Prompt-critical reactivity step insertions are simulated in fast and thermal neutron
spectrum systems. Reactivity equivalent to the total β is inserted at time zero. The
corresponding power amplitude over the first 0.1 seconds is shown in Figure 2-11. Note, in
Figure 2-11 the y-axis is on a logarithmic scale.
Clearly in all cases, for the same reactivity insertion the power excursion occurs
faster and is much higher in the fast spectrum systems than in the corresponding thermal
spectrum simulation, especially in the prompt-critical transients. These simulations
highlight why reactivity insertions in a fast spectrum system, such as the MCFR, are of
concern from a kinetics viewpoint.
2.3.4 Temperature and Fluid Flow
In most commercial nuclear reactor power systems the goal is to produce heat
through fission and then use a working fluid to transport that heat away for generating
electricity. In solid fueled reactors fission occurs within the fuel, which is surrounded by
some kind of cladding, and a coolant is flowed past this cladding to transfer the heat
35
0.00 0.02 0.04 0.06 0.08 0.10Time [s]
100
101
102
103
104
105
106
107
108
Powe
r Am
plitu
de
Thermal = Fast =
Figure 2-11. Comparison of a prompt-critical reactivity insertion in a fast and thermalspectrum core.
away. The rate at which heat is transported away and produced must be balanced. The
selection of materials and fluids for heat transfer play important roles in dictating the size
and operational parameters of a nuclear reactor. As such, the characterization of the heat
transfer and fluid properties are important to understand.
In an MSR the fuel is dissolved into the coolant salt so there is no real distinction
between the fuel and the primary coolant. The fuel salt mixture flows through the core
and to one or more heat exchangers where a secondary fluid transfers the heat produced
from the fuel salt. For the MSR the rate at which the primary fuel salt flows through the
system dictates how much heat is removed from the core. This means the velocity through
the core sets the amount of heat transferred and the power produced in the system.
36
CHAPTER 3SURVEY OF SIMULATION METHODS FOR TRANSIENT ANALYSIS
The study of MSRs poses several challenges compared to conventional solid fueled
reactors. The focus of this work is on the study of time-dependent phenomenon within an
MSR system. Part of the challenge in simulating NSR transients lies in the coupling of the
fluid flow to the neutronic behaviour through the movement of delayed neutron precursors
[13]. Additionally, temperature changes within the core and surrounding materials provide
feedback that changes the neutron production over time. The movement of precursors
in an MSR is a concern as precursors can decay outside of the core thus effectively
reducing the delayed neutron fraction in the core. Since delayed neutrons prevent prompt
critical power excursions it is imperative to carefully study this behavior. The purpose in
surveying the methodologies typically employed for transient analysis is to ascertain what
minimum level of detail is required to test the hypothesis posed in this work.
In every nuclear reactor concept there are a variety of physical phenomena at play, all
of which impact the system on different time scales. Typically, the time scales considered
are broken up as listed below [27].
1. Short: milliseconds to seconds (accident scenarios).
2. Medium: hours to days (build up/decay of important fission products).
3. Long: months to years (build up of fissionable isotopes and long-lived fissionproducts).
Each time scale may require different mathematical techniques or assumptions to simulate.
The focus of this work is on phenomena that occur on the short time scale. The most
desirable computational approach to simulate the physics within nuclear reactors would
solve all of the governing equations simultaneously. However, fully coupled solution of
the neutron transport, fluid dynamics, and heat transfer equations in the full phase space
is a computationally intensive and challenging endeavor. In the early years of nuclear
technology the computational burden was too great to even consider such complex coupled
37
calculations. The lack of computing power motivated the development of equations that
represented the important physics and reduced the dimensionality of the problem such
that the equations are in a tractable form. Specifically, in the study of reactor transients
the so called ‘point kinetics’ equations were developed to calculate the zero-dimensional
(point) evolution of power during accident or simple transient scenarios. The point
kinetics equations can be derived in a consistent manner from the time-dependent form of
the neutron transport equation [29].
In solid fueled reactors, the point kinetics equations were found to be inadequate in
transient scenarios where the spatial distribution of the neutron flux changes significantly
in time. Research to circumvent deficiencies of the point kinetics while maintaining
minimal computational effort led to the development of the quasi-static (QS) and later the
improved quasi-static (IQS) methods for kinetics solutions [30, 31]. Both will be referred
to as quasi-static methods in this discussion. These methods were developed to avoid
fully explicit or implicit time-dependent solves of the full neutron transport or diffusion
equation while still accounting for spatial changes in the neutron flux. In these methods,
the spatial and amplitude evolution of the flux are separated by assuming the spatial
changes to the flux occur slower than the evolution of amplitude of the flux. The spatial
flux equation is a form of the transport equation while the amplitude is determined from
the point kinetics equations. Since the point kinetics equations are easy to solve and
capture the rapid changes in a nuclear system, the point kinetics equations are solved on
a short time scale. Meanwhile, the flux shape is calculated at larger time intervals than
the point kinetics thus saving computational time while still closely following the actual
integral response of the system.
Up to this point the discussion of transient simulations has been restricted to only
consider neutronic behaviour. In a realistic transient, other physics need to be accounted
for in a meaningful way. For instance, increasing the power level in the core will change
the temperature of the fuel; subsequently changes in the temperature of the fuel will
38
affect the neutronic characteristics of the fuel (due to the temperature dependence of
cross sections). Clearly there is a feedback mechanism that needs to be captured. In point
kinetics approaches feedback is incorporated with feedback coefficients, which provide a
way to characterize changes in reactivity based upon a physical change to the system. In
QS approaches, reactivity coefficients can also be used. In a better representation of the
physics in both QS and fully-implicit methods, neutron interaction rates can be adjusted
based on physical changes to the system. This would mean the cross sections present in
the transport equation would be adjusted based on the changes in temperature or density.
Temperature and density changes in a system must be accounted for by heat transfer and
fluid flow equations or an adiabatic approximation of the heat deposition. Solutions of
coupled sets of equations that describe the neutronic, heat transfer, and fluid flow are
often referred to as a multi-physics approach.
To understand the simulation approach developed for MSRs in this work, an overview
of the traditional point kinetics and QS methods will be provided. Then required
modifications to traditional methods to account for fuel flow in an MSR will be discussed.
Finally, with an understanding of the basic approaches for simulating reactor transients,
the methods developed to date for MSR applications will be reviewed and discussed.
3.1 Point Kinetics
The point kinetics equations have been in use for almost 70 years now. The
formulation of these equations can be approached in several ways, from an intuitive
point of view considering an off-critical reactor state and associated production and loss
rates. Alternatively, the point kinetics can be rigorously derived from the diffusion or
transport equation [27].
3.1.1 Overview of the Point Kinetics Equations for Stationary Fuel
For a solid fueled reactor the point kinetics equations are a coupled set of ordinary
differential equations, one governs the power amplitude changes and the remaining
describes the production and losses of each respective precursor group (or family). The
39
equation for the amplitude, which is also referred to in the literature as the power
equation, is provided in Equation 3–1. The equation for the production of a given
precursor group can be found in Equation 3–2.
dN(t)
dt=ρ(t)− β
ΛN(t) +
I∑i=1
F∑f =1
λi ,fCi ,f (t) (3–1)
dCi ,f (t)
dt=βi ,f
ΛN(t)− λi ,fCi ,f (t) (3–2)
For Equations 3–1 and 3–2 the variable definitions are:
N(t) - amplitude,
Ci ,f (t) - total precursor concentration,
ρ(t) - reactivity,
β - total delayed neutron fraction,
Λ - Generation time,
λi ,f - precursor decay rate for isotope i of family f ,
i - precursor isotope,
f - precursor family,
I - total number of precursor isotopes, and
F - total number of precursor families.
The amplitude equation states that amplitude of the initial flux changes occur
primarily due to differences between reactivity and the fraction of delayed neutrons
scaled by the generation time of prompt neutrons. In fast neutron spectrum systems
the generation time is several orders of magnitude lower than for thermal spectrum
systems, meaning any changes in reactivity result in larger and more rapid changes in the
amplitude. The second term in Equation 3–1 states the production of delayed neutrons at
any given time from the decay of precursors will adjust the amplitude of the power as well.
The rate of change of precursors, as described in Equation 3–2, occurs at a rate dependent
on the amplitude minus the losses from previously produced decaying precursors.
40
It is important to note in a solid fueled system these equations are time stable as
the steady state system is defined consistently. This can be observed by assuming no
reactivity is inserted (ρ = 0) and by neglecting the time derivatives in Equations 3–1 and
3–2, which allows these equations can be rearranged as follows:
β
ΛN(t) =
I∑i=1
F∑f =1
λi ,fCi ,f (t) , (3–3)
βi ,f
ΛN(t) = λi ,fCi ,f (t) . (3–4)
Both Equations 3–3 and 3–4 highlight that at steady state there is a balance between the
production and loss terms. Physically, this makes sense as it reinforces that if nothing is
done to perturb a steady state solution then if the system evolves in time then nothing
should change over time. In the case of solid fuel this works out nicely as the precursors
are all born and decay in the core. As discussed, precursors may decay outside of the core
and thus the time stability of Equations 3–1 and 3–2 is questionable and will be addressed
in Section 4.2.2.
3.1.2 Point Kinetics Modification for Molten Salt Reactor Systems
Recently, a review of all kinetics methods developed for MSR applications was
published [32]. This work compares kinetics methods to date and distinguishes between
analysis of thermal and fast spectrum MSR systems. There are several slightly different
approaches for modifying the point kinetics equations for MSR analysis and the
convention to distinguish between each approach will be discussed [32]. The distinctions
are listed as follows:
1. Point Kinetics (PK): This refers to the point kinetics for solid fuel.
2. Delayed Point Kinetics (DPK): The movement of delayed neutron precursors isunderstood through source and sink terms defined by the time spent outside andinside of the core.
3. “I” Point Kinetics (IPK): A fixed mesh is used to calculate the reactivity and fissionpower while a moving mesh is used to track the precursors and the temperature ofthe flowing fluid.
41
4. Modified Point Kinetics (MPK): Point kinetics equations derived starting from thediffusion equation explicitly containing a convective velocity term in the precursorequation.
The DPK was the approach conducted early on for MSRE analysis [33]. In a
companion paper the simulated DPK results were compared to experiment with some
success [34]. The governing equations used are provided in Equations 3–5 and 3–6.
dn
dt=
(ρo − βT
Λ
)n +
(no
Λ
)ρ +
6∑i=1
λici +ρn
Λ(3–5)
ci
dt=βi
Λn − λici −
ci
τC
+ci (t − τL)e(−λiτL)
τC
(3–6)
In Equation 3–6 τC indicates the transit time through the core and τL indicates the transit
time through the external loop. Naturally, in this approach there is a requirement in
knowing the time spent in the external circuit and through the active core. Thus there has
to be some assumption of a velocity through the core. This approach makes it convoluted
to vary the velocity in time due to physical changes in the core. The approach does have
the advantage of having few degrees of freedom and maintaining the point kinetics essence
by not having to explicitly keep track of any spatial quantities.
The approach adopted in Equations 3–5 and 3–6 has been used in the analysis of
the thermal spectrum MSR concept FUJI-12 but only looked at prescribed reactivity
insertions [35, 36]. This approach has been used for analysis of the MSRE in a recently
modified version of RELAP5 [37].
Deriving the MPK for MSR applications was discussed and derived from diffusion
theory [38, 39]. The derivation provides the point kinetics parameters assuming the
precursor concentration can be decoupled into a spatial and time varying component. The
result is a system of equations with a structure like that of the point kinetics but with
different definitions for the parameters within the equations. The details of this approach
are somewhat convoluted and it is not clear how the precursor adjoint is defined nor how
the precursor amplitude function is utilized. Of particular concern is that traditionally the
42
adjoint flux equation used in the weighting of the point kinetics parameters goes to zero
at the reactor boundary. In the case of an MSR the precursors and fuel at the edge of the
active core should not go to zero as precursors and fuel are still present at the core outlet
and have some importance. The conclusions of this study were that the point kinetics
system defined is non-conservative when it comes to predicting the power over time.
However, only test problems were studied with prescribed values and did not consider any
feedback mechanisms.
A review of all kinetics methods suggested that there had been no methods similar
to the MPK that been analyzed on fast spectrum systems [32]. Additionally, most of the
point kinetics-like systems have minimal thermal feedback and have not been used in
conjunction with fast reactor codes for preparing kinetics parameters such as the starting
delayed neutron fraction, decay constants, mean neutron generation time, and reactivity
coefficients.
3.2 Quasi-Static Methods
In quasi-static (QS) methods the objective is to achieve an answer with similar
accuracy to fully solving the time-dependent transport equation with less computational
effort. Since QS methods only rely on the assumption that the spatial shape of the flux
changes much slower than the amplitude changes, one can in principle achieve the same
level of accuracy as the full kinetics methods so long as the assumption holds true during
a given transient. The assumption that the total flux can be broken into the product of
two functions, the amplitude, which provides changes in the magnitude of the flux over
time, and the shape function, which changes on a slow time scale only providing updates
to the spatial change in the neutron flux. This idea of factorizing the flux can be described
by Equation 3–7.
φ(r , Ω, t) = N(t)ψ(r , Ω, t) (3–7)
In 3–7, N(t) represents the amplitude and ψ(x , Ω, t) the flux shape function. Making the
factorization requires a normalization condition as an additional equation is constructed.
43
The normalization holds an integral constraint over the time steps and is tied to the
starting fission source distribution. The entirety of the QS derivation can be found in
many places, here only a few of the steps will be shown to highlight some of the challenges
QS methods have when applied to MSR systems [31]. The first step in traditional QS
methods is to put the factorization, Equation 3–7 into the neutron transport equation.
The time-dependent multi-group transport equation can be compactly described as in
Equation 3–8.
1
vg
d
dtφg(r , Ω, t) +∇ ·Ωφg(r , Ω, t) + Σt,gφg(r , Ω, t) = Sg(r , Ω, t) (3–8)
For Equation 3–8 the source term, Sg(r , Ω, t), contains both prompt and delayed
neutrons. The first step in defining the QS equations is to place the factorization into
Equation 3–8. Then manipulations are made to the the system of equations to eventually
derive an equation for the amplitude and another for the shape. The amplitude or power
equation is the familiar one from the point kinetics and the shape equation is basically the
transport equation with a modified total cross section and source term. The parameters
within the point kinetics equation like the generation time, delayed neutron fraction, and
reactivity are defined as inner products in the QS methodology and weighted with the
steady state adjoint flux [27].
The computational savings in the QS methodology comes from solving the flux shape
over a large time step and the point kinetics equations (and parameter evaluations) many
times between the flux shape updates. The time stepping strategy is represented in Figure
3-1. In Figure 3-1 the ∆t f indicates the largest time step at which the flux shape is found.
A general outline of the QS method as traditionally applied in solid fuel systems
has been provided. The important takeaways from this discussion is requirement of
factorization, requirement of a normalization condition, and the adjoint flux weighting
process to obtain the point kinetics parameters.
44
Δt
Δtk
f
Δtρ
Figure 3-1. Representation of the time scale in a generic QS method.
3.3 Quasi-Static Methods for Molten Salt Reactors
In principal the QS method applied to MSRs would be useful for transient analysis.
The proposed QS method for MSRs factors the precursor concentration into the product
of a spatially dependent and a time-dependent function [40]. The precursor factorization
is questionable as the spatial changes in the precursor concentration are not simple shape
function changes and each group is going to vary on a different time scale dependent on
the respective decay constant. The process for defining the QS method in MSR systems is
very similar to the solid fueled case and is documented elsewhere [40].
A modified QS scheme for MSRs has been derived and implemented in a multigroup
diffusion model with a one-dimensional single channel flow model for the velocity field.
Simple test problems have been analyzed to evaluate the efficiency of this new solution
method [40]. In the implicit QS method employed, recalculation of the shape function
is required to fulfill the normalization constraint and achieve a converged solution.
Recalculation of the flux shape is the most computationally intensive part of the
calculation so the goal of any QS method is to perform as few shape recalculations
over a time step as possible. Otherwise any savings gained by the increased algorithmic
complexity is negated.
Parametric studies looking at the solution quality as a function of the number
of shape recalculations was performed for several MSR transients in this QS MSR
methodology [40]. Transients were simulated and power traces reported showed oscillations
45
and solution quality issues in all cases unless 1000 flux shape recalculations are performed
with 0.1 second time steps [40]. Instead one could directly integrate the diffusion equation
with time steps on the order of 1 ×10−4 with the same computational expense as the QS
results given. The frequent recalculations of the flux shape may indicate the QS method
derived may not be suitable for the transients under study as spatial distortions in the flux
or precursor shape are too large. The transient cases provided indicate poor suitability for
handling MSR transients. It was pointed out in the work that the precursor distortions
are hard to handle because of the difference in decay constants between groups and
recirculation back into the active core [40]. Additionally, it seems problematic to weight
parameters in the point kinetics equations with an adjoint flux, which as traditionally
defined, goes to zero at the boundary of the core.
At this point there is not a compelling case to implement the QS method for MSR
applications as it appears to add additional complexity without achieving computational
savings. It should be pointed out that this conclusion is at odds with a recent review
paper of MSR kinetics transient methodologies, where the review paper suggests a detailed
QS MSR transient code should be developed [32].
46
CHAPTER 4DEVELOPMENT OF A SIMPLE DYNAMICS CODE FOR MOLTEN SALT REACTOR
SAFETY ANALYSIS
To accurately describe the dynamics of a flowing fuel MSR system requires several
modifications to existing approaches developed for solid fuel as highlighted in Chapter
3. To assess the time response to flow perturbations in an MSR a method most closely
resembling the Modified Point Kinetics (MPK) approach is taken. Except in this case the
definitions will be asserted rather than evaluated with modified point kinetics parameter
definitions. A fixed mass flow rate in an assumed single channel is used to set the velocity
field given the cross sectional area and density at every spatial location. In this approach
all spatial quantities are represented throughout the entire domain and the system outlet
is explicitly connected back to the core inlet. To describe the temperature distribution, a
heat equation considering the reactor power as the heat source is employed. The system of
equations containing the modified point kinetics, fluid flow, and temperature distribution
will be spatially discretized using discontinuous finite elements using interpolation
functions of quadratic order. Both explicit and implicit time discretizations are used to
integrate the equations over time. The goal of this section is to derive a set of algebraic
equations governing the dynamic behaviour of an MSR and how these equations are solved
on a computer.
4.1 Prototypical One-Dimensional Molten Salt Reactor Model
To begin an assessment of the MSR dynamics a one-dimensional model with flowing
fuel will be analyzed. The model consists of a single active fuel region where power is
produced. Outside of the core there is a heat exchanger, which pulls heat out of the
system. The pump is placed after the heat exchanger to pump the fuel through the
system. This model is summarized in Figure 4-1. A unique feature with this system is
that the flow of the fuel circulates and thus requires periodic boundary conditions to
connect the out flow to the core inlet. Note, that there is no secondary side explicitly
modeled for current analyses.
47
Active fuel core region
Heat Exchanger
Pump
Flow outFlow in
External piping
External piping
Nodes
Flow circulates, flow out = flow in
Figure 4-1. One-dimensional MSR model with an active core region (fission occurs here),external piping, a heat exchanger, and pump. Note, the flow circulates in thismodel with the flow out becoming the flow in.
4.2 Discontinuous Galerkin Finite Element Method
To solve a set of differential equations for which there is no analytical solution
requires some discretization of the spatial operator and a way to approximate the solution
such that a numerical answer can be obtained on a computer. A variety of spatial
discretization techniques are available such as finite difference, finite volume, continuous
finite element, and discontinuous finite element. Each are chosen depending on the
requirements of the study and the physics in question. In this work the discontinuous
finite element approach is taken to spatially discretize the equations. Specifically,
Galerkin basis functions are employed and the approach is commonly referred to as
the discontinuous Galerkin finite element method (DG-FEM).
The DG-FEM is advantageous because it combines the useful features of the
finite volume and the finite element methods [41]. It allows for high-order spatial
representations and explicit time-integration techniques to be applied, which greatly
aids in developing high-order time approximations. Like in the finite volume approach,
the DG-FEM utilizes a numerical flux to allow for discontinuities between elements,
and employs the local basis function representation to build a global solution like in
the continuous finite element approach. The penalty for the DG-FEMs flexibility is the
48
increase in the total number of degrees of freedom in the problem, since each local element
is decoupled from the rest requiring boundary nodal values to be determined at each
element. From an implementation perspective some of the computational increase in the
DG-FEM can be mitigated due to the sparse nature of the matrix operator compared
to FEM, which becomes especially apparent when high-order spatial approximations are
employed [41].
The goal of the DG-FEM is to approximate a global solution u(x , t) over some
domain Ω with a combination of locally approximated solutions over a discrete domain
of M non-overlapping elements. The local solution of a given element, ue(x , t), can be
expressed as a polynomial of the desired order as shown in Equation 4–1.
ue(x , t) ≈ ~feT
(x) · ~ue(t) (4–1)
In Equation 4–1, ~feT
(x) indicates the local polynomial basis and ~ue(t) indicates the nodal
solution values. In the following derivations quadratic Lagrange approximation functions
will be used resulting in the vector definitions shown in Equation 4–2. Details of the
properties on these interpolation functions can be readily found [42].
ue(x , t) ≈[f 1
e (x), f 2e (x), f 3
e (x)
]·
u1
e (t)
u2e (t)
u3e (t)
(4–2)
The functions in Equation 4–2, assuming the interior node is placed exactly in the center
of the outer nodes, are:
f 1e (x) =
(1− xh
)(1− 2x
h
)(4–3)
f 2e (x) = 4
x
h
(1− xh
)(4–4)
f 3e (x) =
−xh
(1− 2x
h
), (4–5)
49
1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00x
0.0
0.2
0.4
0.6
0.8
1.0
f e(x
)
f1e(x) f2
e(x) f3e(x)
Figure 4-2. Lagrange interpolation functions over an element with a size of 1.0.
where h is the length of the element. The notation f 1e indicates this is a reference to
the first node of a single element, similarly f 2e indicates the second (central) node
of an element and so on. These functions are represented in Figure 4-2. The global
approximate solution can be found as the direct product of the local approximation over
all elements as shown in Equation 4–6 [41].
u(x , t) ≈M⊕
m=1
ue(x , t) . (4–6)
As is typical in finite element analysis the domain will be discretized into a collection of
pre-selected elements and the elemental equations will be derived. As shown in Figure 4-1
four distinct regions are assumed; the active core, external piping, heat exchanger, and
a pump. We begin with derivation of the typical elemental equations for the active fuel
region.
50
4.2.1 Discretization of the Power Amplitude Equation
The power amplitude equation in the classical point kinetics equations, as discussed
in Section 3.1.1, will be modified so that a prescribed spatial dependence of the power will
be introduced. The spatial profile is introduced so the precursor concentration may be
found at any given spatial location as a function of the local power produced and feedback
to the power can be adjusted based on local changes. Note, the spatial profile is fixed over
time. If the power is assumed to have a spatial dependence it will look something like
Equation 4–7.
dP(x , t)
dt=ρ(t)− β
ΛP(x , t) +
I∑i=1
F∑f =1
λi ,fCi ,f (x , t) . (4–7)
Now, the splitting of the spatial profile of the power can be represented as:
P(x , t) = h(x)N(t) , (4–8)
where h(x) is the prescribed spatial function and N(t) is the amplitude. The splitting of
the power in Equation 4–8 is placed in Equation 4–7 as:
dh(x)N(t)
dt=ρ(t)− β
Λh(x)N(t) +
I∑i=1
F∑f =1
λi ,fCi ,f (x , t) . (4–9)
The goal here is to develop a single equation that when solved will yield the amplitude at
any given time. To accomplish that, Equation 4–9 is integrated spatially over the active
core region length designated by Lfuel .
ˆ Lfuel
0
h(x)dN(t)
dtdx =
ˆ Lfuel
0
h(x)ρ(t)− β
ΛN(t)dx+
ˆ Lfuel
0
I∑i=1
F∑f =1
λi ,fCe,i ,f (x , t)dx . (4–10)
Since h(x) will be known it will be possible to evaluate the integral of h(x) in Equation
4–10. Additionally, since the only precursors impacting the power must be in the core, the
integral in Equation 4–10 will be replaced with a summation over the fuel elements in the
remaining equations.
51
For instance, the spatial profile h(x) can be defined as a cosine shape (normalized to
unity) across the active fuel region as in Equation 4–11.
h(x) =
cos
(π2
(x
Lfuel− 1
2
))´ Lfuel
0cos
(π2
(x
Lfuel− 1
2
))dx
. (4–11)
Alternatively, the power profile could be read in from an external solver. To get the total
Figure 4-3. Sample prescribed power profile for a case with 10 nodes and an active fuelregion of 7 nodes.
contribution, the shape function is projected on the solution space and summed over the
active fuel elements.
VLfuel=
ˆ Lfuel
0
h(x) =
Efuel∑e=1
ˆ 1
−1
~feT
(x) · ~h(x)Je(x)dx , (4–12)
where
Je(x) =Ve(x)
2, (4–13)
52
is the Jacobian resulting from the assumed coordinate transformation done here to
integrate from −1 to 1. This transformation has been done to simplify the evaluation of
the approximation functions as will be shown when the implementation details are given.
In Equation 4–13, Ve refers to the volume of the element, which in the 1D case will just be
the length of the element. The 2 in Equation 4–13 comes from the transformation being
imposed on a space with no curvature.
Utilizing Equation 4–12 simplifies the power equation to
dN(t)
dt=ρ(t)− β
ΛN(t) +
1
VL
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCe,i ,f (x , t) . (4–14)
Now we have arrived at a power equation similar to the one typically used in the point
kinetics approach.
4.2.2 Determination of Time Stable Modified Point Kinetics Equations
As discussed in Section 3.1.1 for an MSR system an incongruity arises at steady
state when the fuel is flowing in Equation 4–9. This occurs because the balance of the
precursors and power produced may not be equal as precursors decay out of the core.
To illustrate this point it is helpful to show the slightly rearranged steady state modified
point kinetics as done in Equations 4–15 and 4–16.
β
ΛN(t) =
1
VL
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCe,i ,f (x , t) . (4–15)
βi ,f
Λh(x)N(t) = λi ,fCi ,f (x , t) + u(x , t)
∂Ci ,f (x , t)
∂x(4–16)
The basic problem with determining a time stable solution within the point kinetics
equations is to realize the power equation only considers contributions from the precursors
within the active core while the precursor equations considers precursors throughout the
entire system as observed in the convective term in Equation 4–16. As the flow speed
changes in the core the precursors are redistributed.
53
To correct the incongruity a modification is made to the total β term in the power
equation. Essentially, a βflow term is computed which accounts for the loss in precursors
out of the core for the starting steady state mass flow rate. The loss correction term is
calculated as follows:
DTi ,f (0) =
Efuel∑e=1
λi ,fCi ,f (x , 0) , (4–17)
βflowi ,f = Λ
DTi ,f (0)
PT (0), (4–18)
βflow =
I∑i=1
F∑f =1
βflowi ,f , (4–19)
noting that PT (0) is the total power produced in the system and DT (0) is the delayed
source coming from the production of delayed neutrons due to the distribution of
precursors within the core. The final power equation including the βflow term is provided
in Equation 4–20.
dN(t)
dt=ρ(t)− βflow
ΛN(t) +
1
VL
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCe,i ,f (x , t) (4–20)
4.2.3 Discretization of the Precursor Equation
Next we will examine the precursor concentration equation with a fluid flow term
and develop a matrix-vector system using the discontinuous Galerkin method for spatial
discretization. To begin, the precursor equation is shown again in Equation 4–21.
dCi ,f (x , t)
dt=βi ,f
Λh(x)N(t)− λi ,fCi ,f (x , t)− u(x , t)
∂Ci ,f (x , t)
∂x(4–21)
The total problem domain, Ω = (0,L) is divided into E elements, where a typical element
is denoted as Ωe = (xm, xm+1). For a typical element xm and xm+1 indicate the boundary
of a node and are in terms of a global coordinate.
To simplify notation the following derivation will only consider a single precursor
family and isotope. To begin, consider a single element and multiply by a weight function,
54
denoted as ~w(x), then integrate over the element, which yields:
ˆ xm+1
xm
~w(x)
[dCe(x , t)
dt+ u(x , t)
∂Ce(x , t)
∂x+ λCe(x , t)− β
Λh(x)N(t)
]dx = 0 . (4–22)
Now to distribute the derivative to the approximation space integration-by-parts is applied
to the second integral in Equation 4–22, which has the added benefit of producing a
boundary term as well. At this time the weight function is chosen to be the same as the
approximation function as is the convention of the Galerkin method.
ˆ xm+1
xm
[~fe(x)
dCe(x , t)
dt− ~fe(x)ue(x , t)
dCe(x , t)
dx+ λ~fe(x)Ce(x , t)− β
Λ~fe(x)h(x)N(t)
]dx+[
ue(x , t)~fe(x)Ce(x , t)
]xm+1
xm
= 0 (4–23)
Putting in the finite element approximations for all the dependent variables as
Ce(x , t) ≈ ~f Te (x) · ~ce(t) , (4–24)
yields the following expression
ˆ xm+1
xm
[~fe(x)~f T
e (x) · d~ce(t)
dt− ~f T
e (x) · ~u(t)d~fe(x)
dx~f T
e (x) · ~ce(t)+
λ~fe(x)~f Te (x) · ~ce(t)− β
Λ~fe(x)h(x)N(t)
]dx+[
~ue(x , t)~fe(x)~feT
(x) · ~ce(t)
]xm+1
xm
= 0 . (4–25)
Additionally, since there is allowance for discontinuities at the boundary the flux boundary
term must be evaluated. In this case an upwind technique will be used, meaning the flux
will be determined using information from the previous ‘upwind’ element. In Equation
4–26 we will examine just the boundary term noting L and R have been used to indicate
the left and right hand side of an element, respectively.[~ue(x , t)~fe(xR)~f T
e (xR) · ~ce(t)
]−[~ue−1(x , t)~fe(xL)~f T
e (xL) · ~ce−1,R(t)
](4–26)
55
Note, ue−1,R(t) and ce−1,R(t) indicate the primary variable value at the right hand side of
the previous element. To simplify the previous equations the following matrix definitions
are introduced.
~~Ae =
ˆ xm+1
xm
~fe(x)~feT
(x)dx (4–27)
~~Ue =
ˆ xm+1
xm
ue(x , t)d~fe(x)
dx· ~f T
e (x)dx (4–28)
~qe =
ˆ xm+1
xm
~fe(x)he(x)N(t)dx (4–29)
~~We,R = ue(x , t)~fe(xR)~f Te (xR) = ue(x , t)
0 0 0
0 0 0
0 0 1
(4–30)
~~We,L = ue(x , t)~fe(xL)~f Te (xL) = ue(x , t)
1 0 0
0 0 0
0 0 0
(4–31)
Utilizing the matrix definitions from Equations 4–27 - 4–31 results in Equation 4–32,
which provides the matrix-vector system for a typical element in the domain. At
this point the time derivatives are still included as no time discretization has been
implemented.[~~Ae ·d~ce(t)
dt− ~~Ue ·~ce(t) + λ
~~Ae ·~ce(t)− β
Λ~qe
]+
[~~We,R ·~ce(t)
]−[~~We,L~ce−1(t)
]= 0 (4–32)
4.2.4 Discretization of the Heat Equation
To obtain a temperature profile throughout the domain a modified heat equation is
used. Considering the relative similarity in temperatures and time scales of interest, the
heat conduction is neglected in the fuel salt. The equation describing the temperature due
to power increases and transfer of heat due to movement of the fluid at a given mesh point
56
in time can be described as follows:
ρ(T )Cp(t)dT (x , t)
dt=P(x , t)
V (x)− ρ(T )Cp(T )u(x , t)
dT (x , t)
dx, (4–33)
with variable definitions:
ρ(T ) - density as a function of temperature,
Cp(T ) - heat capacity as a function of temperature,
T (x , t) - spatially- and time-dependent temperature,
V (x) - volume over a given element,
u(x , t) - velocity.
The material properties are assumed to be constant over the element to simplify
the derivation. It is a reasonable assumption as most of the properties are dependent on
temperature, which is changing rather smoothly across sufficiently small elements.
Now to discretize with the discontinuous finite element method, a single element is
examined, and the approximation function is employed throughout Equation 4–33.
~fe(x)dTe(x , t)
dt= ~fe(x)
h(x)P(t)
ρe(Te)Cp,e(Te)V (x)− u(x , t)~fe(x)
dTe(x , t)
dx(4–34)
Utilizing the finite element approximation for temperature as in Equation 4–35.
Te(x , t) ≈ ~f Te (x)~Te(t) (4–35)
Next, placing the approximation in Equation 4–35, and dropping the explicit material
properties variation with temperature results in Equation 4–36.
~fe(x)~f Te (x)
d ~Te(t)
dt= ~fe(x)
h(x)P(t)
ρeCp,eV (x)− u(x , t)~fe(x)
d~f Te (x)
dx~Te(t) (4–36)
Now Equation 4–36 is integrated over the element.
ˆ xm+1
xm
[~fe(x)~f T
e (x)d ~Te(t)
dt− ~fe(x)
h(x)P(t)
ρeCp,eV (x)+ ~ue(t)~fe(x)
d~f Te (x)
dx~Te(t)
]dx = 0 (4–37)
57
Integration by parts is employed such that the derivative is operating on the approximation
space.
ˆ xm+1
xm
[~fe(x)~f T
e (x)d ~Te(t)
dt− ~fe(x)
h(x)P(t)
ρeCp,eV (x)− ~ue(t)~f T
e (x)d~fe(x)
dx~Te(t)
]dx+[
~ue(t)~fe(x)~f Te (x)~Te(t)
]xm+1
xm
= 0 (4–38)
Equation 4–38 can be further simplified if the matrix definitions from Section 4.2.3 are
introduced.
~~Ae
d ~Te(t)
dt= ~qe
1
ρeCp,eVe(x)+~~Ue~Te(t)− ~~We,R
~Te +~~We,L
~Te−1(t) (4–39)
Equation 4–39 can be manipulated to be solved for the temperature across a particular
element in either steady state or over time.
4.2.5 Velocity Field
To solve all of the previously defined matrix-vector equations requires knowing
the velocity throughout the domain. Given a constant mass flow rate with assumed
incompressible fluid we can arrive at a relationship to evaluate the velocity rather easily.
The mass flow rate through a single channel is defined as follows
me = ue(x , t)ae(x)ρe(Te), (4–40)
where ae(x) is the cross sectional area that depends on the spatial location within the
domain. This allows the computation of the velocity anywhere in the domain with
ue(x , t) =me
ae(x)ρe(Te), (4–41)
where ρe(Te) is functionalized and can be evaluated based on the temperature at that
node.
58
4.3 Coupling Approach
A partitioned approach is taken to simulate the multiphysics system where each
subsystem is solved sequentially at every time step. The ordering of the solving steps
comes from considering the dominant physics at play, and the solution strategy is first to
solve for the power amplitude, then determine the temperature and velocity based on the
calculated power level, and finally solve for the precursor distribution. Iteration through
these steps may be required over a given time step if an implicit time integration strategy
is employed.
4.4 Steady State System
Before a transient calculation begins, a suitable steady state solution must be
obtained. The steady state system of equations will be developed for the precursor and
temperature equations. The steady state power solution is prescribed initially, so the
power equation is not solved explicitly in the steady state solution scheme. A complete
overview of the algorithmic approach is provided in Appendix A.
To understand how the precursor distribution is found for steady state, the time
derivative from Equation 4–32 is removed and the terms are rearranged:[− ~~Ue + λ
~~Ae +~~We,R
]· ~ce =
β
Λ~qe + ~we−1,L . (4–42)
In Equation 4–42 the right hand upwind element term has been simplified by introducing
the following:
~we−1,L = ~We,L · ~ce−1 . (4–43)
To solve for ~ce it is helpful to define the left-hand side matrices as:
~~Ge = [− ~~Ue + λ~~Ae +
~~We,R ] , (4–44)
which then allows ~ce to be determined with Equation 4–45.
~ce =~~G−1
e ~qe +~~G−1
e ~we−1,L (4–45)
59
Note, solving Equation 4–45 is valid in the active fuel region where there is a source, the
power, which enables the production of precursors. While the precursors travel outside of
the active fuel region where there is no source they decay away exponentially, as shown in
Equation 4–46.
~ce =~~G−1
e ~we,L (4–46)
In a similar procedure the matrix-vector system shown in Equation 4–39 describing the
temperature can be manipulated to yield a simple expression.
~~Ae
d ~Te(t)
dt= ~qe
1
ρeCp,eVe(x)+~~Ue~Te(t)− ~~We,R
~Te +~~We,L
~Te−1(t) (4–47)
Neglecting the time derivative in Equation 4–47 results in
− ~~Ue~Te +
~~We,R~Te −
~~We,L~Te−1 = ~qe
1
ρeCp,eVe(x). (4–48)
After rearranging Equation 4–48:
[ ~~We,R −~~Ue
]· ~Te = ~qe
1
ρeCp,eVe(x)+~~We,L
~Te−1 . (4–49)
The matrix on the left side of Equation 4–49 is inverted and thus allows ~Te to be readily
found as shown in Equation 4–50.
~Te =[ ~~We,R −
~~Ue
]−1 ·[~qe
1
ρeCp,eVe(x)+~~We,L~Te−1
](4–50)
To close the system, boundary conditions must be imposed. The imposition of boundary
conditions is achieved by performing a spatial sweep through the system (i.e., performing
operations sequentially on small parts of the solution vector and mass matrix) and
explicitly connecting the beginning to the final element such that the nodal value of the
last element is added to the starting node. This is accomplished in an iterative fashion,
starting initially with an upwind flux of zero, as at the beginning of a transient we will
assume no precursors have been created in the system. A sweep, in the direction of fluid
flow, is performed spatially to calculate the solution within an element. After sweeping
60
through the entire domain and arriving back at the beginning the amount added to the
beginning element is determined. Once the contribution from the edge of the system
has gotten sufficiently small, the steady state solution is assumed to have converged.
Convergence is assessed by monitoring the difference in the L2 norm between the current
solution and the previous. For clarity the L2 norm is calculated as:
||~c ||L2 =
√√√√ E∑e=1
c2e (x , t) . (4–51)
The convergence criteria is explicitly determined by the difference in successive L2 norms
with
εconv = ||~c ||jL2 − ||~c ||j−1L2 , (4–52)
where εconv is the tolerance on the convergence and j is the nonlinear iteration counter.
4.5 Time-Dependent System
To form a complete set of algebraic equations to be solved on a computer the time
derivative in all the matrix-vector equations must be discretized. Throughout these
sections j will be used as a nonlinear iteration counter, and k will indicate the current
time step of a given solve. The solution algorithm can be found in detail in Appendix B.
4.5.1 Explicit Euler
As a first step in developing a transient analysis tool a simple forward Euler (explicit)
approach will be implemented to evolve the system over time. The forward Euler method
requires the approximation of the time derivative to use information from the previous
time step to dictate the evolution over the step. This method is perhaps the simplest
to implement but requires small time steps to avoid divergence of the solution. In the
explicit scheme no iterations are performed over the time step. This approximation of the
derivative, for the precursor concentration, is given in Equation 4–53.
d~ce(t)
dt=
~cek − ~ce
k−1
∆t(4–53)
61
In Equation 4–53, k refers to the current time step, so k − 1 indicates the solution at the
previous time step, and ∆t is the time interval. To begin the time discretization, the final
matrix-vector precursor equation from Section 4.2.3 is shown again in Equation 4–54.[~~Ae ·d~ce(t)
dt− ~~Ue ·~ce(t) + λ
~~Ae ·~ce(t)− β
Λ~qe
]+
[~~We,R ·~ce(t)
]−[~~We,L~ce−1(t)
]= 0 (4–54)
Rearranging Equation 4–54 and implementing the forward Euler approximation allows the
solution of the system at a given time step, ~cek , as:
~cek
= ~cek−1
+ ∆t
[~~A−1
e~~Ue − λ
~~I − ~~A−1e~~We,R
]· ~ce
k−1+ ∆t
β
Λ
~~A−1e ~qe + ∆t
~~A−1e~~We,L · ~ck−1
e−1 (4–55)
To simplify, the following matrix definition is introduced in Equation 4–56.
~~He =~~A−1
e~~Ue − λ
~~I − ~~A−1e~~We,R (4–56)
Now Equation 4–55 can be manipulated to solve for be ~cek .
~cek
= ~cek−1
+ ∆t~~He · ~ce
k−1+ ∆t
β
Λ
~~A−1e ~qe + ∆t
~~A−1e~~We,L · ~ck−1
e−1 (4–57)
A similar procedure to what was employed for the time-dependent precursor equation
is done to the temperature equation.
d ~Te(t)
dt=
~Te
k− ~Te
k−1
∆t(4–58)
The matrix-vector system for the temperature equation is shown in Equation 4–59.
~~Ae
d ~Te(t)
dt= ~qe
1
ρeCp,eVe(x)+~~Ue~Te(t)− ~~We,R
~Te +~~We,L
~Te−1(t) . (4–59)
Taking Equation 4–58 and substituting it into Equation 4–59 yields the following:
~~Ae
[~Te
k− ~Te
k−1]= ∆t
[ ~~Ue −~~We,R
]~T k−1
e (t) + ∆t[~qe
1
ρeCp,eVe(x)+~~We,L
~T k−1e−1 (t)
]. (4–60)
62
Inverting the left-hand matrix and arranging other terms in 4–60 yields a final expression
to get the elemental temperature values.
~Te
k(t) = ~Te
k−1(t) + ∆t
~~A−1e
[ ~~Ue −~~We,R
]~T k−1
e (t) + ∆t~~A−1
e
[~qe
1
ρeCp,eVe(x)+~~We,L
~T k−1e−1 (t)
](4–61)
In the case of the power equation, the solution for the explicit Euler case is almost
trivially simple to set up. Utilizing the same forward Euler approximation shown in
Equation 4–62
dN(t)
dt=Nk − Nk−1
∆t(4–62)
The starting modified power equation is given as:
dN(t)
dt=ρ(t)− ρf (t)− βflow
ΛN(t) +
1
VL
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCe,i ,f (x , t) , (4–63)
which has a new term, ρf (t), added to account for the possibility of reactivity feedback
over time.
The time derivative approximation in Equation 4–62 is substituted into Equation
4–63 resulting in the following equation:
Nk = Nk−1 + ∆tρk−1 − ρf (t)− βflow
ΛNk−1 + ∆t
1
VT
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCk−1m,i ,f (x) . (4–64)
4.5.2 Implicit Euler
The backward Euler scheme is implicit in time and first-order accurate. The
approximation of the derivative is similar to forward Euler except the state of the
current solution depends on itself. The implicitness requires iteration to resolve the
nonlinearity introduced in this time discretization. In a similar fashion to the forward
Euler, the semi-discretized form, shown again in Equation 4–65 has all terms except the
time derivative moved to the right-hand side and now depends on the current state of the
63
solution vector.
~~Ae ·d~ce(t)
dt=
[~~Ue · ~ce(t)− λ ~~Ae · ~ce(t) +
β
Λ~qe
]−[~~We,R · ~ce(t)
]+
[~~We,L~ce−1(t)
](4–65)
To better understand how this equation is solved numerically, the l index is introduced,
which represents the nonlinear iteration counter
~cek,l+1
= ~cek−1
+ ∆t
[~~A−1
e~~Ue −λ
~~I − ~~A−1e~~We,R
]· ~ce
k,l+ ∆t
β
Λ
~~A−1e ~qe + ∆t
~~A−1e~~We,L ·~ck,l
e−1 (4–66)
For the temperature equation the time discretization procedure follows almost identically
to the treatment of the precursor equations and starts with Equation 4–67.
d ~Te(t)
dt=
~Te
k− ~Te
k−1
∆t(4–67)
The starting temperature equation is provided in Equation 4–68.
~~Ae
d ~Te(t)
dt= ~qe
1
ρeCp,eVe(x)+~~Ue~Te(t)− ~~We,R
~Te +~~We,L
~Te−1(t) (4–68)
Substituting the temporal approximation into Equation 4–68 yields:
~~Ae
[~Te
k− ~Te
k−1]= ∆t
[ ~~Ue −~~We,R
]~T k−1
e (t) + ∆t
[~qe
1
ρeCp,eVe(x)+~~We,L
~T k−1e−1 (t)
]. (4–69)
For the final equation, nonlinear indices are included for further clarification in Equation
4–70.
~Te
k,l(t) = ~Te
k−1(t) + ∆t
~~A−1e
[ ~~Ue −~~We,R
]~T k,l
e (t) + ∆t~~A−1
e
[~qe
1
ρeCp,eVe(x)+~~We,L
~T k,le−1(t)
](4–70)
Similarly, the power equation can be manipulated to yield the power amplitude at any
time as:
Nk,l+1 = Nk−1 + ∆tρk−1 − ρk−1
f − βflow
ΛNk,l + ∆t
1
VT
Efuel∑e=1
I∑i=1
F∑f =1
λi ,fCk,lm,i ,f (x) . (4–71)
64
4.5.3 Reactivity Feedback
In point kinetics schemes, changes in the system over time are typically incorporated
by increasing or decreasing the reactivity in the system dependent on some physical
change (e.g. temperature). The amount of reactivity introduced varies based on the
reactivity coefficients. These reactivity coefficients are typically calculated by some kind of
perturbation theory code. These coefficients relate a given physical change in the system
to a corresponding change in reactivity. Most of the feedback mechanisms are due to
changes in the temperature of the fuel. The changes to the fuel temperature can lead
to changes in the density of the fuel, which is a dominant factor in MSRs. For the fast
spectrum MSR systems of interest, two reactivity coefficients will be determined, one to
account for Doppler broadening of the cross sections as the temperature changes, and the
other will account for density changes of the fuel.
The Doppler broadening reactivity feedback will be calculated using the following
equation:
ρDoppler =
Efuel∑e=1
γD(x)
(Te(x , t)− T o
e (x)
), (4–72)
where γD(x) is the Doppler reactivity coefficient in units of pcm/K. Temperature increases
in the core lead to an effect known as Doppler broadening, which refers to the broadening
of the resonances of the cross sections. The broadening of these resonances causes the
absorption of neutrons to occur with a greater probability and thus the increased
absorption of neutrons reduces the chain reaction in the core. Doppler broadening is a
dominant shutdown effect in thermal spectrum reactors but is significantly less of a factor
in reactors operating with a fast neutron spectrum.
Equation 4–72 gives the reactivity introduced based on the change in temperature
from the initial steady state value. In this formulation the reactivity coefficient has a
spatial dependence, so the temperature difference at each element is considered, and the
net change in reactivity is the summation across the active fuel region. To account for
65
reactivity changes based on the density the following equation is used:
ρdensity =
Efuel∑e=1
γdensity (x)
(ρe(x , t)− ρo
e (x)
), (4–73)
where γdensity (x) is the density reactivity coefficient in units of pcm/gcm3. The details of
how the reactivity coefficients are found are explained in Section 6.2.
4.6 Computer Implementation
To solve the system of equations developed in the previous sections a Fortran
program was developed. This program reads input files created by the user, sets up the
elemental matrices, assembles the matrices, and solves for the nodal variables. It follows a
prototypical computational approach to solving systems of differential equations developed
with a finite element method [43]. Algorithmic overviews of the solution methods are
given in Appendices A and B for the steady state and transient cases respectively. The
code developed is open source and freely available on GitHub 1 . The code may be cloned
and built only requiring a modern Fortran compiler with dependencies on the LAPACK
and BLAS libraries. Documentation on building and running the code is found online as
well 2 .
1 https://github.com/ZanderUF/MSR 1D
2 https://msr-1d.readthedocs.io/en/latest/
66
CHAPTER 5VERIFICATION OF THE TRANSIENT SOLUTION METHOD
The one-dimensional system presented in Section 4 should reduce to the typical point
kinetics if there is no fuel flow. Since there are many published results on the solutions
to the (solid fuel) point kinetics equations it serves as a preliminary verification that
the solver developed is working as intended. In the following section several transients
are simulated in order to verify the point kinetics like solver developed in this work is
satisfactory. Both backwards and forward Euler time discretizations are used in the
following analysis.
5.1 Step Perturbations
Part of the effort to verify the behaviour of the point kinetics solver implemented is
to examine the system response to idealized step perturbations. Step perturbations are
initiated when reactivity at a predetermined time is added to the system instantaneously.
These perturbations can be insertions or reductions in reactivity. Of course the step
perturbation is an idealization, as in reality every change in the system requires some time
to propagate. To verify the point kinetics solver is satisfactory the fuel in the system will
be assumed stationary so comparisons may be made to published results.
5.1.1 Physics Based Verification
In this first example a step perturbation will be introduced to the system, held at
that reactivity level momentarily, then the reactivity will be reduced back to zero. This
should produce a power profile that rapidly increases when the perturbation is introduced,
continue to increase when reactivity is inserted, and asymptotically return back to the
starting power when the reactivity is removed. In this problem the reactivity inserted is
9.6 × 10−4 pcm and the neutron generation time is 3.85 × 10−7 seconds. The delayed
neutron precursor parameters for this simulation are summarized in Table 5-1. The
reactivity was inserted from 0.0 to 0.2 seconds and then removed. The simulation was
conducted for 100 seconds with time steps of 1.0 × 10−5 seconds. The forward Euler
67
Table 5-1. Decay constant (λ) and delayed neutron fraction (β) values per delayed family(i) for the point kinetics physics based verification problem.
Group 1 2 3 4 5 6λi [s−1] 0.0129 0.0311 0.134 0.331 1.26 3.21βi 8.1×10−5 6.87×10−4 6.12×10−4 1.138×10−3 5.12×10−4 1.7×10−4
time discretization was employed. The normalized power profile is given in Figure 5-1
and shows the rapid rise in power followed by an asymptotic decrease back down to the
original power level.
Figure 5-1. The first 10 seconds of a simulation are shown where a step perturbation isintroduced and maintained for 0.2 seconds.
5.1.2 Step Perturbation Verification
To further verify the solver, results obtained are compared to several transients
from the literature [44]. The method used in the reference calculations was an A-stable
generalized Runge-Kutta algorithm for solving the point kinetics equations [44]. The point
kinetics parameters for this problem are given in Table 5-2. The relative power computed
with the code developed in this work is compared at several times with values computed
in Reference [44]. Several simulations were conducted with step reactivity insertions
68
Table 5-2. Point kinetics parameters for the step perturbations from problems in theliterature [45].
Group 1 2 3 4 5 6λi [s−1] 0.0127 0.0317 0.115 0.311 1.4 3.87βi 2.66×10−4 1.491×10−3 1.316×10−3 2.849×10−3 8.96×10−4 1.82×10−4
occurring at time zero. The normalized power values computed are compared at several
time steps to published results in Table 5-3 [44]. All time steps were 1.0 × 10−5 seconds
except the case where the reactivity was equal to 0.008 pcm and constant time steps of
1.0×10−6 seconds were required. The large reactivity insertion causes a large perturbation
necessitating smaller time steps to capture the rapid increase in power. Clearly, examining
the results in Table 5-3 indicates excellent agreement with published values.
Table 5-3. Comparison of calculated amplitude with forward Euler time discretization(FETD) and backward Euler time discretization (BETD) for several differentstep perturbations.
ρ [pcm] Time [s] P(t) [44] FETD P(t) BETD P(t)0.0030 1.0 2.20985 2.20985 2.20984
10.0 8.01891 8.01925 8.0191920.0 28.2948 28.2977 28.2974
0.0055 0.1 5.21 5.21012 5.210022.0 43.022 43.02664 43.02499
10.0 1.388×105 1.38877×105 1.38859×105
0.0070 0.01 4.50885 4.50886 4.508860.5 5.3445×103 5.34589×103 5.34588×103
2.0 2.05697×1011 2.05919×1011 2.05914×1011
0.0080 0.01 6.20276 6.20269 6.203020.1 1.4101×103 1.41023×103 1.41060×103
1.0 6.1486×1023 6.1556×1023 6.17108×103
5.1.3 Zig-zag Perturbation
Another common perturbation to test in the point kinetics method, which has an
intuitive physical meaning, is the ramp perturbation. In this case the reactivity is linearly
“ramped” up to a prescribed value, which is akin to a control rod withdrawal in a reactor.
69
To verify the ability to perform ramp transients a series of them are constructed to form
a “zig-zag” reactivity pattern described in Table 5-4. In this case the generation time is
5.0 × 10−3 seconds. The results of the zig-zag intersection are given in Table 5-5 and
Table 5-4. Point kinetics parameters for the zig-zag perturbations [45].Group 1 2 3 4 5 6λi [s−1] 0.0127 0.0317 0.115 0.311 1.4 3.87βi 2.85×10−4 1.5975×10−3 1.41×10−3 3.0525×10−3 9.6×10−4 1.95×10−4
shows excellent agreement with previously published results. For this simulation constant
time steps of 1.0 × 10−4 seconds were used. The reactivity (left) and power (right) as a
function of time are provided in Figure 5-2.
Table 5-5. The zig-zag perturbation is described in detail and calculated amplitude valuesare compared with the literature at several time steps.
Time range [s] 0 ≤ t ≤ 0.5 0.5 ≤ t ≤ 1.0 1.0 ≤ t ≤ 1.5 1.5 ≤ t 1.5 ≤ tρ [pcm] slope 7.5×103/s - 7.5×103/s 7.5×103/s 0 0Time [s] 0.5 1.0 1.5 2.0 10.0P(t) [44] 1.72137 1.21109 1.89217 2.52162 12.0465FETD P(t) 1.72168 1.2100 1.89251 2.52174 12.0484BETD P(t) 1.72144 1.21112 1.89225 2.52153 12.0462
0 2 4 6 8 10Time [s]
0.00000
0.00050
0.00100
0.00150
0.00200
0.00250
0.00300
0.00350
Reac
tivity
Figure 5-2. Variation in reactivity for the zig-zag test problem. The reactivity as afunction of time is given on the left and the normalized power amplitude isgiven on the right.
70
5.2 Power Stabilization at New Flow Speed
To ascertain the validity of the point kinetics model with flowing fuel a simple test
is carried out to verify the implementation is physically consistent. In this test the flow
speed is decreased exponentially from one mass flow rate to another with no fuel density
or Doppler feedback. For the starting and ending mass flow rates achieved during this
simulation the delayed neutron fraction (β) for each flow rate is calculated at steady
state prior to running a transient case. The effective difference in the two β values should
cause an insertion of reactivity into the system as the precursor distribution evolves over
time. Throughout this discussion the mass flow rates will be referred to as mA (starting),
and mB (final). The power profile within the core is assumed flat. Additionally, the
velocity throughout the core is constant. The questions this test seeks to answer can be
summarized as:
1. Knowing the precursor distribution and β loss between two flow rates, do weintroduce reactivity by going from one flow rate to another?
2. Can we then stabilize the system by subtracting precisely the amount of reactivitywe know should have been inserted due to the differences in the steady stateprecursor distributions?
For this test the mA is set as 150 kg/s and mB is 125 kg/s with transit times across the
core of 11.6 and 15.4 seconds, respectively. The core length is 0.35 m with a constant
cross sectional area of 0.05 m3. The mean neutron generation time is set at 1 × 10−6
seconds, which is on the order of a typical fast reactor. The ratio of the core size to the
core velocity was chosen to match the ratio found in a realistic core design. The β values
at each mass flow rate and delayed precursor parameters can be found in Table 5-6. Note,
the delayed precursor decay constants were made artificially small so the precursors would
quickly decay and the reactivity insertion would be observed within several seconds. For
the transient simulation the mass flow rate was decreased by 25% over 0.5 seconds. While
this adjustment in the mass flow rate may not be entirely realistic, it was done so the
precursors would quickly transition to their new steady state distribution and reduce the
71
Table 5-6. Delayed precursor parameters for flow transition simulation verification.
Group βnoflow [pcm] βmA[pcm] βmB
[pcm] λ [s−1] ∆β = ∆ρ [pcm]1 500.00 469.56 476.14 0.52 200.00 196.72 197.54 2.0Total 700.00 666.28 673.68 7.40
simulation time needed. The reactivity over time inserted in the system is given in Figure
5-3 where a step change in reactivity equal to -7.40 pcm is introduced at ten seconds and
remains inserted for the remainder of the simulation. The power amplitude is displayed
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time [s]
-0.00007
-0.00006
-0.00005
-0.00004
-0.00003
-0.00002
-0.00001
0.00000
Reac
tivity
Figure 5-3. Reactivity inserted in the system as a function of time.
in Figure 5-4 and clearly shows the increase in amplitude after the mass flow rate is
reduced. However, as the reactivity is compensated for by the step change at ten seconds
the power is promptly reduced and reaches a new steady state level. This result appears
to answer the question that indeed reducing the mass flow rate can introduce a positive
reactivity change. Secondly, this result shows that the simulation is consistent in that it
is possible to transition between mass flow rates and reach a new steady state power level
given the reactivity is compensated for in some way.
72
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time [s]
1.00
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
Powe
r Am
plitu
de
Figure 5-4. Amplitude change over time for the flow transition test problem.
5.3 MSRE Comparison
The Molten Salt Reactor Experiment (MSRE) is the primary source for experimental
data for flowing-fuel systems [46–50]. The experiments were conducted over 50 years
ago and much of the reported data has significant uncertainties. Even so the data from
the MSRE provides the only available experiment to form a basis of comparison to
any simulated results. Of primary interest in this work is the effect delayed neutrons
have on reactivity. Considering this the first comparison to MSRE data will look at the
experimentally found loss in the total delayed neutron fraction to simulations conducted
with the code developed in this work. It is interesting to point out that early theoretical
work conducted at ORNL to determine the delayed neutron fraction was in serious error
compared to experiment and the calculation method was later revised [46, 48].
The effect of fuel circulation on reactivity was experimentally determined by
measuring reactivity differences of the MSRE between circulating flow at the nominal
flow rate and the fuel not circulating [46]. With this experimental method the individual
contributions from precursor groups are not considered. Instead only the total change in
the delayed neutron fraction is inferred from experiment.
73
Table 5-7. Summary of MSRE experimental values as reported.Parameter ValuePower [MW] 8.2Tin [K] 908.0Tout [K] 935.8m [kg/s] 75.71Height [cm] 175Radius [cm] 70.45Heat Capacity [J/kg/K] 1740ρfuel [kg/m3] 2306Time in core [s] 9.37Time out of core [s] 16.45Λ [s] 2.29 × 10−4
Calculation of the steady state precursor distribution is made using the parameters
listed in Tables 5-7 and 5-8. The normalized steady state precursor distribution
considering fuel movement is shown in Figure 5-5.
Table 5-8. Summary of kinetics data used in MSRE theoretical calculations [48].Group 1 2 3 4 5 6λi [s−1] 0.0124 0.0305 0.2153 0.3014 1.1363 3.0137βi 2.11×10−4 1.40×10−3 1.25×10−3 2.53× 10−3 7.40×10−4 2.70×10−4
The loss in the delayed neutrons due to the fuel circulation is determined and
compared to the total loss experimentally derived and other reported calculations in
Table 5-9. In general, the simulated results overestimated the total delayed neutron loss
compared to experiment. Many of the simulated results came from a benchmark exercised
conducted in [51]. Decay data for simulated results came from reported ORNL data, as
given in Table 5-8. The benchmark document did not contain much information on how
the simulated results were determined. Therefore it is difficult to say what the source
of differences are between results. Using the transient analysis code developed in this
work (MSR1D) yielded a prediction of the total delayed neutron fraction about 9% higher
than the experimentally reported value. The over prediction lines up with other reported
simulated values. Overall MSR1D appears to be calculating the precursor distribution due
to fuel flow reasonably well. Again it is difficult to assess with certainty due to the many
74
0 100 200 300 400 500 600Axial Height [cm]
0.0
0.2
0.4
0.6
0.8
1.0
Norm
alize
d Pr
ecur
sor C
once
ntra
tion
Inlet Main Core Outlet
Group 1Group 2
Group 3Group 4
Group 5Group 6
Figure 5-5. Normalized precursor distribution of each group for the steady state condition.
Table 5-9. Comparison of calculated loss in delayed neutron fraction (in units of pcm)between experiment and simulated results.Reference Total 1 2 3 4 5 6MSRE (experimental) 212.0ORNL [48] 301.0ORNL [46] 222.0EDF [52] 228.8 12.0 78.0 62.3 73.7 2.8 0.0ENEA [53] 259.2 14.0 90.5 71.1 80.4 3.2 0.1FZK (a) [54] 262.2 14.1 90.8 70.9 81.9 4.1 0.3FZK (b) [54] 212.2 12.6 77.5 52.8 62.4 5.6 0.9FZR [51] 253.2 13.8 89.2 68.4 77.8 3.9 0.1POLITO [55] 278.0 16.0 100.7 74.1 82.8 4.0 0.2[56] 231.5 13.4 57.1 61.5 66.9 2.5 0.06MSR1D 234.2 13.3 87.1 50.0 77.4 5.7 0.7
differences in modelling approaches. The most beneficial thing to do would have model
preparation done with a standardized approach.
75
CHAPTER 6MOLTEN CHLORIDE SALT REACTOR DESIGN
The initial goal of this work was to assess the time response of an MCFR to
determine if it would be necessary to develop a high-order time integration scheme.
As shown in Section 2.2 there is minimal information on realistic core designs and
inconsistent thermophysical data to use as a starting point for assessing the time response
of a plausible MCFR design. This motivated the investigation into the most reasonable
thermophysical data available, development of a core design for an MCFR, and to
investigate the feasibility of the design. The design efforts presented in this work begin
with the ‘simple tank’ model similar to what has been shown in the literature. It will
quickly become evident the issues with a simple tank model and a revised version will be
presented with defined inlet and outlet fuel flow paths.
6.1 Design Goals
Designing every aspect of a nuclear reactor is a non-trivial goal. The first part of this
analysis focuses on the use of existing fast reactor design tools to determine a plausible
core design and estimate nominal operating parameters. The design goals for this are
summarized in the list below.
• Identify the most up to date thermophysical properties for the materials of interest.
• Select a fuel-salt composition.
• Develop a realistic core size.
• Define the steady state fuel salt mass flow rate.
• Assess reflector and shielding size and lifetimes.
• Determine cooling requirements for the inner reflector and shielding.
• Assess the intermediate heat exchanger sizing.
• Calculate reactivity coefficients for transient simulations.
76
6.2 Design Approach and Tools Used
To develop a core model of an MCFR, the most recent version of the Argonne
Reactor Computation (ARC) set of fast reactor tools are used. A high level overview
of how each code functions in the analysis is shown in Figure 6-1. For steady state
MC2-3Generates neutron and gamma cross sec-
tions at different temperatures [57].
DIF3D Calculates neutron flux, power profile [58].
GAMSOR Calculates power produced due to gamma production [59].
PERSENT Calculates reactivity coefficients for MSR1D [60].
MSR1DSolves modified point kinetics, the fluid flow,
and temperature equations for transient analysis.
Figure 6-1. Overview of each codes role in the analysis of an MCFR.
eigenvalue and flux calculations DIF3D is used. The DIF3D code has steady state
neutron diffusion and transport solvers tailored for fast reactor analysis and has been in
development for over 40 years at Argonne National Laboratory [58]. A variety of simple
Cartesian and hexagonal mesh structures in 1, 2, or 3 dimensions can be specified in a
DIF3D input file. Homogenized cross sections provide the interaction probabilities of the
materials with neutrons and are produced with MC2-3. The cross section processing tool
MC2-3 collapses the continuous neutron energy spectrum into few group cross sections at
a given temperature in the ISOTXS file format.
77
Gamma production is calculated with an extension of DIF3D, referred to as
GAMSOR [59]. The GAMSOR code calculates the gamma flux and heating using gamma
cross sections processed with MC2-3. The process for generating gamma cross sections
is thoroughly explained in the GAMSOR manual [59]. Calculations of the gamma and
neutron flux are done to determine the power produced within the inner reflector and
shielding. This is done to ensure adequate cooling is provided.
Feedback due to changes in the temperature and density of the fuel salt can be
understood with reactivity coefficients. These coefficients relate a change in reactivity
to some change in fuel temperature, density, or other physical parameter. The code
PERSENT is used to calculate these coefficients, as it conveniently makes use of the same
DIF3D model and MC2-3 cross sections [60]. The dominant feedback mechanism in fast
spectrum systems, and especially so for MSRs, is due to changes in the fuel density from
temperature or fluid flow rate changes. Both Doppler and fuel density first order reactivity
coefficients are calculated with PERSENT. The reactivity coefficients for both Doppler
and density changes have a spatial dependence, which PERSENT provides as well. It is
important to note that DIF3D does not explicitly consider any fuel movement or time
dependence. This model is used to provide reasonable starting steady state conditions for
the transient analysis tool developed in this work (MSR1D).
6.3 Material Considerations and Properties
A primary challenge with assessing the simulations of MCFRs is the lack of data on
many fluid and heat transfer properties for the chloride fuel salts of interest. This is due
in large part because the primary MSR reference is the MSRE, which operated with a
thermal neutron spectrum and used fluoride based fuel salt. Unfortunately, the fluoride
properties and analysis does not translate well to chloride systems. For instance, it was
identified in the early 1950s that materials developed to handle molten fluorides are not
necessarily applicable for chloride systems [10]. Parameters such as the density, thermal
78
conductivity, heat capacity, and the respective dependence on temperature is highly
questionable because of minimal or in some cases total lack of experimental data.
6.3.1 Fuel composition
As discussed in Section 2.2, the early analysis on MCFRs proposed using plutonium
chloride salts as fissile material [16–18]. Given the current commercial fleet of LWRs
uses enriched uranium fuel instead of plutonium, it seems most appropriate to select a
uranium chloride salt. The fuel salt composition used in this analysis is a mixture of
NaCl and UCl3. The most important factor in selecting a salt composition is what the
melting and solidification temperatures are for the desired fuel salt mixture. The melting
and solidification temperatures dictate the operating temperatures of the inlet and outlet
of the core. Subsequently, the operating temperature sets the requirements on reactor
materials, evaluations of physical properties, and the overall efficiency of the plant. For
most common elements, the melting temperatures are available and generally understood.
However, when atoms are combined into compounds the temperature at which this
compound will melt or solidify typically changes. Instead of the melting temperature,
what is more useful for an MSR designer is the liquidus temperature. The liquidus
temperature is defined as the temperature at which a material will be completely melted.
In the NaCl-UCl3 system, an understanding of the liquidus temperature for different molar
amounts of UCl3 is of high importance.
Recently, the liquidus curve of several fuel-salt compositions was measured and
compared to the early findings [61, 62]. In Reference [62] the eutectic point agreed with
the earlier findings but did suggest higher liquidus temperatures for molar fractions of
UCl3 below 0.3 compared to results presented from experiments in Reference [61] and
calculations from Reference [63].
To operate at the lowest temperature, the fuel salt compositions at the eutectic molar
composition point is selected. For NaCl-UCl3 of molar percentages of (0.66-0.34) provides
the only eutectic point [61–63]. At this molar composition NaCl-UCl3 has a liquidus
79
temperature of 523 C, or 796 K. A lower bound for the operating temperature of 850 K
is chosen to provide approximately a 50 K margin before parts of the salt start solidifying.
An interesting consideration is the change in liquidus temperature of the fuel salt as
fission products are produced and combine with chlorides. The melting temperatures of
fission products are discussed but never assessed experimentally in detail [19] based on
calculations and measurements from Reference [64]. The variation in melting temperature
as a function of fission product concentration should be investigated by future MCFR
designers.
6.3.2 Density of Fuel Salt
The density of the fuel salt is important as it influences fluid flow properties, and
dictates the effectiveness of the fuel expansion shutdown mechanism. The density and
associated variation with temperature should be evaluated to ensure the fuel expansion is
well understood and subsequent reactivity decreases occur as expected. However for the
molar composition of NaCl-UCl3 of interest there are few correlations for the density as a
function of temperature.
Correlations developed providing the density as a function of temperature was
reported for several molar percentages of UCl3 within NaCl-UCl3 [65]. The temperature
applicability of the correlations listed were not standardized between the reported
correlations. The standard deviations reported were on the order of 10−3 and are plotted
along with the density values for a respective correlations reported temperature range in
Figure 6-2. Note, the standard deviations reported are plotted in Figure 6-2 but are on
the scale of the tick marks in the figure. Considering the eutectic point of the NaCl-UCl3
system was 0.34 molar percent of UCl3, the correlation for 33.3% UCl3, shown in Figure
6-2, is selected for this work. This correlation gives a nominal density value (evaluated at
900K) of 3.112 g/cm3. The correlation, yielding density as a function of temperature in
units of g/cm3, can be written out as:
ρ(T ) = 3.8604− (0.8321T )× (1× 10−3) , (6–1)
80
900 1000 1100 1200 1300 1400Temperature [K]
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Dens
ity [g
/cm
3 ]
UCl3 mole %: 0.0UCl3 mole %: 14.4UCl3 mole %: 24.9UCl3 mole %: 33.3UCl3 mole %: 40.6
UCl3 mole %: 49.5UCl3 mole %: 59.9UCl3 mole %: 70.0UCl3 mole %: 85.0UCl3 mole %: 100.0
Figure 6-2. Reported density values as a function of temperature for several molarcompositions of UCl3 [65].
with a valid temperature range between 892 - 1142 K.
6.3.3 Heat Capacity
The heat capacity plays an important role in dictating the heat transfer properties
within the reactor and the heat exchanger. A primary concern with chloride fuel salts
compared to fluoride salts is that fluoride based salts have a heat capacity nearly twice
that of chloride as highlighted in Table 6-1. Early experiments on fluoride salts indicate
a relatively small dependence on temperature. Furthermore, early experiments indicated
experimental apparatuses were not accurate enough to provide a reasonable temperature
dependence for the heat capacity. This motivated selecting a single value of the heat
capacity for the simulations conducted.
81
Table 6-1. Reported values of the heat capacity in several MSR design studies.Material Mole % Heat [J/kg-K] Temperature [K] Reference
Capacity Evaluation(Pu3-UCl3)-(NaCl-MgCl2) (0.30,0.70) 837 923 [16](PuCl3)-(NaCl) (0.16,0.84) 950 1257 [18](U+TRU1 )-(NaCl) (0.45,0.55) 908 927 [20]LiF-Th4-233UF4 (0.775,0.20,0.025) 1621 983 [66]
Based on the reported heat capacity for molten chlorides typically reported in the
literature a value of 900 J/kg-K is selected.
6.3.4 Overview of Thermophysical Properties Selected
The thermophysical properties of the fuel salt are of paramount importance in
determining operational parameters and making design decisions. As demonstrated
in previous sections, there is a great deal of uncertainty or in some cases lack of
available data for certain parameters. The parameters used throughout this analysis
are summarized in Table 6-2. In Table 6-2 properties of other common reactor coolants
are displayed to give an idea of the differences between molten chloride fuel salt and
conventional coolants. A key difference with molten fuel salt is the very high boiling
temperature and high liquidus temperature. This fact inverts the safety concerns from
melting the fuel to providing mechanisms to avoid solidification. As a consequence the
operating temperatures of MCFRs must be very high, which can be useful from an
efficiency standpoint but creates significant strain on the materials. It is worth noting that
the molten chloride fuel salt is significantly more viscous and has a lower heat capacity
than sodium and water. The result is a large mass flow rate is required to circulate the
fuel salt through the system to remove enough heat.
1 Transuranic materials
2 Liquidus temperature
3 Used value from [16]
4 Used value from [16]
82
Table 6-2. Nominal values selected and compared with typical reactor coolants [67].Property NaCl-UCl3 Na H20 PbAtomic Weight 161.4 23.0 18.0 207.2Melting Point [K] 8062 371 273 600Boiling Point [K] 1973 1165 373 2010Property Temperature [C] 627 300 300 300Density [kg/m3] 3112 880 713 10500Heat Capacity [J/kg-K] 900 1300 5600 160Thermal Conductivity [W/m-K] 0.863 76 0.54 16Viscosity [cP] 4.24 0.34 0.1 2
6.3.5 Vessel and Reflector Materials
In a fast spectrum system a reflector and shield are needed to maintain criticality
and protect the vessel wall from the high neutron flux produced in the core. In a typical
SFR or LWR the fuel is protected in several different ways, the first of which is the metal
cladding that surrounds the fuel rods. Besides providing structure for the fuel rod, the
cladding in part shields the reflector and containment vessel. Given the proximity to
the fuel, the cladding accumulates a large fluence during the lifetime of the fuel. Fuel
rods are changed at regular intervals and therefore so is the cladding surrounding the
fuel. Conversely, in a flowing fuel MSR there is no such cladding and thus the materials
immediately surrounding the fuel salt act as one layer of containment similar to cladding
in a typical SFR. This is potentially a problem if the reflector accumulates a high fluence
and needs to be replaced. Replacing the vessel is more expensive and more complex
than the reflector or shielding, which motivates having a replaceable reflector next to the
flowing fuel salt.
Following similar practices of operating fast reactors the reflector is constructed of 316
stainless steel (SS). It should be noted the upper fluence limit for structural components
for “care-free” operation of 316 SS has been reported as 1.2×1023 n/cm2 based on
experiments performed in the Fast Flux Test Facility (FFTF) [68]. An alternative to SS
for fast reactor cladding is a material referred to as HT9. The HT9 material has some
merit but it is not clear whether HT9 has the necessary strength at the high temperatures
83
over 850 K sustained in an MCFR [68]. Given the operational experience with 316 SS in
operating fast reactors it seems like a reasonable reflector material for this work. Similarly,
the shielding is constructed of 316 SS mixed with B4C as in the demonstration fast
breeder reactor in Japan [1].
Considering the proximity of the reflector and shielding to the active core, gamma
and neutron heating will be significant. To counteract the heating and maintain a
constant reflector and shielding temperature, coolant channels must run through both the
reflector and shielding. For simplicity and to minimize mixing or activation of another
coolant loop, the primary fuel salt will run through the reflector and inner shielding to
provide cooling.
6.3.6 Primary Loop Mass Flow Rate
Based on the desired thermal power output and thermophysical parameters selected
for the fuel salt the design parameters are somewhat constrained. One of the most
important parameters to assess for an MSR is the mass flow rate through the core. The
mass flow rate has implications for the heat transfer, the behaviour of transients, pumping
power requirements, and intermediate heat exchanger performance.
The thermal power produced in the core is set to 3000 MW in order to produce
roughly 1000 MWe to make the electrical output competitive with existing commercial
LWRs and SFRs. The temperature rise over the core is chosen to balance the efficiency of
the system and allow for a manageable mass flow rate through the core. At this point the
secondary side of MCFRs are poorly defined and thus making decisions on the mass flow
rate and temperature rise upon the core are subject to change depending on the needs
or constraints of the secondary side. In large part what these constraints come down to
is the mass flow rate and temperature increase over the core is limited by what the heat
exchangers can remove from the system during normal operating conditions. Again for
these flowing fuel systems the lack of secondary side components makes it difficult to
assess as the heat exchangers as the associated working fluids are not well defined. Based
84
on temperature rises over the core on the order of other MSR designs, a temperature rise
of 100 degrees is selected.
Given an increase in temperature over the core and power produced in the core, a
nominal mass flow rate for the system can roughly be determined with the following:
m =Pth
Cp∆T, (6–2)
where m is the mass flow, Pth is the power over the core, Cp is the heat capacity of the
fuel salt, and ∆T is the temperature rise over the core. Based on Equation 6–2 a nominal
mass flow of 33,000 kg/s is calculated.
6.4 Simple Tank Molten Chloride Fast Reactor Model
Following the typical design ideas observed in the literature for MCFRs, what is
referred to as a ‘simple tank’ model is first analyzed. These designs assume a tank,
which is typically spherically shaped and contains no internal structure or a defined
inlet or outlet plenum. To build upon this tank model a cylindrical core with an inlet
and outlet plenum are provided as a possible flow path for the fuel salt. Initial analysis
began with this model, shown in Figure 6-3. In Figure 6-3 the core is 3.5 m tall with
a 2.5 m by 2.5 m base. The core volume is selected to ensure the system is critical
based upon 16% enrichment of 235U. However, as analysis continued several problems
were realized with this design approach. For instance if the vessel wall is next to the
fuel salt, it will experience high neutron and gamma radiation levels, corrosion, and
high temperatures. Nickel based alloys typically used in nuclear vessels have serious
embrittlement issues, which may be partially negated by higher temperatures but will
likely develop an amorphous crystalline structure leading to a reduction in the vessel’s
strength [69]. An additional concern with nickel based alloys is the formation of helium
bubbles on the grain boundary as shown for the alloys investigated during the MSRE
[69]. During the MSRE titanium was added to nickel based alloys to mitigate helium
embrittlement, but was largely ineffective at temperatures above 700 C [69].
85
Figure 6-3. Cutaway view of a simple tank MCFR model.
Conversely, the reflector and shielding could be placed next to the fuel and the
vessel placed outside, thereby keeping the vessel somewhat safe. Considering the inlet
and outlet plenum are part of the active core and the flux is high in these regions there
would be serious concern with the reflector lifetime. Flow paths through the inlet and
outlet reflector would further complicate construction. Inspecting the inlet and outlet
geometry, such an open core with no defined flow paths opens up the possibility of large
recirculation zones and possible neutron streaming issues through the top and bottom
of the core. Considering the challenges associated with material construction, reflector
lifetime, and fluid flow concerns it seemed prudent to develop an alternative to the core
design approach typical of the MCFR literature.
6.5 Refined Core Design
To mitigate the design problems with the simple tank, a refined core design is
proposed which specifies constrained inlet and outlet fuel flow paths and a simplified
86
reflector construction. Additionally, the reflector and shield are placed within the vessel
to ensure a reasonable vessel lifetime. Placing the reflector and shielding within the vessel
simplifies cooling as fuel salt can flow through both the reflector and shielding to remove
heat. Both the reflector and the shield are envisioned to be removed and replaced as they
reach their fluence limits. A two dimensional representation of the updated core is shown
in Figure 6-4, where the coolant flow paths are approximate and not necessarily drawn to
size. It should be noted that the inlet and outlet flow paths in Figure 6-4 are not straight
Fuel Salt
Reflector
Shield
Vessel
Flow Out
Flow In
Active Core
1.0 m
1.0 m
3.6 m
2.5 m
0.6 m0.4 m
0.2 m
Figure 6-4. Axial view of the updated MCFR design.
but rather helical in order to prevent neutron streaming through the top and bottom of
the core. The nominal design parameters for the analysis are provided in Table 6-3. The
sizing parameters were based on criticality requirements at the selected enrichment level.
6.6 Steady State Analysis with DIF3D
The determination of the nominal parameters for the revised MCFR design begins
with criticality and neutron flux calculations using DIF3D. Using DIF3D requires
87
Table 6-3. Summary of nominal design parameters in the revised MCFR design.Parameter ValueSalt composition UCl3-NaCl (0.34,0.66)Fuel enrichment 15.5%Thermal Power [MW] 3000Core inlet [K] 850Core outlet [K] 950Mass flow [kg/s] 33,300Core height [m] 3.6Core area [m2] 6.25Pipe area [m] 0.5βloss [pcm] 265.5
information about the materials and the interaction probabilities, which come in the form
of neutron cross sections. The neutron cross sections require some care in preparation to
account for temperature and spectrum at different parts of the core and will be discussed
next.
6.6.1 Cross Section Processing
Cross sections used by DIF3D were generated with MC2-3 and collapsed to 33
energy groups. The fuel is enriched to 15.5% 235U. Cross sections for the core materials
were determined at the nominal temperature of 900 K, which is roughly the average
temperature in the core. The temperature increase over the core is assumed to be 100 K.
Cross sections at 850 K, 900 K, and 950 K were calculated for the fuel salt and assigned to
the bottom, middle, and top thirds of the core to roughly mimic the rise in temperature
over the core.
The R-Z flux from the transport code TWODANT is used to collapse the energy
spectrum and account for spectrum differences in the different regions of the core. The
R-Z geometry fed into TWODANT is shown in Figure 6-5.
6.6.2 Core Coolant Paths Assessment Method
The fuel salt coolant must flow through the primary side and into the core, reflector,
and inner shielding regions. These flow paths must be selected to balance several core
parameters. For instance, the pressure drop through the core should roughly match the
88
Core
Reflector
Vessel
Inner Shield
R
Z
Figure 6-5. R-Z core model used in TWODANT flux calculations.
pressure drop across the reflector and inner shielding. Due to gamma and neutron heating
the reflector and shielding must have fuel salt pumped through to remove the heat. Thus
the fuel salt mass flow rate in the reflector and inner shielding must be selected to remove
the power produced in each respective region. Additionally, the fraction of fuel salt in the
upper and lower reflector should be less than 50% fuel to prevent the fuel from becoming
critical as it passes through the upper and lower reflectors. Given the requirement to
balance several parameters an iterative approach is required to meet each constraint. To
understand how the various parameters are balanced, the core, axial reflector, and inner
shielding regions will be discussed separately.
First we begin with the constraint on the total mass flow rate required through the
core. As mentioned the mass flow rate in the core can roughly be determined as follows:
mcore =Pcore
Cp∆T, (6–3)
where mcore is the total mass flow rate through the core. Therefore by conservation
of mass the flow rate in a given channel through the upper and lower reflector can be
89
deduced based on the number of channels and the total mass flow rate through the core as
described in Equation 6–4
mcore =
C∑c=1
mc , (6–4)
where mc is the mass flow rate in a given channel out of a total of C channels. Given the
mass flow rate the velocity, u, in a channel with area Ac can be roughly estimated as:
uc =mc
Acρ, (6–5)
where ρ is the density of the fuel salt.
With a means to evaluate the velocity within a channel feeding into the core or
reflector region it is now possible to assess the pressure drop through these channels. Note,
for these calculations the channels are assumed to be cylindrical. In reality their shape
will be significantly more complex, as they will likely have a helical shape. Such a design
is envisioned to mitigate neutrons streaming through the top and bottom of the core.
To calculate the pressure drop the Reynolds number can be calculated using
Re =udh
µ, (6–6)
where u is the fluid velocity, dh is the hydraulic diameter, and µ is the kinematic viscosity.
The Reynolds number is used to determine the Darcy-Weibach friction factor. Note,
a roughness coefficient is also needed to select a friction factor and for this analysis a
roughness coefficient is chosen based on stainless steel. Now the pressure drop for a
channel, i , can be evaluated with:
∆pifric =
f ρu2L
2dh
, (6–7)
where f is the friction factor, L length of the channel.
At this point the pressure drop over the upper and lower reflector can be evaluated
with Equation 6–7 given the required total mass flow rate. To calculate the mass flow
90
rates needed through the reflector and shielding requires calculation of the heating that
occurs from neutron and gamma production in each of these regions.
6.6.3 Reflector and Shielding Cooling Assessment
To assess the cooling requirements for the inner reflector and shielding the gamma
and neutron heating in each of these regions will be determined. In theory one could have
an additional system that provided non-fuel coolant salt or potentially another coolant
material. However, use of an additional cooling system for the reflector and shielding
region may be cost prohibitive, adds complexity, and there would be serious concern
with fuel salt within the core leaking and contaminating this additional cooling system.
Therefore the simplest option is to have additional flow paths of fuel salt that travel
through the reflector and inner shield, much like in a SFR.
A 21-group gamma library processed with MC2-3 is used for all GAMSOR calculations
with no explicit consideration for secondary gamma production. The gamma cross sections
were evaluated using the detailed R-Z flux profile provided by TWODANT as outlined in
Section 6.6.1.
The goal of these calculations can be summarized as listed below.
1. Determine the required fuel salt mass flow rate in the reflector and inner shield toremove the heat produced.
2. Determine flow path dimensions to balance the pressure drop with that of the maincore.
3. Calculate the fraction of fuel salt in each region for DIF3D calculations.
This calculation process begins by defining the number of flow paths in the reflector and
shielding and their respective sizes. First, the volume of a single channel through the
reflector is defined as:
Vrf = 2πr 2rf h , (6–8)
where rrf is the radius of the channel, and h is the height. Similarly for the shield:
Vsh = 2πr 2shh , (6–9)
91
where rsh is the radius of the channel, and h is the height. Clearly then the total fuel
volume in the reflector and shielding is just the sum of all the channel volumes:
V Trf =
J∑j=1
V jrf , (6–10)
where J is the total number of channels in the reflector. Again the shielding coolant
volume is determined similarly:
V Tsh =
Q∑q=1
V jrf , (6–11)
where Q is the total number of channels in the shielding.
Once the flow paths and fuel fraction are defined, a calculation with GAMSOR can
be performed. This process begins with an initial estimation of the power produced in the
reflector and shield. Typically the amount of power produced due to gamma heating is
a few percent of the core power. Once the GAMSOR calculation is complete the power
produced in the reflector and shield is available. Using the power in each region the mass
flow rate is calculated in the usual way.
Again by conservation of mass, the mass flow rate through a given reflector channel
is
mjrf =
mTrf
J. (6–12)
For the shielding channels a similar set of equations is provided.
mTsh =
Psh
∆TCp
(6–13)
In Equation 6–13 ∆T and Cp are the same values defined for the fuel salt in the core.
Again by conservation of mass, the mass flow rate through a given shielding channel is
mqsh =
mTsh
Q. (6–14)
At this point it is relatively straightforward to calculate the velocity in each channel using
Equation 6–5. With the velocity the Reynolds number can be evaluated with Equation
92
6–6 and therefore the pressure drop with Equation 6–7. The radius of the channels is
varied along with the number of channels in order to match the pressure drop across the
core. Keep in mind the total pressure drop from across the core, reflector, and shielding
must be kept at a level such that a series of reasonably sized pumps can provide the
necessary mass flow rates. A high level representation of the iterative calculation process
is given in Figure 6-6. In the DIF3D model there are no explicitly defined flow paths for
(re)Defineflow paths
GAMSORCalculation
Assess ∆prefl
∆pshield
Does ∆pcore =
∆prefl = ∆pshield
Figure 6-6. Iterative process for determining necessary fuel salt coolant in reflector andinner shield.
the coolant within the reflector and shield. Instead the fuel salt composition is smeared
into the reflector and shield compositions based on the fractional amount of fuel salt
required. Note, that to reproduce the results presented here a renormalization to the
desired power is needed based on the GAMSOR power conversion issue described on page
20 of the GAMSOR manual [59]. Functionally, this amounts to setting a larger power level
in the first two GAMSOR input files.
6.6.4 Coolant Flow Path Results
A nominal coolant path area for the radial reflector, radial shielding, upper and
lower reflector are calculated and described in Tables 6-4, 6-5, and 6-6. For the pressure
drop calculations a constant friction factor for a steel pipe of 0.45 is used along with a
roughness value of 0.015. All other parameters are based on the nominal values described
93
previously in Table 6-3. A major assumption in these calculations is the coolant flow
path is cylindrical through the upper and lower reflector. However, the goal of these
calculations was to ensure that the flow paths required would not take up excessive
volume or have too large of a pressure drop across the channels.
Table 6-4. Calculated parameters for the core inlet and outlet flow paths.Parameter ValueTotal Mass Flow [kg/s] 33,000# Flow Channels 20Mass Flow/Channel [kg/s] 1670Channel Area [m2] 0.126Height Inlet/Outlet Reflector [m] 1.0Height Inlet/Outlet Shield [m] 0.2Fuel Fraction 0.402Reynolds Number (single channel) 1.26×106
Pressure Drop [kPa] 7.63
Table 6-5. Calculated parameters for the radial reflector flow paths.Parameter ValueTotal Mass Flow [kg/s] 713.8# Flow Channels 30Mass Flow/Channel [kg/s] 23.8Channel Radius [m] 0.0439Channel Area [m2] 0.0061Reflector Length [m] 6.0Fuel Fraction 0.0005Reynolds Number 8.22×104
Pressure Drop [kPa] 7.63
Table 6-6. Calculated parameters for the inner shield radial flow paths.Parameter ValueTotal Mass Flow [kg/s] 232.8# Flow Channels 40Mass Flow/Channel [kg/s] 5.82Channel Radius [m] 0.025Channel Area [m2] 0.0028Shield Length [m] 6.0Fuel Fraction 1.41×10−4
Reynolds Number 3.53×104
Pressure Drop [kPa] 7.62
94
Considering the lack of internal structure the overall pressure drop in the core
including contributions from the reflector and shielding flow paths is generally going to
be smaller than solid fueled reactors. Due to the large mass flow rate required, the lack
of internal structure is helpful because significant pumping power will still be required.
Overall it appears the amount of coolant required for cooling is reasonable and the
associated pressure drops are sufficiently small.
6.6.5 Core Component Lifetimes
In solid fuel reactors Zircaloy or steel material surrounds the fuel rods, which is
known as cladding, and prevents fission products from contaminating the coolant. In a
fast spectrum chloride design there is no concept of fuel cladding as there is no internal
structure. This has long been touted as an advantage of MCFRs as the neutron energy
spectrum inside the core can be very hard and opens up the possibility of interesting
fuel cycles. What is of primary concern though is how long the components can last
immediately adjacent to the active core. In the revised design discussed in this work the
reflector and shielding are envisioned to be removed as these components are adjacent to
the core and experience a high fluence during operation.
The reflector surrounding the active core is constructed of 316 stainless steel (SS),
which has a structural fluence limit based work performed at the Fast Flux Test Facility
(FFTF) [68]. In this section there are no considerations for the temperature or chemical
interaction effects on the reflector material, which ultimately will reduce the lifetime of
these materials even further. The goal is merely to point out that irradiation effects are
considerable and care should be given to what material is used as a reflector and how long
it can last in the challenging molten salt environment.
In Figures 6-7 and 6-8 the upper and lower reflector regions fluence values are plotted
as a function of time, respectively. The numbered regions in Figures 6-7 and 6-8 represent
20 cm radial slices of the reflector where the fast neutron flux is calculated in the DIF3D
model. For the fast neutron flux only the contributions from energy groups 1-10 are
95
Region 5
Region 4
Region 3
Region 2
Region 1
Core
2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Years
0
1
2
3
4
Flue
nce
×1023
Region 1Region 2Region 3Region 4Region 5
Figure 6-7. The lower reflector fluence is plotted as a function of time in each radialregion, where the dashed line represents the structural fluence limit.
included, representing energies above 0.1 MeV. In both the top and bottom reflectors the
fluence limit is denoted with dashed line in Figures 6-7 and 6-8. As shown the regions
immediately adjacent to the core in the upper and lower reflectors reach the fluence limit
in about 5.5 years. While this limit is set based on experimental data for structural 316
96
Core
Region 5
Region 4
Region 3
Region 2
Region 1
2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Years
0
1
2
3
4
Flue
nce
×1023
Region 1Region 2Region 3Region 4Region 5
Figure 6-8. The upper reflector fluence is plotted as a function of time in each radialregion, where the dashed line represents the structural fluence limit.
97
SS it still illustrates the significant radiation damage experienced by the reflectors. In the
FFTF other cladding materials were tested such as HT9, which showed promise as being
able to withstand about a 4 times greater fluence than 316 SS [68]. However, the FFTF
work pointed out that the properties of HT9 are highly questionable at temperatures
above 650 C as the strength of HT9 diminishes at high temperatures. Considering the
operating temperatures of MCFRs are near or above 650 C it is questionable how useful
HT9 would be and it is generally advised that other alloys need to be investigated. At this
point here has been no consideration for chemical or temperature effects on the lifetime,
both of which would likely reduce the lifetime of the SS. Even just considering radiation
damage from neutrons it appears that replacement of the material next to the active core
is inescapable.
6.6.6 Core Size
To investigate the effect of the core size on criticality a simple parametric study is
performed. All eigenvalue calculations are performed with the 33 group cross section
library and the nodal diffusion calculation option in DIF3D. The spatial mesh used for
DIF3D calculations is discretized such that no mesh size was above 5 cm. The height of
the core is maintained at a constant value of 3.6 m, while the width of the core is adjusted
in increments of 0.10 m. The inlet and outlet plenum dimensions is maintained in each
simulation. The eigenvalue for each core width is provided in Figure 6-9. In Figure 6-9
the data is fitted to a linear function resulting in a relationship between eigenvalue as a
function of core width, which is given in Equation 6–15.
k(w) = 0.00103w + 0.7481 (6–15)
Similarly, the effect of perturbing the core height was investigated by keeping the
core width constant at 2.5 m and varying the core height by 0.1 m. As the core height
increased additional spatial meshes are added to maintain the same level of spatial
discretization in the z-dimension. The resulting eigenvalues as a function of height are
98
240.0 242.5 245.0 247.5 250.0 252.5 255.0 257.5 260.0Core Width [cm]
0.99500
1.00000
1.00500
1.01000
1.01500
Eige
nval
ue
Linear Fit
Figure 6-9. Variation in eigenvalue as a function of core width.
plotted and the data fit to a linear function in Figure 6-10. The linear fit provides the
350.0 352.5 355.0 357.5 360.0 362.5 365.0 367.5 370.0Core Height [cm]
1.00400
1.00450
1.00500
1.00550
1.00600
1.00650
1.00700
1.00750
Eige
nval
ue
Linear Fit
Figure 6-10. Calculated eigenvalue for different core heights.
eigenvalue as a function of reactor height as shown in Equation 6–16.
k(h) = 0.000162h + 0.9474 (6–16)
99
With Equations 6–15 and 6–16 it is possible to estimate the reactivity change based
on expansion of the core area. During a transient, such as a loss of flow accident, the
temperature of the core is likely to increase and be at a sustained temperature for
some time. The increase in temperature will result in some thermal expansion of the
containment materials. Subsequently, expansion of the MCFR vessel will allow additional
fuel salt to enter the active core and thus may be a positive reactivity feedback effect.
The approximate change in width based on the linear thermal expansion coefficient can be
found with:
w = wo(∆Tα + 1) , (6–17)
where w0 is the nominal width, ∆T is the change in temperature, and α is the linear
thermal expansion coefficient. A linear coefficient of thermal expansion of 19.5× 10−6
cm/cm/C is used, which has a temperature range between 20 and 1000 C [70]. Using
Equation 6–17 for a 100 C increase in temperature would result in a 0.5 pcm increase
in reactivity due to the change in width. Similarly, for the height a 100 C would result
in a 0.11 pcm increase in reactivity. In both these cases the assumption is the thermal
expansion would be outward from the core and thus increase the volume of the core,
which is thought to be a conservative estimate. Even with a conservative estimate the
reactivity increase from vessel expansion does not appear to be a concern given the
negative feedback mechanisms are stronger by two orders of magnitude.
6.6.7 Enrichment of Chlorine-37
Consideration of the nuclear properties of the fuel salt is important for ensuring
an optimal fuel cycle performance and minimizing the amount of fuel salt required for
criticality. In the case of a fuel salt comprised primarily of NaCl it is important to look at
the interactions between constituent isotopes and neutrons in the high energy regime.
Inspecting the plot of the sodium cross section as a function of incident neutron
energy, as shown in Figure C-1 of Appendix C sodium has a very low absorption cross
section ( 1 barn) at energies above 0.1 MeV. Similarly, examining Figure C-2 of
100
Appendix C shows for 37Cl the absorption cross section is reasonably low (< 0.1 barn)
at high energies. However, the other stable isotope of chlorine, 35Cl, has significantly
larger absorption cross section than 37Cl at high energies. Unfortunately, the natural
abundance of chlorine is 76% 35Cl and 24% 37Cl [71]. Thus for an improvement in the
nuclear performance of the fuel salt it would be useful to enrich natural chlorine such that
37Cl is the dominant isotope.
To provide a basic assessment of the impact of increasing the portion of 37Cl and
reducing 35Cl a parametric study is conducted varying the amount of each isotope looking
at the effect on criticality. Criticality calculations are performed with DIF3D using the
nodal diffusion method.
The effect of varying the amount of 37Cl present in the fuel salt on criticality
is observed in Figure 6-11. In all the eigenvalue calculations performed in this 37Cl
enrichment study the 235U enrichment was kept at 16.5% and a core size of 2.5 m by 2.5
m by 3.6 m (x-y-z dimensions). Inspecting Figure 6-11 it is clear using natural chlorine,
30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 90.0% 100.0%37Cl Enrichment
0.88000
0.90000
0.92000
0.94000
0.96000
0.98000
1.00000
Eige
nval
ue
Figure 6-11. Eigenvalue plotted as function of the 37Cl enrichment.
with approximately 24% 37Cl will require increased 235U enrichment or an even larger core
101
volume. Neither increasing 235U enrichment nor core volume is likely to be wise from an
economic point of view so it would be advisable to enrich the chlorine to above 90% 37Cl.
6.6.8 Enrichment of Uranium-235
Enrichment of 235U in operating LWRs is less than 5% because of their operation
with a thermal neutron spectrum and the desire to keep enrichment levels low because of
proliferation concerns. Fast reactors typically require significantly higher levels of fissile
enrichment to maintain criticality as the 235U fission to 238U capture ratio is lower for
high energy neutrons. As such the fast spectrum molten salt reactors too require higher
enrichment of a fissile material like 235U to achieve and maintain criticality. Increased
enrichment will result in higher fuel costs and raises concerns with proliferation of nuclear
material. The enrichment levels in fast reactors must remain below 20% to be classified
as Low Enriched Uranium (LEU) according to the International Atomic Energy Agency
(IAEA) [72].
To investigate the required enrichment level a parametric study was conducted
calculating the eigenvalue at different 235U enrichment levels. For these calculations
a constant 98% enrichment of 37Cl and a core size of 2.5 m by 2.5 m by 3.6 m (x-y-z
dimensions) was maintained. The results of this study are presented in Figure 6-12
where the eigenvalue is plotted as a function of the 235U enrichment. For this core size
enrichment levels around 15.5% will maintain a critical system. However, there is still
significant uncertainty in the 235U cross sections and error accumulated creating the
homogenized cross sections. When designing the core of an MSR these errors should be
considered and instead an operating range for enrichment levels should be developed as
in principle the enrichment of fuel salt can be increased during operation. Furthermore
consideration for the cost of enrichment versus the core size should be considered. As
in principle a smaller core could be created if the enrichment levels are increased. So a
trade-off must be evaluated between core size and 235U enrichment levels.
102
14.0% 15.0% 16.0% 17.0% 18.0% 19.0%235U Enrichment
0.96000
0.98000
1.00000
1.02000
1.04000
1.06000
1.08000
1.10000
1.12000
Eige
nval
ue
Figure 6-12. Calculated eigenvalue as a function of 235U enrichment.
It should be noted that starting a core with different fissile material such as 239Pu has
been proposed in other MCFR concepts. However, considering the proliferation concerns
that would likely be raised, it seems unlikely the first MCFR built would operate with
239Pu as a starting fissile material.
6.7 Heat Exchanger Sizing
The heat generated within the primary loop of an MCFR must be removed and
transferred to a secondary fluid for subsequent electricity generation or a process heat
application. The transfer of heat takes place in an intermediate heat exchanger, which
for an MCFR will have molten fuel salt as the primary fluid and another molten salt
as the secondary fluid for heat removal. For this work LiF-NaF-KF (molar percentages
0.465,0.115,0.42), commonly referred to as ‘FLiNaK’ salt was selected as the working
fluid. The high temperatures of the primary fuel salt limits secondary heat transfer fluid
to be either a gas, lead or another salt. Gases would require pressurization of the vessel
and add further complications, which do not seem useful at this time. The FLiNaK salt
was selected as it has more complete reported thermophysical properties in the literature
103
than other heat transfer salts [73]. The thermophysical properties selected for FLiNaK are
provided in Table 6-7 where all properties are evaluated at 840 K, which is the average
temperature between the inlet and outlet temperatures the FLiNaK salt experiences in the
heat exchanger. For this analysis the common single pass shell-and-tube heat exchanger
Table 6-7. Thermophysical properties selected for FLiNaK.Property ValueHeat Capacity [J/kg-K] 1880Density [kg/m3] 2116Viscosity [cP] 6Thermal conductivity [W/m-K] 0.830
design with straight tubes is utilized. This design is selected as in theory it can be built
to tolerate very high temperatures and pressures, as well as corrosive or hazardous fluids,
provided that suitable materials are chosen [74]. At this time there are no heat exchanger
designs for MCFR systems and as such the goal of this analysis is to obtain a rough idea
for the required size of the heat exchanger. There is still significant work to do to assess
materials, analyze thermal and mechanical stress, specify detailed tube geometry, and
finalize the heat transfer fluid for MCFR heat exchangers. One other interesting constraint
unique to MSR systems is the time spent in the heat exchanger by the fuel salt is going
to impact the loss of neutron precursors. Additionally, radiation damage experienced by
the heat exchanger should be considered for material longevity and tube expansion due to
irradiation.
To begin this analysis it is useful to first define the heat exchanger design parameters
shown in Table 6-8. To assess the sizing of the heat exchanger the log mean temperature
difference (LMTD) correction factor method will be used. This method makes use of a
modified heat transfer rate equation:
q = UAF∆tlm , (6–18)
104
Table 6-8. Heat exchanger design parameters.Parameter ValueDuty [MWth] 3000Fuel salt inlet tube temperature Tf ,i [K] 950Fuel salt outlet tube temperature Tf ,o [K] 850mfuel [kg/s] 33,000Tube inner diameter [cm] 1.9Tube outer diameter [cm] 2.0Tube pitch [cm] 2.5Pitch-to-diameter ratio 1.31FLiNaK inlet shell temperature Ts,o [K] 780FLiNaK outlet shell temperature Ts,i [K] 900mFLiNaK [kg/s] 13,297
where q is the heat transfer rate, U is the overall heat transfer coefficient (htc), A is the
surface area, F is a correction factor dependent on the flow pattern, and ∆tlm is the log
mean temperature difference defined in Equation 6–19.
∆tlm =∆t1 − ∆t2ln(∆t1/∆t2)
(6–19)
In Equation 6–19, ∆t1 = Tf ,i − Ts,o and ∆t2 = Tf ,o − Ts,i . For sizing purposes a F
correction factor of 0.7 is estimated, which is applicable for initial scoping calculations of
the heat exchanger design selected [75].
The overall heat transfer coefficient is dependent on the heat transfer properties of
the fuel salt, the thermal resistance from the tube wall, and the heat transfer properties
of FLiNaK. To calculate the heat transfer coefficient of the fuel salt and FLiNak the
following equation can be used:
htc =Nudbk
dh
, (6–20)
where Nudb is the Nusselt number determined by the Dittus-Boelter correlation, k is the
thermal conductivity, and dh is the hydraulic diameter [76]. In the case the shell fluid the
hydraulic diameter for a fluid flowing through a rectangular lattice can be calculated with
dh = d
[4
π
(p
d
)2
− 1
], (6–21)
105
where d is the diameter of the tube, and p is the pitch. Now, the Nusselt number can be
found using the Dittus-Boelter correlation shown in Equation 6–22.
Nudb = 0.023Re0.8Pr 0.4 (6–22)
In Equation 6–22 Re denotes the Reynolds number and Pr is the Prandtl number. The
Reynolds number can be calculated with:
Re =udh
ν, (6–23)
where u is the fluid velocity, and ν is the kinematic viscosity. Similarly, the Prandtl
number can be found with Equation 6–24.
Pr =Cpν
k(6–24)
Now that the heat transfer coefficient can be determined using Equation 6–20 all that
remains is to evaluate the heat flow resistance of the tube wall. The thermal resistance
can roughly be determined by taking a ratio of the thermal conductivity and the tube
wall thickness. Assuming the tube material is constructed using stainless steel a thermal
conductivity of 14.6 W/m-K is used. Now the overall heat transfer coefficient can be
calculated using Equation 6–25.
U =1
1htcfuel
+ tkss
+ 1htcFLiNaK
(6–25)
In Equation 6–25 t is the tube thickness and kss is the thermal conductivity of stainless
steel.
Now rearranging 6–18 allows the surface area required to be calculated using
Equation 6–26.
A = qUF∆tlm (6–26)
106
Given a heat exchanger surface area and the surface area for a given tube, the total
number of tubes can be determined:
Nt =A
at
, (6–27)
where A is the heat transfer area required, and at is the surface area of a single tube.
The arrangement of these tubes is chosen to be square to reduce the pressure drop and
to more easily enable mechanical cleaning. A triangular tube arrangement could be used
but this choice increases the pressure drop and makes access to the tubes challenging for
mechanical cleaning and inspection [75]. The velocity through a given tube is determined
by assuming the mass flow rate evenly divides between the tubes as shown in Equation
6–28.
mtube =mtotal
Nt
(6–28)
So the velocity for a tube can be determined as:
utube =mtube
Atubeρ(6–29)
Consequently, the velocity through a given tube is based on the number of tubes and
therefore the pressure drop across the tubes is dependent on the number of tubes required.
Care should be taken to ensure the pressure drop through the heat exchanger is not
exorbitant and will require significant pumping power. The pressure drop is calculated
using:
∆p =Lfuρ
2dh
, (6–30)
where L is the tube length, and f is the friction factor. In these calculations a friction
factor of 0.45 is used throughout based fluid flowing in a steel pipe with a high Reynolds
number (> 1000).
Coming up with an estimate for the size of a heat exchanger for this system is now
possible and the results are given in Table 6-9. The pressure drop for both the tube
and shell size appear reasonably small and would not require extraordinary pumping
107
Table 6-9. Estimated heat exchanger thermal design specifications.Tube Shell
Velocity [m/s] 3.0 1.2Prandlt 4.4 12.97Reynolds number 3.47×104 7.03 ×103
Nusselt number 1911 1063Pressure drop [kPa] 111 28.9Heat transfer coefficient [W/m2-K] 8.65 ×104 4.46 ×104
Overall Design ParametersNumber of Tubes 12,376Surface area [m2] 7387LMTD 59.44Diameter [m] 3.3Length [m] 10Overall volume [m3] 85.53Fuel volume [m3] 70.18
requirements. Another important consideration for heat exchangers in MSR systems is
the overall size, as the entire unit will require shielding, and be filled with fuel salt at the
initial loading. So in many ways the smallest heat exchanger is desirable to minimize fuel
loading, fuel time spent outside the core, and to easily fit within a containment structure.
Based on Table 6-9 the overall volume of the heat exchanger system is about four times
greater than the active core. Now the analysis done in this section is preliminary and
other designs are likely possible. In other high temperature reactors it is proposed to
use a Printed Circuit Heat Exchanger (PCHE) [77]. The PCHE systems are compact
but make use of micro-channels and a fin system to maximize heat transfer. While not
thoroughly investigated at this time, radiation induced swelling of these microchannels
may be significant and limit the effectiveness of a PCHE for MSRs.
6.8 Generation of Point Kinetics Data with PERSENT
The point kinetics approach taken in the transient code developed requires the user to
supply the mean neutron generation time, delayed neutron fraction, and precursor decay
constant for each precursor group to be supplied. The mean neutron generation time is
calculated with PERSENT, which uses the adjoint and forward multigroup neutron fluxes
108
from DIF3D. The mean neutron generation time is calculated in PERSENT with:
Λ =〈ψ∗, ν−1Hψ〉〈ψ∗,Fψ〉
, (6–31)
where ψ∗ is the adjoint flux, ν is the neutron velocity, H is an identity-like matrix, F is
the fission source matrix, and ψ is the forward flux [60]. The delayed neutron fraction can
be calculated for each isotope and precursor group using Equation 6–32.
βi ,f =〈ψ∗,Fi ,f ψ〉〈ψ∗,Fψ〉
(6–32)
In Equation 6–32, Fi ,f is the fission source for precursor group f for isotope i . To get
the total delayed neutron fraction Equation 6–32 can be summed over the isotopes and
number of delayed families. The delayed neutron fraction values calculated by PERSENT
can be best thought of as intrinsic to the fuel salt composition of interest. The transient
code determines the fractional loss in delayed neutrons due to the precursor movement
and adjusts the total delayed neutron fraction in the power equation, shown again in
Equation 6–33. In Equation 6–33 the β term is replaced by βflow to stabilize the system as
discussed in Section 4.2.2.
dN(t)
dt=%(t)− βflow
ΛN(t) +
I∑i=1
F∑f =1
λi ,fCi ,f (x , t) (6–33)
However, for the coupled set of equations describing the precursor distribution, shown
again in Equation 6–34, the delayed neutron fractions will remain as the values computed
originally from PERSENT. The values are not altered because the precursor production
in the core is only dependent on the fission events within the core, which will produce the
same fraction of delayed neutrons at any given moment whether or not the fuel is flowing.
dCi ,f (x , t)
dt=βi ,f
ΛN(t)− λi ,fCi ,f (x , t)− u(x , t)
∂Ci ,f (x , t)
∂x(6–34)
Often in the point kinetics approach the delayed neutron fractions are lumped
together between isotopes into coalesced terms. In reality each isotope gives off different
109
fission products resulting in the fraction of delayed neutrons and the decay constants for
each family to vary between isotopes. For the MCFR proposed here the fissile component
is dominated by 235U and therefore only one set of delayed neutron fractions and decay
constants, given in Table 6-10, are used. In Table 6-10 the half-life for each precursor
group is given as denoted by τ1/2.
Table 6-10. Point kinetics parameters generated by PERSENT.Group 1 2 3 4 5 6λ [s−1] 0.0134 0.0325 0.1213 0.3078 0.8684 2.9169τ1/2[s] 51.84 21.33 5.72 2.25 0.80 0.24β 2.12×10−4 1.17×10−3 1.16×10−3 2.78×10−3 1.31×10−4 5.44×10−4
6.9 Reactivity Feedback Coefficients
To carry out transient analysis with a point kinetics approach requires determination
of reactivity coefficients to account for physical feedback effects as discussed in Section
4.5.3. For this work PERSENT is used to calculate spatially dependent reactivity
coefficients using first order perturbation theory [60]. Two reactivity feedback mechanisms
are dominant in the MCFR system, Doppler broadening of the cross sections, and fuel
expansion. Both of these mechanisms occur as the temperature increases in the core and
both are negative reactivity feedback effects with increasing temperature.
To model the Doppler feedback a perturbation of the cross sections is performed. The
nominal cross section set is perturbed to a higher temperature and the cross sections are
reevaluated. Next a perturbation calculation is performed with PERSENT and yields a
change in eigenvalue due to the difference in cross sections. To determine the reactivity
coefficient the change in eigenvalue is divided by the change in temperature as shown in
Equation 6–35.
γDoppler =∆k
∆T(6–35)
To calculate the fuel expansion feedback a percentage change of the fuel density is
carried out with PERSENT. This perturbation results in a reactivity coefficient shown in
110
Equation 6–36.
γdensity =∆k
∆ρ(6–36)
To provide a reactivity coefficient in units of pcm/Kelvin requires correlating a density
change to changes in temperature. The density correlation provided in Equation 6–1 is
used to relate the change in density to a change in temperature.
The calculated reactivity coefficients are shown in Table 6-11 and compared to
reactivity coefficients with the REBUS-3700 MCFR [20]. For the Doppler coefficient a 400
K change in the cross sections is performed. For the fuel expansion a 2% density change
is introduced. In both perturbations the spatial coefficients of reactivity are available,
Table 6-11. Doppler and density coefficients compared to the REBUS-3700 MCFR.This work [20]
Doppler [pcm/K] -0.67 -0.50Fuel expansion [pcm/K] -8.0 -6.0
which provide an axial dependence of the coefficients and is fed into the transient analysis
code. An axial spatial distribution of the reactivity change is provided in Figure 6-13.
The density reactivity feedback calculated is significantly higher than for solid fuel. This
is largely because molten salt changes density with temperature substantially more than
solid fuel. This point is highlighted by comparing the variation in density as a function
of temperature for NaCl-UCl3 to that of solid UO2 fuel, typically found in a LWR.
The comparison is shown in Figure 6-14 where the NaCl-UCl3 and UO2 densities are
normalized by their respective starting value, which is done to facilitate comparison of
the change in density as a function of temperature. Inspecting Figure 6-14 it is evident
that the salt reduces the density by about 10% over the temperature range, while the solid
UO2 fuel changes by about 1% over the same range. The fact that the density varies so
rapidly is a primary physical reason why the reactivity coefficient for the fuel density is so
large.
111
50 100 150 200 250 300 350Core Height [cm]
-0.00045
-0.00040
-0.00035
-0.00030
-0.00025
-0.00020
-0.00015
-0.00010
Reac
tivity
Cha
nge
Fuel expansion Doppler
Figure 6-13. Spatial dependence of Doppler and fuel expansion reactivity changes.
600 650 700 750 800 850 900 950 1000Temperature [K]
0.90
0.92
0.94
0.96
0.98
1.00
Dens
ity [g
/cm
3 ]
NaCl-UCl3UO2
Figure 6-14. Density comparison between NaCl-UCl3 and solid UO2 fuel. Note, in bothcases all values are normalized by the starting density value evaluated at 600K.
112
CHAPTER 7SAFETY ANALYSIS OF THE MOLTEN SALT REACTOR DESIGN
The ultimate goal of developing a modified point kinetics solver for MCFR analysis
is to evaluate the time response of the system during postulated accident scenarios.
Specifically, this work seeks to test whether perturbations introduced will require a
high-order time integration of the coupled fluid flow and neutronic equations to be solved
efficiently. The transient scenarios of interest are listed below.
1. Primary pump failure leading to a gradual reduction of the mass flow rate.
2. Primary pump over speed leading to an increase in the mass flow rate.
3. Loss of coolant feed salt to the intermediary heat exchanger resulting in a decreasein the heat pulled off by the heat exchanger.
4. Increase in coolant feed salt mass flow rate to the intermediary heat exchangercausing an increase in the amount of heat that is pulled off.
Failure of a primary or secondary side salt pump in an MCFR poses an interesting
transient case to study, particularly in systems without any control rods. In either case it
is of high importance to understand the dynamics of the system when the flow speed of
the primary and secondary side salt varies. Ideally there will be a shutdown mechanism
in place to reduce power levels in the core. The shutdown mechanisms in an MCFR
primarily comes from the expansion of the fuel salt as the temperature rises and, to a
lesser extent, a reduction in the fission rate due to the Doppler broadening of the cross
sections. The following sections simulate the previously listed transient scenarios and
assess if high-order time integration is required.
7.1 Primary Fuel Pump Failure Transient Simulations
Failure of one or more of the primary fuel salt pumps will result in the reduction in
flow speed through the system at a gradual rate. The gradual rate assumes centrifugal
pumps are employed and have some rotational inertia [78]. The concern with these types
of transients from a kinetics point of view is the change in pump speed will alter the
distribution of precursors and possibly inject reactivity into the system. The injection
113
of reactivity comes from an increase in the number of delayed neutrons in the core as
precursors remain in the core longer compared to the steady state distribution. If the
negative feedback mechanisms are not ample enough and the change in pump speed is
significant this could lead to a rapid increase in power.
It should be noted that there is only consideration for the power produced in the
core in these transient simulations. Consequently some of the thermal power produced is
deposited in the surrounding materials and the power produced only considering the fuel
is slightly less than the mass flow rate selected in Table 6-3. The starting mass flow rate
in these simulations is 32,474 kg/s and the final mass flow rate is 6,494 kg/s.
Several transients are simulated where each takes a different amount of time for the
mass flow rate to reach a lower level. In these simulations it is assumed the pump fails and
the mass flow rate decreases exponentially to the new mass flow rate. In each simulation
of a pump failure the mass flow rate is decreased to 20 % of the starting mass flow rate.
In Figure 7-1 the power amplitude is plotted as a function of time for the first 100 seconds
of the transient, where each line is differentiated by the time it takes to decay to the final
mass flow rate. In all transient simulations presented here time steps of 1×10−4 seconds
are employed. Spatially the system is discretized such that all elements are 1 cm in length.
Even in extreme cases where the mass flow rate is reduced in 1.6 seconds the reactor
almost immediately begins shutting itself off. To verify the time steps taken are suitably
small and some dynamics are not being missed the time steps are reduced by two orders
of magnitude in the case where the mass flow rate is reduced in 1.6 seconds. The power
profile is compared between simulations conducted with different time steps for the first
1.5 seconds of each simulation is provided in Figure 7-2. No appreciable differences in the
power profile are observed between the nominal (∆t = 1 × 10−4 seconds) and the cases
with increasingly smaller times steps. This is the case as the changes in reactivity are not
large over even the coarsest time step.
114
0 20 40 60 80 100Time [s]
0.2
0.4
0.6
0.8
1.0
Powe
r Am
plitu
de
160 sec64 sec32 sec16 sec11 sec
8.1 sec6.4 sec5.4 sec4.6 sec
4.0 sec3.2 sec2.3 sec1.6 sec
Figure 7-1. Power as a function of time for the first 100 seconds of each simulated pumpcoast down. Each dashed line represents the time it took to reach the lowermass flow rate.
The rapid shutdown comes from a reduction in the flow speed across the core
thereby increasing the temperature at the core outlet. The longer the fuel spends in
the active core the more heat is transferred and therefore resulting in an increase in
core temperature. Increasing the core temperature results in an immediate reduction in
power due to the negative reactivity from the Doppler broadening and expansion of the
fuel. As shown in Figure 7-3 the average temperature across the core increases as the
mass flow rate decreases in all cases. The average core temperature even in cases with a
rapidly decreasing pump speed only increases by 50K. The peak temperature in the core
does reach almost 1030 K (757 C) in the cases where the flow loss occurs in less than 3
seconds but only remains at that peak temperature for several seconds. In all simulations
115
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Time [s]
0.2
0.4
0.6
0.8
1.0
Powe
r Am
plitu
de
t= 1x10 4 t= 1x10 5 t= 1x10 6
Figure 7-2. Different time steps employed in the calculation of the power as a function oftime for a transient where the mass flow rate is reduced in 1.6 seconds.
in Figure 7-3 the temperature increases as the pump speed slows but as the power level
decreases the temperature returns back down. In all cases the average temperature settles
to a new temperature about 23 K higher than the starting value.
The system settles to this new higher temperature to compensate for the reactivity
insertion induced by the change in precursor distribution. To understand why the system
settles to this new temperature it is instructive to look at the difference in the steady
state delayed neutron fraction between the starting mass flow rate and the final one. In
Figure 7-4 the steady state delayed neutron fractions are plotted for different flow rates.
Inspecting Figure 7-4 the difference in delayed neutron fractions between the starting and
final mass flow rates amounts to about a 200 pcm reactivity insertion. Considering the
total reactivity feedback is about 8.7 pcm/K then a 23 K increase would amount to about
200 pcm of negative reactivity feedback.
Overall, even in the most extreme, and likely unphysical flow reduction perturbations,
there is no chance for a large spike in power and the only practical concerns are possible
116
0 50 100 150 200 250Time [s]
910
920
930
940
950
Aver
age
Core
Tem
pera
ture
[K]
160 sec64 sec32 sec16 sec11 sec
8.1 sec6.4 sec5.4 sec4.6 sec
4.0 sec3.2 sec2.3 sec1.6 sec
Figure 7-3. Average temperature across the active core as a function of time for eachsimulated pump coast down.
short term increases in temperatures. Before investigating additional transients the
precursor loss will be quantified and examined in greater detail in Section 7.2.
7.2 Quantification of Precursor Loss
In this section the goal is to quantify the precursor loss to help better understand the
results of Section 7.1 and other transient simulations. First, recall each precursor group
has a different decay constant and contributes a different amount to the total delayed
neutron fraction. Furthermore, when the fuel is flowing the relative percentage of the total
fraction from each group depends on the flow rate through the core and the size of the
core.
To assess the impact of each precursor group as a function of flow rate steady state
calculations are performed with different mass flow rates. For each steady state calculation
117
0 1 2 3 4 5 6 7 8Mass Flow [kg/s] ×104
0.00250
0.00300
0.00350
0.00400
0.00450
0.00500
0.00550
0.00600
0.00650
tota
l
Figure 7-4. Calculated delayed neutron fraction in the core at different steady state massflow rates.
the delayed neutron fractional contribution per group is analyzed. In Figure 7-5 at
each mass flow rate the steady state delayed neutron fraction per precursor group is
determined. At each mass flow rate in Figure 7-5 the fractional contribution of each group
is plotted as described by Equation 7–1.
βi
βtotal (m)(7–1)
For this discussion it is helpful to reiterate the decay constants and half-lives for each
precursor group, which are provided again in Table 7-1. As shown in Figure 7-5 the
Table 7-1. Decay constants and half-lives per precursor group.Group 1 2 3 4 5 6λi [s−1] 0.0134 0.0325 0.1213 0.3078 0.8684 2.9169
τ1/2i [s] 51.84 21.33 5.72 2.25 0.80 0.24
fourth and fifth precursor groups are the biggest contributors to the delayed neutron
fraction for all the mass flow rates. This can be further understood by examining the
time spent across the core at each mass flow rate, also shown on the right hand side
118
0 1 2 3 4 5 6 7 8Mass Flow [kg/s] ×104
0.0%
10.0%
20.0%
30.0%
40.0%
i(m) /
to
tal(m
)
Family 1Family 2
Family 3Family 4
Family 5Family 6
0
5
10
15
20
25
Tim
e Ac
ross
Cor
e [s
]
Time Across Core
Figure 7-5. Fractional contribution of each precursor group to the total fraction of delayedneutrons at each mass flow rate.
y-axis of Figure 7-5. The fourth precursor group contributes the most to the delayed
neutron fraction, and its contribution declines the most as the time across the core
approaches about 2 seconds, which is approximately the half-life of the fourth precursor
group. It is interesting to point out the fourth precursor group actually increases in
influence as the mass flow rate begins to increase from zero to 5×104 kg/s. This can be
understood as nearly all the delayed neutrons produced from this group are decaying
inside the core when the time spent across the core is about 20 seconds, which is about
10 times the half-life. The impact of the sixth precursor group, which has the shortest
half-life, increases as the mass flow rate increases as it has a short half-life and most of the
precursors decay inside the core, while the contribution from other groups varies due to
the longer half-lives.
The takeaway from this analysis is that precursor groups four and five contribute
the most to the delayed neutron fraction and have half-lives on the order of 1-2 seconds.
119
So any influence the redistribution of precursors might have will be felt primarily on
the time scale of seconds even with very high mass flow rates. From a stability point
of view this is a good thing as if a majority of the influence was from group six the
redistribution of precursors would act on the order of tenths to hundredths of seconds.
However, considering the large negative reactivity coefficient due to fuel density changes it
seems likely that any positive reactivity insertion will quickly be compensated for. Since
the precursor influence is felt on the order of seconds and the negative reactivity rapidly
erases any positive reactivity introduced by the precursors this provides evidence that the
hypothesis of this work might be rejected.
7.3 Primary Fuel Pump Over Speed
Another transient scenario postulated considers the case when the primary fuel pump
gradually increases the mass flow rate. Increasing the mass flow rate should result in
precursors being pushed out of the core and decrease the fraction of delayed neutrons
in the core causing a reactivity insertion. As the flow rate increases, the temperature
should decrease through the core as less heat is transferred to the fuel salt. Decreases in
temperature could result in a positive reactivity insertion as the fuel salt density becomes
greater.
In the transients simulated the mass flow rate is exponentially increased to 110% of
the starting flow rate. So for these simulations the starting mass flow rate is 32,474 kg/s
and the ending is 35,721 kg/s. In Figure 7-6 the power as a function of time is plotted for
each transient case. Each transient has a different amount of time for the mass flow rate
to transition to the higher value. Even in the extreme case where the mass flow rate is
increased in 1.6 seconds the peak power is only 12% higher than nominal. In all transients
in Figure 7-6 the power approaches a new level just under 10% of the nominal. To verify
some dynamics were not being missed by a poor time step selection the 1.6 second flow
transition was simulated with smaller time steps. Even as the time steps were decreased
to 1×10−6 seconds the power trace did not deviate from simulations run with larger time
120
0 20 40 60 80 100Time [s]
1.00
1.02
1.04
1.06
1.08
1.10
Powe
r Am
plitu
de
16 sec10.7 sec8.05 sec
6.44 sec5.36 sec4.60 sec
3.2 sec1.6 sec
Figure 7-6. Power amplitude as a function of time for different transient simulations whereeach line represents the time taken to reach the new flow rate.
steps as shown in Figure 7-7. Again, confirming that because of the precursor influence
dominated by groups four and five it is difficult to rapidly introduce reactivity changes
that would require small time steps.
Another consideration is as the mass flow rate increases over the core less heat
is transferred to the fuel resulting in a decrease in the average fuel temperature, as
highlighted in Figure 7-8. The temperature decreases resulting in a positive reactivity
insertion and thus an increase in power. However, at the same time precursors and
delayed neutrons are being pushed outside of the core and contribute less to the change in
power. As the system settles, it does so at a level that balances the loss in precursors with
a decrease in the temperature to compensate.
Overall in these pump over speed transients the main result is the power increases
and approaches a new stable level but not substantially higher than the starting power.
121
0 1 2 3 4 5 6 7Time [s]
1.00
1.02
1.04
1.06
1.08
1.10
Powe
r Am
plitu
de
t= 1x10 4 t= 1x10 5 t= 1x10 6
Figure 7-7. Comparison of the power trace with different time steps for a 10% increase inmass flow rate over 1.6 seconds.
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0Time [s]
902.5
903.0
903.5
904.0
904.5
Aver
age
Core
Tem
pera
ture
[K]
16 sec10.7 sec8.05 sec
6.44 sec5.36 sec4.60 sec
3.2 sec1.6 sec
0 50 100 150 200 250Time [s]
902.5
903.0
903.5
904.0
904.5
Aver
age
Core
Tem
pera
ture
[K]
16 sec10.7 sec8.05 sec
6.44 sec5.36 sec4.60 sec
3.2 sec1.6 sec
Figure 7-8. The average temperature is plotted as a function of time. On the left the first20 seconds of the transients are shown, on the right the first 250 seconds.
122
The resulting temperature increase over the core is only a few degrees and does not pose a
significant safety concern.
7.4 Reduction in Heat Sink Transients
To simulate a reduction in the heat sink, simulations are performed where the
temperature drop across the heat exchanger is reduced as a function of time. As less
heat is removed by the heat exchanger the temperature within the core will increase.
Subsequently, the increase in temperature should shut down the reactor due to the
negative feedback mechanisms. An important consideration is what temperatures will
be achieved and how long the temperatures will be sustained within the core. In these
simulations the heat exchanger performance is not explicitly modelled and only acts by
setting a fixed temperature difference over the heat exchanger domain at any moment
in time. Clearly, to simulate these transients more accurately there would need to be
inclusion of all the factors that lead to the heat exchanger performance as outlined in
discussion of heat exchanger sizing in Section 6.7.
In Figure 7-9 the power profile is given for different temperature increases in the
total heat pulled off by the heat exchanger. Figure 7-9 shows that as the temperature
drop across the heat exchanger increases so too does the power. However, as the power
starts to rise, the average core temperature starts to decrease as the reactivity feedback
is positive. In addition to the reactivity from the feedback there are also reactivity
contributions from the precursors redistributing themselves due to the velocity change in
the core causing some oscillations in the power profile over time. The power decreases as
the temperature in the core is increasing as highlighted by the average core temperature
shown in Figure 7-10. Considering the temperature and power appear to oscillate it
seems useful to see what power level is reached and if the oscillations dampen out. The
10 K reduction in temperature case is run for 1400 seconds, and the results are shown
in Figure 7-11. Figure 7-11 shows that the power oscillates and appears to reach a new
power level approximately 10% lower than the starting level. Similarly, the average power
123
0 50 100 150 200 250 300 350Time [s]
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Powe
r Am
plitu
de
90 K 85 K 80 K 75 K
Figure 7-9. Power profile for different amounts of heat removed from the heat exchanger.
level oscillates in time but appears to return to the starting average temperature after
1400 seconds. The core temperature rises significantly when the temperature drop across
the heat exchanger is reduced by only 25 K. From a safety perspective this seems rather
concerning as sustained temperatures above 1400 K would likely damage the structure
containing the fuel salt.
124
0 50 100 150 200 250 300 350Time [s]
700
800
900
1000
1100
1200
1300
1400
Aver
age
Core
Tem
pera
ture
[K]
90 K 85 K 80 K 75 K
Figure 7-10. Average core temperature over time for different temperature reductionsacross the heat exchanger.
Figure 7-11. Power amplitude (left) and average temperature (right) as a function of timefor a 10 K reduction in the temperature across the heat exchanger.
7.5 Heat Sink Overcool Transients
In the postulated transient scenarios discussed in this section the secondary coolant
pump is assumed to malfunction resulting in an increase in the amount of heat pulled off
125
the heat exchanger. The primary side mass flow rate is assumed to be fixed during each
simulation.
Having the heat exchanger remove a larger amount of heat in turn reduces the
temperature across the core. As the core temperature decreases, the power in the core
begins to increase as the reactivity feedback becomes positive (density increases), as shown
in Figure 7-12. The power becomes larger with temperature decrease, but with some delay
as the velocity through the core varies. While the mass flow rate is fixed in the system
the density changes are induced by the temperature changes. So as the temperature
goes down so too does the density resulting in a decrease in velocity across the core and
subsequently more heat is transferred to the fuel salt. The average core temperature for
each transient is provided in Figure 7-13. An oscillation in the temperature is evident as
the delayed increase in power in turn increases the temperature.
Assuming the heat exchanger can remove enough heat to cause a 50% increase in
the temperature difference across the core seems rather unlikely. Nevertheless if such
an operation is possible with a heat exchanger in an MCFR there is potential for large
power excursions and possible salt solidification. The solidification of the salt is dangerous
as it may causes blockages of flow paths, damage reactor components, and cause large
power increases. In all cases except the 110 K case the average core temperature dips
below the salt liquidus temperature and would present a danger of salt solidification.
The simulations presented here should be verified with a detailed model of the heat
exchanger and the inclusion of decay heat. There is no decay heat model employed in
this work, which almost certainly would increase the temperature in the core towards the
end of these transients. In general any increase in the amount of heat removed presents a
significant concern in an MCFR due to fuel salt solidification.
126
0 50 100 150 200 250 300 350 400Time [s]
1.0
1.2
1.4
1.6
1.8
Powe
r Am
plitu
de
110 K120 K
130 K140 K
150 K
Figure 7-12. Power as a function of time for several heat exchanger temperature drop overcool transients.
127
0 50 100 150 200 250 300 350 400Time [s]
200
400
600
800
1000
1200
1400
Aver
age
Core
Tem
pera
ture
[K]
110 K120 K
130 K140 K
150 K
Figure 7-13. Average core temperature for several heat exchanger temperature drop overcool transients.
128
CHAPTER 8CONCLUSIONS
The MCFR is identified as an interesting advanced reactor candidate for producing
electricity or industrial process heat. The concept is not new as the idea of using molten
chloride mixed with nuclear fuel has been around since the 1950s. However, the majority
of the research focused on different fuel cycle analysis rather than reactor physics and
safety analysis. Reviewing the literature highlighted the large differences in MCFR sizes,
thermophysical properties, and exact fuel compositions.
With no clear starting point for safety analysis and to test the hypothesis of this
work, there was motivation for developing a plausible MCFR core configuration. This
dissertation develops a new MCFR design and investigates the transient behaviour during
several postulated accidents. The study is motivated by the hypothesis that changes in
flow rates of molten fuel salt might inject uncompensated reactivity into the system due
to precursor redistribution. If such flow related changes were possible high-order time
integration techniques would need to be developed for the coupled neutronic and fluid
flow equations to solve them efficiently. Methodologies developed for other MSR transient
analyses have been summarized and it has been shown the Quasi Static modification for
MSR study is not as promising a method as a recent review paper has suggested. Other
modified point kinetics approaches have been developed for MSR study. The typical
‘source’ and ‘sink’ modification to point kinetics for MSR study makes it difficult to
simulate flow related changes. Other modified point kinetics did not clearly demonstrate
correct physics based responses to the flow perturbations of interest or answer succinctly
how to handle the incongruity in the power equation due to the precursor movement.
Considering the need to test the hypothesis of this work and to provide a starting
point for any high-order time integration scheme a code was written. The governing
modified point kinetics, fluid flow, and heat equations were derived in detail and the code
is openly available. Comparisons to experimental results from the MSRE showed good
129
agreement between simulated results and experiment. Simple tests showed responses to
flow perturbations consistent with the expected physical behavior.
Considering the lack of detailed reactor designs in the literature for MCFRs a core
design was developed. The core design provides constrained flow paths for the inlet and
outlet of the active core. Additionally, this design proposes a means to cool the inner
reflector and shielding, which was not shown in any other MCFR design to date. The
design presented provides an evaluation of all thermophysical properties utilized. Modern
fast reactor analysis tools were employed to develop a plausible core size, investigate
variation in salt composition, assess gamma heating in surrounding core materials,
and generate point kinetics data and reactivity feedback coefficients. Additionally, a
conventional tube-and-shell heat exchanger sizing study was conducted. Based on this
initial design work the conclusion drawn is that a large core volume of at least about
22.5 m3 and a heat exchanger volume of around 70 m3 is required. Thus to fill the entire
primary loop with fuel salt will require approximately 290,000 kg of fuel salt. Based on
fluence limits is appears the upper and lower reflectors will need to be replaced every 5-7
years. The large fuel inventory combined with the likely need to frequently replace the
inner reflectors makes the economic case for MCFRs questionable.
With a core design developed it was possible to evaluate the time response to various
flow perturbations. Primary pump failures and malfunctions were simulated with pump
slowdown and speed up transients, respectively. In transients where the pump reduced
the mass flow rate the system rapidly shutdown. The temperature in the core increases
slightly as the flow speed is reduced but even in extreme cases the temperature increase
was only 50 K. The large negative reactivity feedback from the fuel salt expansion quickly
reduced the power in the core before the precursor redistribution could inject reactivity.
A detailed analysis of the precursor influence was conducted. For the MCFR system the
fourth and fifth precursor groups contribute the most to the delayed neutron fraction at
all flow rates and have half-lives on the order of 1-2 seconds. Considering the time the fuel
130
spends flowing through the core is 1-2 seconds the redistribution of precursors is primarily
felt on the 1-2 second scale. Transients where the pump speed was increased by 10% show
a rise in power on the order of the increase in flow speed even when rapidly increased. A
small decrease in temperature was observed in these transients. In general plausible pump
over speed transients did not prove concerning from a safety perspective.
Changes in the secondary side heat exchanger were simulated by increasing or
decreasing the temperature difference removed by the heat exchanger over time. When the
heat exchanger removes less heat the temperature in the core increases substantially for
several minutes before shutting itself down. The power levels decrease as the temperature
increases so from a safety point of view the large temperature excursions are potentially
very problematic. Similarly, if the heat exchanger effectiveness increases then the
temperature in the core drops dramatically and may cause potential salt solidification.
The power produced in the core also increases as the salt becomes more dense but does
so on a large time scale. In general the changes in the system due to variations in the
primary and secondary side flow behavior happen on a relatively long time scale and
very small time steps were not needed to resolve changes in flow rates. The results of
the transient studies and detailed analysis of the precursor group contributions leads to
the rejection of the hypothesis that flow changes in an MCFR might cause large power
excursions. Furthermore, it was not possible to rapidly inject reactivity into the system
thus there is no need for high-order time integration strategies at this time.
131
CHAPTER 9FUTURE WORK
The work ahead for future MCFR designers is substantial. It would be highly
useful to develop an understanding of the secondary side in these systems. To that
end integration of a better representation of the heat exchanger into a code like the
one developed here is of high importance. To verify results presented here with greater
confidence comparisons should be made to transient simulations with higher-fidelity
coupled physics codes. For instance the idealized flow conditions assumed here should
be compared to three-dimensional computational fluid dynamics results. Considering the
large uncertainty in material properties it would be beneficial to ascertain the uncertainty
of various parameters calculated in time dependent simulations. Additional sensitivity
calculation methods would be beneficial for understanding core dynamics. Considering the
operating temperatures in MCFRs is largely dictated by the liquidus temperature it is of
high importance to understand the liquidus temperature as a function of fission product
build up. While in principal the fission products can be removed it seems unlikely in any
commercial plant. Any fission product buildup is likely to change the liquidus temperature
of the fuel salt as fission products bind with NaCl to form other compounds.
132
APPENDIX ASTEADY STATE SOLUTION ALGORITHM
Result: Steady state solution vectors for power, precursor concentration, velocity,
and temperature.
Initialize power profile;
Initialize velocity profile ;
Initialize temperature profile ;
while εconv > ||~c ||nL2 − ||~c ||n−1L2 do
for e = 1 to E do
Loop over entire domain of E elements ;
Calculate spatial matrices via Gaussian integration,~~Ae ,
~~Ue ,~~We,R ,
~~We,L ~qe ;
for i = 1 to I do
Loop over all isotopes ;
for f = 1 to F do
Loop over all precursor families ;
Assemble~~Ge = [− ~~Ue + λ
~~Ae +~~We,R ] ;
Calculate ~we−1,L = ~We,L · ~cn−1e−1,i ,f ;
Calculate~~G−1
e ;
Solve for ~cne,i ,f =
~~G−1e ~qe +
~~G−1e ~we−1,L ;
end
end
end
end
Algorithm 1: Steady state solve of the multiphysics system.
133
APPENDIX BTRANSIENT SOLUTION ALGORITHMS
Result: Solution vectors for power, precursor concentration, velocity, and
temperature over time.
Initialize using steady state solution. ;
for k = 1 to K do
Loop over time while εconv > ||~c ||nL2 − ||~c ||n−1L2 do
Nonlinear iteration ;
for e = 1 to E do
Loop over entire domain of E elements ;
Calculate spatial matrices via Gaussian integration,~~Ae ,
~~Ue ,~~We,R ,
~~We,L
~qe ;
Determine~~A−1
e ;
for i = 1 to I do
Loop over all isotopes ;
for f = 1 to F do
Loop over all precursor families ;
Assemble~~He =
~~A−1e~~Ue · −λ
~~I − ~~A−1e~~We,R ;
Calculate βΛ
~~A−1e · ~qe and
~~A−1e~~We,L · ~ck−1
e−1 ;
Solve for
~cek,n+1
= ~cek−1
+ ∆t~~He · ~ce
k−1,n+ ∆t β
Λ
~~A−1e ~qe + ∆t
~~A−1e~~We,L · ~ck−1
e−1,n
;
end
end
end
Determine total precursor source in fuel domain. ;
Calculate βnew ;
Calculate Nk = Nk−1 + ∆t %k−1−β
ΛNk−1 + ∆t 1
VT
∑Ee=1
∑Ii=1
∑Ff =1 λi ,fC
k−1m,i ,f (x) ;
for e = 1 to E do
Calculate T ke (x , t) = he (x)N(t)
mcp(T k−1e )
+ T k−1e (x , t) ;
Calculate uke (x , t) = me
a(x)ρe (T ke )
end
end
end
Algorithm 2: Implicit Euler solve of the multiphysics system.
134
APPENDIX CCROSS SECTION DIAGRAMS FOR FUEL SALT ATOMS
ENDF/B-VII.1 NA-22Principal cross sections
10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
Energy (MeV)
100
101
102
103
104
105
106
Cro
ss s
ectio
n (b
arns
)
totalabsorptionelastic
Figure C-1. 22Na neutron cross section as a function of energy plot from ENDF/B-VII.1[79].
135
ENDF/B-VII.1 CL-37Principal cross sections
10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
Energy (MeV)
10-3
10-2
10-1
100
101
102
Cro
ss s
ectio
n (b
arns
)
totalabsorptionelasticgamma production
Figure C-2. 37Cl neutron cross section as a function of energy plot from ENDF/B-VII.1[79].
136
ENDF/B-VII.1 CL-35Principal cross sections
10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
Energy (MeV)
10-2
10-1
100
101
102
103
104
Cro
ss s
ectio
n (b
arns
)
totalabsorptionelasticgamma production
Figure C-3. 35Cl neutron cross section as a function of energy plot from ENDF/B-VII.1[79].
137
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BIOGRAPHICAL SKETCH
Zander Mausolff received a Bachelor of Science in physics from the University of San
Francisco in December of 2014. As his interests shifted to nuclear engineering he traveled
across the country to the University of Florida (UF) to pursue a Ph.D. After his first year
at Florida, Zander was awarded a Department of Energy Nuclear Engineering Universities
Program (DoE-NEUP) fellowship to fund his Ph.D research. This fellowship made it
possible to travel and work closely with Idaho National Laboratory and Argonne National
Laboratory.
Apart from research Zander was very involved in the student and local section’s of the
American Nuclear Society (ANS). The highlight of which was leading the successful bid
and hosting of the 2018 ANS Student Conference at UF.
144
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