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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143

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