CORE DESIGN ASSESSMENT AND SAFETY ANALYSIS OF A FAST SPECTRUM MOLTEN CHLORIDE SALT REACTOR By ALEXANDER J. MAUSOLFF A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2019
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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
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
6-7 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
6-11 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
7-5 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
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.
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
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
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.
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.
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
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.
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
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
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.