Parametric Study of Reactivity for the University of Florida Training Reactor
Using MCNP
Requirement for Completing Summa Cum Laude Honors Thesis
Daniel E. Lago, Undergraduate Student
Department of Nuclear and Radiological Engineering, University of Florida, Gainesville, FL 32611
For correspondence contact:
Daniel E. Lago
Undergraduate Student
Department of Nuclear and Radiological Engineering
University of Florida, Gainesville, Florida 32611-8300
Phone: (407) 443-7352 Email: [email protected]
Short Title:
Improved modeling of UFTR reactivity.
Word Count: 3,673
APA Format
Table of Contents
ABSTRACT .................................................................................................................................... 1
INTRODUCTION .......................................................................................................................... 2
BACKGROUND ............................................................................................................................ 2
University of Florida Training Reactor (UFTR) ........................................................................ 2
MCNP Model of UFTR ............................................................................................................... 5
Six-factor Formula ...................................................................................................................... 6
Reactivity..................................................................................................................................... 8
Boron Content of Graphite ......................................................................................................... 9
Neutron Moderation and the Porosity of Graphite................................................................... 10
Fuel Box Size............................................................................................................................. 12
RESULTS AND DISCUSSION ................................................................................................... 13
Benchmark of Original Input .................................................................................................... 13
Boron Content of Graphite ....................................................................................................... 15
Porosity of Graphite ................................................................................................................. 16
Fuel Box Size............................................................................................................................. 18
CONCLUSIONS........................................................................................................................... 19
ACKNOWLEDGMENTS ............................................................................................................ 20
REFERENCES ............................................................................................................................. 21
1
ABSTRACT
This study was catalyzed by the unexpectedly low reactivity observed when the University of
Florida Training Reactor (UFTR) was refueled in the summer of 2010. Three parameters – the
boron content, the porosity of the graphite, and the fuel box size – were investigated with a
Monte Carlo N-Particle (MCNP) model of the UFTR. The results indicated that all three likely
decrease the reactivity, with the most significant factor being the graphite density. It was also
acknowledged that this study is theoretical, and needs to be supported by future experiments.
Keywords: MCNP, Reactor, Reactivity, Boron, Graphite
2
INTRODUCTION
In the summer of 2010, the University of Florida Training Reactor was refueled by the
reactor staff and the students involved in the Digital Upgrade of the UFTR Protection and
Control Systems. The project is an effort led by the University of Florida (UF) Nuclear and
Radiological Engineering (NRE) Department, in conjunction with Areva and Siemens, to license
the first fully digital reactor protection system for a nuclear reactor in the United States. The
project had progressed to the point where the reactor needed to be operational, which required
rebuilding the reactor and reinserting the fuel back into the core.
Computational models were developed to predict criticality as a safety precaution. A
model was created in MCNP for the Highly Enriched Uranium (HEU) to Low Enriched Uranium
(LEU) conversion that took place in 2005. This model was modified for the refueling analysis to
predict criticality for the core arrangement in 2010. Results from the MCNP model implied
criticality with only two safety blades up and the last safety blade and the regulating blade only
partially withdrawn. When all of the fuel was placed in core, the reactor needed all of the safety
blades and a substantial portion of the regulating blade to be fully removed in order to achieve
criticality.
Three parameters were examined for their effects on reactivity: the boron content in the
graphite moderator, the fuel box size, and the porosity of the graphite. The purpose of this study
is to determine which parameters affect the reactivity.
BACKGROUND
University of Florida Training Reactor (UFTR)
The UFTR is a 100 kWth non-power research reactor on the University of Florida
campus. The reactor is graphite and water moderated, water cooled, and based on a modified
Argonaut design; several experimental facilities surround the core, including the three vertical
ports, beam tubes, and the pneumatic rabbit port. Fig. 1 shows a cutaway view of the UFTR
including the experimental port facilities [1].
3
Fig.1. Isometric cutaway view of the UFTR [1].
Graphite moderator surrounds the fuel boxes on all sides; the concrete and shield tank
provide additional shielding outside the core. The fuel is a plate design made of U3Si2-Al powder
with Al cladding and a nominal enrichment of 19.75 wt% 235U. Fuel bundles are comprised of 14
plates; each fuel box is filled with four fuel bundles. Design parameters of the fuel are shown in
Table I. Fig. 2 illustrates the configuration of the fuel boxes in-core [2].
4
Table I Design Parameters of UFTR Core [1]
DESIGN DATA
Fuel Type U3Si2-Al
Fuel Meat Size
Width (cm)
Thickness (cm)
Height (cm)
5.96
0.051
60.0
Fuel Plate Size
Width (cm)
Thickness (cm)
Height (cm)
7.23
0.127
65.1
Cladding 6061 Al
Cladding Thickness (cm) 0.038
Fuel Enrichment (nominal) 19.75%
“Meat” Composition (wt% U) 62.98
Mass of 235U per Plate (nominal) 12.5 g
Number of Plates per Fuel Bundle 14
Number of Full Fuel Bundles (current) 22
Number of Partial Fuel Bundles 1 (10 fuel plates + 3
dummy plates)
Number of Dummy Bundles 1
Fig.2. Horizontal cut-through view of UFTR core [1].
Fuel bundle
Fuel box
Graphite
Graphite
N
5
The UFTR control system consists of 3 safety blades and 1 regulating blade all in-core.
Control blade drives rotate the blades in and out of the core along a vertical arc. Cadmium at the
tip of the control blades provide the neutron absorption needed for control and magnesium
shrouds protect the blades from obstruction.
MCNP Model of UFTR
MCNP is a Monte Carlo continuous energy transport code that simulates the paths and
interactions of neutrons, photons, and electrons in a three-dimensional Cartesian geometry [3].
The Monte Carlo method applies nuclear data to probabilistic methods of tracking a particle. To
achieve adequate statistics, millions of particles are tracked through the geometry created in
MCNP.
The MCNP model used in this study was created by Dr. Gabriel Ghita for use during the
HEU-to-LEU fuel conversion in 2005. The input is a highly detailed representation of the fuel,
boxes, surrounding graphite, and control blades. Fig.3 depicts the reactor geometry of the MCNP
model as provided via the MCNP Visual Editor Computer Code (VisEd). The model was
modified slightly for the reactor refuel, as updated dimensions for the fuel boxes and higher
boron content in the graphite were added. The three parameters under investigation were
manipulated by changing the surface cards (fuel box size), material cards (boron content) and
cell cards (graphite porosity).
6
Fig.3. Vertical cross-sectional view (left) and horizontal cross-section view (right) of UFTR core
model in MCNP [4].
Six-factor Formula
The key for a nuclear reactor to produce power is sustaining a desired eigenvalue, or k;
the variable describing the rate of change of the neutron population is the criticality constant, k,
described as
𝑘 =# 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑖𝑛 𝑛𝑡ℎ 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
# 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑖𝑛 (𝑛−1)𝑠𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛=
𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
𝐿𝑜𝑠𝑠 𝑟𝑎𝑡𝑒 . (1)
The three possible ranges of value for k are as follows:
1. The reactor is sub-critical. k < 1.
2. The reactor is critical. k = 1.
3. The reactor is supercritical. k > 1.
Research and power reactors are designed to be critical (k=1), so the fission reaction is sustained
but controlled. Slight fluctuations below and above 1 are still well controlled. The criticality
constant can be found through the six-factor formula,
𝑘 = 𝜂𝑓𝑝𝜀𝑃𝑇𝑁𝐿𝑃𝐹𝑁𝐿 = 𝑘∞𝑃𝑇𝑁𝐿𝑃𝐹𝑁𝐿 . (2)
Control Blade Graphite Central Hole
Horizontal
Through-port
Fuel Box Control
Blades
7
The terms of the six-factor formula, for a homogenous reactor, are described as followed:
η = average number of neutrons produced per absorption in fuel = νσf/σa
o ν = number of neutrons produced per fission.
o σf = microscopic fission cross section.
o σa = microscopic absorption cross section.
f = thermal utilization = probability neutron is absorbed in fuel, given it is absorbed.
= ΣFa/Σa
o ΣFa = macroscopic absorption cross section of the fuel.
o Σa = macroscopic absorption cross section of the system.
p = resonance escape probability = fraction of fission neutrons that manage to slow down
from fission to thermal energies without being absorbed.
ε = fast fission factor =
Total number of fission neutrons (from both fast and thermal fission)
Number of fission neutrons from thermal fissions
PTNL = Probability of thermal neutrons not leaking.
PFNL = Probability of fast neutrons not leaking
k∞ = criticality constant assuming an infinite assembly (no leakage) [5].
Fig. 4 explicitly details all six factors that determine the critical state of the reactor via an
example of the neutron life cycle with a keff = 1.
The factors η and ε are determined by the fuel (fissile isotope and enrichment) and the energy
distribution of the neutron flux. The two variables highly dependent on the overall design (size,
reflection, etc.) and the fuel-to-moderator ratio are f and p. An increase in the boron content
decreases the thermal utilization, f, since there is another strong thermal neutron absorber in the
system. The fuel-to-moderator ratio directly affects the resonance escape probability, p, since
fission neutrons are less likely to thermalize before being absorbed by the resonance peaks in
238U.
8
Fig. 4. Example neutron life cycle of a nuclear reactor [6].
Reactivity
Reactivity, used to measure the subcritical or supercritical nature of a reactor system, is
mathematically defined as
𝜌 =𝑘−1
𝑘 , (3)
where k is the criticality constant and ρ is the reactivity [7]. This study will investigate reactivity
in terms of Δk/k, which comes out explicitly from Eqn. 3. Most reactor cores are designed to be
inherently supercritical and have excess reactivity – but this reactivity is suppressed through the
use of control rods or blades.
9
Boron Content of Graphite
The raw material used to create the graphite moderator in reactors is laced with trace
amounts of some elements whose nuclei have a very high cross section for absorption of thermal
neutrons [8]. The most prominent of these impurities is 10B. Table II provides the cross sections
of 10B and 235U.
Table II. Comparison of cross sections of 10B and 235U [9].
Total Cross Sections 235U 10B
Thermal (0.0253 eV) 698.2 b 3840 b
Fast (14 MeV) 5.865 b 1.467 b
The 10B poses a serious issue for thermal reactors, especially if the total amount of boron
in the materials of the reactor is unknown. Fig. 5 shows the competing absorption cross sections
of 235U, 238U, and 10B. The 10B clearly dominates at lower energies, allowing its use as a “fission
poison” in a thermal reactor.
Fig. 5. Absorption cross sections for 235U (red), 238U (blue), and 10B (green) [9].
= 10B
= 235U
= 238U
10
Neutron Moderation and the Porosity of Graphite
Neutrons born from fission are produced at a wide range of energies, characterized by the
Watt fission spectrum χ(E), shown in Fig. 6.
Fig. 6. Fission spectrum, χ(E), for thermal neutron-induced fission in 235U [7].
Inspection of Fig. 5 shows that the fission of a 235U nucleus is most efficient at low
energies, specifically thermal energies. The most probable initial energy for a fission-born
neutron is approximately 1 MeV, many orders of magnitude greater than desired for an efficient
thermal reactor. Consequently, neutrons must be slowed down, or “moderated”, via elastic
scattering with a light nucleus. Neutron elastic scattering is characterized by the following
equations:
𝐸′ = 𝐸(1+𝛼)+(1−𝛼)cos𝜃
2 , (4)
𝛼 = (𝐴−1
𝐴+1)
2 , (5)
where E is the initial neutron energy, E’ is the neutron energy after the elastic collision, θ is the
angle of incidence in the center of mass system, and A is the mass number of the scattering
medium [5]. Commercial reactors generally use water as a moderator because of its hydrogen
(A=1) content; on average, neutrons lose half of their energy when colliding with a hydrogen
nucleus. The UFTR uses graphite and water as moderators because of their low absorption and
low mass numbers (A=12 for graphite and A=1 for the hydrogen in water).
11
The probability of a neutron scattering with a nucleus per unit length is characterized by
the macroscopic neutron scattering cross section, Σs. Macroscopic cross sections are derived
from microscopic cross sections, σs, via the relation
𝛴𝑠 = 𝑁𝐶𝜎𝑠 , (6)
𝑁𝐶 =𝜌𝑁𝐴
𝑀 , (7)
where NC is the number density in atoms/cm3, ρ is the mass density in g/cm3, NA is Avogadro’s
number, and M is the molar mass of the atom [7]. Microscopic cross sections, σ, are defined as
the effective area the nuclide presents to the incident neutron in barns (10-24 cm2). The
macroscopic cross section is directly proportional to the mass density of the medium. A decrease
in the density has a significant impact on the reactivity of the core through the resonance escape
probability. Fig. 7 shows the absorption cross section for 238U, which is the more prominent
isotope of uranium in the UFTR fuel. The resonance region spans a significant energy range,
encompassing most of the epithermal region (1eV – 100 keV). Neutrons may be absorbed by the
resonance peaks in 238U before they have the chance to thermalize and fission a 235U nucleus, an
effect enhanced in an undermoderated core.
Fig.7. Total absorption cross section for 238U [9].
Radiation damage to graphite in high temperature reactors (HTR) is generally not an
issue, because high temperatures (~500°C and higher) largely repair damage through thermal
annealing [8]. In the UFTR, the graphite moderator never reaches a high enough temperature to
12
resolve radiation damage. As a result, degradation of the graphite may cause a minute change in
the average density of the material.
Fuel Box Size
Although graphite is considered a moderating medium for the UFTR, water pumped
through the fuel boxes still plays a considerable role in thermalizing the neutrons born from
fission.
During the reactor refuel, a possible issue arose when the fuel was packing tighter than
expected in the fuel boxes. This suggests the fuel boxes are smaller than the recorded
dimensions, or the fuel is bigger than recorded. This poses a neutronics problem, as the effect on
the fuel-to-moderator ratio is non-trivial. Fig. 8 is a visualization of the northwest fuel box in
MCNP. Manipulation of the sizes of the fuel boxes was limited to the error on the specifications
of the boxes, which was less than ± 1/16th of an inch.
Fig. 8. Visual depiction of northwest fuel box in MCNP.
Somewhat analogous to the porosity of graphite discussed earlier, decreasing the size of
the fuel boxes also decreases the resonance escape probability. Unlike the former problem,
where density is the driving factor, shrinking the fuel boxes decreases the moderator-to-fuel
ratio. Although shrinking the fuel boxes likely leads to under-moderation of the system, over-
Fuel Bundle (4) Fuel Box
Water
13
moderation is still possible in a poorly designed core. Fig. 9 illustrates the effects of an over-
moderated and under-moderated system on k. Reduction of k in an under-moderated system is
driven by the resonance escape probability and the thermal utilization factor in an over-
moderated system.
Fig. 9. Effect of over-moderation and under-moderation on k[6].
RESULTS AND DISCUSSION
Benchmark of Original Input
The original input from the HEU-to-LEU conversion and the modified input for the 2010
refuel were run. Results of these runs are shown in Table III.
14
Table III. Results of benchmark criticality runs.
Run Description keff
Standard
Deviation Cycles
Skipped
Cycles
Histories
Per
Cycle
1
Original input from
HEU/LEU conversion 1.00175 0.00013 800 100 50000
2 Modified input (refuel) 1.00162 0.00014 800 100 50000
Although little justification is provided for the changes made, the modified input was
used in the rest of the simulations for this project Fig. 10 is a full, energy independent flux
profile of the UFTR core. The values above, as well the flux profiles agree with the values
presented in the HEU-to-LEU fuel conversion report [1]. As expected, the flux peaks in the
center fuel boxes and follows a cosine shape for a symmetric core.
Fig. 10. Total flux profile of UFTR core from Run 1, showing all six fuel boxes. View is cross-
sectional cut of x-y plane on center boxes.
15
Boron Content of Graphite
Table IV displays the results of the criticality runs with varying boron content in the
graphite. Fig. 11 shows the relationship between the criticality constant and the content of 10B in
the UFTR graphite. Figs. 11 and 12 detail the changes in k and reactivity with varying 10B
content. 1 ppm of 10B is worth 0.0026 Δk/k.
Table IV. Results of criticality runs with varying boron content.
10B Content (ppm) keff
Standard
Deviation Δk/k
0.8 ** 1.00162 0.00014 0.001617
0.5 1.00211 0.00013 0.002106
1 1.00102 0.00013 0.001019
1.5 1.00012 0.00013 0.00012
2 0.99876 0.00013 -0.00124
2.5 0.99761 0.00013 -0.0024 *All runs completed with 800 cycles, 100 skipped cycles, 50000 histories per cycle.
** Original model.
Fig. 11. Relationship between 10B content in graphite and keff. Error bars are encompassed by the
data points.
0.997
0.998
0.999
1.000
1.001
1.002
1.003
0.000 0.500 1.000 1.500 2.000 2.500 3.000
keff
10B (ppm)
16
Fig. 12. Relationship between 10B content in graphite and reactivity. Error bars are encompassed
by the data points.
As expected, the reactivity decreases approximately linearly with increasing boron
content. The core does not reach subcriticality until 2ppm of 10B, a substantial amount for high
purity reactor graphite. Increments of 0.5ppm 10B cause changes in reactivity on the order of
0.1%.
Porosity of Graphite
Table V shows the results of the criticality runs with varying density of graphite. Figs. 13
and 14 detail the changes in k and reactivity with varying density of the surrounding graphite of
the UFTR.
Table V. Results of criticality runs for varying graphite density.
Density of Graphite (g/cm3) keff
Standard
Deviation Δk/k
1.6 ** 1.00162 0.00014 0.001617
1.59 1.00055 0.00013 0.00055
1.58 0.999 0.00013 -0.001
1.57 0.99786 0.00013 -0.00214
1.56 0.99649 0.00013 -0.00352
1.55 0.995 0.00013 -0.00503 *All runs completed with 800 cycles, 100 skipped cycles, 50000 histories per cycle.
-0.003
-0.002
-0.001
0.000
0.001
0.002
0.003
0.000 0.500 1.000 1.500 2.000 2.500 3.000
Rea
ctiv
ity
(Δ
k/k
)
Boron content (ppm)
17
** Original model
Fig. 13. Relationship between the density of the graphite and keff. Error bars are encompassed by
the data points.
Fig. 14. Relationship between the density of the graphite and reactivity. Error bars are
encompassed by the data points.
Decreasing the graphite density has a dramatic effect on the core, causing a subcritical
condition at 1.58 g/cm3, only a 1.25% decrease in density. Despite this decrease being
numerically feasible, an actual decrease of 1% or greater in the density is substantial. Although
0.994
0.995
0.996
0.997
0.998
0.999
1.000
1.001
1.002
1.003
1.540 1.560 1.580 1.600 1.620
keff
Density of Graphite (g/cm3)
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0.000
0.001
0.002
0.003
1.540 1.550 1.560 1.570 1.580 1.590 1.600 1.610
Rea
ctiv
ity (
Δk
/k)
Density of Graphite (g/cm3)
18
1.6 g/cm3 is purely a theoretical density, significant fluctuations from this density should not be
assumed without some experimental measurements.
Fuel Box Size
Table VI shows the results of the criticality runs with varying fuel box sizes. Figs. 15 and
16 show the relationship between the moderator-to-fuel ratio, k, and reactivity. Moderator-to-fuel
ratio was calculated by taking the ratio of the volume of water in each fuel box to the volume of
fuel. The ratio was manipulated by changing the length and width of the fuel boxes.
Table VI. Results of criticality runs for varying moderator-to-fuel ratios.
Moderator-to-Fuel Ratio keff
Standard
Deviation Δk/k
1.18645 ** 1.00162 0.00014 0.001617
1.17710 1.00166 0.00013 0.001657
1.16496 1.00128 0.00014 0.001278
1.15468 1.00121 0.00014 0.001209
1.14441 1.00097 0.00013 0.000969
1.13039 1.00087 0.00013 0.000869 *All runs completed with 800 cycles, 100 skipped cycles, 50000 histories per cycle.
** Original Model
Fig. 15. Relationship between the moderator-to-fuel ratio and keff. The error bars represent one
standard deviation.
1.00060
1.00080
1.00100
1.00120
1.00140
1.00160
1.00180
1.00200
1.100 1.120 1.140 1.160 1.180 1.200
keff
Moderator-to-Fuel Ratio
19
Fig. 16. Relationship between the moderator-to-fuel ratio and reactivity. The error bars represent
one standard deviation.
Large error bars in Fig. 13 are due to the small range of the data. Even with the largest
decrease in fuel box size possible, k only changes a minute amount, much less than 0.1%. The
core never reaches a subcritical condition, which is expected since water is not the primary
moderator. One data point of interest in run 1, the small increase in reactivity, is likely a data
fluctuation well within the recorded uncertainty; it may also indicate some slight overmoderation
with the original fuel box size.
CONCLUSIONS
Of the three parameters investigated, the most dramatic change in reactivity was observed
when the graphite density was changed. The likelihood of this phenomenon being physically
accurate is low, but the effect is significant enough to warrant attention. One way to evaluate the
feasibility of degradation of the graphite is through chemical analysis, specifically an analysis of
indicator elements. The ideal element for such an analysis is boron through the 10B/11B ratio [11].
Investigating this ratio would provide information about the overall accumulated fluence over the
lifetime of the reactor, the condition of the graphite moderator, and the boron content.
The effect on reactivity is likely not due to one variable, but a combination of the three
factors investigated above or others not considered in this report. It is important to recognize the
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.0016
0.0018
0.0020
1.100 1.120 1.140 1.160 1.180 1.200
Rea
ctiv
ity (
Δk
/k)
Moderator-to-Fuel Ratio
20
limitations of a purely theoretical study and the limitations of a tool like MCNP. An MCNP
model is only as accurate as the user makes it, and the physical correspondence of a model with
reality can only be confirmed with experiments. Unfortunately, the UFTR was not operational
for the purposes of this study. Nonetheless, the results suggest all three parameters investigated –
the boron content, the porosity of the graphite, and the fuel box size – are contributing to the
unanticipated loss in reactivity.
ACKNOWLEDGMENTS
The author would like to acknowledge Dr. Alireza Haghighat for his guidance, Dr.
Gabriel Ghita for the use of his original MCNP model of the UFTR, and Dr. Ce Yi for his help in
learning MCNP.
21
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Submittal report to cover analyses of University of Florida Training Reactor (UFTR)
conversion from HEU to LEU, Technical report, University of Florida, 2005.
[2]. PATEL A.D. Detailed neutron flux characterization of the experimental shield tank facility
at the UFTR. University of Florida; 2009.
[3]. “A General Monte Carlo N-Particle Transport Code, Version 5X-5”, MONTE CARLO
TEAM, MCNP5 —, LA-UR-03-1987, Los Alamos National Laboratory, Los Alamos,
NM, (2003).
[4]. R. A. SCHWARZ and L. L. CARTER, “Visual Editor to Create and Display MCNP Input
Files,” Transactions of the American Nuclear Society, 77, 223, (1997).
[5]. DUDERSTADT, J.J., HAMILTON, L.J., 1976 Nuclear reactor analysis / James J.
Duderstadt, Louis J. Hamilton. Wiley, New York.
[6]. DOE FUNDAMENTALS HANDBOOK. NUCLEAR PHYSICS AND REACTOR
THEORY. Volume 2 of 2. DOE-HDBK-1019/2-93.
[7]. STACEY, W.M., 2001 Nuclear Reactor Physics / Weston M. Stacey, Jr . Wiley, New York.
[8]. CARTER, R.L., EGGLESTON, R.R., Moderator Graphite for High Temperature Reactors.
Nuclear Engineering and, Manufacturing, North American Aviation, Inc., Downey,
California. August 1, 1955.
[9]. Table of Nuclides. Web. 17 Feb. 2011. .
[10]. SJODEN, G.E., MOCK, T.A., ENU 4103/6937 Nuclear Reactor Analysis & Computation I
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[11]. GESH, C.J., A Graphite Isotope Ratio Method Primer—A Method for Estimating
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