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Multiphysics Explicit Simulation of Randomly Distributed TRISO Fuel Particles
A. Hegazy, A.J. Novak, and Rizwan-uddin
Department of Nuclear, Plasma, and Radiological Engineering
The Grainger College of Engineering, University of Illinois at Urbana-Champaign
ahegazy2@illinois.edu
INTRODUCTION
High Temperature Gas Reactors (HTGRs) are a notable
advancement in nuclear energy systems, thanks to their in-
trinsic safety qualities and effective heat transfer properties.
A key factor in the design and safety evaluation of HTGRs
is the multiphysics simulation of fuel compacts. These fuel
compacts are comprised of a vast number of millimeter-sized,
multi-layered, TRistructural ISOtropic (TRISO) fuel particles.
The complex and varied composition of TRISO particles influ-
ences the neutronics and heat transfer within the fuel compact.
The explicit modeling of TRISO particles in fuel com-
pacts demands significant computational resources, therefore
researchers have been investigating simplified, homogenized
models of these compacts. This primarily involves using an
effective thermal conductivity to solve the heat conduction
within the fuel element. Conventionally, a fuel element is con-
sidered as a homogeneous material with uniform heat genera-
tion, and the temperature distribution within the fuel compact
is determined using this effective thermal conductivity [
1
,
2
].
In this work we compare the explicit multiphysics modeling of
the fuel compacts including the TRISO fuel particles, against
the homogeneous modeling of the fuel compacts. The neu-
tron transport is coupled to the heat conduction through the
power, which affects the temperature of the material, therefore,
affecting the neutron cross-sections. Cardinal [
3
] is used in
this work to couple neutron transport and heat conduction.
Cardinal is an open-source application that wraps the NekRS,
spectral element Computational Fluid Dynamics (CFD) code
[
4
] and the OpenMC Monte Carlo radiation transport code [
5
]
within the MOOSE framework [6].
The objectives of this work are as follows; 1) compare
the explicit modeling of TRISO fuel particles versus the fully
homogenized modeling in the context of heat conduction, 2)
discuss the effect of the random distribution of the TRISO fuel
particles within the fuel compact, 3) demonstrate the usage of
Cardinal for high-resolution multiphysics analyses of HTGRs,
and 4) illustrate the plug-and-play nature of MOOSE and how
it facilitates multiphysics and multiscale modeling.
COMPUTATIONAL TOOLS
OpenMC
OpenMC is an open-source continuous-energy neu-
tron and photon Monte Carlo code with capabilities for k-
eigenvalue and fixed source calculations. OpenMC supports
two different geometry modes – (i) constructive solid geometry
and (ii) DAGMC Computer-Aided Design (CAD) geometry –
each of which requires unique considerations when applying
multiphysics temperature/density/geometry feedback [5].
Cardinal
Cardinal is an open-source (https://github.com/
neams-th-coe/cardinal) Multiphysics Object-Oriented
Simulation Environment (MOOSE) wrapping of NekRS
(spectral element CFD) and OpenMC (Monte Carlo transport).
Cardinal makes use of the transfer capabilities in MOOSE,
which provides physics-agnostic data transfers through mesh
mirroring, mesh interpolations, nearest node interpolations,
and postprocessors [
3
]. More details on the specific models
used for our analyses will be introduced shortly.
Cubit
Cubit is a software toolkit developed by Sandia National
Laboratory for generating finite element meshes. In this work,
Cubit 2021.11 is used to generate the unit cell problem using
tetrahedral meshes [7].
UNIT CELL PROBLEM
The problem consists of a generic prismatic HTGR unit
cell [
8
] of height 1.0 cm. Note that an extra graphite layer was
added to each TRISO particle mesh for the following reasons:
1.
Allowing for coarser mesh on the matrix side by eliminat-
ing a few occurrences of matrix cell centroids mapping
to the OPyC layer.
2.
Making sure that the TRISO particles are not touching
when generating their random locations,which more ac-
curately represents the actual manufacturing procedure
for the TRISO compacts.
TRISO fuel particles are randomly distributed in a cylindrical
fuel compact within a triangular graphite matrix, which fea-
tures a coolant channel at each corner, occupying one-sixth
of the matrix. The coolant channels are only considered in
the neutronics calculations and are not coupled to the thermal
analysis. This assumption is valid for the following reason;
1) the helium coolant is transparent to neutrons, and 2) the
height of the problem is only 1.0 cm, therefore, the variation
in the coolant temperature is small. The geometry is shown in
Figure 1.
MULTIPHYSICS MODELING AND SIMULATION
Heat conduction is coupled to the neutron transport via
Picard iteration. OpenMC (neutron transport solver) and
MOOSE heat conduction module (heat conduction solver)
are coupled through MOOSE’s hierarchical MultiApp system,
where each application may communicate with one applica-
tion “above” it, some applications “below” it, and side-to-side
Fig. 1. Simulated fuel compact with randomly distributed
TRISO fuel particles (packing fraction 15%).
sibling transfers. Two approaches have been considered to
model the TRISO particles in the matrix within the context of
heat conduction – 1) explicit modeling, and 2) homogeneous
modeling. In all the cases, the neutronics boundary conditions
are all reflective. On the other hand, for the heat conduction
solver, the boundary conditions are Neumann on top and bot-
tom of the matrix, and Dirichlet on the sides of the matrix.
The Dirichlet boundary condition is computed as the average
temperature of the coolant, 760.65 K [8].
Explicit Modeling
In the explicit modeling, two heat conduction sub-apps
are employed to solve for two levels; 1) TRISO fuel particles,
and 2) the graphite matrix surrounding the TRISO fuel parti-
cles which includes the fuel compact and the triangular matrix,
assuming perfect contact. The advantage of having two heat
conduction sub-apps is the ability to separate the mesh of the
TRISO particles from the mesh of the matrix. This separa-
tion implies that the mesh refinement for the TRISO particles
does not influence the refinement of the matrix mesh. The
capability to independently solve heat conduction at these two
interconnected levels showcases the versatility and “plug-and-
play” physics adaptability in Cardinal. Figure 2 shows the data
transfers and MultiApp hierarchy used in the explicit model-
ing. The number of mesh elements for the packing fractions
5%. 7.5%, 10.0%, 12.5%, and 15.0% are 9007035, 13153932,
17288193, 21413291, and 25575317, respectively.
Fig. 2. Explicit modeling data transfers used for coupling
OpenMC with MOOSE heat conduction module. Note that
OpenMC model is colored by cell.
Homogeneous Modeling
In the fully homogenized modeling only the homogenized
matrix is considered with effective thermal conductivity. The
effective thermal conductivity in this work is computed using
the Volume-Weighted homogenization (VWH) method.
CASES CONSIDERED FOR ANALYSIS
To cover a wide variety of ranges of packing fractions
and powers, three different set of cases were considered for
analysis. The amount of fissile material is constant per TRISO.
•
Case A: Five packing fractions (5.0%, 7.5%, 10.0%,
12.5%, and 15.0%) are considered with constant power
of 200 W.
•
Case B: Five packing fractions (5.0%, 7.5%, 10.0%,
12.5%, and 15.0%) are considered with fixed power per
TRISO of 2.1907e-01 W [8].
•
Case C: One packing fraction (15.0%) is considered with
different power range (50, 100, 150, 300, and 400 W).
Additionally, to account for the impact of the random distri-
bution of TRISO fuel particles, Case A (packing fraction of
15.0%), was examined. For this case, ten distinct random
distributions of the TRISO particle locations within the fuel
compact were evaluated.
RESULTS
This section is divided into two parts; 1) Effect of power
per TRISO and packing fraction (
pf
), and 2) Effect of random-
ness of the TRISO particles distribution.
Effect of Power/TRISO and pf
Table I summarizes the
keff
results and the maximum tem-
peratures of cases A, B, and C from the multiphysics compu-
tations. The
keff
obtained from the homogeneous modeling is
higher than that predicted from the explicit modeling in all the
cases, due to the under-prediction of fuel kernel temperatures
and the negative temperature coefficient of reactivity. The
differences between the
keff
values for the different packing
fractions is attributed to the different moderator to fuel ratio.
The homogeneous modeling approach always under-predicts
the maximum temperature, which is an important safety factor.
Case A demonstrates that the difference in maximum tempera-
tures between the two modeling approaches decreases as the
packing fraction increases, due to lower power per TRISO.
In Case B, changing the effective thermal conductivity has
a less significant effect on the comparison method. Further
research is needed to evaluate other homogenization methods
like Chiew-Glandt [
9
]. Case C shows that with higher total
power and power density, the temperature difference between
the models increases, attributed to the increasing TRISO power
density.
TABLE I. Multiphysics results for the considered cases.
Case pf(%) Power (W) keff∆keff(pcm) Max. T(K) ∆Max. T(K)
(Explicit) (Homog.-Explicit) (Explicit) (Explicit - Homog.)
A 5.0 200.0 1.69885±27 770 ±39 1.27652e+03 302.77
A 7.5 200.0 1.66064±28 730 ±40 1.17406e+03 198.34
A 10.0 200.0 1.61481±30 613 ±42 1.11939e+03 141.63
A 12.5 200.0 1.56822±30 680 ±44 1.08715e+03 107.28
A 15.0 200.0 1.52608±31 472 ±45 1.06578e+03 83.71
B 5.0 41.7 1.71104 ±27 210 ±39 8.68153e+02 63.06
B 7.5 62.6 1.67151 ±28 259 ±40 8.90046e+02 62.12
B 10.0 83.4 1.62403 ±30 428 ±42 9.10252e+02 59.05
B 12.5 104.3 1.57760 ±30 385 ±43 9.30798e+02 55.85
B 15.0 125.1 1.53344 ±31 329 ±45 9.51952e+02 52.78
C 15.0 50.0 1.53982 ±31 105 ±44 8.36888e+02 20.88
C 15.0 100.0 1.53531 ±31 300 ±45 9.13576e+02 42.21
C 15.0 150.0 1.53089 ±31 344 ±45 9.90171e+02 63.45
C 15.0 300.0 1.51806 ±31 642 ±45 1.21964e+03 126.86
C 15.0 400.0 1.51046 ±31 868 ±45 1.37214e+03 168.65
Effect of Randomness
TRISO particles are randomly distributed in the fuel com-
pact. OpenMC is used to generate the TRISO particle locations
and place them in a lattice. OpenMC uses random sequential
packing based on a specific packing fraction to generate the
particles locations. Ten different distributions (TRISO config-
urations) were generated using OpenMC and the multiphysics
results are shown in Table II. The table shows that the dif-
ferences between the different configurations are negligible,
which can be attributed to the relatively high packing fraction.
Lower packing fractions are expected to have higher differ-
ences between the local packing fraction in a small control
volume and the average packing fraction of the fuel compact.
The maximum difference between the different configurations
in the peak temperature is 6.054 K. The average peak temper-
ature of all the configurations is 1.0679e+03
±
1.6915 K. The
average
keff
obtained from the explicit modeling is 1.52599
±
44 pcm. Which is within the statistical uncertainty of the
keff
of each configuration. For visualization, the temperature distri-
TABLE II. Multiphysics results of ten different TRISO parti-
cles configurations. Power is 200 W and packing fraction is
15%.
Configuration keff∆keff(pcm) Max. Temp (K) ∆Max. Temp (K)
(Explicit) (Homog.-Explicit) (Explicit) (Explicit - Homog.)
1 1.52581 ±31 551 ±44 1.067743e+03 85.6739
2 1.52528 ±31 578 ±44 1.068782e+03 86.7129
3 1.52613 ±31 501 ±44 1.068056e+03 85.9869
4 1.52568 ±31 573 ±44 1.063895e+03 81.8259
5 1.52616 ±31 493 ±44 1.068166e+03 86.0969
6 1.52627 ±31 527 ±44 1.069790e+03 87.7209
7 1.52604 ±31 503 ±44 1.068068e+03 85.9989
8 1.52640 ±31 464 ±44 1.066988e+03 84.9189
9 1.52612 ±31 510 ±44 1.067473e+03 85.4039
10 1.52604 ±31 492 ±44 1.069949e+03 87.8799
bution and the power density of configuration 1 obtained from
the explicit modeling are shown in Figures 3 and 4. The spatial
self-shielding effect in the fuel kernel can be seen in Figure 4.
Figures 5 and 6 show the average and maximum temperatures
of the different TRISO layers and the matrix, obtained from
all the configurations, respectively. The figures support the
conclusion that the effect of the random distribution of the
TRISO particles in the fuel compact is negligible on the heat
conduction. Figure 7 shows the normalized average power
density for the different configurations as a function of the
radial distance from the center of the fuel compact. Due to the
spatial self-shielding effect of the TRISO particles, the power
Fig. 3. Temperature distribution of configuration 1 obtained
from the explicit modeling.
Fig. 4. Power density of configuration 1 obtained from the
explicit modeling.
Fig. 5. Average temperatures obtained from the different con-
figurations.
Fig. 6. Maximum temperatures obtained from the different
configurations.
inside the fuel compact is less than that outside. The figure
shows that the power density differ the most at the innermost
part of the compact, which can be explained by the statistical
noise.
Fig. 7. Normalized average power density (explicit modeling).
CONCLUSION AND FUTURE WORK
Cardinal, an open-source code developed within the
MOOSE framework, couples NekRS spectral element CFD
and OpenMC Monte Carlo radiation transport, offering de-
tailed multiphysics feedback for various nuclear engineering
applications. This paper employs Cardinal to study a unit cell
problem in a generic HTGR. The unit cell is composed of
TRISO fuel particles embedded in a graphite matrix. OpenMC
is coupled to the MOOSE heat conduction module to model
the multiphysics of this unit cell.
One of the two main objectives of this research was to
compare the common method of modeling TRISO fuel par-
ticles, which employs a homogenized representation of the
fuel compact, with a fully explicit modeling approach. In
the explicit approach, each TRISO particle is resolved and
integrated into the multiphysics framework. The second objec-
tive was to study the effect of the randomness of the TRISO
fuel particles distribution on the multiphysics. The explicit
modeling is computationally expensive compared to the homo-
geneous modeling. The comparison between the explicit and
homogeneous models showed that the homogeneous approach
under-predicts the peak fuel temperature. The maximum dif-
ference between the homogeneous and explicit models occurs
for the case of high power per TRISO. It was also found that
the effect of randomness of the TRISO particles distribution is
negligible on the fuel peak temperature. The effect of random-
ness on the keffis within the statistical uncertainty.
Future work will explore the Heat Source Decomposi-
tion (HSD) method, which is a form of multi-scale approach
[
10
]. HSD is expected to better predict the temperature within
the TRISOs compared to the homogeneous modeling, while
computationally less expensive than the explicit modeling.
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