Conference PaperPDF Available

Multiphysics Explicit Simulation of Randomly Distributed TRISO Fuel Particles

Authors:
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 eective 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
eective 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 eective 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 aects the temperature of the material, therefore,
aecting 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 eect 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 dierent 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 eective thermal conductivity. The
eective 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 dierent 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
dierent 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) Eect of power
per TRISO and packing fraction (
pf
), and 2) Eect of random-
ness of the TRISO particles distribution.
Eect of Power/TRISO and pf
Table I summarizes the
ke
results and the maximum tem-
peratures of cases A, B, and C from the multiphysics compu-
tations. The
ke
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 coecient of reactivity. The
dierences between the
ke
values for the dierent packing
fractions is attributed to the dierent 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 dierence 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 eective thermal conductivity has
a less significant eect 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 dierence between
the models increases, attributed to the increasing TRISO power
density.
TABLE I. Multiphysics results for the considered cases.
Case pf(%) Power (W) keke(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
Eect 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 dierent 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 dierent configurations are negligible,
which can be attributed to the relatively high packing fraction.
Lower packing fractions are expected to have higher dier-
ences between the local packing fraction in a small control
volume and the average packing fraction of the fuel compact.
The maximum dierence between the dierent 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
ke
obtained from the explicit modeling is 1.52599
±
44 pcm. Which is within the statistical uncertainty of the
ke
of each configuration. For visualization, the temperature distri-
TABLE II. Multiphysics results of ten dierent TRISO parti-
cles configurations. Power is 200 W and packing fraction is
15%.
Configuration keke(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 eect in the fuel kernel can be seen in Figure 4.
Figures 5 and 6 show the average and maximum temperatures
of the dierent TRISO layers and the matrix, obtained from
all the configurations, respectively. The figures support the
conclusion that the eect 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 dierent configurations as a function of the
radial distance from the center of the fuel compact. Due to the
spatial self-shielding eect 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 dierent con-
figurations.
Fig. 6. Maximum temperatures obtained from the dierent
configurations.
inside the fuel compact is less than that outside. The figure
shows that the power density dier 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, oering 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 eect 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 eect of randomness of the TRISO particles distribution is
negligible on the fuel peak temperature. The eect of random-
ness on the keis 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.
REFERENCES
1.
N. CHO, H. YU, and J. KIM, “Two-Temperature Homog-
enized Model for Steady-State and Transient Thermal
Analyses of a Pebble with Distributed Fuel Particles, An-
nals of Nuclear Energy,36, 448–457 (2009).
2.
R. LI, Z. LIU, Z. FENG, J. LIANG, and L. ZHANG,
“High-Fidelity MC-DEM Modeling and Uncertainty Anal-
ysis of HTR-PM First Criticality, Frontiers in Energy
Research,9(2022).
3.
A. NOVAK ET AL., “Coupled Monte Carlo and Thermal-
Fluid Modeling of High Temperature Gas Reactors Using
Cardinal,” Annals of Nuclear Energy,177, 109310 (2022).
4.
P. F. ET. AL, “Nek5000: Open source spectral element
CFD solver, (2008).
5.
P. ROMANO ET AL., “OpenMC: A State-of-the-Art
Monte Carlo Code for Research and Development, An-
nals of Nuclear Energy,82, 90–97 (2015).
6.
D. GASTON, C. NEWMAN, G. HANSEN, and
D. LEBRUN-GRANDIÉ, “MOOSE: A Parallel Compu-
tational Framework for Coupled Systems of Nonlinear
Equations,” Nuclear Engineering and Design,239,10,
1768–1778 (2009).
7. COREFORM LLC, “Coreform Cubit,” .
8.
J. STERBENTZ, P. BAYLESS, L. NELSON,
H. GOUGAR, J. KINSEY, G. STRYDOM, and
A. KUMAR, “High-Temperature Gas-Cooled Test
Reactor Point Design,” Tech. Rep. INL/EXT-16-38296,
Idaho National Laboratory (2016).
9.
Y. CHIEW and E. GLANDT, “The eect of structure on
the conductivity of a dispersion, Journal of Colloid and
Interface Science,94,1, 90–104 (1983).
10.
A. NOVAK, S. SCHUNERT, R. CARLSEN,
P. BALESTRA, R. SLAYBAUGH, and R. MAR-
TINEAU, “Multiscale Thermal-Hydraulic Modeling of
the Pebble Bed Fluoride-Salt-Cooled High-Temperature
Reactor, Annals of Nuclear Energy,154, 107968 (2021).
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
A high-fidelity model for the first criticality of pebble-bed reactor HTR-PM is built using Monte Carlo (MC) code RMC and discrete element method (DEM) code LAMMPS. Randomly packed TRi-structural ISOtropic (TRISO) particles and fuel pebbles are modeled explicitly. A cone structure on the top of the pebble bed is also taken into account. Criticality calculation result agrees well with the experiment. Uncertainty analysis is carried out considering three inherent aspects: the randomness of MC code, the randomness of TRISO particle and pebble position, and the randomness of mixed pebbles. Results show that these factors have a significant impact on the uncertainty of effective multiplication factor ( k eff ). And the most influential factor is expected to be the randomness of mixed pebbles. The influence of several configuration factors is studied as well. It is observed that the effects of cross-section library, the heterogeneity of TRISO particles, and the angle of pebble bed cone are nonnegligible contributors. However, the results between randomly and regularly placed TRISO particles are not noticeably different.
Article
Cardinal is an open-source application that couples OpenMC Monte Carlo transport and NekRS computational fluid dynamics to the Multiphysics Object-Oriented Simulation Environment (MOOSE), closing neutronics and thermal-fluid gaps in conducting high-resolution multiscale and multiphysics analyses of nuclear systems. We provide an introduction to Cardinal’s software design, data mapping, and multiphysics coupling strategy to highlight our approach to overcoming common challenges in multiphysics simulation. We then describe an application of Cardinal to prismatic High Temperature Gas Reactors (HTGRs) with various combinations of NekRS, OpenMC, BISON, and THM. A high-resolution coupling of NekRS, OpenMC, and BISON provides a reference solution at the unit cell level and shows excellent agreement with a lower-resolution coupling of THM, OpenMC, and BISON. A full core coupling of THM, OpenMC, and BISON resolving the three-dimensional conjugate heat transfer and sub-pin power distribution then provides detailed predictions of HTGR temperatures and the fission distribution.
Article
The complex core geometry of Pebble Bed Reactors (PBRs) necessitates multiscale techniques for fast-turnaround design and analysis. This paper describes the multiscale model implemented in the Pronghorn PBR simulation tool and demonstrates application to steady-state analysis of the Mark-1 Pebble Bed Fluoride-Salt-Cooled High-Temperature Reactor (PB-FHR). Verification of the pebble model with fully-resolved heat conduction shows that material-wise pebble temperatures are predicted to within 10°C over a wide range in thermal conditions. Anisotropic drag models are correlated for the outer reflector blocks using COMSOL, providing closures for modeling of bypass flows. With a porous media model of the outer reflectors, the core bypass fraction and fuel, reflector, and structural material temperatures are predicted for a number of different inflow conditions. This work demonstrates the full-core analysis capabilities of the Pronghorn application and enables comprehensive reactor analysis with the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework.
Article
In a pebble-bed type very high temperature gas-cooled reactor (VHTGR), a typical fuel pebble consists of over ten thousand five-layer TRISO particles in a graphite-matrix. The high heterogeneity in composition leads to difficulty in explicit thermal calculation of pebble fuels. Thus, a homogenization model becomes essential. Currently, a simple volumetric-average thermal conductivity approach is used. However, this approach is non-conservative and underestimates the fuel temperature.In this paper, we describe a homogenization model that is not only easy to implement but also gives a more realistic temperature distribution in a fuel pebble, providing the fuel-kernel and graphite-matrix temperatures separately. Steady-state and transient thermal analyses are performed using the homogenization model, and the point kinetics model is then coupled with the homogenization model to incorporate fuel-kernel temperature feedback.
Article
The equilibrium hard-sphere fluid is used to model the structure of dispersions of identical impenetrable spheres within a matrix. Pair-correlation functions adjusted to Monte Carlo simulation results, and reported in the literature, are used to compute the contribution of pairs of spheres to the effective thermal (or electrical) conductivity of the dispersions. An improved form of Maxwell's equation is proposed, which is correct to order φ2, where φ is the volume fraction of the dispersed phase. A comparison with experimental measurements shows good agreement over a wide range of conditions. The approach fails for highly concentrated dispersions of very conducting spheres. Alternative models are discussed which are appropriate in this limit.
Article
Systems of coupled, nonlinear partial differential equations (PDEs) often arise in simulation of nuclear processes. MOOSE: Multiphysics Object Oriented Simulation Environment, a parallel computational framework targeted at the solution of such systems, is presented. As opposed to traditional data-flow oriented computational frameworks, MOOSE is instead founded on the mathematical principle of Jacobian-free Newton–Krylov (JFNK). Utilizing the mathematical structure present in JFNK, physics expressions are modularized into “Kernels,” allowing for rapid production of new simulation tools. In addition, systems are solved implicitly and fully coupled, employing physics-based preconditioning, which provides great flexibility even with large variance in time scales. A summary of the mathematics, an overview of the structure of MOOSE, and several representative solutions from applications built on the framework are presented.
Nek5000: Open source spectral element CFD solver
  • P F Et
  • Al
P. F. ET. AL, "Nek5000: Open source spectral element CFD solver," (2008).
OpenMC: A State-of-the-Art Monte Carlo Code for Research and Development
  • P Romano
  • Al
P. ROMANO ET AL., "OpenMC: A State-of-the-Art Monte Carlo Code for Research and Development," Annals of Nuclear Energy, 82, 90-97 (2015).
High-Temperature Gas-Cooled Test Reactor Point Design
  • J Sterbentz
  • P Bayless
  • L Nelson
  • H Gougar
  • J Kinsey
  • G Strydom
  • A Kumar
J. STERBENTZ, P. BAYLESS, L. NELSON, H. GOUGAR, J. KINSEY, G. STRYDOM, and A. KUMAR, "High-Temperature Gas-Cooled Test Reactor Point Design," Tech. Rep. INL/EXT-16-38296, Idaho National Laboratory (2016).