Running the Input File

SAM can be run on Linux, Unix, and MacOS. Due to its dependence on MOOSE, SAM is not compatible with Windows. SAM can be run from the shell prompt as shown below

sam-opt -i pbr.i

Results

There are three types of output files:

1. pbr_csv.csv: this is a csv file that writes the user-specified scalar and vector variables to a comma-separated-values file. The data can be imported to Excel for further processing or in Python using the csv module, Pandas, or other methods.

2. pbr_out_cp: this is a sub-folder where snapshots of the simulation data including all meshes, solutions are saved. Users can restart the run from where it ended using the file in the checkpoint folder.

3. pbr_out.displaced.e: this is an ExodusII file that has all mesh and solution data. Users can use Paraview to open this .e file to visualize, plot, and analyze the data.

Steady-state

The steady-state simulation results are presented in this section. A qualitative comparison of the reflector temperature distribution between the SAM and Griffin/Pronghorn simulations (Stewart et al., 2021) is presented in Figure 1. Note that the temperature distribution in the pebble bed core is not shown here and will be discussed later because the heat structures in each of SAM’s PBCoreChannel appear as a one-dimensional line. Overall, a good agreement is observed between SAM’s and Griffin/Pronghorn’s results where the temperature distributions appear to be similar with the upper half of the core being at a lower temperature compared to the bottom half. The bottom reflectors located under the pebble bed core appear to be the hottest, with a peak temperature of roughly 1100 K in both cases. Furthermore, most of the outer reflectors appear to be in a relatively low temperature compared to the bottom reflectors. This indicates that during steady-state normal operation, only a small amount of heat is conducted radially from the core to the outer reflectors as forced convection is the primary heat removal mechanism in the core.

Figure 1: Comparison of the reflector temperature distribution between the SAM and Griffin/Pronghorn simulation(Stewart et al., 2021).

The radial distributions of the maximum, minimum, and mean fuel and kernel temperatures in the pebble bed from SAM are shown in Figure 2. Note that 'fuel' denotes the overall matrix temperature of the pebbles. The temperature decreases from the innermost to the outermost core channels, following the radial distribution of the power prescribed to the core channels. Additionally, the close proximity of the outer channels to the reflectors allows heat from these channels to escape to the reflectors, further lowering their temperatures. As expected, the kernel temperature is consistently higher than the fuel/matrix temperature.

Figure 2: Radial distributions of fuel and kernel temperatures in the pebble bed from SAM.

The solid and fluid axial temperature profiles in the pebble bed core from SAM and Griffin/Pronghorn are compared in Figure 3 and Figure 4, respectively. Note that SAM’s results are represented by the solid lines while Griffin/Pronghorn’s results are represented by the dashed lines. The overall trends between the two sets of simulations are largely similar. For the solid temperature, a good agreement is observed between SAM and Griffin/Pronghorn in the upper half of the core. However, in the bottom half, SAM predicts a slightly higher solid temperature in the two innermost core channels, namely 'F-1' and 'F-2'. Both models predict a small increase of temperature in the fuel chute, likely due to reduced heat transfer from the fuel pebbles to the surrounding reflectors. The fluid temperature profiles from SAM and Griffin/Pronghorn show good agreement where they have similar trend as the solid temperatures. Lastly, SAM predicts an average coolant outlet temperature of 1020 K, which is 4 K lower than Griffin/Pronghorn’s prediction of 1024 K (Stewart et al., 2021). The discrepancy is likely because the SAM model predicts a higher heat transfer from the core to the side reflectors and subsequently a higher heat loss via the RCCS panels.

Figure 3: Comparison of the steady-state solid temperature axial profiles between SAM and Griffin/Pronghorn.

Figure 4: Comparison of the steady-state coolant temperature axial profiles between SAM and Griffin/Pronghorn.

Transient

For the transient analysis, a load-following case with a varying inlet mass flow rate is selected, similar to the 100-40-100 load following exercise of PBMR-400 (OECDNEA, 2013). This case is chosen for this work because it tests not only the thermal hydraulics modeling of SAM but also its neutronics modeling with the point kinetics equations. The six-group formulation of the PKE is used here with parameters such as the delayed neutron fraction, precursor decay constant, and the local reactivity coefficients obtained from the Griffin/Pronghorn simulation (Stewart et al., 2021).

Temperature reactivity feedbacks are modeled in the pebble bed core and the surrounding reflectors. Note that coolant density and fuel axial expansion feedbacks are not included here. The reactivity coefficients are divided into three groups, namely the coefficients for the fuel, moderator, and reflector regions. The fuel and moderator coefficients contribute primarily to the reactivity feedback in the pebble bed core with a small amount of contribution from the reflector coefficients near the outer edge of the pebble bed core. Meanwhile, the reflector coefficients are responsible for the reactivity feedback in the reflector regions. The reactivity feedback of the fuels is determined based on the fuel kernel temperature, which in SAM is calculated using the model by TINTE (Gerwin et al., 2010) while the reactivity feedbacks for the reflectors and moderators are calculated with the solid temperature. Given the difference in the temperature used for calculating the reactivity feedback in the pebble bed core, the heat structures in the pebble bed core are divided into three layers of fuel, moderator, and reflector where each layer is prescribed with a reactivity coefficient according to its type. Such distinction is not necessary in the reflector region surrounding the pebble bed core because the reactivity feedback in this region depends almost entirely on the reflector coefficients.

As shown in Figure 5, the inlet flow rate is reduced from the nominal value of 78.6 kg/s (100%) to 19.65 kg/s (25%) over the course of 15 minutes. The flow rate is kept at 25% for 30 minutes before it is ramped back up to 100% over 15 minutes.

Figure 5: Coolant mass flow rate for the load-following transient simulation.

The reactivities and reactor power predicted by SAM and Griffin/Pronghorn (Stewart et al., 2021) during transient are shown in Figure 6 and Figure 7, respectively. Note that the SAM results are represented by the solid lines while the Griffin/Pronghorn results are the dashed lines. Despite some differences in both sets of predictions, their overall trends agree relatively well. When flow rate is decreased, the reduction in heat removal from the fuel causes the overall core temperature to rise, leading to negative total reactivity, which in turn decreases the reactor power. As the flow rate is held constant at 25%, the change of temperature decreases and causes the reactivity to be near zero. Conversely, when the flow rate is increased, the overall core temperature decreases suddenly due to improved heat removal. The reactivity shows a sharp increase and becomes positive, raising the reactor power, before gradually decreasing back to zero as the temperature stabilizes.

Figure 6: Reactivities predicted by SAM in the load-following transient simulation.

Figure 7: Total reactor power predicted by SAM in the load-following transient simulation.

During transient, the power decreases linearly from the nominal value of 200 MW to roughly 55 MW as flow rate is decreased. The power maintains at this level for the next 30 minutes as the flow rate is held constant before rising back to the nominal value as the flow rate is ramped back up. The power predicted by SAM is consistently lower than Griffin/Pronghorn's prediction by approximately 2 MW, which is likely caused by the difference in the reactivities predicted by both simulations.

The mean fuel and kernel temperatures, moderator temperature, and reflector temperature are shown in Figure 8, Figure 9, and Figure 10, respectively. During the initial ramp down, despite the increasing fuel temperature, the kernel temperature decreases due to the reduction of reactor power as shown in Figure 7. As the fuel (Doppler) reactivity feedback is calculated with the kernel temperature, a decrease in the kernel temperature causes the fuel reactivity to increase. At the same time, the moderator temperature increases and leads to a decrease of reactivity. On the other hand, given that the reflector reactivity coefficients are primarily positive, an increase in the reflector temperature causes the reflector reactivity to increase. It is also observed that the total reactivity is dominated by the moderator reactivity. During the constant flow rate stage, the kernel, moderator, and reflector temperatures are relatively constant with each experiencing minor changes. As a result, the respective reactivities remain largely constant and produce a total reactivity of approximately zero. Lastly, during the ramp up stage, the fuel and kernel temperatures increase which then cause the reactivity of the fuel to decrease. Conversely, the increased coolant flow reduces the moderator temperature and leads to an increase of moderator reactivity. The reflector temperature also decreases in this stage, resulting in a decrease of reflector reactivity. Overall, the total reactivity shows a sharp initial increase before dropping gradually as the flow rate is increased back to the nominal level. During the course of the transient, the total reactivity is shown to be primarily influenced by the reactivity feedback of the moderator.

The mean fuel and kernel temperatures, moderator temperature, and reflector temperature predicted by SAM are shown in Figure 8, Figure 9, and Figure 10, respectively. Note that the SAM results shown here are the volume-averaged temperature of a particular component. Volume-averaging is necessary to ensure that the mean temperatures are not skewed by the number of sections of a component. For instance, due to its complex geometry, the lower reflector region is comprised of a greater number of sections compared to the side reflector which has a comparatively simpler geometry. Thus, without volume-averaging, the mean temperature will be skewed towards the lower reflector region.

For the SAM prediction, during the initial ramp down, despite the increasing fuel temperature, the kernel temperature decreases due to the reduction of reactor power as shown in Figure 7. As the fuel (Doppler) reactivity feedback is calculated with the kernel temperature, a decrease in the kernel temperature causes the fuel reactivity to increase. At the same time, the moderator temperature increases and leads to a decrease of reactivity. On the other hand, given that the reflector reactivity coefficients are primarily positive, an increase in the reflector temperature causes the reflector reactivity to increase. It is also observed that the total reactivity is dominated by the moderator reactivity. During the constant flow rate stage, the kernel, moderator, and reflector temperatures are relatively unchanged with each experiencing minor changes. As a result, the respective reactivities remain largely constant and produce a total reactivity of approximately zero.

Finally, during the ramp up stage, the fuel and kernel temperatures increase and cause the reactivity of the fuel to decrease. Conversely, the increased coolant flow reduces the moderator temperature and leads to an increase of moderator reactivity. Similarly, the reflector temperature decreases in this stage due to the improved cooling. However, given that the reactivity coefficients of the reflectors are positive, a decrease in reflector temperature results in a decrease in reflector reactivity. Lastly, the resultant total reactivity shows a sharp initial increase before dropping gradually as the flow rate is increased back to the nominal level.

Figure 8: Average fuel and kernel temperatures predicted by SAM in the load-following transient simulation.

Figure 9: Average moderator temperature predicted by SAM in the load-following transient simulation.

Figure 10: Average reflector temperature predicted by SAM in the load-following transient simulation.

The average coolant temperature at the outlet is shown in Figure 11. The overall trend of the outlet temperature matches the the trends of the fuel and moderator temperatures. The coolant temperature increases as the flow rate is reduced. On the other hand, when the flow rate is held constant at 25% of its nominal value, the outlet temperature shows a small decrease of roughly 5 K over a span of 30 minutes. Lastly, when flow rate is increased back to the nominal value, the outlet temperature decreases rapidly before stabilizing at roughly the same value as the temperature at the start of the transient.

Figure 11: Average coolant temperature at the outlet predicted by SAM in the load-following transient simulation.

Given that the SAM and Griffin/Pronghorn models are essentially two different approaches, some differences inevitably exist between the steady-state temperatures predicted by both models. Furthermore, these differences could be further exaggerated during transient. Hence,a direct comparison of temperatures between the two sets of results could be misleading. As a result, it is more insightful to compare the evolution of temperature of different regions in the reactor during transient. To achieve that, the temperature changes of the fuel, moderator, and reflector are shown in Figure 12, Figure 13, and Figure 14, respectively. Note that the temperature change is defined as the difference between the temperature at the start of transient and the temperature at time t during transient. It should be pointed out that a positive value represents a drop in temperature while a negative value represents an increase in temperature with respect to the temperature at the start of transient.

As shown in Figure 12, the change in fuel temperature predicted by Griffin/Pronghorn and SAM are relatively similar during the flow ramp down phase, with the prediction from SAM showing a larger increase of temperature due to a reduction in heat removal by the coolant. However, during the constant-flow phase, the SAM prediction shows a decrease of temperature ranging from 1-4 K while the Griffin/Pronghorn prediction is relatively uniform with only a small increase of temperature of roughly 1-2 K. The difference in trends could be attributed to the difference in power generated by the core where the SAM prediction experiences a drop of roughly 500 kW of core power over the same period, thus leading to a small decrease in fuel temperature. Finally, in the ramp up phase, the differences from both models start to converge and eventually arrive at a reasonably good agreement once the mass flow rate is returned to the nominal level. It should also be pointed out that similar 'spikes' are observed in both sets of results at the termination of flow ramp down (15 min) and the initiation of flow ramp up (45 min).

Figure 12: Comparison of the fuel temperature change predicted by SAM and Griffin/Pronghorn.

The comparison of the change in moderator temperature is shown in Figure 13 where the predictions from both models are observed to have a good overall agreement. In the flow ramp down phase, the increase of moderator temperature predicted by SAM and Griffin/Pronghorn are almost the same with that by SAM being consistently greater by roughly 1 K. In the constant-flow phase, the temperature change predicted by SAM and Griffin/Pronghorn start to diverge as the reactor power predicted by SAM experiences a small decrease. However, the difference diminishes when the flow is ramped up to its nominal value, which is similar to the behavior of the fuel temperature discussed previously. Finally, the change of reflector temperature is shown in Figure 14. It is seen that the predictions from both models agree well with each other in terms of the overall trend and magnitude, with the only difference being the prediction by Griffin/Pronghorn showing a smoother change over time compared to SAM's prediction.

Figure 13: Comparison of the moderator temperature change predicted by SAM and Griffin/Pronghorn.

Figure 14: Comparison of the reflector temperature change predicted by SAM and Griffin/Pronghorn.

Acknowledgements

This work is supported by U.S. DOE Office of Nuclear Energy’s Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. The submitted manuscript has been created by UChicago Argonne LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The authors would like to acknowledge the support and assistance from Dr. Ryan Stewart and Dr. Paolo Balestra of Idaho National Laboratory in the completion of this work.