This section illustrates some of the results obtained using HxF with discrete motion for the HTGR test case. In particular, it discusses the convergency to equilibrium and analyzes the equilibrium parameters, both for incore and discarded pebbles.
A. Statistical considerations
1. Equilibrium state
Determining the convergence criteria is essential when searching for equilibrium in a PBR. Two different metrics were used in this work. First, the evolution of global parameters, such as the multiplication factor keff and the conversion ratio (CR), indicate the reactor's overall state. In this context, the equilibrium state is determined when these parameters have consistent trends with the fine motion step for three complete core cycles. Figure 4 shows how these two parameters have similar behaviors.
A first oscillatory trend is observed from 10 to about 30 passes. Oscillations result from motion steps substantially larger than the diffusion length of neutrons in the core. This first simulation stage with large burnup is beneficial to decrease the computing time to reach equilibrium. Then, a drastic reduction in the burnup step was applied, resulting in smaller motion steps. As soon as this step size reduction happened, the multiplication factor dropped by around 1500 pcm, and the conversion ratio by 7x10− 3. This trend is explained by the lower number of fresh pebbles inserted into the core. Although some oscillatory behavior remains, the core is considered in an equilibrium state when the multiplication factor stays within a band of ± 390 pcm, that is, after 42 passes. Every simulated step after this is regarded as an equilibrium state with a different configuration. In the results presented below, equilibrium average values refer to the average of a given quantity over 297 states (corresponding to 27 passes), whereas representative values for a single state refer to the last equilibrium state simulated. The average equilibrium multiplication factor and conversion ratio are 1.01554 ± 18 pcm and 0.44472 ± 14 pcm, respectively.
Further evidence of achieved equilibrium is sought by analyzing discarded pebbles. Figure 5 shows the evolution of the number of discarded pebbles (thus, of the inserted fresh pebbles) as a function of the total number of passes simulated. The value oscillates around 4,088 pebbles per step, between 3,640 and 4,566, corresponding to about 410 to 515 pebbles/day. Once again, these variations are interpreted as small enough to assume an equilibrium state. The increase of the discarded pebbles at around 50 passes matches the one of the multiplication factor previously observed. Figure 5 also shows the average burnup per pass and how this value, after reaching equilibrium, remains almost constant at around 9.85 MWd/kgHM.
Overall, it is to be observed that the criteria set to determine equilibrium are arbitrary. Given the stochastic nature of PBRs, a core at equilibrium will always present an oscillatory behavior; therefore, it will have to be the responsibility of the modelers to apply their best judgment in determining acceptable criteria for equilibrium.
2. Uncertainties and peaking factors
Performing Monte Carlo calculations in largescale models requires quantifying statistical uncertainties. First, the multiplication factor is not a limiting factor for the test case. The maximum statistical uncertainty obtained during fine steps is 22 pcm, which is small compared to the parameter variations. The main reason for simulating many neutron histories lies in the statistical uncertainty of pebblewise detectors, such as the flux and power tallies. The highest values are found in pebblewise power tallies due to the small size of the TRISO particles in which fissions are scored. The results are summarized in Table 2. Most pebbles (95%) have less than 6% uncertainty on the neutron flux and less than 12% on power.
Nevertheless, as Fig. 6 suggests, the highest uncertainties are, as one can expect, at the bottom of the core, where there is the lowest number of neutron/nuclide interactions. This zone corresponds to where the fuel is the most burned and ready to be discharged, which results in lower power production. In addition, as the histogram shows more clearly, the fraction of pebbles having a very high uncertainty in power and flux is small.
Table 2
Summary of statistical uncertainties in pebblewise detectors.
Detector

Average

Standard Deviation

Minimum

Median

75% Percentile

95% Percentile

Maximum

Thermal flux
(E < 1.86 eV)

2.3%

0.9%

1.0%

2.0%

2.7%

4.3%

6.7%

Epithermal flux
(1.86 eV < E < 0.1 MeV)

2.5%

0.9%

1.1%

2.1%

2.9%

4.5%

8.6%

Fast flux
(E > 0.1 MeV)

3.4%

1.2%

1.5%

3.0%

3.9%

5.9%

12.8%

Power

7.4%

2.1%

3.1%

6.7%

8.3%

12.0%

22.9%

The statistical uncertainty could, instead, have a more significant impact on extreme, maximum, and minimum values. For example, when calculating the pebble power peaking factor, it is impossible to establish to what extent the value for maximum power is a real outlier or a statistical artifact. Nevertheless, Fig. 7 shows that the pebble peaking factor only changes by roughly 5% when calculated using as peak the single highest power value and when using as peak the average 100 highest values. The same is true for neutron flux.
B. ncore equilibrium data
The following subsections provide a summary of the distribution of key incore parameters at equilibrium, such as neutron flux, burnup, and power. It is essential to understand that the data depend highly on the number of times pebbles went through the core. In fact, as the discard criterion is based on the content of 137Cs in the pebble, the number of passes varies depending on the individual history. Table 3 provides the count of pebbles in the core over multiple equilibrium representations grouped by pass number. The number of pebbles is almost evenly distributed between 1 and 9 passes, each accounting for around 10% of the core. Pebbles at the 10th pass, instead, make 8.2% of the total inventory, and an 11th pass is highly improbable. This suggests that pebbles are mostly discarded after 9 and 10 passes, and very few go through the core for 11 passes. Additional discussion on this matter is provided later on concerning discharge burnup.
Table 3
Average incore pebble inventory over multiple equilibrium states.
Pass number

Average number of pebbles

Pebbles fraction [%]

1

46,152

10.2

2

46,206

10.2

3

46,264

10.2

4

46,224

10.2

5

46,076

10.2

6

45,892

10.2

7

45,802

10.1

8

45,811

10.1

9

45,889

10.2

10

37,031

8.2

11

14

3E03

Total

451360

100.0

1. Neutron flux
Figures 8, 9, 10, and 11 show the spatial distribution of thermal (E < 1.86 eV) and fast (E > 0.1 MeV) neutrons in the equilibrium core. As expected, the thermal flux peaks near the radial reflector and toward the top of the core. Indeed, neutrons are thermalized by the reflector, and once they reenter the core, they do not travel long distances before being absorbed. In addition, the hollowcylindrical nature of the core leads to a geometrical peak around the axial and radial centers of the bed while leading to neutrons leakage around the corners. However, since pebbles are inserted from the top and discharged at the bottom and due to the large accumulated burnup per pass, pebbles experience a more significant flux, both thermal and fast, towards the top of the core.
Two observations can be made regarding the statistical distribution of the thermal flux per pass in the core at equilibrium, shown in Fig. 12. Please note that in this plot and all other plots in this section showing per pass information, pass 11 does not appear because the sample size is too small to be visible. On the one hand, the thermal flux distribution is similar regardless of the pass number. On the other hand, the distribution shows clear flux peaks at around 0.2, 0.6, and 1.0 x 1014 n/cm2.s. Each of these peaks can be linked to identifiable core regions and is noticeable in Fig. 8. The low peak corresponds to the bottom region of the core, with the most burned fuel and thermal leakage; pebbles with the median peak value are found at the top of the core and directly above the low peak region; the highest peak (which also has the highest value) corresponds to the central region where pebbles are sufficiently far from the reflector. Finally, a few pebbles at the core's top inner and outer edges experience the highest values. The same distribution for fast neutrons (Fig. 13) shows that the maximum values slightly decrease with the number of passes.
2. Fuel utilization
Figure 14 and Fig. 15 illustrate the spatial incore distribution of burnup. The radial profile shows small peaks around the edges due to pebbles accumulating burnup more rapidly when closer to the reflector, particularly during the first four passes (Table 4). As pebbles are reinserted in a random radial location, the radial burnup profiles flatten with the number of passes. The axial burnup profile shows a monotonic increase behavior as pebbles descend through the core and accumulate burnup. The steplike behavior, noticeable mostly for pebbles in the first few passes, is artificial. It is caused by the discrete motion, moving a bit less than 1/11th of the core active height at each step. In any case, this artifact does not impact the trend and disappears as pebbles are randomly reinserted in different radial positions.
Figure 16 and Table 4 provide statistical data on burnup as a function of the number of passes. Two phenomena are worth noticing. First, the burnup distribution of pebbles during the first pass shows two anomalies: the large peak at zero burnup representing the fresh pebbles inserted in the core and the artificial multipeak behavior due to the discrete motion approach. Second, the two peaks in the cumulative distribution at each pass resulting from the discrete nature (real in this case) of each pass through the core.
Table 4
Burnup statistics as a function of the number of passes
Pass number

1

2

3

4

5

6

7

8

9

10

Core

Avg. cumulative (MWd/kgHM)

7.2

20.3

32.5

43.7

54

63.5

72.1

80.2

87.6

94

54.7

Std (MWd/kgHM)

4.6

4.7

4.5

4.3

4

3.8

3.5

3.3

3

2.6

3.9

Avg. increment (MWd/kgHM)

7.2*

13.1

12.2

11.2

10.3

9.4

8.7

8.0

7.5

6.3

9.5

Minimum (MWd/kgHM)

0

10.1

21.5

32.6

42.7

52.2

61.4

69.9

77.5

84.8

0.0

Maximum (MWd/kgHM)

20.9

37.8

51.2

62.6

73.1

80.7

87.9

94.7

100.5

102.5

102.5

Range (MWd/kgHM)

20.9

27.7

29.7

30

30.3

28.5

26.5

24.8

22.9

17.7

102.5

Peaking factor

2.90

1.86

1.58

1.43

1.35

1.27

1.22

1.18

1.15

1.09

1.87

* The first pass average increment only shows the average burnup, whereas values for other passes correspond to the average burnup difference.
3. Power
Power production in each pebble is a critical incore metric as high power production in a zone of the core leads to hot spots, resulting in lower thermal margins for both fuel and coolant temperatures and increased thermal stress on structural materials and reflectors. Figure 17 and Fig. 18 illustrate the spatial distribution of power per pebble in the core at equilibrium. The radial and axial profiles show similar shapes to the thermal neutron flux, yielding a roughly constant peaking factor. At every pass, power decreases, in line with what was shown for burnup. On average, at the end of life, a pebble generates half of the power produced during the first pass (Table 5). The first four passes account for half of the total core power (202 MW), pass five to eight for 37% (145 MW), and the last two for 13% (53 MW).
The statistical distribution of pebble power for each pass (Fig. 19) shows three peaks representing distinct thermal flux regions, as seen above. It becomes closer to a uniform distribution due, once again, to the random radial reinsertion process.
Finally, it is observed that the peak power per pebble in the core is 3259 W, corresponding to 217 mW per TRISO particle. This information is particularly relevant to assess fuel performance. This work assumes a fixed uniform temperature distribution, but in the future, coupling with a thermalhydraulic model will be implemented to determine the implication of a detailed pebblebypebble power distribution.
Table 5
Pebble power statistics as a function of the number of passes
Pass number

1

2

3

4

5

6

7

8

9

10

Core

Average (W)

1204

1146

1059

970

889

817

753

698

650

618

886

Std (W)

493

475

440

403

369

338

311

288

267

253

366

Minimum (W)

165

131

121

123

106

100

88

83

74

74

74

Maximum (W)

3259

3064

2899

2666

2420

2250

2101

1915

1810

1684

3259

Range (W)

3094

2933

2778

2543

2313

2150

2013

1831

1736

1610

3185

Peaking factor

2.71

2.67

2.74

2.75

2.72

2.75

2.79

2.74

2.78

2.72

3.68

Core power fraction (%)

13.9%

13.2%

12.2%

11.2%

10.2%

9.4%

8.6%

8.0%

7.5%

5.7%

100.0%

4. Data for individual pebbles
A unique capability of HxF is the possibility to pinpoint the history of every single pebble providing further insights. As an example, data are presented for four pebbles to understand why they were discarded after four different numbers of passes. Their history in terms of burnup, power, and spatial position is shown in Fig. 20. The pebble discharged after eight passes travels mostly close to the inner reflector; therefore, it experiences larger flux/power and accumulates burnup more rapidly. At the opposite extreme, the pebble discharged after 11 travels further away from the reflector and, from pass five on, moves more and more into the lower power region. Notably, although pebbles discarded after 8 or 9 passes accumulated a burnup of about 92 MWd/kgHM, the other two reached about 101 MWd/kgHM, in line with an extra pass.
C. Used fuel data
In addition to highfidelity incore data, HxF can be used to collect data on used fuel. First of all, some considerations can be made on the discharge burnup. As explained in the methodology, 137Cs concentration is used as a surrogate for burnup, and pebbles are discharged based on a set threshold. The linear relation between burnup and Cs is confirmed from the discharged pebbles data (Fig. 21). The set threshold of 2.2238x10− 4 mol/pebble corresponds, on average, to a burnup threshold of 92.5 +/ 0.15 MWd/kgHM (ranging from 92.0 to 93.3 MWd/kgHM).
The 137Cs threshold represents the minimum value a pebble must contain in order to be discarded when it is assessed for burnup. Most pebbles are discarded with larger concentrations/burnups, as the assessment occurs only after a full pass. Table 6 summarizes the pebble inventory and burnup information, separated by the number of passes after which the pebbles were discarded. It can be observed that the great majority of pebbles (99.96%) go through the core 9 and 10 times, the average number of passes is 9.8, and the average discarded burnup is 96.5 MWd/kgHM, which is 4% higher than the threshold (understanding this shift is important when determining the threshold value). An extremely low number of pebbles (0.03%) goes through the core for 11 passes and typically reach higher burnup levels or are discarded only after 8 and tend to reach lower burnups. In both extreme cases, the obtained burnup ranges are relatively narrow. The statistical distribution of burnup in used pebbles (Fig. 22) shows the threshold cut around 92.5 MWd/kgHM (with the uncertainty discussed above) and two peaks corresponding to discharge after nine or ten passes.
Table 6
Discarded pebbles inventory and burnup at equilibrium.
Pass number

8

9

10

11

Global

Fraction (%)

0.005%

19.55%

80.41%

0.03%

100%

Avg. discarded burnup (MWd/kgHM)

92.9

93.7

97.1

99.5

96.5

Std (MWd/kgHM)

0.5

1.0

1.4

0.9

1.9

Minimum burnup (MWd/kgHM)

92.3

92.2

92.6

97.7

92.2

Maximum burnup (MWd/kgHM)

94.9

100.7

102.7

102.3

102.7

Burnup range (MWd/kgHM)

2.6

8.5

10.1

4.6

10.5

Information about each individual nuclide can also be obtained. Figure 23 shows a few selected examples (the data are collected at discharge with no decay time). The concentration of 238U monotonically decreases with the number of passes as expected, whereas the fissile isotopes of Pu (239Pu and 241Pu), fission product 135Xe, and 235U exhibit more complex behaviors. This is due to the diversity of neutron spectra a pebble can experience during its lifetime, depending on the location and, thus, on the trajectories in the core. As previously shown, the pebbles discharged after eight passes are exceptional cases in which the pebbles are mainly located near the reflectors and experience a softer spectrum (Fig. 24). Similarly, a softer spectrum leads to a more efficient consumption of 235U and destruction of 135Xe.