## Abstract

In this work, the effect of averaging operating history parameters such as power history, boron concentration and coolant density, and temperature on spent nuclear fuel properties was investigated. The examined properties were assembly activity, decay heat, photon emission rate, spontaneous fission rate, and the concentration of some mobile nuclides and fissile nuclides. Calculations were performed on two similar VVER-440 fuel assemblies irradiated in different positions of the core using Serpent 2. Averaging power history over the entire irradiation history had a significant effect on assembly activity, decay heat, and photon emission rate overestimating these properties approximately 70% right after irradiation. However, the effect quickly died out and after 10 years of cooling, the effect was less than 1%. If the last cycle (third cycle) was modeled accurately and the power density of only the first two cycles was averaged, the differences remained always below 1%. The effect of operating history approximations on spontaneous fission rate and the nuclide concentrations was much smaller remaining mostly below 1.5%. The sensitivity of nuclide concentrations to approximations in individual operating history parameters was dependent on the nuclide in question and no trend applying to all studied nuclides could be observed.

## 1 Introduction

Knowledge of spent nuclear fuel (SNF) properties such as, decay heat, activity, and nuclide inventory is important when planning final disposal, handling, and intermediate storage of SNF. For example, decay heat and reactivity determine how densely SNF canisters can be packed in the repository tunnels [1]. Radiation shielding in transport, intermediate storage, and overall handling of SNF requires knowledge of activity and photon and neutron emission rates. For reactivity calculations, nuclide inventory must be known. Nuclide inventories are needed also for dose estimates from nuclides propagating to the biosphere. According to a POSIVA study [2], nuclides capable of propagating to the biosphere and inflicting dose to humans are at least 14C, 36Cl, 129I, 93Mo, and $108m$Ag. More accurate knowledge of SNF properties and related uncertainties yield cost savings due to reduced margins in, e.g., repository space.

Computational characterization of SNF involves many uncertainty sources such as uncertainties in operating history, impurity concentration, and nuclide data. Uncertainties in operating history can be related, e.g., in the accurate knowledge of operating history parameters such as fuel and coolant temperatures, reactor pressure, power density, and boron concentration or in the intentional approximation of these parameters in order to simplify the calculation. For example, the accurate modeling of a fuel assembly's operating history spanning over three or four years using a Monte Carlo code is not always considered practical and some approximations can be made. Also, accurate weekly or even monthly operating history data are not always available. For example, the operating history data available in SFCOMPO-2.0 [3] often lack such information and only average data are available. Accurate operating history data are also not always available for older fuel assemblies. Such a situation exists, for example, for the VTT research reactor considering the first years of operation [4].

In this work, the operating history of VVER-440 fuel assemblies based on weekly monitoring data is modeled accurately using the Monte Carlo particle transport code Serpent 2 [5]. Then the monitored parameters such as coolant temperature and density, boron concentration, and assembly power are averaged one by one over the simulated operating history. The effect of the averaging on important SNF properties such as, decay heat, activity, and nuclide inventory is investigated. The purpose of the calculations is to determine how rough approximations in the operating history of the investigated parameters can be made without significant effect on SNF properties and to determine which of the investigated parameters are most sensitive to approximations.

## 2 Methods

In this work, two identical VVER-440 fuel assemblies with 126 fuel rods and one instrumentation rod in the middle were modeled through three cycles. The fuel included six fuel rods with Gd2O3 and three different enrichments. The two assemblies were identical but their positions in the core were different and therefore some of the operating history parameters such as water temperature and density and assembly power were slightly different. Two assemblies were modeled in order to see the possible effect of assembly positions in the core. The Serpent model of the fuel assembly and the positions of the assemblies in the core are presented in Fig. 1. In Fig. 1(a), the yellow pins (114 pins) and orange pins (6 corner pins) are fuel pins with 4.4% and 4.2% 235U enrichment, respectively. The red pins (6 pins next to corner pins) are fuel pins with 4.0% 235U enrichment and 3.35% Gd2O3 content. In addition to oxygen and uranium, 200 ppm of 14N and 15 ppm of 35Cl [6] were added in the fuel in order to examine the activation products 14C and 36Cl. The positions of the two modeled assemblies during the three cycles are presented in Fig. 1(b) where the pink hexagons (upper part of the core) mark assembly positions for modeled assembly 1 and the green hexagons (lower part of the core) for assembly 2. The numbers in the hexagons indicate the irradiation cycle.

Fig. 1
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For reference calculations, weekly monitoring data of boron concentration, core pressure, and assembly wise linear power and coolant temperature were used. Core pressure and coolant temperature were used to calculate coolant density. For linear power around maintenance breaks, several measurements were made daily. For operating history approximations, linear power, boron concentration, coolant temperature, and density were averaged one by one keeping the other parameters accurate. Coolant temperature and density approximation were always combined and cladding and shroud tube temperatures were set the same as coolant temperature. The averaging was done separately over the whole operating history and cycle wise. Finally, all of the parameters were averaged in one calculation. The average values of these parameters are presented in Table 1 both over the whole operating history and separately for each modeled irradiation cycle. The table also presents the largest percentual difference in the accurate operating history data to the value averaged over the whole irradiation history and the relative standard deviation. Fuel temperature was at a constant 900 K in all calculations.

Table 1

Operating history parameters averaged over the whole irradiation history (ave) and separately for each cycle (ave 1, ave 2, ave 3) “Diff” gives the largest difference in accurate irradiation history to the average over all cycles and “Rel. Std” is the relative standard deviation over the whole irradiation history. The symbols Tc, ρc, B, and P stand for coolant temperature, coolant density, boron concentration, and assembly linear power, respectively.

Assembly 1Assembly 2
Tc (K)ρc (g/cm3)B (ppm)P (W/cm)Tc (K)ρc (g/cm3)B (ppm)P (W/cm)
Ave5800.70456301905800.7051630189
Ave 15820.69846602035800.7040660199
Ave 25800.70496431915810.7029643194
Ave 35770.71055801745780.7090580171
Diff−10%9.1%163%−73%−11%9.4%163%−73%
Rel. Std1.6%1.6%63%15%1.4%1.4%63%14%
Assembly 1Assembly 2
Tc (K)ρc (g/cm3)B (ppm)P (W/cm)Tc (K)ρc (g/cm3)B (ppm)P (W/cm)
Ave5800.70456301905800.7051630189
Ave 15820.69846602035800.7040660199
Ave 25800.70496431915810.7029643194
Ave 35770.71055801745780.7090580171
Diff−10%9.1%163%−73%−11%9.4%163%−73%
Rel. Std1.6%1.6%63%15%1.4%1.4%63%14%

The 30 deg symmetry of the assembly was utilized in the calculations and periodic boundary conditions were applied. The non-Gd fuel rods were divided in separate depletion zones containing one fuel pin and the fuel rods with Gd were further divided into ten equal size radial depletion zones. The division was done using Serpent's automated depletion zone division. Altogether, 181 burnup steps were used to model the three irradiation cycles with a step length of mostly seven effective full power days (EFPD) regardless of whether approximations were applied in the operating history parameters or not. In the beginning of the cycles, shorter step lengths were used. Doppler broadening rejection correction was applied for some uranium and plutonium isotopes. Neutron cross section data based on JEFF-3.2 data were utilized. For fission yield and decay data, JEFF-3.1.1 libraries were used. At every step, 20,000 neutron histories were run in 200 generations. All calculations were run on a Linux cluster with Intel Xeon 2.2 GHz nodes using OpenMP parallelization. The reference calculation was repeated six times in order to get an indication of the statistical uncertainty caused by the Monte Carlo method. The calculation times were typically slightly under two days, but varied according to how many inputs and restart files were needed.

## 3 Results

This section has been divided into four subsections discussing the effect of averaging on (i) the studied SNF properties: activity, decay heat, photon emission rate, and spontaneous fission rate, (ii) the concentration of some mobile nuclides, (iii) the concentration of fissile nuclides, and (iv) a closer look on the effect of power history approximations on activity, decay heat, and photon emission rate.

In all subsections, SNF characteristics with averaged operating history parameters of a fuel assembly are compared to the reference case where no averaging was done and weekly operating history over three years was used. The results of the reference calculation are an average of the six repetition calculations. In order to conserve space, figures are presented only for assembly 1 and possible difference between the results of assembly 1 and assembly 2 are discussed in the text.

In all figures, the differences between calculations with averaged operating history and the reference are presented on the left–hand-side axis and the absolute value of the SNF property or nuclide concentration in question on the right–hand-side axis. The legends in the figures are as follows: bor, averaging boron concentration; cool, averaging coolant density and temperature; pow, averaging assembly power; all, averaging over all of the above mentioned parameters; std, relative standard deviation of the reference calculation repetitions; last entry, absolute value of the reference calculation.

The differences between averaged and reference calculations are calculated by [(averaged parameter)/reference − 1]*100. In the (a) figures, averaging over the whole operating history is applied (case a) and in the (b) figures, cycle wise averaging is performed (case b). The x-axis and the right-hand y-axis are in logarithmic scale. The left hand y-axis is in logarithmic scale for some of the figures. Because of the logarithmic scale, the x-axis does not start from zero but from 1 day after irradiation. The burnups of assemblies 1 and 2 after the three years of irradiation were 48.5 MWd/kgU and 48.0 MWd/kgU, respectively.

Fig. 2
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Fig. 3
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Fig. 4
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Fig. 5
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Fig. 6
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Fig. 7
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### 3.1 Impact of Averaging on Spent Nuclear Fuel Properties.

The results for assembly 1 are presented in Figs. 26 as a function of cooling time.

Fig. 8
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Fig. 9
Fig. 9
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The differences in Fig. 24 for activity, decay heat, and photon emission rate behave quite similarly. One day after irradiation, the impact of averaging the operating history seems to be significant. Differences to the reference case are around 30% in case a and around 20% in case b. At zero cooling time, the differences are more than 60% (case a) and over 50% (case b). However, the differences rapidly decrease. After one year of cooling, the differences are less than 5% and after 10 years, they are less than 1% for case a. For case b, these limits are reached already in 2 months (< 5%) and 2 years (< 1%). Almost all of the differences in the beginning seem to be originating from averaging the power history. For boron concentration and coolant temperature and density, the differences remain always below 1% except for photon emission rate when averaging boron concentration for which the difference at 1 day cooling time is around 1.4% for both cases a and b. For cycle wise averaging (case b) after about 2–3 months (activity, decay heat) or 30 years (photon emission rate), the differences from averaging coolant properties are of the same order as the statistical uncertainty and can therefore be ignored.

The results suggest that the differences originate from some very short-lived nuclides sensitive to power history approximations. The effect of power history approximation is studied more closely in Sec. 3.4. The effect of the approximations is mostly conservative, overestimating the activity, decay heat, or photon emission rate. Underestimations remain below 1.6%.

Decay heat is one of the limiting factors in final disposal. According to POSIVA study [1], the relevant time period when the bentonite buffer is expected to reach its peak temperature occurs some dozens of years after disposal of the first SNF canister in the repository. Therefore, decay heat is examined more closely in Fig. 5, where Fig. 3 is zoomed between 50 and 300 years cooling time. In both cases a and b, the differences caused by averaging are small during this time period remaining nearly always under 0.5%.

Impact of averaging on spontaneous fission rate in Fig. 6 remains always below 1.3%. In the beginning, most of the spontaneous fissions occur in 242Cm and 244Cm. Later, 246Cm, 240Pu, and 242Pu become dominant, until at 1 × 107 years almost all of the fissions originate from 238U. The shape of the differences in Fig. 6 reflects the dominant periods of some of these nuclides. For example, the importance of averaging power density dies with 242Cm between 1 and 2 years and the effect of boron density approximations changes from over estimation to underestimation with decaying 244Cm. This indicates that the sensitivity of individual nuclides to different operating history parameters may differ. The effect of coolant properties in case b is of the same order as the statistical uncertainty and hence insignificant.

Results for activity, decay heat, and photon emission rate in assembly 2 were very similar to assembly 1. Differences in these values between the assemblies were less than 2%.

Spontaneous fission rate in assembly 2 differed approximately 4% from assembly 1. However, the difference on the effect of the approximations on the two assemblies was rather insignificant. The largest difference was in the absolute values of the impact of power history approximation (0.68% versus 0.84%) before 3 months cooling when averaging over the whole operating history. The curves describing the difference between averaged operating history parameters and the reference case for assembly 2 were otherwise similar to Fig. 6.

### 3.2 Impact of Averaging on the Concentration of Mobile Nuclides.

This section presents the impact of operating history approximations on important mobile nuclides that can inflict dose in the biosphere. The chosen nuclides, 14C, 36Cl, 93Mo, $108m$Ag, and 129I (Figs. 711) are based on a POSIVA study [2]. The study examines, among other things, the migration of nuclides from a deep repository in the bedrock after a copper canister failure and radiation doses to the biosphere. This is the kind of repository planned in Finland for the disposal of SNF.

Fig. 10
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Fig. 11
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Fig. 12
Fig. 12
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Fig. 13
Fig. 13
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Fig. 14
Fig. 14
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In Figs. 710, the differences between approximations and the reference for 14C, 36Cl, 93Mo, and $108m$Ag remain constant in time. This is probably because these nuclides are born mostly only in impurity activation (14C and 36Cl) or as fission products (93Mo and $108m$Ag) and therefore their concentrations after irradiation depend only on half-lives. For 129I in Fig. 11, however, the differences change within the first year after irradiation. Also, there is a visible growth in the concentration of 129I within the first year after irradiation. This is probably related to some fast decay chains of other fission products and their daughter nuclides. For example, 129I is born in the β-decay of 129Te and $129m$Te with half-lives of 69.6 min and 33.6 months, respectively. These nuclides are in turn born in the decay chains of other very short-lived nuclides such as, 129Sb, $129m$Sb, and 129Sn. All these nuclides are present in the irradiated fuel.

The impact of operating history approximations on 14C, 36Cl, and 129I concentrations in Figs. 78 and 11 is rather small, less than 0.3% for 14C and 36Cl. The maximum impact for 129I is 0.4% in case a right after irradiation and is mainly caused by the power history approximation. The impact of approximation quickly decreases and after six months of cooling becomes less than 0.1%.

For 93Mo, the differences appear to be much larger but also the relative standard deviation of the reference calculation is 4.4%. The impact of power history approximation in case a is the only one larger than the relative standard deviation (5.9%). However, when all parameters were averaged, the difference to reference was only 4.4%. This might be the case if the individual approximations of the parameters were to somewhat cancel each other. In the calculations, however, all approximations overestimated the concentration of 93Mo. Therefore, if the calculations were accurate, differences to reference when all parameters were approximated should be larger than the differences when only one parameter is approximated. This suggests that within uncertainties the operating history approximations investigated here did not significantly impact the calculated concentration of 93Mo. Results in the calculation of assembly 2 also support this assumption.

For $108m$Ag, the impact of operating history approximation in all investigated parameters is 1.4% in case a and 0.8% in case b. In both cases, approximation in boron concentration is relatively significant. In case a coolant density and temperature also have a clear effect, but become rather insignificant in case b when approximations are made cycle wise.

Concentrations of 14C, 36Cl, $108m$Ag, and 129I in the two different assemblies are very similar with maximum ∼0.3% differences for $108m$Ag and ∼1% differences for the other three nuclides. The effect of averaging operating history parameters is also quite similar. Some differences occur between Fig. 7 (14C) and the corresponding figure for assembly 2 when averaging coolant history in case a. In Fig. 7, the effect of coolant is about the same as the effect of power history. For assembly 2, the effect of averaging coolant history is smaller than the averaging of the other operating history parameters. Another difference between the two assemblies is visible for $108m$Ag in case a in Fig. 10 when averaging coolant properties and all investigated parameters. In assembly 2, the effect of averaging coolant properties is 0.3% smaller than in assembly 1, i.e., the effect of averaging coolant properties is 0.75% in assembly 1 and 0.44% in assembly 2. Correspondingly, the effect of averaging all investigated parameters in assembly 1 is 0.3% higher than in assembly 2.

### 3.3 Impact of Averaging on Fissile Nuclides.

The impact of averaging on fissile nuclides in Figs. 1215 for 233U, 235U, 239Pu, and 241Pu is always below 1.5% except for 241Pu in Fig. 15 in case a after 200 years of cooling when the concentration of 241Pu begins to fall rapidly. This is true for both assemblies 1 and 2.

Fig. 15
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Fig. 16
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Fig. 17
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Fig. 18
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The concentration of 233U starts from ∼2 mg/tUi at the end of irradiation and becomes almost 1 × 105 times larger at $∼$1 × 106 years after irradiation after which it starts to decrease again. The half-life of 233U is 1.6 × 105 years and it is born in several decay chains like 241Pu → 241Am → 237Np → 233Pa → 233U with half-lives 14 years, 433 years, 2.1 × 106 years, and 30 days, respectively. All these nuclides are also present in the irradiated fuel. The effect of different operating history parameters changes over time reflecting the sensitivity of the various parent nuclides leading to the creation of 233U. In case a between 0 and 30 years after irradiation, the effect of averaging boron concentration and coolant properties is insignificant, but becomes significant relative to the standard deviation after 30 years. For case b, this is true with boron concentration but averaging of coolant properties has no significant effect even after 30 years. In case b, the averaging of none of the investigated parameters differs significantly from the reference below 30 years of cooling time. In case a, power history approximation is the dominant component until around 30 years after irradiation after which boron concentration has a slightly greater effect.

The concentration of 235U in Fig. 13 remains quite static until around 1000 years after which it starts to increase mostly due to the α-decay of 239Pu with a half-life of 24 × 104 years. In the beginning in case a, the averaging of the different operating history parameters has a very similar effect underestimating 235U concentration. When 235U concentration begins to grow, the effect of averaging boron concentration and coolant properties starts to overestimate 235U concentration reflecting their effect on 239Pu concentration in Fig. 14. Similar behavior is seen in case b except that the effect of the different operating history components is not equal to begin with.

Figures 14 and 15 present the differences caused by operating history approximations on 239Pu and 241Pu concentrations. For both nuclides, averaging boron concentration is relatively important. For 239Pu in case b, effect of averaging coolant properties cannot be distinguished from the statistical uncertainty. The same is true also for 241Pu at least before about 300 years cooling. Averaging power history underestimates the concentrations of these plutonium isotopes, but the averaging of the other parameters overestimates the concentrations.

The concentrations of the studied uranium nuclides in the two assemblies agreed within 1% for 233U and within 1.6% for 235U. The agreement for 239Pu was with 0.4% until 1 × 105 years after irradiation after which the differences grew up to 7.5% with decreasing concentration. For 241Pu between 0 and 150 years after irradiation, the two assemblies agreed within 1% after which the differences grew to around 5% with falling 241Pu concentration. The effect of averaging on the fissile nuclides was quite similar in both assemblies although some differences exist in case a for the plutonium isotopes. The effect of averaging all the studied operating history parameters for 239Pu between 10 days and 1 × 105 years is approximately 1.2% for assembly 1 and 0.9% for assembly 2. This is mostly due to the stronger effect of averaging coolant properties in assembly 1. The same difference is present also for 241Pu where the effect of averaging all parameters for assembly 1 is ∼1% with less than 300 years cooling and ∼0.9% for assembly 2. One reason for this may be the slightly smaller deviation in the history of coolant properties in assembly 2 than in assembly 1 as can be seen in Table 1.

### 3.4 Closer examination of the Impact of Power History Approximation.

Because of the significant impact of averaging power history on activity, decay heat, and photon emission rate visible in Figs. 24, power history approximation was studied more closely in Figs. 1618. In this study, power history was modeled accurately for different numbers of depletion steps from the end of the irradiation and the rest of the power history was averaged. For example, if the last five steps were modeled with accurate power history values, the remaining values for assembly power were averaged and this value was used for all other depletion steps except for the last five steps. In the figure, legend “2nd” means that assembly power has been averaged over the first cycle and accurate assembly power has been used from the beginning of the second cycle. Similarly, the legend “3rd” means that assembly power has been averaged over the first two cycles and accurate assembly power has been used from the beginning of the third cycle. The subscripts in the legends mean that these many steps from the end of the cycle have been modeled accurately. For example, 3rd5 means that power in the five steps from the end of the third (and last) cycle have been modeled accurately and the rest of the power values have been averaged. There were 68 steps in the second cycle and 54 steps in the last cycle.

It can be seen in the figures that calculating just the last step (1 EFPD) with accurate power history value (3rd1) brings the uncertainty 1 days after irradiation to around 16% or 18% (Figs. 1618) from around 27% or 31% (Figs. 24). Calculating the last five steps (15 EFPD) accurately reduces the uncertainty at 1 day to 3–5%. Calculating half of the last cycle with accurate power history brings the uncertainty caused by the power history approximation always under 1.5% for activity, decay heat, and photon emission rate. Under 1% is achieved by calculating the last cycle with accurate power history. Calculating the last two cycles with accurate power history values would keep the differences to reference calculation always below 0.053%.

## 4 Summary and Conclusions

The key findings of this paper are summarized in Table 2 when operating history was averaged over the entire irradiation history (case a) and in Table 3 when operating history was averaged cycle wise (case b). The abbreviations used in the tables are listed. Prop, SNF property; Max, maximum difference between approximation and reference; t (days), time after irradiation in days when maximum occurs; Par, the most significant averaged parameter(s); Insig. par, insignificant parameter; Cons., indication whether the difference is conservative (X).

Table 2

Summary of key findings when operating history is averaged over all cycles (case a)

PropMax (%)t (days)ParInsig. parCons.
A660PowX
H790PowX
γ660PowX
SF1.210PowX
14C0.28X
36Cl0.30X
93MoNo significant differences within uncertainties
$108m$Ag1.4X
129I0.390PowBor, Cool
233U1.50PowBora, Coola
235U0.4590
239Pu1.53.65 × 109Bor, coolX
241Pu2.53.65 × 107Bor, coolPowX
PropMax (%)t (days)ParInsig. parCons.
A660PowX
H790PowX
γ660PowX
SF1.210PowX
14C0.28X
36Cl0.30X
93MoNo significant differences within uncertainties
$108m$Ag1.4X
129I0.390PowBor, Cool
233U1.50PowBora, Coola
235U0.4590
239Pu1.53.65 × 109Bor, coolX
241Pu2.53.65 × 107Bor, coolPowX
a

Only between 0 and 20 years after irradiation.

Table 3

Summary of key findings when operating history is averaged cycle wise (case b)

PropMax (%)t (days)ParInsig. parCons.
A530PowCoolX
H640PowCoolX
γ520PowCoolX
SF0.9610PowCoolX
14C0.18CoolX
36Cl0.20CoolX
93MoNo significant differences within uncertainties
$108m$Ag0.821.1 × 104BorX
129I0.200PowCool
233U0.383.65 × 108BorX
235U0.2910Cool
239Pu0.663.65 × 109
241Pu1.21.83 × 107BorCoola, PowaX
PropMax (%)t (days)ParInsig. parCons.
A530PowCoolX
H640PowCoolX
γ520PowCoolX
SF0.9610PowCoolX
14C0.18CoolX
36Cl0.20CoolX
93MoNo significant differences within uncertainties
$108m$Ag0.821.1 × 104BorX
129I0.200PowCool
233U0.383.65 × 108BorX
235U0.2910Cool
239Pu0.663.65 × 109
241Pu1.21.83 × 107BorCoola, PowaX
a

Only between 0 and 200 years after irradiation.

The most significant averaged parameter is mentioned only if averaging this parameter causes significantly larger differences to the reference than the averaging of the other parameters. A parameter is insignificant if its averaging causes differences to reference that are of the same order as the standard deviation of the reference calculation or are otherwise extremely small (< 0.05%).

In Table 3, the largest difference for 233U is actually 0.60% right after irradiation, but the statistical uncertainty is of the same order. For that reason, the largest difference after 30 years of cooling is given when the differences begin to be greater than the Monte Carlo variance.

The effect of averaging assembly power, boron concentration, and coolant temperature and density over three irradiation cycles on total assembly activity, decay heat, and photon emission rate remains below 1% during the period of final disposal ($∼40$ years →). The effect of averaging power history can be significant overestimating these parameters ∼70% right after irradiation, but rapidly decreases below 1% in less than 10 years. If accurate power history values are used during the last (third) cycle of irradiation, the uncertainty caused by the power history approximation is decreased to under 1%.

The effect on spontaneous fission rate showed strong dependence on the nuclides causing the fissions. The overall effect remained always below 1.3%.

The effect of averaging on mobile nuclides 14C, 36Cl, and 129I was always below 0.4%. The maximum effect for $108m$Ag was 1.4%. Operating history approximations on 93Mo were insignificant within the observed Monte Carlo variance.

For the fissile nuclides, the maximum effect of approximations during the first 200 years is 1.5% for 233U right after irradiation. The maximum effect for 235U is 0.45% and is always clearly smaller than for the other fissile nuclides. For Pu-239 and Pu-241, the maximum effect of approximations is mostly around 1%. For Pu-241, the effect increases to 2.5% when Pu-241 begins to strongly decrease after ∼200 years.

Overall, approximation of power history had clearly the largest effect visible in assembly activity, decay heat, and photon emission rate. Averaging coolant properties had the smallest effect on activity, decay heat, photon emission rate, spontaneous fission rate, and many of the studied nuclides. The response on the concentration of different nuclides on different operating history parameters varied and no clear trend applicable on all studied nuclides could be observed.

Observed differences between the two calculated assemblies were small. However, some small differences were present in the plutonium isotopes probably related to the slightly different variance of coolant properties at the two different assembly positions. With larger variations in other operating history parameters such as, power history at different assembly positions, the differences in operating history approximations for different assemblies might be larger. However, commercial power reactor cores are usually designed symmetric and very large deviations in operating history parameters are not so likely.

## Acknowledgment

The author would like to acknowledge the previous work of Antti Rintala with a VVER-440 Serpent input, which was helpful in the construction of inputs for this work.

## Funding Data

• Finnish Research Programme on Nuclear Waste Management (No. KYT2022, Funding Nos. Dnro KYT 17/2020 and Dnro KYT 17/202).

## Nomenclature

• A =

activity, GBq/tUi

•
• B =

boron concentration, ppm

•
• H =

decay heat, kW/tUi

•
• P =

linear power, W/cm

•
• SF =

spontaneous fission rate, fissions/s/tUi

•
• Tc =

coolant temperature

•
• t (days) =

time after irradiation in days when maximum difference between approximation and reference occurs, days

•
• tUi =

tons of initial uranium

### Greek Symbols

Greek Symbols

• γ =

photon emission rate, photons/s/tUi

•
• ρc =

coolant density

### Abbreviations

Abbreviations

• All =

averaging over boron concentration, assembly power and coolant temperature and density

•
• Ave =

average

•
• Bor =

averaging boron concentration

•
• Cons =

indication whether the difference to reference is conservative

•
• Cool =

averaging coolant density and temperature

•
• Diff =

difference

•
• EFPD =

effective full power days

•
• Insig. par. =

parameter whose averaging has an insignificant effect

•
• Max =

maximum difference between approximation and reference

•
• Par =

the most significant averaged parameter

•
• Pow =

averaging assembly power

•
• prop =

spent nuclear fuel property

•
• Rel. Std =

relative standard deviation

•
• SNF =

spent nuclear fuel

•
• Std =

relative standard deviation of the reference calculation repetitions

•
• VVER =

water cooled water moderated reactor

•
• 2nd =

assembly power averaged to the beginning of the second cycle

•
• 3rd =

assembly power averaged to the beginning of the third cycle

•
• 3rdx =

assembly power x steps from the end of third cycle modeled accurately. The rest are averaged

•
• 2ndx =

Assembly power x steps from the end of second cycle modeled accurately. The rest are averaged

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