Rong-Rui Yang ?Yuan Yuan?Chen Hao,?Ji Ma?Guang-Hao Liu

Abstract To benefit from recent advancesin modeling and computational algorithms,aswell astheavailability of new covariance data,sensitivity and uncertainty analyses are needed to quantify the impact of uncertain sources on the design parametersof small prismatic high-temperature gascooled reactors(HTGRs).In particular,the contribution of nuclear data to the keff uncertainty is an important part of the uncertainty analysis of small-sized HTGR physical calculations.In this study,a small-sized HTGR designed by China Nuclear Power Engineering Co.,Ltd.was selected for keff uncertainty analysis during full lifetime burnup calculations.Models of the cold zero power(CZP)condition and full lifetime burnup processwereconstructed using the Reactor Monte Carlo Code RMC for neutron transport calculation,depletion calculation,and sensitivity and uncertainty analysis.For the sensitivity analysis,the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance Characterization(CLUTCH)method was applied to obtain sensitive information,and the ‘sandwich’method was used to quantify the keff uncertainty.Wealso compared the keff uncertainties to other typical reactors.Our results show that 235U is the largest contributor to keff uncertainty for both the CZPand depletion conditions,while the contribution of 239Pu is not very significant because of the design of low discharge burnup.It is worth noting that the radioactive capture reaction of 28Si significantly contributes to the keff uncertainty owing to its specific fuel design.However,the keff uncertainty during the full lifetime depletion process was relatively stable,only increasing by 1.12%owing to the low discharge burnup design of small-sized HTGRs.These numerical results are beneficial for neutronics design and core parameters optimization in further uncertainty propagation and quantification study for small-sized HTGR.

Keywords Small-sized HTGR.SU analysis.Nuclear data.Burnup

1 Introduction

Owing to their significant inherent safety and applicability characteristics,high-temperature gas-cooled reactors(HTGRs)have gradually played indispensable roles in nuclear reactor development[1–3].HTGRs can be split into two types based on their core design:pebble-bed HTGRs,such as the high-temperature reactor pebble-bed module(HTR-PM)developed in China[4],and prismatic HTGRs,such as the modular high-temperature gas-cooled reactor(MHTGR-350),developed in the US[5].Simultaneously,small reactors have become a hotspot in international nuclear energy research.HTGR technology is developing rapidly in China,and a new small-sized prismatic HTGR is under development by the China Nuclear Power Engineering Co.,Ltd.The continued development of HTGRs requires verification of their designs with reliable high-fidelity physics models and efficient accurate codes.The predictive capability of codes for HTGR designs can be assessed using sensitivity and uncertainty(SU)analysis methods.Through advancements in computer modeling and computational algorithms,as well as the accessibility of new covariance data,SU analysis can quantify the impact of uncertainties on the design parameters of small prismatic HTGRs.In particular,the effective multiplication factor(keff)is the most important parameter in reactor physical analysis,its uncertainty propagated by nuclear data isusually indicated as an interval of keffvalue.Because the uncertainty of nuclear data existsnaturally,the contributions of nuclear data on the keffuncertainty are essential for the designer to optimize core lifetime and neutronics design.

For SU analysis of HTGRs,a representative international project is the Coordinated Research Project(CRP)on the HTGR Uncertainty Analysis in Modeling(UAM),supported by the International Atomic Energy Agency(IAEA),which considers the peculiarities of HTGR designs and simulation requirements[6,7].In the CRP,the coupled HTGRsystem calculationsaredivided into several steps,each of which can contribute to the total uncertainty.Simultaneously,the input,output,and assumptions for each step need to beidentified.Theresulting uncertainty in each step is calculated by considering all sources of uncertainties,including related uncertainties from previous steps[8].Some in-depth studies have quantified the contribution of cross-section uncertainties to the eigenvalue uncertainty for some representative but simplified models for both the pebble-bed and prismatic HTGRs[6–10].For SU analysis of the prismatic HTRG,local and global calculations have been performed,including steady-state and depletion calculationsfor thecell and core models.The keffuncertainties due to the nuclear data for the fresh block core and mixed core of the MHTGR-350 have been quantified[11].Although both the small-sized HTGR selected in thisstudy and MHTGR-350 belong to prismatic HTGRs,there are some significant differences.Unlike the mixed core arrangement in MHTGR-350,the small-sized prismatic HTGR only has fresh fuel with burnable poison(BP)in thecoreat thebeginning of life(BOL).At thesame time,thisstudy can enrich the content of the IAEA CRPin HTGRs.

Thisstudy focuseson keffuncertainty analysisdueto the nuclear data during full lifetime burnup calculations of the small-sized prismatic HTGR.We will quantify the different cross-sectional contributions on the keffuncertainty at the CZPcondition and the full lifetime burnup processand analyze the mechanism in-depth.The following section describes the model details of the small-sized prismatic HTGR and SU analysis methods.Based on the first-order perturbation theory[12],we selected the contributionlinked eigenvalue sensitivity/uncertainty estimation via track length importance characterization (CLUTCH)method[13]to perform the sensitivity analysis,and the‘sandwich’rule[14]to quantify the keffuncertainty by using the ENDF/B-VII.1 based covariance data[23].

Therest of thispaper isstructured asfollows.In Sect.2,we introduce the model and method applied in this study,especially the full core burnup model of the small-sized HTGR for neutron transport and depletion calculations.In Sect.3,we present the SU analysis of keffdue to nuclear datain the full lifetimedepletion calculation.In Sect.4,we present the in-depth mechanism analysis of nuclear data contributions on keffduring the full lifetime burnup process.Finally,we present the numerical results and conclusions drawn from SU in Sect.5.

2 Models and methodologies

2.1 Small-sized HTGR model

The small-sized HTGR,which is under development by the China Nuclear Power Engineering Co.,Ltd.,was selected as the research target in this study.This smallsized HTGR is a helium-cooled,graphite-moderated prismatic reactor and has some unique characteristics,such as fuel blocks and a burnable poison rod arrangement[15].A representation of the core layout is shown in Fig.1;30 hexagonal prism fuel blocks and 13 control rod blocks are closely arranged in the core.The seven control rod blocks are surrounded by fuel assemblies,including one center startup control block and six shutdown control blocks.The other six control blocks are on the six corners of the core beside thefuel blocks,and each fuel block contains24 fuel rodsand seven coolant channelswithin thegraphitematrix.Each fuel rod or coolant channel has a hexagonal graphite cladding in the fuel blocks.The radial reflector around the core is also made of graphite,but the density ismuch lower than the hexagonal graphite cladding.The coolant gas and reflector material specifications are the same asthose of the pebble bed reactor,which uses helium as the coolant and graphite as the moderator and reflector.In particular,several cylinder fuel pellets are added to the upper and bottom reflective layers to constitute a fuel rod and TRISO particles[16]are dispersed in the SiC matrix to form a fuel pellet.This is different from the pebble-bed HTGR,in which TRISOparticlesaredispersed in thegraphitematrix.

Fig.1 (Color online)Small-sized prismatic HTGR cross-sectional layout

During the burnup calculation,every fuel kernel in the TRISOparticlesisa basic burnup unit.At thesame time,as the depletion proceeds,fission nuclides are consumed and new fission products,such as Ce,Pr,Pu,and Np,are generated.Some of these will cause fission events again and introduce new uncertainties to the core.Moreover,owing to the arrangement of the reflective layers,fuel blocks,and control rod blocks presented in Fig.1,the discrepancy in the burnup degree in different burnup areas will be gradually evident during the depletion process.Thus,a 24 burnup zone model(four zones in the radial direction and six zones in the axial direction)was established for the depletion calculation,as shown in Fig.2.

Fig.2 (Color online)Schematic depletion areas of the small-sized HTGR

In this study,the reactor Monte Carlo code(RMC),developed by Tsinghua University,was used for calculating the neutron transport and depletion for thehigh-fidelity model of thesmall-sized prismatic HTGR[17].The ENDF/B-VII.1 cross-section library[23]was chosen for the calculation.The setting parameters for the MC critical and burnup calculations are illustrated in Table 1.Based on these parameters,each MC critical calculation for this model can converge and fulfill the accuracy requirements of the calculated results.For the full core depletion calculation,the statistical error-based MC method was lower than 25 pcm in each burnup step.At the same time,fission poisons,such as135Xe and149Sm,can have a huge impact on reactivity at the BOL.Therefore,to study these nuclide reaction contributions to keffuncertainty and its sensitivity variation during the burnup calculation,the time of the burnup steps must be set small at the BOL,as illustrated in Table 1.During the depletion calculation,the RMC produces a large amount of complicated nuclide information,including the nuclide densities for each burnup region in each burnup step.In addition,the predictor correction method wasused for the RMCburnup calculation[18].For the MCdepletion calculation,thenuclidedensitiesused for this burnup step were derived from the results of the previous burnup step.Therefore,the densities of the nuclides as the input parameters for the uncertainty calculation should be the average value of the predicted and corrected densities[19].

Table 1 Monte Carlo neutron transform and depletion calculation setting parameters in RMC

Moreover,because all fresh fuels are input into the core simultaneously,there isa large amount of excessreactivity at the BOL.BPisotopes,such as157Gd,10B,or167Er,have large neutron absorption cross-section of themselves and little absorption cross-section of their products,which usually be chosen to balance the excessive reactivity at BOL to reduce the number of control rods as well as deepen the burnup and flatten the distribution of neutron fluence rate.For thisburnup model,therearesix center fuel blocksaround the center control rod block,each containing three Gd2O3burnable poison rods,each of which is a unique burnup unit during the MC depletion calculation.

2.2 SU analysis method

In this paper,the CLUTCH method[13]based on firstorder perturbation theory was used for keffsensitivity analysis during the Monte Carlo calculations.This method calculates the importance of events during a particle’s lifetime by examining the number of fission neutrons created by that particle after those eventsoccur.The aim is to produce an accurate and efficient method for calculating keffsensitivity coefficientsfor nuclear cross-sectionswith a relatively small computational memory.

The CLUTCH method only calculates the sensitivity information during the forward calculations.Therefore,a fine important weight function F*(r)mesh number should be set to ensure accurate sensitivity.An interval of 1–2 cm mesh istypical for obtaining accurate F*(r)estimates[13].The F*(r)mesh only needs to cover the fissionable regions in thecore;therefore,in thisstudy,themesh only needed to cover the fuel block regions.The F*(r)meshes are also calculated in the inactive generations,so at least 50 to 100 inactive historiesshould be simulated per mesh interval for sufficient F*(r)convergence.In this paper,1.8 cm length meshes in radial,1.86 cm length meshes in axial,and 750 total histories and 600 inactive histories were set for the CLUTCH method calculations.After the sensitivity coefficients were measured by the CLUTCH method,the keffuncertainty can be quantified using the ‘sandwich’rule[14]and the ENDF/B-VII.1 based covariance matrix[23].

3 SU analysis of keff during the full lifetime depletion calculation

3.1 SU analysis of keff at CZP condition

Based on the RMCdepletion calculation,several burnup step results were chosen to investigate the uncertainty of keffdueto nuclear data.At the BOL,thereisonly fresh fuel in the core and no fission products,which is known as the cold zero power(CZP)condition.To observe the sensitivity and uncertainty contribution of key nuclide cross sections in the depletion process,SU analysis at the CZP condition should be considered.Through uncertainty quantification,the relative standard deviation of keffdue to nuclear dataat the CZPcondition was0.6586%.The top 10 most crucial nuclide reaction covariance contributors to the keffuncertainty for small-sized HTGRs under the CZP condition are presented in Table 2,where the numerical results were obtained by RMC.

Table 2 Top 10 nuclide reaction covariance contributors to keff uncertainty at CZP condition

The average number of neutrons emitted per fission event of235U is the main contributor to the keffuncertainty and accounts for nearly 17.40%of the total uncertainty of keff-based nuclear data.This phenomenon is similar to the results of a previously reported uncertainty analysis of HTR-10[10,22].However,the main keffuncertainty contributors from the results of uncertainty analyses of typically pressurized water reactors(PWRs)and boiling water reactors(BWRs)[20,21]are different,which has detailed description in Sect.3.2.2.Moreover,the radioactive capture reaction cross-section of28Si should be considered as a significant factor because it is the second contributor to keffuncertainty.This reaction cross-section has an 11.98%contribution to the total uncertainty of keff.This value is much higher in small-sized HTGRs than in other typical reactors[10,20].In addition,the elastic scattering of C-graphite is the third contributor,and the fission spectrum of235U is the fourth contributor.

It should be noted that the fuel pellet matrix materials are different between the pebble-bed and small-sized HTGRs:C-graphite is used for pebble-bed HTGRs and SiC for small-sized HTGRs.Furthermore,the volume ratio of Si in the small-sized HTGR fuel pellet was 55.87%,which was significantly higher than that in the pebble-bed reactor.For a more in-depth study on the effect of nuclear data in keffuncertainty under the CZP condition,some necessary nuclide reaction energy sensitivity coefficient curves are presented in Fig.3.

Fig.3 (Color online)Important nuclide reaction energy sensitivity coefficients

As the first contributor to keffuncertainty under CZP conditions for small-sized HTGRs,the average number of neutrons emitted per fission event of235U has a large sensitivity coefficient in the thermal neutron energy range(energy less than 1 eV).Based on the ‘sandwich’rule,the large uncertainty contribution of the average number of neutronsemitted per fission event of235U can beattributed to its large sensitivity coefficient.The large keffuncertainty contribution of the radioactive capture reaction of28Si can be explained by the same reason.Additionally,C-graphite and238U as the resonance nuclides are mainly reflected in the resonance energy range(energy less than 0.1 MeV and more than 1 eV).Nevertheless,SU analysis in the CZP condition is just one stage of the depletion calculation.Next,the study focuses on the variation of the important nuclide reaction sensitivity coefficients and cross-section contributions on keffuncertainty during the full lifetime depletion calculation.

3.2 SU analysis of keff in the depletion calculation

3.2.1 keffsensitivity analysis

Generally,the keffsensitivity to important nuclide reactionscan be used to measurethe degree of influence of these reactions on keff.Here,we used RMC to evaluate the variationsin the keffvalueand calculated the keffsensitivity to some important nuclide reactions during the full lifetime depletion process.The specific fuel burnup step times were set as 0,0.5,1.5,5,20,50,100,150,200,250,300,350,400,500,and 600 days.The keffresults for these burnup steps are illustrated in Fig.4.For the keffuncertainty quantification,in a condition of keep keffchange trend during the full lifetime,select as few burnup step results as possible to reduce sensitivity coefficients calculation time.The 0,5,50,150,250,400,and 600 days burnup step results were chosen for analysis.

Fig.4 keff depletion results calculated by RMC

During the depletion process in the RMC,the nuclide density required by the transport calculation was obtained from the solution of the depletion equation.In thisway,the nuclide density is updated in each burnup step.Therefore,the nuclide density variation of certain important nuclides and their effect on keffuncertainty needs to be investigated.In general,some fission elements,fission products,and fission poison nuclides are considered,such as235U,238U,239Pu,135I,135Xe,149Sm,and155Gd.In the RMC Monte Carlo depletion calculation,nuclide density data are obtained from the results of the previous burnup step.Because the poison elements135Xe and149Sm are produced,as shown in Fig.5,the large neutron absorption cross-section of these elements makes the keffdecline steeply from the BOL to 5 days.Owing to the space selfshielding effect of the burnable poison Gd,keffexhibits an upward trend between 5 and 150 days.During the depletion,the burnable poison nuclidE155Gd was consumed rapidly,and after nearly 250 daystheamount in thereactor was very low,as illustrated in Fig.5a.Therefore,without the effect of the burnable poison Gd,the keffvalue decreases with fuel depletion until the end of life.

Fig.5 Important nuclides density variation during the depletion calculation.a 155Gd,10B and 149Sm density variation;b 135Iand 135Xe density variation

According to the sensitivity analysis results,the keffsensitivity coefficients for some vital nuclide reactions wereconsiderable during thedepletion calculation.Table3 lists the 14 main nuclide reactions with average integrated sensitivity coefficients during the small-sized HTGR depletion calculation.It should be noted that the average integrated sensitivity coefficients are calculated by integrating all energy groups for all regions and the sensitivity of the mixture materials through seven burnup steps.Two illustrative line charts of the integrated sensitivity coefficients of these important nuclides and their reaction cross sections are presented in Fig.6.In addition,Fig.7 shows the difference in integrated sensitivity coefficients from the BOL to the end-of-life(EOL).According to Table 3 and Fig.6,the integrated sensitivity coefficients of the average number of neutrons emitted per fission event of235U,elastic scattering of C-graphite,radioactive capture reaction of239Pu,and fission reaction of239Pu have more considerable variations than other nuclide reactions.

Fig.6 (Color online)Integrated sensitivity coefficients of important nuclides and their cross sections.(a)Important nuclides integrated sensitivity coefficients;(b)Important nuclide cross sections integrated sensitivity coefficients

Fig.7 (Color online)Integrated sensitivity coefficients variation(BOL-EOL)of important nuclide reactions

Table 3 14 important nuclide reactions average integrate sensitivity coefficients

Moreover,the radioactive capture reaction of28Si has a relatively high sensitivity,and its sensitivity coefficient value essentially remains unchanged during the full lifetime,as illustrated in Fig.6.At the same time,the integrated sensitivity coefficients of some poison isotopes,such as135Xe,149Sm,and155Gd,have no obvious variation,and these values are quite small during the full lifetime depletion.

3.2.2 keffuncertainty analysis

According to the previous analysis of the important nuclide reaction sensitivity coefficients,the uncertainty contributionsof thereaction crosssectionsto keffmay have significantly different throughout the depletion calculation,which requires further study.The SU results calculated by RMC show that the rank of the top eight contributors during the full lifetime depletion calculation did not differ substantially,however the contribution values varied.There are 12 important nuclide reaction cross-section average contributions to the keffuncertainty for the smallsized HTGR depletion calculation,which are illustrated in Table 4.These contributions to the keffuncertainty are the averagevaluesof theresultsof theseven burnup steps.It is obvious that the elastic scattering of C-graphite,radioactive capture of135Xe,radioactive capture of157Gd,radioactive capture and fission reaction of239Pu,and the average number of neutrons emitted per fission event of235U change substantially throughout the depletion calculation through the standard deviation presented in Table 4.In addition,combined with the results summarized in Tables 3 and 4,the radioactive capture reactions of poison isotopes157Gd and135Xehave relatively largevariationsin contribution values.However,their sensitivity coefficients were mostly stable during the lifetime depletion calculation.

Table 4 12 important nuclide reactions average contributions to the keff uncertainty

The 12 important nuclide reaction cross-sectional contributions to keffuncertainty in small-sized HTGR depletion calculations are shown in Fig.8.The solid lines represent the uncertainty contribution variations of the nuclear reaction for each nuclide that exists at the BOL.The figure shows that the average number of neutrons emitted per fission event of235U contributions decreased with the depletion calculation,but it was still the most significant contributor to the uncertainty of keffacross the full lifetime.Furthermore,all cross sections other than the elastic scattering cross section of C-graphite and the radioactive capture cross section of28Si exhibited a downward trend.In particular,the radioactive capture reaction of157Gd showed a noticeable decline from BOL to nearly 150 days.This was mainly caused by the depletion of157Gd.Simultaneously,this also led to a reduction in the average number of neutrons emitted per fission event of235U,contributing to keffuncertainty.

The dotted lines in Fig.8 express the fission product reaction contribution to keffuncertainty;all reaction contributions have an increasing trend with the nuclides produced during the burnup calculation.As an important fission product,135Xe is produced rapidly at the BOL and reaches equilibrium at 4 to 5 days.Simultaneously,the radioactive capture reaction of135Xe was the main contributor to fission products until nearly 450 days.After 450 days,the fission reaction and radioactive capture cross-section of239Pu became the main contributors.However,the contributions of the radioactive capture reaction of149Sm were low and barely changed.

Fig.8 Important nuclide reaction contribution variations to uncertainty in keff

Figure 9 shows the total variation of the important nuclide reaction contributions to the keffuncertainty.The phenomenon concluded with Tables 3 and 4 can be more intuitively seen in Fig.7 and Fig.9 that the radioactive capture of135Xe and157Gd has only tiny variations in sensitivity coefficients,but the contributions to keffuncertainty differ significantly during the full lifetime.

Fig.9 (Color online)Important nuclide reaction contribution variations(BOL-EOL)to uncertainty in keff

Considering the numerical results in Sect.3.2.1,the fission spectrum of235U makes a large contribution to keffuncertainty after the radioactive capture of135Xe and157Gd.However,the sensitivity coefficients do not noticeably increase during the depletion calculation.However,the nuclide reaction cross-section has uncertainty,which is presented by the covariance matrix based on the nuclear data library[23].Because the ‘sandwich’’rule is used to quantify uncertainty,although the integrated sensitivity of keffto thE235U fission spectrum is only–1.91 × 10–10,the large relative covariance explains why thE235U fission spectrum is the fourth most significant contributor.In addition,the relative covariance of the radioactive capture reaction of28Si is not large in the ENDF/B-VII.1 covariance library[23],but its contribution to the uncertainty of keffis still significant in small-sized HTGRdepletion calculations.Thedensity of28Siremained almost unchanged throughout the lifetime.Thus,it can be concluded that the large volume ratio in the core and large average sensitivity coefficient are the main reasons that the radioactive capture of28Si is the second largest contributor to keffuncertainty.

After analyzing the contribution of some important nuclidereactions,thetotal uncertainty of keffduring the full lifetime depletion calculation was quantified,as shown in Fig.10.According to the numerical results of the seven burnup steps,the uncertainty of keffremained largely constant.Moreover,owing to the fission products constantly generated during the depletion process,the uncertainty has a slightly increasing trend.

Fig.10 Total uncertainty of keff during the full lifetime depletion calculation

Since 2007,there have been many developments in reactor uncertainty analysis modeling,such as the OECD/NEA of Light Water Reactor(LWR)UAM,OECD/NEA of Boiling Water Reactor(BWR)UAM,and IAEA CRP UAM on HTGR[7,9,20,21].The keffuncertainty results of these reactor uncertainty analysis projects are presented in Table 5.The burnup value of the small-sized HTGRwas only 10.789 GWd/tU at EOL.Nevertheless,the burnup of PWRand HTR-10 wasmuch deeper.At the same time,the PWR and BWR all have an extensive increase in keffuncertainty during the full lifetime depletion,and the uncertainty value increased by 28.98% and 40.22%,respectively.However,the uncertainty of keffin smallsized HTGRs only increased by 1.12%in the full lifetime depletion calculation owing to the low discharge burnup design.

After analyzing the keffuncertainty due to the nuclear data during the burnup calculations for different types of reactors,the contributions of the top five most important nuclide reactions to keffuncertainty in different typical reactors under CZP conditions were determined,as illustrated in Table 6.BWR,pebble-bed HTR-10,and smallsized HTGRs were included.It is clear that in BWR,the top contributor to keffuncertainty is the radioactive capture reaction of238U.However,in the two HTGRs,the first contributor to keffuncertainty was the average number of neutrons emitted per fission event of235U.Moreover,the radioactive capture reaction of28Si being the second contributor to keffuncertainty is a novel finding in smallsized HTGRs.

Table 5 Different reactor keff uncertainty due to the nuclear data

Table 6 Different reactor top 5 important reaction contributions to keff uncertainty at CZP condition

4 Mechanism analysis of keff uncertainty

From the above SU analysis during the small HTGR depletion calculation,it was revealed that some cross sections of235U,28Si,157Gd,C-graphite,239Pu,135Xe,and149Sm had high sensitivity coefficients or significant contributions to the uncertainty of keff.The reason that the radioactive capture reaction crosssection of28Sihassuch a large contribution to theuncertainty of keffwasinvestigated in Sect.3.2.2.Additionally,other changes in significant nuclide reaction sensitivity coefficients may directly affect the uncertainty of keffduring the full lifetime depletion process.Therefore,it is necessary to carry out further mechanistic analyses.

Based on the results in Sect.3,the average number of neutrons emitted per fission event,fission spectrum,radioactive capture reaction,and fission reaction of235U all have a significant contribution to keffuncertainty during the depletion calculation.As one of the most important elements in fission reactors,the nuclear data of235U have a significant effect on keffuncertainty and are valuable for further studies.

According to the uncertainty quantification method introduced in Sect.2,the uncertainty of keffdue to nuclear data depends on its covariance data and sensitivity coefficients.The four crucial reactions of thE235U integrated sensitivity coefficients and their contribution to keffuncertainty are presented in Fig.11.From these two histograms,it is note worthy that the high sensitivity of the average number of neutrons emitted per fission event of235Udirectly leads to alarg E235U contribution to keffuncertainty.Although other crucial reactions of235U also significantly contribute to the total uncertainty of keff,the keffsensitivities to these reactions are not very significant and only slightly decrease during the full lifetime.In addition,the sensitivity coefficients of thE235U fission spectrum are too small to be observed in this histogram,however the contribution to keffuncertainty is still large due to the high relative covariance data[23].However,the uncertainties of the average number of neutrons emitted per fission event and fission reaction,which are based on the covariance matrices,are much smaller[23].Therefore,the average number of neutrons emitted per fission event and fission reaction of235U has a large amount of uncertainty,which can be attributed to their large sensitivity coefficients.

Fig.11 (Color online)235U important reactions integrated sensitivity coefficients and contribution to keff uncertainty.(a)235U important reactions integrated sensitivity coefficients;(b)235U important reaction’s contribution to keff uncertainty

Furthermore,the elastic scattering of C-graphite,the radioactive capture reaction,and the fission reaction of239Pu also have considerable impact and variation during the depletion process.The integrated sensitivity coefficients of these two important nuclides are shown in Fig.12.In these two histograms,the elastic scattering of C-graphite has a large basal sensitivity during the full lifetime and even exhibits a slight growth at EOL.This trend also reflects the total uncertainty change in keffto some extent.At the same time,the two important reactions of239Pu sensitivity coefficients have a significant growth at EOL,but their sensitivity coefficient values are still slight compared with those of C-graphite and235U.Based on this finding,the reason that the radioactive capture and fission reactions do not contribute significantly to keffuncertainty at EOL can be explained clearly.Moreover,this phenomenon explains why the total keffuncertainty does not increase significantly at EOL.

Fig.12 (Color online)Integrated sensitivity coefficients of C-graphite and 239Pu.(a)C-graphite elastic scattering integrated sensitivity coefficients;(b)239Pu radioactive capture reaction and fission reaction integrated sensitivity coefficients

Fission poison products,such as135Xe and149Sm,are generated during the full lifetime depletion process.These nuclides dramatically affect keffdue to their substantial absorption cross sections and therefore,the impact of these poison nuclide reaction cross-sections on keffuncertainty should be studied.The integrated sensitivity coefficients of the radioactive capture reaction of135Xe,149Sm,and157Gd during the depletion calculation are shown in Fig.13.Interestingly,based on the rapid production of135Xe at the BOL,the integrated sensitivity coefficients of135Xe increased significantly at 5 days and reached the highest value at 150 days.The integrated sensitivity coefficients of149Sm have slightly increased at 5 daysand also get peak at 150 days,but its integrated sensitivity values are much lower than that of135Xe.The burnable poison material157Gd is input at the BOL.Its sensitivity coefficients show an obvious decrease after 150 days and nearly decrease to zero at EOL.After 150 days,the main contributor of poison elements was135Xe in the small-sized HTGR.However,the integrated sensitivity coefficients of these poison elements are still much smaller than those of C-graphite or235U.Therefore,based on the analysis results and the ‘sandwich’rule,the contributions to keffuncertainty of these main poison nuclide reactions in small-sized HTGRs arise from their relative covariances.

Fig.13 (Color online)Sensitivity coefficients of the poison elements

5 Conclusion

In this study,the keffuncertainties due to the nuclear data in the CZP condition and full lifetime depletion calculation were quantified for a small-sized HTGR.RMC was used to generate the small-sized HTGR high-fidelity model and carry out critical calculations and for depletion and uncertainty calculations.In the depletion calculation,the predictor correction method was applied.In addition,the CLUTCH method was used for sensitivity analysis,and the ‘sandwich ’method was utilized to quantify the keffuncertainty through the ENDF/B-VII.1 covariance data.Our main findings are as follows:

First,the uncertainty of keffdue to nuclear data at the CZPcondition was considered.According to SU results in the CZPcondition,the average number of neutrons emitted per fission event of235U is the most important contributor to the uncertainty of keff.The radioactive capture reaction of28Si is the second largest contributor to the uncertainty in keffbecause of its heavy volume ratio in the fuel pellet.This finding differs from the results of our study of the pebble-bed HTGR[10].The total uncertainty of keffdue to the nuclear data at the CZP condition was approximately 636 pcm.

Second,the 24 fuel zone model was used for the full lifetime depletion calculation.According to the results illustrated in Tables 2 and 4,the top eight most important nuclide reaction contributors themselves to keffuncertainty did not change across the full lifetime.However,the keffuncertainty from the radioactive capture and fission reactions of239Pu increased significantly at EOL.Simultaneously,other types of reactors,such as PWR,BWR,and pebble bed HTGR,were compared with the small-sized HTGR in the full lifetime depletion keffuncertainty quantification.The results showed that the small-sized HTGR had a lower burnup value at EOL,and its keffuncertainty only changed slightly during the full lifetime depletion calculation.In the small-sized HTGR,the uncertainty of keffduring the full lifetime increased by 1.1196%,compared to 28.9855%for PWR and 40.2174%for BWR.

Finally,the variation in keffuncertainty due to nuclear data during the full lifetime depletion was analyzed.The average number of neutrons emitted per fission event of235U and elastic scattering of C-graphite significantly contribute to the uncertainty of keffowing to their large sensitivity coefficients.However,this conclusion is contrary to the fission spectrum of235U,in which the significant contribution to the keffuncertainty is due to the large covariance data of itself.Moreover,239Pu is one of the main fission products,and its important reaction crosssectional contributionsto keffuncertainty increased at EOL,but did not surpass the contributions of235U or C-graphite owing to its small sensitivity coefficient.This is the key reason that the total uncertainty of keffgrew little at the EOL.In addition,the poison elements,135Xe,149Sm,and157Gd,were investigated in the depletion calculation.The keffsensitivity coefficients for the poison element cross sections varied significantly during the full lifetime,as illustrated in Fig.13.These changes have an obvious influence on the keffvalue but did not significantly affect the keffuncertainty.

In general,the keffuncertainty due to nuclear data was quantified,and someimportant nuclidesand reactionswere determined to contribute significantly to the keffuncertainty.These findings are valuable for the design and optimization of new small-sized prismatic HTGRs.However,the nuclear data introduces non-negligible uncertainties to the nuclide density,which further contributes to the uncertainty of keffduring the depletion process.This work isnow in progress,and the uncertainty resultswill be reported in the following papers.

Author contributionsAll authors contributed to the study conception and design.Material preparation,data collection and analysis were performed by Rong-Rui Yang,Yuan Yuan,Chen Hao,Ji Ma and Guang-Hao Liu.The first draft of the manuscript was written by Rong-Rui Yang and Chen Hao,and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.