manufacturing data uncertainties propagation method in...

Research ArticleManufacturing Data Uncertainties Propagation Method inBurn-Up Problems

Thomas Frosio1 Thomas Bonaccorsi1 and Patrick Blaise2

1Reactor Physics and Fuel Cycle Division Reactor Studies Department CAD DEN French Atomic Energy andAlternative Energies Commission (CEA) 13108 Saint Paul-les-Durance France2Experimental Physics Division Reactor Studies Department CAD DEN French Atomic Energy andAlternative Energies Commission (CEA) 13108 Saint Paul-les-Durance France

Correspondence should be addressed toThomas Frosio thomasfrosiogmailcom

Received 25 August 2016 Revised 15 November 2016 Accepted 12 December 2016 Published 26 January 2017

Academic Editor Alejandro Clausse

Copyright copy 2017 Thomas Frosio et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A nuclear data-based uncertainty propagationmethodology is extended to enable propagation ofmanufacturingtechnological data(TD) uncertainties in a burn-up calculation problem taking into account correlation terms betweenBoltzmann andBateman termsThe methodology is applied to reactivity and power distributions in a Material Testing Reactor benchmark Due to the inherentstatistical behavior of manufacturing tolerances Monte Carlo sampling method is used for determining output perturbations onintegral quantities A global sensitivity analysis (GSA) is performed for each manufacturing parameter and allows identifying andranking the influential parameters whose tolerances need to be better controlled We show that the overall impact of some TDuncertainties such as uranium enrichment or fuel plate thickness on the reactivity is negligible because the different core areasinduce compensating effects on the global quantity However local quantities such as power distributions are strongly impactedby TD uncertainty propagations For isotopic concentrations no clear trends appear on the results

1 Introduction

Sensitivity analysis (SA) methods are invaluable tools allow-ing the study of how the uncertainty in the model outputrelies on the different sources of uncertainty in the modelinputs [1] As a ranking method it can be used to determinethe most contributing input variables to an output behavioras the noninfluential inputs or clarify some correlated effectswithin the model The objectives of SA are numerousone can mention model verification and understandingmodel simplification and factor prioritization [2] Finallythe SA helps in the validation of a computer code guidanceresearch efforts or the justification in terms of system designsafety There is a high amount of literature on proceduresand techniques for SA The main outcomes can be foundin [3 4] There are many possible uses of SA describedwithin the categories of decision support communicationincreased understanding or systemquantification andmodeldevelopment Many different approaches to SA are described

elsewhere varying in the experimental design used and inthe way results are processed An example of manufacturinguncertainties propagation is described in [5]

Tolerance analysis is also becoming an important tool fornuclear engineering design This seemingly arbitrary task ofassigning tolerances can have a large effect on the cost andperformance of manufactured products such as fuel designand fabrication However the fact of propagating tolerancesinstead of uncertainties does not lead to a representativeapproach of the errors because in this case only a biasis taken into account It is then imperative to understandwhat kind of physical data creates and propagates uncer-tainties on the neutronics parameters for both safety andperformance reasons In Material Testing Reactors (MTR)the performance parameters can be core fuel cycle or isotopeproduction Therefore it is necessary to calculate isotopicconcentrations uncertainties in the reactor core

We focus in this paper on technological data propagationwith a special attention to uranium enrichment and plate

HindawiScience and Technology of Nuclear InstallationsVolume 2017 Article ID 7275346 10 pageshttpsdoiorg10115520177275346

2 Science and Technology of Nuclear Installations

thickness as example of manufacturing uncertainties prop-agation In a general engineering framework much broaderthan nuclear engineering only as tolerances affect both costand quality of a product tolerancing is now considered asbeing a critical engineering design function As such toler-ance allocation is a significant task that deserves considerableattention The current situation is a compromise betweendesigners who usually specify tight tolerances to ensure highquality and manufacturers who prefer loose tolerances toreduce manufacturing cost [6] So adequate (ie reasonableor well-balanced) tolerances must be achieved in order toboth ensure the desired performance and ease the fabricationprocess

In general there are no specific guidelines for allocatingtolerances for any component but [4] quotes the followingparagraph [7] ldquoThemost commonpractice is to allocate sometolerance that seems appropriate on the basis of experienceor intuition and then conduct an analysis to ensure that theallocated tolerance suits the desired design function In orderto do this the designer must be able to realize all possibleeffects of the tolerances specified especially if universalinterchangeability is one of the design goals The effectsof specified tolerances are generally analyzed by creatingan analytical model that can predict the accumulation oftolerances in an assembly Prediction of tolerance accumu-lation is necessary because critical fits clearances etc areusually controlled by the accumulation of several componenttolerancesrdquo

After a reminder of the theoretical approach and theimplementation of tolerance analysis in the MC propagationmethodology and UQ in coupled BoltzmannBateman prob-lem a practical example is given for complete depletion cal-culation based on a Material Testing Reactor (MTR) bench-mark This latter is described and the associated tolerancedata based on an actual series of manufacturing feedbackare detailed One will focus on two main technologicalparameters uranium enrichment of the plates as well as theirthicknesses

The uncertainty propagation will be performed for twodifferent integral quantities the reactivity that is a moreglobal parameter and the power factor (ie the plate fissionrate distribution) more sensitive to local variations A partic-ular focus on the concentrations of some important isotopeswill also be made

2 Evaluation of the TechnologicalUncertainties

The method used to evaluate the uncertainties comes fromcomplete work performed in [8 9]

The complete evaluation of propagated uncertaintieson neutronics parameters requires a precise knowledge ofboth nuclear data and manufacturing uncertainties If theprimers are relatively well known and characterized throughconsistent covariance matrices such as the latest ENDFB-VII1 [10] or COMAC [11] manufacturing uncertainties aresometimes sparse and often not taken into account in the UQprocess However those values can be built by considering(supposed known) tolerances

The statistical nature of uncertainty analysis naturallyrelies on the use of Monte Carlo sampling methodologyMonte Carlo sampling methods can be used to performuncertainty propagation throughout the whole core calcu-lation process The manufacture of a technological itemis simulated for example by creating a set of componentdimensions with small random changes to simulate naturalprocess variations In this case a Gaussian model can beselected as a statistical distribution of uncertainties andtolerances can be chosen as variances values at 3120590 built byexpert elicitation

Next the resulting assembly dimensions are calculatedfrom the simulated set of component dimensions The num-bers of outliers that fall outside the specification limits arethen counted Sample sizes generally range between 5000 and100000 based on the required accuracy of the simulationThe accuracy of Monte Carlo sampling increases with largersample sizes Obviously the computational effort of largesample sizes can be significant but Monte Carlo samplingoffers many advantages because of its flexibility It alsoallows the generation of a sample of uncertain inputs Wethen obtain a sample corresponding to the outputs of thecalculation code

Of course the best and more rigorous way is to get actualmeasurement of each series of manufacturing parametersthat would allow building the propagated bias on integralparameters between the theoretical core (ie without toler-ance) and the actual (ie as built) core The measurementof each sample enables postulating a statistical model of itsmanufacturing uncertainty This is the methodology used inthe present study

21 Benchmark Description The benchmark used is thepresent paper is a Material Testing Reactor based on 20enriched 235UU3Si2Al fuel plates [9] A unique type ofassembly has been modelled to build the whole core Thebenchmark does not contain absorbing assembly in orderto simplify the calculation the goal being to give orders ofmagnitude of the propagated uncertainties

A fuel assembly is made of 22 127mm thick Zircaloyplates (in green) Each plate contains a 051mm thick U3Si2Alfuel blade called ldquothe meatrdquo The blue elements of Figure 1represent the water The assembly stiffeners are made ofaluminium

The benchmark study is performed in 15 energy groupsusing the APOLLO283 deterministic code [12] based on aMOC (Method of Characteristics) calculation scheme [13]The calculation is performed on a 2D quarter of core withad hoc symmetries The full BoltzmannBateman calculationscheme is described in [8] Each plate is discretized in 8sectors to get a more precise estimation of the local powerfactor and the concentrations

For the sake of the present work two technologicalparameters and their associated tolerances will be studiedIn the following they will be noted UO2MB for ldquoUO2 massbalancerdquo and PTh for ldquoplate thicknessrdquo

Perturbation at the beginning of the calculation enablesassessing global sensitivities

Science and Technology of Nuclear Installations 3

0475 cm

684 cm874 cm

874

cm

820

cm

Figure 1 Geometric representation of the benchmark

1 1

UO2MB PTh

Freq

uenc

y

Freq

uenc

y

Uncertainty average of 0994 standard dev of 02 Uncertainty average of 0977 standard dev of 25

Figure 2 Statistical model of manufacturing uncertainty

22 Statistical Distributions of Manufacturing ParametersThe statistical distributions are built by calculating standarddeviations and average values of manufacturing parame-ters We consider in the following that measurements oftechnological parameters are available An example of thiskind of measurement is presented on Figure 2 The meansmentioned under each plot correspond to the ratio betweentheoretical and measurement averaged values In some casesinconsistencies are observed One way for circumvent thisinconsistency is to perform a calculation of both average andtheoretical cores and compare the results on relevant integralvalues (119896eff power distributions etc) This comparison givesa manufacturing bias (not calculated in this work) which can

further be taken into account in the uncertainty tabulationof the core The red and green curves represent respec-tively the probability density function and the cumulativedistribution function normalizedThe histogram reproducesthe frequency of the values observed by measurement Ifthe distribution function of a manufacturing parameter isa known law the best is to make the propagation of thislaw in the propagation process If it is not we can performthe propagation considering that the parameter follows aGaussian lawAs aGaussian represents themaximumentropylaw the propagation is then conservative In our examplethe UO2MB parameter is not exactly a Gaussian but we willconsider it as such In these two parameters the uncertainty


on PTh is more important (25 at 1120590) than uncertaintyon UO2MB (02 at 1120590) However as we will see in thefollowing the impact of UO2MB is higher on neutronicsuncertainties

As shown in [8 9] the response function is modelled by

119891 R119899 997888rarr R119901

120581 = (1205811 120581119899) 997888rarr 119884 = (1198841 119884119901) = 119891 (120581) (1)

where 120575119891 (119905) = 120575ℎ (119905) +sum119894sub119904

[120575 (ℎ ∘ 119873119894) (119905)] (2)

and different methods can be used to calculate sensitivityindices and uncertainties The method developed in thefollowing does not allow getting the derivative from (7) as120597ℎ120597120581119896 which is the sensitivity of the interest response ℎ tothe input 120581119896 In fact this would be possible if large CPUtime is available on HPC clusters For example to calculatethe sensitivity of reactivity to PTh would require 1078 fullcalculations (ie coupled BoltzmannBateman from 0 to100000MWsdotdT) The 49 assemblies of the benchmark arecomposed each of 22 plates For UO2MB only 8624 fullcalculations should be performed This kind of calculationis impossible today in reasonable time Then calculationsare performed for each parameter by sampling their dis-tribution law Five hundred calculations are performed foreach technological parameter enabling a reconstruction of anestimator of the output distribution laws Hence parametersthat impact the neutronics values can be identified andranked Another difficulty arises here through the coupledcalculation and the associated uncertainty propagation com-ing from direct and transmutation terms Transmutation anddirect effects can be decorrelated by doubling the number ofcalculations with themethods coming from [9] In this paperwe will use theMCmethod described therein with the goal toget a global sensitivity analysis However its major drawbackis that it does not take into account correlations between inputparameters These correlations could be calculated usingPearson coefficients if the measurements are correlated Forexample if the manufacturer simultaneously measures bothfuel size andmass correlations between these two parameterscan be extracted The geometrical perturbations are donefor each assembly independently using the Salome tools(httpwwwsalome-platformorg)

The sampling can be performed using amultidimensionalGaussian law whose probability function is given by

119892 (120577) = 1radic2120587 1det (119872 (120581))

sdot exp [minus12 (120581 minus 120583)119879 [119872 (120581)]minus1 (120581 minus 120583)] (3)

where ΙΕ(120581) = 120583 = (1205831 120583119899) is the manufacturing datamathematical expectancies and119872(120581) is the covariancematrixof inputs 120581

R (httpswwwr-projectorg) has here been used as thetool to perform the sampling made independently of eachtechnological parameter

The ℓ simulations performed by sampling enable buildingthe uncertainties of different quantities 119884 of interest from

(119905) minus 119884ref (119905)= 1ℓsumℓ

119884ℓ (119905) minus 119884ref (119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias

plusmn radic 1ℓ minus 1sumℓ

(119884ℓ (119905) minus 1ℓ sumℓ

119884ℓ (119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty

997888997888997888997888997888997888997888rarrℓ 997891rarr +infinplusmn 120576119884 (119905) forall119905

(4)

The equations from [9] for local isotopic concentrationsuncertainties are still valid then and we resume here thetheory for the sake of completeness

During the calculation process it is possible to extractconcentrations values at each evolution step

Using (4) and if 119884 represents an isotopic concentrationit is then possible to build the uncertainty comparing to areference calculation

We write119872119873(119860 119905) the isotopic covariance matrix in themesh element 119860 at step 119905 (unknown) as follows

119872119873 (119860 119905) = 119885119879 (119860 119905) Ω119885 (119860 119905) (5)

where119885(119860 119905) is a matrix representing sensitivities of isotopicconcentrations ((120597119873119894120597120581119896)(119860 119905))119894119896 and Ω is the correlationmatrix (11990311989611198962)11989611198962 between input manufacturing uncertain-ties

As Ω is a correlation matrix only ones are on itsdiagonal Then we can get the uncertainty to each isotopicconcentration 119894 with the formulation

119885119894119894 (119860 119905) = (radicsum119896

120597119873119894120597120581119896 (119860 119905) 120597119873119894120597120581119896 (119860 119905) 120576119896120576119896)119894119894 (6)

with 119885119894119894(119860 119905) the uncertainty of isotopic concentration in acore mesh The MC estimator of 119885119894119894 is then written as

119894 (119860 119905) minus 119873ref (119860 119905) = 1ℓ sumℓ

119873119894ℓ (119860 119905) minus 119873ref (119860 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias

plusmn radic 1ℓ minus 1 sumℓ

(119873119894ℓ (119860 119905) minus 1ℓ sumℓ

119873119894ℓ (119860 119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty

997888997888997888997888997888997888997888rarrℓ 997891rarr +infinplusmn 120576119873119894 (119905) = 119885119894119894 (119860 119905) forall119905

(7)


Bias (PCM)Uncertainty (PCM)

minus5

0

5

10

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 040e + 00 1e + 058e + 04

(MWmiddotdt)

Figure 3 Uncertainty on reactivity (pcm) induced by PTh

3 Results on Propagation ofTwo Physical Parameters

In this paragraph we discuss the results obtained on bothmanufacturing parameters propagated in the calculationcode In the following the shapes of the outputs follow anormal law Its variance is here called ldquouncertaintyrdquo andmeanis here called ldquobiasrdquo

31 Propagation of Plates Thickness Calculations are per-formed with perturbations of the plate thickness For eachcalculation the thickness is sampled in its distribution law

311 Propagation on Reactivity The results related to reac-tivity uncertainty are shown in Figure 3 We observe a weakimpact on the propagated uncertainty (blue curve) Theimpact is around 5 pcm at 1120590 during all the irradiation Thislow value is justified by the fact that the perturbations arealmost compensated on the whole core geometry For someplates the perturbation increases the thickness as for othersit decreases So because of the important number of plates(more than a thousand) the average of all the perturbationscancels out Moreover the perturbations are made with aconstant mass balance in the plate to get only the geometricalperturbation If the uncertainty on plate mass balance isknown another calculation can be performed to get thisparticular impact

However we observe an important impact on the bias(compared to the standard deviation) This impact is amodel bias coming from the mesh perturbations and fromthe average of measured values compared to theoreticalvalue (Figure 2) In fact when the thickness of the platesis modified the perturbation of the calculation mesh is

0MWmiddotdt

100000MWmiddotdt

002

004

006

008

010

012

014

016

002

004

006

008

010

012

014

016

Figure 4 Uncertainty on power factors () induced by PTh

automatically produced Another part of this bias comes fromthe convergence of the MC estimator

312 Propagation on Power Factors As for the reactivity theperturbation of plate thickness has a relatively weak impacton the power factors (Figure 4) The propagated uncertaintyis of the order of 016 at 1120590 for the maximal value Atthe beginning of life the uncertainties are concentratedprincipally in the centre of the assemblies In fact thethickness perturbation modifies the moderation ratio Thiseffect is most important in the centre of the assemblies wherethe power factor is lower During depletion the uncertaintiesdecrease with uranium consumption So after relatively largeburn-up values the total uncertainty on the perturbed coreconverges to the one of the reference core The uncertaintiesinduced by technological uncertainties on plate thicknessesare then compensated by the fuel consumption


BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

Figure 5 Uncertainty on reactivity (pcm) induced by UO2MB

32 Propagation of UO2 Mass Balance For the UO2MBparameter the results are slightly different and are detailedin the next 2 paragraphs

321 Propagation on Reactivity The propagated uncertaintyvalues for reactivity presented on Figure 5 remain weakfor the same reasons as previously At the beginning of lifethe uncertainty on reactivity is close to 3 pcm at 1120590 Duringirradiation this value increases to reach 15 pcm at 1120590The biasobserved on the reactivity is only the result of the convergenceof MC estimator The bias is weak compared to the totaluncertainty of the reactivity (including nuclear data [8 9])Increasing the number of samples is possible but not efficientbecause of the large CPU time needed for each calculationThe uncertainty increase comes from a flux displacement inthe core during the depletion process During irradiation theareas with highermass balance uncertainties are consumed atfirst hence inducing a flux displacement to the outer regionsThe elements in the outer regions become more importantin the reactivity uncertainty which is then increased conse-quently

322 Propagation on Power Factors For the power factorswe observe a random map (Figure 6) because of the absenceof strong space dependency for mass balance uncertaintyHence no trend on power factors uncertainties can beobserved or pointed out during depletion We can onlyobserve the results of a lot of sampling The maximal valueof uncertainty is 1 at 1120590 With this value being highercompared to uncertainty coming from PTh parameter andPTh uncertainty being ten times lower than uncertaintyon UO2MB this parameter has a higher impact on powerdistribution During irradiation the uranium consumption

0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100

Figure 6 Uncertainty on power factors () coming from UO2MB

in the core centre is responsible for the decrease of theassociated UO2MB propagated uncertainty in this area

4 Computation of Simultaneous NuclearData and Manufacturing Data UncertaintyPropagation

The total propagated uncertainty coming from both nucleardata (fission yield and cross sections) andmanufacturing datais obtained by applying the process previously described in[8 9] The covariance matrices used are the same as thoseused in [9] (read [14 15] for fission yields and [11 16ndash18] forcross sections) The results obtained by the propagation aredescribed for the three parameters reactivity power factorsand isotopic concentrations


Bias (pcm)Uncertainty (pcm)

4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)

Figure 7 Uncertainty on the reactivity during depletion (pcm at1120590)41 Reactivity For the reactivity (cf Figure 7) the totaluncertainty is mainly the uncertainty propagated fromnuclear data as the manufacturing uncertainties propagationis very low So the uncertainty is the result of fission yield andcross section uncertainty propagation The total uncertaintycoming from nuclear data and technological data is herestarting around 400 pcm at the beginning of life of the reactorcore It increases during the life to reach 600 pcm at 1120590at 100000MWsdotdt As detailed in [9] the uncertainty onreactivity coming from fission yield is around 120ndash170 pcmat 1120590 from some tenth of MWsdotdt This has the particularityto modify the bent shape of the curve at the beginning oflife The observed bias is negligible and can come from themesh differences between the input data or the Monte Carloestimator convergence At the end of life the cross sectionuncertainties are themajor contributors to the total reactivityuncertainty the fission yield uncertainty being responsible forabout 20 of the total

42 Power Factors The power factors map uncertaintiesare principally caused by the UO2MB parameter except atthe core-reflector interface We observe in Figure 8 at thebeginning of life an uncertainty of 18 at 1120590 for the fuel atthe interface which corresponds to the value already foundin [9] that is without technological data When one movesaway from this interface the uncertainty drops significantlyfor reaching 10 at 1120590 which is the value coming fromUO2MB only the propagated uncertainty from nuclear databeing almost zero in this area We showed in [9] that nucleardata uncertainty propagation gives 06 at 1120590 in the corecentre and that fission yields were negligible for power fac-tors uncertainties During the depletion process the powerfactor uncertainty decreases everywhere and converges to theUO2MB value

0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt

Figure 8 Uncertainty on power factors ( at 1120590)

43 Isotopic Concentrations For the isotopic concentrationsno clear trend can be extracted from the results The nextfigures reproduce cumulated propagated uncertainties forseveral isotopic concentrations Some present an uncertaintycoming principally from nuclear data while others comefrom manufacturing data We showed in [9] that propagatednuclear data uncertainties on xenon concentration reached8 at 1120590 In Figure 9 the total uncertainty (taking intoaccount nuclear data and manufacturing data) is increaseddemonstrating that manufacturing uncertainties induce anadditional uncertainty on xenon isotopic concentrationHowever for 239Pu and 149Sm build-ups (Figure 10) theuncertainties are slightly reduced indicating that negativecorrelations are generated in the calculation by manu-facturing data and nuclear data perturbation For 235UFigure 11 shows an important heterogeneous uncertaintyessentially coming fromUO2MB hence frommanufacturing


Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11

Figure 9 Uncertainty on local isotopic concentrations for somePu241 and Xe135 at 48000MWsdotdt ( at 1120590)

uncertainty The uncertainties on 155Gd and 241Pu are alsopresented as they are important

Table 1 summarizes some major results obtained forisotopic concentration uncertainties as a function of thelocation

5 Conclusions

A methodology for manufacturing uncertainty propagationbased on Monte Carlo sampling has been presented andtested on a MTR benchmark An adequate use of the manu-facturer information enables simulating different realisationsof the core Those realisations enable getting estimators ofstandard deviation for different neutronics parameters suchas reactivity power factors and local isotopic concentrationsTwo manufacturing parameters are propagated in the study

Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14

Figure 10 Uncertainty on local isotopic concentrations for someSm149 and Pu239 at 48000MWsdotdt ( at 1120590)

Table 1 Summary of isotopic concentrations uncertainties

Position isotope Core centre Core peripheryPu241 20 40ndash45Xe135 115 100ndash115Sm149 35 10ndash15Pu239 10ndash11 10ndash11Gd155 120ndash150 80ndash120U235 08ndash12 08ndash12

One is the UO2 mass balance in the fuel and the other is ageometric characteristic the fuel plate thickness

It is shown that the propagation of these manufacturingparameters uncertainties on reactivity is negligible Thiscomes from the fact that the perturbed core elements arenumerous So globally for the core compensations arise and


U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14

Figure 11 Uncertainty on local isotopic concentrations for someGd155 and U235 at 48000MWsdotdt ( at 1120590)contribute to reducing the reactivity perturbation which is aglobal parameter As an example the propagation of a 25uncertainty on the fuel plate thickness coupled with a 02uncertainty on the UO2 mass balance induces a combinedeffect on the reactivity of less than 20 pcm (10minus5 Δ119896119896) at 1120590

For local parameters the results are different The massbalance strongly impacts the power factors map during allthe depletion cycle Although the uncertainties decreaseduring depletion because of the fuel consumption it remainsimportant (approximately 1 at 1120590 for local power factors)More surprisingly the plate thickness has almost no impacton the power factor It is principally the induced perturbationon themoderation ratio coming from the thickness perturba-tion which impacts the power map This explains the moresignificant uncertainty at the centre of the fuel assembly

In a second step a complete uncertainty propagation of alluncertain data (nuclear data such as cross sections and fission

yields and manufacturing data) through the depletion cyclehas been performed to observe the combination of all theseuncertainties It is shown that even if these manufacturingdata uncertainties can be neglected for reactivity there is noclear trend for isotopic concentrations due to the coupling ofdifferent sources of uncertainties For example contributionsto the 135Xe uncertainty during depletion come from bothnuclear data and manufacturing data From [9] nuclear datapropagated uncertainties were responsible for 8 on 135Xeconcentration for a total propagated uncertainty from bothnuclear data and technological data of 11 at 1120590 On theopposite 235U is mostly sensitive to manufacturing datauncertainties and less sensitive to nuclear data We got (see[9]) less than 01 of propagated uncertainty on 235U con-centration from nuclear data and the present work exhibits a10 at 1120590 due to both nuclear data and technological datauncertainties

Competing Interests

The authors declare that they have no competing interests

References

[1] B Iooss and P Lemaıtre ldquoA Review on global sensitivityanalysis methodsrdquo in Uncertainty Management in Simulation-Optimization of Complex Systems vol 59 of OperationsResearchComputer Science Interfaces Series pp 101ndash122Springer Boston Mass USA 2015

[2] D L Allaire and K E Willcox ldquoA variance-based sensitivityindex function for factor prioritizationrdquo Reliability Engineeringamp System Safety vol 107 pp 107ndash114 2012

[3] A Saltelli K Chan and E M Scott Eds Sensitivity AnalysisWiley Series in Probability and Statistics John Wiley amp SonsChichester UK 2000

[4] C Glancy and K Chase ldquoA second order method for assemblytolerance analysisrdquo in Proceedings of the ASME Design Engi-neering Technical Conferences pp 12ndash15 Las Vegas Nev USASeptember 1999

[5] M Pecchia A Vasiliev H Ferroukhi and A Pautz ldquoCriticalitysafety evaluation of a Swiss wet storage pool using a globaluncertainty analysis methodologyrdquo Annals of Nuclear Energyvol 83 pp 226ndash235 2015

[6] R B Malmiry J-Y Dantan J Pailhes and J-F AntoineldquoFrom functions to tolerance analysis models by means ofthe integration of CTOC with CPMrdquo in Proceedings of the 1stSeminar of the European Group of Research in Tolerancing E-GRT Erlangen Germany June 2015

[7] M Mazur Tolerance analysis and synthesis of assemblies subjectto loading with process integration and design optimization tools[PhD thesis] School of Aerospace Mechanical and Manu-facturing Engineering RMIT University Melbourne Australia2013

[8] T Frosio T Bonaccorsi and P Blaise ldquoNuclear data uncer-tainties propagation methods in BoltzmannBateman coupledproblems application to reactivity in MTRrdquo Annals of NuclearEnergy vol 90 pp 303ndash317 2016

[9] T Frosio T Bonaccorsi and P Blaise ldquoFission yields andcross section uncertainty propagation in BoltzmannBatemancoupled problems global and local parameters analysis with a


focus on MTRrdquo Annals of Nuclear Energy vol 98 pp 43ndash602016

[10] D L Smith ldquoEvaluated nuclear data covariances the journeyfromENDFB-VII0 to ENDFB-VII1rdquoNuclearData Sheets vol112 no 12 pp 3037ndash3053 2011

[11] C De Saint Jean P Archier G Noguere et al ldquoEstimationof multi-group cross section covariances for 235238U 239Pu241Am 56Fe 23Na and 27Alrdquo in Proceedings of the Conferenceon Advances in Reactor PhysicsmdashLinking Research Industry andEducation (PHYSOR rsquo12) Knoxville Tenn USA April 2012

[12] R Sanchez I Zmijarevi M Coste-Delclaux et al ldquoAPOLLO2Year 2010rdquo Nuclear Engineering and Design vol 42 no 5 pp474ndash499 2010

[13] J F Vidal O Litaize D Bernard A Santamarina and CVaglio-Gaudard ldquoNew modeling of LWR assemblies with theAPOLLO2 code packagerdquo in Proceedings of the InternationalConference on Mathematics and Computation (MampC 2007)Monterey Calif USA April 2007

[14] N Terranova O Serot P Archier V Vallet C De Saint andM Sumini ldquoA covariance generation methodology for fissionproduct yieldsrdquo in Proceedings of the 4th InternationalWorkshopon Nuclear Data Evaluation for Reactor applications EPJWeb ofConferences March 2016

[15] N Terranova O Serot P Archier C De Saint Jean and MSumini ldquoCovariance matrix evaluations for independent massfission yieldsrdquo Nuclear Data Sheets vol 123 pp 225ndash230 2015

[16] D Bernard L Leal O Leray A Santamarina and C VaglioldquoU235 covariance matrix associated with JEFF311 evaluationrdquoin Proceedings of the JEFDOC-1360 JEFF Meeting ParisFrance December 2010

[17] C Vaglio-Gaudard A Santamarina D Bernard et al ldquoNew56Fe covariances for the JEFF3 file from the feedback of integralbenchmark analysisrdquo Nuclear Science and Engineering vol 166no 3 pp 267ndash275 2010

[18] C De Saint Jean P Archier P Leconte E Privas G Noguereand O Litaize ldquoCovariances on 239Pu 238U and 235U neutroncross sections with CONRAD coderdquo in Proceedings of theWorkshop NEMEA-7 2013

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of


Journal ofPetroleum Engineering


Industrial EngineeringJournal of


Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of


Journal of


Renewable Energy

Submit your manuscripts athttpswwwhindawicom


StructuresJournal of


RotatingMachinery


EnergyJournal of


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Nuclear InstallationsScience and Technology of


Solar EnergyJournal of


Wind EnergyJournal of


Nuclear EnergyInternational Journal of


High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


thickness as example of manufacturing uncertainties prop-agation In a general engineering framework much broaderthan nuclear engineering only as tolerances affect both costand quality of a product tolerancing is now considered asbeing a critical engineering design function As such toler-ance allocation is a significant task that deserves considerableattention The current situation is a compromise betweendesigners who usually specify tight tolerances to ensure highquality and manufacturers who prefer loose tolerances toreduce manufacturing cost [6] So adequate (ie reasonableor well-balanced) tolerances must be achieved in order toboth ensure the desired performance and ease the fabricationprocess

In general there are no specific guidelines for allocatingtolerances for any component but [4] quotes the followingparagraph [7] ldquoThemost commonpractice is to allocate sometolerance that seems appropriate on the basis of experienceor intuition and then conduct an analysis to ensure that theallocated tolerance suits the desired design function In orderto do this the designer must be able to realize all possibleeffects of the tolerances specified especially if universalinterchangeability is one of the design goals The effectsof specified tolerances are generally analyzed by creatingan analytical model that can predict the accumulation oftolerances in an assembly Prediction of tolerance accumu-lation is necessary because critical fits clearances etc areusually controlled by the accumulation of several componenttolerancesrdquo

After a reminder of the theoretical approach and theimplementation of tolerance analysis in the MC propagationmethodology and UQ in coupled BoltzmannBateman prob-lem a practical example is given for complete depletion cal-culation based on a Material Testing Reactor (MTR) bench-mark This latter is described and the associated tolerancedata based on an actual series of manufacturing feedbackare detailed One will focus on two main technologicalparameters uranium enrichment of the plates as well as theirthicknesses

The uncertainty propagation will be performed for twodifferent integral quantities the reactivity that is a moreglobal parameter and the power factor (ie the plate fissionrate distribution) more sensitive to local variations A partic-ular focus on the concentrations of some important isotopeswill also be made

2 Evaluation of the TechnologicalUncertainties

The method used to evaluate the uncertainties comes fromcomplete work performed in [8 9]

The complete evaluation of propagated uncertaintieson neutronics parameters requires a precise knowledge ofboth nuclear data and manufacturing uncertainties If theprimers are relatively well known and characterized throughconsistent covariance matrices such as the latest ENDFB-VII1 [10] or COMAC [11] manufacturing uncertainties aresometimes sparse and often not taken into account in the UQprocess However those values can be built by considering(supposed known) tolerances

The statistical nature of uncertainty analysis naturallyrelies on the use of Monte Carlo sampling methodologyMonte Carlo sampling methods can be used to performuncertainty propagation throughout the whole core calcu-lation process The manufacture of a technological itemis simulated for example by creating a set of componentdimensions with small random changes to simulate naturalprocess variations In this case a Gaussian model can beselected as a statistical distribution of uncertainties andtolerances can be chosen as variances values at 3120590 built byexpert elicitation

Next the resulting assembly dimensions are calculatedfrom the simulated set of component dimensions The num-bers of outliers that fall outside the specification limits arethen counted Sample sizes generally range between 5000 and100000 based on the required accuracy of the simulationThe accuracy of Monte Carlo sampling increases with largersample sizes Obviously the computational effort of largesample sizes can be significant but Monte Carlo samplingoffers many advantages because of its flexibility It alsoallows the generation of a sample of uncertain inputs Wethen obtain a sample corresponding to the outputs of thecalculation code

Of course the best and more rigorous way is to get actualmeasurement of each series of manufacturing parametersthat would allow building the propagated bias on integralparameters between the theoretical core (ie without toler-ance) and the actual (ie as built) core The measurementof each sample enables postulating a statistical model of itsmanufacturing uncertainty This is the methodology used inthe present study

21 Benchmark Description The benchmark used is thepresent paper is a Material Testing Reactor based on 20enriched 235UU3Si2Al fuel plates [9] A unique type ofassembly has been modelled to build the whole core Thebenchmark does not contain absorbing assembly in orderto simplify the calculation the goal being to give orders ofmagnitude of the propagated uncertainties

A fuel assembly is made of 22 127mm thick Zircaloyplates (in green) Each plate contains a 051mm thick U3Si2Alfuel blade called ldquothe meatrdquo The blue elements of Figure 1represent the water The assembly stiffeners are made ofaluminium

The benchmark study is performed in 15 energy groupsusing the APOLLO283 deterministic code [12] based on aMOC (Method of Characteristics) calculation scheme [13]The calculation is performed on a 2D quarter of core withad hoc symmetries The full BoltzmannBateman calculationscheme is described in [8] Each plate is discretized in 8sectors to get a more precise estimation of the local powerfactor and the concentrations

For the sake of the present work two technologicalparameters and their associated tolerances will be studiedIn the following they will be noted UO2MB for ldquoUO2 massbalancerdquo and PTh for ldquoplate thicknessrdquo

Perturbation at the beginning of the calculation enablesassessing global sensitivities


0475 cm

684 cm874 cm

874

cm

820

cm


1 1

UO2MB PTh

Freq

uenc

y

Freq

uenc

y








119891 R119899 997888rarr R119901

120581 = (1205811 120581119899) 997888rarr 119884 = (1198841 119884119901) = 119891 (120581) (1)

where 120575119891 (119905) = 120575ℎ (119905) +sum119894sub119904

[120575 (ℎ ∘ 119873119894) (119905)] (2)



119892 (120577) = 1radic2120587 1det (119872 (120581))





(119905) minus 119884ref (119905)= 1ℓsumℓ

119884ℓ (119905) minus 119884ref (119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119884ℓ (119905) minus 1ℓ sumℓ

119884ℓ (119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(4)





119872119873 (119860 119905) = 119885119879 (119860 119905) Ω119885 (119860 119905) (5)



119885119894119894 (119860 119905) = (radicsum119896

120597119873119894120597120581119896 (119860 119905) 120597119873119894120597120581119896 (119860 119905) 120576119896120576119896)119894119894 (6)


119894 (119860 119905) minus 119873ref (119860 119905) = 1ℓ sumℓ

119873119894ℓ (119860 119905) minus 119873ref (119860 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119873119894ℓ (119860 119905) minus 1ℓ sumℓ

119873119894ℓ (119860 119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(7)



minus5

0

5

10

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 040e + 00 1e + 058e + 04

(MWmiddotdt)







0MWmiddotdt

100000MWmiddotdt

002

004

006

008

010

012

014

016

002

004

006

008

010

012

014

016





BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)





0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100







4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0475 cm

684 cm874 cm

874

cm

820

cm


1 1

UO2MB PTh

Freq

uenc

y

Freq

uenc

y








119891 R119899 997888rarr R119901

120581 = (1205811 120581119899) 997888rarr 119884 = (1198841 119884119901) = 119891 (120581) (1)

where 120575119891 (119905) = 120575ℎ (119905) +sum119894sub119904

[120575 (ℎ ∘ 119873119894) (119905)] (2)



119892 (120577) = 1radic2120587 1det (119872 (120581))





(119905) minus 119884ref (119905)= 1ℓsumℓ

119884ℓ (119905) minus 119884ref (119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119884ℓ (119905) minus 1ℓ sumℓ

119884ℓ (119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(4)





119872119873 (119860 119905) = 119885119879 (119860 119905) Ω119885 (119860 119905) (5)



119885119894119894 (119860 119905) = (radicsum119896

120597119873119894120597120581119896 (119860 119905) 120597119873119894120597120581119896 (119860 119905) 120576119896120576119896)119894119894 (6)


119894 (119860 119905) minus 119873ref (119860 119905) = 1ℓ sumℓ

119873119894ℓ (119860 119905) minus 119873ref (119860 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119873119894ℓ (119860 119905) minus 1ℓ sumℓ

119873119894ℓ (119860 119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(7)



minus5

0

5

10

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 040e + 00 1e + 058e + 04

(MWmiddotdt)







0MWmiddotdt

100000MWmiddotdt

002

004

006

008

010

012

014

016

002

004

006

008

010

012

014

016





BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)





0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100







4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















119891 R119899 997888rarr R119901

120581 = (1205811 120581119899) 997888rarr 119884 = (1198841 119884119901) = 119891 (120581) (1)

where 120575119891 (119905) = 120575ℎ (119905) +sum119894sub119904

[120575 (ℎ ∘ 119873119894) (119905)] (2)



119892 (120577) = 1radic2120587 1det (119872 (120581))





(119905) minus 119884ref (119905)= 1ℓsumℓ

119884ℓ (119905) minus 119884ref (119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119884ℓ (119905) minus 1ℓ sumℓ

119884ℓ (119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(4)





119872119873 (119860 119905) = 119885119879 (119860 119905) Ω119885 (119860 119905) (5)



119885119894119894 (119860 119905) = (radicsum119896

120597119873119894120597120581119896 (119860 119905) 120597119873119894120597120581119896 (119860 119905) 120576119896120576119896)119894119894 (6)


119894 (119860 119905) minus 119873ref (119860 119905) = 1ℓ sumℓ

119873119894ℓ (119860 119905) minus 119873ref (119860 119905)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Bias


(119873119894ℓ (119860 119905) minus 1ℓ sumℓ

119873119894ℓ (119860 119905))2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Uncertainty


(7)



minus5

0

5

10

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 040e + 00 1e + 058e + 04

(MWmiddotdt)







0MWmiddotdt

100000MWmiddotdt

002

004

006

008

010

012

014

016

002

004

006

008

010

012

014

016





BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)





0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100







4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















minus5

0

5

10

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 040e + 00 1e + 058e + 04

(MWmiddotdt)







0MWmiddotdt

100000MWmiddotdt

002

004

006

008

010

012

014

016

002

004

006

008

010

012

014

016





BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)





0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100







4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















BiasUncertainty

0

5

10

15

Unc

erta

inty

on

reac

tivity

(pcm

)

2e + 04 4e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)





0MWmiddotdt

100MWmiddotdt

060

065

070

075

080

085

090

095

100

060

065

070

075

080

085

090

095

100







4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















4e + 042e + 04 6e + 04 8e + 04 1e + 050e + 00

(MWmiddotdt)

minus200

0

200

400

600

Reac

tivity

unc

erta

inty

(pcm

)



0MWmiddotdt

08

10

12

14

16

18

07

08

09

10

11

100000MWmiddotdt




Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















Xe135

Pu241

20

25

30

35

40

45

7

8

9

10

11




5 Conclusions


Sm149

Pu239

10

15

20

25

30

09

10

11

12

13

14







U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















U235

Gd155

08

10

12

14

16

2

4

6

8

10

12

14





Competing Interests


References

























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of
































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















manufacturing data uncertainties propagation method in...

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