A study of the uncertainty in calculations of the rod ejection accident in a pressurized water reactor is being carried out for the US Nuclear Regulatory Commission. This paper is a progress report on that study. Results are presented for the sensitivity of core energy deposition to the key parameters: ejected rod worth, delayed neutron fraction, Doppler reactivity coefficient, and fuel specific heat. These results can be used in the future to estimate the uncertainty in local fuel enthalpy given some assumptions about the uncertainty in the key parameters. This study is also concerned with the effect of the intra-assembly representation in calculations. The issue is the error that might be present if assembly-average power is calculated, and pin peaking factors from a static calculation are then used to determine local fuel enthalpy. This is being studied with the help of a collaborative effort with Russian and French analysts who are using codes with different intra-assembly representations. The US code being used is PARCS which calculates power on an assembly-average basis. The Russian code being used is BARS which calculates power for individual fuel pins using a heterogeneous representation based on a Green`s Function method

Aronson, Arnold L.

Diamond, David J.

Yang, Chae-Yong

A study of the uncertainty in calculations of the rod ejection accident in a pressurized water reactor is being carried out for the US Nuclear Regulatory Commission. This paper is a progress report on that study. Results are presented for the sensitivity of core energy deposition to the key parameters: ejected rod worth, delayed neutron fraction, Doppler reactivity coefficient, and fuel specific heat. These results can be used in the future to estimate the uncertainty in local fuel enthalpy given some assumptions about the uncertainty in the key parameters. This study is also concerned with the effect of the intra-assembly representation in calculations. The issue is the error that might be present if assembly-average power is calculated, and pin peaking factors from a static calculation are then used to determine local fuel enthalpy. This is being studied with the help of a collaborative effort with Russian and French analysts who are using codes with different intra-assembly representations. The US code being used is PARCS which calculates power on an assembly-average basis. The Russian code being used is BARS which calculates power for individual fuel pins using a heterogeneous representation based on a Green's Function method

Diamond, D. J.

Yang, C. Y.

Aronson, A. L.

UNT Digital Library

BNL-NUREG-66230ESTIMATING THE UNCERTAINTY INREACTIVITY ACCIDENT NEUTRONIC CALCULATIONS’David J, Diamond, Chae-Yong Yang, Arnold L. AronsonBrookhaven National LaboratoryUpton, NY 11973-5000ABSTRACTA study of the uncertainty in calculations of the rod ejectionaccident in a pressurized water reactor is being carried out for theU.S. Nuclear Regulatory Commission, This paper is a progressreport on that study. Results are presented for the sensitivity ofcore energy deposition to the key parameters: ejected rod worth,delayed neutron fraction, Doppler reactivity coefficient, and fuelspecific heat. These results can be used in the future to estimatethe uncertainty in local fuel enthalpy given some assumptionsabout the uncertainty in the key parameters. This study is alsoconcerned with the effect of the intra-assembly representation incalculations. The issue is the error that might be present ifassembly-ayerage power is calculated, and pin peaking factorsfrom a static calculation are then used to determine local fuelenthalpy. This is being studied with the help of a collaborativeeffort with Russian and French analysts who are using codeswith different intra-assembly representations. The U.S. codebeing used is PARCS which calculates power on an assembly-average basis. The Russian code being used is BARS whichcalculates power for individual fuel pins using a heterogeneousrepresentation based on a Green’s Function method.INTRODUCTIONBackgroundThis study (and others) have been carried out for the U.S. Nuclear Regulatory Commission(NRC) to understand fuel behavior at burnups beyond the current licensing limits. Whenbehavior is sufficiently understood, new acceptance criteria may be proposed for design-basisreactivity initiated accidents (RIAs) in high burnup fuel (and perhaps for fuel at burnups alreadyexperienced in operating plants). Acceptance criteria have traditionally been expressed interms of maximum fuel pellet enthalpy, and hence it is of interest to know what is the fuellThis work was performed under the auspices of the U.S. Nuclear Regulatory Commission.oFF~clAL FILE COPYenthalpy during an RIA. “Best-estimate” methods are available to answer this question, butit is necessary to also understand the uncertainty in the calculated fuel enthalpy.The above explains the principal motivation for this study which focuses on the design-basisreactivity accident for a pressurized water reactor (PWR). However, it should also be notedthat NRC licensees may be moving away from the traditional, conservative methods forcalculating fuel enthalpy toward best-estimate methods, and these methods will require anuncertainty analysis. Hence, the study reported upon herein may have a role in regulatorydecisions approving new methods for, and the results of, analyzing RIAs.In a previous study at Brookhaven National Laboratory, the uncertainty in calculating fuelenthalpy for the rod drop accident in a boiling water reactor (BWR) was addressed [1]. Thatstudy indicated that the random error in the calculated fuel enthalpy could be approximately& 75% at the 20 level. It also showed that there could be an additional systematic error of25°A due to the way the intra-assembly power peaking was calculated. This latter error incombination with the random error meant that the calculated fuel enthalpy has to be increasedby approximately 100% to obtain results at the 95-95 confidence level.The current study looks at the RIA for a PWR--namely, the rod ejection accident (REA).Because the intra-assembly calculation of power was important for the BWR, it was felt thatthis should be studied for the PWR and with more rigorous methods than were used for theBWR study. Calculations done as part of a recent Russian study have shown that the peakfuel pin enthalpy during an REA may not be found in the assembly with the peak average fuelenthalpy [2]. In the West, it is typical to use methods that homogenize the assembly and thencalculate the assembly response to the REA. The state-of-the-art is changing as fluxreconstruction methods are being introduced to improve the intra-assembly representation.Nevertheless, the Russian results are another reason to study the effect of the intra-assemblyrepresentation.ObiectiveThe objective of this study is to improve our understanding of the uncertainty in fuel enthalpycalculated for the REA. The approach is twofold. Sensitivity studies are to be carried out todetermine the effect on calculated fuel enthalpy of uncertainties in the important parameterswhich determine the outcome of the REA. The ultimate objective is to use the sensitivity toestimate the random error in the fuel enthalpy due to random errors in these key parameters.The second approach in this study is to compare the results for the REA using a code thattreats the assembly as an homogenized region and a code that represents each pin in theassembly explicitly. This would give an estimate of the uncertainty in results due to the intra-assembly representation. The PARCS code (Purdue Advanced Reactor Core Simulator) [3], likeseveral other nodal codes in use in the West, treats each assembly as an homogenized region.A newer version of PARCS, with a flux reconstruction method, will soon be available and canbe used to obtain additional information about the effect of the intra-assembly representation.The BARS code [2], developed at the Russian Research Centre - Kurchatov Institute (RRC-KI),uses a heterogeneous method wherein each fuel pin is represented explicitly. Hence, it is anobjective of this study to compare results for the REA from PARCS and BARS in order tounderstand the effect of the intra-assembly representation, It should also be noted that sincePARCS is a relatively new addition to the computer tools used by the NRC, the comparison ofPARCS and BARS is likely to contribute to the code assessment carried out for PARCS.Further information on the effect of the intra-assembly representation will come from the sameanalysis being carried out by the French Institute for Nuclear Safety and Protection {IPSN) andfrom calculations expected to be carried out using the version of PARCS with the fluxreconstruction model.ScoRe of this PaperThe results of the sensitivity analysis for the REA are presented in the following section. Inthe future, these results will be applied to give an estimate of the uncertainty in the calculatedfuel enthalpy based on the random errors expected in the key parameters which enter into thecalculation. The comparison being carried out between PARCS and BARS is discussed in thenext section. This includes some of the. limitations of that comparison,SENSITIVITY STUDIESIntroductionAlthough fuel enthalpy is the parameter of interest, the calculations shown in this paper wereof the sensitivity of energy deposition. If the event is adiabatic, then energy deposition andfuel enthalpy are essentially identical. The calculations were for total energy deposition ratherthan for energy deposition at the position of peak power. At a later time, when the editsbecome available in PARCS, local fuel enthalpy will be considered.The sensitivity is the relative change in energy deposition (Q) per relative change in key reactorparameter (x). Hence, the sensitivity to x is~x . (~Q/Q)(6XIX)The parameters of interest are well known from previous studies (e.g., [11) and are related tothe reactivity insertion above the prompt critical condition and the negative reactivity feedbackfrom the energy deposition. There are four key parameters which control these phenomena.The first is the reactivity worth of the ejected control rod, PO. This parameter is determinedby the core design and by the operating procedures which determine the extent of insertionof the rod and the placement of other control rods at any operating condition. The secondparameter is the delayed neutron fraction, ~. For a given core design, this parameter changessignificantly as the fuel burnup changes. The fuel feedback can be expressed in terms of thefuel temperature (or Doppler) reactivity coefficient, a, and the specific heat, CP, of the pelletwhich translates energy into temperature. The Doppler coefficient is determined by coredesign and the time during the fuel cycle whereas the specific heat changes relatively littlewith burnup.Sensitivity to Kev ParametersThe sensitivity to control rod worth is given in Figure 1 where it is plotted versus control rodworth in units of $, i.e., R = p~lp. The data points on the graph come from PARCScalculations of different reactivity insertion events. Only events with rod worths greater than$1 are of interest. The energy deposition is that deposited during the initial power pulse.16.014.012.010.08.06.04.02.00.0rA power puke— S = R/(R-l)AA0.0 1.0 2.0Rod Worth, R ($)Figure 1 Sensitivity of Energy Deposition to Reactivity InsertionIt is well-known that the power excursion resulting from a reactivity insertion above promptcritical consists of an initial power pulse turned around by feedback followed by a slowdecrease in power, due to delayed neutrons, at a level that is still significant as far as energydeposition is concerned. This relatively simple behavior is amenable to simple models.Using the point kinetics model and the Nordheim-Fuchs approximation an expression for thesensitivity to rod worth can be derived for the time up to the end of the initial power pulse.That expression is SP = R/( R-l ) which is plotted on Figure 1. It can be seen that the datapoints are in good agreement with the theory for the range shown. As the rod worthapproaches $1 (from above), corresponding to prompt critical, the simplified expression hasa singularity and is no longer valid, Nevertheless, the sensitivity does increase as rod worthapproaches $1. Although this is true, it is also true that the energy deposition becomessmaller and this sensitivity may become less important. The fact that the energy depositiongets smaller has been shown by many analysts. The corresponding result for the energydeposition from the simplified model introduced above also shows this trend:Q = 2(P0-~)/(~C’)The sensitivity of energy deposition to delayed neutron fraction is shown in Figure 2 as afunction of reactivity insertion. The data points from PARCS are plotted for both the initialpower pulse and by accounting for energy deposition out to three seconds. In addition, thegraph shows the curve obtained from the simplified model which predicts a sensitivity ofSP = -1 /(R-l). Again it is seen that the results for the initial power pulse are in agreement withthe theoretical results for the range shown except that as rod worth approaches $1 fromabove the agreement begins to fail. The sensitivity to 3 s is less than for the initial powerpulse as the energy deposited after the initial power pulse is not dependent on rod worth butonly on the delayed neutron decay.Rod Worth, R ($)0.0 0.5 1.0 1.5 2.00.0k-1.0-2.007= -3.0>.-.=Inc -4.0$-5.0-6.00000AA A Power PulseO T03s— S = -l/(R-l)-7.0 tFigure 2 Sensitivity of Energy Deposition to Delayed Neutron FractionThe sensitivity of energy deposition to fuel heat capacity is shown in Figure 3 as a functionof reactivity insertion. Energy deposition is assumed to be to either the end of the initial powerpulse or to 3 s to obtain the points plotted on the graph. The corresponding sensitivity fromthe simplified model is Sc = 1.0 which is also drawn on the graph.2.0A Power Pulse0T035—s =1.0(n:.-5 1.0a AA AAc: 00000.0 9~0.0 2.0Rod Wo’r&, R ($)Figure 3 Sensitivity of Energy Deposition to Heat CapacityIt is not possible to plot the sensitivity to the Doppler coefficient as the Doppler coefficient isnot a parameter that can easily be extracted from a space-dependent kinetics calculation.Uncertainty AnalvsisThe above results are the first step in obtaining an estimate of the local fuel enthalpy duringan REA. The sensitivity will be different when it is the local fuel enthalpy rather than the coreenergy deposition (as above) that is being assessed. This difference will be quantified in thefuture after the appropriate edits are available in PARCS. An estimate of the uncertainty infuel enthalpy can be made from the sensitivity if the uncertainty in the fundamental parametersis known. For example, from Figure 2, the sensitivity of energy deposition to delayed neutronfraction is as high as -6 at the end of the initial power pulse and declines to -3 at 3 s. This isthe maximum assuming a rod worth just above prompt critical. If we use the -3 value andassume that the uncertainty at the 10 level for the delayed neutron fraction is approximately10OA then the uncertainty in energy deposition is * 30°A due to this key parameter. This typeof analysis would have to be done for all parameters in order to complete the analysis.EFFECT OF DIFFERENT INTRA-ASSEMBLY REPRESENTATIONSThe effect of the intra-assembly representation on the uncertainty in fuel enthalpy is beingassessed by comparing codes with different models. The principal codes involved are PARCSthe code being used by NRC and BARS, a code developed by the RRC-KI. However, inaddition, CRONOS, a code being used by the French Nuclear Protection and Safety Institute(IPSN) and a new version of PARCS with flux reconstruction, will be used to makecomparisons.The PARCS code uses a nodal approximation wherein each assembly is homogenized and thepower (flux) is calculated for each axial region in the assembly (or perhaps in a quadrant of theassembly if the mesh is smaller than an assembly). The assembly is homogenized so thatneutron cross sections are uniform across the assembly and the thermal-hydraulic parametersare calculated for an average channel representing the assembly, The power in individual fuelrods is obtained by overlaying power peaking factors obtained from an auxiliary calculation.Traditionally, this auxiliary calculation is from a static assembly calculation with reflectiveboundary conditions, i.e., without consideration of what is happening in adjacent assemblies.The BARS code uses a Green’s Function approach wherein each fuel pin is representedexplicitly in the time-dependent calculation. The Green’s Functions are based on diffusiontheory. Although each pin is represented explicitly in the neutronics calculation, the fueltemperature for each pin is based on an assembly-average calculation. This model will bebased on a pin-by-pin calculation in the future.-Although this comparison will provide insight into the effect of the intra-assemblyrepresentation, there are other differences between PARCS and BARS which will have abearing on the calculation of fuel enthalpy. Although the same ENDF/B nuclear data files areused, the data processing codes are different and the nuclear data that enters into PARCS andBARS has a different theoretical basis. For PARCS the CASMO-3 code was used to generatetwo-group data whereas for BARS, the TRIFON code is used to produce the necessary lambdamatrices. The data for BARS is for five energy groups rather than the two that are used inPARCS because the heterogeneous representation puts greater demands on representingchanges in spectrum between different regions.Other differences between the two codes include the axial representation, the time integrationmethod, and the thermal-hydraulics model. Each of these are not expected to have a largeeffect but the cumulative effect could be significant.At the time of this paper the specifications of the problem have been completed but nocomparisons have yet been done. The specifications refer to the reactor composition and thelist of calculations that will be compared, Since the codes do not use the same modeling itis not possible to start from a given cross section data set as has been done in severalbenchmark exercises in the past, Rather one must specify the composition and geometry ofevery region within the core: pellet, clad, guide tube, burnable poison rod, control rod, etc.The reactor model is for a core with exposures into the 50 GWd/t range and hence each pellethas a different composition due to burnup.Prior to comparing results for an REA, results will be compared for steady state calculationsof reactivity coefficients, control rod worths, and power distributions. It is only when thedifferences in results for these parameters are assessed that it makes sense to progress tocomparing transient results.1. D. Diamond and L. Neymotin, “Sensitivity Studies for the BWR Rod Drop Accident, ”Letter Report FIN W6382, October 31, 1996.2. A. Avvakumov and V. Malofeev, “Validation of an Advanced Heterogeneous Modelfor LWR Detailed Pin-by-Pin Calculations, ” Proceedings of the InternationalConference on the Physics of Nuclear Science and Technology, Long Island, NY,October 1998.3. H. G. Joo, G. Jiang, D.A. Barber, and T.J. Downar, “NRC-PARCS: V1 .00, A Multi-Dimensional Two-Group Reactor Kinetics Code Based on the Nonlinear AnalyticNodal Method,” PU/NE-98-26, Purdue University, September 1998.

Estimating the Uncertainty in Reactivity Accident Neutronic Calculations

BNL-NUREG-66230 ESTIMATING THE UNCERTAINTY IN REACTIVITY ACCIDENT NEUTRONIC CALCULATIONS' David J. Diamond, Chae-Yong Yang, Arnold L. Aronso Brookhaven National Laboratory Upton, NY 1 1973-5000 ABSTRACT A study of the uncertainty in calculations of the rod ejection accident in a pressurized water reactor is being carried out for the U.S. Nuclear Regulatory Commission. This paper is a progress report on that study. Results are presented for the sensitivity of core energy deposition t o  the key parameters: ejected rod worth, delayed neutron fraction, Doppler reactivity coefficient, and fuel specific heat. These results can be used in the future t o  estimate the uncertainty in local fuel enthalpy given some assumptions about the uncertainty in the key parameters. This study is also concerned with the effect of the intra-assembly representation in calculations. The issue is the error that might be present if assembly-average power is calculated, and pin peaking factors from a static calculation are then used t o  determine local fuel enthalpy. This is being studied with the help of a collaborative effort with Russian and French analysts who are using codes with different intra-assembly representations. The U.S. code being used is PARCS which calculates power on an assembly- average basis. The Russian code being used is BARS which calculates power for individual fuel pins using a heterogenous representation based on a Green's Function method. INTRODUCTION Bac kqround This study (and others) have been carried out for the U.S. Nuclear Regulatory Commission (NRC) to understand fuel behavior at burnups beyond the current licensing limits. When behavior is sufficiently understood, new acceptance criteria may be proposed for design-basis reactivity initiated accidents (RIAs) in high burnup fuel (and perhaps for fuel at burnups already experienced in operating plants). Acceptance criteria have traditionally been expressed in terms of maximum fuel pellet enthalpy, and hence it is of interest t o  know what is the fuel 'This work was performed under the auspices of the U.S. Nuclear Regulatory Commission. Y enthalpy during an RIA. "Best-estimate" methods are available t o  answer this question, but it is necessary t o  also understand the uncertainty in the calculated fuel enthalpy. The above explains the principal motivation for this study which focuses on the design-basis reactivity accident for a pressurized water reactor (PWR). However, it should also be noted that NRC licensees may be moving away from the traditional, conservative methods for calculating fuel enthalpy toward best-estimate methods, and these methods will require an uncertainty analysis. Hence, the study reported upon herein may have a role in regulatory decisions approving new methods for, and the results of, analyzing RIAs. In a previous study at Brookhaven National Laboratory, the uncertainty in calculating fuel enthalpy for the rod drop accident in a boiling water reactor (BWR) was addressed 111. That study indicated that the random error in the calculated fuel enthalpy could be approximately &75% at the 20 level. It also showed that there could be an additional systematic error of 25% due to the way the intra-assembly power peaking was calculated. This latter error in combination with the random error meant that the calculated fuel enthalpy has t o  be increased by approximately 100% to  obtain results at the 95-95 confidence level. The current study looks at the RIA for a PWR-namely, the rod ejection accident (REA). Because the intra-assembly calculation of power was important for the BWR, it was felt that this should be studied for the PWR and with more rigorous methods than were used for the BWR study. Calculations done as part of a recent Russian study have shown that the peak fuel pin enthalpy during an REA may not be found in the assembly with the peak average fuel enthalpy 121. In the West, it is typical t o  use methods that homogenize the assembly and then calculate the assembly response t o  the REA. The state-of-the-art is changing as flux reconstruction methods are being introduced t o  improve the intra-assembly representation. Nevertheless, the Russian results are another reason t o  study the effect of the intra-assembly representation. Obiective The objective of this study is t o  improve our understanding of the uncertainty in fuel enthalpy calculated for the REA. The approach is twofold. Sensitivity studies are t o  be carried out t o  determine the effect on calculated fuel enthalpy of uncertainties in the important parameters which determine the outcome of the REA. The ultimate objective is t o  use the sensitivity t o  estimate the random error in the fuel enthalpy due t o  random errors in these key parameters. The second approach in this study is t o  compare the results for the REA using a code that treats the assembly as an homogenized region and a code that represents each pin in the assembly explicitly. This would give an estimate of the uncertainty in results due t o  the intra- assembly representation. The PARCS code (Purdue Advanced Reactor Core Simulator) 131, like several other nodal codes in use in the West, treats each assembly as an homogenized region. A newer version of PARCS, with a flux reconstruction method, will soon be available and can be used t o  obtain additional information about the effect of the intra-assembly representation. The BARS code 121, developed at the Russian Research Centre - Kurchatov Institute (RRC-KI), uses a heterogeneous method wherein each fuel pin is represented explicitly. Hence, it is an DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liabili- ty or responsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disdosed, or represents that its use would not infringe privately owned rights. Refemnce herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not wcessanly constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. objective of this study to  compare results for the REA from PARCS and BARS in order to  understand the effect of the intra-assembly representation. It should also be noted that since PARCS is a relatively new addition to  the computer tools used by the NRC, the comparison of PARCS and BARS is likely to  contribute to the code assessment carried out for PARCS. Further information on the effect of the intra-assembly representation will come from the same analysis being carried out by the French Institute for Nuclear Safety and Protection (IPSN) and from calculations expected to be carried out using the version of PARCS with the flux reconstruction model. ScoPe of this PaPer The results of the sensiLJity analysis for the REA are presented in the following section. In the future, these results will be applied to  give an estimate of the uncertainty in the calculated fuel enthalpy based on the random errors expected in the key parameters which enter into the calculation. The comparison being carried out between PARCS and BARS is discussed in the next section. This includes some of the limitations of that comparison. SENSITIVITY STUDIES Introduction Although fuel enthalpy is the parameter of interest, the calculations shown in this paper were of the sensitivity of energy deposition. If the event is adiabatic, then energy deposition and fuel enthalpy are essentially identical. The calculations were for total energy deposition rather than for energy deposition at the position of peak power. A t  a later time, when the edits become available in PARCS, local fuel enthalpy will be considered. The sensitivity is the relative change in energy deposition (Q) per relative change in key reactor parameter (x). Hence, the sensitivity to  x is The parameters of interest are well known from previous studies (e.g., [ l  I )  and are related t o  the reactivity insertion above the prompt critical condition and the negative reactivity feedback from the energy deposition. There are four key parameters which control these phenomena. The first is the reactivity worth of the ejected control rod, po. This parameter is determined by the core design and by the operating procedures which determine the extent of insertion of the rod and the placement of other control rods at any operating condition. The second parameter is the delayed neutron fraction, p. For a given core design, this parameter changes significantly as the fuel burnup changes. The fuel feedback can be expressed in terms of the fuel temperature (or Doppler) reactivity coefficient, a, and the specific heat, C, of the pellet which translates energy into temperature. The Doppler coefficient is determined by core design and the time during the fuel cycle whereas the specific heat changes relatively little with burnup. 16.0 14.0 12.0 Q 10.0 v) p .- 8.0 5 6.0 4.0 2.0 0.0 CI .- rn rn Sensitivitv to  Kev Parameters I A Power Pulse -1 - S = R/(R-1) -1 -: ..: -1 -1 -I - The sensitivity to  control rod worth is given in Figure 1 where it is plotted versus control rod worth in units of $, i.e., R = po/p. The data points on the graph come from PARCS calculations of different reactivity insertion events. Only events with rod worths greater than $1 are of interest. The energy deposition is that deposited during the initial power pulse. Figure 1 Sensitivity of Energy Deposition to Reactivity Insertion It is well-known that the power excursion resulting from a reactivity insertion above prompt critical consists of an initial power pulse turned around by feedback followed by a slow decrease in power, due t o  delayed neutrons, at a level that is still significant as far as energy deposition is concerned. This relatively simple behavior is amenable t o  simple models. Using the point kinetics model and the Nordheim-Fuchs approximation an expression for the sensitivity t o  rod worth can be derived for the time up to the end of the initial power pulse. That expression is S, = R/(R-1) which is plotted on Figure 1. It can be seen that the data points are in good agreement with the theory for the range shown. As the rod worth approaches $1 (from above), corresponding to  prompt critical, the simplified expression has a singularity and is no longer valid. Nevertheless, the sensitivity does increase as rod worth approaches $1. Although this is true, it is also true that the energy deposition becomes smaller and this sensitivity may become less important. The fact that the energy deposition gets smaller has been shown by many analysts. The corresponding result for the energy deposition from the simplified model introduced above also shows this trend: The sensitivity of energy deposition t o  delayed neutron fraction is shown in Figure 2 as a function of reactivity insertion. The data points from PARCS are plotted for both the initial power pulse and by accounting for energy deposition out t o  three seconds. In addition, the graph shows the curve obtained from the simplified model which predicts a sensitivity of S, = -1 /(R-I ). Again it is seen that the results for the initial power pulse are in agreement with the theoretical results for the range shown except that as rod worth approaches $1 from above the agreement begins t o  fail. The sensitivity to  3 s is less than for the initial power pulse as the energy deposited after the initial power pulse is not dependent on rod worth but only on the delayed neutron decay. 0 .o - 1  .o - 2  .o elf" .- d - 3 . 0  > u) .- + - 4 . 0  $ -5  .O -6  . O  0 .o R o d  W o r t h ,  R ($)  0 . 5  1 .o 1 .5  2 .o 0 0 0 0 _ _  _ _  A P o w e r  P u l s e  0 T 0 3 s  - S = - 1 / ( R - 1 )  Figure 2 Sensitivity of Energy Deposition to Delayed Neutron Fraction The sensitivity of energy deposition t o  fuel heat capacity is shown in Figure 3 as a function of reactivity insertion. Energy deposition is assumed t o  be t o  either the end of the initial power pulse or t o  3 s t o  obtain the points plotted on the graph. The corresponding sensitivity from the simplified model is S, = 1 .O which is also drawn on the graph. 2 .o 0 T o 3 s  - s = 1.0 v) 0 x c > v) A A A  A E al E 1.0 v) 00 0 0 0 .o 0 .o 1 .o R o d  Worth ,  R ($) 2 .o Figure 3 Sensitivity of Energy Deposition to Heat Capacity It is not possible t o  plot the sensitivity t o  the Doppler coefficient as the Doppler coefficient is not a parameter that can easily be extracted from a space-dependent kinetics calculation. Uncertaintv Analvsis The above results are the first step in obtaining an estimate of the local fuel enthalpy during an REA. The sensitivity will be different when it is the local fuel enthalpy rather than the core energy deposition (as above) that is being assessed. This difference will be quantified in the future after the appropriate edits are available in PARCS. An estimate of the uncertainty in fuel enthalpy can be made from the sensitivity if the uncertainty in the fundamental parameters is known. For example, from Figure 2, the sensitivity of energy deposition to delayed neutron fraction is as high as -6 at the end of the initial power pulse and declines t o  -3 at 3 s. This is the maximum assuming a rod worth just above prompt critical. If w e  use the -3 value and assume that the uncertainty at the l o  level for the delayed neutron fraction is approximately 10% then the uncertainty in energy deposition is f 30% due t o  this key parameter. This type of analysis would have t o  be done for all parameters in order t o  complete the analysis. EFFECT OF DIFFERENT INTRA-ASSEMBLY REPRESENTATIONS The effect of the intra-assembly representation on the uncertainty in fuel enthalpy is being assessed by comparing codes with different models. The principal codes involved are PARCS the code being used by NRC and BARS, a code developed by the RRC-KI. However, in addition, CRONOS, a code being used by the French Nuclear Protection and Safety Institute (IPSN) and a new version of PARCS with flux reconstruction, will be used t o  make comparisons. The PARCS code uses a nodal approximation wherein each assembly is homogenized and the power (flux) is calculated for each axial region in the assembly (or perhaps in a quadrant of the assembly if the mesh is smaller than an assembly). The assembly is homogenized so that neutron cross sections are uniform across the assembly and the thermal-hydraulic parameters are calculated for an average channel representing the assembly. The power in individual fuel rods is obtained by overlaying power peaking factors obtained from an auxiliary calculation. Traditionally, this auxiliary calculation is from a static assembly calculation with reflective boundary conditions, i.e., without consideration of what is happening in adjacent assemblies. The BARS code uses a Green's Function approach wherein each fuel pin is represented explicitly in the time-dependent calculation. The Green's Functions are based on diffusion theory. Although each pin is represented explicitly in the neutronics calculation, the fuel temperature for each pin is based on an assembly-average calculation. This model will be based on a pin-by-pin calculation in the future. Although this comparison will provide insight into the effect of the intra-assembly representation, there are other differences between PARCS and BARS which will have a bearing on the calculation of fuel enthalpy, Although the same ENDF/B nuclear data files are used, the data processing codes are different and the nuclear data that enters into PARCS and BARS has a different theoretical basis. For PARCS the CASMO-3 code was used t o  generate two-group data whereas for BARS, the TRIFON code is used t o  produce the necessary lambda matrices. The data for BARS is for f ive energy groups rather than the t w o  that are used in PARCS because the heterogeneous representation puts greater demands on representing changes in spectrum between different regions. Other differences between the t w o  codes include the axial representation, the time integration method, and the thermal-hydraulics model. Each of these are not expected t o  have a large effect but the cumulative effect could be significant. At the time of this paper the specifications of the problem have been completed but no comparisons have yet been done. The specifications refer t o  the reactor composition and the list of calculations that will be compared. Since the codes do not use the same modeling it is not possible t o  start from a given cross section data set as has been done in several benchmark exercises in the past. Rather one must specify the composition and geometry of every region within the core: pellet, clad, guide tube, burnable poison rod, control rod, etc. The reactor model is for a core with exposures into the 50 GWd/t range and hence each pellet has a different composition due t o  burnup. Prior to  comparing results for an REA, results will be compared for steady state calculations of reactivity coefficients, control rod worths, and power distributions. It is only when the differences in results for these parameters are assessed that it makes sense to  progress to comparing transient results. REFERENCES 1. D. Diamond and L. Neymotin, "Sensitivity Studies for the BWR Rod Drop Accident," Letter Report FIN W6382, October 31, 1996. 2. A. Avvakumov and V. Malofeev, "Validation of an Advanced Heterogeneous Model for LWR Detailed Pin-by-Pin Calculations," Proceedings of the International Conference on the Physics of Nuclear Science and Technology, Long Island, NY, October 1998. 3. H. G. Joo, G. Jiang, D.A. Barber, and T.J. Downar, "NRC-PARCS: V1 .OO, A Multi- Dimensional Two-Group Reactor Kinetics Code Based on the Nonlinear Analytic Nodal Method," PU/NE-98-26, Purdue University, September I 998. i 

Estimating the uncertainty in reactivity accident neutronic calculations

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Estimating the uncertainty in reactivity accident neutronic calculations

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