KR9900066 KAERDTR-1112/98

NUCLEAR DATA UNCERTAINTY ANALYSIS ON A MINOR ACTINIDE BURNER FOR TRANSMUTING SPENT FUEL

August 1998

KOREA ATOMIC ENERGY RESEARCH INSTITUTE 30-46 KAERFTR-1112/98

4

1998 \l£ "DUPIC

1998\i

: DUPIC : 2) tg-

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Nuclear Data Uncertainty Analysis on a Minor Actinide Burner for Transmuting Spent Fuel

by

Hangbok Choi

August 1998

Korea Atomic Energy Research Institute P.O. Box 105, Yuseong Taejon, Korea, 305-600

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NUCLEAR DATA UNCERTAINTY ANALYSIS ON A MINOR ACTINIDE BURNER FOR TRANSMUTING SPENT FUEL

Hangbok Choi

Korea Atomic Energy Research Institute P.O. Box 105, Yuseong Taejon, 305-600, Korea

ABSTRACT

A comprehensive sensitivity and uncertainty analysis was performed on a 1200 MWth minor actinide burner designed for a low burnup reactivity swing, negative doppler coefficient, and low sodium void worth. Sensitivities of the performance parameters were generated using depletion perturbation methods for the constrained close fuel cycle of the reactor. The uncertainty analysis was performed using the sensitivity and covariance data taken from ENDF-B/V and other published sources. The uncertainty analysis of a liquid metal reactor for burning minor actinides has shown that uncertainties in the nuclear data of several key minor actinide isotopes can introduce large uncertainties in the predicted performance of the core. The relative uncertainties in the burnup swing, doppler coefficient, and void worth were conservatively estimated to be 180%, 97%, and 46%, respectively. An analysis was performed to prioritize the minor actinide reactions for reducing the uncertainties.

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TABLE OF CONTENTS

SECTION PAGE

ABSTRACT 3

I. INTRODUCTION 8

II. MINOR ACTINIDE BURNER DESIGN 10 II. 1 Design Characteristics 10 11.2 Fuel Cycle Model 10 11.3 Safety Performance 11

III. SENSITIVITY METHOD FOR PERFORMANCE PARAMETERS 12 III.l Closed Fuel Cycle Sensitivity ;••• 12 III. 1.1 Sensitivity of Charge Density 14 III. 1.2 Sensitivity of Reprocessed Density 14 III. 1.3 Sensitivity of Feed Density 15 III. 1.4 Numerical Examples 16 ni.2 Sensitivity of Safety Performance Parameters 17 111.2.1 Sensitivity of Burnup Reactivity Swing • 17 111.2.2 Sensitivity of Void Worth 18 111.2.3 Sensitivity of Doppler Constant 19 111.2.4 Sensitivity of Number Density 20 111.2.5 Sensitivity of Transmutatioin Flux 21 111.3 Applicution to Minor Actinide Burner 22 111.4 Summary 24

IV. UNCERTAINTY ANALYSIS 26 IV.l Preparation of Covariance Data • 27 IV.1.1 Covariance Data Generation 27 IV.1.2 Additional Covariance Data 28

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IV.2 Uncertainly in Predicted Performance Parameter 29 IV.2.1 Burnup Reactivity Swing ••• 29 IV.2.2 Void Worth 30 IV.2.3 Doppler Constant 30 IV.3 Uncertainty. Reductiion • 31 FV.3.1 Requirement of Reactor Design 31 IV.3.2 Methods for Estimating Required Accuracy of Nuclear Data 32 IV.3.3 Required Accuracy of Nuclear Data 36 IV.4 Summary • 37

V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 39

REFERENCES 41

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TABLES PAGE

Table I. Mass Flow of Closed Fuel Cycle (kg/yr) 45 Table II. Multi-group Structure 46

Table II. Constrained Sensitivity to ^Np y3 for Final MAB Model 47

Table IV. Region-wise Sensitivities to ^Np"^ for Final MAB Model 48 Table V. Sensitivity of Burnup Reactivity Swing for Final MAB Model 49 Table VI. Sensitivity of Void Worth (EOEC) for Final MAB Model 50 Table VII. Sensitivity of Doppler Constant (EOEC) for Final MAB Model 51 Table VIII. Covariance Matrices Affected by Ratio Measurements inENDF/B-V 52 Table IX. Additional Relative Standard Deviations (%) 53 Table X. Uncertainty of Performance Parameters at EOEC (ENDF/B-V Covariance Data) 54 Table XI. Uncertainty of Performance Parameters at EOEC (ENDF/BTV + Additional Covariance Data) 55 Table XII. Uncertainty of Burnup Reactivity Swing (Variance Term) • 56 Table XIII. Uncertainty of Void Worth (Variance Term) 57 Table XIV. Uncertainty of Doppler Constant (Variance Term) 58 Table XV. Uncertainty Requirement of Nuclear Data for Burnup Reactivity Swing 59 Table XVI. Uncertainty Requirement of Nuclear Data for EOEC Void Worth ••• 60 Table XVII. Uncertainty Requirement of Nuclear Data for EOEC Doppler Constant 61

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FIGURES PAGE

Fig.l Mass Flow of MAB Fuel Cycle 62

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I. INTRODUCTION

The transmutation of long-lived minor actinides has been studied as a means of mitigating the high level waste disposal problem.1"3 For the transmutation of minor actinides, the Liquid Metal Reactor (LMR) offers an advantage because of a preferential fission to capture reaction ratio in the harder neutron spectrum4 and a lower spontaneous fission neutron activity.5 In previous work6, the LMR designs for the primary purpose of burning the minor actinide waste from commercial Light Water Reactors (LWR) have been investigated under the condition that it maintains acceptable safety performance as measured by the burnup reactivity swing, the doppler coefficient, and the sodium void worth. As a result, a 1200 MWth Minor Actinide Burner (MAB) core was designed such that it transmutes the annual minor actinide inventory of as many as 16 LWRs and still exhibits acceptable safety characteristics.

One of the principal problems in the design and analysis of an MAB has been the lack of accurate nuclear data for the principal minor actinide isotopes.7 Nuclear data uncertainties have always been a significant source of error in predicting the performance of an LMR.8 Moreover, the problem is exacerbated in an MAB because the more common heavy metal isotopes such as MJ are replaced by isotopes such as ^Pu and ^Np for which nuclear data is much less well known. The effect of nuclear data uncertainty on the design of an MAB is particularly important because the key safety performance parameters are very sensitive to the poorly known minor actinide data. Specifically, the computed values of the burnup reactivity swing, the void coefficient, and the doppler coefficient of a minor actinide burner are highly sensitive to the minor actinide data. Several studies7'9 have shown that it is particularly difficult to maintain acceptable values of these performance parameters with a high minor actinide inventory and therefore it is important to establish the confidence with which they can be computed.

The purpose of the work reported here is to analyze the effect of nuclear data KAERFTR-1112/98 uncertainties on the predicted performance of an MAB. Specifically, we examined the three core performance responses noted above, the burnup reactivity swing, the void coefficient, and the doppler coefficient, as well as the transmutation rates of the various minor actinide isotopes. We also performed a study to prioritize the nuclear data that would provide the greatest reduction in the uncertainty of the various responses. For the uncertainty analysis of the MAB core performance, the MAB design will first be reviewed briefly in Section II and the sensitivity method will be presented in Section III. The sensitivity method uses the sensitivity coefficients obtained from Depletion Perturbation Theory (DPT) and new sensitivities will be defined for the core performance parameters such as burnup reactivity swing, sodium void worth, and Doppler constant. In Section IV, the uncertainly of the most important three performance parameters will be evaluated using the sensitivity coefficient and covariance data processed from ENDF/B-V. The importance of each covariance data will be evaluated when the total uncertainty reduction is required. Section V summarizes the work and presents the conclusions. Some recommendations are also given in the final section.

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EL MINOR ACTINIDE BURNER DESIGN

11.1 Design Characteristics

For the MAB design study, all calculations were performed using the Argonne National Laboratory code REBUS-310 which searches for the equilibrium cycle condition of the core. The MAB was designed as an annular core with two distinct core zones; an inner core consisting of minor actinide fuel and an outer core containing plutonium fuel. There are 114 inner core assemblies, 138 outer core assemblies, and 66 control rod sites. Both the inner and outer boundaries are surrounded with stainless steel reflector and shielded with B4C blocks. The control rod sites are filled with HT-9 rods when control rods are out in order to reduce the spectrum hardening in the event of coolant voiding. The horizontal and vertical configurations of the core are shown in Ref.6.

11.2 Fuel Cycle Model

The MAB was determined to operate under three fuel batches and a 10 month cycle length. The feed material for the MAB is provided by a typical 1000 MWe LWR after cooling for 3 years. At the completion of each bumup cycle, the discharged fuel is recovered and fabricated with the external feed material in the fabrication plant. The external feed material is provided as separate plutonium and minor actinide streams. As a result, the actinide burner accepts 689 kg of minor actinides and 557 kg of plutonium per year, and the net consumption rate of minor actinides is 425 kg/yr. The minor actinides are continuously recycled in the core, however, 70% of the fissile plutonium is surplus material. The mass flow of the important isotopes is summarized in Table I. Assuming the typical 1000 MWe LWR generates about 26 kgs of minor actinides each year, the core design here can transmute the annual minor actinide inventory from about 16 LWRs.

- 10 - KAERI/m-1112/98 n.3 Safety Performance

The decoupled MAB core design enhances a small positive reactivity burnup swing (1.19 %/Jk) because the reactivity gain in the inner core is compensated by the reactivity loss in the outer core. The sodium void worth was also reduced by the aid of HT-9 rods deployed in the control rod sites and the annular core geometry which promotes the neutron leakage upon coolant voiding. During the fuel cycle, the void worth is smaller at the beginning of equilibrium cycle (BOEC) state (0.74% z/k) than at the end of equilibrium cycle (EOEC) state (1.17%z)k). Because the actinide burner does not contain B8U, the Doppler effect is smaller than that of a conventional LMR, but remains negative (-0.35 XI0"3 and -0.32 XlO"3 T8k/dT at BOEC and EOEC, respectively).

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SENSITIVITY METHOD FOR PERFORMANCE PARAMETERS

In order to evaluate the uncertainty of the performance parameters of the conceptual MAB design, a sensitivity method for the closed fuel cycle, which is obtained from Depletion Perturbation Theory (DPT), is used to estimate the effect of nuclear data on reactor performance parameters. The DPT has been developed by several authors. Gandini11 first formally presented adjoint equations for the nuclear transmutation equations, which was later modified by Kallfelz et al. to include the effects of static flux field perturbations. Williams13 and Greenspan et al.14 derived a method which was able to treat burnup problems involving coupled neutron/nuclide field. White15 extended Williams' work to the multicycle problem by incorporating fuel shuffling operator. Yang and Downar16 extended DPT to the constrained equilibrium cycle of a practical fast reactor problem. In this work they developed the DPT equations for the equilibrium cycle of the reactor and then explicitly accounted for the effect of practical core operating constraints such as criticality on the depletion sensitivities. Choi and Downar17 expanded the constrained DPT to include the external fuel cycle and such likely scenarios as the reprocessing and refabrication of some fraction of the discharged fuel.

The following section first reviews the closed fuel cycle sensitivity method developed by Choi and Downar. The general formulations and the sensitivity methods are then demonstrated with the MAB fuel cycle model. The sensitivities of safety performance parameters will be formulated in Section III.B and evaluated in Section III.C for the conceptual MAB model. m.l Closed Fuel Cycle Sensitivity

At the completion of a burnup cycle for a multi-batch core, some portion of the fuel is shuffled and reloaded for the subsequent cycle. The remainder is discharged and sent to the waste stream in the open fuel cycle or reprocessed for subsequent

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reirradiation in the closed fuel cycle. The mass flow of the closed fuel cycle in conjunction with the external material flow is depicted in Fig.l which is well-described in Ref.17. In Fig.l, the portion of the fuel discharged at the end of the cycle has a discharge density iV# which is fed into the reprocessing plant where the portion sent to the fabrication facility has a density iV^ and the remainder is sent to waste. If the makeup material is required at the fabrication facility, the reprocessed material is combined with external feed material having a density JV> , otherwise some portion of the reprocessed material is sold with a sold density of ~Ns . Then the fabricated fuel having a charge density iv£ constitutes the fresh fuel material for the next cycle.

In the depletion sensitivity theory, a general response can be written as a function of the cycle length, T, the cross-section data, a, and the fresh fuel charge density,

R = R(T, a, Tc) (1)

If we introduce the definition18 of the sensitivity coefficient of response R to a parameter a such as

R 8R I OG " ~ R ' a ' the constrained sensitivity ( ^) will be written in terms of other sensitivity coefficients;

Sa = Sa + Sj£- Sa (3)

When the fuel cycle length is fixed, the constrained sensitivity is simply a function of cross-section data and charged density. The effect of cross-section data and charge

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density on the response is an in-reactor problem and no constraints are applied. The effect of constraints is implemented in the charge density change due to perturbation parameter a as shown in the last term of Eq.(3). The practical problem encountered in constrained sensitivity theory is how to evaluate the change in the charge density,

§„ c in Eq.(3), using only the unconstrained sensitivities.

III. 1.1 Sensitivity of Charge Density

In general the charge density is written in terms of distribution operators as below

l % % ~Ng + A£ (4) where Q- and Qf are the distribution operators of reprocessed and feed material,

respectively. lv^ is the recovered density of the discharged fuel and Nf is actually the isotopic split of external feed material. Then the sensitivity of the charge density is written as

N MM. $ * +j_ ME Nc *< Nc

III. 1.2 Sensitivity of Reprocessed Density

The discharged fuel is sent to the reprocessing plant and used as a makeup for refabrication based on the priority and the reactor conditions which are implemented in the operator Q-. Because all quantities are fixed except charge enrichment (e) and recovered density ( Jfr ), the distribution operator of the reprocessed density is written as

Qr= Qr(e, Wr) (6)

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The sensitivity of the reprocessed density is then written as

« % _ a dNR _ g I dQr dNA r + Vr •^ ~ NR da ~ NR [ da " da )

Qr\ Be da dNr da

(7)

III 1.3 Sensitivity of Feed Density

The amount of external feed material and its distribution is described by the operator Qf which is determined by the charge enrichment and the amount of recovered fuel.

Qf= Q/e, Nr) (8)

The sensitivity of feed density is then written as

a dNF _ a I dQf dNA ;~~ ~ ~^{~da~N/+ Qf~da~)

_a_(_dQ/Je , _dQj__dNA a dNf Qf \ de da "*" dNr da j "*" N, da

= S?' SI + S% S/' + Sf' (9)

where §ff ' is zero because the isotopic split does not change for the external feed

- 15 - KAERI/TR-1112/98 source.

III. 1.4 Numerical Examples

The recovered density is a finite fraction of the discharge density such as

X = Qd%> (10)

where Qd is the recovery fraction specified by the users. Therefore the sensitivity of the recovered sensitivity is the same as that of the discharge density.

S."' = $.* (ID

If we define the response as the discharge density ( R = ND ), a series of equations

N can be solved for Sa " using Eqs.(3), (5), (7), and (9) with the unconstrained sensitivities obtained from the depletion perturbation program.

Constrained sensitivities were calculated in 9 energy groups (Table II) and compared

to the result of direct subtraction. For the direct subtraction v3 of ^Np was perturbed by 1%. The sensitivities of concentration of the heavy metal isotopes are given in Table III and show good agreement with direct subtraction. Unlike typical fast reactors the MAB is mostly composed of minor actinides such as Np while the concentration of uranium is very low. The sensitivity of the uranium isotopes have a relatively large error because discharge densities change nonlinearly in direct subtraction when their concentration is very low. The sensitivity of ^Np is small but shows good accuracy because this is the driving isotope in the MAB and the numerical estimation of sensitivity of distribution factor does not effect the final sensitivity.

- 16 - KAERJ/TR-H12/98 m.2 Sensitivity of Safety Performance Parameters

In Section II, a conceptual design of the MAB was described with three important performance parameters: burnup reactivity swing, void worth, and Doppler constant. Because the core is mostly composed of minor actinides, it is expected that these performance parameters will have relatively large uncertainties. In order to estimate the uncertainty, the sensitivities of these performance parameters were computed using the constrained sensitivity methods described in Section III.A. In the following section sensitivities of the burnup reactivity swing, the void worth, and the Doppler constant will be formulated and computed for the MAB model.

III. 2.1 Sensitivity of Burnup Reactivity Swing

The burnup reactivity swing is the difference in the BOEC and EOEC kef :

R = kb - ke (12) where fe and ke are fag- of the BOEC and EOEC core, respectively, which are obtained from single depletion calculation. The sensitivity of reactivity swing can be obtained by applying the chain rule to the response as below8

kt k A R___g_dR_ a dkb g dke _ kb ^ _hs. £ '—ht 6 *» ' R do~ R do R da R *' R *' ~ R i>a where the sensitivity of h can be obtained in the same way as Eq.(3),

dkb

In Eq.(13) the sensitivity of ke, Sa ', was neglected because fc was constrained to

- 17 - KAERI/TR-1112/98 be critical for any kind of perturbation. Thus the sensitivity of reactivity swing is a fraction of the sensitivity of h. The sensitivity of h can be routinely obtained from the sensitivity of charge density derived in Section m.A.

III.2.2 Sensitivity of Void Worth

The void worth is the difference of reactivity for flooded and voided core at certain reactor state (e.g. EOEC state)

R = k"e - kfe (15)

where k"e and h?e are the hff of voided and flooded core at EOEC state.

The void worth is calculated by performing two eigenvalue calculations for the voided and flooded core at EOEC using the number density obtained from the depletion calculation. Since separate calculations are performed at the EOEC state, the sensitivity of the void worth is written as

R e — g dR _ a dk"e _ _a_ R do R do R do

~R " ~R

k = JziKs? - kis?) + ±(Ks i - f/es%)- §/• (16)

where Ne is the number density vector at EOEC state. The sensitivity of EOEC number density, §„ ', will be derived in Section III.B.4.

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The first and second terms in Eq.(16) are called direct and density terms, respectively.

The sensitivity of the EOEC flooded keff, $ff ' , was not neglected in Eq. (16) because different cross-section sets were used in the depletion and the eigenvalue calculations.

Instead, the sensitivity of f/e was separated into both direct and density terms in Eq.(16) in order to evaluate the sensitivity change from the flooded to the voided core.

III. 2.3 Sensitivity of Doppler Constant

The Doppler constant was calculated by taking the difference in keg for the flooded and the hot core at EOEC. In order to get the keff of the hot core, the fuel temperature was doubled in the flooded core.

R = k\ - ki (17) where k\ is the hff of flooded hot core at EOEC state.

The sensitivity of the Doppler constant can be written in terms of the sensitivity of the EOEC number density as Eq. (16)

G dR a dke G dki R da R da R da h k P ki t. R Sa

h k = jt(k e S? - ki S,*) + ^(*i S l - ki SJ) • §/' (18) where the first and second terms refer to the direct and density terms, respectively.

The direct term is the effect of cross section variation to keff without any constraint.

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This sensitivity can be obtained from a DPT static calculation. For the density term, the sensitivity ofA^-to the EOEC number density, SN. and S£. , can also be obtained from a DPT static calculation. The sensitivity of the EOEC number density, 3ff ' , will be discussed in the next section.

III. 2.4 Sensitivity of Number Density

The sensitivities of the void worth and Doppler constant require the sensitivity of the BOEC or EOEC number density, depending on the state at which the performance parameters are evaluated. The number density is obtained by transmuting the charge density for the given time step such as

%= A exv(Mi8i)~Nc (BOEC number density) (19)

Ne = Sj exp(Mi8t) Nc (EOEC number density) (20) where NS is the number of stages and dt is the depletion time.

The sensitivities of the BOEC and EOEC number density have the same form except for the stage numbers. Because the transmutation matrix is a function of the cross-section and the transmutation flux, M = M(a, ), the derivative of EOEC number density with respect to cross-section is as follows,

The Eq.(21) can be rewritten in terms of the sensitivity coefficient

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(22)

where Nc is the charge density matrix which has charge densities in its diagonal terms only.

III.2.5 Sensitivity of Transmutation Flux

It is necessary to calculate the sensitivity of the transmutation flux first in order to get the sensitivity of the EOEC density. The transmutation flux can be obtained from the eigenvalue problem.

B 4 = 0 (23)

Since the matrix operator B is a function of the cross-section and number densities, B = B(a, N ), the system equation can be differentiated as

do (24)

Eq. (24) can be rewritten in terms of the sensitivity coefficient for the BOEC or EOEC state as below

- - {His where 0 is the diagonal matrix of the BOEC or EOEC flux. Eqs.(22) and (25) can be solved simultaneously by assuming the sensitivity of the flux to be zero at the

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first iteration. The inhomogeneous source term, Qin, in Eq. (25) is calculated using the previous sensitivity of the number density, and the new sensitivity of the flux is solved using the Gauss-Seidel method.

The sensitivity of the transmutation flux is simply obtained by taking the average of sensitivities of the BOEC and EOEC flux as below. The new sensitivity of the transmutation flux is used to update the sensitivity of the number density in Eq.(22).

(26)

IQ.3 Application to Minor Actinide Burner

The sensitivity method developed here was applied to the MAB model. Because 9-group cross- section data was used for the depletion calculation, the same number of groups should be used in the sensitivity and in the uncertainty calculation. The sensitivity of the charge density is given in Table IV for the perturbation of the neutron yield per fission of ^Np for group 3, y3. The sensitivities of flooded, voided, and hot keff to the charge density are also shown in Table FV for the typical inner and outer core zones: assembly ring 9 (inner core) and 10 (outer core) for the mid-plane. The sensitivities of the burnup reactivity swing, the void worth, and the Doppler constant are ranked in Tables V, VI, and VII, respectively. The direct and density terms are shown in the same Tables.

The sensitivity of burnup reactivity swing is dominated by the density term which is the effect of the charge density variation to kboec- The charge density changes to meet the reactor constraints in the constrained equilibrium cycle. In the MAB, Np is most abundant (53% of the charge density) and the perturbation (1%) of ^Np neutron yield ( u ) or fission cross-section (

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(class 1) in the outer core region. The sensitivity of keff to 239Pu charge density is very large (0.015) in the outer core compared to other isotopes. Because the sensitivity of the ^Pu charge density is negative, the total effect of charge density variation is negatively large. 23&P\i is very reactive (y =2.46) in the MAB neutron spectrum. Because the concentration is much smaller (3% of the charge density) compared to ^Np, the effect of S8Pu is smaller than ^Np but contributes to the negative component of the sensitivity of the burnup reactivity swing.

237 B The sensitivity of the void worth is dominated by Np and Pu. If the ^Np j/3 is perturbed by +1%, the charge density of Pu-fissile (239Pu) decreases considerably in the outer core. If the core is voided, the effect of 239Pu charge density to ^decreases while that of 237Np charge density in the inner core increases because of spectrum hardening. The net effect of the density term is 5.39 in sensitivity which is comparable in magnitude to the direct term (2.02 in sensitivity). The direct term is positive because the contribution of ^Np y3 to hjf is greater in the harder spectrum.

The sensitivity of the Doppler constant is dominated by the density term. The direct term is the difference of keff in the flooded and hot core after correction by keff. But the cross-section changes little when the fuel temperature is increased because the Doppler effect is very small in the very hard neutron spectrum. In the case of the fin

Np v3 perturbation, which has the largest sensitivity, the direct term is much smaller compared to the density term because of only a small cross-section change in the hot core. Therefore the major contribution to the sensitivity of the Doppler constant comes from the effect of the charge density variation.

Because the reactor condition is not much different in the flooded and hot core, the sensitivities of keff of the charge density are also comparable in the flooded and hot

k core. The sensitivity change due to the charge density ( S Nc) is larger for ^Np and 238Pu. The sensitivity of the 238Pu charge density is even larger than that of a7Np,

- 23 - KAERI/TR-1112/98 and the total density term becomes -3.06 in sensitivity for the perturbation of B7Np 1/3. Though the a8Pu charge density contributes most to the density term, this is a secondary effect, because the charge density of a8Pu changes primarily due to the perturbation of ^Np V3. All three performance parameters are dominated by the B7Np isotope which is the most abundant isotope in the MAB. The neutron yield per fission and the fission cross-section of group 2 and 3 (0.5 ~6.1 MeV) have the dominant contribution to the responses because the neutron spectrum is the strongest in this energy range. m.4 Summary

The sensitivities of performance parameters were formulated using the constrained sensitivities of the closed fuel cycle. The sensitivities of the burnup reactivity swing, the EOEC void worth, and the Doppler constant were calculated for the conceptual MAB design. The results show that the sensitivities are very large for the neutron yield per fission ( 1/ ) and fission cross-section (aj) of Np and Pu. The fast group (0.5—6.1 MeV) cross-section in particular makes a large contribution to the sensitivities because the spectrum is very hard in the MAB. In general, the constrained sensitivity is much larger for the isotopes which are higher up the burnup chain because the sensitivity is transferred from the lower to higher isotope in the burnup chain and the effects are finally accumulated in the highest isotope. This suggests the importance of this work for the sensitivity analyses of advanced reactors with closed fuel cycles.

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IV. UNCERTAINTY ANALYSIS

The uncertainty estimation of the predicted reactor performance parameters has important implications in determining the reactor design margins and evaluating the overall plant safety performance. The uncertainties are inevitably present because of the inaccuracies in the evaluated nuclear data due to the measurement processes and the different corrections made to the micro- scopic cross-sections. In assessing design penalties and establishing a nuclear data improvement program, it is necessary to characterize uncertainties in the nuclear data set.

The uncertainty of a predicted performance parameter due to nuclear data uncertainty can be estimated as

V = S MaS' (27)

where Fis the uncertainty of the solution of diffusion calculation, Mff is the covariance matrix, S is the sensitivity vector, and £ is the transpose of the sensitivity.

The methodology for computing the sensitivities of the performance parameters was discussed in Section III. Covariance data has been compiled from ENDF/B-V for most isotopes and several reaction types: file 31 contains uncertainty data for the neutron yield per fission ( v ), file 32 contains uncertainty for specific resolved resonance parameters, and file 33 contains the relative uncertainty and correlation data among cross-sections at various energies.21

The covariances of nuclear data play a crucial role when the uncertainties in calculated results based on nuclear data are estimated. For example, covariance data are used for the analysis of propagation of the uncertainties to final calculated results, adjustment of data sets by incorporating information from integral measurements22'23, and

- 25 - KAERFTR-1112/98 determination of data accuracies needed to meet the target uncertainties in calculated results.

In the following sections the covariance data will be presented that was processed from the ENDF/B-V. The covariance data will also be discussed which was obtained from the open literature that is important in the uncertainly analysis of the minor actinide burner but not exist in ENDF/B-V. The uncertainties of the safety performance parameters will be evaluated in Section IV.B and the prioritization of the nuclear data for future nuclear data experiments will be discussed in Section IV.C.

IV. 1 Preparation of Covariance Data

IV. 1.1 Covariance Data Generation

Currently ENDF/B-V files have covariance data for most of the neutron cross-sections except for the energy and angular distributions, since these data would require an enormous processing task to generate covariance matrices for group-to-group transfer cross-section as well as for their Legendre moments. The ENDF/B-V covariance files were processed through the NJOY system24 for the actinide isotopes. The module RECONR reads the continuous cross-sections and covariance data from the ENDF/B-V and reproduces the data in PENDF (Point-wise ENDF) format. The module GROUPR produces the average cross-section data in GENDF form for a given temperature and background cross-section. The LANL-18725 group structure was used to collapse the PENDF data.

The fast reactor spectrum, which appears to be the closest to the spectrum of the MAB studied here, was used for the group collapsing and the cross-section which is used to calculate the correlation matrix was generated for 0°K and an infinitely dilute background cross-section. Though the mean values (cross-section) are not exactly the same as those of the MAB, this will not effect the uncertainty of the response since

- 26 - KAERFTR-1112/98 only the relative quantity is used.

The multigroup covariance data was processed through module ERRORR26 with the same energy group structure as was used in the depletion calculation. The output of ERRORR is the multigroup covariance matrix written in an ENDF-like format. A stand-alone utility code called TRIEVER27, supplied by Los Alamos National Laboratory (LANL), has been modified to retrieve the ERRORR output covariances in the appropriate format. A correlation also exists between isotopes as shown in Table VIII because of ratio measurements. The correlation matrix is automatically processed through module ERRORR by compiling together the covariance of the standard isotopes.

IV. 1.2 Additional Covariance Data

In the work here, some isotopes have relatively large sensitivities for the performance parameters of the MAB but do not have covariance data in the ENDF/B-V. The uncertainties of the neutron yields per fission ( v ) and fission cross-sections ( oy) of ^Np, 238Pu, 239Pu, 241Am, 243Am, and 244Cm were obtained from the open

yoii literature as shown in Table IX.

The uncertainties (or standard deviation) of ^Np a/ is available in the ENDF/B-V. The relative uncertainties of ^Np v and a were assumed to be 2% and 50%, 28 respectively. The uncertainty of ^"Np a c was obtained from Nikolsii's work which was performed for the fast reactor using Russian cross-section library BNAB-78.

The fission cross-section of Pu was measured from the nuclear explosion Persimmon.29 The fission cross-section and its uncertainty were measured for the energy range from 17 eV to 3 MeV. The relative uncertainty of 238Pu ~v was assumed to be 1% based on the Stubbins and Wolfe's study30 on the BIFOLD nuclear power source.

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The relative uncertainty of 241Am ~~v was assumed to be 2%. The uncertainties of 243Am ~v and a/ are available in Refs.32-34. The uncertainty of 244Cm

We recognize that these uncertainties are not the real uncertainties of the nuclear data in ENDF/B-V because the studies have been performed under very different conditions.28 However, it is not reasonable to assign 0 or 100% uncertainty to the nuclear data because of their unavailability in ENDF/B-V. Therefore the nuclear data uncertainties estimated here will be used for a "best estimate" uncertainly analysis of the performance parameters.

IV.2 Uncertainty in Predicted Performance Parameter

The uncertainties in the predicted performance parameters were obtained by combining the sensitivities and the covariance data, Eq.(27). The uncertainties were calculated for the burnup reactivity swing, void worth, and Doppler constant at the EOEC state of the minor actinide burner design discussed in Section II. The uncertainties of the performance parameters are listed in Table X when only the ENDF/B-V covariance data is used. In Table XI, the uncertainties are listed when both the ENDF/B-V data and the covariance taken from the literature (Table IX) are used.

W.2.1 Burntq? Reactivity Swing

The uncertainty of the burnup reactivity swing was estimated to be 180% at EOEC state when the ENDF/B-V covariance data was used. The uncertainty increases to 220% if the additional covariance are included. This result implies that the actual reactivity swing may reside in the range of-1.41 to 3.79 %Ak with a 95% confidence level.

- 28 - KAERI/TR-1112/98

If only the variances (diagonal term in the covariance matrix) are used, the total uncertainly is about 180%.

Considering only the variance terms, most of the uncertainty comes from the ^"Np fission and inelastic cross-sections in groups 2 and 3 (0.5—6.1 MeV). These reactions constitute 80% of the diagonal uncertainty, which totals about 160%. Although the ~v of B7Np has the largest sensitivity (12.8 for group 2), its contribution to total uncertainty is less than the inelastic cross- section because the uncertainty of the inelastic cross-section was assumed to be 50%, which is 25 times higher than the uncertainty of u . The remainder of the uncertainty comes from Np v , capture, and the Pu fission cross-sections. The contribution of each nuclear data uncertainty to the total uncertainty is shown in Table XII for the variance terms (diagonal terms).

IV.2.2 Void Worth

The uncertainty of the void worth was estimated to be 97% and 120% for the ENDF/B-V and ENDF/B-V plus additional covariance data, respectively, at the EOEC state. This suggests the EOEC void worth exists between -0.23 and 2.57 %Ak with a 95% confidence level. The uncertainty due to the variance term was 92%. For the void worth, the 237Np fission and inelastic cross-section of groups 2 and 3 dominate the total uncertainty, which is 75% of the uncertainty due to the variance terms. The rest of the uncertainty comes from 237Np capture, ~v , and the 238Pu fission cross-section, similar to the case of the burnup reactivity swing. The contribution of each nuclear data is shown in Table XIII for the variance terms.

IV. 2.3 Doppler Constant

The uncertainty of the Doppler constant is relatively small compared to the reactivity swing and the void worth. The total uncertainty at the EOEC state was 46% and 59%

- 29 - KAERLTR-1112/98 for the ENDF/B-V and ENDF/B-V with the additional covariance data, respectively. If only the variance terms are used, the total uncertainty becomes 48%. The trend of each nuclear data is the same as the case of burnup reactivity swing and the void worth, however the magnitude of the uncertainty is smaller because the sensitivity is smaller. The uncertainty due to the variance term is shown in Table XIV.

IV.3 Uncertainty Reduction

The uncertainty of a predicted performance parameter can be reduced by either differential or integral experiments.36 The amount by which the uncertainty of a particular performance parameter can be reduced depends on the magnitude of the sensitivity to a particular cross-section. In the following section, some of the general requirements on the uncertainty of the predicted performance parameters will be discussed. A technique for identifying the cross-sections which achieve the largest uncertainty reduction will be discussed. Based on this, the priority of each nuclear data will be established if future differential experiments are performed for the purpose of reducing the uncertainty in the predicted performance of the MAB.

IV.3.1 Requirement of Reactor Design

Large uncertainties in the predicted performance parameters reduce confidence in the reactor design. This is particularly true for the minor actinide burner studied here which has very large uncertainties in the performance parameters. The accuracy of the performance parameter depends on the uncertainties in the individual nuclear data. In order to determine the extent to which the uncertainty of each nuclear data should be reduced, it is necessary to first establish requirements on the accuracy of the predicted performance parameters.

The burnup reactivity swing is an important parameter since it is used to determine the control rod worth necessary to suppress the excessive reactivity of the core. The

30 - KAERFTR-1112/98 fast reactor project37 of the United Kingdom proposed several requirements on the uncertainty of the predicted parameters such as kef, power coefficient, control rod worth, and power distribution. The control rod worth was required to be predicted within the limit of ±10% in order to set the mass of absorber required per rod. Because the control rod is used to compensate for the excess reactivity, the same accuracy was used as the target uncertainty of the burnup reactivity swing.

The coolant can be voided in a hypothetical core disruptive accident (HCDA) involving coolant (sodium) vaporization. Butland proposed ± 10% as an accuracy requirement of the void worth for both the non-leakage and total leakage component. The sodium could be removed from the channel due to a temperature change, fuel expansion, assembly bowing, and core compression. The target uncertainty of the void worth was conservatively chosen to be ±10%.

The Doppler effect is an important parameter for predicting the reactor kinetics behavior of the core. It is also important for setting the shutdown control requirement and the energy yield in any prompt-critical excursion resulting from a hypothetical core meltdown accident. Butland39 suggested that an accuracy of ± 15% in the total Doppler constant is sufficient to cover these requirements, assuming that the fuel is a sufficiently homogeneous mixture and therefore the Doppler constants of the individual elements are not necessary.

IV. 3.2 Methods for Estimating Required Accuracy of Nuclear Data

In order to achieve the target uncertainty of each performance parameter, the standard deviation of each nuclear data should be reduced. Because it is more difficult to reduce an already small standard deviation, the required standard deviation of each nuclear data was found by solving an optimization problem. A similar study was performed by Usachev and Bobkov40 to plan the experiments for the microscopic cross-section measurement in the USSR. They formed an equation to find the minimum cost of

- 31 - KAERFTR-1112/98 the experiments by incorporating a global coefficient which includes statistical and system error. But they did not consider the correlation between isotopes and reaction types. Because the ENDF/B-V supplies the total uncertainty of the nuclear data, the optimum nuclear data uncertainty was searched with a simpler equation which is somewhat different from that used by Usachev and Bobkov.

The optimum nuclear data uncertainty was found by minimizing the cost and/or time spent for the experiments. The cost function is usually regarded to be proportional to the statistical weight in the experimental planning40 and, therefore, the cost function (0 was defined as

Q = §C,Uo- x? (28)

where C, is the cost coefficient of nuclear data / for obtaining an unit statistical weight, xio is the present relative uncertainty, and x, is the desired relative uncertainty to be attained from an experiment. The summation is performed for m nuclear data, where m is the product of number of isotopes, reaction types, and energy groups.

The cost coefficient was defined as the reciprocal of the variance of each nuclear data,

C, = -jr • (29)

Therefore the cost is high if the present standard deviation is already small or if the requirement on the uncertainty reduction, (x«> - xf), is great.

The uncertainty of the performance parameter is calculated using Eq.(27) which can be rewritten component-wise as

- 32 - KAERJ/TR-1112/98

(30)

where as = ru Sf Sf

rtj = correlation matrix between nuclear data i and j Sf = sensitivity of performance parameter R to nuclear data /.

The constraint given to determine the cost is the target uncertainty of the predicted performance parameter. If Vexp is the uncertainty to be reached by reducing the standard deviation of each nuclear data, xi or x,-, then a similar equation holds for Vexp as

(31)

This equation implies that the same correlation matrix (nj) exists between nuclear data / and j even if the standard deviation changes.

Note that the cost function becomes zero when x-, = xm . In order to formulate a solution, the constraint equation, Eq.(31), was inserted into the cost function, Eq.(28), with a Lagrangian multiplier^ which is a common technique for solving the constrained problem.41

2 Q = EC( Gc» - x,) + A (Fexp - S S a9 Xi Xj) (32) I — I t — 1 J— 1

The solution equations are obtained by equating the partial derivative of Eq.(32) with respect to x, to zero:

- 33 - KAERI/TR-1112/98 which produces following m equations,

f • • • + fll^J

xm = Xrf) + -2^-(««i*i + a^x2 + • • • + ammxm) (34)

The solution of the above m equations is obtained by the Jacobi iteration with an initial guess for the Lagrangian multiplier A. The algorithm is written as below for the (n+l)th iteration:

Xfi T

Once the iteration is converged, the A is updated to find the solution that satisfies the constraint, Eq.(31). Denoting the uncertainty of the performance parameter as a function of the Lagrangian multiplier, V =X A ), the (outer) iteration algorithm is written as below using Newton's method in which the value of A at (n+l)th iteration is updated by a linear approximation based on previous values (i.e. n and (n-l)th iteration).

(36) where fop is the target uncertainty of the performance parameter.

During the iteration, accuracy required for some nuclear data becomes very small which may not be achievable in a practical experiment. Therefore a boundary value was assigned to the uncertainty x,- based on the ENDF/B-V covariance data. In the

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ENDF/B-V the 241Am thermal (< 3.4 KeV) capture cross-section has the smallest uncertainty, 0.08%, and was used as the lowest bound for any of the nuclear data.

IV. 3.3 Required Accuracy of Nuclear Data

The required accuracy (relative standard deviation) of each nuclear data was found using the target uncertainty described in Section IV.C.l. The results are given in Tables XV, XVI, and XVII for the burnup reactivity swing, void worth, and Doppler constant at EOEC. Several nuclear data uncertainties reached the proposed minimum value (0.08%) because their contribution to the total uncertainty reduction is relatively large. The effects of nuclear data uncertainty reduction on the uncertainty reduction of each performance parameter are given in the last column of Tables XV, XVI, and XVII.

For the bumup reactivity swing it was required to reduce the total uncertainty from 220% to 10% (4.48 to 0.01 in variance), which corresponds to total variance reduction of 4.47. If, for example, the nuclear data uncertainty of Np op (fission cross-section in group 3) is reduced from 10% (present value) to 0.08%, the uncertainty of the reactivity swing will drop to 158% which is 2.49 in variance. Therefore the amount of variance reduction is 1.99 (= 4.48-2.49). The effect is then 44% (= 1.99/4.47) which is the ratio of the variance reduction to the variance reduction required. Because the present uncertainty of the reactivity worth is too large, the uncertainties of the highly ranked nuclear data are required to be very small. The uncertainties of Np fast fission (0.5 —6.1 MeV) cross-sections have the dominant effect on the total uncertainty reduction, followed by the uncertainties of ^Np inelastic and 244Cm fission cross-sections of the higher energy groups (group 2 and 3).

The trend of the nuclear data uncertainty reduction is similar in the uncertainty reduction of the void worth and the Doppler constant. The amount of variance reduction required is 1.384 when the uncertainty of the void worth is to be reduced from 120 to 10%. By reducing the uncertainty of ^Np ap to the minimum value, the uncertainty of

- 35 - KAERI/TR-1112/98 the void worth reduces to 84%, which corresponds to 50% of total uncertainty reduction required. In order to satisfy the requirement, the uncertainties of the first three highly ranked nuclear data, ^Np cp, 0/2, and Oft, need to be reduced to the minimum values.

The requirement on the uncertainty reduction is even smaller for the Doppler constant compared to the reactivity swing or void worth. When the uncertainty is reduced from 59% to 15%, the total variance is reduced by 0.3255 and the uncertainties of the first two nuclear data, 237Np a p and a/2, were required to be reduced to the minimum value. The 237Np op contributes most to the uncertainty reduction of the Doppler constant. The 45% of the total uncertainty reduction requirement is achieved when the uncertainty of ^Np a ft is reduced to the minimum value (0.08%).

IV.4 Summary

An uncertainty analysis was performed for the proposed MAB model. The uncertainties of three important safely performance parameters, the burnup reactivity swing, void worth, and Doppler constant, were evaluated at the EOEC state using the sensitivities obtained for the constrained equilibrium cycle and the covariance data processed from ENDF/B-V. Additional covariance data were obtained from the open literature for the important reaction types which were not available from ENDF/B-V.

It was found that the uncertainties of the predicted performances parameters were very large because the sensitivities were large in the constrained equilibrium cycle and because the covariance is large for the minor actinides which are the most abundant isotopes in the MAB. The uncertainties of the reactivity swing, void worth, and Doppler constant were 220, 120, and 59%, respectively, in the confidence level of 1 standard deviation (STD). The a7Np fission, inelastic scattering, 244Cm fission, and ^Pu fission cross-section of fast energy group (0.5 ~ 6.1 MeV) contribute most to the uncertainty of each of the three performance parameters.

- 36 - KAERI/rR-1112/98

The optimum nuclear data uncertainties were determined for a specified target uncertainty of each predicted performance parameter. It was found that the uncertainty of B7Np fast fission (0.5 ~ 6.1 MeV) needs to be reduced to the minimum value (0.08%) achievable in the nuclear data experiment because its effect on the total uncertainty reduction is the greatest. Though it may not be possible to achieve such a small uncertainty by the experiment, it is still desirable to reduce the uncertainty of the important nuclear data as much as possible.

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V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

The credibility of the predicted Minor Actinide Burner performance parameters was evaluated because the nuclear data uncertainty is known to be large for the minor actinides. In order to estimate the uncertainty of the predicted performance parameters, a sensitivity method was developed by extending the constrained equilibrium cycle sensitivity, which includes fuel reprocessing and refabrication, to the sensitivities of the performance parameters. The sensitivities were calculated for the burnup reactivity swing, EOEC void worth, and EOEC Doppler constant of the proposed MAB. It was found that the sensitivities of these performance parameters are very large for the neutron yield per fission and for the fission cross-section of Np and Pu. The sensitivities of the fast energy group (0.5—6.1 MeV) are especially large because the neutron spectrum is very hard in the MAB.

The uncertainties of the predicted performance parameters were evaluated using the sensitivity coefficients and the nuclear data covariance. The covariance data was obtained from ENDF/B-V and other published data. It was found that the uncertainties of the predicted performance parameters are very large because the constrained sensitivity of the equilibrium cycle and the covariance of the minor actinides are large. The estimated uncertainties of the bumup reactivity swing, EOEC void worth, and EOEC Doppler constant are 220, 120, and 59%, respectively, in 1 STD confidence level. Because the uncertainties of the performance parameters due to the nuclear data were large, the accuracy of each nuclear data was found such that the uncertainties could be reduced to a specified value. The uncertainties of the performance parameters can be reduced the most if high accuracy is achieved for the fast (0.5~6.1 MeV) fission and inelastic scattering cross-sections of ^Np.

As a recommendation, more research is required in the following areas: • Most of the MAB will be loaded with a large amount of minor actinides. The predicted performance parameters of these cores have large uncertainties and it is

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recommended that more nuclear data experiments be performed for the minor actinides such as B7Np fast fission and inelastic scattering cross-sections. • Because the uncertainty of the predicted values of the performance parameters can be reduced by the aid of integral experiments, it is also recommended that such experiments be performed in a facility which has a neutron spectrum similar to the MAB.

- 39 KAERI/TR-1112/98

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8. T. J. Downar and H. S. Khalil, "Uncertainty in the Burnup Reactivity Swing of a Liquid Metal Fast Reactor," Nucl. Sci. Eng., 109, p.278, 1991.

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9. E. Schmidt, "Influence of Nuclear Data Uncertainties on Results When Recycling Actinides other than Fuel", Technical Meeting on the Nuclear Transmutation of Actinides", Ispara, Italy, April 16-18, 1977.

10. B. Toppel, "A Users Guide for the REBUS-3 Fuel Cycle Analysis Capability", ANL-83-2, Argonne National Laboratory, Argonne, 111., 1983.

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15. J. R. White, "On the Implementation, Verification and Application of Multicycle Depletion Perturbation Theory", Proc. American Nuclear Society Topical Meeting, 1980 Advances in Reactor Physics and Shielding, Sun Valley, Idaho, 1980.

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17. H. B. Choi and T. J. Downar, "Sensitivity Theory for the Closed Nuclear Fuel Cycle", Nucl. Sci. Eng., Ill, p.205, 1992.

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18. E. Greenspan, "Sensitivity Functions for Uncertainly Analysis", J. Lewins and M. Becker, Eds., Advances in Nuclear Science and Technology, Vol.14, Plenum Press, New York, NY, 1982.

19. S. Nakamura, Computational Methods in Engineering and Science, John Wiley & Sons, Inc., New York, NY, 1977.

20. D. R. Harris et al., "Characterization of Uncertainties in Evaluated Cross Section Sets", Tran. Am. Nucl. Soc, Vol.16, p.323, Chicago, 111, 1973.

21. F. G. Perey, "The Data Covariance Files for ENDF/B-V", ORNL/TM-5938 (ENDF-249), Oak Ridge Nat. Lab., 1977.

22. J. J. Marable et al., "Uncertainty in Breeding Ratio of a Large Liquid-Metal Fast Breeder Reactor: Theory and Result", Nucl. Sci. Eng., 75, p.30, 1980.

23. T. Takeda et al., "Prediction Uncertainly Evaluation Methods of Core Performance Parameters in Large Liquid-Metal Fast Breeder Reactors", Nucl. Sci. Eng.: 103, p. 157, 1989.

24. R. E. MacFariane, D. W. Muir and R. M. Boicourt, "The NJOY Nuclear Data Processing System, Volume I: User's Manual", LA-9303-M, Vol.1 (ENDF-324), Los Alamos, 1982.

25. R. E. MacFariane and D. W. Muir, "The NJOY Nuclear Data Processing System, Volume III: The GROUPR, GAMINR, and MODER Modules", LA-9303-M, Vol.III (ENDF-324), Los Alamos, 1987.

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26. D. W. Muir and R. E. MacFariane, "The NJOY Nuclear Data Processing System, Volume IV: The ERRORR and COVR Modules", LA-9303-M, Vol.IV (ENDF-324), Los Alamos Nat. Lab., 1985.

27. D. W. Muir and R. J. LaBauve, "COVFILS: A 30-Group Covariance Library Based on ENDF/B-V", LA-8733-MS (ENDF-306), Los Alamos Sci. Lab., 1981.

28. P. B. Nikolskii, "Predicting the Physical Characteristics of Future Fast Reactor Core Designs on the Basis of Analyses Performed with Critical Assemblies and Standard Calculation Models", Proc. of a Symp. on Fast Reactor Physics, Vol.1, Aix-en-Provence, IAEA, 1979.

29. M. G. Silbert, "Fission Cross Sections of ^Pu from Persimmon", LA-4108-MS, Los Alamos Sci. Lab., 1969.

30. W. F. Stubbins and R. A. Wolfe, "Nuclear Data Requirement in the Design of the BIFOLD Nuclear Power Source", Proc. of a Symp. on Nuclear Data in Science and Technology, Vol.1, Paris, IAEA, 1973.

31. K. Wisshak and F. Kappeler, "Neutron Capture and Fission Cross Section of 243Am in the Energy Range from 5 to 250 KeV", Nucl. Sci. Eng., 85, p.251, 1983.

32. J. W. Behrens and J. C. Browne, "Measurement of the Neutron-Induced Fission Cross Section of Americium-241 and Americium-243 Relative to Uranium-235 from 0.2 to 30 MeV", Nucl. Sci. Eng., 77, p.444, 1981.

33. C. Wagemans et al., "Measurement of the Thermal Neutron-Induced Fission Cross Section of 243Am", Nucl. Sci. Eng., 101, p.293, 1989.

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34. H. T. Maguire, Jr. et al., "Neutron-Induced Fission Cross Section Measurements of 244Cm, 246Cm, and 248Cm", Nucl. Sci. Eng., 89, p.293, 1985.

35. Z. Huanqiao et al., "The Average Number of Prompt Neutrons and the Distributions of Prompt Neutron Emission Number for Spontaneous Fission of Plutonium-240, Curium-242, and Curium-244", Nucl Sci. Eng., 86, p.315, 1984.

36. J. R. White and T. F. DeLorey, "Sensitivity and Uncertainty Analysis of Integral Physics Parameters in High-Conversion Reactor Systems", Nuclear Technology, Vol.95, p.129, 1991.

37. C. G. Campbell, "United Kingdom Programme on Fast Reactor Physics", Proc. of a Symp. on Fast Reactor Physics and Related Safety Problems, Vol.1, Karlsruhe, IAEA, 1968.

38. A. T. D. Butland et al., "An Assessment of Methods of Calculating Sodium-Voiding Reactivity in Plutonium-Fueled Fast Reactors", Proc. of a Symp. on Fast Reactor Physics, Vol.1, Aix-en-Provence, IAEA, 1980.

39. A. T. D. Butland et al., "An Assessment of Methods of Calculating Doppler Effects in Plutonium-Fueled Fast Reactors", Proc. of a Symp. on Fast Reactor Physics, Vol.1, Aix-en- Provence, IAEA, 1980.

40. L. N. Usachev and Y. G. Bobkov, "Planning an Optimum Set of Microscopic Experiments and Evaluations to Obtain a Given Accuracy in Reactor Parameter Calculations", INDC(CCP)- 19/U, IAEA, 1972.

41. G. Arfken, Mathematical Methods for Physics, Academic Press, Inc., New York, NY, 1970.

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Table I Mass Flow of Closed Fuel Cycle (kg/yr)

Ratio External External Nuclide / Discharge \ discharge feed I Feed ) 238Pu 6 185 30.8 239P8u 314 199 0.6 241Pu 83 38 0.5 Pu(Total) 552 596 1.1

237Np 337 0 0.0 241Am 159 131 0.8 243Am 158 133 0.8 Cm(Total) 40 0 0.0

Np-chain* 579 169 0.3

Np-chain = + 24IPu + 241Am

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Table U Multi-group Structure

Group Energy 1 19.9710 MeV~6.0653 MeV 2 6.0653 MeV-1.3534 MeV 3 1.3534 MeV~497.87 KeV 4 497.87 KeV~183.16 KeV 5 183.16 KeV~67.379 KeV 6 67.379 KeV-24.788 KeV 7 24.788 KeV~9.1188 KeV 8 9.1188 KeV~3.3548 KeV 9 3.3548 KeV~0.00001 eV

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Table III

Constrained Sensitivity to ^Np j/3 for Final MAB Model

Direct Constrained Isotope Error(%) Subtraction Sensitivity

234U -0.05 -0.06 20.0 235 u 0.01 -0.00 100.0 236 u -0.30 -0.31 3.3 238 u -2.41 -2.44 1.2

237Np -0.02 -0.02 0.0

236Pu 1.35 1.36 0.7 238Pu 1.16 1.16 0.0 239Pu -1.19 -1.18 0.8 240Pu -1.26 -1.26 0.0 24IPu -1.42 -1.41 0.7 242Pu -1.20 -1.20 0.0

241 A 0.83 0.83 0.0 Am 242mAm 1.59 1.58 0.6 243 A 1.05 Am 1.04 1.0 242Cm 1.75 1.74 0.6 243Cm 0.56 0.56 0.0 244Cm 1.05 1.05 0.0 245Cm 0.85 0.84 1.2 246Cm 0.58 0.57 1.7

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Table IV

Region-wise Sensitivities to ^Np v3 for Final MAB Model

Charge Core Isotope fraction « *• region ^ Nc (%) Inner -0.27473 0.00464 0.00569 0.00487 237Np 52 Outer 11.95900 -0.00007 0.00019 -0.00007 Inner 1.19280 0.00443 0.00432 0.00482 238pu 3 Outer 0.95811 0.00235 0.00198 0.00235 Inner 0.66812 0.00027 0.00026 0.00030 239Pu 13 Outer -1.59200 0.01507 0.01168 0.01507 Inner 1.25700 0.00064 0.00063 0.00068 24IAm 5 Outer 0.10606 0.00003 0.00004 0.00003 Inner 0.58992 0.00191 0.00190 0.00202 244Cm 5 Outer 13.15900 0.00009 0.00009 0.00009

Nc = Sensitivity of charge density *> N = Sensitivity to charge density for flooded core

= Sensitivity to charge density for voided core Nc

* Nc = Sensitivity to charge density for hot core

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Table V Sensitivity of Burnup Reactivity Swing for Final MAB Model

Direct Density Rank Reaction Total type term term

237 1 NP U2 11.49 -24.29 -12.80 237 2 Np vs 11.72 -24.51 -12.79 3 B7Np op 8.07 -16.46 -8.39 4 Np oft 7.15 -14.95 -7.79 5 238PU V3 2.22 -6.06 -3.84 6 238PU V2 1.80 -4.87 -3.07 7 238PU V4 1.64 -4.51 -2.87 8 238Pu on 1.55 -3.87 -2.32

9 Np 1/4 2.04 -4.18 -2.14 10 238Pu of, 1.30 -3.37 -2.07

v = neutron yield per fission Of - fission cross-section Of = capture cross-section a m = inelastic cross-section

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Table VI Sensitivity of Void Worth (EOEC) for Final MAB Model

Reaction Direct Density Total Rank type term term

237 1 Np y3 2.02 5.39 7.40 237 2 Np v2 1.46 5.34 6.79

237XT 3 Np op 1.27 3.56 4.83 4 23?Np 0/2 0.87 3.24 4.11

5 238PU .3 0.32 1.29 1.61 238 6 Pu v2 0.17 1.04 1.21 237 7 Np V4 0.25 0.92 1.17 8 238PU 1/4 0.15 0.96 1.12 9 244Cm ,3 0.25 0.78 1.03 10 238Pu ofs 0.20 0.81 1.01

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Table VII Sensitivity of Doppler Constant (EOEC) for Final MAB Model

Direct Density Rank Reaction Total type term term 1 237Np »> -0.12 -3.07 -3.19 2 B7Np vz -0.11 -3.04 -3.15 3 237Np op -0.14 -2.04 -2.18 237 4 Np Of2 -0.11 -1.85 -1.97

5 238PU ,3 -0.01 -0.74 -0.74

6 239PU V4 0.06 -0.67 -0.61 238 7 Pu y2 -0.01 -0.59 -0.60 8 238Pu v* -0.01 -0.55 -0.55 237 9 Np v4 -0.02 -0.52 -0.54 10 239 0.06 -0.55 Pu y3 -0.49

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Table VUI Covariance Matrices Affected by Ratio Measurements in ENDF/B-V

238U 235U 239Pu 239Pu 241Am 242Pu (n, a) (n,7) (n,f) (n,f) (n,f) (n,f) 10B(n, d) * 3 238U(n, y) 2 1 235U(n,f) * 3 3 3 3 239Pu(n,f) 2 1 1 4 4 239Pu(n, ?) 2 1 1 4 4 241Am(n,f) 2 4 4 1 4 242Pu(n,f) 2 4 4 4 1

* standard

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Table IX Additional Relative Standard Deviations (%)

237 Group ^NpC ft,) ^Npt o/») Np(y) 238Pu( oc) 238Pu( Of) ^PuCv)

1 40.0 50.0 2.0 60.0 10.0 1.0 2 40.0 50.0 2.0 60.0 10.0 1.0 3 40.0 50.0 2.0 60.0 10.0 1.0 4 40.0 50.0 2.0 60.0 10.0 1.0 5 40.0 50.0 2.0 60.0 10.0 1.0 6 40.0 50.0 2.0 60.0 10.0 1.0 7 40.0 50.0 2.0 60.0 10.0 1.0 8 40.0 50.0 2.0 60.0 20.0 1.0 9 40.0 50.0 2.0 60.0 20.0 1.0

241 243 243 244 244 244 Group Am( v ) Am( of) AmO) Crn( oc) Crn( of) CmO)

1 2.0 4.1 3.3 50.0 30.0 1.0 2 2.0 2.4 3.3 50.0 30.0 1.0 3 2.0 2.8 3.3 50.0 30.0 1.0 4 2.0 3.8 3.3 50.0 30.0 1.0 5 2.0 2.8 3.3 50.0 30.0 1.0 6 2.0 2.9 3.3 50.0 30.0 1.0 7 2.0 3.7 3.3 50.0 30.0 1.0 8 2.0 5.0 3.3 50.0 30.0 1.0 9 2.0 5.4 3.3 50.0 30.0 1.0

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Table X Uncertainty of Performance Parameters at EOEC (ENDF/B-V Covariance Data)

Reactivity Relative Absolute Parameter Worth Uncertainty Uncertainty (%) (1 STD) Burnup swing (%z/k) 1.19 180 2.14 Void worth (%z/k) 1.17 97 1.13 Doppler (10"3TJk/zJT) -0.32 46 0.15

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Table XI Uncertainty of Performance Parameters at EOEC (ENDF/B-V + Additional Covariance Data)

Relative Absolute Reactivity Parameter Uncertainty Uncertainly Worth (%) (1 STD) Burnup swing (%/}k) 1.19 220 2.60 Void worth (%J k) 1.17 120 1.40 Doppler (10"3T/JkMT) -0.32 59 0.19

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Table XII Uncertainty of Burnup Reactivity Swing (Variance Term)

Relative Reaction Relative Fraction* Rank tyep variance uncertainty (%) (%)

237XT 1 Np 0in2 0.7159 84.6 22.9 2 23?Np 0J3 0.7041 83.9 22.5

237 3 NP 0J2 0.5134 71.7 16.4 4 237Np ow 0.5018 70.8 16.0 5 Cm o/3 0.0828 28.8 2.6 6 Cm o/2 0.0680 26.1 2.2

237 7 Np y2 0.0656 25.6 2.1

237 8 Np y3 0.0654 25.6 2.1 9 238Pu <* 0.0540 23.2 1.7 10 238Pu o/, 0.0430 20.7 1.4

Total 3.1273 176.8 100.0

* ratio of each variance term to total variance

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Table XIII Uncertainty of Void Worth (Variance Term)

Relative Reaction Relative Fraction Rank type variance uncertainty (%) (%)

237Np op 1 0.2336 48.3 27.6 23? 2 Np ow 0.1429 37.8 16.9 3 Np o/? 0.1426 37.8 16.9 4 23?Np OM3 0.1051 32.4 12.4 5 Cm c/3 0.0282 16.8 3.3

6 23?Np 0c4 0.0277 16.6 3.3 7 237Np TB 0.0219 14.8 2.6 8 244Cm ^ 0.0207 14.4 2.4

237 9 NXTp Oc3 0.0188 13.7 2.2 10 0.0185 13.6 2.2 237Np T2 Total 0.8449 91.9 100.0

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Table XIV Uncertainty of Doppler Constant (Variance Term)

Relative Reaction Relative Fraction Rank tyep variance uncertainty (%) (%)

237 1 Np a/3 0.0476 21.8 21.1 2 Np Oin2 0.0474 21.8 21.0 3 23?Np Oln3 0.0351 18.7 15.6 237 4 XT 0.0327 18.1 14.5 Np q/2 5 0.0132 11.5 5.8 237Np o ? 6 c 0.0066 8.1 2.9 244Cm op 7 0.0051 7.1 2.3 8 Cm q/2 0.0041 6.4 1.8 237 9 Np 73 0.0040 6.3 1.8 237 10 Np 72 0.0032 5.7 1.4 237Np o* Total 0.2257 47.5 100.0

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Table XV Uncertainty Requirement of Nuclear Data for Burnup Reactivity Swing

Reaction Present Required Effect Rank type STD(%) STD(%) (%)*

237 1 NP o/s 10.00 0.08 44.49 237 2 Np 0/2 9.20 0.08 39.88 3 Np Oin2 50.00 0.08 14.52 4 Np 0/4 10.00 0.08 10.99 237 r 5 Mx p Omi 50.00 0.08 10.18 6 Cm 0/3 30.00 0.15 1.68

7 23?Np 0/5 10.02 0.08 1.49

8 244Cm 0,2 30.00 0.19 1.38 9 237 2.00 0.08 1.33 Np 72

10 237 2.00 0.08 1.32 Np 73 * Percent effect to total uncertainty reduction

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Table XVI Uncertainty Requirement of Nuclear Data for EOEC Void Worth

Panl' Reaction Present Required Effect IVdllK type STD(%) STD(%) (%)

237 1 Np op 10.00 0.08 49.91 237 2 Np op 9.20 0.08 41.97 37 3 Np op 10.00 0.08 12.32 4 Np Oin2 50.00 0.22 10.31 237 5 NMp Oini 50.00 0.29 7.59

6 Cm Oft 30.00 0.65 2.04

237-KT 7 Np Ocv 40.00 _ . 0.88 2.00 8 Np 1/3 2.00 0.08 1.58 9 244,-, 30.00 0.88 1.49 Cm 0/2 10 237-.T 40.00 1.28 1.36 Np 0c3

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Table XVII Uncertainty Requirement of Nuclear Data for EOEC Doppler Constant

Reaction Present Required Effect Rank type STD(%) STD(%)

237XT 1 Np 0/3 10.00 0.08 44.98 2 Np 0/2 9.20 0.08 39.43 3 Np O/n2 50.00 2.90 14.52

4 Np 0,n3 50.00 3.83 10.73 5 237Np op 10.00 5.24 5.59

6 Np Oc9 40.00 7.25 3.92 7 244.-, 30.00 9.22 1.83 cm 0/3

8 Cm o/2 30.00 10.91 1.36

9 237 2.00 0.84 1.03 Np 73 10 2.00 0.85 1.00 237Np ~,2

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ND External Feed Reprocessing Waste •+ (Of, Or)

Fabrication 1 1 -: Sold

Fig.l Mass Flow of MAB Fuel Cycle

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BIBLIOGRAPHIC INFORMATION SHEET

Performing Org. Sponsoring Org. Standard Report No. INIS Subject Code Report No. Report No. KAERI/TR-1112/98

Title / Subtitle Nuclear Data Uncertainty Analysis on a Minor Actinide Burner for Transmuting Spent Fuel

Project Manager Hangbok Choi (DUPIC Fuel Compatibility Analysis) and Department Researcher and Department

Publication Publication Taejon Publisher KAERI 1998.8 Place Date Page 62 p. 111. & Tab. Yes( O ), No ( ) Size 26 cm.

Note Open ( O ), Restricted ( Classified Report Type Technical Report Class Document Sponsoring Org. Contract No.

Abstract (15-20 Lines) A comprehensive sensitivity and uncertainty analysis was performed on a 1200 MWth minor actinide burner designed for a low burnup reactivity swing, negative doppler coefficient, and low sodium void worth. Sensitivities of the performance parameters were generated using depletion perturbation methods for the constrained close fuel cycle of the reactor. The uncertainty analysis was performed using the sensitivity and covariance data taken from ENDF-B/V and other published sources. The uncertainty analysis of a liquid metal reactor for burning minor actinides has shown that uncertainties in the nuclear data of several key minor actinide isotopes can introduce large uncertainties in the predicted performance of the core. The relative uncertainties in the burnup swing, doppler coefficient, and void worth were conservatively estimated to be 180%, 97%, and 46%, respectively. An analysis was performed to prioritize the minor actinide reactions for reducing the uncertainties. Subject Keywords spent fuel, liquid metal reactor, minor actinide, (About 10 words) neutronics design, void coefficient, doppler constant, transmutation, nuclear data, covariance KAERFTR-1112/98

INIS

KAERI/TR-1112/98

(DUPIC

1998.8 62 p. £( o), si-a-( 3. 7] 26 cm.

1200 MWth

180%, 97%, Zie] jl 46%S.