impact on sensitivity coefficients of typical approximations used in ...

International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011) Rio de Janeiro, RJ, Brazil, May 8-12, 2011, on CD-ROM, Latin American Section (LAS) / American Nuclear Society (ANS) ISBN 978-85-63688-00-2

IMPACT ON SENSITIVITY COEFFICIENTS OF TYPICAL APPROXIMATIONS USED IN SCOPING SENSITIVITY ANALYSES G. Aliberti, W. S. Yang and R. D. McKnight Argonne National Laboratory 9700 S. Cass Ave - Bldg. 208, Argonne, IL 60439 - USA [email protected]; [email protected]; [email protected]

ABSTRACT The design of reactor cores requires confidence in the prediction of core neutronics behavior during normal operation and transient conditions. The uncertainties in computed performance parameters introduced by uncertainties in basic cross section data are typically estimated by propagating the cross section covariance data through sensitivity coefficients. The present study addresses the impact on parameter sensitivities due to the most common approximations (such as transport effects, geometry modeling, group collapsing and the change in the flux distribution during depletion in the case of the burnup reactivity swing) used in the calculations by perturbation theory with deterministic codes. It has been found that the diffusion theory is not adequate for computing sensitivity coefficients in the case of small size assemblies where leakage effects play a significant role. On the contrary, a diffusion calculation can be considered adequate for sensitivity analysis of large cores, even if some transport effects still remain on the parameter values. The common simplifications from a 3D to a RZ model is reasonable for the sensitivity analysis of multiplication factor. However, for control rod worths and, in general, for all parameters that depend on local perturbations of the flux distribution, a 3D model is required. The effects of group collapsing may be non negligible on sensitivities of parameters that are expected to be extremely “sensitive” to the transitional effects at the core-reflector interface, especially for applications where structural material data play a major role. Finally, for burnup dependent parameters the effects on nuclide density evolution through the change in flux distributions due to cross section variations over a burn cycle cannot be neglected. The use of a depletion perturbation theory code is then required. Key Words: Sensitivity Analysis, Generalized Perturbation Theory, Nuclear Data.

1. INTRODUCTION The design of reactor cores requires confidence in the prediction of core neutronics behavior during normal operation and transient conditions. The uncertainties in computed performance parameters introduced by uncertainties in basic cross section data are typically estimated by propagating the cross section covariance data through sensitivity coefficients. The conventional, generalized, and depletion perturbation theory methods based on adjoint formalism [1-8] provide practical means for determining the sensitivities of a performance parameter with respect to a large number of cross sections. However, the perturbation theory codes are relatively less developed compared to the corresponding neutronics codes; currently available codes generally use the diffusion theory approximation or have limitations on geometry modeling and/or applicable integral parameters. In addition, the coupled neutron and nuclide fields are often decoupled for burnup dependent sensitivity analysis [9].

G. Aliberti, W.S. Yang, R.D. McKnight

The present study investigates the impact on integral parameter sensitivities of typical approximations employed in scoping sensitivity analyses. The effects of diffusion theory approximation, simplified geometry modeling, group collapsing, and decoupling of neutron and nuclide fields for burnup dependent sensitivity analysis are analyzed using several fast critical reactor systems. 2. ADJOINT SENSITIVITY FORMALISM The sensitivity function of an integral parameter or response R with respect to a cross section σ is defined as the relative change in R due to a relative change in σ ; that is, S R ( p) =

σ ( p) ∂R , R ∂σ ( p )

(1)

where p denotes the phase space position over which the state variables and cross sections are defined. In discretized computational models, the cross sections are represented by multi-group, region-dependent constants, and thus the sensitivity functions become sensitivity coefficients. The simplest procedure for evaluating the derivative in the definition of the sensitivity function is the brute force approach where direct recalculation with perturbed cross sections are used to obtain finite-difference approximations of the derivative. However, this approach requires high computationally costs when evaluating the effects of a large number of alterations in cross sections for a few responses, which is the usual case in practical applications. In the adjoint sensitivity formalism, the sensitivity functions are evaluated without solving the perturbed system equations by employing adjoint variables. The adjoint system equations depend on the response functional, but not on the input parameters. Thus, by solving the adjoint system equations for a given response, the sensitivity functions with respect to all the cross sections can be obtained at the same time. This section briefly reviews the adjoint-based sensitivity formulas for several responses of practical interest. 2.1. Multiplication Factor and Reactivity Coefficients The Boltzmann transport equation or its diffusion theory approximation can be written in operator form as: BΦ = ( A − λ F ) Φ = 0 ,

(2)

where A is the loss operator, F is the fission production operator, and Φ is the neutron flux. Using the conventional perturbation theory [1-3], the sensitivity coefficients for the multiplication factor, k = λ −1 =< F Φ > / < AΦ > , can be obtained as: Sk = −

kσ Φ ∗ , FΦ

 ∂A 1 ∂F  Φ∗ ,  − Φ ,  ∂σ k ∂σ 

2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

(3)

2/16

Dependence of sensitivity coefficients on common approximations used in deterministic calculations

where < > denotes the integration over space, angle and energy, and the fundamental mode adjoint flux Φ* is determined from the following adjoint equation: B ∗Φ ∗ = ( A∗ − λ F ∗ )Φ ∗ = 0 .

(4)

Here A* and F * are the adjoint operators of A and F , respectively. The reactivity change of a perturbed system from an unperturbed, reference system can be represented as the eigenvalue difference of two systems as ∆ρ = λ − λ ′ = 1/ k − 1/ k ′ , with the primed variables for the perturbed system. Thus the sensitivity coefficients for a reactivity change can be determined as:

S ∆ρ =

σ ∂∆ρ σ  1 ∂k ′ 1 ∂k  1  Sk ′ S k = − −  =  ∆ρ ∂σ ∆ρ  k ′2 ∂σ k 2 ∂σ  ∆ρ  k ′ k

 . 

(5)

2.2. Reaction Rate Ratios or Spectral Indices For a reaction rate ratio, R = Σ n Φ Σ d Φ , defined by macroscopic cross sections Σ n and Σ d , the sensitivity coefficients can be calculated using the generalized perturbation theory [1-3] as:

S=

σ dR σ  ∂R  ∂A 1 ∂F   =  − Γ∗ ,  − Φ  , R dσ R  ∂σ  ∂σ k ∂σ  

(6)

where Γ* is the generalized adjoint flux, which is the solution of the following generalized adjoint equation:

( A∗ − λ F ∗ )Γ∗ =

ΣΦ ∂R = n ∂Φ Σd Φ

 Σ n Σd −  Σd Φ  Σ n Φ

 . 

(7)

The singular inhomogeneous equation in Eq. (7) has solutions since the source term is orthogonal to the flux Φ . Its unique solution is obtained by imposing the bi-orthogonal condition < Γ* , F Φ >= 0 . The first term in the rightmost side of Eq. (6) is called the direct effect and accounts for the variation of the response R directly due to the changes in the cross sections explicitly present in the formula of R . The second term is called the indirect effect and represents the variation of R due to the flux change subsequent to the cross section variation.

2.3. Burnup Dependent Integral Parameters


3/16


Burnup dependent integral parameters are determined by the interaction of the neutron flux field and the nuclide field. The change in the nuclide field is governed by the transmutation equation, which is a system of nonlinear first order differential equations. The behavior of the neutron flux is determined by the transport equation. Since the neutron flux equation is an eigenvalue problem, the magnitude of the flux is specified by the power normalization equation. The depletion equations are nonlinearly coupled due to the nuclide transmutation. For practical problems, solutions to these depletion equations are based on decoupling the flux and the nuclide field equations. The burnup cycle is subdivided into several subintervals; the flux at each time node is calculated by the transport equation in Eq. (2) with the known reactor configuration; and the flux is normalized by the reactor power as P = (κ f σ f + κ cσ c ) ⋅ NΦ ,

(8)

where κ f and κ c are the energy released per fission and capture, respectively. The nuclide density vector over each subinterval is calculated by the transmutation equation:

∂ N = MN . ∂t

(9)

Here the transmutation matrix M consists of neutron flux, microscopic cross sections, and decay constants. Using depletion perturbation theory [4-6], the sensitivity coefficients for a general response of bi-linearly weighted reaction rate ratio can be determined as

S=

σ  ∂R I +∑ R  ∂σ i =1 I

+∑ i =1

∫

ti +1

ti

dt N∗

I ∂B ∂M N + ∑ Γi∗ i Φ i ∂σ ∂σ i =1

I  ∂ (κ f σ f ) ∂ (κ cσ c )  ∂B ∗ Γi i Φ ∗i − ∑ Pi ∗  +  Ni Φi ∂σ ∂σ  i =1  ∂σ

  

,

(10)

where I is the number of time intervals. The adjoint functions in Eq. (10) are obtained from the following system of adjoint equations [6] with appropriate boundary conditions:

∂ ∗ ∂R N = M T N∗ + . ∂t ∂N ti +1 ∂R ∂M Bi*Γ∗i = − − ∫ dt N ∗ N + Pi ∗ (κ f σ f + κ cσ c ) ⋅ N i . t i −1 ∂Φ i ∂Φ i −

Pi ∗ =

ti +1 1  ∂R ∂M + Φ i ∫ dt N∗ N  Φi ti −1 P  ∂Φ i ∂Φ i

Bi Γi = −

 . 

∂R . ∂Φ*i


(11) (12) (13) (14)

4/16


Individual terms in Eq. (10) have distinct physical meanings and are described as the direct, nuclide density, flux, adjoint flux, and power effects, respectively. The direct term denotes the explicit derivative of the response parameter with respect to cross sections; the nuclide density term represents the variation of the response through the nuclide transmutation rate changes; the flux terms represent the variation of the response through the changes in flux distribution in space and energy for fixed flux level; and the power terms represent the variation of the response through the changes in flux level for given power level.

3. SENSITIVITY ANALYSIS RESULTS 3.1. Transport Effects For investigation of the transport effects on sensitivity coefficients, a small size assembly FLATTOP-Pu [10], a medium size experimental core CIRANO ZONA2B [11] and a large size sodium-cooled European Fast Reactor (EFR) [12] were analyzed. Since the goal is to investigate transport effects on sensitivity coefficients, simplified models in RZ geometry with homogenized compositions were used for the analysis of CIRANO and EFR. The FLATTOP-Pu assembly consists of a spherical delta-phase plutonium core (4.5332 cm radius) reflected by normal uranium (24.142 cm radius). The 239Pu content in Pu is about 95%. In the CIRANO ZONA2B configuration, the core loaded with PuO2-UO2 fuel and sodium is surrounded by axial and radial reflectors. The Pu fraction in the heavy metal is about 27%, and the 239Pu content in Pu is about 79%. Minor actinides are less than 1% of the total heavy metal. The EFR is a 3600 MWt reactor having a core loaded with mixed oxide fuel surrounded by axial and radial UO2 blankets. The Pu content in the fuel is about 23% with about 57% of 239Pu in plutonium. The minor actinide content in the heavy metal is 1.2%. The RZ models adopted for analyses of CIRANO ZONA2B and EFR are shown in Figs. 1 and 2, respectively.

Figure 1. RZ model of CIRANO ZONA2B.

Figure 2. RZ model of EFR.


5/16


Transport effects on sensitivity coefficients were investigated for the following integral parameters: the multiplication factor and the spectral index of 238U fission to 239Pu fission (f28/f49) at the core center for the FLATTOP-Pu assembly; and the multiplication factor, the spectral index f28/f49 at the core center and the sodium void worth for CIRANO ZONA2B and EFR. Calculations were performed with the ERANOS code system [13]. Cross sections were generated in 33 energy groups using the ECCO code [14] and ENDF/B-VII.0 nuclear data [15]. Flux and adjoint flux calculations were performed with the BISTRO code [16] both in diffusion and transport theory. The transport calculations were carried out with P1 anisotropic scattering for all assemblies and S16 angular approximation for FLATTOP and S4 for CIRANO and EFR. Table I compares the integral parameter values obtained from the diffusion and transport calculations. The values point out the expected result that treatment of transport effects is very important when analyzing small size assemblies with large leakage effects. For FLATTOP-Pu, where the leakage accounts for 67% of the normalized fission source, the transport correction is more than 14% to the multiplication factor and more than 10% to the threshold fission ratio f28/f49. As shown in Figs. 3 and 4, there are significant changes in both the direct and adjoint flux spectra of FLATTOP-Pu between the diffusion and transport theory calculations. In this assembly there is a distinct increase in the neutron flux and importance above 1 MeV with transport theory (as a result of the decreased leakage). These corrections are much smaller for larger assemblies. For CIRANO ZONA2B, where the leakage accounts for 44% of the normalized fission source, the transport correction is ~1% for the multiplication factor, negligible for the reaction rate ratios, and ~400 pcm for the sodium void worth. For EFR, where the leakage accounts only for 18% of the normalized fission source, the transport correction is only ~0.5% on the reactivity, and negligible on the spectral index and sodium void worth. As shown in Figs. 5 and 6, the changes to the direct and adjoint flux spectra between the diffusion and transport theory calculations are insignificant for both CIRANO ZONA2B and EFR. Clearly it is important that calculated values obtained with lower order methods, such as multigroup diffusion theory, be corrected with the values obtained with more rigorous methods, such as higher order transport theory or continuous energy Monte Carlo. However, based on these very limited comparisons, one might hypothesize that sensitivity coefficients obtained with lower order methods may well be adequate for systems having only small corrections for higher order effects, such as large, well-reflected systems which tend to have low leakage and small transport corrections.

Table I. Integral parameter values from diffusion and transport theory calculations System FLATTOP-Pu Diffusion Transport Parameter 0.710 0.665 Leakage Fraction(a) 0.83811 0.98135 keff 0.109 0.122 f28/f49 Na void worth (pcm) (a)

CIRANO ZONA2B Diffusion Transport 0.443 0.437 0.97349 0.98596 0.040 0.040 -1992 -1603

EFR Diffusion Transport 0.179 0.176 1.10398 1.10927 0.025 0.025 1962 1977

Ratio of leakage from the core to the normalized fission source (fission source / keff).


6/16


0.06

0.15

FLATTOP Pu Diffusion FLATTOP Pu Transport

Normalized Flux Spectra


0.18

0.12 0.09 0.06 0.03 0 1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

0.05 0.04 0.03 0.02 0.01

0 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8

1E+8

Energy [eV]

Energy [eV]

Figure 3. Direct flux spectra at the core center of FLATTOP-Pu.

Figure 4. Adjoint flux spectra at the core center of FLATTOP-Pu. 0.06

ZONA2B Diffusion



0.18 0.15

ZONA2B Transport EFR Diffusion

0.12

FLATTOP Pu Diffusion FLATTOP Pu Transport

EFR Transport

0.09 0.06 0.03 0 1E+2

1E+3

1E+4

1E+5

1E+6

1E+7

0.05 0.04 0.03 0.02

ZONA2B Diffusion ZONA2B Transport EFR Diffusion EFR Transport

0.01

0 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 1E+8

1E+8

Energy [eV]

Energy [eV]

Figure 5. Core average direct flux spectra of CIRANO ZONA2B and EFR.

Figure 6. Core average adjoint flux spectra of CIRANO ZONA2B and EFR.

Sensitivity coefficients (both total values and major isotopic reaction contributions) obtained with diffusion and transport theory calculations are shown in Tables II and III for FLATTOP-Pu, CIRANO ZONA2B and EFR. The sensitivity coefficients were also verified by direct perturbation calculations.

Table II. Sensitivity coefficients for FLATTOP-Pu obtained from diffusion and transport theory calculations Reaction 238

U σc U σf 238 Uν 238 U σel 238 U σinel 239 Pu σf 239 Pu ν 239 Pu σel 239 Pu σinel Overall(a) 238

(a)

keff Diffusion -0.056 0.114 0.139 0.230 0.135 0.635 0.817 0.026 0.029 2.142

Transport -0.042 0.063 0.083 0.133 0.077 0.634 0.877 0.022 0.016 1.923

f28/f49 Diffusion Transport 0.046 0.032 0.963 0.971 -0.062 -0.043 -0.137 -0.113 -0.055 -0.056 -0.766 -0.838 0.071 0.059 -0.001 -0.020 -0.021 -0.095 0.043 -0.102

Overall (total) sensitivities due to all cross sections.


7/16


The sensitivity coefficients for the FLATTOP-Pu assembly show significant differences between the diffusion and transport results on essentially all of the important data. On the contrary, for CIRANO ZONA2B there is generally good agreement between the sensitivity coefficients obtained in diffusion and transport theory. The largest differences occur for the sodium void reactivity, especially for 238U capture, ν and inelastic scattering, 239Pu ν, 23Na and 56Fe inelastic scattering. In the EFR case where the large size further reduces the leakage effects, diffusion and transport theories give practically the same sensitivity coefficients for all considered parameters.

Table III. Sensitivity coefficients for CIRANO ZONA2B and EFR obtained from diffusion and transport theory calculations System

Reaction 238 U 238

CIRANO ZONA2B

EFR

σc U σf 238 Uν 238 U σel 238 U σinel 239 Pu σc 239 Pu σf 239 Pu ν 240 Pu σf 240 Pu ν 56 Fe σel 56 Fe σinel 52 Cr σel 23 Na σel 23 Na σinel 16 O σel Overall 238 U σc 238 U σf 238 Uν 238 U σinel 239 Pu σc 239 Pu σf 239 Pu ν 240 Pu σc 240 Pu σf 240 Pu ν 241 Pu σf 241 Pu ν 56 Fe σinel 23 Na σel 23 Na σinel 16 O σel Overall

keff f28/f49 Na Void Worth Diffusion Transport Diffusion Transport Diffusion Transport -0.107 -0.107 0.122 0.124 -0.545 -0.696 0.061 0.060 0.985 0.984 -0.366 -0.434 0.094 0.095 -0.504 -0.623 -0.037 -0.034 -0.110 -0.106 -0.249 -0.257 0.019 0.098 -0.057 -0.057 0.064 0.065 -0.425 -0.541 0.566 0.564 -0.785 -0.778 -0.531 -0.454 0.811 0.810 -0.207 -0.040 0.038 0.038 -0.227 -0.267 0.055 0.055 -0.287 -0.346 0.069 0.062 -0.064 -0.061 -0.598 -0.613 -0.098 -0.102 -0.045 -0.025 -0.280 -0.299 -0.071 -0.069 1.017 1.153 -0.058 -0.061 -0.104 -0.189 -0.126 -0.125 -0.264 -0.231 1.688 1.659 -0.430 -0.422 -3.911 -4.079 -0.189 -0.188 0.236 0.235 0.447 0.448 0.060 0.059 0.977 0.977 0.104 0.104 0.139 0.136 -0.054 -0.054 -0.295 -0.296 -0.189 -0.190 -0.049 -0.050 0.063 0.063 0.304 0.303 0.496 0.495 -0.800 -0.799 -0.930 -0.930 0.697 0.697 -1.109 -1.106 -0.030 -0.030 0.144 0.143 0.062 0.061 0.093 0.093 0.110 0.111 0.042 0.042 -0.117 -0.117 0.058 0.058 -0.150 -0.150 -0.154 -0.155 0.348 0.352 -0.047 -0.047 0.320 0.323 -0.043 -0.043 -0.064 -0.064 -0.205 -0.203 1.253 1.245 -0.153 -0.148 -0.634 -0.633


8/16


3.2 Effects of Geometry Modeling Due to limitations imposed by the codes in use or by the computational cost, sensitivity coefficients are often calculated for simplified models in RZ geometry. In this section, we discuss the impact on sensitivity coefficients of selected parameters when a RZ model is adopted instead of an original 3D model. For this purpose, an XYZ geometry model was built for the EFR reactor as shown in Fig. 7. Note that the purpose of this analysis is only to investigate the dependence of sensitivity coefficients on the adopted geometry model. Therefore the XYZ configuration presented in Fig. 7 was derived from the RZ model in Fig. 2 by preserving the region volumes.

Figure 7. XYZ model of EFR.

Figure 8. RZ configuration of EFR with control rod insertion.

Two parameters were selected for this study: the multiplication factor and the control rod reactivity worth given by the insertion of B4C absorbers at the rod positions A and C as shown in Figs. 7 and 8. Control rods cannot be represented individually in RZ geometry and it is common to model groups of rods characterized by a comparable distance from the core center in annular rings. Consequently, the simplification from a 3D model to an RZ model is expected to have a large impact on the control rod worth because the absorption effects are more distributed through the core and thereby unrealistically amplified. The calculated values of the selected parameters obtained by diffusion calculations with the RZ and XYZ models are compared in Table IV. It can be seen that use of a simplified RZ model does not introduce a dramatic change in the multiplication factor, showing an effect of ~350 pcm; while the control rod worths obtained in RZ and XYZ geometry are significantly different – by more than a factor of 2. Table V compares the sensitivity coefficients calculated in diffusion theory using the RZ and XYZ models. It can be seen that use of an RZ model relative to an XYZ model makes little 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

9/16


difference for the multiplication factor sensitivities. However the simplified RZ model is not adequate for sensitivity analyses of control rod worth. The same conclusion is expected for other parameters which are “sensitive” to the local effect of control rod insertion such as radial power and reaction rate distributions.

Table IV. EFR parameter values calculated with RZ and XYZ geometry models keff RZ Model 1.10398

XYZ Model 1.10828

Control Rod Worth (pcm) RZ Model XYZ Model -3485 -1241

Table V. EFR sensitivity coefficients obtained with RZ and XYZ geometry models Reaction 238

U σc U σf 238 Uν 238 U σel 238 U σinel 239 Pu σc 239 Pu σf 239 Pu ν 240 Pu σc 240 Pu σf 240 Pu ν 241 Pu σf 238

keff Control Rod Worth keff Control Rod Worth Reaction RZ XYZ RZ XYZ RZ XYZ RZ XYZ 241 -0.189 -0.189 -0.115 -0.295 Pu ν 0.058 0.058 -0.035 -0.019 0.060 0.060 -0.150 -0.154 56Fe σc -0.007 -0.027 56 0.104 0.105 -0.187 -0.181 Fe σel -0.030 -0.171 56 -0.065 -0.209 Fe σinel 0.056 0.008 -0.054 -0.055 0.080 -0.052 52Cr σel -0.007 -0.044 -0.049 -0.049 -0.051 -0.080 58Ni σel -0.019 -0.062 0.496 0.495 -0.628 -0.533 23Na σel 0.008 -0.080 0.697 0.696 -0.574 -0.538 C σel 0.036 0.018 16 -0.030 -0.030 -0.025 -0.035 O σel -0.043 -0.043 -0.062 -0.378 0.062 0.062 -0.130 -0.172 10B σc 0.444 0.249 11 0.093 0.093 -0.145 -0.210 B σel 0.124 0.063 0.042 0.042 -0.043 -0.025 Overall 1.253 1.252 -1.602 -3.139

3.3 Group Collapsing Effects The spectral transition at the core reflector interface of blanket-free fast reactors is a well-known phenomenon. Recent studies [17] showed that conventional procedures for cross section condensation over a small number of energy groups are not suitable for accurately representing the spectral transition effects and that at least about 300 energy groups are required for accurate prediction of reaction rate distributions. As a follow-on study, the impact of group collapsing on sensitivity coefficients was investigated. The analysis was performed for the multiplication factor and the ratio of 239Pu fission rate at a core position to that at a reflector position (7.5 cm in each direction from the interface at the core mid-plane) of CIRANO ZONA2B. Table VI shows the parameter values obtained from transport calculations using cross sections collapsed in 33 and 299 energy groups. The increase of energy groups from 33 to 299 increased the multiplication factor by ~1.4% and the 239Pu fission rate ratio by ~5%. 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

10/16


Sensitivities to cross section variations for the multiplication factor and 239Pu fission rate ratio were obtained from transport calculations with 33 and 299 group cross sections. Table VII shows that the two calculations yield consistent results for multiplication factor. On the contrary, for the 239Pu fission rate ratio that is expected to be “sensitive” to the spectral transition effects at the core reflector interface, noticeable differences are observed in the sensitivity coefficients to structural material cross sections, especially 58Ni, 23Na and 55Mn elastic scattering.

Table VI. CIRANO ZONA2B parameter values obtained from transport calculations with 33 and 299 group cross sections 239

keff 33 Groups 0.98596

299 Groups 0.99957

Pu Fission Reaction Ratio 33 Groups 299 Groups 0.356 0.375

Table VII. CIRANO ZONA2B sensitivity coefficients obtained from transport calculations with 33 and 299 group cross sections Reaction 238

U σc U σf 238 Uν 238 U σel 239 Pu σc 239 Pu σf 239 Pu ν 240 Pu ν 54 Fe σel 56 Fe σc 238

keff 33 Gr. 299 Gr. -0.107 -0.109 0.060 0.059 0.095 0.094 -0.057 0.564 0.810 0.055

-0.058 0.561 0.812 0.054

239

Pu fission ratio keff Reaction 33 Gr. 299 Gr. 33 Gr. 299 Gr. 56 Fe σel 0.062 0.056 52 Cr σc 52 Cr σel 58 0.033 0.031 Ni σc 58 Ni σel 23 Na σel 16 O σel 55 Mn σc 55 -0.045 -0.047 Mn σel 0.276 0.255 Overall 1.659 1.639

239

Pu fission ratio 33 Gr. 299 Gr. -0.511 -0.476 0.035 0.035 -0.065 -0.072 0.041 0.038 -0.171 -0.129 -0.068 -0.082 0.047 0.046 0.069 0.071 -0.049 -0.059 -0.412 -0.401

3.4 Effects of Coupled Neutron and Nuclide Fields In perturbation theory code systems, like ERANOS, the depletion perturbation theory is not always implemented. In such cases for the burnup reactivity swing, only the direct terms (see Section 2.3) of the sensitivity coefficients are calculated. Sometimes, approximate methods are used to account for the density, flux and power terms [12]. In order to evaluate the effects of those approximations, the sensitivity coefficients of the burnup reactivity swing of EFR were calculated using the depletion perturbation theory code DPT [6]. The depletion perturbation method implemented in DPT is based on the non-equilibrium and equilibrium fuel cycle analysis methodologies of REBUS-3 [18] in diffusion theory and RZ geometry. The generalized adjoint 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

11/16


flux equations are solved using DIF3D [19]. Cross sections were processed with the MC2-2 code [20] using ENDF/B-VII.0 nuclear data [15]. The calculated burnup reactivity swing of EFR during a burn cycle of 1700 days was -9261 pcm. Table VIII shows the major component-wise contributions to the EFR burnup reactivity swing sensitivities obtained with DPT. It can be seen that the density, flux and power terms cannot be neglected since they are comparable to the direct term. The direct component of the burnup sensitivity is particularly due to plutonium cross sections, especially fission and ν. 238U capture and 239Pu fission are the major contributors to the density term of the sensitivities. The flux terms show important effects due to 238U capture and inelastic scattering, 239Pu fission and 16O elastic reactions. As expected, the power terms contribute through the fission (especially of 239 Pu) and to a lesser extent the capture cross sections.

Table VIII. Breakdown of DPT sensitivities for the burnup reactivity swing of EFR Reaction U σc 238 U σf 238 Uν 238 U σel 238 U σinel 239 Pu σc 239 Pu σf 239 Pu ν 240 Pu σc 240 Pu σf 240 Pu ν 241 Pu σf 241 Pu ν 23 Na σel 16 O σel Overall 238

Direct Term Density Term Flux Terms -3.056 0.391 -0.081 0.081 -0.072 0.058 0.058 -0.184 0.522 0.066 -0.534 3.234 0.175 -0.382 -0.415 0.039 -0.110 0.195 -0.115 -0.320 0.439 -0.393 0.095 -0.054 0.250 -0.280 -1.351 1.054 0.104

Power Term -0.026 -0.118

-0.685

-0.090 -0.071

-1.049

Sum -2.666 -0.130 -0.084 0.051 -0.125 0.554 2.191 -0.394 -0.372 0.006 -0.109 0.080 -0.380 0.041 -0.031 -1.242

4. IMPACT ON PARAMETER UNCERTAINTIES In the previous sections important corrections were found on integral parameter sensitivities arising from transport effects, geometric modeling, group collapsing and the change in the flux distribution during depletion in the case of the burnup reactivity swing. To investigate the extent which the observed changes in sensitivity coefficients affect the results of their application, the impact on the estimated parameter uncertainties was investigated. Table IX shows the contributions of major single and two correlated cross sections to the total uncertainty estimated with use of the AFCI-2.0 covariance data [21] for those parameters of which sensitivities were affected the most by the investigated approximations or simplifications. 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

12/16


The AFCI-2.0 data include only partial correlation data of two different reactions of the same isotope. Thus, the variation of the parameter R due to the cross section uncertainties of the isotope i can be written as:  (∆R )2   R2 

  = S kT Vk S k + 2 S kT Vkl S l ,  i k k >l l

∑

∑∑

(15)

where Sk is the sensitivity vector of cross section k, Vk is the covariance matrix associated with the cross section k, Vkl is the covariance matrix of cross sections k and l. In Table IX, the square root values of the single cross section contributions in Eq. (15) are presented. For the contribution of two correlated cross sections, the following correlation is given: ρ kl = S kT Vkl S l  S kT V k S k ⋅ S lT Vl S l  . 



(16)

For FLATTOP-Pu the observed discrepancies on the sensitivity coefficients calculated in diffusion and transport theory have a direct impact on the calculated uncertainties. For both multiplication factor and spectral index, the diffusion and transport calculations lead to overall uncertainties which are quite different. Among the major contributions, the largest discrepancies are observed for the elastic and inelastic scattering reactions of 238U and 239Pu. Transport effects appear to be small on the calculated total uncertainty of the sodium void worth of CIRANO ZONA2B. Some effects are however found for specific reactions, as 238U and 23Na inelastic scattering. Similarly, for parameters that are expected to be extremely “sensitive” to the transitional effects at the core-reflector interface, the effects due to group collapsing do not appear to be relevant on the total uncertainties. For the ratio of 239Pu fission rates at two separate locations in core and reflector of CIRANO ZONA2B, the total uncertainty estimated with sensitivity coefficients directly obtained in 33 energy groups show a negligible discrepancy from the uncertainty obtained by collapsing the sensitivities into 33 groups from the 299 group sensitivity calculation. It is however recalled that the effects of group collapsing were particularly observed for the sensitivities to structural material data, such as 58Ni elastic, that do show a significant impact on their uncertainty components too, but these components are generally small. As a consequence, effects of group collapsing could be more important in applications where the sensitivities to structural material data play a major role. For analysis of control rod worth, the calculated results show that the total uncertainties estimated for EFR with the use of an RZ or XYZ model are about the same. However, nonnegligible discrepancies are observed on the uncertainty components of specific reactions, such as 238U inelastic scattering and capture, 56Fe and 16O elastic scattering cross sections. Finally, the obtained uncertainties also confirm that the burnup reactivity swing sensitivities should be determined by taking into account the coupling between neutron and nuclide fields. In the case of EFR, Table IX shows that consideration of density, flux and power sensitivity terms leads to uncertainty estimates (both total and reaction-wise) significantly different than values obtained with the direct sensitivity term only. 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

13/16


Table IX. Contributions (%) of major single and two correlated cross sections to the total uncertainty FLATTOP-Pu keff Diffusion 238 U σel 1.01 238 U σel σinel -93.3 238 U σinel 1.85 239 Pu σf 0.29 239 Pu σel σinel -95.4 239 Pu σinel 0.59 Total 1.12

keff Transport 238 U σel 0.38 238 U σel σinel -79.9 238 U σinel 1.00 239 Pu σf 0.29 239 Pu σel σinel -95.7 239 Pu σinel 0.35 Total 0.83

f28/f49 Diffusion 238 U σf 0.50 238 U σel σinel -25.9 238 U σinel 0.58 239 Pu σf 0.36 239 Pu σel σinel -52.7 239 Pu σinel 0.91 Total 1.17

f28/f49 Transport 238 U σf 0.51 238 U σel σinel -86.3 238 U σinel 0.71 239 Pu σf 0.40 239 Pu σel σinel -53.7 239 Pu σinel 1.83 Total 1.92

CIRANO ZONA2B 239

Sodium Void Worth Diffusion 238

U σc 238 U σel σinel 238 U σinel 239 Pu σf 56 Fe σel 52 Cr σel 23 Na σel 23 Na σinel Total

Transport 1.17 82.5 1.63 1.25 5.42 2.35 4.24 0.96 8.21

238

U σc U σel σinel 238 U σinel 239 Pu σf 56 Fe σel 52 Cr σel 23 Na σel 23 Na σinel Total 238

1.49 85.4 3.08 1.53 5.77 2.55 4.83 1.69 9.63

Pu Fission Ratio (Core/Reflector) 33Gr. Standard Sens. 33Gr. Collapsed Sens. (Transport) (Transport) 239 239 Pu σf 0.73 Pu σf 0.72 56 56 Fe σc 1.17 Fe σc 1.09 56 56 Fe σel 3.49 Fe σel 3.39 52 52 Cr σel 0.38 Cr σel 0.44 58 58 Ni σel 0.50 Ni σel 0.35 23 23 Na σel 0.34 Na σel 0.41 55 55 Mn σc 0.40 Mn σc 0.42 Total 3.89 Total 3.77

EFR Control Rod Worth XYZ 238 238 U σc 0.22 U σc 238 238 U σel 0.14 U σel 238 U σel σinel 69.2 238U σel σinel 238 U σinel 1.59 238U σinel 239 Pu σf 0.30 239Pu σf 240 Pu σf 0.52 240Pu σf 240 Pu ν 0.43 240Pu ν 241 Pu σf 0.61 241Pu σf 56 Fe σel 0.22 56Fe σel 56 Fe σinel 0.39 56Fe σinel 23 Na σel 0.10 23Na σel 16 O σel 0.12 16O σel 10 B σc 0.30 10B σc Total 2.09 Total RZ

0.54 0.49 -36.4 0.35 0.42 0.69 0.62 0.72 1.40 0.10 0.42 0.72 0.17 2.29

Burnup Reactivity Swing Direct Term Only Overall 238 238 U σc 0.05 U σc 238 238 U σel σinel -81.9 U σel σinel 238 238 U σinel 0.76 U σinel 239 239 Pu σc 0.18 Pu σc 239 239 Pu σf 0.23 Pu σf 240 240 Pu σc 0.09 Pu σc 240 240 Pu σf 0.46 Pu σf 241 241 Pu σc 0.17 Pu σc 241 241 Pu σf 2.90 Pu σf 56 56 Fe σel 0.42 Fe σel 23 23 Na σel 0.44 Na σel 16 16 O σel 0.49 O σel Total 3.22 Total


3.76 90.8 1.37 2.90 0.88 3.39 0.02 0.98 0.82 0.20 0.20 0.07 6.29 14/16


5. CONCLUSIONS The present paper addresses the impact on parameter sensitivities due to the most common approximations or simplifications used in the calculations by perturbation theory with deterministic codes. It has been found that the diffusion theory is not adequate for computing sensitivity coefficients in the case of small size assemblies, like FLATTOP Pu, where leakage effects play a significant role. For intermediate size systems, like CIRANO ZONA2B, transport effects are negligible on the sensitivities of multiplication factor and spectral indices at the core center; only in the case of reactivity coefficients, like the sodium void reactivity worth, a transport calculation may be required to perform a sensitivity analysis. With the size of the system further increasing, as in the case of the large sodium-cooled EFR, a diffusion calculation can be considered adequate for calculation of all parameters, even if some transport effects still remain on the parameter values. In the calculation of sensitivity coefficients it is quite common to adopt a simplified RZ model. In the present paper, it has been shown that this simplification is reasonable for the sensitivity analysis of multiplication factor. However, for control rod worths and, in general, for all parameters that depend on local perturbations of the flux distribution, a 3D model is required. The effects of group collapsing on parameter sensitivities have been also analyzed. For parameters that are expected to be extremely “sensitive” to the transitional effects at the corereflector interface, such as for applications where structural material data play a major role, sensitivity coefficients should be calculated in a large number of energy groups (at least 300) and then later collapsed over a broader group structure depending on the application needs. Finally, for burnup dependent parameters the effects on nuclide density evolution through the change in flux distributions due to cross section variations over a burn cycle cannot be neglected. The use of a depletion perturbation theory code, as REBUS-3/DPT, is then required.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Nuclear Energy, under contract DE-AC02-06CH11357.

REFERENCES 1. W. M. Stacy, Variational Methods in Nuclear Reactor Physics, Academic Press, New York (1974) 2. E. Greenspan, “Development in Perturbation Theory,” Advances in Nuclear Science and Technology, Vol. 9, Academic Press, New York (1976). 3. A. Gandini, G. Palmiotti and M. Salvatores, “Equivalent-Generalized Perturbation Theory (EGPT),” Annals of Nuclear Energy, 13, 109 (1986). 4. M. L. Williams, “Development of Depletion Perturbation Theory for Coupled Neutron/Nuclide Fields,” Nucl. Sci. Eng, 70, 20 (1979). 2011 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, 2011

15/16


5. T. Takeda and T. Umano, “Burnup Sensitivity Analysis in a Fast Breeder Reactor – Part I: Sensitivity Calculation Method with Generalized Perturbation Theory,” Nucl. Sci. Eng, 91, 1 (1985). 6. W. S. Yang and T. J. Downar, “Generalized Perturbation Theory for Constant Power Core Depletion,” Nucl. Sci. Eng, 99, 353 (1988). 7. I. Kodeli, “Multidimensional Deterministic Nuclear Data Sensitivity and Uncertainty Code System, Method and Application” Nucl. Sci. Eng, 138, 45 (2001). 8. B. L. Broadhead et al., “Sensitivity- and Uncertainty-Based Criticality Safety Validation Techniques,” Nucl. Sci. Eng., 146, 340 (2004). 9. G. Palmiotti, M. Salvatores, and R. N. Hill, “Sensitivity, Uncertainty Assessment, and Target Accuracies Related to Radiotoxicity Evaluation,” Nucl. Sci. Eng, 117, 239 (1994). 10. International Handbook of Evaluated Criticality Safety Benchmark Experiments, NEA/NSC/DOC(95)03 (DVD), September 2007 Edition. 11. P. Smith, et al., “CIRANO Experimental Program for Plutonium Burning Fast Reactor Physics,” Proceeding of Int. Conf. on Future Nuclear Systems (Global ’97), Yokohama, Japan, Oct. 5-10, 1997, p.384 (1997). 12. G. Aliberti, et al., Uncertainty and Target Accuracy Assessment for Innovative Systems Using Recent Covariance Data Evaluation, OECD/NEA Report No. 6410, OECD (2008). 13. G. Rimpault, et al., “The ERANOS Code and Data System for Fast Reactor Neutronic Analyses,” Proceeding of Int. Conf. PHYSOR 2002, Seoul, South Korea (October 2002). 14. G. Rimpault, “Algorithmic Features of the ECCO Cell Code for Treating Heterogeneous Fast Reactor Assemblies,” Proceeding of Int. Topical Meeting on Reactor Physics and Computation, Portland, Oregon, May 1995. 15. M. B. Chadwick et al., “ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Science and Technology,” Nuclear Science and Engineering, 107, 2931 (2006). 16. G. Palmiotti, J. M. Rieunier, C. Gho, M. Salvatores, “BISTRO Optimized Two Dimensional Sn Transport Code,” Nuclear Science and Engineering, 104, 26 (1990). 17. G. Aliberti, et al., “Methodologies for Treatment of Spectral Effects at Core-Reflector Interfaces in Fast Neutron Systems,” Proceeding of Int. Conf. PHYSOR 2004, Chicago, USA, Apr. 25-29, (2004). 18. B. J. Toppel, “A User’s Guide to the REBUS-3 Fuel Cycle Analysis Capability,” ANL-83-2, Argonne National Laboratory (1983). 19. K. L. Derstine, “DIF3D: A Code to Solve One-, Two-, and Three-Dimensional FiniteDifference Diffusion Theory Problems,” ANL-82-64, Argonne National Laboratory (1984). 20. H. Henryson II, B. J. Toppel, and C. G. Stenberg, “MC2-2: A Code to Calculate Fast Neutron Spectra and Multigroup Cross Sections,” ANL-8144, Argonne National Laboratory (1976). 21. M. Herman et al., “AFCI-2.0 Covariance Library: BNL & LANL Report FY2010,” BNL and LANL Joint Report, October 2010.


16/16