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Method of Calculating Sensitivity Coefficients

A sensitivity coefficient is defined as the ratio of the variation in concentration of the target activation product to the variation in the initial composition or cross section. The sensitivity coefficient of the initial composition and cross section is expressed by the following equations, respectively:

S ΔW=W0

ΔX=X0

S ΔW=W0

Δσ=σ0 ð20:1Þ ð20:2Þ

W0: Concentration of the target activation product under normal condition ΔWW0 W0): Variation in concentration of the target activation product X0: Initial concentration of the element in the material under normal condition ΔXX0 X0): Variation in initial concentration of the element in the material σ0: Cross section under normal condition Δσ(¼σ0 σ0): Variation in cross section

For the calculation of concentration of activation products, ORIGEN2.2 was used with ORLIBJ40 [2], which is a set of the one-group cross-section libraries based on JENDL-4.0 [3]. The sensitivity coefficients are evaluated by executing two different burn-up calculations under normal condition and under compositionchanged or cross-section-changed condition. In the former, the ORIGEN2.2 input files are changed; in the latter, the cross-section library files are changed. Utility programs to evaluate the sensitivity coefficients were prepared and used in these analyses.

Sensitivity Analyses

Analyses Conditions

As stated in Sect. 20.1, activations of in-core structure materials, such as cladding tubes, end plugs, and spacers of fuel assemblies and channel boxes, were investigated in this study. The materials of the in-core structures of PWR and BWR are shown in Table 20.1. The compositions of Zircaloy-2, Zircaloy-4, SUS304 stainless steel, and INCONEL alloy 718 are shown in Table 20.2. In Table 20.2, the average value of the upper and lower limits of the standard specification was applied to the calculation condition for additive elements and the upper limit was applied for impurity elements. The effect of impurity elements that are not specified in the standard are investigated in Sect. 20.3.4.

Typical conditions of BWR were assumed for the cross-section libraries and the irradiation condition, because the difference between the conditions of PWR and BWR is not so significant for the purpose of this study, which is clarifying the dominant generation pathways of activation products.

The cross-section libraries used in these analyses (Table 20.3) were chosen to correspond to the condition of the void ratio in the axial direction. A library made with an average void ratio (40 %) was applied to cladding tubes, spacers, and channel boxes for which the void ratio varies from 0 % to 70 %.

A BWR typical irradiation history consists of four cycles of irradiation of about

377 days with constant flux and 90 days of cooling time in the intervals of irradiation (Fig. 20.1). Considering the period for processing of radioactive wastes,

Table 20.1 Materials of in-core structure

BWR

PWR

Cladding tube

Zircaloy-2

Zircaloy-4

Top end plug

SUS304

Bottom end plug

SUS304

Spacer

Plate: Zircaloy-2

Zircaloy-4 or

Spring: INCONEL alloy 718

INCONEL alloy 718

Channel box

Zircaloy-4

Table 20.2 Compositions of materials

Specification (wt%)

Value in analysis (wt%)

(a) Zircaloy-2 (JIS H 4751)

H

0.0025

Max.

0.0025

B

0.00005

Max.

0.00005

C

0.027

Max.

0.027

N

0.008

Max.

0.008

Mg

0.002

Max.

0.002

Al

0.0075

Max.

0.0075

Si

0.012

Max.

0.012

Ca

0.003

Max.

0.003

Ti

0.005

Max.

0.005

Cr

0.05

0.15

0.10

Mn

0.005

Max.

0.005

Fe

0.07

0.20

0.135

Co

0.002

Max.

0.002

Ni

0.03

0.08

0.055

Cu

0.005

Max.

0.005

Zr

Balance

98.1456

Nb

0.01

Max.

0.01

Mo

0.005

Max.

0.005

Cd

0.00005

Max.

0.00005

Sn

1.20

1.70

1.45

Hf

0.01

Max.

0.01

W

0.01

Max.

0.01

U

0.00035

Max.

0.00035

(b) Zircaloy-4 (JIS H 4751)

H

0.0025

Max.

0.0025

B

0.00005

Max.

0.00005

C

0.027

Max.

0.027

N

0.008

Max.

0.008

Mg

0.002

Max.

0.002

Al

0.0075

Max.

0.0075

Si

0.012

Max.

0.012

Ca

0.003

Max.

0.003

Ti

0.005

Max.

0.005

Cr

0.07

0.13

0.10

Mn

0.005

Max.

0.005

Fe

0.18

0.24

0.21

Co

0.002

Max.

0.002

Ni

0.007

Max.

0.007

Cu

0.005

Max.

0.005

Zr

Balance

98.1186

Nb

0.01

Max.

0.01

(continued)

Table 20.2 (continued)

Specification (wt%)

Value in analysis (wt%)

Mo

0.005

Max.

0.005

Cd

0.00005

Max.

0.00005

Sn

1.20

1.70

1.45

Hf

0.01

Max.

0.01

W

0.01

Max.

0.01

U

0.00035

Max.

0.00035

(c) SUS304 stainless steel (JIS G 4303)

C

0.08

Max.

0.08

Si

1.00

Max.

1.00

P

0.045

Max.

0.05

S

0.030

Max.

0.03

Cr

18.00

20.00

19.00

Mn

2.00

Max.

2.00

Fe

Balance

68.595

Ni

8.00

10.50

9.25

(d) INCONEL alloy 718 (UNS N07718)

B

0.006

Max.

0.006

C

0.08

Max.

0.08

Al

0.20

0.80

0.50

Si

0.35

Max.

0.35

P

0.015

Max.

0.015

S

0.015

Max.

0.015

Ti

0.65

1.15

0.90

Cr

17.00

21.00

19.00

Mn

0.35

Max.

0.35

Fe

Balance

16.809

Co

1.00

Max.

1.00

Ni

50.00

55.00

52.50

Cu

0.3

Max.

0.30

Nb

4.75

5.50

5.125

Mo

2.80

3.30

3.05

Table 20.3 Cross-section libraries

Specification in cross-section library

Cladding tubes, spacers, channel boxes

BWR STEP-III, void ratio 40 %

Top-end-plugs

BWR STEP-III, void ratio 70 %

Bottom-end-plugs

BWR STEP-III, void ratio 0 %

10 years of cooling time after irradiation was assumed in these analyses. The flux intensities at the center, top, and bottom in the axial direction are shown in Table 20.4. The flux intensity at the center corresponds to the average power in typical BWR fuel assemblies. The flux intensities at the top and bottom were

Fig. 20.1 Irradiation history

Table 20.4 Flux intensity at center, top, and bottom in axial direction

determined to be 5 % of that at the center, based on flux distribution evaluated by the one-dimensional neutron diffusion calculation.

 
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