A macroscopic cross-section model for BWR pin-by-pin core analysis

Abstract

A macroscopic cross-section model used in boiling water reactor (BWR) pin-by-pin core analysis is studied. In the pin-by-pin core calculation method, pin-cell averaged cross sections are calculated for many combinations of core state and depletion history variables and are tabulated prior to core calculations. Variations of cross sections in a core simulator are caused by two different phenomena (i.e. instantaneous and history effects). We treat them through the core state variables and the exposure-averaged core state variables, respectively. Furthermore, the cross-term effect among the core state and the depletion history variables is considered. In order to confirm the calculation accuracy and discuss the treatment of the cross-term effect, the k-infinity and the pin-by-pin fission rate distributions in a single fuel assembly geometry are compared. Some cross-term effects could be negligible since the impacts of them are sufficiently small. However, the cross-term effects among the control rod history (or the void history) and other variables have large impacts; thus, the consideration of them is crucial. The present macroscopic cross-section model, which considers such dominant cross-term effects, well reproduces the reference results and can be a candidate in practical applications for BWR pin-by-pin core analysis on the normal operations.

Keywords:

• pin-by-pin
• fine mesh
• BWR core analysis
• cross section
• macroscopic depletion model
• instantaneous and history effects
• cross-term effect

1. Introduction

A cross-section set, which is provided for a boiling water reactor (BWR) core simulator, depends on many core state and depletion history variables (e.g. exposure, void fraction, fuel temperature, control rod insertion, and related histories of them). In order to exactly reproduce appropriate cross sections for a core simulator, all possible combinations of the core state and depletion history variables would be tracked in the cross-section preparation through lattice physics calculations. However, such calculation procedures would be impractical due to the limitation on computation time since the number of possible combinations of core state variables becomes huge. Thus, in current core analysis, assembly-averaged cross sections are calculated for the limited combinations of the core state and the depletion history variables and they are tabulated for the subsequent use by core calculations. In the core calculations, therefore, the cross sections are reconstructed for a node with a particular combination of the core state and the depletion history variables from the tabulated cross-section set.

In the current BWR core analysis using the advanced nodal method, calculations are carried out through two subsequent steps. First, the heterogeneous structure of fuel assemblies is homogenized through lattice physics calculations and then core calculations are carried out using the assembly-averaged cross sections. Since fuel assemblies are treated in a homogeneous manner in core calculations, the pin-power reconstruction method, which synthesizes the global smooth power distribution in a homogeneous node and the heterogeneous pin-power distribution obtained by the lattice physics calculations, is used to estimate the detail pin-power distribution [1,2]. However, the accurate estimation of the detail pin-power distribution in a highly heterogeneous core (e.g. a mix-oxide fuel loaded, a heavily poisoned, or a large enrichment splitting core) might be difficult. Therefore, various efforts (e.g. submeshing of fuel assembly, on-the-fly boundary condition adjustment through iteration of lattice physics calculations) have been developed for this issue [3–5].

Recently, as a candidate of the next-generation core analysis method, the pin-by-pin fine mesh core calculation method, in which the cross sections are homogenized in each pin-cell, has been studied [6–9]. It is possible to directly estimate the detail pin-power distribution in the pin-by-pin fine mesh core calculation method. Since the spatial homogenization is carried out only for a pin-cell, it is expected that the error associated with the spatial homogenization of cross sections will be reduced. Therefore, crucial core parameters (e.g. power peaking factor) would be more accurately calculated with the pin-by-pin core analysis method.

In the pin-by-pin fine mesh core calculation method, the cross-section set used for core analysis is a crucial point from the viewpoints of the calculation accuracy, the required memory, and the computation time. The pin-by-pin fine mesh core calculation method naturally requires larger size of the cross-section set since the cross sections for each pin-cell should be independently tabulated in the cross-section set. Therefore, an efficient cross-section model (i.e. a cross-section tabulation method) is very important for a practical pin-by-pin core simulator. Pin-by-pin core analysis codes have been developed for pressurized water reactors (PWRs) and thus there are practical cross-section models for them [8]. However, BWR fuel assemblies are much more complicated than those of PWRs since they have large water rods, a large enrichment splitting, large water gaps between fuel assemblies, and a control rod insertion. In this context, we should newly develop a cross-section model for the BWR pin-by-pin fine mesh core calculation method, which has not been investigated so far.

The present paper focuses on the fundamental development of the cross-section model on BWR normal operations, which is based on a macroscopic depletion model, used in the BWR pin-by-pin core calculation method. Section 2 is devoted to the description of the cross-section model used in the present study. Verifications of the present cross-section model are given in Section 3 through numerical benchmark calculations. Finally, concluding remarks are given in Section 4.

2. Cross-section model

2.1. Overview of the cross-section model

In the pin-by-pin core calculation method, pin-cell averaged cross sections are calculated for many combinations of the core state and depletion history variables, which have influences on the cross sections, and are tabulated prior to the core calculations. The appropriate cross sections for the target core state are interpolated, synthesized, and reconstructed during the core calculations. In the conventional method, the cross sections are homogenized for a fuel assembly, as shown in Figure 1. On the other hand, in the pin-by-pin fine mesh core calculation method, the cross sections are homogenized for each pin-cell, as shown in Figure 2.

Figure 1. Homogenization of cross sections in the conventional advanced nodal method.

Figure 2. Homogenization of cross sections in the pin-by-pin fine mesh core calculation.

The cross sections for each pin-cell are different due to the variations of exposure and neutron flux in each position within fuel assemblies even if fuel rod types are the same. When the variations among fuel rods are not very large, the cross sections for a particular fuel rod type may be tabulated in a unified set and it can be applied to all fuel rods of the same type in a fuel assembly. However, in BWR fuel assemblies, heterogeneity is much larger than that of PWRs due to the existence of the channel box, the water rods, the large enrichment splitting, and the control rod insertion. Therefore, in the present study, the cross sections for each pin-cell are independently tabulated, even if there are several identical types of fuel rods in fuel assemblies.

There are two major approaches of the cross-section model (i.e. the macroscopic depletion and the microscopic depletion models) [3,10,11]. In the macroscopic depletion model, since the macroscopic cross sections needed as one of the input data to carry out the core calculations are mainly tabulated, the computational load can be saved. On the other hand, in the microscopic cross-section model, the differences between nuclide compositions in each fuel pin can be directly estimated through the depletion of nuclides. However, the larger size of the computational memory is necessary since the microscopic cross sections and the number densities of major nuclides (e.g. ²³⁵U, ²³⁸U, ²³⁹Pu, ¹⁵⁵Gd, and ¹⁵⁷Gd) should be tabulated and maintained in the computational memory. Therefore, in the present study, we focus on the macroscopic depletion model since the pin-by-pin fine mesh core calculation method naturally requires a larger size of the cross-section set (e.g. for an 8×8 fuel assembly, the number of homogenized meshes in the pin-by-pin fine mesh core calculation method is about 50 times larger than that in the conventional core calculation method) and the application of the microscopic cross-section model for the pin-by-pin fine mesh core calculation method might be difficult from the viewpoint of the required memory, especially for routine (or production) design calculations.

The macroscopic cross sections in BWR cores have the dependence on many core state and depletion history variables (e.g. exposure, void fraction, fuel temperature, moderator temperature, control rod insertion, and related history of them) as (1) where, Σ, EXP, VOI, TFU, TMO, CR, HVOI, HTFU, HTMO, and HCR are the macroscopic cross section, the exposure, the void fraction, the fuel temperature, the moderator temperature, the control rod insertion, the void history, the fuel temperature history, the moderator temperature history, and the control rod history, respectively. These core state and depletion history variables continuously change during the reactor operation; thus, the macroscopic cross sections are also varied.

Note that the following relationships between the void fraction and the moderator temperature were assumed in the present study:

1. The void fraction in the subcooled boiling condition is not considered; thus, the void fraction is constant at 0% when the moderator temperature is below the saturation temperature (559 K) at the reactor pressure on BWR normal operation (about 70 MPa).

2. When the void fraction is higher than 0%, the moderator temperature is constant at 559 K.

The variations of cross sections in core calculations are classified into two different types of phenomena (i.e. the instantaneous and the history effects). The instantaneous effect is caused by the instantaneous changes of the core state variables. For example, the variations of cross sections are caused when the void fraction changes. In this effect, the fuel composition is fixed. Therefore, the instantaneous effect is captured by the branch calculations in lattice physics calculations, in which the fuel composition is fixed and particular core state variables (e.g. the void fraction and the fuel temperature) are changed. The history effect is caused by the variation of the fuel composition due to the changes in the core state variables during the burnup. For example, when fuel rods are depleted under a high void fraction condition, the harder neutron spectrum promotes the production of plutonium generation; thus, the reduction of the k-infinity during the burnup becomes smaller. In BWR core calculations, it is very important to accurately capture the history effect due to the large variation of the void fraction and the control rod insertion. Therefore, the history effect is captured by a series of history depletion calculations, in which particular core state variables are changed from the base calculation case (e.g. the void fraction in the base calculation case is 40% while that in the history depletion calculation is 80%).

As shown in Equation (1), the macroscopic cross sections have the dependence on many core state and depletion history variables. However, it is difficult to directly tabulate the macroscopic cross sections in this form since it would require the calculations on all possible combinations of the core state and depletion history variables. Thus, several approximations are necessary to efficiently tabulate the macroscopic cross sections.

In the present study, we tabulate the macroscopic cross sections in the following manner:

1. The macroscopic cross sections are independently tabulated for different exposure points since the exposure has a dominant effect on the macroscopic cross sections through the variations of number densities of fuel materials. This is described in detail in the next subsection.

2. The macroscopic cross sections for each pin-cell are independently tabulated, even if there are identical types of fuel rods in fuel assemblies, since the heterogeneity of BWR fuel assemblies is much larger than that of PWRs due to the existence of the channel box, the water rods, the large enrichment splitting, and the control rod insertion.

3. The macroscopic cross sections are approximated by a linear combination of the macroscopic cross sections on a specific condition (i.e. a base state condition) and the variations of macroscopic cross sections for the instantaneous and the history effects.

4. The cross-term effect among the instantaneous and the history effects is not considered since the variations of microscopic cross sections and number densities of moderator and structure materials are mainly caused by the instantaneous effect, while the variations of those of fuel materials are mainly caused by the history effects.

Their validity is discussed in Section 3.

In the above tabulation manner, the macroscopic cross sections shown in Equation (1) are approximated as (2) where the subscripts “base”, “inst”, and “hist” represent the base condition, the instantaneous, and the history effects, respectively. The first, second, and third terms on the right-hand side denote the macroscopic cross section on the base condition, the variation of macroscopic cross section for the instantaneous effect, and that for the history effect, respectively. Note that each term on the right-hand side incorporates the dependence on the exposure and is separately tabulated for the exposure since the exposure has a dominant effect on the macroscopic cross sections through the variations of number densities of nuclides.

In the present study, the following condition is considered as the base state condition, which corresponds to BWR normal operations:

–	Void fraction is 40%, which is the typical core average value.
–	Fuel temperature is 800 K, which is the typical core average value.
–	Moderator temperature is 559 K, which is the saturation temperature at the reactor pressure (about 70 MPa).
–	Control rod is withdrawn since the majority of control rods are withdrawn during full power operations.

The second term is the correction term for the instantaneous effect and is calculated by the polynomial interpolations of the macroscopic cross sections for the core state variables. Note that this term has complicated dependences for the core state variables (i.e. the cross-term effects among several core state variables are caused). For example, when the void fraction changes from 40% to 50%, the variations of the macroscopic cross sections due to the change of the void fraction would also depend on the fuel temperature or the control rod insertion. The treatments of the cross-term effect are discussed in detail in Section 2.3.

The third term is the correction term for the history effect and is calculated by the polynomial interpolations of the macroscopic cross sections for the depletion history variables. Note that the cross-term effects among several depletion history variables occur in this term similar to the correction term for the instantaneous effect. The treatments are discussed in detail in Sec- tion 2.4.

2.2. Interpolation for exposure

The exposure has a dominant effect on the macroscopic cross sections through the variations of number densities of nuclides. For example, the macroscopic absorption cross sections in a Gd-bearing fuel pin-cell rapidly change until most of Gd is depleted. Thus, in the present macroscopic cross-section model, the instantaneous and the history effects and the xenon feedback are considered for each exposure point used in the lattice physics calculations.

The pin-wise exposures depend on the depletion histories. Namely, the pin-wise exposures calculated in core calculations are different from those in the lattice physics calculations to generate the pin-wise cross sections. Moreover, the pin-wise exposures, which are obtained by various depletion calculations in a case matrix calculation in a lattice physics code, would be significantly different from each other. Therefore, the pin-wise cross sections obtained by lattice physics calculations should be tabulated on standard exposure points.

The macroscopic cross sections are interpolated for standard exposure points by utilizing the cubic spline interpolation (CSI), which is one of piece-wise polynomial approximations [12]. Since the CSI requires some boundary conditions on limits of interpolation, second-order differential coefficients of macroscopic cross sections with respect to exposure are assumed to be zero, which is called the natural boundary condition, in the present macroscopic cross-section model.

The verification of the interpolation of macroscopic cross sections for the exposure, which utilizes the CSI, is described in Section 3.2.1.

2.3. Instantaneous effect

The core state variables that have the influence on the macroscopic cross sections continuously change during the operation of reactor cores. The instantaneous effect, which is caused by the instantaneous changes of the core state variables, is incorporated by the variations of macroscopic cross sections. The variations of macroscopic cross sections due to the variation of fuel composition are not considered in the instantaneous effect.

The correction term for the instantaneous effect is described as the second term on the right-hand side of Equation (2). This term is estimated by the branch calculations from several depletion points of the depletion calculation on the base condition. The concept of the branch calculation is shown in Figure 3. In the branch calculations, a (or a few) core state variable(s) is (or are) changed while fixing the fuel composition, which depends on the exposure.

Figure 3. Concept of branch calculation.

The macroscopic cross sections after the instantaneous variations of core state variables are estimated in lattice physics calculations. When one of the core state variables instantaneously changes from the base condition, the variations of macroscopic cross sections (i.e. the delta cross sections) are estimated by (3) where X is one of the core state variables and the subscript branch represents the branch calculation from the base depletion calculation. Note that, in the present study, we independently tabulate the macroscopic cross sections for each pin-cell mesh. Therefore, EXP is the pin-wise exposure if the pin-cell mesh includes a fuel rod; otherwise, EXP is the assembly-averaged exposure. In Equation (3), the left-hand side denotes the delta cross section due to the instantaneous change of the core state variable X from the base condition X_base to the perturbed condition X′. The first term on the right-hand side denotes the macroscopic cross section calculated by the branch calculation with X′. The second term denotes the macroscopic cross section calculated by the depletion calculation on the base condition with X_base.

Note that the variations of macroscopic cross sections depend not only on the corresponding core state variable, but also on other core state variables, as described in the previous subsection. Therefore, the cross-term effect should be considered if it considerably affects the macroscopic cross sections. In the present study, we consider the dependences of macroscopic cross sections on the void fraction, the fuel temperature, the moderator temperature, and the control rod insertion, which are the typical core state variables in BWR cores. However, it is impractical to directly treat such dependences of macroscopic cross sections (i.e. a four-dimensional tabulation of macroscopic cross sections) since the macroscopic cross-section model would become complicated. Therefore, it is necessary to approximate the dependences of macroscopic cross sections on the core state variables in order to simplify the macroscopic cross-section model.

On BWR normal operation, we can separate the cross-term effect for the following conditions:

1. The moderator temperature is constant at 559 K, which is the bulk boiling condition. In this condition, the void fraction, the fuel temperature, and the control rod insertion would change.

2. The moderator temperature is lower than 559 K. In this condition, we assume that the void fraction is constant at 0% and the fuel temperature, the moderator temperature, and the control rod insertion would change.

By separating the cross-term effect as described above, we can tabulate the macroscopic cross sections for the instantaneous effect in the following two types of three-dimensional tabulations:

1. Void fraction, fuel temperature, and control rod insertion (moderator temperature: fixed to 559 K).

2. Fuel temperature, moderator temperature, and control rod insertion (void fraction: fixed to 0%).

Then, we can approximate the second term on the right-hand side of Equation (2) as (4) or (5)

If the moderator temperature is constant at 559 K, we use Equation (4) to approximate the second term on the right-hand side of Equation (2). The term on the right-hand side of Equation (4) considers the cross-term effect among the void fraction, the fuel temperature, and the control rod insertion, in which the moderator temperature is fixed to 559 K. If the moderator temperature is lower than 559 K, we use Equation (5) instead of using Equation (4). The term on the right-hand side of Equation (5) considers the cross-term effect among the fuel temperature, the moderator temperature, and the control rod insertion, in which the void fraction is fixed to 0%.

In order to accurately estimate the variations of macroscopic cross sections due to the instantaneous effect, several branch calculations, in which some core state variables are simultaneously changed from the base condition, are carried out and the cross-term effects among the core state variables are considered. The branch calculation cases used in the present study are shown in Table 1.

Table 1. Branch calculation cases used in the present study.

No.	Type	CR	VOI [%]	TFU [K]	TMO [K]	Depletion/branch
0	BASE	Out	40.0	800.0	559.0	Depletion
1	VOI	Out	0.0	800.0	559.0	Branch from No. 0
2	VOI	Out	30.0	800.0	559.0	Branch from No. 0
3	VOI	Out	50.0	800.0	559.0	Branch from No. 0
4	VOI	Out	80.0	800.0	559.0	Branch from No. 0
5	VOI/TFU	Out	0.0	600.0	559.0	Branch from No. 0
6	VOI/TFU	Out	0.0	1100.0	559.0	Branch from No. 0
7	VOI/TFU	Out	30.0	600.0	559.0	Branch from No. 0
8	VOI/TFU	Out	30.0	1100.0	559.0	Branch from No. 0
9	VOI/TFU	Out	50.0	600.0	559.0	Branch from No. 0
10	VOI/TFU	Out	50.0	1100.0	559.0	Branch from No. 0
11	VOI/TFU	Out	80.0	600.0	559.0	Branch from No. 0
12	VOI/TFU	Out	80.0	1100.0	559.0	Branch from No. 0
13	CR/VOI	In	0.0	800.0	559.0	Branch from No. 0
14	CR/VOI	In	30.0	800.0	559.0	Branch from No. 0
15	CR/VOI	In	50.0	800.0	559.0	Branch from No. 0
16	CR/VOI	In	80.0	800.0	559.0	Branch from No. 0
17	CR/VOI/TFU	In	0.0	600.0	559.0	Branch from No. 0
18	CR/VOI/TFU	In	0.0	1100.0	559.0	Branch from No. 0
19	CR/VOI/TFU	In	30.0	600.0	559.0	Branch from No. 0
20	CR/VOI/TFU	In	30.0	1100.0	559.0	Branch from No. 0
21	CR/VOI/TFU	In	50.0	600.0	559.0	Branch from No. 0
22	CR/VOI/TFU	In	50.0	1100.0	559.0	Branch from No. 0
23	CR/VOI/TFU	In	80.0	600.0	559.0	Branch from No. 0
24	CR/VOI/TFU	In	80.0	1100.0	559.0	Branch from No. 0
25	VOI/TMO	Out	0.0	800.0	540.0	Branch from No. 0
26	VOI/TFU/TMO	Out	0.0	600.0	540.0	Branch from No. 0
27	VOI/TFU/TMO	Out	0.0	1100.0	540.0	Branch from No. 0
28	CR/VOI/TMO	In	0.0	800.0	540.0	Branch from No. 0
29	CR/VOI/TFU/TMO	In	0.0	600.0	540.0	Branch from No. 0
30	CR/VOI/TFU/TMO	In	0.0	1100.0	540.0	Branch from No. 0

In Table 1, the case of No. 0 is the depletion calculation on the base condition. The cases from No. 1 to No. 30 are the branch calculations from the case of No. 0 and the core state variables described in the “Type” column are changed. The core state variables of the cases from No. 1 to No. 12 and from No. 13 to No. 24, and from No. 25 to No. 27 and from No. 28 to No. 30 are the same except for the control rod insertion. The cases from No. 0 to No. 24 in Table 1 are used to estimate the second term on the right-hand side of Equation (2) with Equation (4). On the other hand, the cases of No. 0 and from No. 25 to No. 30 in Table 1 are used to estimate the second term on the right-hand side of Equation (2) with Equation (5).

The verifications and the discussions of the treatment of cross-term effects, among the core state variables discussed in this subsection, are described in Section 3.2.2.

2.4. History effect

The core state variables continuously change during the burnup and have influence on the variations of fuel composition as described in Section 2.3. Thus, the macroscopic cross sections may be different due to the variations of fuel composition even if the core state variables are the same at one exposure point. Such cumulative effects of core state variables are called the history effects.

In order to capture the history effect, some exposure-averaged core state variables are considered as the indices as (6) where is the exposure-averaged value of the core state variable X. In particular, the void fraction, the fuel temperature, the moderator temperature, and the control rod insertion depend on the exposure [13]. In Equation (6), w is the weight function, which reflects the decay effect of the core state variable X, especially for the control rod insertion. Note that we have not yet considered the decay effect; thus, we use w(EXP′, EXP) = 1 in the present study. Improvement of the weight function in Equation (6) would be an open issue to be addressed in future work.

The neutron spectrum also depends on the exposure. Therefore, in the conventional advanced nodal methods and the pin-by-pin core calculations for PWRs, the spectral history (SH) has been successfully used as the history indices to capture the history effect. The SH is defined as (7) where the subscripts “nominal” and “off-nominal” represent the depletion calculations on the base and other conditions, respectively [1,2]. SI is the spectral index and is defined as (8) where the subscripts “fast” and “thermal” represent the fast and the thermal energies, respectively, in which 0.625 eV has been traditionally used as the energy boundary between the fast and the thermal energies. From Equations (7) and (8), the SH is the exposure-averaged value of the SI.

In the conventional advanced nodal methods and the pin-by-pin core calculations for PWRs, the variations of macroscopic cross sections due to the history effect are treated by a unified approach using the exposure and the SH (e.g. the variations of macroscopic cross sections are expressed as a function of the exposure and the SH) [1,2,8]. In this manner, the third term on the right-hand side of Equation (2) is described as (9) However, in the BWR pin-by-pin fine mesh core calculations, the unified approach using the SH may have some difficulties described as follows. In BWR cores, there is a large variation of the neutron spectrum due to the void fraction and the exposure distributions within fuel assemblies and the control rod insertion into the water gaps between fuel assemblies.

In nature, several history effects (e.g. the void history, the fuel temperature history, the moderator temperature history, and the control rod history) are included in the SH. Therefore, we can consider the variation of macroscopic cross sections for the history effect, which corresponds to the third term on the right-hand side of Equation (2), as a function of the cumulative effects of core state variables, which are represented by the exposure-averaged core state variables as (10) where , , , and are the exposure-averaged void fraction, fuel temperature, moderator temperature, and control rod insertion, respectively, and they are calculated by Equation (6).

The correction term for the history effect, which is the third term on the right-hand side of Equation (2), is estimated by the depletion and the branch calculations. The overview of the relationship between the correction term for the history effect and the depletion and the branch calculations is shown in Figure 4. In Figure 4, the core state variables of the depletion calculation on other conditions and the branch calculation are the same.

Figure 4. Overview of the relationship between the correction term for history effect and depletion and branch calculations.

When one of the cumulative effects of core state variables is different from the base condition, the variations of macroscopic cross sections (i.e. the delta cross sections) are estimated by (11) where the subscripts “depletion” and “branch” represent the depletion calculation and the branch calculation from the base depletion calculation, respectively. The left-hand side of Equation (11) is the delta cross section due to the variation of the cumulative effect of the core state variable X from the base condition X_base. The first term on the right-hand side is the macroscopic cross section calculated by the depletion calculation with the core state variable X′. The second term is the macroscopic cross section calculated by the branch calculation with the core state variable X′.

In BWR cores, the void and the control rod histories are the most dominant history effects. The cumulative effects of core state variables depend not only on the history of the corresponding core state variables, but also on that of other core state variables. Namely, the cross-term effect is also considered for the history effect. Therefore, in order to estimate the variations of macroscopic cross sections for the history effect, several depletion calculations, in which some core state variables are simultaneously changed from the base condition, are carried out in addition to the branch calculations shown in Table 1. The depletion calculation cases used in the present study are shown in Table 2.

Table 2. Depletion calculation cases used in the present study.

No.	Type	CR	VOI [%]	TFU [K]	TMO [K]	Depletion/branch
0	BASE	Out	40.0	800.0	559.0	Depletion
1	HVOI	Out	0.0	800.0	559.0	Depletion
2	HVOI	Out	30.0	800.0	559.0	Depletion
3	HVOI	Out	50.0	800.0	559.0	Depletion
4	HVOI	Out	80.0	800.0	559.0	Depletion
5	HVOI/HTFU	Out	0.0	600.0	559.0	Depletion
6	HVOI/HTFU	Out	0.0	1100.0	559.0	Depletion
7	HVOI/HTFU	Out	30.0	600.0	559.0	Depletion
8	HVOI/HTFU	Out	30.0	1100.0	559.0	Depletion
9	HVOI/HTFU	Out	50.0	600.0	559.0	Depletion
10	HVOI/HTFU	Out	50.0	1100.0	559.0	Depletion
11	HVOI/HTFU	Out	80.0	600.0	559.0	Depletion
12	HVOI/HTFU	Out	80.0	1100.0	559.0	Depletion
13	HCR/HVOI	In	0.0	800.0	559.0	Depletion
14	HCR/HVOI	In	30.0	800.0	559.0	Depletion
15	HCR/HVOI	In	50.0	800.0	559.0	Depletion
16	HCR/HVOI	In	80.0	800.0	559.0	Depletion
17	HCR/HVOI/HTFU	In	0.0	600.0	559.0	Depletion
18	HCR/HVOI/HTFU	In	0.0	1100.0	559.0	Depletion
19	HCR/HVOI/HTFU	In	30.0	600.0	559.0	Depletion
20	HCR/HVOI/HTFU	In	30.0	1100.0	559.0	Depletion
21	HCR/HVOI/HTFU	In	50.0	600.0	559.0	Depletion
22	HCR/HVOI/HTFU	In	50.0	1100.0	559.0	Depletion
23	HCR/HVOI/HTFU	In	80.0	600.0	559.0	Depletion
24	HCR/HVOI/HTFU	In	80.0	1100.0	559.0	Depletion
25	HVOI/HTMO	Out	0.0	800.0	540.0	Depletion
26	HVOI/HTFU/HTMO	Out	0.0	600.0	540.0	Depletion
27	HVOI/HTFU/HTMO	Out	0.0	1100.0	540.0	Depletion
28	HCR/HVOI/HTMO	In	0.0	800.0	540.0	Depletion
29	HCR/HVOI/HTFU/HTMO	In	0.0	600.0	540.0	Depletion
30	HCR/HVOI/HTFU/HTMO	In	0.0	1100.0	540.0	Depletion

In Table 2, the case of No. 0 is the depletion calculation on the base condition. The cases from No. 1 to No. 30 are also the depletion calculations; however, their core state variables are the same as those of the cases from No. 1 to No. 30 in Table 1. The core state variables of the cases from No. 1 to No. 12 and from No. 13 to No. 24, and from No. 25 to No. 27 and from No. 28 to No. 30 are the same except for the control rod insertion.

The treatment of the cross-term effect for the history effect is similar to that for the instantaneous effect described in the previous subsection. Namely, we separate the cross-term effect among the depletion history variables for the following conditions:

1. The exposure-averaged moderator temperature is constant at 559 K. In this condition, the exposure-averaged void fraction, the exposure-averaged fuel temperature, and the exposure-averaged control rod insertion would change.

2. The exposure-averaged moderator temperature is lower than 559 K. In this condition, we assume that the exposure-averaged void fraction is constant at 0% and the exposure-averaged fuel temperature, the exposure-averaged moderator temperature, and the exposure-averaged control rod insertion would change.

By separating the cross-term effect as described above, we tabulate the macroscopic cross sections for the history effect in the following two types of three-dimensional tabulations:

1. Void, fuel temperature, and control rod histories (exposure-averaged moderator temperature: fixed to 559 K).

2. Fuel temperature, moderator temperature, and control rod histories (exposure-averaged void fraction: fixed to 0%).

The verifications and the discussions of the treatment of the cross-term effect, among the depletion history variables discussed in this subsection, are also described in Section 3.2.2.

2.5. Simplified xenon feedback

The accumulations of the particular fission product nuclides, which have short half-life and large absorption cross sections, have large impacts on the accuracies of core calculations. In the conventional macroscopic cross-section model, the cumulative effects of ¹³⁵I, ¹³⁵Xe, ¹⁴⁹Pm, and ¹⁴⁹Sm are treated in a microscopic manner.

We consider steady operation conditions (i.e. core power level is constant) in the present study. Under the steady operation conditions, the accumulation of ¹³⁵Xe can be considered as an equilibrium condition in each exposure point. However, since the absorption cross section of ¹³⁵Xe is large and ¹³⁵Xe has a large impact on core characteristics, the cumulative effect of ¹³⁵Xe is treated in a microscopic manner. Namely, the microscopic absorption cross section of ¹³⁵Xe is tabulated in addition to the macroscopic cross sections and the number density of ¹³⁵Xe is approximately calculated from the absolute fission rate density.

The effective fission yield of ¹³⁵Xe is estimated as (12) where , N, λ, σ_a, Σ_f, and φ are the effective fission yield, the number density, the decay constant, the microscopic absorption cross section, the macroscopic fission cross section, and the neutron flux, respectively. The subscript X represents ¹³⁵Xe. The effective fission yield of ¹³⁵Xe calculated by Equation (12) is also tabulated.

Note that the microscopic absorption cross section and the effective fission yield of ¹³⁵Xe are tabulated similarly to the macroscopic cross sections. They are also tabulated for the core state and the depletion history variables (i.e. the exposure, the void fraction, the fuel temperature, the moderator temperature, the control rod insertion, the void history, the fuel temperature history, the moderator temperature history, and the control rod history) and the treatments of these variables are the same with the macroscopic cross sections described in Sections 2.2–2.4.

3. Calculation

3.1. Calculation conditions and verification approaches

In order to verify the present macroscopic cross-section model for BWR pin-by-pin core analysis, which is described in the previous section, benchmark calculations are carried out in typical BWR fuel assemblies shown in Figure 5 [14,15]. The fuel rod enrichment distributions within each fuel assembly are decided by the Ref. [16]. The pin-cell averaged cross sections used in the present study are calculated by the HELIOS code in eight energy groups, which are collapsed from 47 energy groups [17]. Note that the eight-group structure shown in Table 3 is slightly modified in order to be consistent with the energy boundaries described in the Refs. [6] and [7]. The exposure points, which are used in the depletion calculations by the HELIOS code, are given in Table 4. Note that the exposure points, which are given in Table 4 as the shaded area, are used to establish the present macroscopic depletion model (i.e. used as the standard exposure points as described in Section 2.2).

Table 3. Energy group structure.

Group	Upper [eV]	Lower [eV]
1	2.0000E+07	2.2313E+06
2	2.2313E+06	8.2085E+06
3	8.2085E+06	9.1188E+03
4	9.1188E+03	1.3007E+02
5	1.3007E+02	3.9279E+00
6	3.9279E+00	6.2506E – 01
7	6.2506E – 01	1.4572E – 01
8	1.4572E – 01	1.0000E – 04

Table 4. Exposure points for HELIOS and verification calculations.

Exposure points for HELIOS and verification calculations
0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0	5.5
6.0	6.5	7.0	7.5	8.0	8.5	9.0	9.5	10.0	10.5
11.0	11.5	12.0	12.5	13.0	13.5	14.0	14.5	15.0	15.5
16.0	16.5	17.0	17.5	18.0	18.5	19.0	19.5	20.0	21.0
22.0	23.0	24.0	25.0	26.0	27.0	28.0	29.0	30.0	31.0
32.0	33.0	34.0	35.0	36.0	37.0	38.0	39.0	40.0	41.0
42.0	43.0	44.0	45.0	46.0	47.0	48.0	49.0	50.0	51.0
52.0	53.0	54.0	55.0	56.0	57.0	58.0	59.0	60.0	61.0
62.0	63.0	64.0	65.0	66.0	67.0	68.0	69.0	70.0	71.0
72.0	73.0	74.0	75.0	76.0	77.0	78.0	79.0	80.0	[GWd/t]

Figure 5. Geometries of typical BWR fuel assemblies.

In order to prepare the pin-cell averaged macroscopic cross sections to various conditions, various combinations of core state variables should be covered by depletion and branch calculations. The case matrix (i.e. the list of depletion and branch calculations), which is used to establish the present macroscopic depletion model, is already shown in Tables 1 and 2. The pin-cell averaged macroscopic cross sections are generated according to the case matrix and they are edited and tabulated as the base and the delta cross sections described in Section 2. In order to reduce the spatial homogenization and the energy-collapsing errors, the superhomogenization (SPH) method is applied for each pin-cell averaged cross section [18].

The SPH factor also depends on the core state and the depletion history variables as the cross sections and it is very important to appropriately estimate such dependencies from the viewpoint of the calculation accuracy. Thus, the SPH factor should be independently considered for each pin-cell. However, the tabulation of the SPH factor requires an additional computational memory. Therefore, in order to reduce memory requirement, the cross sections and the SPH factor are not independently tabulated, but the SPH-corrected cross sections, which are calculated by multiplying the cross sections by the SPH factor, are directly tabulated in the present study.

The accuracy of the present macroscopic cross-section model described in Section 2 is confirmed by comparing the calculation results obtained by the tabulated and the reference cross sections. The tabulated cross sections are reconstructed by the present macroscopic cross-section model. The reference cross sections are obtained by the lattice physics calculations, in which the values of the core state variables on the benchmark calculation cases are directly used. The benchmark calculation cases, which are used for the verification of depletion calculations, are discussed in Sections 3.2.1, 3.2.2, and 3.2.3, respectively.

In the benchmark calculations, we directly treat the exposure distribution within fuel assemblies but the void fraction and the fuel temperature distributions are assumed as uniform. From the Ref. [19], the effect of the pin-by-pin fuel temperature distribution is considered to be small. Thus, the necessity of direct treatment of the pin-by-pin fuel temperature distribution would be low. On the other hand, an in-channel void distribution has a large effect on the calculation results (e.g. the k-infinity and the pin-by-pin fission rate distribution) [20]. Consideration of an in-channel void distribution is beyond the scope of the present paper, but it is an important issue to be addressed in future work.

The pin-by-pin (cell-homogenized) calculations are carried out by the SUBARU code, which is a pin-by-pin core analysis code for BWRs under development [21]. The SUBARU code utilizes the semi-analytic nodal method with the simplified P3 (SP3) theory and explicitly models each homogenized pin-cell. The k-infinities, the pin-by-pin fission rate distributions, and the pin-wise exposure distributions within the fuel assembly, which are obtained with the tabulated and the reference cross sections, are compared. The absolute value of the relative difference of k-infinities, the root-mean-square (RMS) difference of pin-by-pin fission rate distributions, and the RMS difference of the pin-wise exposure distributions are calculated as (13) (14) (15) where Δk_inf, ΔR_f,RMS, ΔEXP_RMS, R_f,i, EXP_i, and N_fuelmesh are the absolute value of the relative difference of k-infinities, the RMS difference of pin-by-pin fission rate distributions, the RMS difference of pin-wise exposure distributions, the fission rate on fuel mesh i, the exposure of fuel mesh i, and the number of fuel meshes, respectively. The superscripts “tabulated” and “reference” represent the calculations with the tabulated and the reference cross sections, respectively.

The target accuracies for the absolute value of the relative difference of k-infinities and the RMS difference of pin-by-pin fission rate distributions in the present study are as follows:

1. Absolute value of the relative difference of k-infinities is less than 0.1%dk/k.

2. RMS difference of pin-by-pin fission rate distributions is less than 0.3%.

In conventional BWR core analysis, the target accuracy of k-infinity is approximately 0.4%dk/k from the viewpoint of the prediction accuracy of the reactor shutdown margin. Similarly, the target accuracy of power distribution is generally considered as 10%. In this paper, the verification calculations are carried out in single assembly geometries without thermal-hydraulic feedback. Therefore, we set the target accuracies as described above, which are sufficiently smaller than those for general BWR core analysis.

In this paper, we focus on the absolute value of the relative difference of k-infinities and the RMS difference of pin-by-pin fission rate distributions, which are obtained by the SUBARU code with the tabulated and the reference cross sections, in order to discuss only the accuracy of the present macroscopic cross-section model. However, we also compared the calculation results of the HELIOS code and the SUBARU code. The absolute value of the relative difference of k-infinities and the RMS difference of pin-by-pin fission rate distributions, which are estimated by the HELIOS code and the SUBARU code with the reference cross sections, are less than approximately 0.02%dk/k and 0.02%, respectively, on the “BASE” depletion calculation cases, as given in Tables 1 and 2. Therefore, the SUBARU code with the reference cross sections well reproduces the calculation results obtained by the HELIOS code.

3.2. Calculation results and remarks

3.2.1. Calculation accuracies on the treatment of exposure

Since the exposure has a dominant effect on the macroscopic cross sections, the macroscopic cross sections are independently tabulated for different exposure points. The macroscopic cross sections are interpolated for exposure points, which are calculated in macroscopic depletion calculations, by utilizing the CSI as described in Section 2.2.

In order to confirm the calculation accuracies of the exposure distribution and the interpolation of macroscopic cross sections, we use the following three types of cross sections and compare the calculation results:

1. The reference cross sections. cross sections are directly obtained by the HELIOS code. The present cross sections give most accurate (reference) results.

2. The tabulated cross sections with the reference exposure distribution. cross sections are reconstructed from the present macroscopic cross-section model with the reference exposure distribution. The present cross sections will reveal the impact of errors due to the CSI for the exposure by comparing the calculation results by “the reference cross sections”.

3. The tabulated cross sections. cross sections are reconstructed from the present macroscopic cross-section model with the exposure distribution calculated in the macroscopic depletion calculation. The present cross sections will reveal the impact of errors due to the macroscopic depletion calculations, in which the exposure distribution is calculated, by comparing the calculation results by “the tabulated cross sections with the reference exposure distribution”.

The benchmark calculation cases are shown in Table 5.

Table 5. Benchmark calculation cases (exposure).

No.	Type	CR	VOI [%]	TFU [K]	TMO [K]	Depletion/branch
1	HVOI/HTFU	Out	80.0	1100.0	559.0	Depletion
2	HVOI/HTFU/HTMO/HCR	In	0.0	600.0	540.0	Depletion

In Table 5, the core state and the depletion history variables of the benchmark calculation cases are the same as those of the depletion calculation cases, which are shown in Table 2, and are used to establish the present macroscopic cross-section model. Namely, the errors due to the polynomial interpolations for the core state and the depletion history variables do not exist. The benchmark calculation case of No. 1 corresponds to the bulk boiling condition in the core top region on BWR normal operation, in which the void fraction and the fuel temperature are high and the control rod is withdrawn. The benchmark calculation case of No. 2 corresponds to the non-boiling condition in the core bottom region on BWR normal operation, in which the fuel temperature and the moderator temperature are low and the control rod is inserted. They are the limiting cases covered in the present macroscopic cross-section model. Through these benchmark calculation cases, we estimate the impact of the interpolation errors for exposure on the calculation accuracies, and discuss the treatment of exposure in the present macroscopic cross-section model.

The absolute value of the relative difference of k-infinities, the RMS difference of pin-by-pin fission rate distributions, and the RMS difference of pin-wise exposure distributions are shown in Figures 6, 7, and 8, respectively. In Figures 6–8, the calculation results obtained by the tabulated cross sections with the reference exposure distribution and those with the calculated exposure distribution are represented as “Case # (Reference exposure distribution)” and “Case # (Calculated exposure distribution)”, respectively. In Figures 6 and 7, the lines indicating the target accuracies are also shown.

Figure 6. Calculation results of k-infinity. (Calculation cases: Table 5, Target accuracy line: 0.1%dk/k).

Figure 7. Calculation results of the pin-by-pin fission rate distribution. (Calculation cases: Table 5, Target accuracy line: 0.3%).

Figure 8. Calculation results of the pin-wise exposure distribution. (Calculation cases: Table 5).

From Figures 6–8, on an 8×8 fuel assembly, the differences of k-infinities and pin-by-pin fission rate distributions obtained by the tabulated cross sections with the reference and the calculated exposure distributions are almost the same. On 9×9 and 10×10 fuel assemblies, the differences of k-infinities and pin-by-pin fission rate distributions obtained by the tabulated cross sections with the calculated exposure distribution have several peaks related to the depletion of Gd-bearing fuel pin-cells around 10 GWd/t, since they have Gd-bearing fuel pin-cells and then have almost the same accuracies as those obtained by the tabulated cross sections with the reference exposure distribution after 20 GWd/t. Common to all types of fuel assemblies, the differences of k-infinities and pin-by-pin fission rate distributions obtained by the tabulated cross sections with the reference and the calculated exposure distributions have several peaks around 0 GWd/t. Errors due to the CSI tend to appear around limits of interpolation since the CSI requires some boundary conditions on limits of interpolation (i.e. 0 and 80 GWd/t) as described in Section 2.2, and the setup of the boundary condition has an effect on the interpolation accuracy around limits of interpolation. Therefore, such interpolation errors due to the boundary condition cause several peaks of the differences of k-infinities and pin-by-pin fission rate distributions around 0 GWd/t. However, the above peak values of the differences of k-infinities and pin-by-pin fission rate distributions are less than about 0.03%dk/k and about 0.03%, respectively. Therefore, the error due to the interpolation for exposure using the CSI is small and the target macroscopic cross sections can be well reproduced.

Compared to the differences of the pin-by-pin fission rates and the exposure distributions, they show almost the same tendencies on all types of fuel assemblies. This result indicates that the errors of the pin-by-pin fission rate distribution cause the errors of the exposure distribution which in turn cause the errors of the macroscopic cross sections.

3.2.2. Calculation accuracies on the treatment of instantaneous and history effects

The macroscopic cross sections have complicated dependences on the core state and the depletion history variables. Thus, in the present macroscopic cross-section model, we approximate the macroscopic cross sections as a linear combination of the macroscopic cross section on the base condition and the correction terms for the instantaneous and the history effects. Then, we separately tabulate the cross-term effect for the instantaneous and the history variables according to the moderator temperature and the exposure-averaged moderator temperature, as described in Sections 2.3 and 2.4.

Using the above approximations, the macroscopic cross sections for the instantaneous effect are tabulated in the following two types of three-dimensional tabulations:

1. Void fraction, fuel temperature, and control rod insertion (moderator temperature: fixed to 559 K).

2. Fuel temperature, moderator temperature, and control rod insertion (void fraction: fixed to 0%).

Similarly, those for the history effect are tabulated in the following two types of three-dimensional tabulations:

1. Void, fuel temperature, and control rod histories (exposure-averaged moderator temperature: fixed to 559 K).

2. Fuel temperature, moderator temperature, and control rod histories (exposure-averaged void fraction: fixed to 0%).

In order to confirm the calculation accuracies for the dependences on the core state and the depletion history variables and evaluate the impact of the treatment of the cross-term effect, the following five types of cross sections are used and then the calculation results are compared:

1. The reference cross sections. Cross sections are directly obtained by the HELIOS code. The present cross sections give most accurate (reference) results.

2. The tabulated cross sections. Cross sections are reconstructed by the present macroscopic cross-section model with the cross terms for the instantaneous and the history effects.

3. The tabulated cross sections without cross terms for the history effect. Cross sections are reconstructed by the present macroscopic cross-section model without and with the cross terms for the instantaneous and the history effects, respectively. The present cross sections will reveal the impact of cross term for the history effect by comparing the calculation results by “the tabulated cross section”.

4. The tabulated cross sections without all cross terms. Cross sections are reconstructed by the present macroscopic cross-section model without all cross terms for the instantaneous and the history effects. The present cross section will reveal the impact of the cross term for the instantaneous effect by comparing the calculation results by “the tabulated cross sections without cross terms for history effect”.

5. The tabulated cross sections with a specific cross term. Cross sections are reconstructed by the present macroscopic cross-section model with one of the cross-term effects among the core state and the depletion history variables (e.g. the void fraction and the fuel temperature, the void fraction and the control rod insertion, or the void fraction and the control rod histories). The present cross sections will reveal the contribution of each cross term (i.e. the breakdown of cross-term effects).

In the following parts, the cross-term effects for the two independent three-dimensional cross-section tables (i.e. (a) the void fraction, the fuel temperature, and the control rod insertion, (b) the fuel temperature, the moderator temperature, and the control rod insertion) are investigated.

Cross-term effects among the void fraction, fuel temperature, control rod insertion, and their histories.

First, we verify the treatment of the cross-term effects among the void fraction, the fuel temperature, the control rod insertion, and their histories through the benchmark calculation cases shown in Table 6.

Table 6. Benchmark calculation cases (void fraction, fuel temperature, control rod insertion, and their histories).

No.	Type	CR	VOI [%]	TFU [K]	TMO [K]	Depletion/branch
3	HVOI/HTFU	Out	60.0	950.0	559.0	Depletion
4	HVOI/HTFU/HCR	In	20.0	700.0	559.0	Depletion

The benchmark calculation case of No. 3 corresponds to the bulk boiling condition on BWR normal operation, in which the void fraction and the fuel temperature are high and the control rod is withdrawn. The benchmark calculation case of No. 4 also corresponds to the bulk boiling condition; however, the void fraction and the fuel temperature are low and the spatial and energetic tilt of the neutron spectrum due to the control rod insertion is caused. Since errors due to polynomial interpolations tend to be larger at intermediate values between interpolation points, the core state and the depletion history variables of the benchmark calculation cases in Table 6 are determined as intermediate values between those of the branch and the depletion calculation cases in Tables 1 and 2, which are used to establish the present macroscopic cross-section model.

The absolute value of the relative difference of k-infinities and the RMS difference of pin-by-pin fission rate distributions are shown in Figures 9 and 10, respectively. In Figures 9 and 10, the calculation results obtained with and without cross terms are represented as “Yes” and “No”, respectively. The lines indicating the target accuracies are also shown.

Figure 9. Calculation results of k-infinity. (Calculation cases: Table 6, Target accuracy line: 0.1%dk/k).

Figure 10. Calculation results of the pin-by-pin fission rate distribution. (Calculation cases: Table 6, Target accuracy line: 0.3%).

From Figures 9 and 10, in the benchmark calculation cases of No. 3 and No. 4, the calculation results obtained by the tabulated cross sections with all cross terms achieve the target accuracies for the differences of k-infinities and pin-by-pin fission rate distributions. The differences of k-infinities and pin-by-pin fission rate distributions become larger as the exposure becomes higher for all benchmark calculation cases. On the other hand, those shown in the previous subsection are mostly constant, while the exposure becomes higher except for the effect of Gd. From these comparisons, the polynomial interpolations for the core state and the depletion history variables cause the errors of the macroscopic cross sections, and then they turn out the errors of the pin-by-pin fission rate distribution. Note that, given the later discussions in the following subsections, the errors due to the polynomial interpolations for the void fraction and the void history especially have large impacts on the reconstruction of macroscopic cross sections.

In the benchmark calculation case of No. 3 without the control rod history, the calculation accuracies obtained by the tabulated cross sections without cross terms for the history effect and those without all cross terms are worse than those obtained by the tabulated cross sections with all cross terms. The differences of k-infinities and pin-by-pin fission rate distributions obtained by the tabulated cross sections without cross terms for the history effect and those without all cross terms are shown within 0.03%dk/k and 0.005%, respectively. These comparison results indicate that the impact of the cross-term effect among the void fraction and the fuel temperature is small; thus, such cross-term effects could be negligible in the macroscopic cross-section model. On the high-exposure points, the differences of k-infinities obtained by the tabulated cross sections without cross terms for the history effect and those without all cross terms exceed the target accuracies. Thus, the cross-term effect among the void and the fuel histories is necessary to consider in the present macroscopic cross-section model.

In the benchmark calculation case of No. 4 with the control rod history, the calculation accuracies obtained by the tabulated cross sections without cross terms for the history effect and those without all cross terms become much worse than those obtained by the tabulated cross sections with all cross terms for all types of fuel assemblies. The differences of k-infinities and pin-by-pin fission rate distributions exceed the target accuracies on most of exposure points. These comparison results indicate that the cross-term effect among the control rod insertion, the control rod history, and other core state and depletion history variables (i.e. the void fraction, the fuel temperature, and their histories) is extremely large.

In order to evaluate and discuss the necessity of the treatment of the cross-term effect, the benchmark calculation case of No. 4 is calculated with the tabulated cross sections without all cross terms and those with a specific cross term. The evaluation results are shown in Table 7. In Table 7, the values are calculated by subtracting the calculation results obtained by the tabulated cross sections with a specific cross term from those obtained by the tabulated cross sections without all cross terms.

Table 7. Impact of the cross-term effect among the void fraction, fuel temperature, control rod insertion, and their histories.

		Instantaneous cross terms			History cross terms
	Exposure [GWd/t]	VOI-TFU	VOI-CR	TFU-CR	HVOI-HTFU	HVOI-HCR	HTFU-HCR
8×8 fuel assembly
Difference of k-infinity	0	−0.03	0.30	−0.02	−0.02	−0.01	−0.02
[%dk/k]	10	−0.07	0.24	−0.07	−0.09	0.93	0.02
	20	−0.10	0.22	−0.09	−0.13	1.47	0.04
	30	−0.07	0.26	−0.06	−0.11	1.84	0.11
	40	−0.05	0.28	−0.04	−0.09	2.07	0.15
	50	−0.03	0.29	−0.03	−0.09	2.22	0.18
	60	2.69	2.69	2.69	2.69	2.69	2.69
Difference of pin-by-pin	0	−0.01	0.09	−0.01	−0.01	0.00	0.00
fission rate distribution [%]	10	−0.04	0.06	−0.04	−0.03	0.51	0.01
	20	−0.08	0.04	−0.08	−0.06	0.82	0.01
	30	−0.07	0.06	−0.07	−0.05	1.16	0.05
	40	−0.05	0.09	−0.05	−0.03	1.44	0.10
	50	−0.04	0.12	−0.04	−0.01	1.65	0.14
	60	2.16	2.16	2.16	2.16	2.16	2.16
9×9 fuel assembly
Difference of k-infinity	0	−0.05	0.04	−0.03	−0.04	−0.03	−0.03
[%dk/k]	10	−0.15	−0.05	−0.13	−0.15	−0.04	−0.11
	20	−0.96	−0.81	−0.95	−1.00	0.04	−0.87
	30	−0.89	−0.73	−0.88	−0.96	0.55	−0.76
	40	−0.93	−0.75	−0.91	−1.02	0.96	−0.76
	50	1.57	1.57	1.57	1.57	1.57	1.57
	60	2.03	2.03	2.03	2.03	2.03	2.03
Difference of pin-by-pin	0	−0.05	0.00	−0.05	−0.06	0.02	−0.05
fission rate distribution [%]	10	−0.42	−0.34	−0.42	−0.43	0.29	−0.40
	20	0.29	0.31	0.29	0.30	0.73	0.33
	30	−0.28	−0.26	−0.28	−0.26	0.27	−0.23
	40	−0.43	−0.41	−0.43	−0.39	0.33	−0.35
	50	0.72	0.72	0.72	0.72	0.72	0.72
	60	0.99	0.99	0.99	0.99	0.99	0.99
10×10 fuel assembly
Difference of k-infinity	0	0.02	−0.01	0.03	0.03	−0.01	0.04
[%dk/k]	10	−0.22	−0.11	−0.20	−0.23	−0.09	−0.18
	20	−0.50	−0.36	−0.48	−0.54	−0.10	−0.43
	30	0.09	0.09	0.09	0.09	0.09	0.09
	40	0.26	0.26	0.26	0.26	0.26	0.26
	50	0.40	0.40	0.40	0.40	0.40	0.40
	60	0.53	0.53	0.53	0.53	0.53	0.53
Difference of pin-by-pin	0	−0.12	−0.04	−0.12	−0.13	0.01	−0.11
fission rate distribution [%]	10	−0.24	−0.18	−0.24	−0.27	0.06	−0.21
	20	−0.18	−0.12	−0.18	−0.20	0.26	−0.12
	30	0.51	0.51	0.51	0.51	0.51	0.51
	40	0.46	0.46	0.46	0.46	0.46	0.46
	50	0.55	0.55	0.55	0.55	0.55	0.55
	60	0.67	0.67	0.67	0.67	0.67	0.67

From Table 7, the impacts of the cross-term effect among the void fraction and the fuel temperature represented as “VOI-TFU” and that among the fuel temperature and the control rod insertion represented as “TFU-CR” are small or almost zero for all types of fuel assemblies; thus, such cross-term effects could be negligible in the present macroscopic cross-section model. On the other hand, other cross-term effects have large impacts, especially for the cross-term effect among the void and the control rod histories represented as “HVOI-HCR”. The cross-term effect among the void and the control rod histories has the largest impact and this impact becomes larger as the exposure becomes higher on all types of fuel assemblies. Therefore, the treatment of this cross-term effect is essential to accurately reproduce the target cross sections. The cross-term effect among the void fraction and the control rod insertion represented as “VOI-CR” has the second largest impact after that among the void and the control rod histories, especially on the low-exposure points, which is almost the same as that among the void and the control rod histories and the impact of this is constant for the exposure. The cross-term effect among the void and the fuel temperature histories represented as “HVOI-HTFU” and that among the fuel temperature and the control rod histories represented as “HTFU-HCR” have small impacts on the low-exposure points, which are about or less than a tenth part of that of the cross-term effect among the void and the control rod histories. However, they become larger as the exposure becomes higher and are not be negligible from the viewpoints of the target accuracies, especially on the high-exposure points

Therefore, the following cross-term effects have large impacts on the reconstruction of macroscopic cross sections and are necessary to consider in the present macroscopic cross-section model in the given priority order:

1. Void and control rod histories (HVOI-HCR).

2. Void fraction and control rod insertion (VOI-CR).

3. Fuel temperature and control rod histories (HTFU-HCR).

4. Void and fuel temperature histories (HVOI-HTFU).

The treatment of them is important in order to accurately reproduce the target cross sections and estimate the k-infinity and the pin-by-pin fission rate distribution.

Cross-term effect among the fuel temperature, moderator temperature, control rod insertion, and their histories.

Next, the treatment of the cross-term effect among the fuel temperature, the moderator temperature, the control rod insertion, and their histories is verified through the benchmark calculation cases shown in Table 8.

Table 8. Benchmark calculation cases (fuel temperature, moderator temperature, control rod insertion, and their histories).

No.	Type	CR	VOI [%]	TFU [K]	TMO [K]	Depletion/branch
5	HVOI/HTFU/HTMO	Out	0.0	950.0	550.0	Depletion
6	HVOI/HTFU/HTMO/HCR	In	0.0	700.0	550.0	Depletion

The benchmark calculation case of No. 5 corresponds to the non-boiling condition, in which the fuel temperature is high and the control rod is withdrawn. The benchmark calculation case of No. 6 also corresponds to the non-boiling condition; however, the fuel temperature is low and the spatial and energetic tilt of the neutron spectrum due to the control rod insertion is caused. The core state and the depletion history variables of the benchmark calculation cases in Table 8 are also determined as intermediate values between those of the branch and the depletion calculation cases in Tables 1 and 2, similarly to the benchmark calculation cases in Table 6.

The absolute value of the relative difference of k-infinities and the RMS difference of pin-by-pin fission rate distributions are shown in Figures 11 and 12, respectively. In Figures 11 and 12, the calculation results obtained with and without cross terms are represented as “Yes” and “No”, respectively. The lines indicating the target accuracies are also shown.

Figure 11. Calculation results of k-infinity. (Calculation cases: Table 8, Target accuracy line: 0.1%dk/k).

Figure 12. Calculation results of the pin-by-pin fission rate distribution. (Calculation cases: Table 8, Target accuracy line: 0.3%).

From Figures 11 and 12, in the benchmark calculation cases of No. 5 and No. 6, the calculation results obtained by the tabulated cross sections also achieve the target accuracies for the differences of k-infinities and pin-by-pin fission rate distributions. The differences of k-infinities and pin-by-pin fission rate distributions in Figures 11 and 12 are smaller than those in Figures 9 and 10. Unlike to the differences of k-infinities and pin-by-pin fission rate distributions of the benchmark calculation cases of No. 3 and No. 4, shown in Figures 9 and 10, those of the benchmark calculation cases of No. 5 and No. 6 are fairly constant as the exposure becomes higher for the benchmark calculation cases except for the effect of Gd. Combined with the discussions in the previous subsection, the errors due to the polynomial interpolations for the void fraction and the void history have dominant effects on the calculation accuracies of the present macroscopic cross-section model.

In the benchmark calculation case of No. 5, three types of the differences of k-infinities and pin-by-pin fission rate distributions are shown within 0.01%dk/k and 0.01%, respectively (i.e. three types of calculation results are almost the same). On the other hand, in the benchmark calculation case of No. 6, the calculation results obtained by the tabulated cross sections without cross terms for the history effect and those without all cross terms become larger than those obtained by the tabulated cross sections with all cross terms. From these comparisons, the cross-term effect among the fuel temperature, the moderator temperature, and their histories is small. Furthermore, the cross-term effects among the control rod insertion, the control rod history, and other core state and depletion history variables (i.e. the fuel temperature, the moderator temperature, and their histories) are extremely large, which shows the same tendencies as described in the previous subsection.

Similar to the previous subsection, the benchmark calculation case of No. 6 is also calculated out in several cross-term conditions in order to evaluate and discuss the necessity of the treatment of the cross-term effect. The evaluation results are shown in Table 9. The values are calculated in a similar manner to those in Table 7. Note that the cross-term effects among the fuel temperature, the control rod insertion, and their histories are omitted in Table 9 since they are already discussed in the previous subsection.

Table 9. Impact of the cross-term effect among the fuel temperature, moderator temperature, control rod insertion, and their histories.

		Instantaneous cross terms		History cross terms
	Exposure [GWd/t]	TFU-TMO	TMO-CR	HTFU-HTMO	HTMO-HCR
8×8 fuel assembly
Difference of k-infinity	0	0.03	0.03	0.03	0.03
[%dk/k]	10	0.15	0.15	0.15	0.15
	20	0.26	0.26	0.26	0.26
	30	0.33	0.33	0.33	0.33
	40	0.36	0.36	0.36	0.36
	50	0.38	0.38	0.38	0.38
	60	0.40	0.40	0.40	0.40
Difference of pin-by-pin	0	0.06	0.06	0.06	0.06
fission rate distribution [%]	10	0.10	0.10	0.10	0.10
	20	0.18	0.18	0.18	0.18
	30	0.27	0.27	0.27	0.27
	40	0.33	0.33	0.33	0.33
	50	0.37	0.37	0.37	0.37
	60	0.39	0.39	0.39	0.39
9×9 fuel assembly
Difference of k-infinity	0	0.04	0.04	0.04	0.04
[%dk/k]	10	0.03	0.03	0.03	0.03
	20	0.01	0.01	0.01	0.01
	30	0.09	0.09	0.09	0.09
	40	0.18	0.18	0.18	0.18
	50	0.29	0.29	0.29	0.29
	60	0.38	0.38	0.38	0.38
Difference of pin-by-pin	0	0.02	0.02	0.02	0.02
fission rate distribution [%]	10	0.06	0.06	0.06	0.06
	20	0.14	0.14	0.14	0.14
	30	0.09	0.09	0.09	0.09
	40	0.12	0.12	0.12	0.12
	50	0.16	0.16	0.16	0.16
	60	0.24	0.24	0.24	0.24
10×10 fuel assembly
Difference of k-infinity	0	0.04	0.04	0.04	0.04
[%dk/k]	10	0.04	0.04	0.04	0.04
	20	0.03	0.03	0.03	0.03
	30	0.01	0.01	0.01	0.01
	40	0.05	0.05	0.05	0.05
	50	0.09	0.09	0.09	0.09
	60	0.13	0.13	0.13	0.13
Difference of pin-by-pin	0	0.02	0.02	0.02	0.02
fission rate distribution [%]	10	0.05	0.05	0.05	0.05
	20	0.10	0.10	0.10	0.10
	30	0.10	0.10	0.10	0.10
	40	0.13	0.13	0.13	0.13
	50	0.15	0.15	0.15	0.15
	60	0.19	0.19	0.19	0.19

From Table 9, the impacts of the cross-term effect among the fuel temperature and the moderator temperature represented as “TFU-TMO” and that among the fuel temperature and the moderator temperature histories represented as “HTFU-HTMO” are small or almost zero for all types of fuel assemblies. Thus, similar to that among the void fraction and the fuel temperature and that among the fuel temperature and the control rod insertion in Table 7, such cross-term effects could be negligible in the present macroscopic cross-section model. The cross-term effect among the moderator temperature and the control rod histories represented as “HTMO-HCR” has a large impact and this impact becomes larger as the exposure becomes higher for all types of fuel assemblies. The impact of this cross-term effect is larger than that of the cross-term effect among the fuel temperature and the control rod histories in Table 7. Even if the cross-term effect among the moderator temperature and the control rod insertion represented as “TMO-CR” is neglected in the present macroscopic cross-section model, the calculation results would achieve the target accuracies. However, the impact of the cross-term effect among the moderator temperature and the control rod insertion for the k-infinity is about a third part of the target accuracy. Therefore, it is desirable to include the cross-term effect among the moderator temperature and the control rod insertion in the present macroscopic cross-section model.

Combined with the discussions in the previous subsection, the impacts of the cross-term effect among the control rod history (or the void history) and other depletion history variables are generally large and the consideration of such cross-term effects is crucial. The following cross-term effects are necessary to consider in the present macroscopic cross-section model in the given priority order:

1. Void and control rod histories (HVOI-HCR).

2. Void fraction and control rod insertion (VOI-CR).

3. Moderator temperature and control rod histories (HTMO-HCR).

4. Fuel temperature and control rod histories (HTFU-HCR).

5. Void and fuel temperature histories (HVOI-HTFU).

6. Moderator and control rod insertion (TMO-CR).

Their treatment is important in order to accurately reproduce the target cross sections and estimate the k-infinity and the pin-by-pin fission rate distribution.

4. Conclusions

In the present paper, we have developed a macroscopic cross-section model for BWR pin-by-pin core analysis, in which the macroscopic cross sections are basically approximated by a linear combination of the macroscopic cross section on the base condition and the correction terms for the instantaneous and the history effects.

For the instantaneous effect, in order to capture the cross-term effect among the core state variables, several branch calculations, in which some core state variables are simultaneously changed from those on the base condition, are carried out and are used to calculate the variations of cross sections. For the history effect, the unified approach using the SH, which has been successfully used in the conventional advanced nodal method and the pin-by-pin core calculation for PWRs, would be difficult to use since there is a large variation of the neutron spectrum due to the void fraction distribution and the control rod insertion within BWR fuel assemblies. Therefore, we use the exposure-averaged core state variables as the history indices. In order to capture the cross-term effect among the depletion history variables, several depletion calculations, in which some depletion history variables are simultaneously changed from those on the base condition, are carried out and are used to calculate the variations of cross sections in combination with several branch calculations prepared for the instantaneous effect.

Verification calculations are carried out for typical BWR fuel assemblies. The pin-by-pin fine mesh calculations are carried out by the SUBARU code, which is a pin-by-pin core analysis code for BWRs. Calculation results obtained by the reference and the tabulated cross sections are compared. The reference cross sections are obtained by the lattice physics calculations, in which core state variables are directly specified. The tabulated cross sections are reconstructed through the present macroscopic cross-section model. In order to verify the present macroscopic cross-section model, several benchmark calculations are carried out from the viewpoints of the treatment of exposure and the cross-term effect for the instantaneous and the history effects.

First, in order to confirm the calculation accuracies for the treatment of exposure in the present macroscopic depletion model, several benchmark calculations, in which the polynomial interpolations for the core state and the depletion history variables do not have any influence on calculation accuracies, are carried out. From the calculation results, though the differences of k-infinities and pin-by-pin fission rate distributions have several peaks related to the depletion of Gd-bearing fuel pin-cells, the differences are sufficiently small from the viewpoints of the target accuracies. Therefore, the interpolation for exposure using the CSI in the present macroscopic cross-section model can well reproduce the target cross sections.

Then, in order to confirm the validity of the treatment of the instantaneous and the history effects, several benchmark calculations are carried out. The calculation results obtained by the tabulated cross sections, which are reproduced with or without the cross-term effect among the core state and the depletion history variables, are compared. The calculation results obtained by the tabulated cross sections, which consider the cross-term effect among the core state and the depletion history variables, achieve the target accuracies on the differences of k-infinities and pin-by-pin fission rate distributions. Therefore, the present macroscopic cross-section model well reproduces the target cross sections. The impacts of some cross-term effects (e.g. the cross-term effect among the void fraction and the fuel temperature) are small or almost zero; thus, such cross-term effects could be negligible in the present macroscopic cross-section model. However, the cross-term effects among the control rod history (or the void history) and other depletion history variables have large impacts; thus, the consideration of such cross-term effects is crucial in order to accurately reproduce the target cross sections and estimate the k-infinity and the pin-by-pin fission rate distribution.

Consequently, the present cross-section model for BWR pin-by-pin core analysis can be a candidate in practical applications on normal operation.