Evaluation of Pin-cell Homogenization Techniques for PWR Pin-by-pin

calculation

Bin Zhanga, Hongchun Wua, Yunzhao Lia, Liangzhi Caoa, Wei Shenab

a School of Nuclear Science and Technology

Xi’an Jiaotong University

b Canadian Nuclear Safety Commission

jsszwtzb@stu.xjtu.edu.cn, hongchun@xjtu.edu.cn, yunzhao@mail.xjtu.edu.cn,

caolz@mail.xjtu.edu.cn, wei.shen@canada.ca

27 Pages

9 Figures

4 Tables

Corresponding Author: Yunzhao Li

School of Nuclear Science and Technology

Xi’an Jiaotong University

28 West Xianning Road

Xi’an, Shaanxi 710049, China

Email: yunzhao@mail.xjtu.edu.cn

Tel: +86 29 8266 8692

Fax: +86 29 8266 7802

ABSTRACT

In general, the spatial homogenization, energy-group condensation and angular approximation are

all included in the homogenization process. For the traditional Pressurized Water Reactor (PWR)

two-step calculation, the assembly homogenization with assembly discontinuity factors (ADF) plus

2-group neutron-diffusion calculation have been proved to be a very efficient combination. Howev-

er, it becomes different and unsettled for the pin-by-pin calculation. Thus, this paper evaluates

pin-cell homogenization techniques by comparing with the two-dimensional one-step whole-core

transport calculation. For the homogenization, both the Generalized Equivalence Theory (GET) and

the Super-homogenization (SPH) Methods are studied. Considering the spectrum interference effect

between different types of fuel pin-cells, both 2- and 7-group structures are condensed from the

69-group WIMS-D4 library structure. For practical reactor-core applications, the low-order angular

approximations, including the diffusion and the SP3 methods, are compared with each other to de-

termine which one is accurate enough for the PWR pin-by-pin calculation. Numerical results have

demonstrated that both of the GET and the SPH methods work effectively in pin-cell homogeniza-

tion. In consideration of the spectrum-interference effect, 7-group structure is sufficient for the

pin-by-pin calculation. Compared with the diffusion method, the SP3 method can decrease the errors

dramatically.

Key Words: Pin-cell homogenization, energy-group condensation, angular approximation

I. INTRODUCTION

For Pressurized Water Reactor (PWR) simulation, the computational cost of one-step calculation

with fully detailed description is too expensive using either stochastic or deterministic methods even

with the currently most advanced computing powers. Therefore, approximations in spatial, neutron

energy and angular spaces have been developed to provide efficient solutions with an acceptable

accuracy. With the advantages of the small storage requirement and high computing speed, the tra-

ditional two-step calculation scheme has successfully been applied to PWR analysis.

The traditional two-step calculation scheme contains the 2D lattice neutron-transport calculation

and the 3D whole-core neutron- diffusion calculation. The 2D lattice neutron-transport calculations

are carried out for each type of assemblies with the reflective boundary condition to provide assem-

bly-homogenized cross-sections (including diffusion coefficients, and discontinuity factors) and

pin-power form functions. The whole-core diffusion solver determines the nodal power shape and

then reconstructs the 3D pin power distribution.

However, there has been a concern that the traditional two-step calculation scheme developed for

the uranium-fueled PWRs does not perform satisfactorily for the cores with strongly heterogeneous

fuel loading such as the Mixed Oxide (MOX)-fueled core [1]. Thus, an alternative scheme with more

detailed core modeling named pin-by-pin calculation becomes attractive in recent years [2, 3]. Dif-

ferent from the traditional two-step calculation, the heterogeneous structure in pin-cells (i.e. pellet,

clad and moderator) is homogenized into a node in the pin-by-pin calculation scheme. The

whole-core pin-by-pin simulator eliminates the pin-power reconstruction and is expected to provide

a more accurate pin-power distribution with a reasonable cost.

In general, the spatial homogenization, energy-group condensation and angular approximation are

all included in the homogenization theory. All of them are the error sources of the homogenization.

For the traditional two-step calculation, the assembly homogenization with assembly discontinuity

factors (ADF) plus the 2-group neutron-diffusion method have been proved to be a very efficient

combination in reducing the errors of homogenization for the uranium-fueled PWRs. However, the

pin homogenization with the Pin Discontinuity Factors (PDF) plus the 2-group neutron-diffusion

method just like the traditional two-step calculation may be unsuitable for the pin-by-pin calculation

[4, 5]. When pin-cell homogenization is applied in a strongly heterogeneous core, more than two en-

ergy groups are necessary to adequately capture the spectrum effect. What’s more, several research-

ers have indicated that higher-order approximations to the transport equation, such as SP3 [6-7], are

necessary for the pin-by-pin calculation. What’s more, in the process of the homogenization, it is

assumed that the assembly-homogenized few-group constants are functions of the assembly type

without considering the effect of the environment. It neglects the assembly neighboring effect and

hence introduces additional error. The most rigorous approach to obtain the homogenization con-

stants is to carry out the lattice calculation with the exact boundary condition [8-9].

Because of these approximations, the Flux-Volume Weighted (FVW) cross-section cannot recover

the heterogeneous solution even with the exactly neutron flux. Then the homogenization methods

are developed to deal with these problems. With a determined boundary condition, if the reference

solution is known, both the Generalized Equivalence Theory (GET) and the Super-homogenization

(SPH) Method can eliminate the errors aroused by the homogenization completely [10].

In this paper, the pin-cell homogenization techniques were evaluated via comparing with the

two-dimensional whole-core transport calculation. Considering the spatial homogenization, the

spectrum interference and the low-order angular approximations, the GET and SPH method togeth-

er with either 2- and 7-group structures, diffusion and SP3 methods were all analyzed. Note that the

Exponential Function Expansion Nodal (EFEN) method was employed as the low-order method for

both the homogeneous diffusion and SP3 core calculations [11].

The rest of this paper is organized as following. Section.Ⅱ describes the implementations of the

SPH and the GET for pin-cell homogenization for diffusion and SP3 calculations, followed by a de-

tailed discussion on the spatial homogenization, the energy-group condensation, and the angular

approximation. In Section Ⅲ, numerical results are listed and analyzed to assess the error sources

during pin-cell homogenization. Finally, Section Ⅳ summarizes the paper.

II. THEORATICAL MODELS

This section provides an introduction of the SPH and GET methods aiming at both P1 and SP3 for

pin-cell homogenization. Both of the methods are based on well-established Equivalence Theory to

generate appropriate group constants. It should be noted that the derivation of the following equa-

tions is not energy-group-structure dependent.

II.A. Homogenization Methods

II.A.1. The Super-Homogenization Method

The SPH method developed by Hebert in 1990s [12-14] is a homogenization technique to reduce the

errors brought in by the homogenization. In 2000, it is successfully applied to the pin-by-pin core

analysis by A. Yamamoto et al. [15-16]. Its basic idea is to adjust the homogeneous cross-sections used

in low-order calculation to reproduce the integral reaction rate inside every node of the heterogene-

ous calculation via a set of correction factors called SPH factors. For pin-by-pin calculation, a

pin-cell is regarded as a node.

The SPH factor make the homogenized flux over a macro region and coarse energy group redefined

as the flowing equation:

1hom ref

i,g i,g

i,g

(1)

where, i,g is the SPH factor, the superscript ref and hom of the average flux represent the het-

erogeneous and homogeneous calculations and the subscript i and g represent the homogenized

node region i and the coarse group g.

It is an iterative calculation to determine the SPH factors [17] in which the node homogenization

calculation method used has to be exactly the same with the down-stream calculation. What’s more,

which kind of equivalent macroscopic cross-section should be multiplied by a SPH factor depends

on the type of the homogeneous calculation methods [18].

1. When the EFEN diffusion method serves as the homogeneous calculation method, the neutron

balance determines that the SPH correction is applied as follows:

' '

' ' '

' '' '

2

, ,1 1

g g g g g

g g g a g fg g s g gg gg g eff g g

r rr rD

k

(2)

Then the absorption cross-section, the fission and scatter cross-section together with the diffu-

sion coefficient should be multiplied by the SPH factor.

2. When the EFEN SP3 method serves as the homogeneous calculation method, the SPH factor of

the 2nd flux are assumed to be equal to that of the 1st flux. So the neutron balance determines

that the SPH correction is applied as follows:

0 2 0 2 2

2

, ,

2 2 0

2

, ,

2 22

27 2

35 5

g g g g g

g g g r g g g r g

g g g

g g g

g g g t g g r g g

g g g

r r r r rD S

r r rD S

(3)

where,

' '

' ' '

' '' '

'

0 0

,1 1

g g g

g fg g s g gg geff g g

g g

r rS

k

(4)

Then the removal cross-section, the fission and scatter cross-section together with the diffusion

coefficient would be multiplied by the SPH factors of the 1st flux.

II.A.2. The Generalized Equivalence Theory

The generalized equivalence theory has been developed in 1980s and successfully applied to the

PWR core analysis. In GET, a new interface relationship between the nodes was introduced to pre-

serve the integral reaction rate within each node and the leakage rate on each interface. The hetero-

geneous surface flux would be continuous via the discontinuity factors which are the original

equivalence factors.

The discontinuity factor is defined as the ratio of the heterogeneous over the homogeneous surface

fluxes:

, ,

, ,

, ,

ref

s i g

s i g hom

s i g

f

(5)

where, subscript s stands for the surface.

For single assembly calculation with a specific boundary conditions (zero current), the heterogene-

ous information including ,

s,ref

i g ,,

s,ref

i gJ and ,

ref

i g can be provided, while the homogeneous infor-

mation ,

s,hom

i g is supposed to be obtained from the homogeneous neutron diffusion calculation. To

get rid of the non-linear relationship between the homogeneous surface flux and the PDF, a proce-

dure is proposed to estimate the homogeneous surface flux accurately for EFEN P1 or SP3 method

just using the heterogeneous information.

I. In the diffusion method, the partial current response relation based on the EFEN method is

showed as following:

'

, , , ,+homhom,out hom hom,in hom hom hom

i,gx i,g x i,g x ,i,g x i,g x i,g i,gJ J S

(6)

, , , , , , , ,'

,

, ,

hom hom hom hom

y i g y i g z i g z i ghom hom

i,g i g

i y i z

J J J JS S

h h

(7)

where, subscript x, y, z stand for the coordinates, h is the nodal dimension, ,

hom

x i,g , ,

hom

x i,g , ,

hom

x i,g

are determined by the cross-sections and the nodal dimensions, ,

hom

i gS refers to the sum of the

outer scattering source and the fission source.

According to the Fick’s low, the relationship between the partial current, the net current and the

surface flux is as follows:

, , ,

, , ,

, ,

, , ,

2s hom hom out hom in

x i,g x i,g x i,g

hom hom out hom in

x i,g x i,g x i,g

J J

J J J

(8)

Substituting Eq. (6) into Eq. (8), the homogeneous nodal surface flux can be expressed as the

following:

, '

, , , , ,

,

21 2 2

1

homs hom hom hom hom hom hom

i,gx i,g x i,g x i,g x i,g x i,g i,ghom

x i,g

J S

(9)

Since the volumetric flux and the surface net current should be preserved during the homoge-

nization process, the homogeneous quantities can be replaced by the corresponding heteroge-

neous ones.

, '

, , , , ,

,

21 2 2

1

refs hom hom ref hom hom ref

i,gx i,g x i,g x i,g x i,g x i,g i,ghom

x i,g

J S

(10)

II. For SP3 method, different from diffusion method, it contains a 2nd order flux moment. Through

the same derivation as that in the diffusion method, the homogeneous nodal surface flux can be

obtained as following:

0, 0 0, 0 0, 2, 0 '1

, , , , , , , ,0

,

2, 2 2, 2 2,

, , , , , ,2

,

21 2 2 2

1

21 2 2

1

sur,hom ,hom ref ,hom ref ref ,hom ,ref

x i,g x i,g x i,g x i,g i g i g x i,g i g,hom

x i,g

sur,hom ,hom ref ,hom ref

x i,g x i,g x i,g x i,g i g x i,hom

x i,g

J S

J

2 '2

,

,hom ,ref

,g i gS

(11)

The single assembly calculation is usually done by a high order angular approximation method

such as MOC or Sn which can provide the heterogeneous value of 0,

,

ref

i g and 0,

,

ref

x i,gJ but not

the 2nd order flux moment. Then approaches would be required to obtain the 2nd order flux

moment.

This work just simply solves the second equation of the SP3 equations as in Eq. (12) by con-

structing the source term which is the leakage of the scalar flux.

2

1 2 2

27

35trD S

(12)

where

2

1 0

2

5S D (13)

This method can determine the 2nd order flux moment via utilizing the heterogeneous transport

solutions. It has been applied successfully in the mesh-centered finite difference method by

Kozlowski et al. [19-20]. However, Eq. (13) poses the problem of computing the divergence of a

current at every point in the domain. In the pin-cell homogenization, only the pin cell surface

currents are needed, then the S is to be calculated only on a pin cell basis, divergence theorem

can be used to calculate S using surface integrals. The 2nd order flux moment generated in this

manner is not the exact solution of the heterogeneous transport problem, but rather only ap-

proximate value. However, even though these are approximations to the heterogeneous problem,

they do provide equivalence between the heterogeneous transport and homogeneous SP3 meth-

ods for scalar flux and surface current necessary to satisfy GET through discontinuity factors.

This method makes use of the lowest order component of the reference neutron current from

the transport calculation. Yu et al. [4] and Yamamoto [5] propose the methods to consider the

second order component of the neutron current which are more correct methods. What’s more,

a further work about a unified method of calculation discontinuity factors for diffusion, SP3 and

transport calculation was done by Yu and Chao [21]. These different methods are alternatives to

be applied for the calculation of the discontinuity factors for SP3 method. They are expected to

provide numerical results with similar accuracy for pin-cell homogenization.

However, the PDF for the 2nd order flux still cannot be obtained even with the heterogeneous

and homogeneous surface flux known. Different from 0 , the value of 2 is small or even

negative. Very large or even negative PDF would be obtained if the definition of the 2nd order

flux has the same form as that of the scalar flux. Unfortunately, negative PDFs may cause the

divergence of the EFEN SP3 solution iteration. To avoid that from happening and make the

methodology more robust, the definition of PDF for the 2nd order flux is modified:

2, , 2, ,2

s, ,

2, , 2, ,

s,het

i g i g

i g s,hom

i g i g

f

(14)

In the work of Yu et. al. [4], the PDF for the 2nd order flux is introduced for the function 0, ,

s

i g

defined as 0, , 2, ,2s s

i g i g , which means the extra term is equal to half of the scalar flux:

2, , 0, ,

1

2i g i g

(15)

If Eq. (15) is adopted in the 2nd order PDF, the corresponding interface boundary condition in

the SP3 calculation would be changed as Eq. (16). In 2014,

0 0

, , 0, , , 1, 0, 1,

2 2

, , 0, , 2, , , 1, 0, 1, 2, 1,2 2

u u

u i g i g u i g i g

u u u u

u i g i g i g u i g i g i g

f f

f f

(16)

It is important to note that the PDF is not exactly correct on the assembly outer interface be-

cause of the reflective boundary condition which is different from the ambient core conditions.

It is the PDF ratio shown in Eq. (17), not the PDF, plays an important role in modifying the

calculation, so there is an extra freedom for the PDF ratio. The extra freedom will be cancelled

on the assembly inner interface. When it comes to the assembly outer interface, it is not the

case. This is an intrinsic feature of PDF that has important implications for multi-assembly

problems where discontinuities between assemblies are a significant source of error. The PDF

is 1.0 on the reflective interface, even though the value of PDF itself is not 1.0.

0

0, , , 1,

0

0, 1

1

1, , ,

hom ref

u ,i,g u ,i ,g

hom ref

u ,i ,g u+,i,

u

i g u i g

u

i g gu i g

f

f

(17)

II.B. Spatial homogenization

For the pin-cell homogenization, the spatial homogenization effect is one of the main error sources.

Different from the coarse mesh nodes, the pin-cells are optically thin and their group constants are

more environment-dependent. The surroundings of each pin-cell will strongly affect the target pin,

especially when the target pin is adjacent to special pin-cells like strong absorbers. Then the cell

heterogeneous calculation in a larger geometry (an assembly) than a single pin with reflective

boundary condition should be carried out to do the spatial homogenization.

II.C. Energy-group condensation

The second source of the error is the group collapsing effect. It occurs when there is a significant

difference between the spectrum in the single lattice calculation and the spectrum in the actual core.

For uranium-fueled PWRs, the interference effect between the coarse-mesh-homogenized nodes is

small and the 2-group structure is accurate enough for the core simulation. However, for the strong-

ly heterogeneous or Mixed Oxide (MOX) fueled cores, the spectrum interference would be very

strong. When the pin-by-pin calculation scheme is applied, more than two energy groups are neces-

sary to adequately capture this important physical effect.

II.D. Angular approximation

The angular approximation is another main error source. Different from the coarse mesh nodes in

the traditional two-step calculation, the heterogeneity of the pin-cell layout in the pin-by-pin calcu-

lation is much stronger. The angular flux even the scalar flux changes dramatically, especially where

there are control rods or burnable poison rods. The low-order angular approximated diffusion

method is not good at dealing with the large gradient of the neutron flux. It is because that the diffu-

sion assumptions are inapplicable for the locations where strong neutron sources or strong absorbers

appear. Thus, higher-order approximation to the transport equation is necessary for the pin-by-pin

calculation.

III. NUMERICAL RESULTS

All the reference heterogeneous calculations were carried out by an in-house developed PWR lattice

code Bamboo-Lattice, which uses a two-dimensional MOC transport solver [22].

III.A. Homogenization Methods

In order to verify the homogenization methods applied in the pin-by-pin calculation, the modified

KAIST fuel assembly problems were calculated [23] to test the performance of SPH factor and PDF.

In this benchmark suit, there are 6 UOX and 3 MOX assembly problems with different number of

burnable absorbers and control rods. The UOX and MOX assembly geometries are shown in Fig. 1.

a) UOX fuel assembly

b) MOX fuel assembly

Fig.1 Geometries of the KAIST benchmark assemblies [23]

Eigenvalues and pin powers of the 2G/7G diffusion/SP3 calculation of different single-assembly

types with SPH/ADF-based pin-cell-homogenized cross-sections are compared with the reference

results in Table 1. It can be found that, given the reference solution, either SPH or PDF method can

reproduce reference results even with a few-group low-order angular approximation such as the 2G

diffusion method. This indicates the effectiveness of the SPH and the PDF methods.

Assembly Type Reference

Eigenvalue

Eigenvalue Difference (pcm)

Pin Power %RMS Error

Diffusion Method SP3 Method

2G 7G 2G 7G

SPH PDF SPH PDF SPH PDF SPH PDF

UOX-1 1.10014 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

UOX-1 CR 0.67819 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

UOX-1 BA16 0.87656 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

UOX-2 1.25161 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

UOX-2 CR 0.82172 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

UOX-2 BA16 1.04488 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

MOX-1 1.17176 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

MOX-1 CR 0.90381 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

MOX-1 BA16 1.08811 0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

0

0.00

Table 1 Single-Assembly Results with SPH/PDF-Based Pin-cell-Homogenized Cross-Sections

III.B. Spatial homogenization effect

In order to analysis the error caused by the spatial homogenization, the eigenvalues and pin powers

of the full-energy-group MOC calculations of different single-assembly types with FVW-based

pin-cell-homogenized cross-sections are compared with the reference results in Table 2. The results

show that the errors on eigenvalues and pin powers are very large, and the errors increase with the

enhancement of the heterogeneity.

Assembly

Type UOX-1

UOX-1

CR

UOX-1

BA16 UOX-2

UOX-2

CR

UOX-2

BA16 MOX-1

MOX-1

CR

MOX-1

BA16

Reference

kinf(pcm) 1.10014 0.67819 0.87656 1.25161 0.82172 1.04488 1.17176 0.90381 1.08811

kinf Er-

ror(pcm) 42 -3219 -2005 49 -3247 -1795 -42 -1413 -500

Pin-Power 0.21% 1.45% 1.75% 0.26% 1.30% 1.53% 0.34% 0.43% 0.77%

RMS Error

Table 2 Single-Assembly Results with Pin-cell-Homogenized Assembly Results without PDF/SPH

III.C. Energy Group condensation effect

In order to analysis the impact of different energy-group structures on the results alone with the

SPH or PDF method, four color-set problems with reflective boundary condition have been calcu-

lated. The configurations of the problems are shown in Fig. 2.

Fig.2 KAIST color-set problems [21]

The spectrum interference between the assemblies of the four problems is very strong. The neutron

flux profile between the assemblies in the “Unposisoned” case is shown in Fig. 3. The values of the

particular pins are obtained from the locations surrounded by the blue line showed in Fig. 3a. We

can find that the values of normalized neutron flux of the pins which are close to the interface

boundary change dramatically for the thermal groups, even the flux of the pins which are far away

from the interface boundary varies intensely when it comes to the water rods. So the spectrum in-

terference effect between the UO2 and MOX is very strong for the 5-7 groups.

(a) the particular pins surrounded by blue line in the “Unposisoned” case

(b) the normalized neutron flux;

Fig.3 The normalization neutron flux distribution

Table 3 compares the spectrum interference effect for both 2- and 7-group calculations. For SP3

calculation, the increase of the energy groups improves the solution accuracy for these four prob-

lems. With 7-group structure, the eigenvalue error and the pin-power RMS error of the SP3 calcula-

tion are almost less than 100 pcm and 1.0% respectively, while the error of the 2-group structure is

much larger. However, for the diffusion calculations, sometimes the errors of the 7-group structure

are larger than those of the 2-group structure, especially when the calculation was carried out with

the SPH factors. This is mainly because of the error cancellation between the group condensation

and the angular approximation effect. In order to better understand this effect, for the “UO2 Poi-

soned” case, the solution of the 7-group SP3 calculation with SPH factor is regarded as the reference

result. The pin-power error distribution of the 2-group SP3 calculation with SPH factor represents

the error brought in by the energy group condensation, and that of the 7-group diffusion calculation

with SPH factor represents the error brought in by the angular approximation are shown on Fig. 4.

We can find that at the particular locations, especially for the left bottom assembly, the errors

brought in by the energy group condensation and the angular approximation have a different sign

and they will cancel with each other in the 2-group diffusion calculation with SPH factor.

The SPH factor is applied to all the pin-cell homogenized cross-sections to preserve the cell reaction

rate. Since reaction rate is the product of cross-section and flux, the SPH factor can be regarded as a

uniform PDF applied to flux. The equivalent PDF is surface independent because that there is only

one SPH factor per cell (per energy group). Therefore the calculation using PDF is expected to be

more accurate than the calculation using SPH factor. While the results of 7-group SP3 calculation

using SPH factor is almost the same as that of using PDF shown in table 3. The similar result was

also appeared in the work of Yu [21]. The main reason is that the pin cells in the analyzed problems

are all symmetric cells. For the 7-group diffusion calculation, however, PDF appears better than

SPH. Our explanation is that the diffusion calculation only has one SPH factor alone, while the SP3

calculation has two “equivalent” SPH factors. For these problems, the calculation with two parame-

ters is sufficient, while one is insufficient.

Color-set

problem

Reference

Eigenvalue

Eigenvalue Difference (pcm)

Pin Power %RMS Error

Diffusion Method SP3 Method

SPH PDF SPH PDF

2G 7G 2G 7G 2G 7G 2G 7G

Unpoisoned 1.18211 123

0.78%

-18

1.38%

100

1.33%

-29

0.88%

121

0.86%

13

0.79%

105

1.66%

1

0.86%

MOX

Poisoned 1.10910

103

1.66%

-167

2.60%

159

1.92%

-130

1.66%

142

1.37%

-50

1.28%

188

2.29%

-16

0.96%

UO2

Poisoned 1.05279

38

0.85%

-237

2.01%

84

1.00%

-167

1.26%

151

0.97%

24

0.37%

160

1.66%

73

0.92%

Heavily

Poisoned 0.98422

-13

1.93%

-209

2.65%

27

1.73%

-186

2.02%

53

1.46%

-22

0.94%

25

1.76%

-21

0.89%

Table 3 Results of color-set problems

(a) the 7G diffusion calculation with SPH factor

(b) the 2G SP3 calculation with SPH factor;

Fig.4 The pin-power error distributions

III.D. Angular approximation effect

The diffusion method has been proved to be a very efficient tool for the coarse-mesh nodal calcula-

tion. But it cannot handle the problems with a large gradient of the neutron flux very well. Due to

the heterogeneity of the pin-by-pin calculation, several researchers have indicated that higher-order

approximations to the transport equation are necessary for the pin-by-pin calculation. The results

shown in Table 2 and Table 3 indicate that the application of the SP3 method can improve the accu-

racy of the calculation compared with the diffusion theory, it is because the SP3 transport approxi-

mation is higher than the low-order diffusion method. For practical core analysis applications, the

SP3 method has the advantage of computation cost compared with the low-order discrete ordinate

calculation.

III.E. Two-dimensional PWR whole core problem

In order to evaluate all the combinations in a heterogeneous core problem, the modified KAIST core

calculation was performed. The core showed in Fig. 5 consists of 2.0% UO2 (UOX-1), 3.3% UO2

(UOX-2) and MOX (MOX-1) assemblies.

Fig.5 The configuration of the 1/4 PWR [23]

Several schemes were carried out to test the performances of the homogenization effects in the

two-dimensional 1/4 PWR core:

(1) The reference solution was provided by a 2D whole-core one-step transport calculation.

(2) The traditional two-step calculation scheme: Single-assembly calculations with the reflective

boundary condition determine the 2-group assembly-homogenized parameters including the

ADF, followed by the coarse-mesh core-diffusion calculation [24].

(3) Pin-by-pin schemes: Single-assembly calculations with the reflective boundary condition de-

termine the pin-cell-homogenized 2-/7-group parameters including the PDF and the SPH fac-

tors, followed by the pin-by-pin core-diffusion/SP3 calculations.

Case Spatial Energy

Groups Angular keff

keff Error

(pcm)

Pin-Power%

RMS Error

Max Relative Pin-Power

Error% (Peak

Pin-Power Relative Er-

ror%)

Reference - 69 MOC 0.97948 - - -

Two-step ADF 2 Diffusion 0.98167 219 2.11 9.98(1.16)

Pin-by-pin

PDF 2

Diffusion

0.98064 116 1.27 3.83(0.02)

SPH 0.98075 127 1.16 3.87(0.85)

PDF 7

0.97679 -269 1.98 6.29(2.68)

SPH 0.97723 -225 2.28 8.20(3.72)

PDF 2

SP3

0.98133 185 1.51 5.13(1.03)

SPH 0.98166 218 1.27 5.60(0.62)

PDF 7

0.97902 -46 0.59 1.64(0.40)

SPH 0.97914 -34 0.70 3.29(0.04)

Table 4 The eigenvalue results of the 2D whole core problem

As in Table 4, better results could be obtained from the pin-by-pin calculations compared with the

traditional two-step calculation except the 7-group diffusion calculation due to the error cancellation

explained above. The reference pin-power distribution is showed in Fig. 6 and the relative recon-

structed pin-power error of the traditional two-step calculation is showed in Fig. 7. The maximum

error, about 10.0%, appeared in Fig.7 locates in the corner of the MOX-1 assembly adjacent to the

UOX-1 assembly. The pin-power error distributions of the diffusion pin-by-pin calculations are in

Fig. 8, while the SP3 ones are in Fig. 9. It can be found that PDF and SPH schemes can provide

similar accuracy via different approaches. Also, SP3 is more accurate than diffusion for pin-by-pin

calculation. 7-group structure is superior to the 2-group one mainly due to the consistency require-

ment between space, angle and energy.

Fig.6 The reference pin-power distribution

Fig.7 Relative pin-power error of two-step calculation

(a) 2G diffusion with PDF

(c) 7G diffusion with PDF

(b) 2G diffusion with SPH factor

(d) 7G diffusion with SPH factor

Fig.8 Relative pin-power error of the diffusion pin-by-pin calculation

(a) 2G SP3 with PDF

(c) 7G SP3 with PDF

(b) 2G SP3 with SPH factor

(d) 7G SP3 with SPH factor

Fig.9 Relative pin-power error of the SP3 pin-by-pin calculation,

IV. CONCLUSIONS

After analyzing the sources of errors in PWR pin-cell homogenization, the spatial homogenization,

the spectrum interference effect and angular approximation are all different in the pin-by-pin calcu-

lation compared with the traditional two-step calculation.

The implementations of the SPH method and the GET for pin-cell homogenization are described

and the derivation of the formulation to get the SPH factor and the PDF is proposed for the EFEN

method. In consideration of the spectrum interference effect, the 2-group structure is severely insuf-

ficient, while coarse-group (7-group) has been proved to be sufficient enough. For the low-order

angular approximations, the SP3 method will decrease the errors dramatically in the pin-by-pin cal-

culation. The pin-by-pin calculation scheme is more accurate than the traditional two-step calcula-

tion scheme for the strongly heterogeneous fuel loading or Mixed Oxide (MOX) fueled cores.

By looking at the relative error distributions in Fig. 8 and Fig. 9, it can be found that the relatively

large errors locate at the interface of two assemblies even they are smaller than the ones in the leg-

acy two-step scheme. The main reason is fact that the reflective boundary condition before the

pin-cell homogenization is different from the active boundary condition in the core. It is usually

called the environment effect. Thus, more investigation and improvements would be required to re-

duce or even eliminate it.

ACKNOWLEDGMENTS

This research was carried out under the financial support by the National Natural Science Founda-

tion of China (No. 11305123). The first four authors would like to thank Yung-An Chao for the

helpful discussions.

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