Effect of including corner point fluxes on the pin power reconstruction using nodal point flux scheme

Annals of Nuclear Energy 69 (2014) 25–36

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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene

Effect of including corner point fluxes on the pin power reconstruction using nodal point flux scheme F. Khoshahval ⇑, A. Zolfaghari, H. Minuchehr Engineering Department, Shahid Beheshti University, G.C, P.O. Box 1983963113, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 9 October 2013 Received in revised form 29 December 2013 Accepted 8 January 2014 Available online 19 February 2014 Keywords: Pin power reconstruction Intra nodal NEM Corner point

a b s t r a c t Although there have been well established transport based codes for core neutronics analysis, it is yet impractical to implement them in the real core treatment because their performance is not so great on ordinary server computers. For this reason, most of neutronics codes for core calculation are subject to two steps calculation procedure which consists of homogenized group constant generation and flux distribution generation which is the main concern of this work. This paper brings out a 2 dimensional nodal code based on point flux algorithm and implements two schemes for pin power reconstruction. In the first scheme, pin power reconstruction is obtained without considering corner point fluxes in the fuel assemblies but in the second method corner fluxes are included to assess their effect on pin power reconstruction. To obtain corner point fluxes, Smith’s procedure and the method of successive smoothing are used. Improvement in pin power reconstruction by including fuel assembly corner fluxes is illustrated in this paper and assessed by Monte Carlo simulation. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Fuel pin power information is important for the safety assessment of a core fuel loading since it is required in the determination of the peak linear heat generation rate and the minimum departure from nucleate boiling ratio (DNBR). Nodal expansion method (NEM) is one of the most widely used methods in modern neutronic codes. NEM provides a fast and accurate method of calculating flux and power distribution in a reactor core. The core is divided into large homogenized nodes (Typically the size of a node is a 20 cm  20 cm). A large core may typically be represented in half-core geometry by using 10,000 nodes. The NEM solution gives the node average flux, power etc., but no information about the detailed structures inside the nodes (Hojerup, 1990). A reconstruction method can be used for rebuilding pin powers from reactor core calculations performed with a coarse-mesh finite difference diffusion approximation and single-assembly lattice calculation. This method assumes that the detailed flux shape in an assembly can be approximated by superposing detailed inner assembly form function on a smoother intra-nodal shape function. The assembly form functions are obtained from single-assembly lattice calculation and the intra-nodal flux distribution are computed using polynomial shapes constrained to satisfy the ⇑ Corresponding author. Fax: +98 21 29902546. E-mail addresses: Khoshahval).

[email protected],

http://dx.doi.org/10.1016/j.anucene.2014.01.012 0306-4549/Ó 2014 Elsevier Ltd. All rights reserved.

[email protected]

(F.

nodal information approximated from the node-average fluxes (Na Gyun et al., 2001). Several researches have been focused on the pin power reconstruction method (Koebke and Wagner, 1977; Boer and Finnemann, 1992; Bahadir and Lindahl, 2006; Joo et al., 2009; Dahmani et al., 2011). In this paper, we try to investigate the effect of including the corner point flux on the pin power prediction. Doubtless, the accuracy of the pin power predictions depends on the accuracy of the intra-nodal flux and cross section values and form factors. In this work we develope a NODAL code in FORTRAN programming language capable of implementing pin power reconstructions in square structures. Two different methods for intranodal flux approximation are presented (one without considering corner point fluxes and the other with considering corner point fluxes). In addition, two different schemes for computing corner point flux approximation are implemented (Smith’s method and method of successive smoothing). The procedure is applied on PWR reactor core fuel assemblies (see Section 4) and results are compared to those attained by Monte Carlo calculations. 2. Nodal expansion method During the operation of nuclear reactors, fast calculation of the neutronic parameters is necessary. Nodal methods are fast tools for reactor calculation. These methods were developed in the 1970s for numerical reactor calculations, especially for neutron diffusion applications, Lawrence (1986). Nodal methods now have taken a

26

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

firm place in the current production codes for reactor design as a main computational engine. These methods use coarse meshes with dimensions as large as fuel assemblies approximately resulting in dramatic reduction in computing time compared to the finite difference methods. They attain very high accuracy by careful treatment in discretizing the diffusion equations to enforce neutron balance, Cho (2005). Nodal equations are obtained by integrating the multi-group diffusion equation over a homogeneous region or node and then relating the net currents across the surfaces to the outgoing and incoming partial currents. Spatial coefficients are then used to relate the average fluxes and the average outgoing partial currents on surfaces. Alternatively, the spatial coupling coefficients can be defined in terms of the net currents across a surface and the average fluxes in two adjacent nodes. The elimination of the interface current in favor of the coupling coefficients yields a 5 point equation in two dimensions for nodal fluxes. The nodal expansion method, NEM, is a class of nodal techniques in which the average interface partial currents are treated explicitly. Integrating the multi-dimensional diffusion equation over transverse directions will lead to a coupled set of one dimensional equation from which additional equations to relate the partial currents on the surfaces of a node to the flux within the node using polynomial expansion technique are obtained. Weighted residual procedures are used to calculate the coefficient expansions, Bennewitz et al. (1975a,b), Turinsky (1994). Procedures in this scheme involve approximation of the one dimensional equations obtained by integrating over transverse directions. The fluxes expanded in quadratic polynomials with coefficients being interpreted as the nodal flux and the average partial currents on the surfaces. The flux expansion method Langenbuch et al. (1977b), is developed by integrating the neutron diffusion equation over a node and evaluating the resulting integral by expanding the flux in products of higher order polynomials. In this work we use the NEM method which has been developed by the Putney (1984). This method enables the nodal equations to be written in terms of the average node fluxes. Specially, we focused on the point flux technique which has been described in detail in Section 2.1.

u m; hu

u ¼ x; y

ð2Þ

The local Cartesian axes form the basis for the following notations and a schematic view of definitions is illustrated in Fig. 1 where: m Cm us , left (s = l)/right (s = r) u-surface of node P , u = x, y m Um , average flux for group g in P g m Wm gus , average flux for group g at Cus m Wgu , one dimensional spatially averaged flux in the u-direction of node Pm m km gus , value of boundary condition at Cus Pmus, node adjacent to surface Cm (of node Pm) us m hu ; thickness of node Pm in the u direction

In this paper the point flux method is implemented as a core calculation module. 2.2. Point flux method This method is the coarse mesh flux expansion method of Langenbuch et al. (1975, 1977a,b), later investigated by Rydin and Sullivan (1978). The point flux method is similar to the average flux method in that it is also based on the nodal integrated neutron diffusion equation, but employs a nodal expansion which is fitted to the center point flux of the node and the center point fluxes of its surfaces. The nodal equations are derived in detail in (Putney, 1984), which a quadratic polynomial flux expansion in each node is fitted to the node and surface center point fluxes to lead a ‘‘nodal balance equation’’ of the form:

" # m m X Wm gul  2Ug þ Wgur m

 4Dg

m2

hu

u¼x;y

2 3 m X X 1 6 m m7 þ 4 W þ 2Ug 5 6 rg u¼x;y gul s¼l;r

2 3 G m X 1X 6X m m7 ¼ 4 W 0 þ 2Ug0 5 6 gg0 u¼x;y g us g 0 ¼1 s¼l;r

2

In the nodal expansion methods the multi-group neutron diffusion equation is effectively solved by representing the neutron flux in each node by a polynomial expansion, and using a combination of weighted residual equations, i.e.: ( ) Z G G X vg X 0 0 W ½k div D r / þ R /  R /  t R / dPm ¼ 0 0 0 g tg g gg g g gu keff g0 ¼1 fg g Pm g 0 ¼1 k ¼ 0; 1; 2; . . . ; K;

nu ¼

g 0 –g

2.1. Derivation of nodal equations

m ¼ 1; 2; . . . ; M;

The derivation can be further simplified if we define on these axes the local dimensionless variables:

3

G m X m vg X 1 X m7 þ m 6 4 Wg0 us þ 2Ug0 5 m ¼ 1; 2; . . . M;

K eff

g 0 ¼1

6

fg 0

g ¼ 1; 2; . . . ; G;

u¼x;y s¼l;r

u ¼ x; y

g ¼ 1; 2; . . . ; G; u ¼ x; y

ð1Þ

where m, g, u and k are node number, energy group, Cartesian axes and order of weighted diffusion equation respectively along continuity conditions to determine its coefficients. In the zeroth order of the method, the nodal flux expansions is chosen to be quadratic along setting W ½0 gu ¼ 1. The necessary coefficients can be determined by forcing the expansions to satisfy the zeroth order (k = 0) weighted diffusion Eq. (1), i.e. the integral neutron balance in the nodes, and the continuity of neutron flux and net normal neutron current on their surfaces. In order to expand the neutron flux in each node, it is necessary to introduce a set, or sets, of axes in its dependent variables. For the case of 2D rectangular geometry, the most convenient approach is to reference the flux expansion in each node to the local Cartesian axes (x, y).

Fig. 1. Some defined notations for derivation of methods.

ð3Þ

27

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36 m If surfaces Cm ur and Cul are internal surfaces, imposing the conditions of continuity of point neutron flux and point net normal neutron current, one obtains (Putney, 1984):

mur Wm gur ¼ Wgul mul Wm gul ¼ Wgur

h Dm g m

hu

h Dm g m

hu

3W

m gul

3W

m gur

m g

m gur

m g

m gul

þ 4U  W

þ 4U  W

i

i

PM PG

h i Dmul g mul  Wmul ¼ mul 3Wmul gur  4Ug gul hu ¼

h Dmur g mur

hu

mur gul

3W

mur g

 4U

mur gur

W

m¼1

m¼1

h

W

Wm gus0

4

i

Um g

¼ h m u 3 1 þ hhmus u

Dmus u Dm u

h m g

iU þ h 3 1þ

m mus

m

¼ kgus Um g þ kgus

Dm u Dmus u

ð4Þ

with kgus

m us ,

mus g

iU

m mus

Umus g ;

For boundary surface C

Dm g m hu

g hmus u m hu

mus

¼ kgus0

ð5Þ

the requirement is

m m m m ½3Wm gus þ 4Ug  Wgus0  ¼ kgus Wgus ;

ðs0 ¼ l; r; s0 –sÞ

ð6Þ

Which yields

h m gus

W

4

Wm gus0

i

Um g

m m i m ¼ h m Ug ¼ kgus Ug m hu 3 1 þ kgus Dm

ð7Þ

g

It should be noted, for perfect reflector boundary km gus is zero, large values of km leads to boundaries with zero flux (Putney, gus 1984). Using these equations to eliminate the surface fluxes appearing in the nodal balance equation (See Eq. (3)), leads to a pseudo finite difference equation,

2 3 G m X m mus X X X 1 6 m m7 bgus Ug þ bg Um 4 W 0 þ 2Ug0 5 g ¼ 6 gg0 u¼x;y g us u¼x;y g 0 ¼1 s¼l;r

s¼l;r

g 0 –g

2

3

G m X m vg X 1 X m7 þ m 6 4 Wg0 us þ 2Ug0 5

K eff

g 0 ¼1

6

m ¼ 1; 2; . . . ; M;

g ¼ 1; 2; :::; G

¼

( 4Dm g m2

hu

) m 1X mmus kgus ; u ¼ x; y; þ 6 rg

s ¼ l; r

m mus

with kgus m

¼ 0 if Cm us 2 C; ( ) m X 2  kgul  km gur m

bg ¼ 4Dg

u¼x;y

m2 hu

þ

1 6

"

m X rg

Xn

o m m kgul þ kgur þ 2

#

ð9Þ

u¼x;y m

3.1. Reconstruction method The pin power reconstruction (PPR) process involves a fundamental assumption; that is, detailed pin by pin distribution within an assembly can be estimated by multiplication of a global intra nodal distribution and a local heterogeneous form function. The form function accounts for assembly heterogeneities caused by water holes, burnable absorber pins, enrichment zoning, etc. and it is generated for each fuel assembly type by a lattice physics code along the homogenized nodal cross sections generation. In other words, the form functions are needed for dealing with local heterogeneities in material composition within form assembly. The assumption of separability of the global intra-nodal flux and the local form function is commonly adopted in various pin power reconstruction methods that have been extensively investigated in the past two decades (Boer and Finnemann, 1992; Rempe et al., 1988; Girardi et al., 2008; Tatsumi and yamamoto, 2000; Lee et al., 2002; Dahmani et al., 2011). The flux is obtained by combining these two functions as folows:

/ðx; yÞ ¼ /hom ðx; yÞ  /het ðx; yÞ

Clearly, if the coupling coefficients kgus were known we would have achieved our objective and Eq. (8) could be solved by well known method ‘‘fission source iteration scheme (power method)’’. Unfortunately, the coupling coefficients are not known but depend on the ratio of surface to node center point fluxes in the coupled nodes. In the absence of any further information therefore, the only way to pursue this approach is to develop a procedure to iteratively generate the coupling coefficients within the power method scheme. Thus in the iteration number one, we guess the initial

ð11Þ

where /hom ðx; yÞ is ‘‘homogeneous’’ intra cells flux and is calculated by nodal method at core level; and /het ðx; yÞ is the ‘‘heterogeneous’’ form function which isevaluated at lattice level. It is common in the modern nodal method to approximate the two dimensional intra-nodal flux distributions by a non-separable expansion with polynomial or hyperbolic functions Dahmani et al. (2011). The procedure is used in this work as follows: 2 X cgij xi yj

ð12Þ

i;j¼0

ð8Þ

where, m bgus

3. Pin power reconstruction method

/hom ðx; yÞ ¼

u¼x;y s¼l;r

fg 0

ð10Þ

where n is the iteration number.

i

i

Wmus

gus 4  Umus

P

m mfg Um;nþ1 g Pm m;n g¼1 m fg Ug

g¼1

n K nþ1 eff ¼ K eff PM PG

According to the above equations, the surface flux equations may be cast as:

m gus

m value of the surface and node average fluxes (Wm gus and Ug ), then insert these values in the Eq. (5) and Eq. (7) to modify the surface fluxes, finally the modified surface fluxes will be inserted into the Eq. (3) to obtain node average fluxes and this procedure repeated to reach converged fluxes. Furthermore, we applied the fission source weighting in the power method scheme for the accelerating of eigenvalue convergence (Ackroyd, 1997),

where the homogeneous flux is expanded on polynomials in each square cell (side length equal to 2a). In the above expansion five coefficients, cgij , must be determine (in the case of the full expansion, i, j = 0,1,2 nine coefficients are needed). Having known cell average flux, the surface average fluxes and the corner point fluxes, one may specify cgij coefficients. In other words 5 or 9 coefficients for each group are determined with constraints given by four surface average fluxes and/or four surface net currents, four corner point fluxes and one node average flux. The surface average fluxes and the node average flux are directly obtained from nodal solution. Average fluxes along the four surfaces are obtained as: (See Fig. 2)

¼ 1 / 4a2

Z

a

a

dx

Z

a

a

X

cij xj yj dy

ij

1 X 1 1 ¼ 2 cij ðaiþ1  ðaÞiþ1 Þ  ðajþ1  ðaÞjþ1 Þ 4a ij iþ1 jþ1

ð13Þ

28

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

 ¼ aÞ ¼ 1 /ðy 2a

Z

a

a

X cij ðaÞj xi ij

1 X 1 ¼ cij ðaÞj ðaiþ1  ðaÞiþ1 Þ 2a ij iþ1  ¼ aÞ ¼ 1 /ðx 2a

Z

a a

ð14Þ

X cij ðaÞj yi ij

1 X 1 ¼ cij ðaÞi ðajþ1  ðaÞjþ1 Þ 2a ij jþ1

ð15Þ

Similar calculations are carried out to derive the fluxes at the rest of the surfaces. In order to compute fewer coefficients; a reduced basis of polynomials is used. One can select another reduced basis of polynomials. By using the form function (See Section 3.2) and the homogeneous intra-nodal flux determined at a pin position, the pin power can be obtained as:

pðx; yÞ ¼

G X

mRfg /ghom ðx; yÞ/ghet ðx; yÞ

ð16Þ

Table 1). 16 guide tubes are normally empty and light water flows through inside tubes. In some fuel assemblies, the burnable poison rods or control elements can be replaced with empty guide tubes. In those fuel assemblies with burnable absorber, 12 or 8 of empty guide tubes are occupied by absorber rods. In the fuel assemblies with control rods, 16 empty guide tubes would be occupied by control rods during reactor operation or shutdown condition. One of the guide tubes may be used to locate the instrumentation devices. The schematic view of one fuel assembly and the reactor core is demonstrated in Figs. 3 and 4 respectively. The fuel rods within a fuel assembly consist of enriched UO2 fuel pellets with the height of 11 mm that are bounded with a zirconium clad filled with helium gas. The general characteristics of the fuel rods are shown in Table 2. Burnable absorber rods are located in the guide tubes of fuel assemblies. These rods consist of B2O3 + SiO2 cladding with a SS-304. The general characteristics of the burnable absorber rods are shown in Table 3. 4.2. PPR without using corner point fluxes

g¼1

3.2. The form functions Form functions or form factors are relative pin powers and are calculated by 2D transport codes during fuel assembly cross section homogenization; they are the power of pins which are divided by average pin power of the assembly. The assembly form function calculation usually can be obtained by solving a single assembly criticality calculation along a zero current boundary condition. 4. Implementation of pin power reconstruction method 4.1. Description of the test case core The accuracy of the described method tested against results that have been published for BEZNO reactor. BEZNO reactor is made of 121 fuel assemblies with three different enrichments (2.44%, 2.78% and 3.48%). The fuel assemblies are array type 1414. Of these 179 are fuel rods and 17 are guide tubes (See

The corner flux cannot be obtained by the nodal expansion methods;, we try to calculate intra-nodal flux in specific fuel assembly with just the known values from nodal expansion method (average flux and surfaces fluxes). The resulting system of equation is obtained by combining Eq. (12), Eq. (13), Eq. (14) and Eq. (15),i.e.:

2

1

0

a2 3

6 6 1 a a2 6 6 6 1 þa a2 6 6 6 1 0 a2 3 4 2 1 0 a3

3

3 2 3 2  a00 / 7 7 7 6 6  ¼ aÞ 7 0 7 6 a01 7 6 /ðy 7 7 6 7 6 7 2 7 6 7 6 0 a3 7 7  6 a02 7 ¼ 6 /ðy ¼ þaÞ 7 7 7 7 6 6  4 a10 5 4 /ðx ¼ aÞ 5 a a2 7 5  a20 /ðx ¼ þaÞ þa a2 0

a2 3 a2 3

ð17Þ

It is worth noting that, the accuracy of the pin power predictions depends on the accuracy of the intra-nodal flux. In the following subsections, we use corner points to investigate the influence of these points approximation on the result of the intra-nodal flux approximation and in turn on the pin power distribution. 4.3. PPR using corner point fluxes Since the corner point fluxes are not nodal known during the nodal solution process, one needs additional approximations to estimate the corner points which in turn may be participated in the intra-nodal flux determination. Assuming the flux is separable in x and y directions, the corner flux can be reconstructed. Sometimes, the corner point flux discontinuity factor is used to assure that the heterogeneous corner flux is continuous. The corner point flux discontinuity factor is approximated by the assembly corner discontinuity factor (CDF), which is calculated as the ratio of the heterogeneous corner flux to the average flux in the single assembly spectrum calculation. In the following subsections, two methods which are used in this paper for corner point flux approximation are discussed. After

Table 1 General characteristics of the fuel assembly.

Fig. 2. The coordinate system with side length equal to 2a.

Number of fuel rods Lattice type Fuel rod array Fuel assembly pitch, cm Number of guide tube Guide tube material

179 Square 1414 19.8 17 SS-304

29

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36 Table 2 General characteristics of the fuel rods. Material Enrichment, % Pellets height, cm Pellets diameter, cm Density, g/cm3 Clad material Clad ID, cm Clad OD, cm

UO2 2.44, 2.78, 3.48 1.1 .93 10.3 Zr .948 1.072

Table 3 General characteristics of the burnable absorber rods. Material Density, gr/cm3 Inside diameter, cm Outside diameter, cm Clad material Outer clad ID, cm Outer clad OD, cm Inner clad ID, cm Inner clad OD, cm

B2O3(12%) + SiO2(88%) 2.2 .627 .988 SS-304 .999 1.056 .567 .600

Fig. 3. Fuel assembly layout for 14  14 FA configuration (red for regular fuel pin, blue for guide tube). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

extracting corner point flux values, the full expansion, i, j = 0,1,2, is applied and one may obtain: 2

1

0

a2 3

6 6 1 a a2 6 6 6 1 a a2 6 6 6 1 0 a2 3 6 6 6 1 0 a2 3 6 6 1 a a2 6 6 6 1 a a2 6 6 4 1 a a2 1 a a2

0

0

0

a2 3

0

0

0

0

a2

0

0

0

a2 3

a3 3

a

0

 a3

a2

0

a2 3

a2 a2 a2 a2 a2

0 a3 a3 a3 a3

a 0 a a2 a a2 a a2 a a2

3

2

3

a a3 a3 a3



a3 3

3

3 2 3 2  a00 / 7 7 6a 7 6  7 6 01 7 6 /ðy ¼ aÞ 7 3 7 7 6 7 6 7 a4 7  ¼ þaÞ 7 6 a02 7 6 /ðy 7 6 7 6 3 7 7 7 6 6  ¼ aÞ 7 a4 7 7 6 a10 7 6 /ðx 3 7 7 6 7 6 7 6 7 6 7 a4 7  6 a11 7 ¼ 6 /ðx ¼ þaÞ 7 3 7 7 6 7 6 7 6a 7 6 FA a4 7 7 6 12 7 6 7 7 7 7 6 6 a F B 7 6 20 7 6 a4 7 7 6 7 6 7 5 4 a21 5 4 F 47 C a 5 a22 F 4 D a a4 9

4.3.1. Corner point flux approximations using Smith’s method An approximation for the corner flux can be derived by simply expressing the 2-D flux distribution with multiplication of two one-dimensional fluxes (x and y). This can be formulated as:

a4

ð18Þ

The FA, FB, FC, FD are corner points of the node (See Fig. 2).

uc ¼

 xs  u  ys u  u

ð19Þ

 xs and u  ys are the surface average flux for the x and y direcwhere u  is the node average tion edge from the corner respectively and u  xs and u  ys are homogeneous surface flux attained using flux. Indeed, u nodal program. In the other words, Smith’s method is based on the assumption that the flux can be separated into x and y directions. For each corner, normally four estimates of flux are available, one from each node surrounding the corner. The final corner flux

Fig. 4. The schematic view of test case reactor core.

30

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

Table 4 Fuel assemblies group constants. Fuel assembly

Group number

D

SIGA

NUSIGF

SIGR

FA-2.44

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1.41933E+00 3.64900E01 1.41275E+00 3.70928E01 1.40942E+00 3.73714E01 1.43412E+00 3.65441E01 1.42753E+00 3.71320E01 1.42419E+00 3.74045E01 1.45013E+00 3.65230E01 1.44021E+00 3.73332E01

9.81040E03 8.36958E02 1.02043E02 9.21246E02 1.03977E02 9.61429E02 9.90570E03 8.84085E02 1.02954E02 9.68752E02 1.04868E02 1.00918E01 1.02245E02 9.84028E02 1.07935E02 1.11070E01

6.15811E03 1.10812E01 6.14685E03 1.10045E01 6.14041E03 1.09740E01 6.51052E03 1.21513E01 6.49869E03 1.20769E01 6.49197E03 1.20477E01 7.30506E03 1.43668E01 7.28273E03 1.42779E01

1.67184E02 0.00000E+00 1.58849E02 0.00000E+00 1.54699E02 0.00000E+00 1.65793E02 0.00000E+00 1.57502E02 0.00000E+00 1.53373E02 0.00000E+00 1.62756E02 0.00000E+00 1.50463E02 0.00000E+00

FA-2.44BP8 FA-2.44BP12 FA-2.78 FA-2.78BP8 FA-2.78BP12 FA-3.48 FA-3.48BP12

Table 5 Form factor of the fuel assembly type 2.44.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

0.9196 0.9342 0.9439 0.9550 0.9607 0.9637 0.9591 0.9582 0.9656 0.9561 0.9520 0.9502 0.9306 0.9191

0.9359 0.9608 0.9995 0.9864 0.9931 1.0138 0.9911 0.9907 1.0160 0.9962 0.9869 0.9997 0.9581 0.9323

0.9465 0.9981 0.0000 1.0379 1.0505 0.0000 1.0408 1.0357 0.0000 1.0474 1.0399 0.0000 1.0004 0.9509

0.9512 0.9848 1.0394 1.0341 1.0640 1.0588 1.0212 1.0234 1.0627 1.0665 1.0420 1.0381 0.9869 0.9511

0.9556 0.9874 1.0418 1.0594 0.0000 1.0488 1.0115 1.0218 1.0574 0.0000 1.0696 1.0446 0.9923 0.9563

0.9592 1.0100 0.0000 1.0552 1.0474 1.0142 1.0143 1.0397 1.0368 1.0578 1.0636 0.0000 1.0121 0.9607

0.9544 0.9877 1.0292 1.0116 1.0032 0.9968 1.0295 0.0000 1.0413 1.0187 1.0221 1.0398 0.9905 0.9552

0.9545 0.9870 1.0332 1.0132 1.0001 0.9909 1.0037 1.0279 1.0127 1.0141 1.0235 1.0369 0.9870 0.9545

0.9590 1.0106 0.0000 1.0581 1.0472 1.0082 0.9908 0.9971 1.0152 1.0517 1.0591 0.0000 1.0132 0.9592

0.9557 0.9900 1.0455 1.0605 0.0000 1.0474 1.0024 1.0041 1.0502 0.0000 1.0691 1.0476 0.9931 0.9587

0.9515 0.9880 1.0393 1.0392 1.0636 1.0560 1.0145 1.0163 1.0573 1.0651 1.0388 1.0367 0.9881 0.9511

0.9475 0.9985 0.0000 1.0399 1.0465 0.0000 1.0351 1.0341 0.0000 1.0463 1.0367 0.0000 0.9991 0.9472

0.9351 0.9626 1.0008 0.9890 0.9909 1.0148 0.9951 0.9895 1.0166 0.9886 0.9847 0.9986 0.9591 0.9327

0.9202 0.9346 0.9502 0.9534 0.9583 0.9643 0.9589 0.9587 0.9650 0.9556 0.9544 0.9496 0.9331 0.9182

Table 6 Form factor of the fuel assembly type 2.44BP8.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1.0321 1.0378 1.0405 1.0143 0.9856 0.9523 0.9455 0.9501 0.9545 0.9840 1.0169 1.0370 1.0385 1.0314

1.0437 1.0620 1.0830 1.0327 0.9708 0.8986 0.9213 0.9253 0.9022 0.9733 1.0355 1.0819 1.0620 1.0382

1.0428 1.0860 0.0000 1.0606 0.9490 0.0000 0.8847 0.8825 0.0000 0.9470 1.0592 0.0000 1.0831 1.0398

1.0216 1.0369 1.0561 1.0450 1.0282 0.9467 0.9608 0.9645 0.9501 1.0291 1.0440 1.0630 1.0381 1.0188

0.9829 0.9661 0.9454 1.0243 0.0000 1.0566 1.0300 1.0360 1.0647 0.0000 1.0288 0.9450 0.9747 0.9867

0.9532 0.9010 0.0000 0.9457 1.0561 1.0568 1.0698 1.1015 1.0821 1.0636 0.9475 0.0000 0.9037 0.9546

0.9501 0.9226 0.8811 0.9542 1.0205 1.0556 1.1039 0.0000 1.0997 1.0377 0.9620 0.8834 0.9246 0.9500

0.9501 0.9234 0.8825 0.9534 1.0151 1.0441 1.0725 1.0972 1.0673 1.0262 0.9607 0.8886 0.9235 0.9510

0.9542 0.8965 0.0000 0.9419 1.0455 1.0442 1.0428 1.0514 1.0547 1.0540 0.9475 0.0000 0.9014 0.9549

0.9823 0.9707 0.9476 1.0257 0.0000 1.0466 1.0176 1.0182 1.0526 0.0000 1.0270 0.9460 0.9725 0.9853

1.0164 1.0376 1.0618 1.0434 1.0231 0.9435 0.9562 0.9526 0.9426 1.0275 1.0447 1.0585 1.0309 1.0162

1.0428 1.0866 0.0000 1.0599 0.9479 0.0000 0.8795 0.8802 0.0000 0.9490 1.0607 0.0000 1.0857 1.0390

1.0419 1.0613 1.0835 1.0356 0.9738 0.9016 0.9245 0.9217 0.8985 0.9700 1.0323 1.0857 1.0614 1.0399

1.0345 1.0419 1.0420 1.0153 0.9859 0.9557 0.9479 0.9491 0.9542 0.9847 1.0151 1.0384 1.0410 1.0285

Table 7 Form factor of the fuel assembly type 2.78BP8.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1.0189 1.0324 1.0363 1.0180 0.9815 0.9553 0.9490 0.9489 0.9502 0.9804 1.0113 1.0362 1.0320 1.0244

1.0320 1.0556 1.0863 1.0359 0.9705 0.9003 0.9248 0.9220 0.8977 0.9695 1.0323 1.0834 1.0561 1.0302

1.0374 1.0866 0.0000 1.0622 0.9499 0.0000 0.8876 0.8880 0.0000 0.9535 1.0600 0.0000 1.0809 1.0337

1.0144 1.0361 1.0597 1.0464 1.0329 0.9502 0.9611 0.9665 0.9531 1.0363 1.0488 1.0626 1.0340 1.0122

0.9843 0.9696 0.9498 1.0314 0.0000 1.0581 1.0267 1.0388 1.0695 0.0000 1.0356 0.9538 0.9715 0.9849

0.9534 0.8979 0.0000 0.9478 1.0558 1.0590 1.0669 1.0992 1.0763 1.0672 0.9526 0.0000 0.9023 0.9532

0.9450 0.9196 0.8850 0.9575 1.0180 1.0505 1.1048 0.0000 1.0982 1.0353 0.9635 0.8882 0.9218 0.9491

0.9468 0.9225 0.8854 0.9571 1.0160 1.0438 1.0693 1.1039 1.0652 1.0298 0.9627 0.8900 0.9277 0.9506

0.9549 0.9022 0.0000 0.9478 1.0532 1.0472 1.0444 1.0537 1.0554 1.0589 0.9480 0.0000 0.9007 0.9521

0.9812 0.9703 0.9511 1.0299 0.0000 1.0529 1.0162 1.0204 1.0576 0.0000 1.0326 0.9506 0.9678 0.9851

1.0123 1.0364 1.0641 1.0493 1.0325 0.9485 0.9547 0.9592 0.9476 1.0360 1.0502 1.0637 1.0333 1.0136

1.0346 1.0816 0.0000 1.0654 0.9512 0.0000 0.8853 0.8860 0.0000 0.9489 1.0645 0.0000 1.0836 1.0323

1.0363 1.0558 1.0851 1.0321 0.9705 0.9022 0.9243 0.9231 0.9012 0.9713 1.0323 1.0818 1.0571 1.0327

1.0242 1.0344 1.0376 1.0156 0.9825 0.9546 0.9506 0.9491 0.9544 0.9815 1.0130 1.0355 1.0319 1.0217

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

Fig. 8. Intra nodal flux approximated in the fuel assembly number 1. Fig. 5. Form factor of the fuel assembly 2.44.

Fig. 6. Form factor of the fuel assembly 2.44BP8%.

Fig. 9. Intra nodal flux approximated in the fuel assembly number 2.

Fig. 7. Form factor of the fuel assembly 2.78BP8%.

Fig. 10. Intra nodal flux approximated in the fuel assembly number 3.

31

32

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

Fig. 11. Error between reconstructed pin power and MCNP result in the fuel assembly number 1.

Fig. 12. Error between reconstructed pin power and MCNP result in the fuel assembly number 2.

is determined by averaging the four available estimations. This method is the most economical method due to computational cost.

 xs þ u  ys  u  uc ¼ u

ð20Þ

 ys

 xs and u are the surface average flux for the x and y direcwhere u  is the node average tion edge from the corner respectively and u flux. For cases which Eq. (20) yields negative flux Eq. (19) is used. This method is used in PARCS code. To sum up, the final corner flux is determined by averaging the four corner flux estimates.

4.3.2. Corner point flux approximations using method of successive smoothing For the estimation of the corner point fluxes, Boer and Finnemann used the method of successive smoothing (MSS) approach (Boer and Finnemann, 1992). The MSS approach is based on assumption of linear flux variation around a corner and the resulting expression for the corner flux involves only 4 node averages and 4 surface average fluxes around the corner point. The MSS method is an efficient method but would lead to non-negligible errors in the estimated corner flux due to low order approximation, particularly in the regions where the flux variation is severe, such as at the corner point of a UO2/MOX checkerboard (Downar et al., 2004). Based on the above scheme corner flux is assumed as:

5. Results 5.1. Pin power distribution without using corner point fluxes In order to have a sight to know how much the corner fluxes can influence reconstructed pin power, we calculated homogenized

Table 8 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

8.356 5.367 3.784 3.335 3.012 2.587 0.913 1.780 3.128 0.284 3.314 2.221 2.689 4.562

8.379 7.502 6.625 8.236 4.566 4.678 2.936 4.403 3.635 5.006 3.051 5.747 4.449 4.742

7.598 6.331 0.000 4.535 6.952 0.000 2.295 5.611 0.000 2.298 2.210 0.000 3.437 1.549

6.670 5.996 5.808 7.937 6.747 5.244 4.924 6.290 2.839 5.105 2.780 5.966 4.694 3.023

6.548 3.919 6.193 8.539 0.000 4.166 5.947 6.950 3.918 0.000 2.948 2.811 3.059 4.011

4.559 4.891 0.000 4.527 2.909 3.627 4.456 6.189 6.622 5.055 3.369 0.000 6.269 5.922

5.180 5.888 6.455 6.923 3.253 4.888 6.248 0.000 2.894 2.510 6.261 3.919 4.381 5.250

4.190 3.097 3.044 3.990 4.534 1.100 5.551 4.963 2.390 2.489 4.296 3.192 4.971 3.656

4.280 3.473 0.000 3.558 2.088 5.282 3.212 2.569 2.602 1.062 3.779 0.000 0.862 1.864

1.451 1.095 0.599 1.737 0.000 1.145 0.480 0.475 2.738 0.000 0.787 0.545 0.680 0.777

1.388 0.377 0.784 1.124 0.407 1.057 0.692 0.877 0.811 5.320 0.116 0.798 0.919 2.784

1.544 2.745 0.000 5.616 3.210 0.000 2.141 1.437 0.000 2.901 2.835 0.000 2.166 0.961

3.812 1.645 3.247 0.134 1.172 2.590 0.348 1.217 0.568 0.126 1.721 1.347 1.214 1.319

6.250 2.760 2.819 1.860 1.047 2.672 3.602 1.678 0.522 2.989 3.363 1.623 0.203 4.034

Table 9 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

10.201 8.956 11.575 12.831 14.932 13.335 12.969 11.351 13.949 11.116 10.635 9.324 8.188 5.831

9.779 14.083 10.281 8.337 12.191 7.300 9.519 8.531 10.541 9.842 8.664 8.481 8.737 8.680

9.219 8.112 0.000 9.276 8.833 0.000 6.631 9.723 0.000 7.066 7.494 0.000 6.133 1.710

11.562 9.502 11.237 7.034 9.015 9.611 9.141 8.484 8.485 5.488 8.179 6.376 5.189 5.348

10.663 10.188 10.526 7.906 0.000 9.889 8.400 6.334 7.834 0.000 7.662 6.552 8.938 7.311

9.457 12.198 0.000 6.502 8.694 7.789 6.831 7.943 9.347 6.963 7.186 0.000 2.606 6.397

8.830 7.318 8.687 6.429 6.660 9.107 7.721 0.000 10.089 6.566 6.927 5.965 2.180 1.853

9.337 8.413 8.448 6.808 6.746 6.999 6.118 6.280 8.256 6.586 4.599 3.922 1.721 1.865

8.109 8.609 0.000 7.869 9.283 5.216 4.722 5.493 1.288 6.063 8.274 0.000 4.450 2.693

5.416 4.667 5.971 8.850 0.000 7.619 5.611 1.899 2.696 0.000 6.419 5.145 0.148 2.763

3.543 6.115 7.027 5.372 5.525 5.038 2.247 5.117 2.351 4.622 1.210 5.338 6.136 0.347

0.995 6.088 0.000 6.418 6.102 0.000 2.436 4.462 0.000 3.096 5.781 0.000 2.671 2.252

2.861 3.031 3.321 6.049 4.585 6.486 5.062 2.009 4.585 7.358 3.863 2.303 1.349 3.586

1.753 2.836 3.060 5.157 5.621 7.520 7.314 7.067 7.114 5.949 1.212 0.755 0.431 3.602

33

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

Fig. 13. Error between reconstructed pin power and MCNP result in the fuel assembly number 3.

cross section (See Table 4) and the form factors of each assembly using DRAGON code which is a cell calculation code (Marleau et al., 2010). For example the form function for the fuel assembly of 2.44%, 2.44B8% and 2.78B8% have been given in Table 5, Table 6 and Table 7 respectively. In addition normalized form factors are illustrated in Figs. 5–7 for the aforementioned fuel assemblies respectively. Having obtained form factor, we calculate the intra nodal flux using the average and surface fluxed from nodal calculation (See Section 2). The calculated intra nodal flux for fuel assemblies of number 1 and 2 and 3 which have been selected for pin power calculation (See Fig. 4) are shown in Figs. 8–10 respectively. The heterogeneous pin power distribution in the desired fuel assemblies are achieved by multiplying the heterogeneous form functions by the homogeneous intra-nodal flux distribution.

Table 10 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 3.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

19.86 10.49 4.16 4.06 5.36 7.35 11.38 16.05 15.33 13.15 13.94 15.54 17.30 21.94

8.74 6.56 0.00 0.36 8.86 11.71 9.88 15.44 17.57 19.36 13.94 18.81 23.24 22.75

10.93 0.89 0.00 0.52 8.57 0.00 14.07 11.88 0.00 18.77 16.30 0.00 20.75 23.89

8.16 1.68 3.49 10.55 10.92 10.51 12.15 19.66 19.70 19.15 16.71 21.90 21.55 23.62

10.50 7.82 4.70 11.48 0.00 10.22 15.59 17.86 19.69 0.00 21.96 22.23 21.44 25.25

2.73 11.13 0.00 12.15 11.78 10.52 13.80 17.37 18.84 21.63 23.66 0.00 25.14 24.32

6.95 11.83 13.65 12.59 12.85 13.09 18.09 0.00 22.18 20.30 20.09 22.30 25.27 28.07

4.08 10.26 15.59 13.32 14.74 18.11 18.63 18.48 18.94 22.28 20.81 27.50 29.27 30.76

0.25 9.61 0.00 13.26 14.51 18.53 16.89 18.14 18.97 19.37 21.95 0.00 26.09 30.78

7.08 10.05 14.40 15.34 0.00 21.29 21.96 21.10 18.93 0.00 23.20 20.32 28.86 29.06

8.50 11.21 16.13 13.64 19.37 22.37 22.02 20.94 20.29 22.13 22.34 24.98 26.69 27.94

6.71 12.69 0.00 16.37 19.02 0.00 24.24 21.04 0.00 22.99 25.40 0.00 25.44 27.26

8.26 15.06 14.76 19.58 20.41 23.49 24.20 24.80 20.13 25.36 23.58 24.69 25.02 27.79

10.35 13.59 17.52 18.59 22.45 24.29 26.27 27.41 29.97 28.18 24.05 22.51 24.69 23.43

Table 11 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 1 using Smith’s method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

7.249 5.843 4.211 3.709 3.333 2.856 0.689 1.948 3.245 0.355 3.338 2.200 2.625 4.458

6.370 5.911 6.957 7.129 4.830 4.906 3.133 4.565 3.766 5.105 3.123 5.790 4.466 4.735

6.982 6.622 0.000 4.781 7.169 0.000 2.478 5.769 0.000 2.426 2.322 0.000 3.519 1.622

4.857 6.224 6.022 7.199 6.931 5.420 5.090 6.445 2.992 5.248 2.921 6.099 4.826 3.157

6.737 4.104 6.367 7.160 0.000 4.327 6.103 7.103 4.075 0.000 3.111 2.978 3.231 4.188

4.715 5.043 0.000 4.675 3.060 3.778 4.609 6.342 6.779 5.222 3.546 0.000 6.460 6.125

5.317 6.022 6.588 7.057 3.395 5.031 6.394 0.000 3.060 2.685 6.439 4.113 4.586 5.467

4.327 3.234 3.182 4.127 4.673 1.248 5.698 5.116 2.555 2.663 4.477 3.387 5.174 3.876

4.434 3.625 0.000 3.706 2.238 5.429 3.364 2.725 2.764 1.233 3.953 0.000 1.060 2.072

1.645 1.281 0.778 1.907 0.000 1.308 0.641 0.635 2.894 0.000 0.950 0.712 0.505 0.596

1.134 0.142 1.001 1.326 0.215 0.875 0.860 1.035 0.961 5.457 0.025 0.934 1.051 2.913

1.216 2.441 0.000 5.357 2.981 0.000 2.316 1.275 0.000 3.022 2.722 0.000 2.082 0.890

3.382 1.265 2.901 0.164 1.430 2.358 0.155 1.375 0.696 0.027 1.650 1.390 1.232 1.313

5.694 2.279 2.394 1.494 0.736 2.408 3.386 1.516 0.634 2.919 3.339 1.642 0.263 4.137

Table 12 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 2 using Smith’s method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

12.02 12.15 12.15 12.03 12.27 12.21 12.24 12.26 12.17 11.46 11.84 10.10 8.54 5.76

12.17 12.25 12.23 12.09 12.15 10.65 11.83 11.18 11.76 11.83 10.38 9.91 9.87 9.54

12.09 12.13 0.00 11.84 12.12 0.00 9.57 11.87 0.00 9.41 9.65 0.00 7.99 3.49

12.09 11.89 11.95 10.22 12.01 12.17 11.92 11.19 11.11 8.11 10.66 8.84 7.62 7.73

11.55 11.45 11.87 10.76 0.00 11.96 11.14 9.12 10.57 0.00 10.41 9.35 11.69 10.14

12.05 11.66 0.00 9.19 11.33 10.48 9.58 10.70 12.10 9.85 10.12 0.00 5.84 9.59

11.26 9.81 11.17 9.01 9.28 11.71 10.42 0.00 12.23 9.52 9.96 9.12 5.57 5.37

11.75 10.87 10.93 9.38 9.36 9.66 8.86 9.08 11.07 9.53 7.70 7.14 5.12 5.37

10.73 11.21 0.00 10.50 11.89 7.97 7.52 8.31 4.28 8.96 11.16 0.00 7.60 5.99

8.48 7.70 8.90 11.65 0.00 10.39 8.42 4.80 5.57 0.00 9.19 7.96 3.14 5.71

7.22 9.55 10.30 8.57 8.60 8.02 5.21 7.90 5.12 7.25 3.85 7.81 8.53 2.16

5.51 10.12 0.00 9.96 9.44 0.00 5.46 7.23 0.00 5.52 7.96 0.00 4.59 0.40

8.17 7.94 7.83 10.07 8.31 9.80 8.09 4.80 6.98 9.39 5.66 3.83 2.60 2.58

8.16 8.62 8.30 9.76 9.70 11.04 10.37 9.67 9.26 7.67 2.56 0.15 0.02 3.60

34

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

In Fig. 11 and Table 8 the reconstructed pin power distribution for FA number-1 are compared with those attained directly from the MCNP calculations (Errori ¼ ðPiMCNP  Pirecon: Þ=P i mcnp in which Pi mcnp and Pi recon: is the pin power attained directly from the MCNP calculations and from reconstructed method respectively for the pin number i). In order to avoid making paper long, the Table of MCNP results along reconstructed power and average of errors is not illustrated. It is found from the Table 8, the maximum error is 8.54%. To evaluate the average error, following equation, Eq. (21), is used where i is the fuel pin number and N is the total number of fuel pins in a fuel assembly. For fuel assembly number-1, this value is equal to .0018. In addition as illustrated in Fig. 11 maximum errors have been detected near guide tubes and in the periphery of the fuel assembly.

Error av e ¼

N X 2 ðPirecon:  P iMCNP Þ =N

is about 3% lower than the one without corner point flux consideration (See the Table 9) and Errorav e is equal to .005. Table 13 illustrates that the maximum error for the predicted pin power for FA number 3 is 27.55% which is about 4% lower than the one without corner point flux consideration (See the Table 10). In addition, Errorav e is equal to .065. Thus, as expected adding corner point flux approximation attained using Smith’s method improved the results. In addition, the greater error for FA-3 is due to errors related to nodal calculation which leads to less accurate results for periphery assemblies.

ð21Þ

i¼1

In Fig. 12 and Table 9, the reconstructed pin power distribution for FA number-2 has been compared with those attained directly from the MCNP calculations; the maximum error is 14.9% and Errorav e is equal to .009. In addition as illustrated in Fig. 12 the maximum errors again is appeared near the guide tubes and in the periphery of the fuel assembly. In Fig. 13 and Table 10 the reconstructed pin power distribution for FA number-3 has been compared with those attained directly from the MCNP calculations. As can be seen from the Table 10, the maximum error is 30.8% and Errorav e is equal to .087. In addition as illustrated in Fig. 13 maximum errors again has been occurred near guide tubes and in the periphery of the fuel assembly. It can be found that, the errors for the FA No.3 is greater that FA No.1 and FA No.2. At the periphery of the core, large errors in the prediction of fine mesh powers (pin power distribution) are due to either low order intra-nodal flux shape function which assumed to be second order polynomials or the assembly form function calculations with zero current boundary condition or both.

Fig. 14. Error between reconstructed pin power and MCNP result in the fuel assembly number 3 using Smith’s method.

5.2. Pin power distribution using corner point fluxes 5.2.1. Results of Smith’s method In Table 11, Table 12 and Table 13 the difference between the predicted PPR using Smith’s method and MCNP result have been given for FA number 1, 2 and 3 respectively. To have better comparison of errors, the errors versus position of rods are plotted in Figs. 14–16. As can be seen from the Table 11, the maximum error is 7.25% and Errorav e is equal to .0016. In comparison to one without considering corner point flux (See the Table 8), the maximum error has reduced about 1.3%. Table 12, show that the maximum error for the predicted pin power for FA number 2 is 12.3% which

Fig. 15. Error between reconstructed pin power and MCNP result in the fuel assembly number 3 using Smith’s method.

Table 13 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 3 using Smith’s method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15.20 18.22 19.65 23.15 21.52 20.70 21.91 24.03 21.48 17.64 16.70 16.71 17.03 20.46

16.39 15.60 18.62 16.25 22.18 23.01 19.89 23.48 24.21 24.75 18.62 22.29 25.74 24.52

9.14 15.82 0.00 14.27 20.09 0.00 23.05 20.28 0.00 25.22 22.38 0.00 25.64 27.39

7.16 14.71 15.48 21.01 20.75 19.86 20.90 27.32 27.08 26.35 23.95 26.81 27.30 27.45

1.98 17.80 14.64 20.44 0.00 18.93 23.69 25.70 27.36 0.00 27.35 27.06 27.53 27.22

11.89 19.35 0.00 20.15 19.84 18.76 21.86 25.26 26.79 27.41 26.87 0.00 26.81 27.52

14.88 19.32 21.01 20.13 20.49 20.90 25.66 0.00 27.32 26.78 26.16 27.41 26.91 27.11

12.27 17.90 22.80 20.80 22.23 25.47 26.16 26.27 26.97 27.30 27.42 27.14 27.23 27.55

9.66 17.99 0.00 21.19 22.34 26.05 24.68 25.96 26.90 27.50 27.35 0.00 27.39 27.12

17.45 19.71 23.29 23.89 0.00 26.86 27.17 27.47 26.69 0.00 27.36 27.29 27.48 27.22

21.02 22.64 26.31 23.56 27.22 27.50 26.90 27.55 26.70 27.45 27.55 26.56 27.31 26.89

22.51 26.34 0.00 27.47 27.32 0.00 27.18 26.56 0.00 27.54 26.49 0.00 26.75 27.41

27.42 27.29 27.53 27.13 26.66 27.37 27.23 27.34 26.77 26.86 27.43 26.99 27.41 26.87

26.59 27.41 26.88 27.47 26.71 27.51 26.99 26.80 27.44 27.41 27.33 24.67 25.70 23.43

35

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

Fig. 16. Error between reconstructed pin power and MCNP result in the fuel assembly number 3 using Smith’s method.

5.2.2. Results of MSS The previous computations are carried out with the method described in Section 4.3.2. In Table 14, Table 15 and Table 16 the difference between the predicted PPR using method of successive smoothing and MCNP result have been given for FA number 1, 2 and 3 respectively. As can be seen from the Table 14, the maximum error is 5.27% and Errorav e is equal to .0011. In comparison to the one without considering corner point flux (See the Table 8), the error has reduced about 3.3%. In comparison with the one using Smith’s method (See Section 5.2.1), the error has decreased about 2%. It is found from in Table 15, the maximum error for the predicted pin power for FA number 2 is 9.87% which is about 4.4% lower than the one without corner point flux consideration (See the Table 9). In contrast with the one using Smith’s method, the error is decreased about 2.4% (See the Table 12). In addition, and Errorav e is equal to .0036.

Table 14 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 1 using MSS method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

5.27 4.85 4.30 3.79 3.40 2.91 0.64 1.98 3.27 0.37 3.34 2.20 2.61 3.32

4.67 4.85 5.03 4.60 4.89 4.95 3.18 4.60 3.79 5.13 3.14 3.79 4.47 4.73

4.88 4.70 0.00 4.35 4.40 0.00 2.52 4.84 0.00 2.45 2.35 0.00 3.54 1.64

3.88 4.27 2.96 5.11 3.94 4.76 5.13 1.29 3.02 5.18 2.95 3.57 4.85 3.19

4.75 4.14 4.66 4.25 0.00 4.36 4.21 5.25 4.11 0.00 3.15 3.01 3.27 4.23

4.75 5.08 0.00 4.71 3.09 3.81 4.64 4.52 4.96 5.26 3.12 0.00 5.17 3.16

4.33 4.57 4.76 3.38 3.42 5.06 5.12 0.00 3.09 2.72 4.51 3.21 4.63 3.80

4.36 3.26 3.21 4.16 4.70 1.28 2.47 5.15 2.59 2.70 4.51 3.43 5.22 3.92

4.47 3.66 0.00 3.74 2.27 3.56 3.40 2.76 2.80 1.27 3.99 0.00 1.10 2.12

1.69 1.32 0.82 1.94 0.00 1.34 0.67 0.67 2.93 0.00 0.98 0.75 0.47 0.56

1.08 0.09 1.05 1.37 0.17 0.84 0.89 1.07 0.99 4.59 0.05 0.96 1.08 2.94

1.14 2.38 0.00 3.26 2.93 0.00 2.35 1.24 0.00 3.05 2.70 0.00 2.06 0.87

3.29 1.18 2.83 0.23 1.48 2.31 0.11 1.41 0.72 0.01 1.63 1.40 1.24 1.31

4.41 2.18 2.30 1.41 0.67 2.35 3.34 1.48 0.66 2.90 3.33 1.64 0.27 4.16

Table 15 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 2 using MSS method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

9.85 9.41 9.61 9.13 9.87 9.12 9.20 9.22 9.68 9.61 9.81 9.79 8.45 5.80

9.73 9.86 8.99 9.72 9.79 9.70 9.71 9.03 9.29 9.66 9.49 9.49 9.55 9.32

9.47 9.58 0.00 9.36 9.58 0.00 8.69 9.03 0.00 8.70 9.00 0.00 7.46 3.01

9.26 9.69 9.54 9.29 9.57 9.63 9.45 9.33 9.78 7.31 8.96 8.11 6.93 7.08

9.75 9.56 9.37 9.61 0.00 9.37 9.31 8.25 9.72 0.00 9.58 8.52 9.39 8.96

9.04 9.63 0.00 8.37 9.56 9.63 8.71 9.83 9.41 8.95 9.23 0.00 4.91 8.71

9.50 9.08 9.84 8.21 8.45 9.37 9.56 0.00 9.28 8.60 9.04 8.19 4.60 4.40

9.26 9.05 9.03 8.58 8.53 8.81 7.98 8.19 9.25 8.62 6.76 6.19 4.14 4.40

9.66 9.78 0.00 9.69 9.75 7.09 6.63 7.41 3.33 8.06 9.24 0.00 6.70 5.08

7.65 6.84 8.04 9.31 0.00 9.52 7.53 3.88 4.66 0.00 8.34 7.12 2.28 4.89

6.24 8.60 9.36 7.63 7.68 7.11 4.29 7.03 4.26 6.43 3.04 7.07 7.83 1.46

4.33 9.03 0.00 8.95 8.46 0.00 4.54 6.38 0.00 4.77 7.29 0.00 4.03 0.92

6.81 6.65 6.61 8.96 7.26 8.84 7.20 3.96 6.25 8.76 5.11 3.36 2.22 2.86

6.55 7.13 6.92 8.52 8.58 9.56 9.50 8.91 8.62 7.15 2.14 0.14 0.13 3.60

Table 16 The comparison of the reconstructed and MCNP pin power calculation for the test case problem in FA. 3 using MSS method.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

7

8

9

10

11

12

13

14

16.38 19.31 20.68 20.06 22.44 21.60 22.77 21.54 22.29 18.47 17.52 17.50 17.81 21.19

17.59 16.74 19.65 17.25 22.28 22.49 20.69 21.80 22.87 21.08 19.28 22.89 22.78 21.50

10.44 16.94 0.00 15.26 20.94 0.00 18.64 22.24 0.00 22.54 22.88 0.00 22.55 22.67

8.47 15.82 16.49 21.87 21.54 20.59 21.55 22.20 20.56 22.91 22.40 22.72 23.10 22.44

3.31 18.83 15.62 21.26 0.00 19.60 20.17 23.01 23.08 0.00 22.58 22.44 22.99 22.22

13.04 20.30 0.00 20.91 20.52 19.36 22.36 22.85 22.49 23.06 22.29 0.00 22.21 22.87

15.91 20.20 21.79 20.82 21.10 21.42 22.63 0.00 22.94 22.36 21.04 21.02 22.14 22.51

13.24 18.71 22.24 21.41 22.75 22.79 22.75 22.96 22.64 22.45 22.74 22.51 22.10 23.15

10.54 18.70 0.00 21.71 22.78 22.57 22.78 22.47 22.18 22.45 21.67 0.00 22.54 22.23

18.13 20.29 22.64 22.25 0.00 22.40 22.34 22.44 22.36 0.00 22.95 22.35 22.43 22.27

21.52 23.07 22.17 21.25 21.90 22.62 22.25 22.89 21.02 22.41 22.39 21.77 21.23 20.77

22.82 22.62 0.00 22.94 22.01 0.00 21.30 22.56 0.00 20.57 21.35 0.00 23.05 21.98

22.11 23.06 22.29 22.88 21.93 21.97 23.00 22.55 22.86 22.55 22.66 22.63 21.63 22.57

22.56 22.74 22.97 22.73 21.96 22.88 23.10 22.82 22.83 23.09 22.60 23.11 22.73 23.03

36

F. Khoshahval et al. / Annals of Nuclear Energy 69 (2014) 25–36

The maximum error for the predicted pin power for FA number 3 is 23.15% which is about 8% lower than the one without corner point flux consideration (See the Table 10). In comparison with the one using Smith’s method, the error has been decreased about 4% (See the Table 13). In addition, Errorav e is equal to .049. Thus, it can be concluded that the outcome of MSS is better than Smith’s method in all three considered assemblies. 6. Conclusion An efficient nodal point flux solution was formulated to build up pin power calculation. The form factors were extracted from DRAGON code. To evaluate pin power construction, the effect of including corner fluxes to average point flux of surface nodes was assessed. From the results it was emerged that taking into account the corner point fluxes improve pin power reconstruction. Furthermore, two schemes were implemented for corner flux generation which MSS predictions were preferable. The maximum errors in pin power reconstruction occur in the peripheral water region. The errors are significantly less in the fuel region. The high error in the periphery of the core is due to inefficient result of nodal method for regions with high flux gradient; to overcome this issue, one needs to use nodal method based on transport instead of diffusion formulation and improve the form factor calculation. Acknowledgments The authors would also like to extend their gratitude to Dr. sanchez, Dr. Grgic and Dr. Akbari for their valuable guidance. The authors also would like to express their appreciation to unknown reviewers of the manuscript for their valuable comments. References Ackroyd, R.T., 1997. Finite Element Methods for Particle Transport Applications to Reactor and Radiation Physics. Research Studies Press Ltd., England. Bahadir, T., Lindahl, S.-Ö., 2006. SIMULATE-4 pin power calculations. In: Proc. of PHYSOR2006, Vancouver, Canada, September 10–14, 2006, on CD-ROM. Bennewitz, F., Finnemann, H., Moldaschl, H., 1975a. Solution of the multidimensional neutron diffusion equation by nodal expansion. In: Conf. 750413, Proc. Conf. Comput. Methods Nucl. Eng., vol. 1, Charleston, South Carolina, p. 99. Bennewitz, F., Finnemann, H., Wagner, M.R., 1975b. Higher-order corrections in nodal reactor calculations. Trans. An. Nucl. Soc. 22.

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