Development of 3D neutron noise simulator based on GFEM with unstructured tetrahedron elements

Annals of Nuclear Energy 97 (2016) 132–141

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

Development of 3D neutron noise simulator based on GFEM with unstructured tetrahedron elements Seyed Abolfazl Hosseini ⇑, Naser Vosoughi Department of Energy Engineering, Sharif University of Technology, Tehran Zip code: 8639-11365, Iran

a r t i c l e

i n f o

Article history: Received 24 May 2015 Received in revised form 9 June 2016 Accepted 6 July 2016

Keywords: Galerkin Finite Element Method Unstructured tetrahedron elements Gambit Neutron flux Neutron noise Adjoint

a b s t r a c t In the present study, the neutron noise, i.e. the stationary fluctuation of the neutron flux around its mean value is calculated based on the 2G, 3D neutron diffusion theory. To this end, the static neutron calculation is performed at the first stage. The spatial discretization of the neutron diffusion equation is performed based on linear approximation of Galerkin Finite Element Method (GFEM) using unstructured tetrahedron elements. Using power iteration method, neutron flux and corresponding eigen-value are obtained. The results are then benchmarked against the valid results for VVER-1000 (3D) benchmark problem. In the second stage, the neutron noise equation is solved using GFEM and Green’s function method for the absorber of variable strength noise source. Two procedures are used to validate the performed neutron noise calculation. The calculated neutron noise distributions are displayed in the different axial layers in the reactor core and its variation in axial direction is investigated. The main novelty of the present paper is the solution of the neutron noise equations in the three dimensional geometries using unstructured tetrahedron elements. Ó 2016 Published by Elsevier Ltd.

1. Introduction The knowledge of the neutron noise, i.e. the difference between the time-dependent neutron flux and its steady state value with the assumptions that the processes are stationary and ergodic in time, is very useful for core monitoring, surveillance and diagnostics (Rácz and Pázsit, 1998; Hosseini and Vosoughi, 2014). Noise analysis method might be applied to determine dynamical parameters of the core, like the Moderator Temperature Coefficient (MTC) in a Pressurized Water Reactor (PWR) (Demazière and Pázsit, 2004) or the Decay Ratio in a Boiling Water Reactor (BWR). It can be applied to identify and localize the noise source such as unseated fuel assemblies, absorber of variable strength, vibrating absorber or vibrations of core internals (Hosseini and Vosoughi, 2013, 2014; Demazière and Andhill, 2005; Williams, 1974). One of the main advantages of the noise-based methods is Abbreviations: FEM, Finite Element Method; ug(r), neutron flux in the energy group g; keff, forward neutron multiplication factor; vg, neutron spectrum in energy group g; Dg, diffusion constant in the energy group g; Rr,g, macroscopic removal cross section in the energy group g; Rf,g, macroscopic fission cross section in the energy group g; Rs,g0 ?g, macroscopic scattering cross section from energy group g0 to g; m, fission neutron yield; d/(r, x), forward noise; r, the nabla operator; 3D, 3 dimensional; 2G, 2 energy groups. ⇑ Corresponding author. E-mail address: [email protected] (S.A. Hosseini). http://dx.doi.org/10.1016/j.anucene.2016.07.006 0306-4549/Ó 2016 Published by Elsevier Ltd.

that they can be used on-line without disturbing reactor operation (Sunder et al., 1991). Such monitoring techniques are of interest for the extensive program of power upgrades around the world. The other important advantage of noise analysis is that the calculations of neutron noise are performed in the frequency domain. This leads to reducing the dimension of the variable space of the noise equations. Namely, instead of solving the diffusion equations in both the time and space, after the Fourier transform one needs to solve them only in space whereas the frequency variable acts as a parameter in such a case (i.e. no any derivatives with respect to frequency). Since the system of noise equations is not an eigenvalue problem, it is solved easier than the static ones. To calculate the neutron noise, firstly the neutron noise distribution induced by a point-like noise source has to be determined. Although many commercial codes for static calculations are available, there is no dedicated commercial code that is able to calculate the dynamical reactor transfer function. Recently, several researchers have tried to develop convenient methods for calculations of the dynamical reactor transfer function and noise analysis. Some selected researches are presented in the following: 1. Demazie‘re proposed a 2D, 2Group neutron noise simulator, which calculates the neutron noise and the corresponding adjoint function in the 2D heterogeneous systems. The simulator can only be applied to the western LWRs (PWR and

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2.

3.

4.

5.

BWR cases). Since the used spatial discretisation based on finite difference scheme was derived for the rectangular mesh boxes, the VVER-type, some of the Gen-IV reactor cores and totally all the hexagonal-structured reactor cores are not included in its applications (Demazière, 2004). Larsson et al. calculated neutron noise in 2Group diffusion theory using the Analytical Nodal Method (ANM). The same spatial discretisation scheme was applied for the static calculation (Larsson et al., 2011). Demazie‘re has developed a multi-purpose neutronic tool, in which both critical and subcritical systems with an external neutron source can be studied, and static and dynamic cases in the frequency domain can be considered. This tool has the ability to determine the different eigenfunctions of any nuclear core. For each situation, the static neutron flux, the different eigenmodes and eigenvalues, the first order neutron noise, and their adjoint functions are estimated (Demazière, 2011). Larsson and Demazie‘re have developed a coupled neutronics/ thermal-hydraulics tool for calculating fluctuations in PWRs. The fluctuations in neutron flux, fuel temperature, moderator density and flow velocity in PWRs are estimated by coupling a dynamic thermal hydraulic module and a dynamic neutron kinetic module. There is also a possibility to directly define the perturbations in the macroscopic cross sections and to supply them to the neutronic part of the model (Larsson and Demazière, 2012). Tran and Demazie‘re have developed a neutronic and kinetic solver for hexagonal geometries based on the diffusion theory with multi-energy groups and multi-groups of delayed neutron precursors. The Tool is utilized to forward/adjoint problems of static and dynamic states for both thermal and fast system with hexagonal geometries. The spatial discretisation of the equations is based on FDM (Tran and Demaziere, 2012).

In the aforementioned researches, the spatial discretisation of the noise equations has been performed based on FDM or ANM in 2D geometries. In our recently published work (Hosseini and Vosoughi, 2012), development of 2D noise simulator based on GFEM with unstructured triangle elements was reported. The knowledge of the how to change the neutron noise distribution in the different distances in the axial direction helps to perform better noise analysis. Therefore, it is of interest to develop a 3D noise simulator for noise analysis in the reactor cores. To this end, the neutron diffusion equation and neutron noise equation should be solved in 3D geometries. The authors did not find any research in which neutron noise due to noise source of type absorber of variable strength has been calculated in 3D geometries. In the present study, DYNamical Galerkin Finite Element Method (DYN-GFEM) computational code is developed to calculate neutron noise due this type of noise source in the typical VVER-1000 (3D) reactor core (Schulz, 1996). The GFEM and unstructured tetrahedron elements generated by Gambit are used to discretize the equations. The advantage of the unstructured tetrahedron elements is their superiority in mapping the any three dimensional geometries. The motivation of the present study is the investigation of the variation of induced neutron noise by source type of absorber of variable strength in axial direction of the reactor core. An outline of the remainder of this contribution is as follows: In Section 2, we briefly introduce the mathematical formulation used to solve the neutron diffusion equation in static calculation. In Section 3, the method of noise calculation is described. Section 3 presents the main specification of the VVER-1000 (3D) benchmark problems. The Main specification of the benchmark problem is explained in Section 4. Numerical results and discussion on results are presented in Section 5. Section 6 gives a summary and concludes the paper.

2. Static calculation In the absence of external neutron source, the multigroup neutron diffusion equation is as Eq. (1) (Duderstadt and Hamilton, 1976; Lamarsh, 1966):

 Dg r2 uexa;g ðrÞ þ Rr;g uexa;g ðrÞ ¼

G X vg X mRf ;g0 uexa;g0 ðrÞ þ Rs;g0 !g uexa;g0 ðrÞ g ¼ 1; 2; . . . ; G:

keff

g 0 –g

g 0 ¼1

ð1Þ where, all quantities are defined as usual. Removal cross section is P expressed as Rr;g ¼ Ra;g þ Gg0 ¼1;g0 –g Rs;g!g0 . Eq. (1) is a linear partial differential equation which may be solved by different numerical methods. All of these methods transform the differential equation into a system of algebraic equations. Here, GFEM, a weighted residual method, is used to discretize the neutron diffusion equation (Hosseini and Vosoughi, 2013). To start the discretization, the whole solution volume is divided into the unstructured tetrahedron elements as shown in Fig. 1. These elements have been generated using Gambit mesh generator. In the linear approximation of shape function, the neutron flux in each element might be considered as Eq. (2) (Zhu et al., 2005):

/ðeÞ ðx; y; zÞ ¼ L1 ðx; y; zÞ/1 þ L2 ðx; y; zÞ/2 þ L3 ðx; y; zÞ/3 þ L4 ðx; y; zÞ/4 ; ð2Þ where Li ; i ¼ 1; 2; 3; 4 are the components of the shape function in Eq. (3):

NðeÞðx; y; zÞ ¼ ½ L1 ðx; y; zÞ L2 ðx; y; zÞ L3 ðx; y; zÞ L4 ðx; y; zÞ 

ð3Þ

The shape function components are defined as Eq. (4):

Li ðx; y; zÞ ¼

ai þ bi x þ ci y þ di z ; 6V

i ¼ 1; 2; 3; 4:

ð4Þ

with

2

y1 y3

z2 7 7 7 z3 5

1 x4

y4

z4

y2

z1

3

1 x1 61 x 2 6 6V ¼ det 6 4 1 x3

Fig. 1. The unstructured tetrahedron element.

ð5Þ

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in which, incidentally, the value V represents the volume of the tetrahedron. By expanding the other relevant determinants into their cofactors we have:

2

3 2 3 x2 y2 z2 1 y2 z2 4 5 4 b1 ¼  det 1 y3 z3 5 a1 ¼ det x3 y3 z3 x4 y4 z4 1 y4 z4 2 3 2 3 x2 1 z 2 x1 y 2 1 c1 ¼  det 4 x3 1 z3 5 d1 ¼  det 4 x2 y3 1 5 x3 y 4 1 x4 1 z 4

ð6Þ

keff

ð7Þ

In the weighted residual methods, the purpose is the minimizing the residual by multiplying with a weight WðrÞ and integrate over the domain as Eq. (8):

V

RðrÞWðrÞdv ¼ 0

LeastSquares, Galerkin) according to choices for W T . Since the value of W T is 1 (one) in the Galerkin method, the weighting function is considered as Eq. (9):

WðrÞ ¼ NðrÞ;

ð9Þ

where NðrÞ is the global shape function. Multiplying Eq. (1) by the weighting function and integrating the results over the solution space, Eq. (10) is obtained: G vg X dv WðrÞðDg r /g ðrÞ þ Rr;g /g ðrÞ  mRf ;g0 /g0 ðrÞ 2

V



X

keff

Rs;g0 !g /g0 ðrÞÞ ¼ 0

g 0 ¼1

ð10Þ

g 0 –g

In the above equation, the differential part may be transformed by applying the Divergence’s theorem:

Z

dv WðrÞðDg r /g ðrÞÞ ¼ 2

V

Z

dv rWðrÞ  r/g ðrÞ Z  dv r  ðWðrÞr/g ðrÞ V Z Z @/g ðrÞ ¼ dv rWðrÞ  r/g ðrÞ  dsWðrÞ @n V A ð11Þ V

where,

@/g ðrÞ ¼ r/g ðrÞ  n: @n

ð12Þ

Here, n is the normal unit vector on the volume V. Two types of boundary conditions (B.C.) are considered. The first B.C. is no incoming neutrons at vacuum boundaries (Marshak B.C.) which is expressed as Eq. (13):

@/g ðrÞ /g ðrÞ : ¼ @n 2Dg

V

Z þ

ðeÞ

dsN ðrÞN

ð13Þ

TðeÞ

A

/gðeÞ ðrÞ 2

#

Z V

dv N ðeÞ ðrÞNTðeÞ ðrÞ/ðeÞ g

2 Z E G X v X 4 g mRf ;g0 dv NðeÞ ðrÞNTðeÞ ðrÞ/ðeÞ ¼ g0 keff g0 ¼10 V e¼1 # Z g1 X Rg0 !g dv NðeÞ ðrÞN TðeÞ ðrÞ/gðeÞ0 þ

ð15Þ

V

When element matrices have to be evaluated it will follow that we are faced with integration of quantities defined in terms of volume coordinates over the tetrahedron region. It is useful to note in this context the following exact integration expression:

Z

V

ð8Þ

The weighting function may be considered as WðrÞ ¼ W T NðrÞ. There are (at least) four sub-methods (Collocation, Sub-domain,

Z

e¼1

ðeÞ dv Dg rNðeÞ ðrÞrN TðeÞ ðrÞ/gðeÞ þ Rr;g

g 0 ¼1

g 0 –g

g 0 ¼1

ð14Þ

Substituting Eqs. (9), (11), (13) and (14), and converting the integration on the reactor domain to sum of the integration on finite elements, the final form of Eq. (10) is obtained as Eq. (15):

G X vg X mRf ;g0 ug0 ðrÞ  Rs;g0 !g ug0 ðrÞ

¼ Rg ðrÞ g ¼ 1; 2; . . . ; G:

Z

@/g ðrÞ ¼ 0: @n

E Z X

with the other constants defined by cyclic interchange of the subscripts in the order 1, 2, 3, 4. Here, the linear approximation for shape function in each element is considered (Zhu et al., 2005). The GFEM is a weighted residual method in which the purpose is to minimize the residual integral. The approximate solution of the neutron flux function, uðx; y; zÞ, gives a residual as Eq. (7):

 Dg r2 ug ðrÞ þ Rr;g ug ðrÞ 

The second B.C. is zero net current or perfect reflective boundary condition which is described by Eq. (14):

a!b!c!d! 6V ða þ b þ c þ d þ 3Þ!

La1 Lb2 Lc3 Ld4 dxdydz ¼

ð16Þ

Evaluating the integrals of Eq. (15) for each element, using Eq. (16), and assembling the local matrix into the global matrix, the system of equations which is an eigenvalue problem is obtained. Here, the problem is solved using the power iteration method (Booth, 2006) and the neutron flux distribution in each energy group and the multiplication factor are calculated. 3. Neutron noise calculation 3.1. Forward noise calculation Here, the first-order forward noise approximation of two-group forward diffusion equation is applied to noise calculation. The global form of two-group noise equation which the neutron noise sources is due to the variations of scattering, absorption and fission macroscopic cross sections, is given in Eq. (17) (Demazière, 2004; Hosseini and Vosoughi, 2012):

h

i

r:DðrÞr þ Rdyn ðr; xÞ 



d/1 ðr; xÞ



d/2 ðr; xÞ   dRa;1 ðr; xÞ ¼ /s;1!2 ðrÞ dRs;1!2 ðr; xÞ þ /a ðrÞ dRa;2 ðr; xÞ   dm1 Rf ;1 ðr; xÞ ; þ /f ðr; xÞ dm2 Rf ;2 ðr; xÞ

ð17Þ

where duðr; xÞ, dRs ðr; xÞ, dRa ðr; xÞ and dmRf ðr; xÞ denote to neutron noise and perturbations in scattering, absorption and fission cross sections, respectively. Also, the above mentioned matrices and vectors are expressed as Eqs. (18)–(23):



D¼

D1 ðrÞ

0



D2 ðrÞ 2 R1 ðr; xÞ Rdyn ðr; xÞ ¼ 4 Rs;1!2 ðrÞ

ð18Þ

0

 /s;1!2 ðrÞ ¼

/1 ðrÞ /1 ðrÞ

m2 Rf ;2 ðrÞ keff

ð1 

3

ixbeff Þ ixþk 5

ðRa;2 ðrÞ þ vix2 Þ

;

ð19Þ

 ;

ð20Þ

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 /a ðrÞ ¼

/1 ðrÞ 0 0 /2 ðrÞ "

/f ðr; xÞ ¼

 ;

ð21Þ

   # ixbeff ixbeff /1 ðrÞ 1  ixþk /2 ðrÞ 1  ixþk 0

0

:

ð22Þ

The coefficient R1 ðr; xÞ applied in Eq. (19) is defined as Eq. (23):

R1 ðr; xÞ ¼ Rr;1 ðrÞ þ

ix

v1



m1 Rf ;1 ðrÞ



keff

1

 ixbeff : ix þ k

ð23Þ

In Eqs. (19)–(23), m1 and m2 , x, beff and k denote to neutron velocity in energy group 1 and 2, frequency of occurrence of the noise source, effective delayed neutron fraction and decay constant, respectively. Here, it is assumed that the neutron noise source type is absorber of variable strength. To solve the forward noise equation (Eq. (17)), the Green’s function technique is applied. In the mentioned method, the Green’s function (neutron noise due to the point noise source with strength 1) is calculated from Eq. (24):

h

"

i

r  DðrÞr þ Rdyn ðr; xÞ  "

dðr  r0 Þ

¼

#

 or

0

g¼1

Gg!1 ðr; r 0 ;xÞ

#

Gg!2 ðr; r 0 ; xÞ  0 : dðr  r0 Þ g¼2

ð24Þ

The same applied discretisation method for the static calculation is used to the dynamical calculation. By considering the neutron noise source in group 2, the discrete form of Eq. (20) is obtained as Eqs. (26) and (27):

e¼1

X ðeÞ

 R1 þ

Z Z X

keff

@ XðeÞV

X

ðeÞ

ðeÞ

þ Rs;1!2

dXNðeÞ ðrÞNðeÞT ðrÞG2!1

Z Z ixbeff ðeÞ dXNðeÞ ðrÞNðeÞT ðrÞG2!2 ix þ k X ðeÞ # ðeÞ G ðeÞ dsN ðrÞNðeÞT ðrÞ 2!1 ¼ 0; 2 1

ðeÞ

Z Z X

ðeÞ

ðeÞ

v2

Z @ XðeÞV

ð26Þ

dXD2 rNðeÞ ðrÞrNðeÞT ðrÞG2!2 dXN ðeÞ ðrÞNðeÞT ðrÞG2!1

 Z Z ix ðeÞ  Ra;2 þ

þ



d/1 ðr; xÞ d/2 ðr; xÞ

"R

 ¼

 # G2!1 ðr; r0 ; xÞS2 ðr0 ; xÞ dr0  R

: G2!2 ðr; r0 ; xÞ S2 ðr0 ; xÞ dr0

ð29Þ

The estimation of the neutron noise and the corresponding adjoint function has some practical applications. Hence, it is highly important to calculate the adjoint function of the neutron noise. Considering a point-like neutron source located at the position r0, the adjoint noise equations may be derived by a similar process used to obtain the forward noise equations. The matrix form of adjoint noise equations might be expressed as Eq. (30) (Hosseini and Vosoughi, 2012):

h

i

r:DðrÞ þ Rydyn ðr; xÞ 

"

d/y1 ðr; xÞ d/y2 ðr; xÞ

#

 ¼

dðr  r0 Þ



dðr  r0 Þ

ð30Þ

where d/y1 ðr; xÞ and d/y2 ðr; xÞ are the adjoint fast and thermal noise distribution, respectively. Rydyn ðr; xÞ is the transpose of the dynamical matrix Rdyn ðr; xÞ: In addition, dðr  r0 Þ refers to Dirac delta function at the position r0 . Like the forward noise calculations, Green’s function technique and GFEM are applied to adjoint noise calculations. The fast and thermal adjoint noise distributions are obtained from the adjoint noise calcualtion. 4. Main specification of the benchmark problem

ðeÞ

ðeÞ

 m2 RðeÞ f ;2

"Z Z E X e¼1

In this study, it is assumed that the occurred perturbation is located in the thermal macroscopic absorption cross section. In such a case, there is only thermal neutron noise source (Demazière, 2004; Hosseini and Vosoughi, 2012) and Eq. (28) reduces to Eq. (29):

ðeÞ

dXD1 rN ðeÞ ðrÞrNðeÞT ðrÞG2!1

ðeÞ

Z þ

ð28Þ

3.2. Adjoint noise calculation

in which, Gg!1 ðr; r 0 ;xÞ and Gg!2 ðr; r 0 ; xÞ are the Green’s function components in groups 1 and 2 due to neutron noise source in group g, respectively. The neutron noise source may be considered to be in fast or thermal energy group. The general form of neutron noise source is expressed as Eq. (25): "  " #  #   S1 r 0 ; x dRa;1 r 0 ; x  ¼ /s;1!2 r 0 dRs;1!2 r 0 ; x þ /a ðrÞ  Sðr 0 ; xÞ ¼ S2 r 0 ; x dRa;2 r 0 ; x "  #  dm1 Rf ;1 r 0 ; x 0  : ð25Þ þ /f r x dm2 Rf ;2 r 0 ; x

"Z Z E X

where @ XðeÞV refers to boundary length with vacuum boundary condition in element e. The integrals in Eqs. (26) and (27) may be solved using Eq. (16). Finally, the fast and thermal neutron noise distributions may be calculated through Eq. (28):  # R

  "R d/1 ðr; xÞ G1!1 ðr; r0 ; xÞS1 ðr 0 ; xÞ þ G2!1 ðr; r 0 ; xÞS2 ðr0 ; xÞ dr 0  R

: ¼ R d/2 ðr; xÞ G1!2 ðr; r0 ; xÞS1 ðr 0 ; xÞ þ G2!2 ðr; r0 ; xÞS2 ðr0 ; xÞ dr 0

ðeÞ

X

ðeÞ

ðeÞ

dXNðeÞ ðrÞNðeÞT ðrÞG2!1

dsN ðrÞNðeÞT ðrÞ

ðeÞ G2!2

2

#

3 ðeÞ N ðrÞ 7 6 i 7 6 ¼ 6 NjðeÞ ðrÞ 7; 5 4 ðeÞ Nk ðrÞ

The VVER-1000 is considered to validate the calculation against the available data for the mentioned benchmark problem (Schulz, 1996). The radial fuel assembly lattice pitch is 24.1 cm. This corresponds to the prototype VVER-1000 and is slightly different from the actual FA pitch of 23.6 cm in VVER-1000/V320, however it is acceptable for a mathematical benchmark. The core height is 355 cm, covered with axial and radial reflectors. The total height is 426 cm including 35.5 cm thick axial reflectors. Fig. 2 shows the 1/12 of the benchmark core configuration. The boundary conditions of the reactor core comprise of the no incoming current for the external boundaries and perfect reflective for the symmetry line boundaries. The material cross section of each assembly for the VVER-1000 is given in Table 1. 5. Numerical results and discussion

2

5.1. Results of static calculation

ð27Þ

The developed program (STA-GFEM (3D)) is used for static simulation of the VVER-1000 reactor core. Table 2 shows the calculated neutron multiplication factors with their Relative Percent

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Fig. 2. The 1/12 of the VVER-1000 reactor core.

Table 1 The material cross section of each assembly for the VVER-1000 reactor core.

D2 (cm) D1 (cm) mRf,1 (cm1) mRf,2 (cm1) Ra,1 (cm1) Ra,2 (cm1) Rs,1?2 (cm1)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

1.375 0.383 0.005 0.084 0.008 0.066 0.016

1.409 0.388 0.005 0.084 0.010 0.075 0.014

1.371 0.380 0.006 0.115 0.009 0.080 0.015

1.394 0.385 0.006 0.126 0.010 0.095 0.014

1.369 0.379 0.006 0.130 0.009 0.088 0.015

1.000 0.333 0.000 0.000 0.016 0.053 0.025

1.369 0.379 0.006 0.126 0.009 0.086 0.015

Table 2 The effective neutron multiplication factor with corresponding RPE in the VVER-1000 reactor core. Number of elements

Number of unknowns

keff

RPE (%)

44,006 75,040 171,964

7979 13,340 30,133

1.04825 1.04893 1.04906

0.1220 0.0572 0.0448

The reference effective multiplication factor is keff = 1.04953 (Schulz, 1996).

Error (RPE) (defined as Eq. (31)) vs. number of the unstructured tetrahedron elements.

RPEð%Þ ¼

calculated v alue  reference v alue  100: reference v alue

ð31Þ

Fig. 3 shows the arrangement of unstructured tetrahedron elements in the VVER-1000 reactor core. The power distribution (corresponding to 171964 elements in core modeling) in 1/12th of the reactor core is compared to the reference data (Schulz, 1996) in Table 3. The axial distribution of relative power in the different layers of the reactor core is also given in the mentioned table. As shown, the results of STA-GFEM (3D) and reference values are in a good agreement. As shown in Table 3, the calculations are repeated for three different numbers of the elements in order to analyze the sensitivity of the calculations to number the elements. As expected, differences between the calculated neutron multiplication factor and reference value decreases as the number of elements is increased. The calculated RPEs for neutron multiplication factor and power

Fig. 3. A view of arrangement of unstructured tetrahedron elements in the VVER-1000 reactor core.

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S.A. Hosseini, N. Vosoughi / Annals of Nuclear Energy 97 (2016) 132–141 Table 3 The distribution of the relative power in the 1/12th of the VVER-1000 reactor core. F.A.

Layer 1

2

3

4

5

6

7

8

9

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19

0.891 0.654 0.668 0.649 0.880 0.701 0.691 0.703 0.873 0.762 0.665 0.659 0.416 0.627 0.692 0.567 0.364

1.914 1.411 1.451 1.409 1.918 1.528 1.512 1.537 1.904 1.668 1.437 1.442 0.911 1.370 1.514 1.243 0.796

2.369 1.798 1.900 1.835 2.542 2.019 2.015 2.045 2.531 2.232 1.921 1.926 1.221 1.836 2.029 1.664 1.068

1.838 1.712 1.925 1.843 2.617 2.070 2.097 2.119 2.620 2.344 2.011 2.013 1.288 1.935 2.137 1.753 1.129

1.465 1.394 1.569 1.509 2.100 1.677 1.724 1.707 2.139 2.014 1.701 1.693 1.125 1.684 1.856 1.521 0.993

1.052 0.985 1.040 1.030 1.007 1.049 1.074 0.859 1.318 1.447 1.160 1.140 0.839 1.245 1.365 1.118 0.749

0.703 0.652 0.675 0.670 0.781 0.663 0.684 0.529 0.819 0.964 0.758 0.736 0.570 0.842 0.920 0.752 0.513

0.469 0.426 0.437 0.433 0.506 0.426 0.438 0.338 0.519 0.616 0.484 0.468 0.365 0.539 0.589 0.481 0.330

0.361 0.268 0.258 0.258 0.293 0.248 0.252 0.195 0.299 0.354 0.278 0.269 0.209 0.309 0.338 0.276 0.189

0.158 0.113 0.104 0.105 0.117 0.099 0.100 0.078 0.119 0.140 0.110 0.107 0.083 0.123 0.134 0.109 0.075

Qmean

Qmean (Ref.)

RPE (%)

1.1219 0.9413 1.0026 0.9741 1.2761 1.0479 1.0587 1.0109 1.3142 1.2542 1.0516 1.0453 0.7027 1.0509 1.1572 0.9484 0.6207

1.1199 0.9475 1.0138 0.9821 1.3112 1.0571 1.0603 1.0091 1.3135 1.2495 1.0481 1.0409 0.6981 1.0452 1.1508 0.9434 0.6161

0.1786 0.6544 1.1048 0.8146 2.6769 0.8703 0.1509 0.1784 0.0533 0.3762 0.3339 0.4227 0.6589 0.5454 0.5561 0.5300 0.0075

Fig. 4. The fast neutron flux distribution in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

Fig. 5. The thermal neutron flux distribution in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

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Fig. 6. The magnitude of the right hand side of Eq. (31) in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

Fig. 7. The phase of the right hand side of Eq. (31) in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

distribution in the present study are in the range of reported results in the similar work (Schulz, 1996). 5.2. Validation of the noise calculation The comparison between static calculations and noise calculations in zero frequency is the first benchmarking process of noise calculations. As expected, at zero frequency and without the neutron noise source the fast and thermal forward noises approach to static corresponding fluxes (Hosseini and Vosoughi, 2012). Figs. 4 and 5 show the obtained fast and thermal forward noises in the middle layer of the VVER-1000 reactor core at zero frequency without the neutron noise source. The static fast and thermal neutron fluxes have the same distribution displayed in Figs. 4 and 5. Considering the features of the inner product of two vectors, the calculated adjoint noise can be benchmarked against the formerly calculated forward noise. As mentioned in the other similar works (Demazière and Pázsit, 2004; Hosseini and Vosoughi, 2012), if the neutron noise source is assumed to be a point source located in the position rs and adjoint noise source as a Dirac function located in the position r0, Eq. (31) could be applied to benchmark the forward noise calculation:

c2 d/y2 ðrs ; r0 ; xÞ ¼ d/1 ðr0 ; rs ; xÞ þ d/2 ðr0 ; rs ; xÞ

ð32Þ

in which, c2 stand for /20 ðr s Þ, respectively. Therefore, the calculated adjoint noise can be benchmarked against the forward noise by comparing the calculated amplitude and phase of these two approaches in all the elements inside the reactor core. Figs. 6 and 7 show the magnitude (absolute value) and phase of the right hand side of Eq. (31) in the layer with Z = 106.5 cm, respectively. Also, the absolute and phase of the left hand side of Eq. (31) are displayed in the Figs. 8 and 9, respectively. As shown in the Figs. 6–9, the magnitude and phase of the left and right hand sides have a good agreement with each other. The neutron noise and corresponding adjoint distributions have the same agreement in the other layers. To avoid the duplication, the results obtained from Eq. (31) for the other layers have not been presented. 5.3. Results of noise calculation To investigate the variation of calculated neutron noise by DYN-GFEM (3D) in axial direction, the noise calculation is performed for source type of absorber of variable strength located

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Fig. 8. The magnitude of the left hand side of Eq. (31) in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

Fig. 9. The phase of the left hand side of Eq. (31) in the layer with Z = 106.5 cm of the VVER-1000 reactor core.

Fig. 10. Magnitude of neutron noise in layer with Z = 35.5 cm.

Fig. 11. Magnitude of thermal neutron noise in layer with Z = 71 cm.

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Fig. 15. Magnitude of thermal neutron noise in the layer with Z = 213 cm. Fig. 12. Magnitude of thermal neutron noise in the layer with Z = 106.5 cm.

Fig. 13. Magnitude of thermal neutron noise in the layer with Z = 142 cm. Fig. 16. Magnitude of thermal neutron noise in the layer with Z = 248.5 cm.

Fig. 14. Magnitude of thermal neutron noise in the layer with Z = 177.5 cm.

at X = 7.5312 cm, Y = 2.6089 cm and Z = 9.516 cm. The variation in the macroscopic cross section should be such that it still is P assumed as perturbation. In the first approximation, the d is P d considered as perturbation when P be negligible (in the present Fig. 17. Magnitude of thermal neutron noise in the layer with Z = 284 cm.

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noise source was calculated using the developed simulator. The results of static simulator were benchmarked against the valid reported data for the VVER-1000 reactor core. The variation of neutron noise in axial direction was investigated. It was concluded that the induced neutron noise have a different distribution for the layers far from the noise source. Overall, a reader can conclude that the developed computer code is a more reliable tool for static and dynamical calculations in 3D geometry. STA-GFEM (3D) is applicable to neutronic calculation of the reactor core in the design studies of nuclear reactors. DYN-GFEM (3D) may be used as a reliable tool for neutron noise analysis in 3Dgeometries. References

Fig. 18. Magnitude of thermal neutron noise in the layer with Z = 390.5 cm.

P d study, P ffi 104 ). The small variation in the neutron flux due to perturbation is the neutron noise. Figs. 10–18 display the magnitude of neutron noise distribution in different layers of the VVER-1000 reactor core. As shown in these figures, the magnitudes of neutron noise in the layers Z = 35.5 cm and Z = 71 cm have a similar behavior of neutron noise in 2D geometry (Demazière and Andhill, 2005; Hosseini and Vosoughi, 2012). By increasing the distance between the selected layer and noise source, the calculated neutron noise takes the different distribution and its values decreases remarkably. 6. Conclusion In the present study, 3D, 2G static and dynamic simulators based on GFEM was developed using tetrahedron unstructured elements. The advantage of unstructured tetrahedron elements to structured ones is the possibility of implementing smaller elements in the boundary layers, regions with high flux gradient or fuel assembly containing neutron noise source. This leads to obtaining the desired accuracy with low cost of calculations. Due to finite element method capability, the STA-GFEM (3D) and DYN-GFEM (3D) is applicable for both rectangular and hexagonal reactor cores. In the present study, the results of calculation for the VVER-1000 reactor core were only reported. To solve the neutron noise equation, the Green’s function technique was applied. The neutron noise induced by the absorber of variable strength

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