School of Nuclear Science and Engineering, North China Electric Power University, Beijing, China

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1 Hindawi Publishin Corporation Science and Technoloy of Nuclear Installations Volume 06, Article ID , paes Research Article Development and Validation of a Three-Dimensional Diffusion Code Based on a Hih Order Nodal Expansion Method for Hexaonal-z Geometry Daoan Lu and Chao Guo School of Nuclear Science and Enineerin, North China Electric Power University, Beijin, China Correspondence should be addressed to Chao Guo; pumuchao@ncepu.edu.cn Received April 06; Revised 4 July 06; Accepted July 06 Academic Editor: Euenijus Ušpuras Copyriht 06 D. Lu and C. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the oriinal wor is properly cited. A three-dimensional, multiroup, diffusion code based on a hih order nodal expansion method for hexaonal-z eometry (HNHEX) was developed to perform the neutronic analysis of hexaonal-z eometry. In this method, one-dimensional radial and axial spatially flux of each node and enery roup are defined as quadratic polynomial expansion and four-order polynomial expansion, respectively. The approximations for one-dimensional radial and axial spatially flux both have second-order accuracy. Moment weihtin is used to obtain hih order expansion coefficients of the polynomials of one-dimensional radial and axial spatially flux. The partially interated radial and axial leaaes are both approximated by the quadratic polynomial. The coarsemesh rebalance method with the asymptotic source extrapolation is applied to accelerate the calculation. This code is used for calculation of effective multiplication factor, neutron flux distribution, and power distribution. The numerical calculation in this paper for three-dimensional SNR and VVER 440 benchmar problems demonstrates the accuracy of the code. In addition, the results show that the accuracy of the code is improved by applyin quadratic approximation for partially interated axial leaae and four-order approximation for one-dimensional axial spatially flux in comparison to flat approximation for partially interated axial leaae and quadratic approximation for one-dimensional axial spatially flux.. Introduction It has been nearly 30 years since the initial implementation of finite-difference method (FDM) and finite element method (FEM) in computer codes desined to solve the few-roup neutron diffusion equations in more than one spatial dimension. A lare number of mesh points were required for solvin few-roup neutron diffusion equations by traditional finite-difference techniques in order to represent accurately the spatial variation of the neutron flux. Due to the lare number of unnowns involved, these calculations were very expensive, particularly for fuel manaement studies which require repeated solution of the diffusion equation. In order to solve this problem, the development of nodal methodshadenabledcomputintimesforlwranalysisto be sinificantly reduced. The nodal methods consisted of two main approaches which are polynomial nodal method 3 and Analytic Nodal Method (ANM) 4. Two nodal methods have one feature in common which is that they are based on approximations to one-dimensional equations derived by interatin the three-dimensional equation over the two directions transverse to each coordinate axis. The essential difference of the two methods suests classification of the methods developed for the solution on the basis of whether information obtained from an analytic solution of the diffusion equation within the node is incorporated into the numerical scheme. In the polynomial nodal method, the basic scheme is that the one-dimensional fluxes are approximated by a polynomial without the use of analytic information. A well-nown example of the polynomial methods is the nodal expansion method (NEM) 5, 6. Its basis is the solution of equations for the averae flux on a coarse spatial mesh, combined with a hih order local polynomial flux expansion, which is used to determine a relationship

2 Science and Technoloy of Nuclear Installations between the mean interface partial currents and the mesh averae fluxes. Accuracies of both ANM and NEM are comparable. However, the ANM method, due to its complexity, is restricted to not more than two enery roups. NEM is an example of first-eneration polynomial nodal method. One of its main advantaes is that there is no restriction on number of enery roups. Recently, the capacity to compute accurate numerical solutions to the neutron diffusion equation in hexaonal-z eometry is required for the physics and safety analysis of hexaonal fuel assembly (FA) desins. However, the NEM, which is initially developed for LWR, is formulated in rectanular eometry. The analysis of hexaonal-z eometry requires an extension to hexaonal eometry of the transverse interation procedure widely used in the development of Cartesian-eometry nodal schemes. The success of these Cartesian-eometry nodal methods has prompted the development of analoous techniques 7, 8 for fast reactor calculations in hexaonal eometry. Extension of AFEN for hexaonal-z eometry was also employed by Cho et al. 9. Besides, a mathematical basis for an extension of nodal methods to hexaonal eometry usin conformal mappin of the hexaon into a rectanle has been provided by Chao and Shatilla 0. In the traditional nodal expansion method for hexaonal-z eometry, partially interated axial flux is approximated by quadratic polynomial and flat approximation is used for partially interated axial leaae. The accuracy of this traditional method can be improved by usin hih order approximation. In this paper, we developed a 3D nodal code based on a hih order nodal expansion method for hexaonal-z eometry (HNHEX). This method exploits quadratic polynomial expansion for partially interated radial flux usin which is based on the NEM and four-order polynomial expansion for partially interated axial flux. Moment weihtin is used for radial direction and axial direction to obtain hih order expansion coefficients of the polynomials. The partially interated radial and axial leaaes are both approximated by the quadratic polynomial. Numerical results of HNHEX demonstrate the efficiency and ood accuracy of method. The oranization of this paper is as follows. Section presents the summary of hih order nodal expansion method for hexaonal-z eometry. Section 3 presents calculation of the leaae moments. In the next section, the proposed iterative solution alorithm is iven and the results of HNHEX aainst three-dimensional SNR and VVER 440 benchmars are presented. Section 6 presents the conclusion.. A Hih Order Nodal Expansion Method for Hexaonal-z Geometry In this section, we present a hih order nodal expansion method for hexaonal-z eometry. The multiroup neutron diffusion equation is solved by usin a nodal scheme with one node per hexaonal assembly. One-dimensional radial spatially flux of each node and enery roup is defined as the quadratic polynomial expansion. One-dimensional axial spatially flux of each node and enery roup is defined as the four-order polynomial expansion. The polynomial for v y h Fiure : Hexaonal coordinate system. the radial spatially flux is equivalent to a quadratic approximation and has second-order accuracy. The polynomial for the axial spatially flux is a four-order approximation and has second-order accuracy. The partially interated radial and axial leaaes are both approximated by the quadratic polynomial.thefinalequations,whicharecastinresponse matrix form, involve spatial moments of the node flux distribution plus surface-averaed partial currents across the faces of the node. The coarse-mesh rebalance method with the asymptotic source extrapolation is applied to accelerate the calculation... Nodal Coordinate Systems. The hexaonal-z eometry only has difference with rectanular eometry in the radial direction. There is no difference in the axial direction. Thus, a combination of a hexaonal eometry analysis in the radial direction with a D rectanular eometry analysis in the axial direction is required to the derivate of a nodal expansion method for hexaonal-z eometry. The hexaonal axes which consist of x-direction, u- direction, and v-direction are employed in the hexaonal eometry as shown in Fiure. The u-direction is rotated 60 counterclocwise with respect to the x-axis and the v- direction is rotated 0 from the x-axis. Usin the orientation shown in Fiure, with the oriin (in local coordinates) taen as the center of the hexaon, a homoeneous node volume of th node can be defined as shown in () in the local coordinate: V :(x,y,z) x h, h (), y y (x),y(x), z Δz, Δz, u y (x) = (h x ). () 3 h is thelatticepitch andthex-axis is taen as perpendicular to one pair of opposite faces of the hexaon. Analoous partially interated fluxes and currents are defined for the two opposite faces. x

3 Science and Technoloy of Nuclear Installations 3 ( h,y( h Δz ), ) ( h, y ( h Δz ), ) (0, y (0), Δz ) ( h,y( h Δz ), ) (0, y (0), Δz ) (0, 0, 0) ( h, y ( h Δz ), ) ( h,y( h ), Δz ) (0, y (0), Δz ) ( h,y( h ), Δz ) ( h, y( h ), Δz ) (0, y (0), Δz ) ( h, y ( h ), Δz ) Fiure : A three-dimensional hexaonal-z eometry... Hih Order Nodal Expansion Method for Hexaonal-z Geometry. In this method, reactor is divided into number of homoeneous hexaonal nodes. Cross sections within thenodesareassumedtobeconstant.letusconsidera homoeneous node shown in Fiure, the multiroup neutron diffusion equation for th node, and th roup can thenbewrittenintheformshownin D φ (x,y,z)σr, φ (x, y, z) = Q (x,y,z), =,...,K, =,...,G, (3) d dw J w (w) Σr, φ w (w) =Q w (w) 3 J wy (w, y (w)) J wy (w, y (w)) L z (w), φ w (w) = Δz / w = x,u,v, y(w) dz dyφ (x,y,z), y(w) (5) Q (x, y, z) = χ G υσ f, φ eff = G = < (x,y,z) Σ s, φ (x, y, z). (4) The solution of (3) requires additional relationships between the surface-averaed leaaes and the nodal fluxes in the th node and its immediate neihbors. The additional relationships are iven by approximations to onedimensional equations derived by interatin the threedimensional equation over the two directions transverse to each coordinate axis. The one-dimensional radial equation can be obtained by interatin the three-dimensional equation over y and z directionsandtaetheformshownin J w (w) = Δz / Q w (w) = Δz / y(w) dz y(w) dy D w φ y(w) dz dyq (x, y, z), y(w) L Δz / z (x, y) = dz D z φ (x,y,z), (x, y, z), w=x,u,v. The one-dimensional axial equation can be obtained by interatin the three-dimensional equation over x and y directions and tae the form shown in (6) d dz J z (z) Σr, φ z (z) =Q z (z) L xy (z), (7)

4 4 Science and Technoloy of Nuclear Installations φ z (z) = h/ J z (z) = h/ Q z (z) = h/ y(x) dx dyφ (x,y,z), y(x) y(x) dx y(x) dy D z φ (x,y,z), y(x) dx dyq (x,y,z), y(x) (8) The approximation of the partially interated axial fluxes taes the form shown in φ z (z) = Δz φ z (z) V = φ a z (z) a z (z) a z3 3 (z) a z4 4 (z), () L xy (z) = h/ y(x) dx dy y(x) D x y φ (x,y,z). In order to solve (5) and (7), the partially interated radial and axial fluxes are defined as quadratic polynomial expansion and four-order polynomial expansion shown in (9) and (), respectively. The approximation defined in (9) for the partially interated radial fluxes has second-order accuracy due to the fact that this latter procedure is equivalent to enforcin a neutron balance over each of the half-nodes. The approximation defined in () for the axial direction has second-order accuracy. The approximation of the partially interated radial fluxes taes the form shown in φ w (w) = φ w (w) y (w) = φ a w f (w) a w f (w) a w3 f 3 (w) (9) (z) = z Δz, (z) =( z Δz ), 3 (z) =( z z )(( Δz Δz ) 4 ), 4 (z) =(( z Δz ) z )(( 4 Δz ) 0 ). () In order to determine the zeroth expansion coefficients, thepolynomialsof(9)and()arefittedtotheaveraeflux in the node and the averae fluxes on its surfaces. Thus, we impose the conditions shown in φ w (w = h )=φ a w f ( h )a w f ( h ) a w4 f 4 (w), w = x,u,v, a w3 f 3 ( h )a w4 f 4 ( h ) φ = V Δz / f (w) = w h, h/ dz y(x) dx dyφ (x,y,z), y(x) = φ w a w a w =φ w φ, w=x,u,v, f (w) = 36 3 (w h ) 5 6, w h f 4 (w) =( w h )( w h ), f 3 (w) = 0 3 (w h ) 3 5, w=x,u,v. (0) φ w (w = h )=φ a w f ( h ) a w f ( h )a w3 f 3 ( h ) a w4 f 4 ( h )=φ w a w a w =φ φ w, f (w) and f (w) are obtained by usin orthoonal characteristics of f n (w) polynomials. Hiher order approximations are obtained by first addin a basis function f 3 (w) which has a first derivative discontinuity at w = 0 and then addin an additional basis function f 4 (w) which provides a quadratic approximation over the half-intervals < w < 0 and 0<w<h/. φ z (z = Δz )=φ a z ( Δz ) a z ( Δz ) w=x,u,v,

5 Science and Technoloy of Nuclear Installations 5 a z3 3 ( Δz ) a z a z =φ z φ, a z4 4 ( Δz )=φ z φ z (z = Δz )=φ a z ( Δz ) a z ( Δz ) a z3 3 ( Δz ) be continuous at w=0and the imposed condition is shown in lim D x=ε d ε 0 dx φ x (x) x= ε x=ε = D y (x) (x) a x3 f 3 (x)y y (x) φ x (x) (5) x= ε = D 3 a x3 φ x (x). x=0 Apply the two-step leaae approximation and the final form of a w3 (w = x, u, V) is iven by a x3 = 6 h J u 89 J u J V J V D a z4 4 ( Δz )=φ z a z a z =φ φ z. (3) We solve (3) to et the zeroth expansion coefficients shown in a w = φ w φ w, 8 φ x φ x φ, a u3 = 6 h J x 89 J x J V J V D 8 φ u φ u φ, a V3 = 6 h J x 89 J x J u J u D 8 φ V φ V φ, (6) a w = φ w φ w φ, a z = φ z φ z, (4) J w± = J w (±h ), w=x,u,v. (7) a z = φ z φ z φ, w=x,u,v. The coefficients a w3 (w = x, u, V) are determined by requirin the y-interated net current J w (w) (w = x, u, V) to Hiher order coefficients a w4 (w = x, u, V) are determined by applyin a moment weihtin scheme to the onedimensional balance equation (5). The weihted residual equation is obtained by weihtin equation (5) with a weiht function h i (w) and require the result to be zero when interated over the interval w, h/ (w = x, u, V) as shown in h/ V h i (w) { d dw J w (w) Σr, φ w (w) Q w (w) L zw (w) 3 J (w, y (w)) J (w, y (w))}dw=0, w=x,u,v. (8) For moment weihtin, h (w) = f (w) = w/h (w = x, u, V); interate and rewrite (8) to et flux moment equation in three radial directions shown in Σ r, 3D h φ x 3h Γ x Γ u Γ V 40D 9h a x =Q x Δz L zx, Σ r, 3D h φ u 3h Γ x Γ u Γ V

6 6 Science and Technoloy of Nuclear Installations 40D 9h a u =Q u Δz L zu, Σ r, 3D h φ V 3h Γ x Γ u Γ V 40D 9h a V =Q V Δz L zv, (9) φ w = h/ V dwf (w) φ w (w), Q w = h/ V dwf (w) Q w (w), L zw = Δz V h/ y(w) dwf (w) dyl z (w, y), y(w) (0) For moment weihtin, i=, h z (z) = (z) = z/δz, interate and rewrite (3) to et one-order flux moment equation in axial direction shown in Σ r, φ z (J z J z ) D Δz =Q z 3h L xyz, (Δz ) a z φ z = Δz / V dz (z) φ z (z), Q z = Δz / V dz (z) Q z (z), L xyz = 3h Δz V / dz (z) L xy (z), (4) (5) Γ x = J x J x, Γ u = J u J u, w=x,u,v, () J z± = J z (±Δz ). For moment weihtin, i =, h z (z) = (z) = (z/ Δz ) /, interate and rewrite (3) to et second-order flux moment equation in axial direction shown in Γ V = J V J V. Now substitute φ w (w) in φ w with (9) and then interate to et a w4 as shown in φ w = h/ V dxf (w) φ w (w) = 9 a w 4 a w4 a w4 = 4φ w 6 3 a w = 4φ w 6 3 φ w φ w, w=x,u,v. () Hiher order coefficients a z3 and a z4 are determined by applyin a moment weihtin scheme to the one-dimensional balance equation (7). The weihted residual equation is obtained by weihtin equation (7) with a weiht function h zi (z) (i =, ) and requirin the result to be zero when interated over the interval z, Δz / / Δz V h zi (z) { d dz J z (z) Σr, φ z (z) Q z (z) (3) L xy (z)} dz = 0, i =,. Σ r, φ z 6Δz (J z J z ) =Q z 3h L xyz, D 3(Δz ) a z φ z = Δz / V dz (z) φ z (z), Q z = Δz / V dz (z) Q z (z), L xyz = 3h Δz V / dz (z) L xy (z). (6) (7) Substitute φ z (z) into φ z and φ z with () and then interate to et (8a) and (8b): φ z = Δz / V (z) φ Δz z (z) dz = / V Δz / ( z V )( Δz Δz )φ a z f z (z) a z f z (z) a z3 f z3 (z) a z4 f z4 (z)dz = a z 0 a z3, (8a)

7 Science and Technoloy of Nuclear Installations 7 φ z = Δz / V (z) φ Δz z (z) dz = / V φ u = (Σ r, 3D /h ) (Q u Δz L zu ) Δz / (( z Δz ) )φ a z f z (z) (8b) 3h (Σ r, 3D /h ) Γ x Γ u Γ V (34) a z f z (z) a z3 f z3 (z) a z4 f z4 (z)dz = 80 a z 00 a z4. Thus, a z3 and a z4 are shown in a z3 = 0φ z 0a z 40D 9h (Σ r, 3D /h ) a u, φ V = (Σ r, 3D /h ) (Q V Δz L zv ) 3h (Σ r, 3D /h ) Γ x Γ u Γ V (35) = 0φ z 0φ z φ z, a z4 = 00φ z 35 3 a z = 00φ z 35 3 φ z φ z φ. (9) The 3D multiroup nodal balance equation is obtained by interatin over a homoeneous node volume V 3h J x J x J u J u J V J V (30) Δz (J z J z )Σr, φ = Q, Q = V Δz / h/ dz y s (x) dx dyq (x,y,z). (3) y s (x) φ is derived from (30) and shown in (3). φ x, φ u,andφ V are derived from (9) and shown in (33), (34), and (35). φ z and φ z are derived from (4) and (6) and shown in (36) and (37): φ = Σ r, Q J 3hΣ r, x J x J u J u (3) J V J V (J Δz Σ r, z J z ), φ x = (Σ r, 3D /h ) (Q x Δz L zx ) 3h (Σ r, 3D /h ) Γ x Γ u Γ V 40D 9h (Σ r, 3D /h ) a x, (33) 40D 9h (Σ r, 3D /h ) a V, φ z = (Q Σ r, z 3h L xyz ) (J Δz Σ r, z J z ) D (36) (Δz ) a Σ r, z, φ z = (Q Σ r, z 3h L xyz ) (J 6Δz Σ r, z J z ) D 3(Δz ) Σ r, a z. (37) The partial net current and one-dimensional averae spatiallyfluxcanbewrittenintheformofoutoinand incomin partial currents as shown in ±J w± = Jout, w± Jin, w±, φ w± =Jout, w± Jin, w±, w=x,u,v,z. (38) By usin (9) and (38), outoin partial currents throuh radial-directed surfaces of th node in terms of flux expansion coefficients are shown in w = J w Jin, w = D d dw φ w (w) y (w) D y (w) E w (w) x=h/ J in, w = D h a w 36 3 a w 7 6 a w3 a w4 E w J in, w, w = x,u,v,

8 8 Science and Technoloy of Nuclear Installations w = J w Jin, w = D d dw φ w (w) y (w) D y (w) E w (w) x= J in, w = D h a w 36 3 a w 7 6 a w3 a w4 E w E x J in, w, w = x,u,v, (39) = h {7 J u 85 J V J V J u } D 35 79φ x 49φ x 8φ 30 a x3, E u = h {7 J V 85 J x J x J V } E V D 35 49φ u 79φ u 8φ 30 a u3, = h {7 J x 85 J u J u J x } D 35 49φ V 79φ V 8φ 30 a V3. (40) By usin () and (38), outoin partial currents throuh axial-directed surfaces of th node in terms of flux expansion coefficients are shown in E u = h {7 J V 85 J x J x J V } E V D 35 79φ u 49φ u 8φ 30 a u3, = h {7 J x 85 J u J u J x } E x D 35 79φ V 49φ V 8φ 30 a V3, z z = J z Jin, z = d D dz φ z (z) z=δz / = D Δz d V dz φ z (z) J in, z z=δz / J in, z = D Δz a z a z a z a z4 Jin, z, = J z Jin, z = D d dz φ z (z) z= J in, z = D Δz d V dz φ z (z) J in, z z= = D Δz a z a z a z a z4 Jin, z. (4) = h {7 J u 85 J V J V J u } D 35 49φ x 79φ x 8φ 30 a x3, We use (4), (6), (), and (9) for expansion coefficients alon with (3), (33), (34), (35), (36), and (37) for node averaed flux and its moments and (49), (50), (5), (56), and (58) for node leaae moments. One-dimensional averae spatially flux is expressed by partial currents with (38). Eiht outoin partial currents can be expressed in terms of eiht

9 Science and Technoloy of Nuclear Installations 9 incomin partial currents, source, and its moments and leaae moments. Matrix elements (a ij,b ij,c ij ) are iven below. These elements are characterized by node dimensions and homoenized material of the node. a a a 3 a 4 a 5 a 6 a 7 a 8 a a a 3 a 4 a 5 a 6 a 7 a 8 a 3 a 3 a 33 a 34 a 35 a 36 a 37 a 38 a 4 a 4 a 43 a 44 a 45 a 46 a 47 a 48 a 5 a 5 a 53 a 54 a 55 a 56 a 57 a 58 a ( 6 a 6 a 63 a 64 a 65 a 66 a 67 a 68 ) a 7 a 7 a 73 a 74 a 75 a 76 a 77 a 78 ( ( a 8 a 8 a 83 a 84 a 85 a 86 a 87 a 88 ) ( b b x u V x u V z z Q b 0 b Q x b b b = 4 b Q u b 5 0 b Q V b ( b ) ( Q z b b 75 b 76 ( b b 85 b 86 ) ( Q z c c c 3 c 4 c 5 c 6 c 7 c 8 c c c 3 c 4 c 5 c 6 c 7 c 8 c 3 c 3 c 33 c 34 c 35 c 36 c 37 c 38 c 4 c 4 c 43 c 44 c 45 c 46 c 47 c 48 c 5 c 5 c 53 c 54 c 55 c 56 c 57 c 58 c ( 6 c 6 c 63 c 64 c 65 c 66 c 67 c 68 ) c 7 c 7 c 73 c 74 c 75 c 76 c 77 c 78 ( A Jout, ( c 8 c 8 c 83 c 84 c 85 c 86 c 87 c 88 ) ( =B {Q L }C Jin, =A B {Q L }A C Jin, =P {Q L }R Jin,, ) ) Δz L zx Δz L zu Δz L zv 3h L xyz 3h L xyz J in, x J in, u J in, V J in, x J in, u J in, V J in, z J in, z Q = col Q,Q x,q u,q V,Q z,q z, ) ) ) ) L = col 0, Δz L zx, Δz L zu, Δz L zv, 3h L xyz, 3h L xyz. (4) (43) a =a =a 33 =a 44 =a 55 =a 66 = D h D } 3(h Σ r, 3D ) } } 8D (h Σ r, 3D ), 4D 7h Σ r, { 468 { 33 { a =a 3 =a 34 =a 45 =a 56 =a =a 3 =a 43 =a 54 =a 65 =a 6 =a 6 = 4D 7h Σ r, 8D (h Σ r, 3D ) 3 33, a 3 =a 4 =a 35 =a 46 =a 3 =a 4 =a 53 =a 64 =a 5 =a 6 =a 5 =a 6 = 4D 7h Σ r, 6 33, a 4 =a 5 =a 36 =a 4 =a 5 =a 63 = D h D } 3(h Σ r, 3D ) } } 8D (h Σ r, 3D ), 8D (h Σ r, 3D ) { 6 { 33 { 4D 7h Σ r, a 7 =a 7 =a 37 =a 47 =a 57 =a 67 =a 8 =a 8 =a 38 36D =a 48 =a 58 =a 68 =, 7Δz hσ r, a 7 =a 7 =a 73 =a 74 =a 75 =a 76 =a 8 =a 8 =a 83 =a 84 =a 85 =a 86 = 595D (Δz ), Σ r, D 3Δz hσ r, 43

10 0 Science and Technoloy of Nuclear Installations a 77 =a 88 = D Δz D (Δz ) 3 (Σ r, ) 75D (Δz ) Σ r,, a 78 =a 87 = D Δz 595D, (Δz ) 3 (Σ r, ) 475D (Δz ) Σ r, b =b =b 3 =b 4 =b 5 =b 6 = 36D, hd b =b 3 =b 34 =, h Σ r, 3D hd b 4 =b 53 =b 64 =, h Σ r, 3D b 7 =b 8 = D Δz 43 b 75 = 60D, Δz Σ r, b 85 = 60D, Δz Σ r, Σ r, b 76 =b 86 = 785D, Δz Σ r, 7hΣ r, 57 3Δz Σ r, 39 3Δz Σ r, 595D, (Δz ) (Σ r, ) c =c =c 33 =c 44 =c 55 =c 66 = D h D } 3(h Σ r, 3D ) } } 8D (h Σ r, 3D ), 4D 7h Σ r, { 468 { 33 { c =c 3 =c 34 =c 45 =c 56 =c =c 3 =c 43 =c 54 =c 65 =c 6 =c 6 = 4D 7h Σ r, 8D (h Σ r, 3D ) 3 33, c 3 =c 4 =c 35 =c 46 =c 3 =c 4 =c 53 =c 64 =c 5 =c 6 =c 5 =c 6 = 4D 7h Σ r, 8D c 4 =c 5 =c 36 =c 4 =c 5 =c 63 = D h D } 3(h Σ r, 3D ) } } 8D (h Σ r, 3D ), (h Σ r, 3D ) 6 33, { 6 { 33 { 4D 7h Σ r, c 7 =c 7 =c 37 =c 47 =c 57 =c 67 =c 8 =c 8 =c 38 =c 48 36D =c 58 =c 68 =, 7Δz hσ r, c 7 =c 7 =c 73 =c 74 =c 75 =c 76 =c 8 =c 8 =c 83 =c 84 =c 85 =c 86 = D 3Δz hσ r, c 77 =c 88 = D Δz 57 3Δz Σ r, 75D (Δz ) Σ r, D (Δz ), Σ r, 87, c 78 =c 87 = D Δz 39 3Δz Σ r, 475D (Δz ). Σ r, 595D 3(Δz ) 3 (Σ r, ) 595D (Δz ) 3 (Σ r, ) (44) 3. Calculation of the Leaae Moments Leaae moments in the response matrix are derived in this section. The partially interated axial leaae L z (x) and partially interated radial leaae L xy (z) are expanded in terms of polynomials, but here it is restricted up to second order as the accuracy achieved in calculation is insinificant with respect to computational effort involved in hiher order expansion. 3.. The Radial Moment of the Partially Interated Axial Leaae. The x-direction moment of the partially interated axial leaae L zx is calculated by usin the approximation shown in (45) in which f (x) and f (x) are as iven in (0). The

11 Science and Technoloy of Nuclear Installations partially interated axial leaae δ z (x) is iven by (45) and the total axial leaae is shown in (46): y s (x) dyl z (x, y) = L z (x) y s (x) (45) L z = Δz V h/ = L z δ z f (x) δ z f (x), y s (x) dx L z (x, y) y s (x) (46) = Δz h/ V dxδ z (x, y) = J z J z. The coefficients δ z and δ z are calculated in the followin manner. Let x and x denote the neihborin nodes in the minus and plus x-directions and the quadratic polynomial extends over the three nodes x,, andx. The expansion coefficients in (45) are calculated such that the total axial leaaes shown in (46) in the nodes x and x are preserved as shown in (47). The expansion coefficients δ z and δ z are calculated by usin the constraints iven in (47) and the results are shown in (48): L x z L x z = Δz V = Δz V V (L x δ z = z Δz h dxδ z (x), h (47) h/h dxδ z (x), h/ Lx z ) δ z = 3 V (L x z Lx z ) (48) L 73 Δz z. h The required leaae moment L zx is calculated by substitutin (45) into (0) and performin the necessary interations and the total axial leaaes L x z and Lx z shown in (46) can be computed by axial direction net currents. The final form of the x-direction moment of the partially interated axial leaae L zx is shown in (49). Similar equation for L zu and L zv can be obtained as shown in (50) and (5): L zx = Δz V h/ = Δz h V 46, y s (x) dxf (x) dyl z (x, y) y s (x) (L z δ z h 6 )=Δz (L V z V (L x z Δz h Lx z ) L z ), (49) L zu = Δz h V 46 L zv = Δz h V 46 (L z V (L u z (L z Δz h V (L V z Δz h Lu z ) LV z ) L z L z ), ). (50) (5) 3.. The Axial Moment of the Partially Interated Radial Leaae. The axial moment of the partially interated radial leaae L xyz is calculated by usin the approximation equation (5). (z) and (z) areivenin().thetotal radial leaae L xy can be written in terms of the averae leaaes in the three radial directions as shown in (53): L xy (z) = L xy δ xy (z) δ xy (z), (5) L xy = 3h Δz V / L xy (z) = L x L u L V. (53) The coefficients δ xy and δ xy arecalculatedinthe followin manner. Let z and z denote the neihborin nodes in the minus and plus z-directions and the quadratic polynomial extends over the three nodes z,, andz. Interatin(54)overz and z, respectively,andthe expansion coefficients δ xy and δ xy is calculated by usin the constraints iven in (54) and the results are shown in (55). L z xy = 3h L z xy = 3h / Δz V dzl z Δz z xy (z), Δz V z Δz / /Δz z dzl xy (z), (54) δ xy = V 3h d {Δzz Δz Δz z Δz L z xy L xy Δz Δz z Δz Δz z L xy Lz xy }, δ xy = Δz 3h V d {Δzz Δz L z xy L xy Δz Δz z L xy Lz xy },

12 Science and Technoloy of Nuclear Installations d = Δz z Δz Δz z Δz Δz z Δz z Δz. (55) The required leaae moment L xyz is calculated by substitutin (5) into (5) and performin the necessary interations to et the final form for L xyz shown in L xyz = 3h ξ z = ξ z = = h 8 Δz V / Δz V δ xy dz (z) L xy (z) =ξ z L xy Lz xy ξz L xy Lz xy, Δz Δz z Δz Δz z Δz Δz z Δz Δz z Δz Δz z Δz z Δz, Δz Δz z Δz Δz z Δz Δz z Δz Δz z Δz Δz z Δz z Δz. (56) (57) The required leaae moment L xyz is calculated by substitutin (5) into (7) and performin the necessary interations to et the final form for L xyz shown in ξ z = ξ z = L xyz = 3h = h 8 Δz V / dz (z) L xy (z) Δz V (δ xy 5 δ xy ) =(ξ z ξ z )L xy Lz xy (ξ z ξ z )L xy Lz xy, (Δz ) Δz z Δz 80 Δz z Δz Δz z Δz Δz z Δz z Δz, (Δz ) Δz z Δz 80 Δz z Δz Δz z Δz Δz z Δz z Δz. 4. Numerical Solution of the Nodal Equations (58) (59) 4.. Overview of the Solution Procedure. The effective multiplication factor, neutron flux, and fission power distribution are obtained by usin iteration technique to solve the neutron diffusion equation. The nodal equations are solved by usin a conventional fission source iteration procedure accelerated by coarse-mesh rebalance in combination with the asymptotic source extrapolation technique. At each fission source iteration, the solution of the interface partial currents for each roup is accomplished via a series of sweeps throuh the spatial mesh. The alorithm used to solve the nodal equations is shown in Fiure 3 and steps are briefly described here: () We calculate nodal couplin coefficients shown in (4). This calculation is performed only once before the iteration starts. () initialize the node flux moment vector φ,eienvalue (0) eff, rebalance factors f m and partial currents vector (0).0, eff and J in, =.0, to initial value (assumin φ = =J in, = 0.0) forallenery roups and all nodes. Theinitialtotalfission source vector for th node is calculated by F (0) = G Σ f, φ (0) =. (60) (3) At the beinnin of outer iteration n, the roup partial currents are scaled by rebalance factors f m calculated by coarse-mesh rebalance procedure as shown in (n) =f m (n). (6) (4) Fission source vector for th nodes and th roup, which is defined in (6), is calculated from flux moment vector and eienvalue obtained in previous outer iteration (n ) (or, it will be uess value if it is the first outer iteration) QF (n ) = χ (n ) eff = G Σ f, φ (n ) = eff (n ) χ F (n ). (6) (5) From this step, inner iteration (roup iteration) starts. Scatterin source vector, as iven in (63), is calculated with the most recently computed flux moment vector QS (n) = G = < Σ s, φ(n). (63) Then, construct roup source vector for th node and th roup due to fission source vector and scatterin source vector by Q (n) =QF (n ) QS (n). (64)

13 Science and Technoloy of Nuclear Installations 3 Calculate nodal couplin coefficients (P and R matrix) For all enery roups and all nodes, initialize node flux moment vector φ, eienvalue, eff (0) and partial currents ( and J in, ) to initial value (assumed to be.0). Calculate the initial total fission source vector F (0). n=0 n=n Apply rebalance factor f m to roup partial currents Calculate fission source vector for th roup by QF (n ) = (/ (n ) G Σ f, eff )χ φ(n ) = Calculate scatterin source vector for th roup by QS (n) s, = G = Σ < φ(n) Construct roup source terms due to fission and scatterin for th roup by Q (n) =QF (n ) QS (n) Calculate leaae moment vector for th roup L Solve response matrix equation for partial currents Calculate flux moment vector for th roup φ Yes More roups? Solve coarse-mesh rebalance equations for rebalance factors and eienvalue No Apply rebalance factor f m to total fission source F (n) No Chec converence of fission source and eienvalue Yes Exit Fiure 3: Overview of the nodal solution alorithm.

14 4 Science and Technoloy of Nuclear Installations (6) Leaae moment vector for th nodes and th roup defined in (49), (50), (5), (56), and (58) is calculated by usin the most recently computed incomin and outoin partial currents. (7) The response matrix defined in (4) for th nodes and th roup is calculated by usin latest incomin currents (these outoin partial currents are assined to the incomin partial currents of neihborin nodes throuh respective surfaces). (8) The flux moment vector defined in (3), (33), (34), (35), (36), and (37) is calculated by usin the most recently computed partial currents, total source vector calculated in step (5), and leaae moment vector calculated in step (6). (9) Steps (5) (8) are repeated for all nodes and all enery roups with eepin fission source vector calculated in step (4)constant. (0) Once all enery roups have been processed, the coarse-mesh rebalance equations are solved to et coarse-mesh rebalance factors and eienvalue. The total fission source vector for th nodes is scaled by rebalance factors prior to checin the converence ofthefissionsourceasshownin F (n) =f m F (n), V m. (65) () A new eienvalue eff (n) is calculated from eff (n ), the total fission source F (n ) obtained in previous outer iteration (n ), and the total fission source F (n) obtained in present outer iteration n as shown in F n (n) (n ) eff = eff, (66) F (n ) F (n) = G Σ f, φ (n) =. (67) The outer iterations are terminated when three converence criteria shown in (68) are satisfied: (n) (n ) eff eff <ε =.0 0 7, F (n) F (n ) max <ε F (n) =.0 0 5, K K F(n) F (n ) / <ε = F (n) 3 = (68) 4.. An Improved Coarse-Mesh Rebalance Method. The outer (fission source) iterations are accelerated by usin an improved coarse-mesh rebalance method in combination with the asymptotic source extrapolation technique. The basic idea of the method is to scale the fluxes calculated at each outer iteration on the fine-mesh by rebalance factors such that a neutron balance is enforced over each cell (reion) of a coarse-mesh superimposed on the fine mesh. This approach is nonlinear since the fine mesh fluxes are used to compute the coefficients of the coarse-mesh equations. In this section, the traditional coarse-mesh rebalance method improved by Wielandt method and vector normalization procedure is described. The outer iterations are accelerated by usin coarse-mesh rebalancin in which each rin of hexaons corresponds to a hex-plane coarse-mesh rebalance zone. The coarse-mesh equations can be combined in the form shown in M f m = λ P fm, (69) f m is coarse-mesh rebalance factor vector. Equation (69) is constructed and solved followin each outer iteration. The solution to this eienvalue problem can be obtained usin thewielandtmethod.forproblemsinwhichthemmatrix can be inverted directly, the Wielandt method is often more efficient. This approach is based on the application of the power method to the shifted eienvalue problem obtained by rewritin (69) to M f m = λ P f m, (70) M = M λ e P, λ = λ λ e. (7) The converence rate of the power method is determined by the dominance ratio of the matrix M P;thecloserthis ratio is to, the slower the converence rate will be. It can be shown that, for λ e >λ 0,thedominanceratioof M P is smaller than that of M P. HencetheWielandtmethod, whichisobtainedbyapplyinthepowermethodto(70),will convere faster than the power method applied directly to (69). However, the application of power method to calculate eienvalues of a matrix and correspondin eienvectors may lead to diverence of calculation. Thus, it is necessary to improve the power method by normalizin vector. The Wielandt method improved by vector normalization procedure, which is obtained by applyin the power method to (70), will convere faster and be more stable than the power method applied directly to (69). We solve (70) by usin the followin iterative procedure as shown in S 0 =f m =0, 0 S = M P f m, λ = max {S }, f m = S λ, =,,... (7)

15 Science and Technoloy of Nuclear Installations 5 Numerical calculations to date have demonstrated that only two power iterations are required at each power iteration ( in (7) is equal to ) due to the efficiency of the Wielandt method.thecomputedrebalancefactorsareusedtoscalethe partial currents and fission source moments in accordance with (6) and (65). The fission source is scaled prior to checin the converence of the fission source, while the roup partial currents are scaled at the beinnin of the loop over roups in the next outer iteration in order to avoid an additional roup loop followin the rebalance procedure. It is not necessary to scale the flux moments φ (n) because, as shown in (65), they do not enter into the calculation of the roup source term Q (n) due to the assumption that there is no upscatter. The final estimate for the eienvalue at the nth outer iteration λ (n) is obtained from (73). The coarse-mesh rebalance is performed once every five outer iterations: 5. Validation of HNHEX λ (n) = λ λ e. (73) In order to validate the effective multiplication factor, neutron flux, and fission power distribution calculation of code HNHEX, two 3D benchmar problems are selected. First problem is a three-dimensional LMFBR benchmar problem which is a simplified model of MARK-I core desin of SNR 300 prototype LMFBR. The second problem is a threedimensional VVER-440 benchmar problem. It should be noted that all calculations are performed by a destop computer with CPU 3. GHz. 5.. The Three-Dimensional SNR Benchmar Problem. The 3D SNR benchmar problem was introduced by Lawrence in ANL-746. This benchmar problem is derived from the trianular-z modelbyalterintheouterradialboundary (while preservin the total volume of the core) to allow imposition of boundary conditions on surfaces of hexaons. Thismodificationthuspermitssolutionbybothhexaonaleometry and trianular-eometry codes. Fiures 4 and 5 show the three-dimensional layout of the model reactor. The model consists of a two-zone core surrounded by radial and axial blanets without a reflector. The heiht of the active core is 95 cm and each axial blanet is 40 cm thic. A total of rins of hexaons (includin the central hexaon) are included in the model with a lattice pitch of. cm. Vacuum boundaryconditionsareimposedontheoutersurfacesof the blanets. Materials and are the inner and outer zone consistin of a PuO UO mixture as fuel as materials 3 and 4 represent axial and radial blanet consistin of U35-depleted uranium. Finally, materials 5 and 6 stand for absorber-material and follower. The whole core model includes a total of 8 control rods. In order to simulate a realistic three-dimensional problem, each of the absorber rods on the inner rin is withdrawn to the upper core/blanet boundary as the outer rin of control rods is halfinserted. More details about four-roup homoenized cross sections for assemblies are from /6 5/ /6 5/ /6 5/ /6 5/ /6 5/ /6 5/ /6 5/ /6 5/ /6 5/ Inner core reion (M) Outer core reion (M) Radial blanet reion (M3) 5 6 Absorber reion (M5) Follower reion (M6) Fiure 4: Cross section of SNR benchmar calculation model for hexaonal-z eometry Inner core reion (M) Outer core reion (M) Radial blanet reion (M3) Axial blanet reion (M4) Absorber reion (M5) Follower reion (M6) Fiure 5: Vertical section of SNR benchmar calculation model for hexaonal-z eometry.

16 6 Science and Technoloy of Nuclear Installations Table : HNHEX results for whole core 3D SNR benchmar in hexaonal-z eometry. Code Number of axial planes eff eff error (pcm) ε IC (%) ε OC (%) ε RB (%) ε AB (%) ε CR (%) CPU time (sec) DIF a HNHEX b HNHEX c HNHEX c DIF3D Nodal DIF3D Nodal DIF3D(6) DIF3D(4) AFEN AFEN a Reference. b Applyin flat approximation for partially interated axial leaae and quadratic approximation for one-dimensional axial spatially flux. c Applyin quadratic approximation for partially interated axial leaae andfour-orderapproximationforone-dimensionalaxialspatiallyflux. The HNHEX of traditional method and hih order method both have been used to solve the diffusion equation with four enery roups for whole core calculation. The hexaonal node size of the radial direction is a fuel assembly dimension. The reference solution is obtained by Richardson extrapolation of DIF3D (6 trianular meshes per assembly) and DIF3D (4 trianular meshes per assembly) solutions. Several codes were used to calculate whole core of 3D SNR problem and Table shows the numerical results of 3D SNR problem includin effective multiplication factor and its relative percentae error ε ; relative percentae error of reion-averaed fluxes in comparison to reference solution; and the CPU times. ε IC, ε OC, ε RB, ε AB,andε AO are the errors of the roup- and reion-averaed fluxes for the inner core, outer core, radial blanet, axial blanet, and absorber reions, respectively. DIF3D nodal conducts diffusion calculation with usin nodal expansion method. DIF3D(6) and DIF3D(4) perform finite-difference calculations with 6 and 4 trianular mesh cells per hexaonal assembly, respectively. ItshouldbenotedthatcalculationsofHNHEXareperformed by a destop computer with CPU 3. GHz. The calculations of DIF3D Nodal and DIF(6, 4) were performed usin the IBM 370/95 computer. The calculations of AFEN were performed usin the SUN Sparc/ computer 9. Accordin to Table, the hiher accuracy of results is belonin to the solution with usin quadratic approximation for partially interated axial leaae and four-order approximation for one-dimensional axial spatially flux with less effective multiplication factor and reion-averaed fluxes errors relative to the flat approximation for partially interated axial leaae and quadratic approximation for one-dimensional axial spatially flux. The results calculated by HNHEX with 8 axial planes are more accurate than that with 8 axial planes. Table shows the reion-averaed fluxes calculated by HNHEX with a hih order nodal expansion method. These fluxes are normalized to a total power of 3 watts over the whole-core model usin a power conversion factor of fissions/watt-sec. It canbeseenfromtablethatthemajorfluxerrorsappearin R6 R5 CR CR CR Cores,, and 3 R5 0 R6 Layer Fiure 6: Axial core cut for three-dimensional VVER 440 benchmar problem. the absorber reions and blanet assemblies and it is due to hih flux radient existence in these reions. 5.. The Three-Dimensional VVER 440 Benchmar Problem. Three-dimensional VVER 440 benchmar 4 models a VVER-440 core in 30-deree symmetry usin two-roup diffusion approximation and iven cross sections. The core is50cmhihandcoveredwithaxialandradialreflectors and contains fuel elements of 3 different enrichments. The radialfuelassemblypitchis4.7cm.thecontrolrodsarelare control elements with fuel followers. Rod Ban 6 is partially inserted, as shown in Fiure 6. Vacuum boundary conditions are applied to the entire reflector outside boundaries. The two-roup diffusion parameters are iven in Table 3. The tas is to calculate the effective multiplication factor and the 3D power distribution. The power distribution should z (cm)

17 Science and Technoloy of Nuclear Installations 7 Table : Reion-averaed fluxes calculated by HNHEX for 3D SNR benchmar in hexaonal-z eometry. Group 3 4 Reion Averae fluxes Calculated Reference Relative error (%) Inner core E E Outer core.4888e E Radial blanet E E Axial blanet 3.663E E Rods.73E E Rod followers.3874e E Inner core.906e E Outer core.0584e E Radial blanet.665e 06.09E Axial blanet 3.475E E Rods 6.578E E 06.0 Rod followers.544e E Inner core.763e E Outer core 9.373E E Radial blanet 3.86E E 05.9 Axial blanet 5.894E E Rods E E Rod followers.409e E Inner core 3.760E E Outer core.5468e E Radial blanet 9.536E E 04. Axial blanet.979e E Rods E E Rod followers 3.4E E Table 3: Fuel assembly cross sections of three-dimensional VVER-440 benchmar. Parameter FA type D (cm) D (cm) Σ R, (cm ).555E E E E E 0.930E 0 Σ R, (cm ) 6.477E E 0.000E E E E 0 Σ (cm ).6893E 0.59E E 0.64E 0 3.6E 0.748E 0 Σ f, (cm ).676E 03.79E E Σ f, (cm ) E E E υσ f, (cm ) E E E υσ f, (cm ) E 0.058E E be normalized to core averae power density of unity and presented for axial node size of 5 cm. The reference solution was extrapolated from DIF3D-FD runs with 6 and 94 trianle/hexaon subdivisions and.5 cm axial mesh spacin. The node size used in HNHEX for 3D VVER 440 benchmar is one node per a fuel assembly in radial shape and axial sections with 5.0 cm heiht. Several codes were used to calculate 3D VVER 440 problem and Table 4 ives the numerical resultsof3dvver-440testcase.itshouldbenotedthat calculations of HNHEX are performed by a destop computer with CPU 3. GHz. The calculations of ACNECH 5 were performed usin a laptop computer with CPU.4 GHz. The CPU times are referred to HP9000/735-5 for ANC-H and AFEN. The HNHEX of traditional method and hih order method both have been used to solve the diffusion equation with two enery roups for whole-core calculation. Accordin to Table 4, implementin quadratic approximation for

18 8 Science and Technoloy of Nuclear Installations Table 4: Numerical results of three-dimensional VVER 440 benchmar. Code Number of axial planes Eienvalue eff error (pcm) ε max (%) CPU time (sec) DIF a.03 HNHEX b HNHEX c HNHEX c ACNECH AFEN ANC-H a Reference. b Applyin flat approximation for partially interated axial leaae and quadratic approximation for one-dimensional axial spatially flux. c Applyin quadratic approximation for partially interated axial leaae andfour-orderapproximationforone-dimensionalaxialspatiallyflux / 4/ 3 5 Fiure 7: Radial core map and assemblies identification for threedimensional VVER 440 benchmar. partially interated axial leaae and four-order approximation for one-dimensional axial spatially flux, the accuracy of results has been improved relative to flat approximation for partially interated axial leaae and quadratic approximation for one-dimensional axial spatially flux by decreasin the maximum power error from 8.6% to 3.7%. There is no obvious increase in execution time with quadratic approximation for partially interated axial leaae and four-order approximation for one-dimensional axial spatially flux. The eff error of 4 axial plane decreases by 5 pcm comparin to that of axial planes for VVER 440. At last, Table 5 presents the three-dimensional power distribution obtained by HNHEX of hih order method based on the assembly identification numbers as shown in Fiure 7. Table 6 shows relative power error of the HNHEX solution of hih order method to DIF3D reference solution for 3D VVER-440 problem. 6. Conclusion In this wor, a three-dimensional, multiroup, diffusion code based on a hih order nodal expansion method for hexaonal-z eometry (HNHEX) is developed. In this method, one-dimensional radial and axial spatially flux of each node and enery roup are defined as quadratic polynomial expansion and four-order polynomial expansion, 5 respectively. The partially interated axial leaae is approximated by the quadratic polynomial instead of flat approximation to improve the accuracy for three-dimensional problem. HNHEX has been tested aainst two popular benchmar problems. The numerical results confirm that the hih order nodal expansion method for hexaonal-z eometry can obtain the adequate accuracy for the neutronic calculation of nuclear reactor core with fuel assemblies of hexaonal shape. Besides, obtained results present that the accuracy of four-order polynomial expansion for one-dimensional axial spatially flux and quadratic approximation for partially interated axial leaae is enhanced relative to quadratic approximation for one-dimensional axial spatially flux and flat approximation for partially interated axial leaae. The improved coarse-mesh rebalance method effectively accelerates the computation. In future wor, HNHEX will continue to be improved to increase the accuracy and calculation speed with our efforts. Furthermore, HNHEX can be extended for calculation of the time dependent nuclear problems in order to conduct simulation of the realistic reactor core neutron inetic treatments. Nomenclature φ w (w): One-dimensional radial spatially flux (w = x, u, V) for th nodes and th roup φ z (z): One-dimensional axial spatially flux for th nodes and th roup h: Lattice pitch of the hexaonal assembly Δz : Axialheihtofth node Σ f, : Macroscopic fission cross section of th node and th roup Σ s, : Macroscopic scatterin cross section from roup to roup Σ r, : Macroscopic removal cross section of th node and th roup D : Diffusion coefficient for th node and th roup L z (x): Partially interated axial leaae for th nodes and th roup L xy (z): Partially interated radial leaae for th nodes and th roup

19 Science and Technoloy of Nuclear Installations 9 Table 5: Power distribution obtained by HNHEX for three-dimensional VVER-440 problem. Assembly identity Axial layer (bottom) (top) φ w± : One-dimensional face-averaed spatially fluxonthew-directed face S w and S w for th nodeand th enery roup (w = x, u, V,z) S ws : Left(s = )/riht (s = ) w-surface of node, w=x,u,v,z, (the left surface is x=, u=, V =,and z=;therihtsurfaceisx=h/, u=h/, V =h/,andz=δz /) J w± : Face-averaed net current on the w-directed face S w and S w for th node and th enery roup (w = x, u, V,z) F : Total fission source vector for th node QF : Fission source vector for th node and th roup QS (n) :Scatterinsourcevectorforth node and th roup

20 0 Science and Technoloy of Nuclear Installations Table 6: Relative power error of the HNHEX power distribution solution to DIF3D reference solution for three-dimensional VVER-440 problem (unit: %). Assembly identity Axial layer (bottom) (top) Q :Groupsourcevectorforth node and th roup L : Leaae moment vector for th nodes and th roup φ : Volume-averaed flux for th nodes and th roup. Competin Interests The authors declare that there is no conflict of interests reardin the publication of this paper. Acnowledments ThisworissupportedbytheFundamentalResearchFunds for the Central Universities (JB0663). References S. V. G. Menon, D. C. Khandear, and M. S. Trasi, Iterative solutions of finite difference diffusion equations, Tech. Rep. BARC- 9, 98. V. Jaannathan, Evaluation of the finite-element-synthesis model usin the 3-D finite-element technique, Annals of Nuclear Enery,vol.0,no.,pp ,983.

21 Science and Technoloy of Nuclear Installations 3 M.L.Zerle,Development of a Polynomial Nodal Methods with Flux and Current Discontinuity Factors, Department of Nuclear Enineerin, Massachusetts Institute of Technoloy, Cambride, UK, K. S. Smith, An Analytic Nodal Method for Solvin Two-Group Multidimensional Static and Transient Diffusion, MIT Press, New Yor, NY, USA, H. Finnemann, A consistent nodal method for the analysis of space-time effects in lare LWRs, in Proceedins of the Joint NEACRP/CSNI Specialists Meetin on New Developments in Three-Dimensional Neutron Kinetics and Review of Kinetics Benchmar Calculations, MRR 45, pp. 3 7, Munich, Germany, January H. Finnemann, F. Bennewitz, and M. R. Waner, Interface current techniques for multidimensional reactor calculations, Atomernenerie,vol.30,no.,pp.3 8, R. D. Lawrance, A nodal interface current method for multiroup diffusion calculations in hexaonal eometry, Transactions of the American Nuclear Society,vol.39,p.46,98. 8 M. Maai and J. Aruszewsi, A hexaonal coarse-mesh proram based on symmetry considerations, Transactions of the American Nuclear Society,vol.38,p.347,98. 9 N.Z.Cho,Y.H.Kim,andK.W.Par, Extensionofanalytic function expansion nodal method to multiroup problems in hexaonal-z eometry, Nuclear Science and Enineerin, vol. 6,no.,pp.35 47, Y. A. Chao and Y. A. Shatilla, The theory of ANC-H: a hexaonal nodal diffusion code usin conformal mappin, in Proceedins of the International Meetin on Advances in Reactor Physics, pp. 3 34, American Nuclear Society, Knoxville, Tenn, USA, 994. R. D. Lawrence, The DIF3D Nodal Neutronics Option for Twoand Three-Dimensional Diffusion-Theory Calculations in Hexaonal Geometry, Aronne National Laboratory, Lemont, Ill, USA, 983. ANL, Benchmar Problem Boo, Report ANL-746, supplement 3, Aronne National Laboratory, Aronne, Ill, USA, G.Bucel,K.Kuefner,andB.Stehle, Benchmarcalculations for a sodium-cooled breeder reactor by two- and three-dimensional diffusion methods, Nuclear Science and Enineerin, vol. 64,no.,pp.75 89, F. Seidel, Diffusion Calculations for VVER-440 D and 3D Test Problem, in Proceedins of the 4th Symposium of Temporary International Collective (TIC 85),vol.,p.6,Warsaw,Poland, N. Poursalehi, A. Zolfahari, and A. Minuchehr, Three-dimensional hih order nodal code, ACNECH, for the neutronic modelin of hexaonal-z eometry, Annals of Nuclear Enery, vol. 68, pp. 7 8, 04.

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