Journal of Nuclear Energy Science & Power Generation Technology
Khan et al., J Nucl Ene Sci Power Generat Technol 2016, 5:6 DOI: 10.4172/2325-9809.1000170
Research Article
Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code Suhail Ahmad Khan1*, Jagannathan V2 and Umasankari Kannan1
Abstract The advances in computer processing power have made it possible to perform a detailed pin by pin calculation of the whole core. The methods based on response matrix are being used to perform whole core transport calculations. This includes the current coupled methods based on 2D collision probability (CP) and method of characteristic (MOC). The basic approach in the whole core transport theory methods is not to homogenize the lattice cells and subdivide each cell location in the fuel assembly (FA) into finer regions. The coupling of lattice cells within the assembly and assembly to assembly coupling can be achieved using interface currents. Due to very fine discretisation of the lattice structure and large core size, the physical memory requirements for the whole core simulations are huge. This requirement is compounded if ultra-fine discretisation of energy domain is also considered. When there is an inherent symmetry one can solve for the symmetric portion of the core, thereby save both memory and computational time. Rotational symmetry boundary condition in the whole core is normally considered. Application of this boundary condition gets very complicated when the whole core is modeled by a pin by pin approach. The present paper describes the methodology to apply the rotational symmetry boundary condition in the core discretized with complex microstructures of various heterogeneous cells of the problem. Keywords Whole core transport calculation; Response matrix method; Rotational symmetry
Introduction The reactor physics calculations of nuclear reactor core are traditionally performed in two steps. First, the isolated heterogeneous fuel assembly (FA) is treated in detail using multigroup transport theory. This calculation is performed with reflective boundary or zero leakage current boundary condition. Few group homogenized parameters of FA are generated as a result of this calculation. These parameters are used to perform core calculations using traditional finite difference or nodal methods employing diffusion theory. This approach of calculation, however, suffers with the well-known *Corresponding author: Suhail Ahmad Khan, Computer Lab No. 2, 1st floor, New Training School Complex, HRDD, Anushaktinagar, Mumbai - 400094, Maharshtra, India, E-mail:
[email protected] Received: September 01, 2016 Accepted: October 06, 2016 Published: October 10, 2016
International Publisher of Science, Technology and Medicine
a SciTechnol journal assembly homogenization errors and does not take into account the inter assembly transport effects present in the actual problem. Also, the pin level heterogeneous information in final core calculation is lost and the core results represent average core behaviour. In the recent times, there is a phenomenal improvement in computer processing power. Massive parallel computer with thousands of processors are now available. This has encouraged the development of accurate models for whole core pin by pin calculation. Response matrix methods can be used to perform whole core transport theory pin by pin calculation. In the pin by pin analysis based on this approach, the heterogeneous lattice structure of fuel rod and absorber rod cells are subdivided into finer regions. The coupling between the cells in the same FA and cells of different FAs is achieved by interface currents. These interface currents can be obtained using methods based on 2D collision probability (CP) or method of characteristic (MOC). In the response matrix method based on 2D CP [1-3], the zone to zone coupling in the lattice cell is achieved using region to region CPs. The interface currents are obtained by expanding the angular flux leaving or entering a lattice cell into a finite set of linearly independent functions. In the response matrix method based on MOC [4,5], standard characteristics equation is solved along an integration chord of discretized direction. The interface current along each region boundary is obtained by integrating the angular flux using the weights and directions of characteristics. The scalar flux in the regions of lattice cells is calculated by solving the linear balance equations using these currents. Since in the present approach, the spatial discretisation extends up to single lattice cell level, the whole core pin by pin calculation has huge physical memory requirements, even for isotropic angular flux expansion. This memory requirement increases manifold when ultra-fine energy discretisation grid and higher order of angular flux expansion is considered. The huge memory requirement can be alleviated by utilizing the inherent symmetry present in the problem under consideration. The most common symmetry present in the core is rotational symmetry. For hexagonal geometry, the core may have a rotational symmetry of 60°, 120° or 180° whereas the square cores can have symmetry of 45°, 90° or 180°. The present paper describes the methodology to use the 60° rotational symmetry boundary condition in the whole core pin by pin code in hexagonal geometry. Its extension to other angles of symmetry and to square geometry can be similarly derived. To illustrate the methodology and for completeness, brief theory of response matrix method based on 2D CP & MOC is described in Section 2. Section 3 describes the methodology to implement 60° rotational symmetry in detail. Section 4 gives the application to an OECD 1000 VVER MOX computational benchmark [6]. Section 5 gives summary remarks.
Breif Theory Response matrix method based on 2D CP The integral transport equation in one energy group can be written as [1,2] (group index ‘g’ is omitted for convenience),
( ) ( )
φ r , Ω= φ rs , Ω e −τ +
Where,
rS =r − RS Ω
S
1 4π
∫
RS
0
τ R' dR ' q ( r ') e − ( )
(1)
is an arbitrary point on the line passing
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Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170
through r in the direction Ω on the surface S, where boundary conditions will be applied and r ' =r − R ' Ω τ and τ are the optical r ' distances between r and and between r and rs respectively. The total source density q ( r ) for a group of energy E is isotropic and defined as χ (E) q ( r )= ∫ dE ' ∑ S ( E ' → E )φ ( r , E ') + dE ' v ∑ f ( E ') φ ( r , E ') (2) K eff ∫ S
The equation for scalar flux is obtained by integrating Eq. (1) over all angles. This gives
φ (r= )
∫ S
e −τ S e −τ ( R ) q ( r ') dr ' Ω.nˆ _ φ _ rs , Ω dS + ∫ 2 V RS 4π R 2
(
) (
)
(3)
Where, φ _ rs , Ω is the incoming angular flux at surface S.
(
)
The outgoing flux at surface S can be obtained from Eqn. (1) as it is valid at any point. The outgoing flux is given by
(
( )
)
φ+ r 'S = , Ω φ _ rs , Ω e−τ + S
1 4π
∫
RS
0
dR ' q ( r ') e −τ ( R ')
(4)
For a given incoming angular flux to region under consideration, the system of Eqns (3) and (4) gives an exact description of the flux distribution inside the region as well as the outgoing angular flux. In order to solve these equations we have to make some numerical approximations for the scalar fluxes inside the cells and for the angular fluxes leaving and entering the cell surfaces. The scalar flux inside the cells can be expanded into a set of linearly independent functions. Inside the cell regions, flat flux approximation for the scalar flux is generally used i.e. the flux inside each zone is assumed constant. The angular flux at the cell surface is expanded into properly orthonormalized double PN expansion functions. Using these approximations, Eqns. (3) and (4), for group ‘g’ (the group index ‘g’ is omitted for simplicity), when discretised over a region consisting of N V zones and NS surfaces reduces to a set of linear flux and current equations. = ∑ j V jφ j v + ,α
= Sα J
∑α ∑ ∑β ∑µ NS
Nv
= 1= v 0
NS
Nµ
= 1= 0
v jα
P Sa J
v − ,α
+= ∑ i 1 Pji qi
(5)
Pαβ J − , β S β += ∑ i 1 P q vµ
µ
J −v,α = = ∑ β S1= ∑ µ µ 0 Aαβvµ J +µ,β N
NV
N
NV
NV i= 1 i
(6)
(7)
The summation over ν in above equations represents the order of expansion of angular flux at pincell boundary. = qi ∑ S Viφi + SiVi is the total source in region i, ∑ S is the Here, self scattering cross section within the group and Si is the fission and scattering source in a group given by i
i
G χ g '→ g = Si ∑ g '=1 ∑ Si φig ' + g g '≠ g K eff
∑
G g '=1
g' v ∑ fi φig '
(8)
Here Pji gives the probability of a neutron emitted uniformly and isotropically in region i and having its first collision in region j and Pjαv gives the probability of neutron entering through surface α in mode ν and having first collision in region j. Pαiv is the probability that neutrons emitted uniformly and isotropically in region i will escape through surface α in mode ν and Pαβv𝜇 is the probability that neutrons entering through surface β in mode μ will be transmitted through the cell and out through surface α in mode ν without making a collision 𝜇.
Eqns. (5) to (7) are the required discretized neutron transport equation for a cell under consideration. In Eqn. (5), the two terms Volume 5 • Issue 6• 1000170
on the right are the contributions to the collision rate in a region of cell from the neutrons entering through all the surfaces of the cell and sources within all the regions respectively. Similarly, in Eqn. (6), the two terms on right give the contribution to the outward current through a surface of cell from the inward currents from all other surfaces of the cell plus the sources within all regions of the cell. In Eqn. (7) Aαβνµ is the boundary condition matrix which gives a relation between the outgoing current on a given surface and the incoming current on different surfaces. Eqn. (7) describes the continuity and boundary condition to link the lattice cells through the interface currents. This describes the linking between the lattice cells of same FA as well as the cells of different FAs. This is achieved in the form of a relation between the average outgoing current on surface S β and the average incoming current on a different surface S α . In matrix form this can be written as
J − = AJ +
(9)
The elements of matrix A can be written as νµ Aαβ = βα δαβ δ νµ
(10)
Where βα is the reflection coefficient at surface α. Aαβ is zero except for those surfaces, belonging to neighboring cells, that share a common area with surface α. The albedo boundary condition at the surface α belonging to the lattice cells located at the reflector boundary is achieved by setting, βα=0. Eqns. (5) & (6) are iteratively solved using the conventional inner-outer iteration scheme to calculate partial currents across the surfaces and collision rates in each region.
Response matrix method based on MOC The current coupled MOC method is based on the so called method of ‘short characteristics’. This method has been implemented in the lattice code HELIOS [4] and APOLLO2 [5]. In this method, the standard characteristics equation along integration chord k with the constant cross section, in the discretized region i is given by [4].
(
)
( )
φi ,k r , Ω= φiin,k Ω e − ∑ + ti
r
∑
Si
φ i + Qi
4π Vi ∑ ti
(1 − e ∑ ) −
ti
r
(11)
Where, r is the distance along the direction Ω. This equation is obtained with flat flux and constant source approximations in region under consideration. The angular variable is discretized with each chord k having a directional angle θ with cosθ=μ and weight ω. The interface currents at each region boundary j is obtained by integrating the angular flux using the weights and directions of the characteristics and is given by J ij = ∑ k ∩ j ω µkφiout ,k k
(12)
Where, Jij is the outgoing current of region i through boundary j and the summation is over all chords k intersecting boundary j. The region-averaged flux is found from the following balance equation
∑ V φ= ∑ i
i i
φ + Qi + J iin − J iout
Si i
(13)
with the total in and out currents defined as J iin − J iout=
∑
j
J ij − ∑ j J ji
(14)
In Eqn. (14), Jij is the current entering region i through boundary j. Eqns (12) & (13) are iteratively solved in multigroup formalism to • Page 2 of 9 •
Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170 calculate the outgoing currents and region averaged response fluxes. The albedo boundary condition can be considered as described.
Rotational Symmetry Methodology A transport theory code TRANPIN has been developed to perform a whole core pin by pin calculation in 2D hexagonal geometry. This code uses the response matrix method based on 2D CP described above. To describe the steps for implementing the 60° rotational symmetry in whole core transport theory code, we consider the OECD VVER-1000 MOX Core Computational Benchmark problem [6]. This problem is chosen because it is realistically close to practical problems. In the present paper, the methodology to use rotational symmetry is described by considering the response matrix method based on 2D CP.
Description of benchmark The expert group at NEA has proposed a computational benchmark investigating the physics of a whole VVER-1000 reactor core using two-thirds low-enriched uranium (LEU) and one-third MOX fuel. The full details of the benchmark can be found in [6]. Since the rotational symmetry pertains to the geometry of the problem, we describe here only the geometric part of the benchmark problem in detail. The benchmark model consists of a full-size 2-D VVER-1000 core with heterogeneous 30% MOX-fuel loading. The core consists of 28 FAs considered in 60° rotational symmetry with a FA pitch of 23.6 cm. The system has an infinite axial dimension and vacuum boundary condition on the side surface. The core map is shown in Figure 1. The core consists of fresh and burned fuel assemblies (FA) of two types - graded UOX FA with U-Gd burnable absorber (BA)
rods and profiled MOX FA with U-Gd BA rods. The UOX and MOX FA configurations are given in Figures 2 and 3. Each FA consists of 331 hexagonal locations with 312 fuel pins, 18 guide tube cells with or without absorber and one central guide tube cell. The pin pitch within the FA is 1.275 cm. The core is surrounded by a reflector. The reflector is a very complicated structure consisting of a thin film of water gap, steel baffle with water holes, steel barrel, down comer water and steel pressure vessel. The water gap of 3 mm width is located between fuel assemblies and steel baffle.
Conventions considered Before describing the methodology for rotational symmetry, we will describe the broad conventions considered. The surfaces of the hexagonal fuel assembly are considered in anti-clockwise direction as shown in Figure 4. As seen in Figure 2 and 3, there is a regular hexagon structure within the fuel assembly. Beyond this regular structure, there is a thin layer of moderator. In this layer, two distinct geometric meshes are encountered which are designated as side meshes (Figure 5B) and corner mesh (Figure 5C). The surface numbering in these meshes is shown in Figure 5. The surface numbering convention for (Figure 5B) & (Figure 5C) is fixed and doesn’t change on rotation of these meshes. The assemblies in the solution domain of 1/6th core are numbered from centre to outward in a spiral manner. In the core map of Figure 1, there are a total of 46 FAs after including two layers of reflector with the same hexagonal assembly pitch. The numbering scheme of FAs in the (core + reflector) is shown in Figure 6. The meshes inside each FA are also numbered from centre to outwards in a spiral manner. The numbering scheme considered inside each FA is as shown in Figure 7.
Figure 1: Core Map of the Benchmark Problem with 30% MOX Loading.
Volume 5 • Issue 6• 1000170
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Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170
Figure 2: Graded UOX Fuel Assembly.
Figure 3: Graded MOX Fuel Assembly.
Volume 5 • Issue 6• 1000170
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Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170
Figure 4: Surface numbering for hexagonal fuel assembly.
Figure 5: Surface numbers in (A) hexagonal, (B) side and (C) corner meshes.
Volume 5 • Issue 6• 1000170
• Page 5 of 9 •
Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170
Figure 6: FA numbering in Core.
Figure 7: Spiral numbering of meshes in each FA.
Linking of meshes under rotational symmetry In the interface current method, as seen from Eq. (9), the meshes are linked through interface currents as follows: Volume 5 • Issue 6• 1000170
J in = J out of neighbour
(15)
i.e. the incoming current into the mesh is computed from the outgoing current from the neighbouring mesh through a common • Page 6 of 9 •
Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170 surface. The meshes within a fuel assembly are connected to their neighbours through one of the common surfaces shown in Figure 5. At the FA periphery, only meshes shown in Figure 5B and 5C appear. The inter assembly coupling is achieved through these meshes only. This is illustrated in Figure 8. Due to the numbering convention described here, the side meshes are connected only through common surface 1 and the corner meshes are connected through surfaces 1 & 2 to the corresponding meshes in neighbouring FAs. Table 1 gives the neighboring meshes for a representative non boundary assembly no. 14 shown in core map in Figure 6 on each side of the FA. Table 1 holds true for all the FAs not present at the two rotational boundaries shown in Figure 6. For the rotational boundary FAs, the FAs present on the left boundary have to be linked to those on the right boundary and vice versa. For the left boundary FAs, the boundary surfaces are 1 & 6 whereas boundary surfaces for right boundary FAs are 4 & 5. The meshes on these surfaces cannot be linked directly because the surfaces tend to change on account of 60° rotation. The surfaces on the left boundary will undergo 60° rotation whereas on the right boundary will undergo +60° rotation. Table 2 gives the surface number after rotation by ±60° for each of the surfaces. As seen from Table 2, the surfaces 1 & 6 of FAs on left
rotational boundary will be linked to surfaces 5 & 4 of FAs on the right boundary respectively. Similarly, the surfaces 4 & 5 of FAs appearing on the right rotational boundary will linked to surfaces 6 & 1 of FAs on left rotational boundary respectively. Table 3 gives the modified linking of peripheral meshes of a representative FA no. 12 appearing on left boundary with the meshes of FAs on right rotational boundary. This modified coupling of meshes is obtained after considering the rotation of surfaces given in Table 2. Similarly, Table 4 gives the modified coupling of meshes of representative FA no. 29 appearing on right rotational boundary with the meshes on left boundary. The meshes appearing on non boundary FA surfaces of rotational symmetry lines will follow the linking provided in Table 1. It is noted from Table 3 and 4 that only one surface of the two corner meshes appearing on the start and end of the boundary surfaces undergo modification of neighbors. The unmodified surfaces of these corner meshes are marked with asterisk in Tables 3 and 4. This modification of neighbors and neighboring surfaces is performed in geometry processing routine and stored at the start of calculation. Once this linking of the meshes is modified the incoming currents in these meshes through appropriate surfaces is calculated using Eq. (9) during the iteration process. Eqns. (5) and (6) give the required flux and current values in the meshes of 1/6th core problem shown in Figure 1.
Figure 8: Mesh Connectivity between neighbouring fuel assemblies.
Volume 5 • Issue 6• 1000170
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Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170 Table 1: Neighbors of Peripheral meshes of FA No. 14. Surface No.
Meshes of
Meshes in the Neighbouring FAs
of FA 14
FA 14
(Numbers in bracket are FA numbers)
365
387 (9) &343 (13)
366 to 375
342 to 333 (13) in order
376
332 (13) & 354 (19)
377 to 386
353 to 344 (19) in order
387
343 (19) & 365 (20)
388 to 397
364 to 355 (20) in order
332
354 (20) & 376 (15)
333 to 342
375 to 366 (15) in order
343
365 (15) & 387 (10)
344 to 353
386 to 377 (10) in order
354
376 (10) & 332 (9)
355 to 364
397 to 389 (9) in order
1 2 3 4 5 6
identified by differentiating on the basis of the geometric shape or material present in cells. Four CP matrices in Eqns (5) & (6) are calculated only for these distinct cells in the whole core. The present benchmark problem provides the material composition of all the required materials of the problem. For the purpose of illustrating the application of rotational symmetry, the calculations are performed in five energy groups. Five group cross sections were prepared for fuel, clad and coolant regions present in the distinct lattice cells. The five group cross sections are obtained by performing a 2D lattice cell calculation using a cross section library ‘htemplib’ in WIMS format for each distinct cell. This library has cross section for 185 elements in 172 groups. The macroscopic cross sections of fuel, clad and coolant regions are obtained by collapsing the 172 group cross sections to five groups by volume and energy weighting using the flux obtained in single cell calculation.
Surface No.
Meshes of
Meshes in the Neighbouring FAs
Using the five group macroscopic cross sections, the core calculation is performed for full core and 1/6th core configurations. Table 5 gives the comparison of eigenvalue with MCNP & MCU results provided in the benchmark report. The present results are obtained using a convergence criterion of 10-7 for multiplication factor and 10-5 for flux respectively. Each hexagonal lattice cell is subdivided into three fuel rings, one clad and seven coolant regions. The identical eigenvalue for the two types of simulations demonstrate that the 60° rotational symmetry boundary condition as implemented in TRANPIN code system is reliable. The TRANPIN shows a deviation of 8.8mk from MCNP and 4.8 mk from MCU. Here it should be noted that both MCNP & MCU use a different cross section set in continuous energy format.
of FA 12
FA 12
(Numbers in bracket are FA numbers)
365
332 (16) &354 (22)
Conclusion
366 to 375
353 to 344 (22) in order
376
343(22) & 354* (17)
354
376* (8) & 343 (16)
355 to 364
342 to 333 (16) in order
Table 2: Effect of Rotation on the surfaces of FA. -60° Rotation
+60° Rotation
(Anticlockwise)
(Clockwise)
Surface No.
Surface after Rotation
Surface No.
Surface after Rotation
1
2
1
6
2
3
2
1
3
4
3
2
4
5
4
3
5
6
5
4
6
1
6
6
Table 3: Neighbors of Peripheral meshes of FA No. 12 on left boundary.
1 2 6
Table 4: Neighbors of Peripheral meshes of FA No. 29 on right boundary. Surface No.
Meshes of
Meshes in the Neighbouring FAs
of FA 14
FA 14
(Numbers in bracket are FA numbers)
332
354* (37) & 365 (23)
333 to 342
364 to 355 (23) in order
343
354 (23) & 376 (17)
344 to 353
375 to 366 (17) in order
354
365 (17) & 332* (22)
4 5 6
Note: *Meshes connected without rotational symmetry. Table 5: Comparison of Core keff. TRANPIN
Core Configuration Full Core
1/6th Core
MCNP
All Rods OUT 1.028315 1.028315 1.03770±0.007%
MCU 1.03341±0.013%
Results and Discussion The present benchmark problem is studied using the whole core analysis code TRANPIN. In TRANPIN, CP matrices for all the lattice cells present in the problem domain are not calculated. First, FA cells within the core and reflector are identified which are materially distinct. Afterwards, within each FA cell, unique lattice cells are Volume 5 • Issue 6• 1000170
The whole core transport theory codes are being increasingly developed due to the availability of computational resources. The response matrix methods based on 2D CP or MOC can be used for performing whole core calculation without homogenization. In response matrix method, the coupling between the cells in the same FA and cells of different FAs is achieved using interface currents. Due to the fine mesh division to the level of a single lattice cell, the whole core transport theory codes have huge physical memory requirements. These memory requirements can be reduced using the inherent symmetry present in the core. The methodology to use 60° rotational symmetry for the whole core calculation in hexagonal geometry is described in the present paper. A whole core transport theory code TRANPIN in 2D hexagonal geometry has been developed. The code performs the full core calculation using response matrix method based on 2D CP. The code TRANPIN is used to analyze full core and 1/6th core of the OECD VVER-1000 MOX Core Computational Benchmark problem. The use of rotational symmetry makes the calculation feasible even on a desktop computer. References 1. Hébert A (2010) The collision probability method. In: Handbook of Nuclear Engineering. Springer, US. 2. Marleau G (2001) Dragon Theory Manual Part 1: Collision Probability Calculations, IGE-236 Rev. 1. 3. Khan SA, Mathur A, Jagannathan V (2016) Incorporation of Interface Current Method Based on 2D CP Approach in VISWAM Code System for Hexagonal Geometry. Annals of Nuclear Energy 92: 161-174.
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Citation: Khan SA, Jagannathan V, Kannan U (2016) Rotational Symmetry Boundary Condition in Current Coupled Whole Core Pin by Pin Transport Theory Code. J Nucl Ene Sci Power Generat Technol 5:6.
doi: 10.4172/2325-9809.1000170 4. Wemple CA, Gheorghiu HNM, Stamm’ler RJJ, Villarino EA (2008) Recent advances in the HELIOS-2 lattice physics code. In Proceedings of the international conference on the Physics of Reactors (PHYSOR2008), Interlaken, Switzerland.
5. Masiello E, Sanchez R, Zmijarevic I (2009) New Numerical Solution with the Method of Short Characteristics for 2-D Heterogeneous Cartesian Cells in the APOLLO2 Code: Numerical Analysis and Tests. Nucl Sci Eng 161: 257. 6. VVER-1000 MOX Core Computational Benchmark. NEA/NSC/DOC 17.
Author Affiliations
Top
Fuel Cycle Studies Section, Reactor Physics Design Division, BARC, India 2 Light Water Reactor Physics Section, Reactor Physics Design Division (Retired), BARC, India 1
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