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ISSN 1810-5408 Nuclear Science and Technology Volume 3, Number 4, December 2013 Published by VIETNAM ATOMIC ENERGY SOCIETY NUCLEAR SCIENCE AND TECHNOLOGY Volume 3, Number 4, December 2013 Editorial Board Editor-in-chief Tran Huu Phat (VINATOM) Executive Editors Vuong Huu Tan (VARANS) Le Van Hong (VINATOM) Cao Đình Thanh (VINATOM) Editors Phan Sy An (HMU) Cao Chi (VINATOM) Nguyen Nhi Dien (VINATOM) Bui Dieu (NCI) Le Ngoc Ha (Tran Hung Dao Hospital) Duong Ngoc Hai (IOM) Le Huy Ham (VAAS) Nguyen Quoc Hien (VINATOM) Bui Hoc (HUMG) Nguyen Phuc (VINATOM) Nguyen Tuan Khai (VINATOM) Hoang Anh Tuan (VAEA) Ngo Quang Huy (HUI) Le Hong Khiem (IOP) Dao Tien Khoa (VINATOM) Do Ngoc Lien (VINATOM) Dang Duc Nhan (VINATOM) Nguyen Mong Sinh (VINATOM) Le Xuan Tham (DOST of Lamdong) Tran Duc Thiep (IOP) Le Ba Thuan (VINATOM) Huynh Van Trung (VINATOM) Dang Thanh Luong (VARANS) Nguyen Thi Kim Dung (VINATOM) Pham Dinh Khang (VINATOM) Foreign Editors Pierre Darriulat (INST) Myung Chul Lee (WFNM) IU.E.Ponionzkevich (DUBNA , Russia) Hideki Namba (JAEA, Japan) Philippe Quentin (CENBG, CNRS, France) Yang (KAERI, Korea) Kato Yasuyoshi (TIT, Japan) Managing Secretary Nguyen Trong Trang (VINATOM) Science Secretary Hoang Sy Than (VINATOM) .................................................................................................................................................................................................................... Copyright: ©2008 by the Vietnam Atomic Energy Society (VAES), Vietnam Atomic Energy Institute (VINATOM). Pusblished by Vietnam Atomic Energy Society, 59 Ly Thuong Kiet, Hanoi, Vietnam Tel: 84-4-39420463 Fax: 84-4-39424133 Email: nuscitech@vinatom.gov.vn Vietnam Atomic Energy Institute, 59 Ly Thuong Kiet, Hanoi, Vietnam Tel: 84-4-39420463 Fax: 84-4-39422625 Email: infor.vinatom@hn.vnn.vn .................................................................................................................................................................................................................... Contents Development of a multi-group neutron noise simulator for fast reactors Tran Hoai Nam.....................................................................................................................1 Adsorption of U(VI) ion by modifiled chitosan flakes: optimization of process parameters and thermodynamic studies Ho Thi Yeu Ly, Nguyen Mong Sinh ................................................................................12 Validation of neutronics libraries through benchmarks and critical configurations of The Dalat Nuclear Research Reactor using low enriched uranium fuel by monte carlo method Nguyen Kien Cuong, Huynh Ton Nghiem, Luong Ba Vien, Le Vinh Vinh ......................... 20 Study on decontamination of microorganisms in garlic powder by gamma Co-60 radiation Vo Thi Kim Lang, Nguyen Thuy Khanh, Doan Thi The, Nguyen Thi Tu Trinh .............29 Study on gamma-irradiation degradation of chitosan swollen in H2O2 solution and its antimicrobial activityfor E. coli Dang Xuan Du, Bui Phuoc Phuc, Tran Thi Thuy, Le Anh Quoc, Dang Van Phu, Nguyen Quoc Hien ……………………………………………………………………….............33 Relations between phia oac granitoids and uranium in Nguyen Binh area - Cao Bang province Nguyen Chien Dong, Duong Hong Son, Bui Dinh Cong, Ho Huu Hieu, Nguyen Thi Hoang Linh……………………………………………………………………………………...40 Parallel radon and thoron measurements in mineral sand mining area of Ha Tinh province Bui Dac Dung, Trinh Van Giap, Tibor Kovacs, Tran Ngoc Toan, Le Dinh Cuong, Tran Khanh Minh, Nguyen Huu Quyet, Nguyen Van Khanh…………………………….49 Nuclear Science and Technology, Vol. 3, No. 4 (2013), pp. 1-11 Development of a multi-group neutron noise simulator for fast reactors Tran Hoai Nam Division of Nuclear Engineering, Department of Applied Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden Email: nam@nephy.chalmers.se (Received 7 Octorber 2013, accepted 11 February 2014) Abstract: The paper presents the development of a neutron noise simulator for fast reactors based on diffusion theory with multi-energy groups. The noise sources are modelled via small stationary fluctuations of macroscopic cross sections, and the induced first order noise is solved fully in the frequency domain. The tool is expected to be applicable for monitoring of fast reactor cores and also for other reactor types with hexagonal fuel assemblies. Some numerical calculations of the neutron noise induced by localized perturbations in a sodium-cooled fast reactor are demonstrated. Keywords: Neutron noise, fast reactor, hexagonal geometry, ESFR I. INTRODUCTION Neutron noise analysis has been early considered as a powerful technique in reactor diagnostics and core monitoring [1][2]. Numerical simulation of the neutron noise still remains a challenge for describing detector signals and improving core surveillance. Recently, several attempts have been carried out for the development of numerical simulations of the space- and frequency-dependent neutron noise for LWRs based on two-group diffusion theory, such as the CORE SIM code using a finite difference method in Cartesian geometries [3], an analytical nodal method [4], and hexagonal geometries [5][6][7]. The neutron noise is solved in the frequency domain, while the noise source is modelled via the fluctuations of macroscopic cross sections. Various types of the noise, e.g. perturbations of cross sections and/or vibrating absorber or fuel assemblies, can be simulated through defining the fluctuations of cross sections as input parameters. Measurement of the neutron noise in fast reactors and a test facility has also been done [8][9][10]. However, the experience and knowledge of the noise properties in fast reactors are much less than that in LWRs. Therefore, it is highly desirable to extend the noise calculation method to fast systems. The development of a new noise simulator for fast reactors was also motivated by the needs to support the safety in design and operation of future SFRs. The tool is expected to give a possibility to investigate the noise behaviour in the SFR cores and support the development of the neutron instrumentation. This paper presents the numerical development of a neutron noise simulator for fast reactors with hexagonal fuel assemblies. The numerical implementation was based on diffusion theory with multi-energy groups for solving both the static and noise equations, where the neutron noise equation was solved fully in the frequency-domain. Numerical calculation and verification have been performed for a large SFR core. In the present paper, calculations of the space- and frequency- ©2013 Vietnam Atomic Energy Society and Vietnam Atomic Energy Institute DEVELOPMENT OF A MULTI-GROUP NEUTRON NOISE SIMULATOR FOR FAST REACTORS dependent noise induced by the perturbation of the absorption cross section near the core center have been demonstrated. II. NEUTRON NOISE SIMULATOR p is expressed via the prompt,  g and delayed,  gd, j spectra as follows:  g  (1   )  gp    j  gd, j . A. Static and noise equations (3) j In order to solve the neutron noise equation, it is necessary to define a noise source which is usually modelled through the fluctuations of static cross sections. For the noise source definition, it is necessary to know the static state of the system such as the keff and the static fluxes. This means that the solution of the static equation is also required. Thus, in this simulator, the numerical implementation is for solving both the static and noise equations. The multi-group diffusion equation for the static state is written as follows:    Dg g (r )   t , gg (r )  χg is the fission energy spectrum, which 1  g  f , g 'g ' (r ) keff g ' Further, j= 1, 2, ... J denotes the group of delayed neutron precursors, and β is the total fraction of delayed neutrons   j . All cross sections in Eqs. (1) and (2) are space-dependent but for the sake of brevity, the space-dependence is dropped. The multi-group noise equation is obtained from the space- and time-dependent diffusion equations by assuming that all timedependent terms, X (r , t ) , can be split into a stationary    s , g ' gg ' (r ) (4) j component, X0 (r ) , which corresponds to the value at the steady state, plus a small fluctuation,  X (r , t ) as g ' g (1) where g = 1, 2, ... G denotes the energy group, ϕ g is the neutron flux in group g, Dg is the diffusion coefficient in group g, νΣf,g is the production cross section in group g, Σs,g’→g is the scattering cross section from group g' to group g, Σt,g is the total cross section, which is X (r , t )  X0 (r )   X (r , t ) . (5) By assuming that the fluctuations are small so that only the first order noise needs to be taken into account, products of fluctuating terms can be neglected and the result is a linear equation for the fluctuation of the flux. Subtracting the static equation and after performing a Fourier transform of all timedependent terms,  X (r , t ) , as   X (r ,  )    X (r , t )eit dt ,  (6) defined as  t , g   a , g    s , g ' g , (2) g ' g Σa,g is the absorption cross section in group g, 2 with the assumption that the system was in the unperturbed (critical) state at t   , the first order space- and frequency-dependent neutron noise in multi-group diffusion theory is written as follows TRAN HOAI NAM    Dg g (r ,  )   t , g ( )g (r ,  )  The last term on the r.h.s of Eq. (7) is the noise source, which is calculated from the fluctuations of macroscopic cross sections as 1  g ( ) f , g 'g ' (r ,  )   s , g ' gg ' (r ,  ) keff g' g ' g Sg (r ,  )  a , g ( )g (r )  Sg (r ,  )    s , g  g ' ( )g (r )    s , g ' g ( )g ' (r ) (7) g ' g where, g (r ,  ) denotes the space- and  frequency-dependent noise in group g. The frequency-dependent total cross section in Eq. (7) is written as t , g ( )   a , g ( )    s , g ' g , (8) (9) The keff in the noise equation is the eigenvalue obtained from the static calculation.  g ( ) denotes the frequency-dependent fission energy spectrum, which is obtained from the equation of delayed neutron as  g ( )   g    gd, j j i j .  j  i  g ( )   f , g ' ( )  g ' (r ) g' In this model, the fluctuation of the diffusion coefficient is neglected. The effect of the fluctuation of the diffusion coefficient on the space-dependent noise will be investigated in a separate work.The static equation (1) is an eigenvalue problem, where the keff and the static fluxes correspond to the fundamental mode, while the balance equation for the neutron noise, as given by Eq. (7), is an inhomogeneous equation with an external source. Another important difference is that all quantities in Eq. (7) are frequency-dependent, i.e. complex quantities. with i . g keff (11) g ' g a , g ( )  a , g  1 g ' g (10) Fig. 1. Triangular discretization of a hexagonal system in a 60 degrees domain. ( i/j) coordinates are used to handle the triangular meshes, while (u/v) coordinates are used to handle the hexagonal meshes. The number in each triangular mesh indicates its j coordinate. 3 DEVELOPMENT OF A MULTI-GROUP NEUTRON NOISE SIMULATOR FOR FAST REACTORS a n,n '  B. Numerical solution methods Numerical solutions for both the static and noise equations are based on Matlab(c), which can easily handle complex quantities in the noise equation. Finite differences are used for the spatial discretization and a power iterative solution is implemented for solving both the static and noise equations. A hexagonal assembly is divided radially into six triangular right prisms. Therefore, in a 3D model, each fundamental node has five interfaces including two equilateral triangular bases and three rectangular sides. For higher accuracy, the six triangular right prisms can be continuously subdivided radially into 24 finer triangular meshes, where the finer meshes are also equilateral triangles. Fig. 1 shows the spatial discretization of a hexagonal system in the 60 degrees domain. With this discretization scheme, the balance equation in each group in a node n is written as where h is the distance between the centers of two neighboring nodes. In the radial direction, the distance is h  L / 3 with L being the triangular edge of a node. If the neighboring node n' is a boundary, the current is calculated as J n,n '  '1 , ' , ' (1   )2 Dn  h  4 Dn n (14) where a vacuum boundary has   0 and a reflective boundary has   1 . This boundary condition allows the modeling of a half or one-sixth of a symmetrical system. The same boundary condition (14) is used for both the static and noise calculations. A power iterative solution procedure was implemented with outer and inner iteration sweeps for solving the balance equations of both the static equations and the noise equations with a given noise source, where the noise equations are fully solved in the frequency domain. In the outer iteration sweep, 5 An n J n n  t nV  QnV  n 2 Dn Dn ' h Dn  Dn ' (12) where J n,n ' is the surface-averaged net current from node n through the interface with the keff and static flux g in case of static a neighboring node n', and An ,n ' is the area of solution or the noise g are updated after the interface of the two nodes.  n represents each iteration, and the source term in the static and noise equations, Qn , is calculated. The the static flux or the neutron noise, and Qn is inner iteration sweep solves the balance equation iteratively throughout the number of energy groups and the triangular meshes on each axial plane. When solving the balance equation through the i- or j-coordinates, the diffusion terms related to the neighboring nodes on the two perpendicular coordinates on the l.h.s of the balance equations are moved to the r.h.s. By using this expression, when written in a matrix form, the l.h.s. has a the total source on the r.h.s of Eq. (1) or (7). Since Eq. (12) is group-independent, one simply drops the group symbol. In the finite difference approximation the surface-averaged net current J n,n ' is calculated as J n,n '  an,n ' (n  n ' ) . (13) Here, the coupling coefficient a n ,n ' is given as 4 TRAN HOAI NAM tridiagonal form and is easy to be handled numerically. With the homogenized cross sections and the correction of the surface-averaged net current, the solution for a coarse mesh system C. CMFD acceleration hom can be performed to obtain the keff and the Coarse mesh finite difference (CMFD) acceleration has been implemented for accelerating convergence of both the static and noise solutions, where a coarse mesh is radially defined as a hexagonal assembly. Fig. 1 illustrates the discretization method to handle the triangular fine mesh system in the (i/j) coordinates and the hexagonal coarse mesh system in the (u/v) coordinates. In order to solve a coarse mesh system, homogenization of cross sections in a coarse mesh N is needed as follows: hom ,g  flux,  g , N . Then, a re-balance factor for each coarse mesh is calculated as fg , N    g (r )dV unaccelerated fine mesh fluxes which needs to be renormalized wit  g , N The solution procedure is summarized as follows: (15) (1) Estimate an initial solution by starting with the coarse mesh solution, so that a good estimation of the initial keff and flux shape for the static solution can be obtained. This step is optional for the noise calculation. where,  , g represents Dg ,  a , g , s , g g ' or   f , g . The integration is taken over the coarse mesh N, to which the fine mesh n belongs. In order to preserve the conservation of the surface-averaged net current through each interface of a coarse mesh N, the correction of the net current is calculated as 1  AN , N ' nN ,n 'N ' J n,n ' An,n ' (2) Estimate a fine mesh solution using the keff and the flux shape or the noise shape obtained in step (1) as a previous solution. (3) Calculate the average flux, the homogenized cross sections and the correction of the surface-averaged net current according to Eqs. (15) - (16). (16) (4) Estimate the solution for a coarse mesh system using the homogenized cross sections and the correction of the surfacedaveraged net current until convergence is obtained. The coupling finite difference coefficient for a coarse mesh system is now written as âN, N '  a N, N '  ã N, N ' (17) where, ã N , N ' is the correction factor to (5) Calculate the rebalance factor as given in Eq. (19). preserve the surface-averaged net current ã N,N '  J N , N '  a N , N ' ( N   N ' ) . N  N ' in the static calculation. N J N,N '  (19) where,  g , N is the average using the   , g (r ) g (r )dV N g ,N  g ,N (6) Update the flux or the neutron noise using the rebalance factor as (18) 5 DEVELOPMENT OF A MULTI-GROUP NEUTRON NOISE SIMULATOR FOR FAST REACTORS g ,n  fg , Ng ,n . The static calculations of the ESFR core [11] at the beginning of cycle with 33 energy groups have been performed using the noise simulator and compared with ERANOS results. The cross section data used in the noise simulator were generated by ERANOS [11]. The calculated domain of the 1/6th core is constructed by 128 assemblies. Each assembly is radially divided into 6 triangular meshes. The axial core height is divided into 60 meshes, so that the axial height of each mesh in the fuel region is 5 cm. Fig. 2 shows the radial configuration of 1/6th of the ESFR core model in the noise simulator. (20) In the static problem, keff is also updated as keff  keffhom . (21) (7) Calculate the source term, Qn , in the heterogeneous system using the updated keff and fluxes or the neutron noise in step (6). (8) Repeat steps (1) to (7) until convergence is obtained. III. NUMERICAL CALCULATIONS A. Comparison of static calculations with ERANOS Fig. 2. Configuration of 1/6th of the ESFR core. (u/v)-coordinates shows the locations of the assemblies. For verification of the static module of the noise simulator, comparison with the results from ERANOS has been done for the keff and the space-dependent neutron flux. The integral production rates over each fuel assembly have also been calculated and compared with ERANOS. The keff of 1.00623 is obtained using this simulator. Compared to 6 TRAN HOAI NAM the value of 1.00792 obtained with ERANOS, Fig. 4 shows the comparison of the production rate distribution in the core, where the production rate was integrated over each fuel assembly. One can see that the results obtained with the two codes have a good agreement. The relative deviation between the two calculations is within 0.5% near the core center and around outer regions. The same deviation or even less than 0.5% is found in the middle core. At some fuel assemblies near the periphery, the deviation is larger but within 1.5% except a few assemblies at the outer periphery where the relative deviation is found to be 3.0%. This agreement is adequate since the two calculation codes are based on different methods, the one in this work based on diffusion theory with a finite difference method and six triangular meshes per hexagon, while the neutronics module of ERANOS solves the transport equation using the nodal method. This agreement is acceptable for assuring the noise calculations and investigating the noise properties. the difference of 170 pcm in reactivity is found between the two calculations. It is noteworthy that the two codes are based on two different neutronics solvers: a nodal transport module of ERANOS was used, meanwhile the static module of the present simulator solves the diffusion equations using a finite difference method. Comparison of neutron spectra has been performed at several locations in the fuel region and the reflector as displayed in Fig. 3. The spectra calculated using this simulator has a good agreement with that obtained with ERANOS. In the reflector region the difference of the flux is larger in the thermal and epithermal range and in some groups in the fast energy range, especially with energy greater than 1.0 MeV. This may result in some differences at fuel assemblies near the outer periphery. Fig. 3. Comparison of the neutron spectrum at the midplane of assemblies (2/2) and (14/14). 7