Legendre expansion in multilayer slab geometry

The monoenergetic integral transport equation for a multilayer slab geometry has been solved by the Legendre expansion method. The method utilizes an expansion of the flux density in each layer in Legendre polynomials of the position co-ordinate. The use of these polynomials makes it possible to calculate most of the resulting matrix by means of recurrence formulae. These formulae have been obtained by a procedure which is an extension of Carlvik's method for a homogeneous slab. A code (MULREG) has been written for this purpose. Using MULREG a series of calculations have been performed for a homogeneous slab with vacuum boundary conditions. The slab has been divided into NR number of regions and in each region flux is expanded into Legendre polynomials of order NSI. For a particular value of NR, the NSI varies from 0 to 4. The effective multiplication factor k_eff of the slab is calculated. By comparing the computational time for all the cases, it is studied as how severe it is to consider the flat flux approximation (conventional collision-probability approach) as compared to a case when more terms in the flux expansion are considered per layer.     

Improved nodal expansion method for solving neutron diffusion equation in cylindrical geometry

The challenges encountered in the development of nodal expansion method (NEM) in cylindrical geometry and the method to circumvent these difficulties are introduced and discussed in this paper. Due to the fact that the azimuthal term contains a factor 1/r², the traditional transverse integration fails to produce a 1D transverse integrated equation in θ-direction; a simple but effective approach is employed to obtain the θ-directional transverse integration equation. When the traditional polynomials are used to solve the 1D transverse integral equation in r-direction, some additional approximations, which may undermine the precision of the method, are required in the derivation of the moment equations; in order to preserve the accuracy of calculations, the special polynomial approximation is used to solve the 1D transverse integrated equations in r-direction. Moreover, the Row-Column iterative scheme, which is considered to be the more efficient and convenient schemes in cylindrical geometry, is used to solve the partial currents equations. An improved NEM for solving the multidimensional diffusion equation in cylindrical geometry is implemented and tested. And its accuracy and efficiency are demonstrated through several benchmark problems.     

The finite-element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry

Proof-tests on 1-D multigroup neutron transport problems are reported for strong anisotropic scattering. These tests have been undertaken as part of the validation of the 3-D multigroup finite-element transport code fel tran for ansisotropic scattering media. To illustrate the treatment of within-group and intergroup anisotropic scattering in the finite-element method the relevant theory is outlined. Ingroup scattering is checked using the backward-forward-isotropic (BFI) scattering law for source and eigenvalue problems. With this law anisotropic scattering problems can be transformed into equivalent isotropic scattering problems. In this way the well-validated isotropic scattering version of fel tran is used to validate the anisotropic version. Intergroup scattering effects are checked by solving few-group source problems for P₁ and P₃ scattering and the BFI scattering law. For P₁ and P₃ scattering checks are made with the discrete-ordinate finite-difference code anisn and the spherical harmonics finite-difference code marc/pn. For the BFI scattering law comparison is made with two-group exact solutions of Williams (1985) for 1-D systems.     

Flux expansion nodal method for solving multigroup neutron diffusion equations in hexagonal-z geometry

A flux expansion nodal method (FENM) has been developed to solve multigroup neutron diffusion equations in hexagonal-z geometry. In this method, the intranodal fluxes are expanded into a set of analytic basis functions for each group. In order to improve the nodal coupling relations, a new type of nodal boundary conditions is proposed, which requires the continuity of both the zero- and first-order moments of partial currents across the nodal surfaces. The response matrix technique is used for the iterative solution of the nodal diffusion equations, which greatly improves the computational efficiency. The numerical results for a series of benchmark problems show that FENM is a very accurate and efficient method for the prediction of criticality and nodal power distributions in the reactors with hexagonal assemblies.     

Sub-cell balanced nodal expansion methods using S4 eigenfunctions for multi-group SN transport problems in slab geometry

A highly accurate S₄ eigenfunction-based nodal method has been developed to solve multi-group discrete ordinate neutral particle transport problems with a linearly anisotropic scattering in slab geometry. The new method solves the even-parity form of discrete ordinates transport equation with an arbitrary S_N order angular quadrature using two sub-cell balance equations and the S₄ eigenfunctions of within-group transport equation. The four eigenfunctions from S₄ approximation have been chosen as basis functions for the spatial expansion of the angular flux in each mesh. The constant and cubic polynomial approximations are adopted for the scattering source terms from other energy groups and fission source. A nodal method using the conventional polynomial expansion and the sub-cell balances was also developed to be used for demonstrating the high accuracy of the new methods. Using the new methods, a multi-group eigenvalue problem has been solved as well as fixed source problems. The numerical test results of one-group problem show that the new method has third-order accuracy as mesh size is finely refined and it has much higher accuracies for large meshes than the diamond differencing method and the nodal method using sub-cell balances and polynomial expansion of angular flux. For multi-group problems including eigenvalue problem, it was demonstrated that the new method using the cubic polynomial approximation of the sources could produce very accurate solutions even with large mesh sizes.     

Spectral Green's function method for neutron transport: isotropic,forward, and backward scattering in 1-D slab geometry

Numerical solutions of one-group and one-dimensional neutron transport problems are reported for isotropic, forward, and backward scattering. Numerical solution is carried out by using two different methods, the SGF “ spectral Green's function ” method and the DD “ diamond-difference” scheme, to test the accuracy of the results. Results of cell-edge scalar fluxes obtained for both methods are presented in the tables.     

Attenuation of pile neutron flux in polyethylene

The relaxation lengthsofresearch reactor neutrons were measured in polyethyIene. The findings are in excellent accord with data computed theoretically using the method of moments. The relaxation lengths are 15% less than in the case of water.Translated from Atomnaya Énergiya, Vol. 15, No. 1, pp. 17–20, July, 1963     

加速器中子源的中子注量测量方法

在用静电加速器中子源标定探测器的中子灵敏度实验中,采用“BF3长计数管 定标器”系统过渡,用^197Au中子活化分析方法达到了对中子注量在线、绝对监测的目的。这种方法给出与加速器束流不同角度、不同距离处的中子注量。介绍了这种中子注量测量方法。     

Development of an innovative neutron flux mapping system

An innovative in-core flux mapping system has been developed and applied successfully for service in commercial pressurized water reactors. With the benefit of double indexing path selector mechanism, the reliability of the detector drive system has been improved five times higher than that of a conventional system. The problems caused by strong friction generated between the detector cable and guide tubes have been resolved by the double indexing path selector architecture enabling the detector guide tubes to have smaller curvature and shorter length in nature. The system normally exerts two-thirds of force to make the detector reach the top of the fuel position, compared with conventional ones. In addition, simple and fast maintenance is realized by optimizing the number of components in the detector drive system and providing easy access to the components, thereby guaranteeing minimum radiation exposure to maintenance personnel. The programmable logic controller-based digital controller with Windows^®-based operator console provides fully automated and user-friendly operation and maintenance support means. The developed in-core flux mapping systems have been deployed at the Kori nuclear units 1–4.     

Preliminary study of high neutron flux fusion heating

A heating scheme for nuclear fusion is proposed based on the availability of a high flux, low energy neutron source. The heat is derived in the reaction ⁶Li (n, T) ⁴He resulting from the incidence of a low energy neutron beam on a sample of ⁶Li D. The energy release per reaction, Q = 4.6 MeV, is converted through electron Coulomb collisions thereby quickly dissociating the solid sample to the plasma state. For 10⁻³ eV neutrons it is estimated that this dissociation occurs in 7 ms for an incident flux of 10¹⁷ cm⁻² · s⁻¹. The possibility of further driving the heated fuel to fusion is also discussed.     

Evolution of the neutron sensor characteristics in the RBMK-1500 reactor neutron flux

Variation of characteristics of the RBMK-1500 reactor radial neutron flux sensors with the HfO₂ emitter during long-term maintenance was investigated. The influence of nuclear fuel enrichment and burnable erbium admixtures on the energy neutron spectrum, neutron absorption, and hafnium isotopic composition variation was considered. The dependences of corrective factors of the neutron sensor signal on the nuclear fuel burnup depth and the integral current accumulated by the sensor for different enrichment nuclear fuel are presented in the work. The experimental verification of the calculated dependence of the sensor corrective factor on the accumulated integral current was performed.     

能谱变化对239Pu裂变室测量快中子注量的影响

热中子和共振区的中子在快中子临界装置中所占的份额很小,但是由于其相对大的截面,在慢化物存在的情况下,热中子和共振中子份额的微小变化,对^239Pu裂变室测量中子注量的结果影响很大。通过测量^239Pu裂变电离室在包镉和包硼、周围有无慢化物等情况下的反应率,Au、In活化片的镉比,S活化片在能谱变化下与^239。Pu的反应率比等,分析了快中子临界装置中热中子和共振区中子的分布,讨论了中子能谱变化对^239Pu裂变室测量快中子注量的影响及解决办法。     

A critique of inverse solutions to slab geometry transport problems

An inverse transport problem requires determination of the angular scattering and absorption coefficients of the medium using measurements of the intensity. Methods for solving such a problem for monoenergetic transport in a thick homogeneous (i.e. multiple-scattering) slab medium are critiqued. The methods include those that require local measurement of the intensity inside the slab plus remote measurement of the angular distributions entering and leaving (the local-&-remote methods) and those (remote methods) that require only the surface angular distributions. The possible use of these methods to determine the properties of a multi-layer slab medium is also examined.     

Cellular neural networks (CNN) simulation for the TN approximation of the time dependent neutron transport equation in slab geometry

This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T_N) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T_N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T_N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space–time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision).     

Direct measurement of mechanical momentum of a neutron flux

Translated from Atomnaya Énergiya, Vol. 66, No. 1, pp. 48–49, January, 1989.     

Development of self-powered neutron detectors for neutron flux monitoring in HCLL and HCPB ITER-TBM

Self powered neutron detectors (SPND) have a number of interesting properties (e.g. small dimensions, capability to operate in harsh environments, absence of external bias), so they are attractive neutron monitors for TBM in ITER. However, commercially available SPNDs are optimized for operation in a thermal nuclear reactor where the neutron spectrum is much softer than that expected in a TBM. This fact can limit the use of SPND in a TBM since the effective cross sections for the production of beta emitters are much lower in a fast neutron spectrum.This work represents the first attempt to study SPNDs as neutron flux monitors for TBM. Three state-of-the-art SPND available on the market were bought and tested using fast neutrons at TAPIRO fast neutron source of ENEA Casaccia and with 14 MeV neutrons at the Frascati neutron generator (FNG).The results clearly indicate that in fast neutron spectra, the response of SPNDs is much lower than in thermal neutron flux. Activation calculations were performed using the FISPACT code to find out possible material candidates for SPND suitable for operation in TBM neutron spectra.     

Polynomial approximation of neutron flux in first-collision probability method

Conclusions The generalized first-collision method may prove useful in calculating channels with complicated geometry — e.g., RBMK channels. Then polynomials of different order may be used in different zones, taking into account, where necessary, the curvature of the neutron distribution by means of quadratic terms.Translated from Atomnaya Énergiya, Vol. 48, No. 2, pp. 80–84, February, 1980.