Perturbation theory for nuclear fuel depletion calculations with predictor–corrector method

The perturbation theory for nuclear fuel depletion calculations with the predictor–corrector method is derived. This theory is implemented to a reactor physics code system CBZ, and the theory itself and its implementation are numerically verified. Sensitivities of nuclide number densities after fuel depletion with respect to nuclear data calculated with this theory are compared with reference sensitivities calculated by numerical differentiation, and good agreements are obtained. Importance of accurate angle integration on product of neutron flux and generalized adjoint neutron flux is also pointed out. Sensitivities in a 3×3 multi-cell system including a gadolinium-bearing fuel pin are calculated, and it is demonstrated that the derived theory yields accurate sensitivities even if coarse depletion time step division is adopted. The present work drastically increases the applicability of the depletion perturbation theory to actual problems.     

A rigorous weight function for neutron kinetics equations of the quasi-static method for subcritical systems

The purpose of the present work is to develop an efficient solution method to solve the time dependent multi-group diffusion equations for subcritical systems with external sources using the quasi-static method.Usually, the k-eigenfunction for an adjoint criticality equation is used as a weight function to derive a one-point neutron kinetics equation for the amplitude function in the quasi-static method. It is shown that the use of this k-eigenfunction introduces a first order error due to the change of the flux, when the systems are not close to the critical state. It is shown also that the use of the ω-eigenfunction for the adjoint time dependent equation as the weight function can eliminate such first order error resulting from ignoring the term of first order change of the shape function to solve subcriticality problems, and it gives more accurate results than the use of conventional k-eigenfunctions of the critical adjoint equation.     

HGPT based sensitivity time-dependent methods for the analysis of subcritical systems

In recent years an increasing interest is observed with respect to subcritical, accelerator driven systems (ADS). Considering the attention being given to these systems for their supposed ability to play a major role as actinides incinerators, as well as power production plants, the application of the heuristically-based generalized perturbation theory (HGPT) methodology for the cycle life analysis of these systems is reviewed and commented. It is discussed in particular the role of the importance function associated with the power control, and the definition of the concept of “generalized reactivity”, merging into the standard concept of reactivity with the system approaching criticality. Basing on these results, a formulation is also described of a point kinetic equation, with physically significant coefficients, similar to those presented by Usachev in 1955 (Usachev, L.N., 1955. Atomnay a Energiya, 15, 4726), using the standard adjoint flux as weighting function.     

Adjoint Sensitivity and Uncertainty Analyses in Monte Carlo Forward Calculations

The adjoint-weighted perturbation (AWP) method, in which the required adjoint flux is estimated in the course of Monte Carlo (MC) forward calculations, has recently been proposed as an alternative to the conventional MC perturbation techniques, such as the correlated sampling and differential operator sampling (DOS) methods. The equivalence of the first-order AWP method and first-order DOS method with the fission source perturbation taken into account is proven. An algorithm for the AWP calculations is implemented in the Seoul National University MC code McCARD and applied to the sensitivity and uncertainty analyses of the Godiva and Bigten criticalities.     

A perturbation theory for the non-linear system of neutron transport and burnup equations

A perturbation theory for use in nuclear reactor burnup analysis is derived. An important characteristic function is defined, and the related adjoint equation is obtained simply by using the variational principle. The adjoint matrix operator is evaluated directly from its defining differential expression. Responses at the end of cycle due to changes in initial material inventory, in nuclear data, and in power demands can be calculated using the previously determined forward and adjoint solutions. The method has additional applications, notably in selecting beginning of cycle conditions so as to achieve a particular end of cycle condition.     

Methods for Solving Adjoint Equations in Nonlinear Conditionally-Critical Problems

The special features of methods for calculating the neutron value functions with respect to functionals which are determined by solving a nonlinear conditionally-critical neutron transport equation are examined. The adjoint equations for these functions are written in an abstract operator form and contain additional, compared with linear problems, terms which are orthogonal to the neutron flux. These terms take account of the contribution due to the change in the properties of the medium when neutrons are introduced into the reactor into the balance of the values. The operators of the adjoint problem are analyzed and criteria ensuring that the proposed iteration methods converge are formulated. The satisfaction of the criteria is checked for the solution of problems where the nonlinearity of the emission equations is due to the dependence of the concentration of fuel nuclei on the neutron flux. It is noted that the iteration processes converge rapidly.     

Coolant void reactivity adjustments in advanced CANDU lattices using adjoint sensitivity technique

Coolant void reactivity (CVR) is an important factor in reactor accident analysis. Here we study the adjustments of CVR at beginning of burnup cycle (BOC) and k_eff at end of burnup cycle (EOC) for a 2D Advanced CANDU Reactor (ACR) lattice using the optimization and adjoint sensitivity techniques. The sensitivity coefficients are evaluated using the perturbation theory based on the integral neutron transport equations. The neutron and flux importance transport solutions are obtained by the method of cyclic characteristics (MOCC). Three sets of parameters for CVR-BOC and k_eff-EOC adjustments are studied: (1) Dysprosium density in the central pin with Uranium enrichment in the outer fuel rings, (2) Dysprosium density and Uranium enrichment both in the central pin, and (3) the same parameters as in the first case but the objective is to obtain a negative checkerboard CVR-BOC (CBCVR-BOC). To approximate the EOC sensitivity coefficient, we perform constant-power burnup/depletion calculations using a slightly perturbed nuclear library and the unperturbed neutron fluxes to estimate the variation of nuclide densities at EOC. Our aim is to achieve a desired negative CVR-BOC of −2 mk and k_eff-EOC of 0.900 for the first two cases, and a CBCVR-BOC of −2 mk and k_eff-EOC of 0.900 for the last case. Sensitivity analyses of CVR and eigenvalue are also included in our study.     

An explicit solution for a renewal process with waiting time and its variational principle

The forward and backward equations for the conditional probability density are derived for a reliability system consisting of a single component whose repair is subject to a delay time in providing a spare part but whose mean rate of repair is otherwise constant and whose time to failure is exponentially distributed. Exact solutions are quoted. These equations are then shown to be an adjoint pair that provide stationary conditions for a variational principle, in elementary form, from which all properties of the systems can be predicted with an accuracy greater than that implied by the trial functions or approximations used. A second or specific form of variational principle provides specific estimates to questions at hand. The second or adjoint field in the first elementary principle is the backward Kolmogorov solution and the in the specific form is the importance function, as used in nuclear reactor theory. The solutions are given for long-time and in a recurrence relation form valid for all times so that approximate solutions can be checked. Approximations suitable for variational trial functions are given. Two examples give the effect of a change of delay time for a steady state and an initial transient, respectively.     

A response-based time-dependent neutron transport method

An efficient response-based solution to the time-dependent neutron transport equation in a semi-infinite slab is derived. The solution is based on polynomial expansions of the source terms and neutron flux in the time domain. The expansion coefficients of the flux solution are computed in terms of response functions, which are special cases of Green’s functions for arbitrary in-volume and surface sources. The resulting response equation, which is a convolution integral equation in time, is reduced to a linear algebraic system of equations in the expansion coefficients. Two example problems are solved using the response-based method, and the extension of the method to general (finite, heterogeneous) problems is discussed.     

Applicability of Avery's Coupled Reactor Theory to Estimate Subcriticality of Test Region in Two Region System

The author examined the validity to estimate the subcriticality of a test region in a coupled reactor system using only measurable quantities on the basis of Avery's coupled reactor theory. For the purpose, we analyzed coupled reactor experiments performed at the Tank-type Critical Assembly in Japan Atomic Energy Research Institute by using two region systems and evaluated the subcriticality of the test region through a numerical study. Coupling coefficients were redefined at the quasi-static state because their definitions by Avery were not clear. With the coupling coefficients obtained by the numerical calculation, the multiplication factor of the test region was evaluated by two formulas; one for the evaluation using only the measurable quantities and the other for the accurate evaluation which contains the terms dropped in the former formula by assuming the unchangeableness for the perturbation induced in a driver region. From the comparison between the results of the evaluations, it was found that the estimation using only the measurable quantities is valid only for the coupled reactor system where the subcriticality of the test region was very small within a few dollars in reactivity. Consequently, it is concluded that the estimation using only the measurable quantities is not applicable to a general coupled reactor system.     

Development of analysis method for sodium void reactivity of step type FBR cores by using group-wise Monte Carlo code

A new method is proposed to separate the sodium void reactivity of step type FBR cores to components including non-leakage terms and a leakage term by using a newly developed perturbation code MCPERT where fluxes and adjoint fluxes are derived from a group-wise Monte Carlo code. The step type FBR core is a core where the height of the inner core is smaller than that of the outer core and a large sodium plenum region is located above the core so as to decrease the sodium void reactivity. The conventional diffusion perturbation method cannot treat such a large void region due to the diffusion approximation, while the Monte Carlo code can treat it exactly. In this study, a group-wise Monte Carlo code GMVP with a 70-group constant set JFS-3-J3.3 is employed to evaluate the neutron fluxes and adjoint fluxes which are used as inputs to the MCPERT code to evaluate the non-leakage terms. The leakage term is derived from the difference of the total sodium void reactivity evaluated by the direct calculation of GMVP and the summation of the non-leakage terms. It is found that the proposed method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method.     

Nodal analysis of density-wave oscillations in boiling water nuclear reactors

This work presents a model for density-wave oscillations in boiling water nuclear reactors (BWRs). A nodal approach is adopted to describe the power generation in the core. The nodal equations are derived within the framework of Avery's coupled-core kinetics theory. This method enables us to evaluate easily the internodal coupling coefficients using steady-state flux and importance distributions. Particular care is taken in treating the heater wall dynamics by using a distributed parameter model for the fuel rod. The flow is described by the homogeneous equilibrium conservation equations.A steady-state calculation is performed to determine the flow distribution and the neutronic coupling parameters in a core constituted by three hydrodynamic channel types and two neutronic radial nodes. The dynamics solution is based on linearization and use of the Nyquist stability criterion. The stability margin is found to be significantly affected by the coupling effect. The importance of a detailed representation of the neutronic behavior in BWR stability studies is demonstrated.     

Three-Dimensional First-Order Perturbation Calculation Method for Reactivity Changes Due to Core Deformations of Fast Breeder Reactors

A three-dimensional diffusion calculation method has been proposed to rapidly and accurately calculate reactivity changes of LMFBRs caused by assembly displacements in accidental events. The method requires shorter computation times and provides almost the same accuracy as a conventional direct eigenvalue calculation method. In this method, changes in macroscopic neutron cross-sections and diffusion coefficient are defined so that changes in both region volume and material composition can be treated in a mesh-centered finite-difference program under the same coarse mesh division as used for the normal, non-deformed core. Reactivity changes are calculated from the above-mentioned changes by the first-order perturbation method using normal and adjoint neutron fluxes calculated beforehand for the normal core.

The method was applied to deformations of a 1,000-MWe LMFBR core. Reactivity changes calculated by the method agreed within 0.4% with those by a conventional direct eigenvalue calculation method, while computation time was less than 1/35.     

Rossi-α and Feynmann Y functions for non-Poissonian pulsed sources of neutrons in the stochastic pulsing method: Application to subcriticality monitoring in ADS and comparison with the results of Poissonian pulsed neutron sources

In this paper, analytical expressions for the Rossi-α and the Feynmann Y functions are deduced for the case of Poissonian and non-Poissonian neutron sources when the stochastic pulsing method is used. These analytical expressions are used to fit the experimental data and to obtain the prompt neutron time constant. Also we perform in this paper a comparison of the results obtained for the Rossi-α and Feynmann Y functions with Poissonian and non-Poissonian neutron sources, and we study how much change the shape of these functions when the fission probability decreases and the capture probability increases due to the depletion with time of the fuel, and the increase of the fission products. Some comparisons with experimental data and with the results of other authors have been performed. Another important question analyzed in this paper and that it is interesting from an academic point of view is that the average number of detected counts induced by one single neutron injected in the system at an arbitrary time t′, should obey in point kinetics theory an adjoint equation in the time domain. Also the cross-factorial moment of the number of counts induced by one neutron in two counting intervals should obey also an adjoint equation in the time domain with a source term that depends on the first moments. These results are a consequence of more general results that have been obtained using stochastic transport theory for the one particle probability generating function or Kernel generating function.     

Higher Order Perturbation Method in Reactor Calculation

The higher order perturbation theory is applied to the one-dimensional multi-group diffusion equation.

The method of calculating the eigenvalue, eigenfunction and adjoint function of the higher mode is established for a non-Hermetian matrix.

The diffusion equation is solved by the higher order perturbation method, through the use of the eigenvalues, eigenfunctions and adjoint functions of the higher modes. The newly developed method is applied to the estimation of reactivity effect in a reactor.     

Sensitivity analysis of multiplication rate for subcritical systems

Sensitivity coefficients of the fission neutron multiplication rate k_f with respect to the change of cross-sections and an external source are derived for subcritical systems using the generalized perturbation theory (GPT). It is confirmed that the sensitivity coefficients derived using the GPT for a simple geometry where analytic solutions can be derived, give the same values as those derived by the direct numerical differentiation. These equations may be used to analyze the experimental values for the subcritical systems such as acceleration driven subcritical systems.     

连续能量核数据敏感性分析及其在AP1000中的应用

AP1000是典型的第三代核电技术,对AP1000反应堆进行核数据的敏感性分析是不确定度量化分析的基础,对AP1000后续的安全分析有重要作用。本文基于反复裂变几率方法在蒙特卡罗前向计算中求解共轭通量,并根据一阶微扰理论得到keff对核数据的灵敏度系数。针对反复裂变几率方法普遍存在占用内存大的问题,采用稀疏矩阵的存储方式降低内存。针对计数效率低、统计涨落大的问题,采用重叠块法提高计数效率。通过在蒙特卡罗程序NECP-MCX中开发连续能量核数据敏感性分析功能模块,并对AP1000进行连续能量核数据灵敏度系数的计算,得到了对keff的灵敏度系数影响较大的核数据,同时将计算结果与MCNP6进行了比较。结果表明,NECP-MCX和MCNP6的计算结果吻合较好。     

Effect of Group Collapsing in Perturbation Theory for Sample Worth Analysis

The effect of group collapsing applied to the perturbation theory for sample worth analysis in fast reactor systems is theoretically and numerically examined assuming the validity of the thin sample approximation.

As a result, the calculated worths of scattering predominant materials placed at the center of core are found to be strongly influenced by the group collapsing. The effect on the sample worth when the sample is placed in positions off the core center decreases with increasing distance from the center. It is noted that the reactivity perturbation due to inelastic scattering is also affected significantly by group collapsing especially near the core-blanket interface.

Based on the above observations, it is concluded that the perturbation theory with about 70 energy groups appropriately arranged is necessary to reproduce the experimental values of Na, O and Fe sample reactivity worths with accuracy efficient for ordinary purposes.     

Uncertainty quantification of neutronic parameters of light water reactor fuel cells with JENDL-4.0 covariance data

Neutronic parameter uncertainty induced by nuclear data uncertainty is quantified for several light water reactor fuel cells composed of different combinations of fissile/fertile nuclides. The covariance data given in JENDL-4.0 are used as the nuclear data uncertainty, and uncertainty propagation calculations are carried out using sensitivity coefficients calculated with the generalized perturbation theory for burnup-related neutronic parameters.

It is found that main contributors of nuclear data uncertainty to the neutronic parameter uncertainty are the uranium-238 capture cross section in a uranium-oxide fuel cell, and the plutonium-240 and plutonium-241 capture cross sections and fission spectrum of fissile plutonium isotopes in a uranium–plutonium mixed-oxide fuel cell. It is also found that thorium-232 capture cross section uncertainty is a dominant source of neutronic parameter uncertainty in thorium–uranium and thorium–plutonium mixed-oxide fuel cells. It should be emphasized that precise and detail information of component-wise uncertainties can be obtained by virtue of the adjoint-based sensitivity calculation methodology. Furthermore, cross-correlations are evaluated for each fuel cell, and strong correlations among the same parameters at the beginning of cycle and at the end of cycle and among different parameters are observed.     

Stoichiometry of Fully Oxidized PuO2-NdO1.5 and PuO2-YO1.5 Solid Solutions

A new method has been established to calculate sensitivity coefficients of cell parameters based on generalized perturbation theory using the collision probability method. The proposed method does not require the calculation of the changes of collision probabilities due to cross section changes, so it is as powerful as the commonly used generalized perturbation theory in diffusion theory, We demonstrate the validity of the method by comparing the calculated sensitivity coefficients with those obtained from the direct cell calculations. AS an application, we calculate the sensitivity coefficients of neutronic properties in cells with different moderator to fuel volume ratios, and discuss the physical meaning of the difference between the sensitivity coefficients.