Improved Quasistatic Research Articles

Forward sensitivity analysis procedure (FSAP) with improved quasi static method for the estimation of dynamic uncertainty in reactor transients

Forward sensitivity analysis procedure (FSAP) with improved quasi static (IQS) method is developed and presented here for the estimation of dynamic uncertainty in reactor transients. The propagation of uncertainty in the macroscopic cross section data with time and its effect on reactor power transient is discussed in this paper. The incorporation of the IQS method in FSAP creates a new kind of point kinetics equations and its solution with the shape function gives the dynamic uncertainty in the flux during the transient. This method is applied to estimate the dynamic uncertainty in the power transients in 3D homogeneous reactor, TWIGL-2D and CANDU-2D benchmark reactors. In TWIGL-2D reactor, a Doppler temperature feedback is introduced and this creates an additional internal uncertainty in the core. The core power, core regional temperature and their uncertainties are estimated for the combination of external and internal uncertainties. It is also shown that as the uncertainty in the macroscopic cross section data is increased, the uncertainty in the core power increases. In CANDU-2D benchmark reactor, the dynamic uncertainty in the core power is estimated with uncertainty in the delayed neutron fraction. The estimated uncertainties in the power transient and core temperature are compared with that of direct solution method. The results are found to agree to a good accuracy. From the comparison of results, it is established that this computational method is capable of estimating the dynamic uncertainty in reactor power transient, caused by uncertainties in macroscopic cross section data. The advantage of this new computational method is that the dynamic uncertainty in the power transient can be estimated in a much faster way as compared to the direct solution method and the dynamic uncertainty estimation is represented as a forward problem in time. It is also shown here that using this computational method, both the reactor transient and its uncertainty can be estimated simultaneously.

Semi wavelet-based improved quasi static method for the analysis of PHWR transients

A new computational method, by incorporating wavelets in the improved quasi static (IQS) scheme, is developed and presented here for the analysis of CANDU PHWR and TWIGL reactor transients. According to this method the time dependent neutron diffusion equation is split into amplitude function and shape function. The amplitude function is solved using the Haar wavelet and its solution is used in the estimation of shape function. The advantage of this computational method is that the concept of micro time step and macro time step, adopted in the IQS scheme, does not arise here at all and the transient can be estimated with only one time step, i.e. single time step. In this method, the transient estimation methodology is different from the improved quasi static scheme. This computational method is applied to estimate the transients in CANDU 3D-PHWR benchmark, CANDU-2D and TWIGL reactors. In TWIGL reactor, a Doppler temperature feedback in introduced and the transient is estimated with feedback. The estimated transients are compared with the reference. From the comparison of results, it is observed that this method is capable of estimating the reactivity initiated transients in heterogeneous reactors to a good accuracy, by adopting only one time step. This computational method is first of its kind in the estimation of reactor transients. It is also shown that as the order of resolution of wavelet is increased, the error in the estimation of transient is reduced.

Position-dependent delayed-neutron fractions for IQS calculations and application to PHWR kinetics calculations

Abstract The core of a Pressurized Heavy-Water Reactor (PHWR) contains fuel bundles with burnups varying between zero and the discharge burnup. Because the delayed-neutron fraction varies (decreases) with burnup, the spatial variation of the fuel burnup causes the delayed-neutron fraction to also depend on position. High-fidelity kinetics calculations need to account for such spatial dependence of the delayed-neutron fraction. This work documents the implementation of the capability to model position-dependent delayed-neutron fractions in the multi-group diffusion code DONJON which performs kinetics calculations using the Improved Quasi-Static (IQS) method. Consistent with the IQS method, the effect of the spatial variation of the delayed-neutron fraction is accounted for both in the overall effective delayed neutron fraction and in the spatial dependence of the delayed-neutron source. Consequently, both the amplitude and the shape of the neutron flux are impacted by the position-dependence of the delayed-neutron fraction. The functionality of the newly-implemented capability is tested for a stylized coolant-voiding transient in a PHWR core and differences in reactor power and core neutron flux shape are shown to exist between the results of calculations that use position-dependent delayed-neutron fractions and those that only use spatially uniform ones.

Enhanced geometric capabilities for the transient analysis code T-ReX and its application to simulating TREAT experiments

Advances in computational architecture have prompted a resurgence in the simulation of reactor transients from first principles. Most codes are unable to simulate transient events with complex models, and require numerous approximations. The code T-ReX (Transient-Reactor eXperiment simulator), an extensive update to TDKENO, has been developed as a transient analysis tool with few geometric limitations, and minimal theoretical approximations. T-ReX achieves this by employing the Improved Quasi-Static (IQS) method to solve the time-dependent Boltzmann transport equation with explicit representation of delayed neutrons. The primary change in T-ReX relative to TDKENO is the incorporation of a modified version of the Monte Carlo code KENO-VI to calculate the flux shape and model the geometry of a problem. Using KENO-VI to model systems allows exact representation of the geometry. The changes to T-ReX are verified by comparison of solutions to computational benchmark problems found with a previous version of TDKENO that made use of KENO V.a, and several other codes with time-dependent capabilities. In addition, a three-dimensional KENO-VI model of the Transient Reactor Test Facility (TREAT) core is used in simulations of several temperature-limited transient experiments from the M8 Calibration series. T-ReX produces results that agree with benchmark problems and are in better agreement with TREAT experimental data than TDKENO.

Development of a Nodal Expansion Method based space-time kinetics code for hexagonal geometry

A two and three dimensional, multigroup, neutron space-time kinetics theory based computer code IQSHEX (Improved Quasi Static method based kinetics code for HEXagonal geometry) is developed in order to carry out reactivity initiated transient in large size thermal (i.e. neutronically loosely coupled reactor core) and fast breeder reactors with hexagonal lattice. For such reactors, neutron flux is considered as function of both space and time and neutron space-time kinetics equations are solved for obtaining the dependence of flux on space and time. This code uses Improved Quasi Static (IQS) method to solve the equations. In this method, it is assumed that shape function changes slowly relative to amplitude function and product of these two functions gives the complete behavior of neutron flux both in space- and time. Multigroup neutron diffusion equation is solved over macro (or larger) time step by advanced Nodal Expansion Method (NEM) to find out shape function, and point kinetics equations are solved over micro (or smaller) time step by Piecewise Constant Approximation (PCA) method to find out amplitude function. The code has been validated against several international benchmark problems with and without temperature feedback. Results corroborate well with that of the benchmark problems.

The modified exponential time differencing method with IQS model for the neutronic analysis of PHWR transients

A new method of analyzing reactor transients using the modified exponential time differencing method with Improved Quasi Static (IQS) model is developed and presented here. According to this method, the neutron flux is factored into amplitude function and shape function. The amplitude function is solved using the modified exponential time differencing method and the shape function is solved using implicit scheme. Using this method, a large time step can be chosen for solving the amplitude function and the shape function can also be solved by proportionately increasing the time step. This method is applied to estimate the transients in the well known CANDU 3D-PHWR kinetics benchmark, 3D-TWIGL Heterogeneous and 3D Homogeneous Reactors. These reactors contain different kinds of heterogeneities and the reactivity perturbations are also different. The transients in CANDU 3D-PHWR, simulated by LOCA, are estimated by this method for various time steps and the results are compared with other standard methods. From the comparison of results, it is established that this method is capable of estimating the transients using large time steps with good accuracy. A novel scheme to choose the time step for solving the amplitude function under constant and varying reactivity perturbations is also presented and applied here in the estimation of CANDU 3D-PHWR and 3D Homogeneous Reactor transients. It is also shown that by using large time steps, the transient computation time is reduced without affecting the accuracy and the error remains bounded.

Solution of PHWR 3D kinetics benchmark using TRIKIN code

Scaling up of core sizes beyond present generation has rendered renewed topicality to space-time kinetics and relevant safety research. Detailed computational codes to solve transient neutron diffusion equation for the highest level of approximation including core thermal as well as system feedbacks are being established worldwide. Consistent with this objective a general purpose 3D kinetics code ‘TRIKIN’ has been developed in the Atomic Energy Regulatory Board (AERB) and the Indian Institute of Technology Bombay, India to carry out transient analysis for a variety of power reactors. Code has been validated against a series of international benchmark problems for light water reactors. This paper presents an extension of the validation array of the TRIKIN code by solving a 3D kinetics PHWR benchmark problem (AECL-7236). A brief overview of space time kinetics methods is presented. Detailed numerical scheme of improved quasistatic (IQS) method to solve transient neutron diffusion equations, adopted in TRIKIN code has been described. Description of 3D kinetics PHWR benchmark problem and its solution using TRIKIN code has been presented. TRIKIN results have shown good agreement with benchmarks results, demonstrating excellent space-time kinetics capability of the code for PHWR type reactors.

Burnup-dependent effect of lattice-level homogenization and group condensation on calculated kinetics parameters for CANDU-type lattices

Modern analysis of nuclear reactor transients uses space-time reactor kinetics methods. In the Canadian nuclear industry, safety analysis calculations use almost exclusively the Improved Quasistatic (IQS) flux factorization method. The IQS method, like all methods based on flux factorization, relies on calculating effective point kinetics parameters, which dominate the time behavior of the flux, using adjoint-weighted integrals. The accuracy of the adjoint representation influences the accuracy of the effective kinetics parameters. Routine full core calculations are not performed using detailed models and transport theory, but rather using a cell-homogenized model and two-group diffusion theory. This work evaluates the effect of homogenization and group condensation at different burnups, for three fuel types: natural-uranium (NU) fuel, low-void reactivity (LVR) fuel and Advanced CANDU Reactor (ACR) fuel. Results show that the use of a two-group lattice-homogenized adjoint consistently overestimates the effective delayed neutron fraction by approximately 5% for all three fuel types and over a wide burnup range. The use of a two-group lattice-homogenized adjoint also introduces errors in the effective neutron generation time, but these are at most 1.3% (and their sign changes with burnup). Errors tend to vary with burnup by approximately 1% (of the individual parameter value). If a 69-group lattice-homogenized adjoint is used, the errors drop to approximately 2% for the effective delayed neutron fractions and 0.5% for the effective neutron generation time, for all three fuel types and over the entire studied burnup range. The variation of errors with burnup is also reduced.

Effect of Lattice-Level Homogenization and Group Condensation on Calculated Kinetics Parameters of a Natural-Uranium-Fueled Equilibrium CANDU Core

Modern analysis of nuclear reactor transients uses space-time reactor kinetics methods. In the Canadian nuclear industry, safety analysis calculations use almost exclusively the improved quasi-static (IQS) flux factorization method. The IQS method, like all methods based on flux factorization, relies on calculating effective point-kinetics parameters, which dominate the time behavior of the flux, using adjoint-weighted integrals. The accuracy of the adjoint representation influences the accuracy of the effective kinetics parameters.Routine full-core calculations are not performed using detailed models and transport theory, but rather using a cell-homogenized model and two-group diffusion theory. This work evaluates the effect of homogenization and group condensation on the calculated effective kinetics parameters of an equilibrium CANDU core.Results show that homogenization combined with group condensation introduces a positive bias of ˜5% in the effective delayed neutron fraction over a wide range of discharge burnups. Homogenization alone induces a positive bias of only ˜2%.The bias in the effective generation time is <1% for all studied discharge burnups, and its effect on the results of a positive-reactivity transient is found to be negligible, with differences being caused solely by the effective delayed neutron fraction bias. The fractional delayed neutron fraction bias for the equilibrium core is found to be very close to that for a fresh-fuel core. However, because of the lower effective delayed neutron fraction of the equilibrium core, the effects of the bias are larger for the equilibrium core than for the fresh-fuel core. For a sample positive-reactivity transient, the maximum power is found to be underestimated by 9% for the fresh core and by 14% for the equilibrium core as a consequence of homogenization and group condensation.

A unified approach for numerical calculation of space-dependent kinetic equation

A unified numerical method based on the factorization approach is developed to solve the space-dependent neutron kinetic equation. Various numerical methods for solving the space-dependent kinetic equation have been developed so far. These methods can be classified into two categories, i.e., the direct and the factorization methods. The factorization method is known as an effective numerical method. In the present study, a new factorization method named the multigrid amplitude function (MAF) method is developed. Unlike the improved quasi-static (IQS) method, an independent amplitude function is assigned for each spatial region and energy group in the MAF method. The MAF method is a generalization of conventional methods, e.g., the frequency transform, IQS, and Theta methods. To evaluate the amplitude function in the MAF method, the time-dependent coarse-mesh finite difference (TCMFD) method is developed. The MAF method is implemented into a space-dependent kinetic code on the basis of the analytical polynomial nodal method. In order to verify the effectiveness of the MAF method, the TWIGL, Langenbuch, Maurer, and Werner (LMW), and Laboratorium für Reaktorregelung and Anlagensicherung (LRA) benchmark problems are analyzed. The calculation results show the effectiveness of the present method.

Effect of lattice-level adjoint-weighting on the kinetics parameters of CANDU reactors

Space–time kinetics calculations for CANDU ® 1 CANDU ® ( CAN ada D euterium U ranium) is a registered trademark of Atomic Energy of Canada Limited (AECL). 1 reactors are routinely performed using the Improved Quasistatic (IQS) method. The IQS method calculates kinetics parameters such as the effective delayed-neutron fraction and generation time using adjoint weighting. In the current implementation of IQS, the direct flux, as well as the adjoint, is calculated using a two-group cell-homogenized reactor model which is inadequate for capturing the effect of the softer energy spectrum of the delayed neutrons. Additionally, there may also be fine spatial effects that are lost because the intra-cell adjoint shape is ignored. The purpose of this work is to compare the kinetics parameters calculated using the two-group cell-homogenized model with those calculated using lattice-level fine-group heterogeneous adjoint weighting and to assess whether the differences are large enough to justify further work on incorporating lattice-level adjoint weighting into the IQS method. A second goal is to evaluate whether the use of a fine-group cell-homogenized lattice-level adjoint, such as is the current practice for Light Water Reactors (LWRs), is sufficient to capture the lattice effects in question. It is found that, for CANDU lattices, the generation time is almost unaffected by the type of adjoint used to calculate it, but that the effective delayed-neutron fraction is affected by the type of adjoint used. The effective delayed-neutron fraction calculated using the two-group cell-homogenized adjoint is 5.2% higher than the “best” effective delayed-neutron fraction value obtained using the detailed lattice-level fine-group heterogeneous adjoint. The effective delayed-neutron fraction calculated using the fine-group cell-homogenized adjoint is only 1.7% higher than the “best” effective delayed-neutron fraction value but is still not equal to it. This situation is different from that encountered in LWRs where weighting by a fine-group cell-homogenized adjoint is sufficient to calculate the correct effective delayed-neutron fraction. It is concluded that lattice-level weighting with a detailed fine-group heterogeneous adjoint is desirable for the correct calculation of the effective delayed-neutron fraction in CANDU lattices.

Parallelised Krylov subspace method for reactor kinetics by IQS approach

Nuclear reactor kinetics involves numerical solution of space–time-dependent multi-group neutron diffusion equation. Two distinct approaches exist for this purpose: the direct (implicit time differencing) approach and the improved quasi-static (IQS) approach. Both the approaches need solution of static space-energy-dependent diffusion equations at successive time-steps; the step being relatively smaller for the direct approach. These solutions are usually obtained by Gauss–Seidel type iterative methods. For a faster solution, the Krylov sub-space methods have been tried and also parallelised by many investigators. However, these studies seem to have been done only for the direct approach. In the present paper, parallelised Krylov methods are applied to the IQS approach in addition to the direct approach. It is shown that the speed-up obtained for IQS is higher than that for the direct approach. The reasons for this are also discussed. Thus, the use of IQS approach along with parallelised Krylov solvers seems to be a promising scheme.

Development and Verification of an Efficient Spatial Neutron Kinetics Method for Reactivity-Initiated Event Analyses

A space/time nodal diffusion code based on the nodal expansion method (NEM), EPISODE, was developed in order to evaluate transient neutron behavior in light water reactor cores. The present code employs the improved quasi-static (IQS) method for spatial neutron kinetics, and neutron flux distribution is numerically obtained by solving the neutron diffusion equation with the nonlinear iteration scheme to achieve fast computation. A predictor-corrector (PC) method developed in the present study enabled to apply a coarse time mesh to the transient spatial neutron calculation than that applicable in the conventional IQS model, which improved computational efficiency further. Its computational advantage was demonstrated by applying to the numerical benchmark problems that simulate reactivity-initiated events, showing reduction of computational times up to a factor of three than the conventional IQS. The thermohydraulics model was also incorporated in EPISODE, and the capability of realistic reactivity event analyses was verified using the SPERT-III/E-Core experimental data.

Variational Estimates for Use with the Improved Quasistatic Method for Reactor Dynamics

A new variational estimate for dynamic reactivity that accounts for delayed neutron holdback is introduced and adapted for use with the improved quasistatic (IQS) method. Numerical tests on a large light water reactor model indicate that the computational effort required with the IQS method can be reduced by a factor of 3 to 4 by using this new variational estimate of dynamic reactivity. With comparable computational effort, the accuracy of the standard IQS method, which uses the flux shape interpolation/recomputation procedure and a first-order estimate of static reactivity, can be improved by using a standard variational estimate without the flux shape interpolation/recomputation procedure, and it can be further improved by using the new variational estimate of dynamic reactivity.

A New Variational Functional for Space-Time Neutronics

A new variational functional for space-time neutronics is presented. This functional is stationary about the integro-differential form of the diffusion equation, in which the delayed neutron source is expressed as a convolution integral of the flux, and an integro-differential adjoint flux equation. The new functional is used to derive a quasi-static method that is similar to the improved quasistatic (IQS) method, except that the equation for the flux shape uses a different expression for the delayed neutron source. In a one-dimensional sub-prompt critical test problem, the new variational quasi-static method was slightly more accurate than the IQS method.

Analysis of super delayed-critical transients in thermal reactors using 2-D and 3-D adiabatic and improved quasistatic models

A computer model 3 d-fast is developed for solving the space-time kinetics equations in 1-D, 2-D or 3-D using the adiabatic and improved quasistatic (IQS) methods. Using this model, some super delayed-critical transients in 2-D and 3-D are analysed for varying reactivity insertion rates. It is shown that for transients where the reactivity insertion rate is small and change in flux shape is slow, the results obtained both by adiabatic and IQS models are in good agreement, while for fast reactivity insertion rates where changes in flux shapes are also rapid, the results obtained by the adiabatic method are grossly in error, while those obtained by the IQS method are more satisfactory.