• Nuclear Engineering and Technology
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• v.31 no.2
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• pp.172-181
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• 1999

Series expansion methods to compute the exponential of a matrix have been compared by applying them to fuel depletion calculations. Specifically, Taylor, Pade, Chebyshev, and rational Chebyshev approximations have been investigated by approximating the exponentials of bum matrices by truncated series of each method with the scaling and squaring algorithm. The accuracy and efficiency of these methods have been tested by performing various numerical tests using one thermal reactor and two fast reactor depletion problems. The results indicate that all the four series methods are accurate enough to be used for fuel depletion calculations although the rational Chebyshev approximation is relatively less accurate. They also show that the rational approximations are more efficient than the polynomial approximations. Considering the computational accuracy and efficiency, the Pade approximation appears to be better than the other methods. Its accuracy is better than the rational Chebyshev approximation, while being comparable to the polynomial approximations. On the other hand, its efficiency is better than the polynomial approximations and is similar to the rational Chebyshev approximation. In particular, for fast reactor depletion calculations, it is faster than the polynomial approximations by a factor of ∼ 1.7.

• Nuclear Engineering and Technology
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• v.55 no.7
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• pp.2516-2533
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• 2023

Several practical methods for accelerating the depletion calculation in a GPU-based Monte Carlo (MC) code PRAGMA are presented including the multilevel spectral collapse method and the vectorized Chebyshev rational approximation method (CRAM). Since the generation of microscopic reaction rates for each nuclide needed for the construction of the depletion matrix of the Bateman equation requires either enormous memory access or tremendous physical memory, both of which are quite burdensome on GPUs, a new method called multilevel spectral collapse is proposed which combines two types of spectra to generate microscopic reaction rates: an ultrafine spectrum for an entire fuel pin and coarser spectra for each depletion region. Errors in reaction rates introduced by this method are mitigated by a hybrid usage of direct online reaction rate tallies for several important fissile nuclides. The linear system to appear in the solution process adopting the CRAM is solved by the Gauss-Seidel method which can be easily vectorized on GPUs. With the accelerated depletion methods, only about 10% of MC calculation time is consumed for depletion, so an accurate full core cycle depletion calculation for a commercial power reactor (BEAVRS) can be done in 16 h with 24 consumer-grade GPUs.

• Nuclear Engineering and Technology
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• v.46 no.3
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• pp.407-412
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• 2014

In this research, we investigated the burnup characteristics and the conversion of fertile $^{232}Th$ into fissile $^{233}U$ in the core of a Sodium-Cooled Fast Reactor (SFR). The SFR fuel assemblies were designed for burning $^{232}Th$ fuel (fuel pin 1) and $^{233}U$ fuel (fuel pin 2) and include mixed minor actinide compositions. Monte Carlo simulations were performed using Serpent Code1.1.19 to compare with CRAM (Chebyshev Rational Approximation Method) and TTA (Transmutation Trajectory Analysis) method in the burnup calculation mode. The total heating power generated in the system was assumed to be 2000 MWth. During the reactor operation period of 600 days, the effective multiplication factor (keff) was between 0.964 and 0.954 and peaking factor is 1.88867.

• Nuclear Engineering and Technology
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• v.50 no.7
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• pp.1043-1050
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• 2018

For an efficient Monte Carlo (MC) burnup analysis, an accurate high-order depletion scheme to consider the nonlinear flux variation in a coarse burnup-step interval is crucial accompanied with an accurate depletion equation solver. In a Seoul National University MC code, McCARD, the high-order depletion schemes of the quadratic depletion method (QDM) and the linear extrapolation/quadratic interpolation (LEQI) method and a depletion equation solver by the Chebyshev rational approximation method (CRAM) have been newly implemented in addition to the existing constant extrapolation/backward extrapolation (CEBE) method using the matrix exponential method (MEM) solver with substeps. In this paper, the quadratic extrapolation/quadratic interpolation (QEQI) method is proposed as a new high-order depletion scheme. In order to examine the effectiveness of the newly-implemented depletion modules in McCARD, four problems in the VERA depletion benchmarks are solved by CEBE/MEM, CEBE/CRAM, LEQI/MEM, QEQI/MEM, and QDM for gadolinium isotopes. From the comparisons, it is shown that the QEQI/MEM predicts ${k_{inf}}^{\prime}s$ most accurately among the test cases. In addition, statistical uncertainty propagation analyses for a VERA pin cell problem are conducted by the sensitivity and uncertainty and the stochastic sampling methods.

• Journal of Nuclear Fuel Cycle and Waste Technology(JNFCWT)
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• v.20 no.2
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• pp.171-183
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• 2022

The parent and daughter nuclides in a radioactive decay chain arrive at secular equilibrium once they have a large half-life difference. The characteristics of this equilibrium state can be used to estimate the production time of nuclear materials. In this study, a mathematical model and algorithm that can be applied to radio-chronometry using the radioactive equilibrium relationship were investigated, reviewed, and implemented. A Bateman equation that can analyze the decay of radioactive materials over time was used for the mathematical model. To obtain a differential-based solution of the Bateman equation, an algebraic numerical solution approach and two different matrix exponential functions (Moral and Levy) were implemented. The obtained result was compared with those of commonly used algorithms, such as the Chebyshev rational approximation method and WISE Uranium. The experimental analysis confirmed the similarity of the results. However, the Moral method led to an increasing calculation uncertainty once there was a branching decay, so this aspect must be improved. The time period corresponding to the production of nuclear materials or nuclear activity can be estimated using the proposed algorithm when uranium or its daughter nuclides are included in the target materials for nuclear forensics.

• Nuclear Engineering and Technology
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• v.54 no.12
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• pp.4722-4730
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• 2022

This paper introduces a new point depletion-based source term calculation code named BESNA (Bateman Equation Solver for Nuclear Applications), which is aimed to estimate nuclide inventories and source terms from spent nuclear fuels. The BESNA code employs a new modified CE/CM (Constant Extrapolation - Constant Midpoint) predictor-corrector scheme in depletion calculations for improving computational efficiency. In this modified CE/CM scheme, the decay components leading to the large norm of the depletion matrix are excluded in the corrector, and hence the corrector calculation involves only the reaction components, which can be efficiently solved with the Talyor Expansion Method (TEM). The numerical test shows that the new scheme substantially reduces computing time without loss of accuracy in comparison with the conventional scheme using CRAM (Chebyshev Rational Approximation Method), especially when the substep calculations are applied. The depletion calculation and source term estimation capability of BESNA are verified and validated through several problems, where results from BESNA are compared with those calculated by other codes as well as measured data. The analysis results show the computational efficiency of the new modified scheme and the reliability of BESNA in both isotopic predictions and source term estimations.

• Nuclear Engineering and Technology
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• v.52 no.9
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• pp.1907-1916
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• 2020

In this work, the depletion capability implemented in Monte Carlo code MCS is investigated to predict the isotopic compositions of spent nuclear fuel (SNF). By comparison of MCS calculation results to post irradiation examination (PIE) data obtained from one pressurized water reactor (PWR), the validation of this capability is conducted. The depletion analysis is performed with the ENDF/B-VII.1 library and a fuel assembly model. The transmutation equation is solved by the Chebyshev Rational Approximation Method (CRAM) with a depletion chain of 3820 isotopes. 18 actinides and 19 fission products are analyzed in 14 SNF samples. The effect of statistical uncertainties on the calculated number densities is discussed. On average, most of the actinides and fission products analyzed are predicted within ±6% of the experiment. MCS depletion results are also compared to other depletion codes based on publicly reported information in literature. The code-to-code analysis shows comparable accuracy. Overall, it is demonstrated that the depletion capability in MCS can be reliably applied in the prediction of SNF isotopic inventory.