• Journal of Nuclear Fuel Cycle and Waste Technology(JNFCWT)
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• v.16 no.2
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• pp.211-221
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• 2018

Nodal transport methods are proposed for solving the simplified even-parity neutron transport (SEP) equation. These new methods are attributed to the success of existing nodal diffusion methods such as the Polynomial Expansion Nodal and the Analytic Function Expansion Nodal Methods, which are known to be very effective for solving the neutron diffusion equation. Numerical results show that the simplified even-parity transport equation is a valid approximation to the transport equation and that the two nodal methods developed in this study also work for the SEP transport equation, without conflict. Since accuracy of methods is easily increased by adding node unknowns, the proposed methods will be effective for coarse mesh calculation and this will also lead to computation efficiency.

• Nuclear Engineering and Technology
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• v.32 no.6
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• pp.605-617
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• 2000

Nodal transport methods are studied for the solution of two dimensional discrete-ordinates, simplified even-parity transport equation(SEP) which is known to be an approximation to the true transport equation. The polynomial expansion nodal method(PEN) and the analytic function expansion nodal method(AFEN）which have been developed for the diffusion theory are used for the solution of the discrete-ordinates form of SEP equation. Our study shows that while the PEN method in diffusion theory can directly be converted without complication, the AFEN method requires a theoretical modification due to the nonhomogeneous property of the transport equation. The numerical results show that the proposed two methods work well with the SEP transport equation with higher accuracies compared with the conventional finite difference method.

• Nuclear Engineering and Technology
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• v.37 no.6
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• pp.525-536
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• 2005

The interfacial area transport equation dynamically models the changes in interfacial structures along the flow field by mechanistically modeling the creation and destruction of dispersed phase. Hence, when employed in the numerical thermal-hydraulic system analysis codes, it eliminates artificial bifurcations stemming from the use of the static flow regime transition criteria. Accounting for the substantial differences in the transport mechanism for various sizes of bubbles, the transport equation is formulated for two characteristic groups of bubbles. The group 1 equation describes the transport of small-dispersed bubbles, whereas the group 2 equation describes the transport of large cap, slug or chum-turbulent bubbles. To evaluate the feasibility and reliability of interfacial area transport equation available at present, it is benchmarked by an extensive database established in various two-phase flow configurations spanning from bubbly to chum-turbulent flow regimes. The geometrical effect in interfacial area transport is examined by the data acquired in vertical fir-water two-phase flow through round pipes of various sizes and a confined flow duct, and by those acquired In vertical co-current downward air-water two-phase flow through round pipes of two different sizes.

• Journal of Soil and Groundwater Environment
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• v.8 no.3
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• pp.8-12
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• 2003

The centrifuge test result on capped sediment was compared to the advection- dispersion equation proposed for one layered to predict contaminant transport parameters. The fitted contaminant transport parameters for the centrifuge test results were one to three orders of magnitude greater than the estimated parameters from the advection-dispersion equation. This indicates that the centrifuge model over estimated the contaminant transport phenomena. Thus, the centrifuge provides a non-conservative approach to modeling contaminant transport. It should be also noted that the advection-dispersion equation used in this study is a one layered model. Two layered modeling approaches are more appropriate for modeling this data since there are two layers with different partitioning coefficients. Further research is required to model the centrifuge test using two-layered advection-dispersion models.

• Nuclear Engineering and Technology
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• v.52 no.3
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• pp.485-498
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• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3181-3204
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• 2022

The variational nodal method for solving the neutron transport equation has evolved over 40 years. Based on a functional form of the Boltzmann neutron transport equation, the method now comprises a complete set of variants that can be employed for different problems. This paper presents an extensive review of the development of the variational nodal method. The emphasis is on summarizing the whole theoretical system rather than validating the methodologies. The paper covers the variational nodal formulation of the Boltzmann neutron transport equation, the Ritz procedure for various application purposes, the derivation of boundary conditions, the extension for adjoint and perturbation calculations, and treatments for anisotropic scattering sources. Acceleration approaches for constructing response matrices and solving the resulting system of algebraic equations are also presented.

• Journal of Energy Engineering
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• v.16 no.1 s.49
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• pp.46-52
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• 2007

The piecewise constant angular approximation is developed to replace the conventional angular quadrature sets in the solution of the second-order, multi-dimensional $S_{N}$ neutron transport equations. The newly generated quadrature sets by this method substantially mitigate ray effects and can be used in the same manner as the conventional quadrature sets are used. The discrete-ordinates and the piecewise-constant approximations are applied to both the first-order Boltzmann and the second-order form of neutron transport equations in treating angular variables. The result is that the mitigation of ray effects is only achieved by the piecewise-constant method, in which new angular quadratures are generated by integrating angle variables over the specified region. In other sense, the newly generated angular quadratures turn out to decrease the contribution of mixed-derivative terms in the even-parity equation that is one of the second-order neutron transport equation. This result can be interpreted as the entire elimination or substantial mitigation of ray effect are possible in the simplified even-parity equation which has no mixed-derivative terms.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.1
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• pp.1-11
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• 1998

For the analysis of phonon heat transfer within short time and spatial scales, conventional macroscopic heat conduction equations with jump boundary conditions are tried and the results are compared to those of equation of phonon radiative transport(EPRT), which is one of microscopic transport equation. In transient state the macroscopic temperatures show far different behavior from EPRT. In steady state the hyperbolic temperatures with temperature jump at the wall from time relaxation model agrees well with EPRT temperatures. Since EPRT is also an approximate form of microscopic transport equation and there are no experimental results to verify the proposed model in this study, we can not conclude whether the approaching method from this study is valid or not. To the authors' knowledge, there are no experimental results available which can be used to test the validity of these models. Such an experiment, while difficult to conduct, would be invaluable.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 2001.05c
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• pp.164-167
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• 2001

The electron transport coefficients, the electron drift velocity W, the longitudinal diffusion coefficient $ND_L$ and $D_L/{\mu}$, in pure $CO_2$ were calculated over the wide E/N range from 0.01 to 500 Td at 1 Torr by two-term approximation of the Boltzmann equation for determination of electron collision cross sections set and for quantitative characteristic analysis of $CO_2$ molecular gas. And for propriety of two-term approximation of Boltzmann equation analysis, the calculated results compared with the electron transport coefficients measured by Nakamura.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 1995.10a
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• pp.76-83
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• 1995

Estimation of blasting behavior of explosives is prerequisite in the numerical analysis of blasting works. In this study, blasting pressure is estimated by the finite difference method using the Flux-Corrected Transport Algorithm. To formulate the behavior of blasting gas, the mass conservation equation, the moment conservation equation, the energy conservation equation and the ideal gas state equation are used. The simplified species conservation equation is included to simulate the behavior of reacting explosives. To verify the calculation, the Sod's shock tube problem, the strong shock problem and the reacting problem we used. Numerical results show that the shock wave can be captured by means of the FCT algorithm in the reacting and nonreacting states.