• Nuclear Engineering and Technology
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• v.52 no.4
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• pp.681-688
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• 2020

The telegrapher's theory was used to develop a new formulation for the neutron noise equation. Telegrapher's equation is supposed to demonstrate a more realistic approximation for neutron transport phenomena, especially in comparison to the diffusion theory. The physics behind such equation implies that the signal propagation speed is finite, instead of the infinite as in the case of ordinary diffusion. This paper presents the theory and results of the development of a new method for calculation of the neutron noise using the telegrapher's equation as its basis. In order to investigate the differences and strengths of the new method against the diffusion based neutron noise, a comparison was done between the behaviors of two methods. The neutron noise based on SN transport considered as a precision measuring point. The Green's function technique was used to calculate the neutron noise based on telegrapher's and diffusion methods as well as the transport. The amplitude and phase of Green's function associated with the properties of the medium and frequency of the noise source were obtained and their behavior was compared to the results of the transport. It was observed, the differences in some cases might be considerable. The effective speed of propagation for the noise perturbations were evaluated accordingly, resulting in considerable deviations in some cases.

• Journal of Energy Engineering
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• v.17 no.4
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• pp.233-240
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• 2008

Conventionally, the second-order self-adjoint neutron transport equations have been studied using the even parity and the odd parity equations. Recently, however, the SAAF(self-adjoint angular flux) form of neutron transport equation has been introduced as a new option for the second-order self-adjoint equations. In this paper we validated the SAAF equation mathematically and clarified how it relates with the existing even and odd parity equations. We also developed a second-order SAAF differencing formula including DSA(diffusion synthetic acceleration) from the first-order difference equations. Numerical result is attached to show that the proposed methods increases accuracy with effective computational effort.

• Nuclear Engineering and Technology
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• v.53 no.2
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• pp.399-413
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• 2021

The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.

• Nuclear Engineering and Technology
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• v.49 no.1
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• pp.29-42
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• 2017

The purpose of the present study is the presentation of the appropriate element and shape function in the solution of the neutron diffusion equation in two-dimensional (2D) geometries. To this end, the multigroup neutron diffusion equation is solved using the Galerkin finite element method in both rectangular and hexagonal reactor cores. The spatial discretization of the equation is performed using unstructured triangular and quadrilateral finite elements. Calculations are performed using both linear and quadratic approximations of shape function in the Galerkin finite element method, based on which results are compared. Using the power iteration method, the neutron flux distributions with the corresponding eigenvalue are obtained. The results are then validated against the valid results for IAEA-2D and BIBLIS-2D benchmark problems. To investigate the dependency of the results to the type and number of the elements, and shape function order, a sensitivity analysis of the calculations to the mentioned parameters is performed. It is shown that the triangular elements and second order of the shape function in each element give the best results in comparison to the other states.

• 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.55 no.6
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• pp.2172-2194
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• 2023

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations. Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (k_eff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2-3.

• 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.51 no.5
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• pp.1181-1194
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• 2019

A new moving-mesh Finite Volume Method (FVM) for the efficient solution of the two-dimensional neutron diffusion equation is introduced. Many other moving-mesh methods developed to solve the neutron diffusion problems use a relatively large number of sophisticated mathematical equations, and so suffer from a significant complexity of mathematical calculations. In this study, the proposed method is formulated based on simple mathematical algebraic equations that enable an efficient mesh movement and CV deformation for using in practical nuclear reactor applications. Accordingly, a computational framework relying on a new moving-mesh FVM is introduced to efficiently distribute the meshes and deform the CVs in regions with high gradient variations of reactor power. These regions of interest are very important in the neutronic assessment of the nuclear reactors and accordingly, a higher accuracy of the power densities is required to be obtained. The accuracy, execution time and finally visual comparison of the proposed method comprehensively investigated and discussed for three different benchmark problems. The results all indicated a higher accuracy of the proposed method in comparison with the conventional fixed-mesh FVM.

• Nuclear Engineering and Technology
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• v.53 no.5
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• pp.1391-1402
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• 2021

The aim of this work is to prepare a neutron noise calculator based on the second order of average current nodal expansion method (ACNEM). Generally, nodal methods have the ability to fulfill the neutronic analysis with adequate precision using coarse meshes as large as a fuel assembly size. But, for the zeroth order of ACNEM, the accuracy of neutronic simulations may not be sufficient when coarse meshes are employed in the reactor core modeling. In this work, the capability of second order ACNEM is extended for solving the neutron diffusion equation in the frequency domain using coarse meshes. For this purpose, two problems are modeled and checked including a slab reactor and 2D BIBLIS PWR. For validating of results, a semi-analytical solution is utilized for 1D test case, and for 2D problem, the results of both forward and adjoint neutron noise calculations are exploited. Numerical results indicate that by increasing the order of method, the errors of frequency dependent coarse mesh solutions are considerably decreased in comparison to the reference. Accordingly, the accuracy of second order ACNEM can be acceptable for the neutron noise calculations by using coarse meshes in the nuclear reactor core.

• Nuclear Engineering and Technology
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• v.48 no.1
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• pp.43-54
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• 2016

In the present paper, development of the three-dimensional (3D) computational code based on Galerkin finite element method (GFEM) for solving the multigroup forward/adjoint diffusion equation in both rectangular and hexagonal geometries is reported. Linear approximation of shape functions in the GFEM with unstructured tetrahedron elements is used in the calculation. Both criticality and fixed source calculations may be performed using the developed GFEM-3D computational code. An acceptable level of accuracy at a low computational cost is the main advantage of applying the unstructured tetrahedron elements. The unstructured tetrahedron elements generated with Gambit software are used in the GFEM-3D computational code through a developed interface. The forward/adjoint multiplication factor, forward/adjoint flux distribution, and power distribution in the reactor core are calculated using the power iteration method. Criticality calculations are benchmarked against the valid solution of the neutron diffusion equation for International Atomic Energy Agency (IAEA)-3D and Water-Water Energetic Reactor (VVER)-1000 reactor cores. In addition, validation of the calculations against the $P_1$ approximation of the transport theory is investigated in relation to the liquid metal fast breeder reactor benchmark problem. The neutron fixed source calculations are benchmarked through a comparison with the results obtained from similar computational codes. Finally, an analysis of the sensitivity of calculations to the number of elements is performed.