Fractional-space neutron point kinetics (F-SNPK) equations for nuclear reactor dynamics

doi:10.1016/j.anucene.2016.08.007

Annals of Nuclear Energy

Volume 107, September 2017, Pages 136-143

https://doi.org/10.1016/j.anucene.2016.08.007 Get rights and content

Highlights

•
Fractional-space neutron point kinetics (F-SNPK) is developed.
•
F-SNPK has a new term called anomalous diffusion source.
•
The anomalous diffusion source is a function of fractional geometric buckling.
•
F-SNPK considers neutron leakage.
•
The neutron leakage is greater with decreasing anomalous diffusion exponent.

Abstract

The aim of this paper is the mathematical derivation, and numerical analysis of the fractional-space neutron point kinetics (F-SNPK) equations for nuclear reactor dynamics. The classic neutron point kinetics (CNPK) equations are one of the most important models of reduced order in nuclear engineering, that been the subject of countless studies and applications to understand the reactor dynamics and its effects. The F-SNPK derived in this work is an extended model respect to CNPK. The F-SNPK model was derived considering a fractional-space law for the neutron density current where the differential operator is the fractional order (FO), also known as anomalous diffusion exponent. The physical meaning is related with non-Fickian effects (anomalous diffusion), which in F-SNPK gives rise to a new term called in this work anomalous diffusion source that considers neutron leakage, and it depends on the geometric buckling and anomalous diffusion exponent. The numerical analysis shows that the neutron leakage is greater with decreasing anomalous diffusion exponent.

Introduction

The fractional-space law for the neutron current density proposed by Espinosa-Paredes et al. (2013) to obtain the neutron spatial-fractional diffusion equation (NFDE), allows to extend the scope and to improve the classical diffusion theory (normal diffusion or Fickian approximation). The diffusion theory provides a strictly valid mathematical description of the neutron flux when the assumptions made in its derivation are satisfied (Stacey, 2004): (1) Absorption much less likely than scattering, (2) Linear spatial variation of the neutron distribution, and (3) Isotropic scattering. The first assumption is satisfied for most of the moderating and structural materials found in a nuclear reactor, but not for the fuel and control elements. The second condition is satisfied within a few mean free paths away from the boundary of large homogeneous media with relatively uniform source distribution. The third condition is satisfied for scattering from heavy atomic mass nuclei. The non-fulfillment of any of these assumptions represents clear evidence that the standard Fick’s law, needs to be modified. Specifically, when the highly heterogeneous configuration in nuclear reactors, in presence of strong neutron absorbers in the fuel, control rods and chemical shim in the coolant. The dynamics of these absorbers radically changes the local energy generation and in turn the re-distribution of the same absorber, frequently requiring a more accurate treatment, for example the neutron transport, than that provided by the classical diffusion theory.

The (NFDE) equation has been the subject of studies and applications to understand the anomalous diffusion phenomena in nuclear reactors (e.g., Moghaddam et al., 2014, Moghaddam et al., 2015a, Moghaddam et al., 2015b, Moghaddam et al., 2015c). The developed of one-dimension code (NFDE-1D), which can simulate the reactor core using arbitrary exponent of the fractional order of differential operators, where a numerical solution of the NFDE is using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame, is presented by Moghaddam et al. (2014). The numerical solution of the NFDE in two-dimension (NFDE-2D) for multigroups of energy was developed by Moghaddam et al. (2015a), and the three-dimension (NFDE-3D) version is presented in the work of Moghaddam et al. (2015b). The NFDE-3D version is validated against experimental measurements of DIMPLE reactor in the work of Moghaddam et al. (2015c).

The NFDE is starting point to mathematical derivation of fractional-space neutron point kinetics (F-SNPK) equations for the nuclear reactor dynamics, presented in this work. The F-SNPK model is an extended model respect to classic neutron point kinetics (CNPK). The F-SNPK considers neutron leakage that depends on the geometry and dimensions of the reactor, and the anomalous diffusion exponent being of the fractional order. This extended term is called in this work anomalous diffusion source that is made of the difference of the product between buckling and the diffusion coefficient of integer order, and the corresponding of fractional order. In this work the numerical experiment considered a transient due to reactivity insertion positive and negative for different values of anomalous diffusion coefficient and two reactors geometries.

The rest of this work is accomplished as follows. In Section 2, the motion of neutron as process diffusion in the multiscale system and fundamental ideas are presented. In Section 3, a brief introduction to fractional calculus is presented. In Section 4, the neutron fractional diffusion equation (NFDE) is introduced. In Section 5, the mathematical derivation of fractional-space neutron point kinetics (F-SNPK) equations for nuclear reactor dynamics is developed. In Section 6, the analysis of new term of F-SNPK is analyzed and discussed.

In Section 7, the comparison between F-SNPK and FNPK is presented. In Section 8, numerical experiments, results and discussed applying F-SNPK model are presented. The conclusions with a brief summary of the main findings are reported in Section 9.

Section snippets

Multiscale system

The process of neutron diffusion that takes place in a highly heterogeneous hierarchical configuration as is illustrated in Fig. 1. The scaling considered in this figure are: (Fig. 1a) nuclear reactor core, (Fig. 1b) fuel assembly which is an array of the fuel cell, and (Fig. 1c) the array of the fuel rod. Normally these arrays (assemblies, cells and rods) are periodic with an anisotropy characterized by the nominal array geometry (Todreas and Kazimi, 1990). In order to describe the neutron

Fractional calculus

In order to achieve a fractional order of differentiation and integration operator, we should effort to interpolate the operators between two integer order operations, so in the limit sense when the order of operator approaches to integer. Some popular relations of fractional derivatives are explained briefly:

Riemann-Liouville defined the fractional differential as follows: $D^{κ} ϕ (x) = \frac{d^{m}}{{dx}^{m}} [\frac{1}{Γ (m - κ)} \int_{0}^{x} \frac{ϕ (τ)}{{(x - τ)}^{κ + 1 - m}} d τ], for (m - 1) ⩽ κ < m$

Caputo formulated the fractional differentials as follows: $D^{κ} ϕ (x) = \frac{1}{Γ (m - κ)} \int$

Neutron fractional diffusion equation (NFDE)

Consider the processes of collision and reaction in a reactor core with the characteristic length scales given in Fig. 1, where the material fuel (σ) is dispersed in lumps within the moderator (γ). Then, the conservation equation that governs the neutron collision and reaction processes in the moderator (γ) in this system, as well as the initial conditions and boundaries at interfaces are given by (Duderstadt and Hamilton, 1976):

γ-Moderator

\frac{1}{υ} \frac{\partial ϕ_{γ}}{\partial t} + \nabla \cdot J_{γ} (r, t) + Σ_{a γ} (r) ϕ_{γ} (r, t) = S_{γ} (r, t)

\frac{1}{υ} \frac{\partial J_{γ} (r, t)}{\partial t} + \frac{1}{3} \nabla ϕ

Fractional-space neutron point kinetics (F-SNPK) equations

The derivation of the F-SNPK equation is obtained with one-speed applying NFDE given by Eq. (20), which is the fractional order (FO). The one-group source term S(r, t), is given by (Glasstone and Sesonske, 1981): $S_{γ} (r, t) = (1 - β) k_{\infty} Σ_{a γ} ϕ (r, t) + \sum_{i = 1}^{m} λ_{i} {\hat{C}}_{i} (r, t)$ where the count i indicates the ith precursor of delayed neutron, given by $\frac{d {\hat{C}}_{i} (r, t)}{dt} = β_{i} k_{\infty} Σ_{a γ} ϕ_{γ} (r, t) + λ_{i} {\hat{C}}_{i} (r, t), 1, 2, \dots, m$

The term β is the total fraction of delayed neutron, k_∞ is the infinite medium multiplication factor, and $λ$ is the decay

Analysis of f(κ)

The anomalous diffusion in the spatial coordinate results in an extended equation (given by Eq. (33), called in this work F-SNPK) respect to classical point reactor kinetics (CNPK) equation, through of f(κ), which can be consider as a source term, and therefore we can refer to this term as anomalous diffusion source. This term was derived rigorously in Section 5 given by Eq. (38). We rewrite this equation in this section again for purposes discussion and analysis: $f (κ) = υ (\underset{IO term}{\underset{︸}{D_{γ} B_{g}^{2}}} - \underset{FO}{\underset{︸}{D_{κ γ} B_{g}^{κ + 1}}}$

Comparison between models: FNPK and F-SNPK

The fractional neutron point kinetics (FNPK) developed by Espinosa-Paredes et al. (2011) was obtained from time-fractional diffusion equation (Espinosa-Paredes et al., 2008): $\frac{τ^{κ}}{υ} \frac{\partial^{κ + 1} ϕ}{\partial t^{κ + 1}} + τ^{κ} Σ_{a} \frac{\partial^{κ} ϕ}{\partial t^{κ}} + \frac{1}{υ} \frac{\partial ϕ}{\partial t} = \hat{S} - Σ_{a} ϕ + D_{γ} \nabla^{2} ϕ + τ^{κ} \frac{\partial^{κ} \hat{S}}{\partial t^{κ}}$ where the fractional (time) constitutive equation of the current density J for one-speed case: $τ^{κ} \frac{\partial^{κ} J (r, t)}{\partial t^{κ}} + J (r, t) = - D_{γ} \nabla ϕ (r, t)$ was applied (If this equation is used in Eq. (12), can be obtained Eq. (44)). These equations physical interpretation of this model is that the

Numerical experiments applying F-SNPK model

The numerical experiments with fractional-spatial neutron point kinetics (F-SNPK) are compared with classical point reactor kinetics (CNPK), which is obtained with κ = 1 to obtain f(κ) = 0.

Since the new model derived in this work depends on the buckling geometry, in this work we present the numerical analysis of two reactors:

•
Slab reactor.
•
Cylindrical reactor.

F-SNPK equations for nuclear reactor dynamics, derived in Section 5, for six neutron precursors are given by (re-written the equations to

Conclusions

The fractional-space neutron point kinetics (F-SNPK) equation for nuclear reactor dynamics was derived (Eq. (39)) in this work. The fundamental consideration to derived F-SNPK is the application of fractional (space) constitutive equation of the current density (Eq. (1)) where the differential operator is the fractional order, i.e., the anomalous diffusion (non-Fickian) effects are considered.

The F-SNPK contains a new term with respect to the classical equation of neutron point kinetics (CNPK),

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¹: Sabbatical leave at Facultad de Ingeniería of the Universidad Nacional Autónoma de México through Programa de Estancias Sabáticas del CONACyT.

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Fractional-space neutron point kinetics (F-SNPK) equations for nuclear reactor dynamics

Highlights

Abstract

Introduction

Section snippets

Multiscale system

Fractional calculus

Neutron fractional diffusion equation (NFDE)

Fractional-space neutron point kinetics (F-SNPK) equations

Analysis of f(κ)

Comparison between models: FNPK and F-SNPK

Numerical experiments applying F-SNPK model

Conclusions

Nucl. Eng. Des.

Ann. Nucl. Energy

Ann. Nucl. Energy

Ann. Nucl. Energy

Ann. Nucl. Energy

Nucl. Eng. Des.

Ann. Nucl. Energy

Appl. Math. Model.

Commun. Nonlinear Sci. Numer. Simul.

Nuclear Reactor Analysis

Time-fractional telegrapher’s equation (P1) approximation for the transport equation

Nucl. Sci. Eng. J.