Numerical simulation of two-dimensional fractional neutron diffusion model describing dynamical behaviour of sodium-cooled fast reactor

Roul, Pradip; Rohil, Vikas; Espinosa-Paredes, Gilberto; Obaidurrahman, K.

doi:10.1016/j.anucene.2021.108709

Cited by 16 publications

(4 citation statements)

References 32 publications

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“…Since fractional calculus is rapidly growing recently with the old models replaced by fractional ones in the light of a diverse choice of fractional derivative definitions, we mainly concentrate on recent studies. For instance, anomalous diffusion processes were investigated by means of fractional models in oil pollution [5], in tumor growth and oncological particularities [6,7], in antioxidant vegetable [8], in the voltage regulator of the power industry [9], in nuclear neutron transport [10], in enhancing low-frequency signal [11], in computer vision [12], in radioactive and transmutation linear chains [13], in optimizing current sequences in lithium-ion batteries [14,15], in structural analysis creep [16], in the transmission dynamics of Nipah virus [17], in chronic hepatitis B-related liver fibrosis [18], in cytokeratin [19], in the link formation of temporal networks [20], in slow decay phenomena of the Tesla Model S battery [21], in the slip flow of nanoparticles [22], and in the ultrasonic propagation of wave in a fractal porous material [23], among many others.…”

Section: Introductionmentioning

confidence: 99%

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Turkyilmazoglu,

Alofi

2024

Fractal Fract

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This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.

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Section: Introductionmentioning

confidence: 99%

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Turkyilmazoglu,

Alofi

2024

Fractal Fract

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“…Antioxidant properties of banana slices were investigated in Granella et al (2022) on the grounds of anomalous drying kinetics and moisture diffusion. Neutron transport in cylindrical nuclear reactor core was modelled in Roul et al (2022) and numerically solved via alternating direction implicit scheme.…”

Section: Introductionmentioning

confidence: 99%

Transient and passage to steady state in fluid flow and heat transfer within fractional models

Türkyılmazoğlu

2022

HFF

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Purpose The classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided. Design/methodology/approach In this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models. Findings From the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration. Originality/value With the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.

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“…Among the numerous applications in the various fields of science and engineering, we recall solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments [19]. In fact, field experiments demonstrate that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called "heavy tails", i.e., effects which cannot be predicted via the standard mass transport equation [20][21][22][23] but via fractionalorder differential equations that may be viewed as long-time and long-space limits of a continuous time random walk (CTRW) [24].…”

Section: Introductionmentioning

confidence: 99%

“…In case of time-fractional neutron diffusion models with delayed neutrons [19,29,30], non-local effects are established and sub-diffusive phenomena caught, coming from the heterogeneity of nuclear reactors. Anyway, analytical solutions for this problem are generally not available and only a few efficient numerical techniques have been developed in the literature to approximate the solution of even a 2D fractional [10,31] or non-fractional [32] diffusion models, even in the presence of a reaction term [26,33].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Salomoni

Marchi

2021

Fractal Fract

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Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations.

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Numerical simulation of two-dimensional fractional neutron diffusion model describing dynamical behaviour of sodium-cooled fast reactor

Cited by 16 publications

References 32 publications

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Transient and passage to steady state in fluid flow and heat transfer within fractional models

Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

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