Fractional Neutron Point Kinetics Equations Research Articles

Fractional neutron point kinetics equation with Newtonian temperature feedback effects

In this paper we present the numerical analysis of the fractional neutron point kinetics (FNPK) equations with temperature feedback effects. The FNPK model considers a relaxation time associated with a rapid variation in the neutron flux density considered in the differential operator of fractional order known as anomalous diffusion exponent. Different values of the relaxation time with one-group delayed neutron precursor were used for this analysis. Results for neutron flux density dynamic behavior with temperature feedback for different values of anomalous diffusion exponent are shown and compared with the classical neutron point kinetics equations.

Numerical simulation for solving fractional neutron point kinetic equations using the multi-step differential transform method

In this work, we implement the most popular transformation technique, the differential transform method (DTM), to compute numerical solutions for fractional neutron point kinetic equations in a nuclear reactor. The DTM (Alomari 2011 Comput. Math. Appl. 61 2528–34, Arikoglu and Ozkol 2007 Chaos Solitons Fractals 34 1473–81, Darania and Ebadian 2007 Appl. Math. Comput. 188 657–68) is one of the semi-numerical analytical solutions used to solve a wide variety of differential equations and it normally gets the solution in a series form. Here, a fractional neutron point kinetic equation has been solved using the new algorithm that is known as the multi-step differential transform method (MDTM). The MDTM is treated as an algorithm in a sequence of intervals that is used to find simple and approximate solutions. Numerical examples with variable step reactivities, ramp reactivity and sinusoidal reactivity are used to illustrate the precision and effectiveness of the proposed method.

Numerical analysis of startup PWR with fractional neutron point kinetic equation

In this paper we present the numerical analysis of the neutron density behavior when the nuclear reactor power is increased during startup of a PWR. The fractional neutron point kinetic (FNPK) equation with one-group delayed neutron precursor and external neutron source was used for this analysis. It is considered that there is a relaxation time associated with a rapid variation in the neutron flux and this effect is considered with the FNPK which have a physical interpretation of the fractional order is related with the sub-diffusive process, i.e., non-Fickian effects from the neutron diffusion equation point of view. In order to study the relaxation time effects during start-up of a PWR, a numerical analysis with FNPK is carried out, which it is assumed that during the ith step of control rod withdrawal the way of reactivity insertion is step to step, where the neutron source strength was defined as a constant in terms of a known initial stable sub-criticality and the neutron signal from a steady state condition. The results of the FNPK were compared with the classical neutron point kinetics (CNPK), for different values of the anomalous relaxation time.

Application of the fractional neutron point kinetic equation: Start-up of a nuclear reactor

In this paper we present the behavior of the variation of neutron density when the nuclear reactor power is increased using the fractional neutron point kinetic (FNPK) equation with a single-group of delayed neutron precursor. It is considered that there is a relaxation time associated with a rapid variation in the neutron flux and its physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. We analyzed the case of increase the nuclear reactor power when reactor is cold start-up which is a process of inserting reactivity by lifting control rods discontinuously. The results show that for short time scales of the start-up the neutronic density behavior with FNPK shows sub-diffusive effects whose absorption are government by control rods velocity. For large times scale, the results shows that the classical equation of the neutron point kinetics over predicted the neutron density regarding to FNPK.

Sensitivity and uncertainty analysis of the fractional neutron point kinetics equations

The aim of the present work is to evaluate the sensitivity and uncertainty of the anomalous diffusion coefficient in the Fractional Neutron Point Kinetics (FNPK) equations. This analysis was carried out through Monte Carlo simulations of sizes up to 65,000; the size of 50,000 was considered as valid for routine applications. The sensitivity was evaluated in terms of 99% confidence intervals of the mean to understand the range of mean values that may represent the entire statistical population of performance variables. The regression analysis with anomalous diffusion coefficient as the predictor variable showed statistically valid quadratic relationship for neutronic density and the delayed neutron precursor concentration. The uncertainties were propagated as follows: in a 1% change in the anomalous diffusion exponent the responses for neutron density, and precursor density changed by 0.017% and 0.0000125% for short times, and for long times by 0.012% and 0.000267%, respectively.

An Explicit Finite Difference scheme for numerical solution of fractional neutron point kinetic equation

In the present article, a numerical procedure to efficiently calculate the solution for fractional point kinetics equation in nuclear reactor dynamics is investigated. The Explicit Finite Difference Method is applied to solve the fractional neutron point kinetic equation with the Grunwald–Letnikov (GL) definition ( Podlubny, 1999; Oldham and Spanier, 1974). Fractional Neutron Point Kinetic Model has been analyzed for the dynamic behavior of the neutron motion in which the relaxation time associated with a variation in the neutron flux involves a fractional order acting as exponent of the relaxation time, to obtain the best operation of a nuclear reactor dynamics. Results for neutron dynamic behavior for subcritical reactivity, supercritical reactivity and critical reactivity and also for different values of fractional order have been presented and compared with the classical neutron point kinetic (NPK) equation as well as the results obtained by the learned researchers Espinosa-Paredes et al. (2011).

Fractional neutron point kinetics equations for nuclear reactor dynamics

The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. The fractional model retains the main dynamic characteristics of the neutron motion in which the relaxation time associated with a rapid variation in the neutron flux contains a fractional order, acting as exponent of the relaxation time, to obtain the best representation of a nuclear reactor dynamics. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. The numerical approximation to the solution of the fractional neutron point kinetics model, which can be represented as a multi-term high-order linear fractional differential equation, is calculated by reducing the problem to a system of ordinary and fractional differential equations. The numerical stability of the fractional scheme is investigated in this work. Results for neutron dynamic behavior for both positive and negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Additionally, a related review with the neutron point kinetics equations is presented, which encompasses papers written in English about this research topic (as well as some books and technical reports) published since 1940 up to 2010.