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The main purpose of this paper is to perform a numerical analysis of the Neutron Spatial Kinetic Equations, subject to transients of the External Neutron Source, by applying the Implicit Euler Method as well as the Runge-Kutta Method in order to check which methods are best applicable in transients caused by External Neutron Source. For this purpose, a one-dimensional ADS reactor with a constant external source was simulated based on the geometry of ANL-BSS-6 reactor for benchmark effects.

One of the society main concerns refers to the management of nuclear waste, which are generated at every stage of the fuel cycle. High-Activity Waste (HLW) are composed of fission products and transuranic elements, generated in the reactor core, and they can last a half-life of thousand years. However, the advent of the hybrid reactor concept, also known as “Accelerator-Driven System” (ADS), has opened the possibility that such waste can be reused in the future, after being reprocessed [

The hybrid systems [

The protons are injected onto a spallation target, producing a source of neutrons to propel the subcritical nucleus. The target is made of solid heavy metal or of liquid-metal. The reactions of the spallation on the target issue from ten to twenty neutrons per incident proton, which are introduced into the subcritical nucleus inducing future nuclear reactions. Except for the subcritical state, the core of the reactor is very similar to that of a critical one [

Hybrid reactors, such as ADSs, have attracted world attention and are objects of research and development in many countries [

Hybrid reactors consist of intrinsically safe systems, so that the chain reaction inside them is not self-sustainable, and it can be interrupted simply by shutting off the proton accelerator simply, which demonstrates a straightforward relationship between the external source and the reactor control.

Therefore, this article brings forward a numerical analysis of the spatial kinetic equations subject to transients caused by the external source of neutrons, that of a one-dimensional ADS reactor.

A model of one-dimensional multi-group diffusion dependent on the time considering delayed neutrons is used to study the kinetic of the ADS reactor subject to transients caused by external source of neutrons. The spatial kinetic neutron diffusion equations, for two energy groups, six delayed neutron precursor groups and with the presence of an external source are written as follows:

where

The spatial discretization scheme adopted is based on classical formulation of finite differences, implemented in the box schema [

where

where

with

In order to solve time dependent equation system, the analytical integration procedure [

In the analytical integration, it was assumed that the term fission rate varies linearly between times

where coefficients

The implicit Euler method applied to the matrix equation, Equation (3), leads to the following expression:

Replacing Equation (7) in Equation (9) the result is the following system of linear equations:

where the blocks of matrix are given for:

The solution to Equation (7) is given, by Rosenbrock method by:

where the correction vectors

where _{s},

During the implementation of the automatic time step size control two solutions of Equation (18) are used: a third order solution,

and the equation applied to automatically adjust time step size is:

where

In order to test the related numerical methods, computational codes programmed in the FORTRAN language were implemented. For the implicit method of Euler, a computational code called KDF1D2GIE was developed, whereas for the Runge-Kutta method a computational code was developed called KDF1D2GRK. Both codes solve the spatial kinetics equations with or without external neutron source for a one-dimensional, multi-region, and two energy groups. In addition, a computational code called DF1D2G was developed to solve the stationary diffusion equation, providing the neutron fluxes and the multiplication factor. For purposes of comparison the implicit Euler method was considered the reference method. Before simulating the transients associated with the external source of an ADS reactor, codes DF1D2G, KDF1D2GIE and KDF1D2GRK were validated considering a known benchmark, as follows in the next section.

To test the presented methods, the ANL-BSS-6 benchmark [

At first a stationary calculation was performed by using the DF1D2G code to solve the neutron diffusion equation for two energy groups, thus obtaining the fast and thermal neutron fluxes, which will be used as the initial condition of the transient problems, and a multiplication factor ^{2}.

The ANL-BSS-6 benchmark presents two cases different from transients: A1 and A2. In both cases, the KDF1D2GIE and KDF1D2GRK codes were performed considering the same spatial discretization with a 1 cm mesh. In the simulation of the transient with the KDF1D2GIE code a time step size of 0.001 s was adopted, while the KDF1D2GRK code considered the two options of numerical parameters: Kaps-Rentrop (KR) and Shampine (S) and was used in Equation

Parameters | Region 1 and 3 | Region 2 |
---|---|---|

1.5 | 1.0 | |

0.5 | 0.5 | |

^{−1}) | 0.026 | 0.02 |

^{−1}) | 0.18 | 0.08 |

^{−1}) | 0.015 | 0.01 |

^{−1}) | 0.01 | 0.005 |

^{−1}) | 0.2 | 0.099 |

Group of Precursors | Region 1 and 3 | Region 2 |
---|---|---|

1 | 0.00025 | 0.0124 |

2 | 0.00164 | 0.0305 |

3 | 0.00147 | 0.1110 |

4 | 0.00296 | 0.3010 |

5 | 0.00086 | 1.1400 |

6 | 0.00032 | 3.0100 |

Speeds:

(21) a tolerance equal to 0.001. Moreover, the neutron fluxes obtained by the implicit Euler method and the Runge-Kutta method were also compared at time 1, 2, 3 and 4 s, considering the definition of relative percentage error is given by:

In this case the thermal absorption cross section in the first region is increased linearly in 3% up to 1 s, and maintained constant up to 4 s.

the reference solution presented in [

In the second case the thermal absorption cross section in the first region is reduced linearly in 1% up to 1 s, and maintained constant up to 4 s. Figures 5-7 show, respectively, the behavior of the neutron fluxes in the instants in 1 s and 4 s and the evolution in the time of the power per unit area during the simulation of 4 s. As in case A1, it can be seen from these graphs that the methods obtained very close results and that they are also in agreement with the reference solution presented in [

Methods | BSS-6-A1 Case | BSS-6-A2 Case |
---|---|---|

Implicit Euler | 28.95 | 45.52 |

Runge-Kutta (KR) | 63.80 | 31.47 |

Runge-Kutta (S) | 42.39 | 26.24 |

With respect to the processing time, according to

In this section, the Implicit Euler method and the generalized Runge-Kutta method were used to analyze some types of transients caused by the external neutron source in a one-dimensional ADS reactor, in order to verify which methods are more efficient in convergence and computation time.

The one-dimensional ADS reactor has its geometry and nuclear and kinetic parameters based on the ANL-BSS-6 benchmark reactor, in which case an external source of neutrons located geometrically in the center of the reactor and with a length of 4 cm, as shown in the ^{14} neutrons/s was used.

Using the KDF1D2GIE and KDF1D2GRK codes, three types of transients associated with an ADS reactor will be simulated and will focus on the proton accelerator perturbations, causing variations in the intensity of the proton beam and consequently the intensity of the external source of neutrons. The first transient concerns the activation of the proton accelerator when the ADS reactor is in zero power level condition. The second transient corresponds to the interruption in the proton beam for a short period of time and the third transient to be addressed describes the occurrence of a power peak in the proton beam. These last two transients were based on the cases studied in [

The switching on of the proton accelerator to start the ADS reactor can be considered an operational transient. The external source of neutrons begins to emit

neutrons at the initial instant, t = 0 s, and after some time the generated neutron flux reaches an asymptotic behavior. The simulation was performed for 20 s and Figures 9-11 show, respectively, the behavior of the neutron fluxes at these instant and the evolution in the time of the power per unit area during the simulation.

In the time in 20 s, the highest percentage relative error, when comparing the results of KDF1D2GIE with KDF1D2GRK using the Kaps-Rentrop parameters was 0.163%, in the fast and thermal fluxes, whereas for the Shampine parameters it was 0.189% also in the fast and thermal fluxes. Regarding the processing time,

Methods | Source Start Case | ABI Case | ABO Case |
---|---|---|---|

Implicit Euler | 73.13 | 91.23 | 41.20 |

Runge-Kutta (KR) | 42.97 | 126.97 | 50.41 |

Runge-Kutta (S) | 66.72 | 68.72 | 32.68 |

In this transient the external source of neutrons is switching on instantaneously and this impacts, practically, in the same way in the power per unit area. As can be seen in ^{2}, which corresponds to 75% of the nominal value, equal to 87 KW/cm^{2}. Whereas, with the code KDF1D2GRK, at the same time, it reached 93% of the nominal value.

In this transient the reactor is operating critically and the proton beam of the accelerator is interrupted in the instant in 1 s and after 2 s over the beam is reconnected.

In this transient the reactor is operating critically and the intensity of the proton beam of the accelerator is increased by 100% instantaneously and after 2 s over the beam has its intensity restored to the initial level.

show the behavior of the fast and thermal neutron fluxes at the instant in 1 s, at the beginning of ABO, and

The behavior of the power per unit area in the transient ABO, considering a simulation with the duration of 10 s can be verified in

observes a variation of power, going from 87 KW/cm^{2} to around 170 KW/cm^{2}. With the accelerator operating at normal intensity, the ADS reactor operates at criticality and therefore, with an increase in beam intensity, the reactor starts operating on super criticality. Thus, a gradual increase in power between the instants of 1 s and 3 s can be observed. Figures 17-20 also show the corresponding increase in neutron flux intensity between these instants.

In this work the solution of the spatial kinetics equations for ADS reactors was presented. The spatial kinetics equations were discretized in the spatial variable considering the finite difference method. In order to solve the time-dependent part, the implicit Euler method and the Runge-Kutta method were used, which were implemented in computational codes based on the Fortran language. The implicit method of Euler did not consider an automatic adjustment in the time step. While the code developed for Runge-Kutta was developed considering a truncation error monitoring scheme to automatically adjust the size in the time step. The codes were tested and validated in a well-known benchmark for one- dimensional transients. Both codes were satisfactory in the transient simulations for the ADS reactor involving fluctuations in the external neutron source, and the Runge-Kutta method using the numerical parameters of Shampine proved to be the most efficient in the processing time.

It is intended to implement the Runge-Kutta method in more complex geometries for the ADS reactors, using a three-dimensional geometry and considering a more detailed description of the spallation source. It is also relevant in the future to consider the effects of thermohydraulic feedback because it has been found that the transients of the external neutron source cause strong variations in the power of the reactor in milliseconds and this is likely to impact on the reactivity coefficients.

The authors are grateful for the support provided by the Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Brazil.

de Abreu, W.V., Gonçalves, A.C. and de Lima, Z.R. (2017) Numerical Analysis for Transients in External Source Driven Reactors. World Jour- nal of Nuclear Science and Technology, 7, 103-120. http://doi.org/10.4236/wjnst.2017.72009