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This paper proposes an approximation for the modified point kinetics equations to obtain a mathematically simpler set of equations similar to those of the classic point kinetics. The new method, which is called the adjusted point kinetics method, discards the least important terms in the modified kinetics equations and adjusts the mean generation time parameter to tentatively incorporate the influence of the discarded terms. To validate the proposed model, it is compared to the modified kinetics model in numerical calculations of two very simple reactivity insertion cases.

The Doppler Broadening Phenomenon provides a good description for the effects of thermal agitation on the cross-section of the neutron-nucleus interaction and has its theoretical basis on Quantum Mechanics through the Single Level Breit-Wigner Formalism, as well as on Statistical Mechanics, with the Maxwell Boltzmann distribution. In this paper, the consequences of replacing the Maxwell Boltzmann speed distribution by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension is discussed. The Tsallis speed distribution introduces a deformation parameter q that measures the deviation from the Maxwellian speed distribution and, by taking the limit q → 1 , the deformation is removed. In this context, and considering the Tsallis distribution, an integral form is obtained for a deformed Doppler Broadening Function in the scope of the single-level Breit-Wigner formalism and Bethe-Placzek approximations. This deformed Doppler Broadening Function reproduces the well-established conventional Doppler Broadening Function at the limit when q → 1 in which the deformation is removed. In order to study the range of values where the effect is significant, numerical tests were carried out considering several values for the q parameter.

Several works have been done for the development of models that generalize the Maxwell-Boltzmann distribution, aimed at encompassing physical phenomena that lie outside the thermal equilibrium. Amongst these, there are distributions that result from the non-extensive statistics of Tsallis and Kaniadakis. Starting from these generalized distributions, a Doppler broadening function was proposed in recent papers, using the deformed Kaniadakis distribution, which was numerical, evaluated with the use of a Gauss-Legendre quadrature. From this perspective, this paper presents an analytical solution for the generalized Doppler broadening function through obtaining a partial differential equation, considering the Kaniadakis distribution. This equation is solved analytically using the methods of Frobenius and variation of parameters, in order to obtain a generalized solution for the Doppler broadening function, containing a deformation parameter κ , that measures the deviation in relation to the Maxwell-Boltzmann distribution. Finally, the results were produced considering several values for κ , with the intent of making a comparison with the reference values. For the validation of the deformed Doppler broadening function’s analytical solution, a numerical solution of the partial differential equation was generated. It was possible to use this numerical solution as a benchmark for the analytical solution that was derived. It was demonstrated that the analytical solution obtained is consistent, because when κ tends to zero, the solution falls in the conventional form, when the Maxwell-Boltzmann distribution is considered. Apart from this, the results were shown to be good, especially when we consider the temperature and power ranges for practical applications, as the maximum error obtained was smaller than 1%.

In a Molten Salt Reactor, its circulating fluid fuel introduces some important phenomena in the study of the reactor neutron physics. Prominent among them is the decaying of a fraction of the delayed neutron precursors outside the core, which depends on their recirculation time through the external loop, also known as transit time. In this paper we analyze situations where the transit time cannot be trivially calculated (such when flow velocity varies with time) and derive analytical solutions for some specific cases, along with a numerical method useful for more complex cases, which is then validated by comparing its solutions with the analytical ones. In order to check the influence of this detailed calculation of transit time, we compare solutions of the point kinetics equations calculated using the analytical expression for transit time with approximated ones, to verify which cases are most sensitive to the exact calculation of transit time.

This paper presents a method that establishes a relation of equivalence between the Doppler broadening function ψ x , ξ using the Maxwell-Boltzmann distribution and the Deformed Doppler broadening function ψ κ ( x , ξ ) , according to the generalised Kaniadakis distribution, through the effective temperature of the medium T ef . Although the Kaniadakis distribution is more precise, its implementation in existing computer systems, in reactor design calculation is a complex undertaking due to the amount of computing time. To overcome this hurdle, the main objective of this paper is to present a simple methodology that allows calculation of the Deformed Doppler broadening function ψ κ ( x , ξ ) , with the Kaniadakis distribution based on the idea of a mapping of the Deformed Doppler broadening function, ψ x , ξ via a polynomial multidimensional parametrization. Deformed Doppler broadening function ψ κ ( x , ξ ) is approximated by the Doppler broadening function ψ ( x , ξ ~ ) , where ξ is associated to the temperature of medium T and ξ ~ relates to the effective temperature of medium T ef . That is, in calculation systems aimed at determining multigroup cross sections, it will only be necessary to replace T with T ef .

Much work has been done in the recent decades on the fundamentals and applications of generalized statistical theories based on the quasi-Maxwellian distribution of probabilities. In this paper the effect of taking a non-Gaussian statistics – well known as Kaniadakis statistics – into consideration, based on a κ parameter which accounts for the non-Maxwellian behaviour in the study of neutron-nuclei interaction is assessed. In this context, an integral form for a generalized Doppler Broadening Function is obtained in the scope of the single-level formalism given by the Beth-Plackzec approximations. This new function reproduces the well-established conventional Doppler Broadening Function on the limit when κ → 0 . Numerical tests were carried out and by varying the κ parameter it was possible to study the range of values where the effect is relevant.

The activation technique allows much more precise measurements of neutron intensity, relative or absolute. This technique requires the knowledge of the Doppler Broadening Function to determine the resonance self-shielding factors in the epithermal range. In this technical note the influence of considering the enlarged Doppler Broadening Function proposed by Palma et al. in the calculation of the self-shielding factors in the epithermal range in round disks is evaluated. Although the enlarged Doppler Broadening Function version shows considerable deviations from the conventional Doppler Broadening Function when ξ ⩽ 0.06 it has been shown that the effect is not fully transmitted to the calculation of the self-shielding factors in the epithermal range of 232 Th, 240 Pu, and 238 U isotopes. Thus, it is possible to conclude that there is no measurable gain in using enlarged Doppler Broadening Function instead of conventional Doppler Broadening Function in the activation technique and the conventional version is recommended in this application. For applications involving higher temperatures and consequently lower values of parameter ξ , there may be a gain depending on the degree of precision required in the calculations.

The accidental drop of a control rod may cause the reactor to operate unsafely. In this type of event there is a distortion in the power distribution in the core which may exceed safe operating limits of the reactor. This work aims to develop an alternative model capable of detecting an inadvertent drop event of a control rod and then identifying it using the in-core and ex-core nuclear instrumentation. The proposed model has two stages: The first aims to detect in an online form the inadvertent drop of a control rod at the core through the readings of the ex-core detectors and the second is to identify the dropped control rod through the in-core detectors readings, using a Delaunay triangulation method. The results showed that it is possible to detect the event of a control rod drop using an established limit on the asymmetry of the radial power distribution measured by ex-core detectors and the identification method of dropped control rod depends exclusively of the in-core nuclear instrumentation readings, because it is based on a physical premise that higher power variations occur in the position where the control rod is inserted and its surroundings.

The accidental dropping of a control rod may cause the reactor to operate unsafely. In this type of event, there is a distortion in the distribution of power and temperature in the core may exceed operating limits reactor safe. This work aims to develop an alternative model capable of identifying, at any time of the cycle, the control rod that has accidentally dropped at the core of a PWR reactor, using the readings of the thermocouples in order to minimize possible losses. The model assumes that in a possible drop of a control rod, the largest temperature change occurs in the position where the control rod is inserted. Considering the fact that there are no temperature gauges in all control rod positions, the proposed model uses radial basis function (RBF) neural networks to make a reconstruction of temperatures in these positions from the measurements of the thermocouples at the time of the accidental drop. The study found that the predictions of the temperatures made by the RBF neural networks showed good results, which enables the identification of the control rod dropped accidentally in the core, by simple inference of the fuel assembly of lowest temperature among temperatures reconstructed.

In all nuclear reactors some neutrons can be absorbed in the resonance region and, in the design of these reactors, an accurate treatment of the resonant absorptions is essential. Apart from that, the resonant absorption varies with fuel temperature due to the Doppler broadening of the resonances. The thermal agitation movement in the reactor core is adequately represented in the microscopic cross-section of the neutron-core interaction through the Doppler broadening function. This function is calculated numerically in modern systems for the calculation of macro-group constants, necessary to determine the power distribution of a nuclear reactor. It can also be applied to the calculation of self-shielding factors to correct the measurements of the microscopic cross-sections through the activation technique and used for the approximate calculations of the resonance integrals in heterogeneous fuel cells. In these types of application we can point at the need to develop precise analytical approximations for the Doppler broadening function to be used in the calculation codes that calculate the values of this function. However, the Doppler broadening function is based on a series of approximations proposed by Beth–Plackzec. In this work a relaxation of these approximations is proposed, generating an additional term in the form of an integral. Analytical solutions of this additional term are discussed. The results obtained show that the new term is important for high temperatures.

The goal of this work is to develop a method to identify the accidental drop of a control rod in the core of a pressurized water reactor based on the readings obtained from thermocouples. The method is based on the physical premise that, due to the local effect that the event produces on the distribution of power and temperature, the greatest variations take place at the fuel assembly position where the control rod is inserted, as well as in its neighbouring regions. In order to identify the control rod, the method presented here analyses the profiles on temperature in the thermocouples and calculates virtual temperature at the positions of the fuel assemblies of the control rods via inverse distance interpolation. The results have shown the feasibility of the method, which has the advantage of not relying of previously simulated data to train the neural networks as seen in previous works.

In thermal nuclear reactors, the calculation of the Doppler broadening function and interference term are necessary for the precise determination of cross-sections. These functions are calculated numerically in systems for the calculation of macro-group parameters, which are necessary to know the power distribution of a nuclear reactor. Most of these calculations consider the standard model of Maxwell-Boltzmann statistics. More recently, some works proposed numerical and analytical solutions for the Doppler broadening function using quasi-Maxwellian distributions. In this work, new analytical formulations are proposed for the Doppler broadening function and interference terms. With these new approximations, the scattering cross-section was calculated. The results obtained show that the proposed approximations have good accuracy, with values of percentage deviation above 1% in regions near the resonance peak. For the values far from the peak, the new approximation presented better results when compared with previous analytical solutions.

A control rod drop event in PWR reactors leads to an unsafe operating condition. It is important to quickly identify the rod to minimise undesirable effects in such a scenario. The goal of this work is to develop an online method to identify control rod drops in PWR reactors. The method entails the construction of a tool based on ex-core detector responses. It proposes to recognize patterns in the neutron ex-core detectors responses and thus to make an online identification of a control rod drop in the core during the reactor operation. The results of the study, as well as the behaviour of the detector responses demonstrated the feasibility of this method.

This paper provides a new integral representation for the Doppler broadening function ψ(ξ, x), which is interpreted as being a Fourier cosine transform. This integral form allows the obtaining of an analytical solution in a simple and accurate functional manner as regards the elementary functions. The solution obtained through the new integral representation can be widely used in several applications such as the calculation of self-shielding factors and measurement corrections for the microscopic cross section through the activation technique.

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.