Science and Technology of Nuclear Installations

Volume 2016, Article ID 6980547, 8 pages

http://dx.doi.org/10.1155/2016/6980547

## A Simple Formula for Local Burnup and Isotope Distributions Based on Approximately Constant Relative Reaction Rate

Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai, Guangdong 519082, China

Received 28 September 2015; Revised 31 December 2015; Accepted 21 January 2016

Academic Editor: Tim Haste

Copyright © 2016 Cenxi Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple and analytical formula is suggested to solve the problems of the local burnup and the isotope distributions. The present method considers two extreme conditions of neutrons penetrating the fuel rod. Based on these considerations, the formula is obtained to calculate the reaction rates of ^{235}U, ^{238}U, and ^{239}Pu and straightforward the local burnup and the isotope distributions. Starting from an initial burnup level, the parameters of the formula are fitted to the reaction rates given by a Monte Carlo (MC) calculation. Then the present formula independently gives very similar results to the MC calculation from the starting to high burnup level but takes just a few minutes. The relative reaction rates are found to be almost independent of the radius (except of ^{238}U) and the burnup, providing a solid background for the present formula. A more realistic examination is also performed when the fuel rods locate in an assembly. A combination of the present formula and the MC calculation is expected to have a nice balance between the numerical accuracy and time consumption.

#### 1. Introduction

To increase the efficiency of the fuel, one possible way is to increase the burnup of the fuel before discharge. The local burnup on the edge of the UO_{2} fuel rod is much higher than the average burnup. Thus it is of great importance to investigate the properties of the rim of the fuel rod when considering the increment of the average burnup. Many investigations show that the mechanical structure close to the surface is rather different from that at the center of the fuel rod. In the high burnup range, a microstructure change was found on the rim of the fuel rod through transmission electron microscopy [1]. One explanation of the formation of the high burnup structure supposed that the bubbles in high burnup region are nucleated and stabilized by fission fragments, which depends on the fission rate [2]. The small pores at the high burnup region are calculated to be highly overpressurized [3]. Recently, the UO_{2} fuel in both light and heavy water reactor is investigated to understand the high burnup structure of the fuel [4, 5].

In a thermal reactor, the neutrons generated from fission need to be slowed down in the moderator to be able to induce the next fission. When the low energy neutrons go from the moderator to the fuel rod, they are firstly absorbed by the fuel close to the surface. In general, the reaction rate is higher in the rim when induced by slowed neutrons, such as the reaction of ^{235}U and Pu induced mainly by the thermal neutron. In contrast, the reaction of U is mainly induced by the resonance neutrons. The local burnup phenomenon is mainly caused by the reaction of U, which has large cross section for resonance neutrons, especially some high peaks of cross section at certain energies. Such reaction produces much more Pu near the surface of the fuel rod, thus giving origin to the increased local burnup. As a consequence of the different reaction rates along the radial direction, the U and Pu isotopes also have different radial distributions. In a fast reactor, the Pu isotopes are redistributed along the radial direction [6]. A large local burnup is not expected because the neutrons do not need to be slowed down in the moderator.

In nuclear science, theoretical models are very important and powerful to solve various problems. Our previous works have shown that the mass, levels, electromagnetic moments and transitions, and other properties of the nuclei can be described with high precision in the frame of the nuclear shell model even for very unstable nuclei, such as C and Al [7–9]. In nuclear reactor, the simulation is even more important because the experimental data are relatively difficult to be obtained, sometimes with high risk. For the present problem, various models can be used to calculate the local burnup and the isotope distributions along the radial direction. The TRANSURANUS model is very successful in the description of the local burnup and related properties in various reactors including light and heavy water reactors [10–12]. In TRANSURANUS model, a fixed neutron flux is assumed except for the absorption reaction of U and Pu [12], while the RAPID model considers detailed properties of the neutron flux at each burnup [13]. Recently, an empirical formulation is suggested to describe the local burnup and high burnup properties based on DIONISIO code [14–16].

Here we suggest an analytical and simple formula to calculate local burnup and isotope distributions based on the constant relative reaction rates at the different radius (except for U) and burnup. The present method considers the radial distribution of the neutron flux through two extreme conditions. The parameters are fitted to the reaction rates of the starting burnup level. From this level, the present method is shown to have independent and similar results to the Monte Carlo (MC) simulation, but just a few minutes is needed because of its simplicity.

#### 2. The Description of the Model

The rate of certain reaction type () between neutrons and a given nuclide () at an average burnup (bu) can be written as a function of the radius () and the energy of neutron ():where is the concentration of the nuclide at the radial position , is the neutron flux at the energy and the radius , and is the microscopic cross section between neutron and for reaction type at the energy . The and reactions are assumed to be the two more important reaction types in the fuel rod. In the present work, the main purpose is to examine the validity of the present model. Thus only ^{235}U, U, and Pu isotopes are considered to constrain the model testing to a simple case.

In the present study, (1) is assumed to be two terms for and of ^{235}U and Pu, and of U: and three terms for of U:where the coefficients are specified for each nuclide and the reaction type. The coefficient is expected to be the same for all three nuclides. The coefficient in of U is much larger than .

The following discussion concentrates on the explanations of the above assumptions of the reaction rates. In a thermal reactor, two kinds of the neutron reactions are most important: the scattering reaction in the moderator and the absorption reaction in the fuel rod. When the slowed down neutrons penetrate into the fuel rod, we mainly concentrate on the neutron flux at the thermal and resonance region which are important to the absorption reaction in the fuel rod. The average energy of fission neutron is around 2 MeV, which is much larger than that of the thermal and resonance neutron. Thus no multiplication of the neutron flux is considered at the energy of thermal and resonance region in the following discussion. Two extreme conditions can be considered for certain energy of the neutron. The first one is that the neutrons are strongly absorbed by the fuel, which indicates that the mean free path (MFP) of the absorption, , is much smaller than the size of the fuel rod at the energy . The second one is that few neutrons are absorbed in the fuel at the energy .

For the first extreme condition, one can further simplify that the velocities of the neutrons are all perpendicular to the surface of the fuel rod. Such assumption is acceptable because the velocity is almost symmetric around the radial directions in the real case. Then the transportation of the neutrons at certain energy is the solution of the Boltzmann equation in the cylindrical coordinate without the terms of the scattering and the source. The radial part of the equation is as follows:where the negative sign comes from the opposite direction between the velocity and the radial vector, and the solution is as follows: where is the radius of the fuel rod. Because of the assumption of the strong absorption, the neutron flux is meaningful near the surface of the fuel rod. The divergence at should not be considered. Because is much smaller than the size of the fuel rod, , the neutron flux decreases very quickly to zero at small . The neutron flux is approximately near the surface and zero at the other radial region. The reaction rate per nucleus is proportional to , resulting in the third term in (3). One example of such situation corresponds to the reaction of U at certain energy. The magnitude of the atomic concentration of U is around /cm^{3} in the fuel rod. For reaction of U, some peaks of cross section at the resonance region can achieve barns [17], resulting in at the magnitude of cm, which is much smaller than the size of the fuel rod.

For the second extreme condition, is much larger than the size of the fuel rod . is almost unchanged in the fuel rod and is also approximately constant, resulting in the first term in (2) and (3). The magnitude of the concentrations of ^{235}U and ^{235}Pu is around or less than /cm^{3} in the fuel rod. If and are much smaller than barns and barns, respectively, then is much larger than .

For the situations between these two limits, the second term in (2) and (3) is assumed with the same magnitude of . Please note that the above discussion is restricted in the fuel rod. If the neutrons go out of the fuel rod, they may be slowed and reenter the fuel rod with a different energy.

In principle, , in (2) and (3) can be calculated through the cross section data and the neutron flux at . But in some energy regions, the cross sections change dramatically, and hence it is difficult to obtain the coefficients. Spatially Dependent Dancoff Method can be used to calculate the cross sections in the resonance region [18]. In the present work, these coefficients are fitted using a MC calculation. An initial burnup level is assumed and with it a MC calculation is done for one fuel rod to calculate the reaction rates at different radial positions. The coefficients are then fitted to these reaction rates and used to calculate the burnup levels after . After , the MC calculation and the present formula are performed independently for all burnup levels, , , and so on. The concentrations can be calculated throughby assuming that the reaction rates do not change during time . The time duration corresponds to the burnup change between two levels:where is the average energy released by fission and is the initial mass of U isotopes in volume . One can transformIn MC calculation, the reaction rates in the above equations are simulated with the concentration of all three isotopes at each burnup level. The present formula calculates all reaction rates through the coefficient fitted to the reaction rate at . The reaction rate is neglected in the above equations in both MC and analytical calculation because the present model does not include Pu. It is acceptable as the present work concentrates on the examination of the formula not on a real burnup problem. The next section shows that the present formula can obtain results for quite close to those obtained with the MC calculation.

The above equation looks similar to the formula in TRANSURANUS model [10, 12], in which the relationship between and bu can be generally written as where is one kind of nuclide. For U, there are no second terms. For U, Np, and Pu, are U, Np, U, and Pu, respectively. and are the one-group effective cross sections for the total neutron absorption and neutron captured, respectively. The cross sections are obtained differently for UO_{2} and MOX fuel because of the very different initial concentrations in each fuel and the corresponding different neutron spectrum. is a conversion constant. is the radial form factor, which is for U and Pu and unit for all other nuclides. comes from the resonance absorption and the parameters are determined by comparison with measurements [10, 12]. The local burnup and isotope distributions can be calculated through (9). More details can be found in [10, 12]. The present work does not consider unit (actually the different radial neutron flux) for all nuclei.

#### 3. Calculations and Discussions

In the present work, the continuous energy Monte Carlo code TRIPOLI-4 [19] is used for the MC calculations as the starting point of the analytical formula and the reference after calculations. The geometry of the fuel rod in the present investigation is set to be cm, with cladding between 0.4127 and 0.4744 cm. The moderator is in an outside box with the length 1.2647 cm. All neutrons are reflected back when colliding with the six surfaces of the box. The fuel rod is divided to seven parts in the MC simulation, with the dividing point located at . The corresponding center of each part is listed in Table 1. For a normal UO_{2} fuel, there is no Pu isotopes at the beginning. It is reasonable to start the present calculation at a certain burnup with Pu included. The MC calculation is done with 3.3% enrichment U fuel in the fuel cell to obtain and reaction rates of U and U. With the reaction rates, the number of U, U, and Pu in a certain burnup can be calculated by assuming that the reaction rates do not change in this period. The concentration of U, U, and Pu at MWd/kgU is given in Table 1 as the starting point of the following calculations.