Depletion Calculation

As for the Boltzmann Equation, the set of burn-up equations (3.18) can only be solved analytically for simple problems. For more complex geometries and material compositions, iterative approaches based on numerical methods are used. The general procedure is explained in the following and afterward, the two applied programs, VESTA and MCMATH, are described in more detail.

4.2.1 Solution Approach for Depletion Calculations

To solve the set of equations describing the neutron population in the core, the problem is discretized in time and space. The space dependency of equation 3.18 is removed by assuming a spatially constant neutron flux and concentration of certain isotopes for certain volumes within the problem. This calculated neutron flux is set to be constant during the following time step. In doing so, the equations form a set of linear first order differential equations with constant coefficients that can be solved. The change in concentration for each time step is then determined depending on reaction cross sections and decay constant for each considered nuclide. With the new material composition, a new neutron flux is calculated which is then used for a subsequent time step. This iteration process continues until the whole period that should be analyzed is covered. Information connected to the isotopic concentration of the fuel can be derived from the output files. Beside the obvious material composition, they might also include the decay heat, the calculated effective cross sections or the activity of the spent fuel.

For the accuracy of the results, the choice of time steps and burn cells is crucial. The assumption of a constant neutron flux will not hold true neither in space nor in time, because every change in isotopic composition also influences the neutron flux. The time steps should be chosen in order to keep changes in the neutron flux as small as possible between subsequent time steps, at best not exceeding few percent. As an illustration, for a light water reactors the effect of xenon poisoning must be taken into account by adding a small time step at low burn-up to allow for xenon-135 with a half-life of nine hours to build-up to equilibrium concentration. The material in the burn cells should be as homogeneously spread as possible.

Meanwhile, splitting the calculation period into too many time steps might neglect the effect of delayed gammas and, more importantly, proportionally increases computing time. There are similar issues concerning burn cells: the more cells are defined for which the neutron flux is calculated separately, the better is the varying neutron flux distribution in the core mirrored. At the same time model complexity and computing time are increased.

There are several possibilities to improve the results with only a small penalty on computing time. One example is the predictor corrector approach, in which the neutron flux calculated at the end of one time step for the next time step is once again used to deplete the material for a second time. For the actual nuclide concentration, an average value from both depletion calculations is taken. Another option that can be considered is the reduction of the number of nuclides for which the Bateman Equations are actually solved. Typically, the neutron flux is dominated by merely a handful of isotopes. Deciding which isotopes must be treated requires good knowledge of the relevant reactions in the core.

Naturally, the above procedure does not allow for the analysis of rapidly changing conditions, such as transient calculations. Also, changes in core geometry are only possible to a very limited scale. For safety analysis including possible core deformations, other modelling tools are needed.

4.2.2 The Depletion Codes MCMATH and VESTA

There are different code systems that allow the approximation of the solution of the burn-up equations in the previously described manner. For this work, the code VESTA is used in most cases. It was developed by the French Institute for Radiological Protection and Nuclear Safety (IRSN) and is still maintained. In the long term, VESTA is planned to be used as a generic code that can couple an arbitrary Monte Carlo transportation code to any desired depletion code. With the current version of VESTA, any MCNP(X) version can be used for neutron transport. For depletion calculations, either the build-in PHOENIX module or the depletion code ORIGEN-2.24 can be selected (Haeck 2008).

The ORIGEN-2.2 depletion code was originally developed in the 1980s and has several limitations, such as the small memory due to the age of the code. To overcome these limitations, the PHOENIX depletion module is under development at IRSN. Unlike ORIGEN-2.2, PHOENIX can choose between three different methods to solve the transition matrix determined by the Bateman Equations for the problem. One obstacle in solving the matrix exponential is the wide range of entries. To ease the numerical problems associated with this, the very short-lived isotopes are treated separately in the PHOENIX module (Haeck 2011).

A second option to perform depletion calculations is the MCMATH code which was developed in the IANUS working group by Glaser (1998), Pistner (1998), Pistner (2006), Kütt (2007), Englert (2009), and Kütt (2011). It is written in Mathematica5and relies on MCNPX 2.7 for the determination of the neutron flux. The depletion module is included in Mathematica. MCMATH allows for a better control of the different parameters, but it is not applicable to problems where material is shuffled within the reactor core.

One major difference between the two code systems is the determination of the effective cross sections for the depletion step. The effective cross sectionsΣ needed for the depletion step is given by an integral of the form

σe f f = R∞ 0 d Eσ(E, t)Φ(E, t) R∞ 0 d EΦ(E, t) . (4.1)

This integral can directly be calculated in MCNPX using the average flux in one cell. This approach is named one-group binning of the energy, because only one cross section for all energies is provided as MCNPX output. MCMATH uses this method. It can be seen as the convolution of the energy distribution and the cross section over the total energy range. VESTA is based on the multi-group binning approach: the integral is converted into a sum over several small energy ranges. For each energy bin, the energy and the microscopic cross sections are assumed to be constant. Consequently, instead of a quasi-continuous neutron flux, only a fine multi-group spectrum of the neutron flux has to be calculated by MCNPX. This is then used by VESTA to calculate the reaction rates using implemented multi-group cross section data. The default structure in VESTA consists of 43,000 energy bins but can be changed to the user’s preference in accordance with the problem’s requirements. It is for example possible to account for exotic resonance regions if necessary. Another difference between the two systems is the determination of the neutron flux used for the depletion step. In MCMATH, the neutron flux for the subsequent time steps is calculated using the isotopic composition at the end of the previous time step. For better results, after the first round of calculations called the direct run, a second run with twice the time steps is performed. The cross sections for the center calculations are determined via linear interpolation of the values from the first run. In doing so, no new Monte-Carlo calculations are necessary. The same procedure

4 _{ORIGEN-2.2, Isotope Generation and Depletion Code Matrix Exponential Method, Nuclear Energy Agency Date}

Bank, CCC-0371/17, 2002.

can be followed by VESTA, which it is called the predictor-corrector approach there. Using VESTA, other methods to improve the result for the neutron flux are also possible. Beside the default predictor-corrector treatment, there are also options that often require additional flux calculations using interpolated reaction rates.

VESTA and MCMATH have different concepts in regard to the definition of burn cells: in MCMATH discrete volumes filled with any material are defined as burn cells. This material can be removed from the core and continues decaying but cannot be inserted into the core at another position. In VESTA, burn materials are defined. Every cell in which this material is used in MCNP is then part of a burn cell. The burn materials can be removed from the core or placed at another position. Limited changes in the geometry are explicitly possible in VESTA. In MCMATH, additional burn cells must be defined where the material is replaced accordingly, for example coolant instead of absorbers.