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A fast numerical method for the calculation in a zero-dimensional approach of the equilibrium isotopic composition of an iteratively used transmutation system in an advanced fuel cycle, based on the Banach fixed point theorem, is described in this paper. The method divides the fuel cycle in successive stages: fuel fabrication, storage, irradiation inside the transmutation system, cooling, reprocessing, and incorporation of the external material into the new fresh fuel. The change of the fuel isotopic composition, represented by an isotope vector, is described in a matrix formulation. The resulting matrix equations are solved using direct methods with arbitrary precision arithmetic. The method has been successfully applied to a double-strata fuel cycle with light water reactors and accelerator-driven subcritical systems. After comparison to the results of the EVOLCODE 2.0 burn-up code, the observed differences are about a few percents in the mass estimations of the main actinides.

The implementation of the advanced technologies of partitioning and transmutation (P&T) depends on the particular topics that energy policy makers of a country or region choose to optimize because of its singular constraints or motivations. Each of the different possible P&T configurations, that is, the advanced fuel cycle scenarios, will have different objectives that can impact technology choices and performance expected from such systems.

However, the full potential of any P&T policy can be exploited only if the advanced fuel cycle strategy is utilized for a minimum time period of about a hundred years [

The calculation of the equilibrium isotopic composition of a transmutation system in an advanced fuel cycle is a matter of special interest since this composition is often used as representative of the nuclear reactor through the whole fuel cycle [

In this paper, we describe a fast numerical method for the calculation of the equilibrium isotopic composition in a zero-dimensional approach, taking advantage of the Banach fixed point theorem [

With the aim of validating the methodology, a simulation of the fuel cycle with the advanced burn-up code EVOLCODE 2.0 [

EVOLCODE 2.0 is an in-house development to solve the burn-up problem, that is, the coupled problem of neutron transport and isotopic evolution. Its cycle data flow is shown in Figure

EVOLCODE 2 cycle data flow scheme.

The neutron transport stage is solved by the MCNPX code [

One of the proposed advanced fuel cycle scenarios for waste reduction is the so-called double-strata scenario, whose scheme is shown in Figure

Scheme of the double-strata fuel cycle scenario with LWR and ADS.

In this particular specification of the double-strata scenario, the first stratum of the fuel cycle consists in the irradiation of UO_{2} in light water reactors (LWRs) and the later advanced Purex reprocessing of the irradiated fuel for the Pu reutilization, only once, as MOX, again in LWR. Recovered MA coming from the first partitioning process and Pu and MA coming from the reprocessing of the MOX irradiated fuel are reutilized as fuel for the accelerator-driven subcritical system (ADS) in the second stratum. This second stratum is based on a fast spectrum ADS, which operates with continuous recycling of the main actinides (U, Pu, Np, Am, and Cm). This recycling is supposed to be a pyro-metallurgical process.

The isotopic composition of the ADS fuel depends on the LWR park since the LWR recovered material (from both UO_{2} and MOX reprocessed fuels) becomes a part of the ADS fuel content in each new ADS irradiation (a 15% approximately). The other part of the ADS fuel comes from the pyro-reprocessing of the preceding ADS irradiated fuel. In different cycles, the ADS fuel isotopic composition would change (and being consumed), but maintaining similar ADS operating conditions and a constant additional supply per cycle of actinides from the first LWR-stratum, the fuel approaches an equilibrium (as it is demonstrated below) in which the fuel isotopic composition of each cycle is equal to the following one.

The iterative second stratum of this fuel cycle scenario can be divided in a succession of stages: fuel fabrication, storage, irradiation inside the ADS, cooling, reprocessing, and incorporation of the external material (from the first stratum) into the new fresh fuel. Each process can be described in a matrix formulation as an operator over the fuel composition vector

Equilibrium isotopic composition of the ADS fuel, before the irradiation, for cycles

Isotope* | ADS fresh fuel equilibrium composition for the | ADS fresh fuel composition for the | Ratio (EVOLCODE 2-Method)/ EVOLCODE 2(%) |
---|---|---|---|

U234 | |||

U235 | |||

U236 | |||

U238 | |||

Np237 | |||

Pu238 | |||

Pu239 | |||

Pu240 | |||

Pu241 | |||

Pu242 | |||

Am241 | |||

Am242m | |||

Am243 | |||

Cm242 | |||

Cm243 | |||

Cm244 | |||

Cm245 | |||

Cm246 | |||

Cm247 | |||

Total |

*The short-lived isotopes U-237, U-239, Np-238, Np-239, Pu-243, Am-242, and Am-244 were also used in the calculations.

Coming back to the previous notation, we can find the solution directly from the equilibrium equation

The solution of this equation is

In order to reach this solution, it is necessary to calculate

In the base of eigenvectors of the matrix

In order to estimate the flux level and the one-group zero-dimensional equivalent cross-sections, a fully detailed transport simulation was performed. In this simulation, the exact geometry, the required total power, and an initial guess of the equilibrium composition of the reactor materials were used. Careful weighting had to be used in the procedure of averaging the cross-sections over the whole reactor volume, which was divided in different cells with isotopic evolution. For every one of these cells, one set of one-group effective cross-sections were calculated using the EVOLCODE 2.0 simulation system. The final reactor-averaged cross-sections were calculated weighting the different one-group cross-sections of each fuel cell by its fuel mass. The initial guess of the fuel composition was set to the same vector as the vector of actinides from the external refuelling,

Since the initial ADS fuel composition, chosen for the calculation of matrices

The isotopic composition of the different stages of the iterative stratum of the fuel cycle has been calculated using the matrix equation method. Table

The deviation between the equilibrium isotopic compositions obtained from the matrix equation method and from the irradiation (and the successive decay and reprocessing as indicated in the fuel cycle scenario) using EVOLCODE 2.0 can be seen in the fourth column in Table

The objective of the fuel cycle scenario will define the maximum allowed deviation between one cycle and the following one and also the final decision concerning the necessity of a second iteration on the matrix equation method if the equilibrium isotopic composition has not been reached. For instance, for a fuel cycle scenario where the objective is the study of a representative set of the different (primary and secondary) waste streams, the obtained equilibrium isotopic composition is a good approximation so no iterative calculations are needed.

A fast numerical method, based on an arbitrary precision arithmetic solution to the matrix equations, for the calculation in a zero-dimensional approach of the equilibrium isotopic composition of a transmutation system in an advanced fuel cycle has been developed and successfully applied to a double-strata fuel cycle with LWR and ADS.

The observed differences between the matrix equation method and the EVOLCODE 2.0 results for the successive cycle are smaller than a few percents in the mass estimations of the main actinides. The precision of the calculation can be improved, if needed, by performing a second iteration on the matrix equation method, using the one-group cross-sections of the ADS with the equilibrium composition obtained in the first iteration.