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1. Introduction

The drawings are the authoritative source of dimensions, tolerances, and material composition of components important to criticality safety. This section should provide a summary table of criticality analysis results for the package for the following cases, as described in Sections 6.

The maximum value of the effective neutron multiplication factor k eff , any stochastic uncertainty, the biases and associated uncertainties, and the number of packages evaluated in the arrays should be specified in the table. The table should also show that the sum of k eff , two standard deviations, and the biases with their associated uncertainties adjustment does not exceed 0. Therefore, a package is considered subcritical, under the regulatory conditions, if it can satisfy the following relationship:.

The value of N, which is the number of packages allowed for transport, should be specified. This section should describe in detail the fissile material in the package. Specifications for the contents used in the criticality evaluation e. Specifications relevant to the criticality evaluation should include fissile material mass, dimensions, enrichment, physical and chemical composition, density, moisture, and other characteristics dependent on the specific contents. Because a partially filled container may allow more room for moderators e.

Therefore, a minimum mass may have to be specified.

Evaluated Nuclear Data

If the package is designed for multiple types of contents, the application may include a separate criticality evaluation and propose different criticality controls for each content type. Any assumptions that certain contents need not be evaluated because they are less reactive than evaluated contents should also be properly justified. The conditions of the SNF assemblies, including missing or replaced fuel rods, should be described. In general, the description of the contents should be sufficient to permit a detailed criticality evaluation of each type or to support a conclusion that certain types are bounded by the evaluations performed.

If the contents include damaged fuel, the maximum extent of damage should be specified.

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Any cans or canisters used as part of the content for the packaging should be described. This section should address general considerations used to evaluate the criticality of the package. These may apply to the criticality evaluations of a single package and arrays of packages under both normal conditions of transport and hypothetical accident conditions. This section should describe and provide sketches of the calculation model used in the calculations.

The sketches should identify the materials used in all regions of the model. Any differences between the model and the actual package configuration should be identified and justification given that the model is conservative. In addition, the differences between the models for normal and accident conditions of transport should be clearly identified and justified. Within the specified tolerance range, dimensions should be selected to result in the highest reactivity. For example, cavity sizes and poison thickness should be considered in the manner that maximizes reactivity.

Deviations from nominal design configuration should be considered. For example, the contents of a powder packaging can be positioned at varying locations and densities, the fuel assemblies might not always be centered in each basket compartment, and the basket might not be exactly centered in a spent fuel package. The relative location and physical properties of the contents within the packaging should be justified as those resulting in the maximum multiplication factor.

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For fuel assembly packages, the fully flooded scenario should address preferential flooding and include flooding of the fuel-cladding gap. In addition, variable water density should be considered for possible system reactivity peaks. Due to the capabilities of modern computer codes, homogeneous modeling should not be used.

This section should provide the appropriate mass densities and atomic number densities for materials used in the models of the packaging and contents. Materials relied on for criticality control must remain in place and be effective. The differences in material condition between normal conditions of transport and hypothetical accident conditions should be clearly identified. Materials relevant to the criticality design, such as poisons, foams, plastics, and other hydrocarbons, should specifically be addressed.

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Neutron absorbers and moderators e. A fuel cycle simulation is usually based on a complex computer code that models material irradiation in reactors, cooling phases and exchanges in facilities. Most important effort concerns, in the neutron physics point of view, the fresh fuel composition needed to reach reactor requirements and the calculation of the composition according to the irradiation conditions. The fresh fuel determination is usually based on a fuel loading model FLM that aims to provide fractions of materials needed to satisfy reactor requirements maximal Burn-Up, regeneration rate, etc.

This could be achieved for instance by a simple formula, by the use of Pu equivalent methods [ 14 ] or with neural network based predictors [ 15 ].

An additional algorithm is then needed to determine the appropriate composition built from available stocks. Once the fresh fuel is built, the model calculates the evolution of materials under neutron irradiation. The calculation scheme is based on the resolution of the two following equations: the Boltzmann equation provides the neutron spectrum which leads to the mean cross sections calculation;. In practice, the coupling between those two equations leads to precise calculated inventory.

Indeed, when the composition is evolving, neutron spectrum has to be frequently updated in order to use correct mean cross sections. To precisely simulate fuel evolution in a fuel cycle simulator, two methods are usually used. The first one consists to build a coupling with a neutron transport code which includes also a Bateman equations solver. This solution is very accurate but requires a high computing time.


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The second solution consists in building and using neutron data predictors in the fuel cycle simulation. The advantages of predictors is that computing time is very fast compared to neutron code coupling. Nevertheless, a minimum bias in comparison with the reference calculation must be guaranteed. A fuel cycle simulator integrates several spatial and temporal scales connected to different physics phenomena: nuclear reactions induced by neutrons flux;. As a consequence, a dynamic fuel cycle simulation output may be characterized by a high uncertainty, with several origins that could be classified as follow: nuclear data used to calculate neutron flux and mean cross sections;.

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The nuclear data include microscopic cross sections, fission and decay data used in the transport code. Effect of nuclear data uncertainties on a fuel cycle output has been investigated in [ 12 ].

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Reactor simulation simplifications are coming from the difficulties to simulate precisely a full core reactors taking into account all the specificities irradiation history, reactivity control with boron or control rods, fuel loading patterns, etc. The common methodology for fuel cycle simulations consists in using assemblies with mirror conditions, which leads to increase uncertainty. The impact of using assembly calculation has been investigated in [ 16 ]. Fuel cycle simplifications could be part of the model or decided by the user during the scenario construction.

Among them, we can point for instance first loadings after reactor operation starting date, shutdown of unit duration during fresh fuel loadings, etc. The impact of those kind of simplifications should be quantified in the future. Operational assumptions are operational data that are user-defined in a fuel cycle simulation. That could be reactor Burn-Up, thermal power, loading factor, deployment date or other facility characteristics.

Those kind of data could not be determined since fuel cycle simulators aims to model future trajectories. The utilization of neutron data predictors and the increase of computing capacities provide new opportunities to perform nuclear scenario studies since a very high number of calculations is currently reachable. In this new vision, operational assumptions are not unknown data but become scenario results obtained by applying optimization criteria on scenario outputs.

In a longer-term perspective, this methodology based on a multi-criteria analysis that would take into account technical, economical and even sociological criteria could be considered. Massive fuel cycle simulation requires a suitable mathematical framework and GSA fulfills perfectly this role. GSA is used in many research fields involving modeling of complex physical phenomena. Each field applies GSA according to its specific needs. A lot of relevant bibliographical sources are available, for instance [ 17 — 19 ]. According to [ 20 ], GSA provides relevant answers for following applications: test if the model is in agreement with the simulated process;.

For fuel cycle simulators applications, GSA could also help understanding the physics from input and output variables relations highlight. In addition, it could provide relevant informations for detecting and correcting errors in the code algorithms.


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  • For application involving a lot of variables with potential interactions, variance based methods are powerful and suitable tools. A lot of sensitivity indices may be used. Since fuel cycle studies have a high input data number and spread, output observables such as plutonium mass at a given time may have a high variability. The fuel cycle simulation used to illustrate the methodology describes a simplified nuclear fleet composed by PWR loaded with uranium oxide UOx and mixed oxide MOx fuels. Since the goal of the work is to study impact of reactor parameters on fuel cycle simulation outputs, one reactor of each type UOx and MOx is defined.

    It has been shown [ 1 ] that this kind of simplifications produced results that could be extrapolated to complex fleet simulation with reasonable accuracy. Total thermal power at beginning of scenario is 2. During the calculation, reactors power and mass can be modified but the specific thermal power remains constant. Simulations schematic representation is shown in Figure 1. An infinite stock feeds with natural uranium a fabrication plant that provides enriched uranium to the PWR-UOx reactor.

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    Uranium enrichment is calculated from reactor required Burn-Up. An irradiation cycle is done and the spent UOx fuel is sent to the pool. After a cooling time defined by the user, the spent fuel is sent to a stock that is used by the MOx fabrication plant to build PWR-MOx fresh fuel. The fabrication plant separates the plutonium needed and add depleted uranium. During the separation process, the reprocessing losses are 0. The plutonium fraction in the MOx fuel is calculated to satisfy the required Burn-Up.

    The MOx fuel is irradiated in the reactor and sent to a pool and a stock after the cooling time. The power of the fleet is supposed to be constant during the scenario that end after years of operation. Nevertheless, this condition can not always be realized because of the availability of the plutonium, as discussed in Section 4. In practice to simplify calculations, reactors lifetime has been set at the duration of the scenario.

    If this is not realistic from the technical point of view and if we could have defined more reactors, this has no impact on the inventory evolution calculation.