\section{Fuel assembly model output}
This section aims to characterize the 2D and 3D fuel assembly model neutronic parameters. The evolution of $k_{inf}$ and the neutron spectrum during irradiation are presented.

\subsection{2D fuel assembly model output}
$k_{inf}$ and spectral plots are shown for the accurate 2D model, respectively in Fig. \ref{fig:2d_kinf} and Fig. \ref{fig:2d_spectrum}.
\begin{figure}
  \centering
  \includegraphics[width=0.95\linewidth]{images/plots/FA/2D_fa/k_inf.png}
  \caption{$k_{inf}$ evolution in 2D fuel assembly with reflective BCs. Statistical uncertainty associated is shown in the zoom.}
  \label{fig:2d_kinf}
  \end{figure}
  
\begin{figure}
  \centering
  \includegraphics[width=0.85\linewidth]{images/plots/FA/2D_fa/spectrum.png}
  \caption{Neutron spectrum in the fuel assembly at different burnup level}
  \label{fig:2d_spectrum}
\end{figure}
The multiplication factor reduces linearly with burnup and the uncertainty is on the order of 7 pcm. The curve has a slope:
$$\frac{dk}{dBU} \ = \ -2.75e-3$$
The spectral variation as function of burnup is low. There is global reduction of the neutron flux due to the progressive reduction of the power but the shape stays constant during FA irradiation.\\
% This provides an evidence of negligible variation of the fissioning system in the fuel assembly.\\
The main reason for FA model development is the possibility to perform uncertainty analysis. The 2D fuel assembly model is further simplified with the purpose of performing statistical sampling lowering the computational cost in term of time and resources. The simplifications do not concern the geometry but only the simulation parameters: the number of particles simulated is reduced of a factor 10 and the predictor-corrector was switched off. It is verified that the simplifications introduced have a very limited impact on the inventory of the nuclides analyzed. Largest deviation of around 3\% occurs in $^{246}$Cu, meaning that the simplified model is suitable to perform uncertainty analysis being representative of the accurate one.\\
\begin{center}
\fbox{
\begin{minipage}{0.98\textwidth}
The 2D fuel assembly model is suitable for uncertainty quantification of the nuclide vector. The preliminary analysis of the neutronic parameters shows that the $k_{eff}$ reduces almost linearly with burnup and the shape of the neutron spectrum stays the same increasing the burnup of the assembly.
\end{minipage}}
\end{center}
\subsection{3D fuel assembly models output}
The 3D fuel assembly simulation is performed introducing depletion zones uniformly distributed in axial direction. In this way each region is independently depleted and the local neutron spectrum is taken into account when estimating the reaction rates used to solve the burnup equations. The variation of the $k_{inf}$ and the nuclide inventory as function of the number of depletion zones is assessed in this section.\\

\textit{Multiplication factor}\\
$k_{inf}$ evolution is presented in Fig. \ref{fig:k3d}. The simulation without divisions is taken as a reference. The reduction of multiplication factor is linear with burnup as in 2D case and the value at specific burnup steps is shown in Tab. \ref{tab:k3}.
\begin{table}
\begin{minipage}{\textwidth}
\caption{$k_{inf}$ in the homogeneous 3D fuel assembly model (without depletion zones) as function of burnup.}\label{tab:k3}
\begin{center}

\begin{tabular}{lrrr}

\toprule

\textit{burnup}  MWd/kg &  0 & 34.7 &  52.8 \\
\textbf{$k_{inf}$} &   1.36 & 1.28 & 1.24\\
\textit{Statistical error}  pcm &  22 & 25 & 25\\



\bottomrule
\end{tabular}
\end{center}

\end{minipage}
\end{table}
The impact of axial depletion zones on $k_{inf}$ is investigated in Fig. \ref{fig:k3d} where the deviation with respect to homogeneous model is assessed.\\
\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{images/plots/FA/3D_fa/k_inf.png}
\caption{$k_{inf}$ variation with respect to homogeneous 3D FA model introducing axial depletion zones.}
\label{fig:k3d}
\end{figure}
The values match for all the models with depletion zones considered in the range of $2\,\sigma$. A slight reduction of $k_{inf}$ with respect to the homogeneous model is noticed at high burnup levels.
This reduction is limited to 200 pcm in the last irradiation cycle, induced by a variation in the material composition.\\

\textit{Nuclide vector}\\
The deviation in the nuclide vector introduced by the fuel material division is negligible for most of fission products, plutonium and americium isotopes. The most affected nuclides are shown in Fig. \ref{fig:discr3d}, where the \acrshort{EoI} concentrations are compared in the model with depletion zones with respect to the homogeneous one.
\begin{figure}[h]
      \centering
    \begin{minipage}{0.78\textwidth}
        \includegraphics[width=\linewidth]{images/plots/FA/3D_fa/discr_zones.png} 
    \end{minipage}
    \hfill
\caption{Deviation on EoI concentrations evaluated in 3D FA simulations: comparison of axial depletion zones with respect to the homogeneous model.}
\label{fig:discr3d}
\end{figure}
The largest discrepancies are observed for higher curium isotopes and they tend to grow increasing the number of zones. For all the nuclides analysed the deviations are lower than 1\% except for $^{246}$Cm ( $t_{\frac{1}{2}} = $ 4.730e3 y) that is underestimated up to 5\% not considering axial depletion.\\
\begin{center}
\fbox{
\begin{minipage}{0.98\textwidth}
The axial fuel material sub-division in the 3D fuel assembly model has a limited effect on the neutronic parameters and the nuclide vector. 
\end{minipage}}
\end{center}