\chapter{MYRRHA reference model}
\section{MYRRHA reference model description}\label{myrrha model}
%The reference simulation of MYRRHA \cite{LucaFiorito2022NovelCore} for the study of the source term in the LBE and MOX fuel comprises the core of the reactor, up to the reflector. The external LBE pool is neglected in the simulation since the observables relevant for the LBE are mainly spallation products whose production is negligible in the outer part outside the core.\\
The model of MYRRHA design revision 1.8, presented in \cite{LucaFiorito2022NovelCore}, represents a detailed 3D neutronic model reproducing the latest design of the reactor. In this work it is used as a reference to study the source term in the LBE and the MOX fuel present in the reactor.\\

\textit{Core layout}\\
The core is modeled both in critical and sub-critical configuration. The cross sectional view is shown in Fig. \ref{core_xs}.
\begin{figure}
    \hfill
    \centering
    \begin{minipage}{0.45\textwidth}
    
        \includegraphics[width=\linewidth]{images/plots/pics/core_sc.png} 
    \end{minipage}
    \hfill
    \centering
    \begin{minipage}{0.45\textwidth}
    \includegraphics[width=\linewidth]{images/plots/pics/core_cr.png}
    \end{minipage}
\caption{Schematic cross sectional view of the core in sub-critical (left) and critical configuration (right).}
\label{core_xs}
\end{figure}
The names of the assemblies follows the scheme:
\begin{itemize}
    \item 101-608: fuel assemblies grouped in batches;
    \item LBE: dummy assemblies filled with coolant present in sub-critical condition to fill the empty core spaces;
    \item CR: control rod bundle;
    \item SR: safety rod bundle;
    \item IPS/ThIPS: in-pile section where to perform the irradiation experiments of the samples, Th stands for thermal, a softer spectrum is achieved. Used for the production of radioisotopes;
    \item MgO: external ring of reflector assemblies to reduce the neutron leakage from the radial boundaries.
\end{itemize}
The plots show the layout of the core at \acrfull{BoC}, i.e. when the equilibrium condition is achieved. The concept of equilibrium is discussed thoroughly in the dedicated section, Sect. \ref{eqCycleDef}.
This is the layout always taken into account for the calculation, when the dynamic steady state is achieved.
The same core layout is present at the end of the irradiation cycle, that is called EoI, even if the neutronic conditions vary during the cycle because of fuel depletion.\\
The core configuration at startup is called \acrfull{BoL} and it comprises more LBE dummy assemblies in both configurations. During operation, from cycle to cycle, the empty spaces are filled with fuel assemblies. At \acrshort{BoL} all fuel loaded in the core are fresh so less assemblies give the necessary neutronic multiplication.\\

\begin{table}[h]
  \caption{Core configuration summary}
  \label{tab:core_sum}
  \centering
  \begin{tblr}{
      colspec={lllll},
      row{1}={},
      column{1}={font=\itshape},
      row{even}={bg=gray!10},
    }

    \toprule

        
        & BoL  & BoC  & BoL  & BoC  \\
    \toprule
    Fuel assemblies & 54 & 78 & 69 & 105 \\
    Dummy assemblies & 54 & 30 & 36 & - \\
    Reflector assemblies & 42 & 42 & 42 & 42 \\
    Control rods & 3 & 3 & 6 & 6 \\
    Safety rods & - & - & 3 & 3 \\
    Fast IPSs & 6 & 6 & 1 & 1 \\
    Thermal IPSs & 3 & 3 & 6 & 6 \\
    Spallation Target & 1 & 1 & - & - \\
    \midrule
    Total & 163 & 163 & 163 & 163 \\
    \end{tblr}
\end{table}

\textit{FA reshuffling procedure}\\
Fuel assemblies are reshuffled in the core following a IN-to-OUT procedure in which two batches of three assemblies are loaded and unloaded every cycle. The core presents 1/3 azimuthal symmetry so each assembly of the batch undergoes the same irradiation history. The assemblies are unloaded after a variable number of cycles depending on the batch and the mode; the discharge burnup is around 50 MWd/kg for the fuel assemblies in normal operation. In normal operations the fuel assembly unloading happens in the core periphery, after the FA is reshuffled in all the available core positions. Early unloading is envisaged in case of off-normal situations.\\

\textit{Operational history}\\
MYRRHA operational period comprises irradiation cycles of 90 days followed by a maintenance period of 30 or 90 days in case of short or long maintenance as depicted in Fig. \ref{irr_hist}.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\textwidth]{images/plots/pics/irradiation_history.png}
\caption{Example of irradiation history in MYRRHA, showing one period made of three irradiation and maintenance cycles.}\label{irr_hist}
\end{figure}
To assess the source term in the fuel the irradiation of a FA being reshuffled in the core was considered, while the irradiation of LBE lasts the whole operational period of the reactor, that is assumed to be 40 years since no coolant replacement is scheduled during the reactor life. The end of the last irradiation cycle before permanent shut-down of the reactor is called \acrfull{EoL}.
The acronyms \acrshort{BoL}, \acrshort{BoC}, \acrshort{EoI}, \acrshort{EoL} will be used in the following to define the moment in the operational history.
\section{Equilibrium cycle definition}\label{eqCycleDef}
%The equilibrium core is characterized by stable reactivity and power density profile as well as isotopic composition for each fuel assembly position.
The equilibrium condition provides the steady state pattern of fuel assemblies in the core, that stays constant from cycle to cycle performing in-core fuel assemblies reshuffling, loading of fresh FAs in the center and unloading of burnt assemblies from the periphery \cite{EqYamamoto}. The particle flux distributions and spectra vary during the equilibrium cycle as well as the the $k_{eff}$ because of burnup increase associated to fuel depletion, but they come back to the initial condition at the beginning of following cycle since the same fuel assembly pattern is restored. Two separate states are considered for the equilibrium: \acrshort{BoC} and \acrfull{EoC}; they are identical in each irradiation cycle.
\subsection{Initial conditions}
The equilibrium condition can be achieved for any core configuration and it is fixed by input and design parameters: 
\begin{itemize}
    \item Configuration at \acrshort{BoL}: position of IPS, safety and control rods SR, CR; 
    \item Normalization factor - Core power;
    \item Number of fuel assemblies in the reactor core;
    \item (Un)Loading and shuffling scheme: number of assemblies loaded and discharged from the core in each cycle and the pattern they follow along the core (IN-to-OUT reshuffling);
\end{itemize}
The parameters are defined according to design constraints, to achieve criticality in critical mode and to optimize the fuel utilization.
The core layout and the reshuffling procedure were already discussed in the previous section, as regard the core power in critical mode it is 63 MW thermal power, while it is 55 MW for the sub-critical configuration. 
\subsection{Equilibrium core modeling}
\begin{comment}
The equilibrium condition is modeled starting from the neutronic core model at \acrshort{BoC}, but with all the fuel assemblies being inserted with the fresh fuel composition.
From the initial configuration a number of irradiation cycles, comprising the reshuffling of FAs, is performed with the loading of fresh fuel assemblies in the core center and the removal of a constant number of burnt FAs from the periphery. After a sufficient number of irradiation cycles the FAs with a certain burnup level are in the same position between consecutive cycles. Thus, the main characteristics, such as discharge burnup,  power maps, do not change from cycle to cycle. This core layout establishes the \acrshort{BoC}, in which each fuel assembly has a certain burnup level and composition, function of its position in the core.
\end{comment}
The core fuel assemblies pattern that characterizes equilibrium is attained performing a number of consecutive burnup cycles. Fresh fuel assemblies are present throughout the core at the beginning of the first cycle, then the specified fuel reshuffling, loading and unloading procedure is carried out for a certain number of cycles. A steady state condition is reached when the fuel assemblies pattern does not change anymore from cycle to cycle the FAs with a certain burnup level have always the same location in the core.
\subsection{Convergence analysis}
The achievement of the equilibrium cycle is practically assessed 
tracking the evolution of the relevant neutronic parameters during successive irradiation cycles to verify their convergence to an equilibrium value.
The study is carried out for the multiplication factor, the most important parameter for a critical reactor.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{images/plots/k_convergence.png}
\caption{Convergence analysis of $k_{eff}$ in the critical core model.}\label{fig:k_conv}
\end{figure}
Fig. \ref{fig:k_conv} shows $k_{eff}$ value stabilizes after 36 irradiation cycles and this core configuration is taken as the reference for the condition at \acrshort{BoC}.
\begin{center}
\fbox{\begin{minipage}{0.98\textwidth}
The outputs of equilibrium cycle provide valuable pieces of information employed both as input and output of the simplified depletion models, developed to avoid to perform the depletion calculations directly on a full core model that would require larger computational cost.
The behaviour at \acrshort{BoC}, the fluxes profiles and particle spectra are fed to the LBE irradiation model as described in the following chapter. The FAs nuclide concentration at \acrshort{EoC} is investigated and the power profiles are employed as input of FA models aiming at determining fuel source term and associated uncertainty. 

\end{minipage}
}
\end{center}

