\chapter{Fuel source term - Equilibrium core model}
\begin{comment}
The definition of equilibrium cycle implies neutron flux profile stays constant after convergence is achieved. Other quantities dependent by the flux as power and burnup behave the same way as show in Eq. \eqref{eq1}.

\begin{gather*}
P_A = C \sum_i Q_{fiss, i} \int_{V} d\Vec{r} \int_{E} dE \, \sigma_{f,i}(E) N_i(\Vec{r}) \phi( \vec{r},E)\\
BU = \frac{\int_{t_{irr}} P_A(t) dt }{m_{HM}}
\label{eq1}
\end{gather*}

The irradiated fuel nuclide vector obtained at equilibrium. The analysis of nuclide vector in each assembly position is carried out to determine the buildup of concerning radionuclides among actinides and fission products. The decay of the irradiated FAs can be performed to obtain relevant parameters as decay heat activity associated to a certain fuel assembly after its unloading from the core.
\end{comment}
The output of the equilibrium model, briefly discussed in Sect. \ref{eqCycleDef}, is the nuclide inventory of the irradiated nuclear fuel contained in every FA (or FA batch) loaded in the core. The nuclide concentrations are homogenized over the fuel volume of each FA batch, as shown in eq. \eqref{N_batch} and no spatial gradient internal to the FA is considered.
\begin{equation}
N_{i,batch}(t) = \frac{1}{V} \int_{V} d\Vec{r} \ N_{i,batch}(\Vec{r},t) 
\label{N_batch}
\end{equation}
The nuclide concentration in a FA of the equilibrium core is fed to a radioactive decay model to simulate the time evolution of the radionuclide inventory once the FA is unloaded from the core.

According to the normal operation strategy, FAs are discharged from the core after a fixed number of irradiation cycles that depends on the considered reactor operation mode (critical or sub-critical) and that corresponds to a fixed pre-calculated burnup level as discussed in \cite{FioritoFuelIrr}. The burnup level at discharge is around 55.1 and 47.0 MWd/kg in the critical and sub-critical configuration, respectively. The burnup lever in the critical mode is higher because of the larger number of irradiation cycles, 18 with respect to 13 in sub-critical.
Discharged FAs are temporarily stored in the In-Vessel Fuel Storage (IVFS) in the reactor pool for further cooling before being extracted from the reactor. 
Predictions of the residual decay heat load and activity of a discharged FA act as a boundary condition to optimize the FA extraction process.

In this work we investigate the possibility that partly-irradiated FAs --- i.e., FAs with an average burnup that is lower than the pre-calculated average discharge burnup --- located in the IVFS are extracted from the reactor.
Because of the MYRRHA IN-to-OUT in-core FA shuffling strategy, FAs with a lower burnup level generally experience higher neutron fluxes in the irradiation cycles prior to being moved to the IVFS.
This condition is generally associated to a larger amount of short-lived fission products and actinides that significantly affect the FA decay heat and activity in the short term.

Given that the characterization of the FA source term --- activity and deay heat --- for any possible burnup level and irradiation option is unrealistic, in this work we computed the source term derived quantities for each FA in the equilibrium core at EoC, as if they were immediately unloaded into the IVFS and later on extracted from the reactor.
The analyzed decay period extends to 70 years after discharge.

The governing depletion equation of this study is the exponential radioactive decay law since no irradiation is involved (the irradiation was simulated to derive the equilibrium core).
The system is described by the Bateman equations and it is shown in eq. \eqref{eqDecay}:
\begin{equation}
\frac{dN_i(t)}{dt} = - \lambda_i N_i(t) + \sum_j \lambda_{j \rightarrow i} N_j(t)
\label{eqDecay}
\end{equation}
where $N_i$ is the concentration of the nuclide $i$, depleted because of its own radioactive decay governed by the decay constant $\lambda_i$ and produced by decay of the other atomic species whose concentration is $N_j$ and decay constant $\lambda_{j \rightarrow i}$.  
An example of relevant nuclides in the case study involves $^{241}$Pu ( $t_{\frac{1}{2}} = $ 14.33 y) and $^{241}$Am ( $t_{\frac{1}{2}} = $ 432.81 y). The latter comes from the $\beta^-$ decay of plutonium 241, that is not produced during decay unless in negligible amount by decay of $^{245}$Cm ( $t_{\frac{1}{2}} = $ 8.25e3 y).
This is important because $^{241}$Am is the greatest contributor to the decay heat of a FA at 70 years of cooling.
The equations governing the $^{241}$Pu-$^{241}$Am evolution are reported in eq. \eqref{eq_decayEx}.
\begin{equation}
\begin{gathered}
N_{Pu1}(t) = N_{Pu1}(0) e^{- \lambda_{Pu1} t} \\
N_{Am1}(t) = N_{Pu1}(0) \frac{\lambda_{Pu1}}{\lambda_{Am1}-\lambda_{Pu1}} e^{- \lambda_{Pu1} t}
\label{eq_decayEx}
\end{gathered}
\end{equation}
$N_{Pu1}$ and $N_{Am1}$ are the concentrations of $^{241}$Pu and $^{241}$Am while $\lambda_{Pu1}$ and $\lambda_{Am1}$ are the corresponding decay constants. 
Rather than to nuclide concentrations, the main focus of this chapter moves to the decay heat and to the activity of the FAs after the unloading.
Knowing the nuclide inventory, both quantities can be by eqs. \eqref{act_dh}.

\input{content/chapters/Fuel source term - Equilibrium cycle/sections/Concentrations}
\input{content/chapters/Fuel source term - Equilibrium cycle/sections/Activity}
\input{content/chapters/Fuel source term - Equilibrium cycle/sections/Decay heat}
\input{content/chapters/Fuel source term - Equilibrium cycle/sections/Production Mechanisms}
