From 7dfcc480ba1e19bd3232349fc733caef94034292 Mon Sep 17 00:00:00 2001
From: stainer_t <thomas.stainer@oecd-nea.org>
Date: Mon, 8 Sep 2025 13:48:49 +0200
Subject: Initial commit from Polytechnique Montreal

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+\subsection{The \moc{PKINI:} module}\label{sect:pkini}
+
+\vskip 0.2cm
+The \moc{PKINI:} module is used to initialize the point kinetics parameters, to define the delayed neutron information
+and to set the global feedback parameters. The point kinetics equations are solved for a {\sl time stage} using module
+\moc{PKINS:} (See Sect.~\ref{sect:pkins}) as a function of a fixed set of global parameters recovered from the \dds{map} object \dusa{MAPFL}.
+Modules \moc{PKINI:} and \moc{PKINS:} are intended to be used with thermal-hydraulics module \moc{THM:} (See Sect.~\ref{sect:thm})
+to simulate a single reactor channel.
+
+\vskip 0.08cm
+
+\noindent
+The \moc{PKINI:} module specification is:
+
+\begin{DataStructure}{Structure \moc{PKINI:}}
+\dusa{MAPFL} \moc{:=} \moc{PKINI:} \dusa{MAPFL} \moc{::} \dstr{descpkini}
+\end{DataStructure}
+
+\noindent where
+
+\begin{ListeDeDescription}{mmmmmmmm}
+
+\item[\dusa{MAPFL}] \texttt{character*12} name of the \dds{map} 
+object containing fuel regions description and global parameter informations.
+
+\item[\dstr{descpkini}] structure describing the input data to the \moc{PKINI:} module. 
+
+\end{ListeDeDescription}
+
+\vskip 0.2cm
+
+\subsubsection{Input data to the \moc{PKINI:} module}\label{sect:pkinistr}
+
+\begin{DataStructure}{Structure \dstr{descpkini}}
+$[$ \moc{EDIT} \dusa{iprint} $]$ \\
+\moc{POWER} \dusa{power} \\
+$[$ \moc{EPSILON} \dusa{epsilon} $]$ \\
+\moc{TIME} \dusa{t0} \dusa{dt} \\
+\moc{LAMBDA} \dusa{lambda} \\
+\moc{NGROUP} \dusa{ngroup} \\
+\moc{BETAI} (\dusa{beta}(i),i=1,\dusa{ngroup}) \moc{LAMBDAI} (\dusa{dlambda}(i),i=1,\dusa{ngroup}) \\
+\moc{ALPHA} $[[$ \dusa{PNAME} $\{$ \moc{DIRECT} $|$ \moc{DERIV} $|$ \moc{SQDERIV} $\}$ \\
+~~~~$[~\{$ \moc{LINEAR} $|$ \moc{CUBIC} $\}~]$ \dusa{nalpha} (\dusa{x}(i) \dusa{y}(i),i=1,\dusa{nalpha}) $]]$ \moc{ENDA} \\
+$[$ \moc{PTIME} $[[$ \dusa{PNAME} $[~\{$ \moc{LINEAR} $|$ \moc{CUBIC} $\}~]~\{$\dusa{ntime} (\dusa{t}(i) \dusa{x}(i),i=1,\dusa{ntime}) $|$ \\
+~~~~\moc{T-DELT} $[$ \dusa{nxy} $]$ \dusa{t1} \dusa{t2} \moc{P-VALV} \dusa{gamma} \dusa{p1} \dusa{p2} \dusa{tb1}
+\dusa{bval1} $[$ \moc{RESET} \dusa{p3} \dusa{tb2} \dusa{bval2} $]~\}~]]$ \moc{ENDP} $]$ \\
+;
+\end{DataStructure}
+
+\noindent where
+\begin{ListeDeDescription}{mmmmmmmm}
+
+\item[\moc{EDIT}] keyword used to set \dusa{iprint}.
+
+\item[\dusa{iprint}] integer index used to control the printing on screen:
+= 0 for no print; = 1 for minimum printing; larger values produce
+increasing amounts of output.
+
+\item[\moc{POWER}] keyword used to set the initial reactor power.
+
+\item[\dusa{power}] reactor power in MW.
+
+\item[\moc{EPSILON}] keyword used to set the epsilon of the Runge-Kutta solver used to adjust the internal time step. The default value is $\epsilon=1.0\times 10^{-2}$.
+
+\item[\dusa{epsilon}] user-selected epsilon.
+
+\item[\moc{TIME}] keyword used to set the initial time and stage duration. It is possible to readjust the stage duration in module \moc{PKINS:}.
+
+\item[\dusa{t0}] initial time (s).
+
+\item[\dusa{dt}] stage duration (s).
+
+\item[\moc{LAMBDA}] keyword used to set the prompt neutron generation time.
+
+\item[\dusa{lambda}] prompt neutron generation time (s).
+
+\item[\moc{NGROUP}] keyword used to set the number of delayed precursor groups.
+
+\item[\dusa{ngroup}] number of delayed precursor groups.
+
+\item[\moc{BETAI}] keyword used to set the delayed neutron fraction vector.
+
+\item[\dusa{betai}] value of the delayed neutron fraction in a delayed precursor group.
+
+\item[\moc{LAMBDAI}] keyword used to set the delayed neutron time constant vector.
+
+\item[\dusa{lambdai}] value of the delayed neutron time constant (s) in a delayed precursor group.
+
+\item[\moc{ALPHA}] keyword used to set the feedback parameters.
+
+\item[\moc{PNAME}] character*12 name of a feedback parameter. A feedback parameter should be a global parameter defined in the fuelmap.
+
+\item[\moc{DIRECT}] keyword indicating that the reactivity is a direct function of the global parameter.
+
+\item[\moc{DERIV}] keyword indicating that the reactivity is a function of the variation of the local parameter with respect to its initial value.
+
+\item[\moc{SQDERIV}] keyword indicating that the reactivity is a function of the variation of the square root of the local parameter with respect to its initial value.
+This option is generally used to represent the Doppler effect due to the fuel temperature.
+
+\item[\moc{LINEAR}] keyword indicating that interpolation of the reactivity effect uses linear Lagrange polynomials (default option).
+
+\item[\moc{CUBIC}] keyword indicating that interpolation of the reactivity effect uses the Ceschino method
+with cubic Hermite polynomials, as presented in Ref.~\citen{Intech2011}.
+
+\item[\dusa{nalpha}] number of values in the table describing reactivity effects for feedback parameter \moc{PNAME}.
+
+\item[\dusa{x}] value of the global parameter.
+
+\item[\dusa{y}] corresponding reactivity coefficient.
+
+\item[\moc{PTIME}] keyword used to set the time laws for some feedback parameters.
+
+\item[\dusa{ntime}] number of values in the time law for feedback parameter \moc{PNAME}.
+
+\item[\dusa{t}] time (s)
+
+\item[\dusa{x}] corresponding value of the global parameter.
+
+\item[\moc{T-DELT}] keyword used to set an analytical time law between two times.
+
+\item[\dusa{nxy}] number of points used to construct the discrete time law. The default value is \dusa{nxy} $=1001$.
+
+\item[\dusa{t1}] initial time (s) for the analytical time law domain.
+
+\item[\dusa{t2}] final time (s) for the analytical time law domain.
+
+\item[\moc{P-VALV}] keyword used to define a time law corresponding to the depressurization of a gas reservoir. The time-vatiation of the pressure $P(t)$ in a
+reservoir is given by a relation of the form
+\begin{equation}
+P(t)=\begin{cases} P_{\rm max} & {\rm if} \ t \le {\rm \dusa{tb1}}\\
+\max{\left(P_{\rm min},{\displaystyle P_{\rm max}\over\displaystyle (1+Bt)^\alpha}\right)} & {\rm otherwise}\end{cases}
+\label{eq:pkin1}
+\end{equation}
+
+\noindent where $P_{\rm max}$ is the pressure of the reservoir at time $t\le$ \dusa{tb1}, before depressurization and $P_{\rm min}$ is the final pressure after
+depressurization. $B$ (s$^{-1}$) is the time constant of the exhaust pipe and $\alpha$ is given by relation
+\begin{equation}
+\alpha={2\gamma\over \gamma-1}
+\label{eq:pkin2}
+\end{equation}
+
+\noindent where $\gamma$ is a constant related to the gas capacitance ($=1.66$ for Helium).
+
+\item[\dusa{gamma}] value of $\gamma$ parameter in Eq.~(\ref{eq:pkin2}).
+
+\item[\dusa{p1}] value of pressure $P_{\rm max}$ in Eq.~(\ref{eq:pkin1}).
+
+\item[\dusa{p2}] value of pressure $P_{\rm min}$ in Eq.~(\ref{eq:pkin1}).
+
+\item[\dusa{tb1}] time (s) when the exhaust pipe is open with value \dusa{bval1}. We must have \dusa{t1}$\le$\dusa{tb1}$<$\dusa{t2}.
+
+\item[\dusa{bval1}] value of $B$ parameter in Eq.~(\ref{eq:pkin1}) after time \dusa{tb1}.
+
+\item[\moc{RESET}] optional keyword used to reset the opening of the exhaust pipe after a fixed period of time.
+
+\item[\dusa{p3}] value of pressure $P_{\rm min}$ in Eq.~(\ref{eq:pkin1}) after reset.
+
+\item[\dusa{tb2}] time (s) when the exhaust pipe is open with value \dusa{bval2}. We must have \dusa{tb1}$<$\dusa{tb2}$<$\dusa{t2}.
+
+\item[\dusa{bval2}] value of $B$ parameter in Eq.~(\ref{eq:pkin1}) after time \dusa{tb2}.
+
+\end{ListeDeDescription}
+\clearpage
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