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diff --git a/doc/IGE335/Section3.09.tex b/doc/IGE335/Section3.09.tex new file mode 100644 index 0000000..1c74f3b --- /dev/null +++ b/doc/IGE335/Section3.09.tex @@ -0,0 +1,554 @@ +\subsection{The {\tt EVO:} module}\label{sect:EVOData} + +The \moc{EVO:} module performs the burnup calculations. The depletion equations +for the various isotope of the {\sc microlib} are solved using the burnup chains +also present in the {\sc microlib}. Both in-core and out-of-core calculations +can be considered. For in-core depletion calculations, one assumes linear flux variation +over each irradiation period (time stage). The initial (and possibly final) flux +distributions are recovered from previous \moc{FLU:} calculations. In-core depletion can +be performed at constant flux or constant power (expressed in MW/Tonne of initial heavy +elements) but these values can undergo step variations from one time stage to another. +All the information required for successive burnup calculation is stored on the PyLCM +\dds{burnup} data structure. Thus it is possible at any point in time to return to a previous +time step and restart the calculations. + +\vskip 0.2cm + +In each burnup mixture of the unit cell, the depletion of $K$ isotopes over a time +stage $(t_0,t_f)$ follows the following equation: + +\begin{equation} +{dN_k \over dt} + N_k(t) \ \Lambda_k(t)=S_k(t) \ \ \ ; \ {k=1,K} +\label{eq:depletion} +\end{equation} + +\noindent with + +\begin{equation} +\Lambda_k(t)= \lambda_k + \langle \sigma_{{\rm a},k}(t) \phi(t) \rangle \ , +\end{equation} + +\vskip 0.2cm + +\begin{equation} +S_k(t)=\sum^L_{l=1} {Y_{kl} \ \langle \sigma_{{\rm f},l}(t) \phi(t) \rangle } \ N_l(t) + +\sum^K_{l=1} m_{kl}(t) \ {N_l(t)} \ , +\end{equation} + +\vskip 0.2cm + +\begin{equation} +\langle \sigma_{{\rm x},l}(t) \phi(t) \rangle = \int_0^\infty {\sigma_{{\rm x},l}(u) \phi(t,u) du} +\end{equation} + +\noindent and + +\begin{equation} +\sigma_{{\rm x},k}(t,u)\phi(t,u)= \sigma_{{\rm x},k}(t_0,u)\phi(t_0,u)+ +{\sigma_{{\rm x},k}(t_f,u)\phi(t_f,u)-\sigma_{{\rm x},k}(t_0,u)\phi(t_0,u) \over t_f-t_0}(t-t_0) +\end{equation} + +\noindent where +\begin{eqnarray} +\nonumber K &=& \hbox{number of depleting isotopes} +\\ +\nonumber L &=& \hbox{number of fissile isotopes producing fission products} +\\ +\nonumber N_k(t) &=& \hbox{time dependant number density for {\sl k}-th isotope} +\\ +\nonumber \lambda_k &=& \hbox{radioactive decay constant for {\sl k}-th isotope} +\\ +\nonumber \sigma_{{\rm x},k}(t,u) &=& \hbox{time and lethargy dependant microscopic cross section for +nuclear reaction x on} +\\ +\nonumber &~& \hbox{{\sl k}-th isotope. x=a, x=f and x=$\gamma$ respectively stands for absorption, fission and} +\\ +\nonumber &~& \hbox{radiative capture cross sections} +\\ +\nonumber \phi(t,u) &=& \hbox{time and lethargy dependant neutron flux} +\\ +\nonumber Y_{kl} &=& \hbox{fission yield for production of fission product {\sl k} by fissile +isotope {\sl l}} +\\ +\nonumber m_{kl}(t) &=& \hbox{radioactive decay constant or $\langle \sigma_{{\rm x},l}(t) +\phi(t) \rangle$ term for production of isotope {\sl k} by} +\\ +\nonumber &~& \hbox{isotope {\sl l}.} +\end{eqnarray} + +Depleting isotopes with $\Lambda_k(t_0)\left[t_f-t_0\right]\geq$\dusa{valexp} and +$\Lambda_k(t_f)\left[t_f-t_0\right]\geq$\dusa{valexp} are considered to be at saturation. They are +described by making ${dN_k \over dt}=0$ in \Eq{depletion} to obtain + +\begin{equation} +N_k(t)={S_k(t)\over\Lambda_k(t)} \ \ \ ; \ {{\rm if} \ k \ {\rm is \ at \ saturation.}} +\label{eq:sat1} +\end{equation} + +If the keyword \moc{SAT} is set, beginning-of-stage and end-of-stage Dirac contributions are +added to the previous equation: + +\begin{equation} +N_k(t)={1\over\Lambda_k(t)}\left[a \delta(t-t_0) +S_k(t)+b \delta(t-t_f)\right] \ \ \ ; \ {{\rm +if} \ k \ {\rm is \ at \ saturation}} +\label{eq:sat2} +\end{equation} + +\noindent where $a$ and $b$ are chosen in order to satisfy the time integral of \Eq{depletion}: + +\begin{equation} +N_k(t_f^+)-N_k(t_0^-) + \int_{t_0^-}^{t_f^+}{N_k(t) \ \Lambda_k(t) \ dt} = +\int_{t_0^-}^{t_f^+}{S_k(t) \ dt} +\end{equation} + +It is numerically convenient to chose the following values of $a$ and $b$: + +\begin{equation} +a=N_k(t_0^-)-{S_k(t_0^+) \over \Lambda_k(t_0^+)} +\end{equation} + +\noindent and + +\begin{equation} +b={S_k(t_0^+) \over \Lambda_k(t_0^+)}-{S_k(t_f^+) \over \Lambda_k(t_f^+)} +\end{equation} + +\vskip 0.2cm + +The numerical solution techniques used in the \moc{EVO:} module are the following. +Very short period isotopes are taken at saturation and are solved apart from non-saturating +isotopes. If an isotope is taken at saturation, all its parent isotopes, other than fissiles +isotopes, are also taken at saturation. Isotopes at saturation can procuce daughter isotopes +using decay {\sl and/or} neutron-induced reactions. + +\vskip 0.2cm + +The lumped depletion matrix system containing the non-saturating isotopes is solved +using either a fifth order Cash-Karp algorithm or a fourth order Kaps-Rentrop +algorithm\cite{recipie}, taking care to perform all matrix operations in sparse matrix algebra. +Matrices $\left[ m_{kl}(t_0) \right]$ and $\left[ m_{kl}(t_f) \right]$ are therefore +represented in diagonal banded storage and kept apart from the yield matrix +$\left[ Y_{kl}\right]$. Every matrix multiplication or linear system solution is obtained +via the LU algorithm. + +\vskip 0.2cm + +The solution of burnup equations is affected by the flux normalization factors. DRAGON can +perform out-of-core or in-core depletion with a choice between two normalization techniques: + +\begin{enumerate} + +\item Constant flux depletion. In this case, the lethargy integrated fluxes at +beginning-of-stage and end-of-stage are set to a constant $F$: + +\begin{equation} +\int_0^\infty{\phi(t_0,u) du}=\int_0^\infty{\phi(t_f,u) du}=F +\end{equation} + +\item Constant power depletion. In this case, the power released per initial heavy element at +beginning-of-stage and end-of-stage are set to a constant $W$. + +\vskip -0.5cm + +\begin{eqnarray} +\nonumber \sum^K_{k=1} \big[ \kappa_{{\rm f},k} \ \langle \sigma_{{\rm f},k}(t_0) \phi(t_0) \rangle +\kappa_{\gamma,k} \ \langle +\sigma_{\gamma,k}(t_0) \phi(t_0) \rangle \big] \ N_k(t_0) &=& \\ +\sum^K_{k=1} \big[ \kappa_{{\rm f},k} \ \langle \sigma_{{\rm f},k}(t_f) \phi(t_f) \rangle +\kappa_{\gamma,k} \ \langle \sigma_{\gamma,k} +(t_f) \phi(t_f) \rangle \big]\ N_k(t_f) &=& C_0 \ W +\end{eqnarray} + +\noindent where +\begin{eqnarray} +\nonumber \kappa_{{\rm f},k} &=& \hbox{energy (MeV) released per fission of the fissile isotope $k$} +\\ +\nonumber \kappa_{\gamma,k} &=& \hbox{energy (MeV) released per radiative capture of isotope $k$} +\\ +\nonumber C_0 &=& \hbox{conversion factor (MeV/MJ) multiplied by the mass of initial heavy +elements} +\\ +\nonumber &~& \hbox{expressed in metric tonnes} +\end{eqnarray} + +The end-of-stage power is function of the number densities $N_k(t_f)$; a few iterations will +therefore be required before the end-of-stage power released can be set equal to the desired +value. Note that there is no warranties that the power released keep its desired value at every time +during the stage; only the beginning-of-stage and end-of-stage are set. + +\end{enumerate} + +Whatever the normalisation technique used, DRAGON compute the exact burnup of the unit cell +(in MW per tonne of initial heavy element) by adding an additional equation in the depletion +system. This value is the local parameter that should be used to tabulate the output cross +sections. + +\vskip 0.2cm + +The general format of the data which is used to control +the execution of the \moc{EVO:} module is the following: + +\begin{DataStructure}{Structure \dstr{EVO:}} +\dusa{BRNNAM} \dusa{MICNAM} \moc{:=} \moc{EVO:} \\ +~~~~~$[$ \dusa{BRNNAM} $]~\{$ \dusa{MICNAM} $|$ \dusa{OLDMIC} $\}~[~\{$ \dusa{FLUNAM} \dusa{TRKNAM} $|$ \dusa{POWNAM} $\}~]$\\ +~~~~~\moc{::} \dstr{descevo} +\end{DataStructure} + +\noindent where + +\begin{ListeDeDescription}{mmmmmmmm} + +\item[\dusa{BRNNAM}] {\tt character*12} name of the \dds{burnup} data +structure that will contain the +depletion history as modified by the depletion module. If \dusa{BRNNAM} appears +on both LHS and RHS, it is updated; otherwise, it is created. + +\item[\dusa{MICNAM}] {\tt character*12} name of the \dds{microlib} containing +the microscopic cross sections at save point {\sl xts}. \dusa{MICNAM} is modified +to include an embedded \dds{macrolib} containing the updated macroscopic cross +sections at set point {\sl xtr}. If \dusa{MICNAM} appears on both LHS and RHS, +it is updated; otherwise, the internal library \dusa{OLDMIC} is copied in +\dusa{MICNAM} and \dusa{MICNAM} is updated. It is possible to assign different +\dds{microlib} to different save points of the depletion calculation. In this +case, the microscopic reaction rates will be linearly interpolated/extrapolated +between points {\sl xti} and {\sl xtf}. + +\item[\dusa{OLDMIC}] {\tt character*12} name of a read-only \dds{microlib} +that is copied in \dusa{MICNAM}. + +\item[\dusa{FLUNAM}] {\tt character*12} name of a read-only \dds{fluxunk} at save point +{\sl xts}. This information is used for in-core depletion cases. This information is not required for +out-of-core depletion cases. Otherwise, it is mandatory + +\item[\dusa{TRKNAM}] {\tt character*12} name of a read-only \dds{tracking} +constructed for the depleting geometry and consistent with object \dusa{FLUNAM}. + +\item[\dusa{POWNAM}] {\tt character*12} name of a read-only \dds{power} object (generated by DONJON) at save point +{\sl xts}. This information is used for micro-depletion cases. + +\item[\dstr{descevo}] structure containing the input data to this module +(see \Sect{descevo}). + +\end{ListeDeDescription} + +For the in-core depletion cases, the tracking \dds{tracking} data structure on which +\dusa{FLUNAM} is based, is automatically recovered in read-only mode from the +generalized driver dependencies. + +\subsubsection{Data input for module {\tt EVO:}}\label{sect:descevo} + +\begin{DataStructure}{Structure \dstr{descevo}} +$[$ \moc{EDIT} \dusa{iprint} $]$ \\ +$[$ $\{$ \moc{SAVE} \dusa{xts} $\{$ \moc{S} $|$ \moc{DAY} $|$ \moc{YEAR} $\}~\{$ +\moc{FLUX} \dusa{flux} $|$ \moc{POWR} \dusa{fpower} $|$ \moc{W/CC} \dusa{apower} $\}~|$ +\moc{NOSA} $\}$ $]$ \\ +$[$ \moc{EPS1} \dusa{valeps1} $]~~[$ \moc{EPS2} \dusa{valeps2} $]~~[~\{$ \moc{EXPM} \dusa{valexp} $|$ \moc{SATOFF} $\}~]$ \\ +$[$ \moc{H1} \dusa{valh1} $]~[$ $\{$ \moc{RUNG} $|$ \moc{KAPS} $\}$ $]$ \\ +$[~\{$ \moc{TIXS} $|$ \moc{TDXS} $\}~]~[~\{$\moc{NOEX} $|$ \moc{EXTR} $[$ \dusa{iextr} $]~\}~]$ \\ +$[~\{$ \moc{EDP0} $|$ \moc{NOGL} $|$ \moc{GLOB}$\}~]~[~\{$\moc{NSAT} $|$ \moc{SAT}$\}~]~[~\{$\moc{NODI} $|$ \moc{DIRA}$\}~]$ \\ +$[~\{$ \moc{FLUX\_FLUX} $|$ \moc{FLUX\_MAC} $|$ \moc{FLUX\_POW} $\}~]~[~\{$ \moc{CHAIN} $|$ \moc{PIFI} $\}~]$ \\ +$[$ \moc{DEPL} $\{$\dusa{xti} \dusa{xtf} $|$ \dusa{dxt} $\}~\{$ \moc{S} $|$ \moc{DAY} $|$ \moc{YEAR} $\}$ $\{$ \moc{COOL} $|$ +\moc{FLUX} \dusa{flux} $|$ \moc{POWR} \dusa{fpower} $|$ \moc{W/CC} \dusa{apower} $|$ \moc{KEEP} $\}$ $]$ \\ +$[$ \moc{SET} \dusa{xtr} $\{$ \moc{S} $|$ \moc{DAY} $|$ \moc{YEAR} $\}$ $]$ \\ +$[$ \moc{MIXB} $[[$ \dusa{mixbrn} $]] ~]~~~[$ \moc{MIXP} $[[$ \dusa{mixpwr} $]] ~]$ \\ +$[$ \moc{PICK} {\tt >>} \dusa{burnup} {\tt <<} $]$ \\ +{\tt ;} +\end{DataStructure} + +\noindent +where + +\begin{ListeDeDescription}{mmmmmmm} + +\item[\moc{EDIT}] keyword used to modify the print level \dusa{iprint}. + +\item[\dusa{iprint}] index used to control the printing of the module. The +amount of output produced by this tracking module will vary substantially +depending on the print level specified. + +\item[\moc{SAVE}] keyword to specify that the current isotopic concentration +and the microscopic reaction rates resulting from the last transport calculation +will be normalized and stored on \dusa{BRNNAM} in a sub-directory corresponding +to a specific time. By default this data is stored at a time corresponding to +\dusa{xti}. + +\item[\moc{NOSA}] keyword to specify that the current isotopic concentration +and the results of the last transport calculation will not be stored on +\dusa{BRNNAM}. By default this data is stored at a time corresponding to +\dusa{xti}. + +\item[\moc{SET}] keyword used to recover the isotopic concentration already +stored on \dusa{BRNNAM} from a sub-directory corresponding to a specific time. By +default this data is recovered from a time corresponding to \dusa{xtf}. + +\item[\moc{DEPL}] keyword to specify that a burnup calculation between an +initial and a final time must be performed. In the case where the \moc{SAVE} +keyword is absent, the initial isotopic concentration will be stored on +\dusa{BRNNAM} on a sub-directory corresponding to the initial time. If the +\moc{SET} keyword is absent, the isotopic concentration corresponding to the +final burnup time will be used to update \moc{MICNAM}. + +\item[\dusa{xti}] initial time associated with the burnup calculation. The +name of the sub-directory where this information is stored will be given by +`{\tt DEPL-DAT}'//{\tt CNN} where {\tt CNN} is a {\tt character*4} variable +defined by {\tt WRITE(CNN,'(I4.4)') INN} where {\tt INN} is an index associated +with the time \dusa{xti}. The initial values are recovered from this +sub-directory in \dusa{BRNNAM}. + +\item[\dusa{xtf}] end of time for the burnup calculation. The results of the +isotopic depletion calculations are stored in the tables associated with a +sub-directory whose name is constructed in the same manner as the \dusa{xti} +input. + +\item[\dusa{dxt}] time interval for the burnup calculation. The initial time \dusa{xti} in +this case is taken as the final time reached at the last depletion step. If this is the first +depletion step, \dusa{xti} $=0$. + +\item[\dusa{xts}] time associated with the last transport calculation. The +name of the sub-directory where this information is to be stored is constructed +in the same manner as the for \dusa{xti} input. By default (fixed default) +\dusa{xts}=\dusa{xti}. + +\item[\dusa{xtr}] time associated with the next flux calculation. The name of +the sub-directory where this information is to be stored is constructed in the +same manner as for the \dusa{xti} input. By default (fixed default) +\dusa{xtr}=\dusa{xtf}. + +\item[\moc{S}] keyword to specify that the time is given in seconds. + +\item[\moc{DAY}] keyword to specify that the time is given in days. + +\item[\moc{YEAR}] keyword to specify that the time is given in years. + +\item[\moc{COOL}] keyword to specify that a zero flux burnup calculation is to +be performed. + +\item[\moc{FLUX}] keyword to specify that a constant flux burnup +calculation is to be performed. + +\item[\dusa{flux}] flux expressed in $cm^{-2}s^{-1}$. + +\item[\moc{POWR}] keyword to specify that a constant fuel power depletion +calculation is to be performed. The energy released outside the fuel (e.g., by +(n,$\gamma$) reactions) is {\sl not} taken into account in the flux normalization, +unless the \moc{GLOB} option is set. + +\item[\dusa{fpower}] fuel power expressed in $KW\;Kg^{-1}=MW\;{\it tonne}^{-1}$. + +\item[\moc{W/CC}] keyword to specify that a constant assembly power depletion +calculation is to be performed. The energy released outside the fuel (e.g., by +(n,$\gamma$) reactions) is always taken into account in the flux normalization. + +\item[\dusa{apower}] assembly power density expressed in $W/cm^3$ (Power per +unit assembly volume). + +\item[\moc{KEEP}] keyword to specify that the flux is used without been normalized. +This option is useful in cases where the flux was already normalized before the call to +\moc{EVO:} module. + +\item[\moc{EPS1}] keyword to specify the tolerance used in the algorithm for +the solution of the depletion equations. + +\item[\dusa{valeps1}] the tolerance used in the algorithm for the solution of the +depletion equations. The default value is \dusa{valeps1}=$1.0\times 10^{-5}$. + +\item[\moc{EPS2}] keyword to specify the tolerance used in the search +algorithm for a final fixed power (used if the \moc{POWR} or \moc{W/CC} option is activated). + +\item[\dusa{valeps2}] the tolerance used in the search algorithm for a final +fixed power. The default value is \dusa{valeps2}=$1.0\times 10^{-4}$. + +\item[\moc{EXPM}] keyword to specify the selection criterion for non-fissile +isotopes that are at saturation. + +\item[\dusa{valexp}] the isotopes for which $\lambda \times($\dusa{xtf}$-$ +\dusa{xti})$ \ge $\dusa{valexp} will be treated by a saturation approximation. Here, +$\lambda$ is the sum of the radioactive decay constant and microscopic neutron +absorption rate. The default value is \dusa{valexp}=80.0. In order to remove the +saturation approximation for all isotopes set \dusa{valexp} to a very large number +such as $1.0\times 10^{5}$. On the other way, the saturation approximation can be set +for a specific isotope by using the keyword \moc{SAT} in Sect.~\ref{sect:descmix1} +(module \moc{LIB:}). + +\item[\moc{SATOFF}] keyword to remove the saturation approximation for all isotopes +even if \moc{SAT} keyword was set in Sect.~\ref{sect:descmix1} (module \moc{LIB:}). + +\item[\moc{H1}] keyword to specify an estimate of the relative width of the +time step used in the solution of burnup equations. + +\item[\dusa{valh1}] relative width of the time step used in the solution of +burnup equations. An initial time step of +$\Delta_{t}=$\dusa{valh1}$\times ($\dusa{xtf}$-$\dusa{xti}$)$ +is used. This value is optimized dynamically by the program. The +default value is \dusa{valh1}=$1.0\times 10^{-4}$. + +\item[\moc{RUNG}] keyword to specify that the solution will be obtained using +the $5^{th}$ order Cash-Karp algorithm. + +\item[\moc{KAPS}] keyword to specify that the solution will be obtained using +the $4^{th}$ order Kaps-Rentrop algorithm. This is the default value. + +\item[\moc{TIXS}] keyword that specified that time independent cross sections will be used. +This is the default option when no time dependent cross sections are provided. + +\item[\moc{TDXS}] keyword that specified that time dependent cross sections will be used if available. +This is the default option when time dependent cross sections are provided. + +\item[\moc{NOEX}] keyword to supress the linear extrapolation of the +microscopic reaction rates in +the solution of the burnup equations. + +\item[\moc{EXTR}] keyword to perform an extrapolation of the microscopic reaction rates, using +the available information preceding the initial time \dusa{xti}. This is the +default option. + +\item[\dusa{iextr}] extrapolation order ($=1$: linear (default value); $=2$: parabolic). + +\item[\moc{EDP0}] keyword to compute the burnup using the energy released by heavy isotopes in +fuel only using the Serpent empirical formula ({\tt edepmode} $= 0$ in Serpent).\cite{edep} +In this mode, all energy is deposited locally at fission sites and the energy deposition per +fission for fissile nuclide $i$ is calculated as +$$E_{{\rm fiss},i}={Q_i \over Q_{\rm 235}} H_{\rm 235}$$ +\noindent where $Q_i$ is the fission pseudo-Q value for fissile nuclide $i$; $Q_{\rm 235}$ is the fission +pseudo-Q value for U235 and $H_{\rm 235} = 202.27$ MeV is an estimate for the energy deposition per +fission (including the additional energy released in capture reactions) in a typical light water reactor. + +\item[\moc{NOGL}] keyword to compute the burnup using the energy released by all isotopes present in +fuel only. This is the default option. + +\item[\moc{GLOB}] keyword to compute the burnup using the energy released in +the complete geometry. This option has an effect only in cases where some +energy is released outside the fuel (e.g., due to (n,$\gamma$) reactions). +This option affects both the meaning of \dusa{fpower} (given after the +key-word \moc{POWR}) and the value of the burnup, as computed by {\tt EVO:}. + +\item[\moc{NSAT}] save the non--saturated initial number densities in the {\sc burnup} +object \dusa{BRNNAM} (default value) + +\item[\moc{SAT}] save the saturated initial number densities in the {\sc burnup} +object \dusa{BRNNAM} + +\item[\moc{NODI}] select \Eq{sat1} to compute the saturated number densities +(default value) + +\item[\moc{DIRA}] select \Eq{sat2} to compute the saturated number densities + +\item[\moc{FLUX\_FLUX}] recover the neutron flux from \dusa{FLUNAM} object (default option) + +\item[\moc{FLUX\_MAC}] recover the neutron flux from embedded macrolib present in \dusa{MICNAM} or \dusa{OLDMIC} +object. This option is useful to deplete in cases where the neutron flux is obtained from a Monte Carlo +calculation. + +\item[\moc{FLUX\_POW}] recover the neutron flux from the \dds{power} object named \dusa{POWNAM} generated in DONJON. This option is useful in +micro-depletion cases. The neutron flux recovered from \dusa{POWNAM} is generally normalized to the power of the full core. It is therefore +recommended to use the \moc{KEEP} option in \moc{DEPL} data structure. + +\item[\moc{CHAIN}] recover the fission yield data from {\tt 'DEPL-CHAIN'} directory of \dusa{MICNAM} or \dusa{OLDMIC} +object (default option). With this option, the fission yield data is the same in all material mixtures. + +\item[\moc{PIFI}] recover the fission yield data from {\tt 'PIFI'} and {\tt 'PYIELD'} records present in isotopic directories +of \dusa{MICNAM} or \dusa{OLDMIC} object. With this option, the fission yield data is mixture-dependent. This option is useful +in micro-depletion cases. + +\item[\moc{MIXB}] keyword to select depleting material mixtures. By default, all mixtures +with depleting isotopes are set as depleting. + +\item[\dusa{mixbrn}] indices of depleting material mixtures. + +\item[\moc{MIXP}] keyword to select material mixtures producing power. By default, +\begin{itemize} +\item if \moc{MIXB} is not set, all mixtures with isotopes producing power are set as producing power +\item if \moc{MIXB} is set, the same mixtures \dusa{mixbrn} are set as producing power. +\end{itemize} + +\item[\dusa{mixpwr}] indices of material mixtures producing power. + +\item[\moc{PICK}] keyword used to recover the final burnup value (in MW-day/tonne) in a CLE-2000 variable. + +\item[\dusa{burnup}] \texttt{character*12} CLE-2000 variable name in which the extracted burnup value will be placed. + +\end{ListeDeDescription} + +\subsubsection{Power normalization in {\tt EVO:}}\label{sect:powerevo} + +Flux-induced depletion is dependent of the flux or power normalization factor +given after key-words \moc{FLUX}, \moc{POWR} or \moc{W/CC}. The depletion +steps, given after key-words \moc{SAVE}, \moc{DEPL} or \moc{SET}, are set +in time units. Traditionally, the power normalization factor is given in +${\it MW}\;{\it tonne}^{-1}$ and the depletion steps are given in +${\it MWday}\;{\it tonne}^{-1}$. However, a confusion appear in cases where +some energy is released outside the fuel (e.g., due to (n,$\gamma$) reactions). + +\vskip 0.2cm + +The accepted rule and default option in {\tt EVO:} is to compute the burnup +steps in units of $MWday\;{\it tonne}^{-1}$ by considering only the energy +released in fuel (and only the initial mass of the heavy elements present +in fuel). However, it is also recommended to provide a normalization power +taking into account the {\sl total} energy released in the global geometry. +The \moc{GLOB} option can be use to change this rule and to use +the energy released in the complete geometry to compute the burnup. However, +this is not a +common practice, as it implies a non-usual definition of the burnup. +A more acceptable solution consists in setting the normalization power +in power per unit volume of the complete geometry using the key-word +\moc{W/CC}. The value of \dusa{apower} can be computed from the linear +power $f_{\rm lin}$ (expressed in ${\it Mev}\;{\it s}^{-1}\;{\it cm}^{-1}$) +using: + +\begin{equation} +{\it apower}={f_{\rm lin} \ 1.60207 \times 10^{-13} \over V_{\rm assmb}} +\label{eq:eq1} +\end{equation} + +\noindent where $V_{\rm assmb}$ is the 2--D lumped volume of the assembly expressed in $cm^2$. + +\vskip 0.2cm + +The corresponding normalization factor $f_{\rm burnup}$ in +${\it MW}\;{\it tonne}^{-1}$ is given as + +\begin{equation} +f_{\rm burnup}={ {\it apower} \over D_{\rm g} \ F_{\rm power}} +\label{eq:eq2} +\end{equation} + +\noindent where $D_{\rm g}$ is the mass of heavy elements per unit volume +of the complete geometry ($g\; {\it cm}^{-3}$) and $F_{\rm power}$ is the +ratio of the energy released in the complete geometry over the energy +released in fuel. Numerical values of $D_{\rm g}$ and $f_{\rm power}$ are +computed by {\tt EVO:} when the parameter \dusa{iprint} is greater or +equal to 2. The burnup $B$ corresponding to an elapsed time $\Delta t$ is +therefore given as + +\begin{equation} +B=f_{\rm burnup} \ \Delta t +\label{eq:eq3} +\end{equation} + +\noindent where $B$ is expressed in ${\it MWday}\;{\it tonne}^{-1}$ and $\Delta t$ +is expressed in ${\it day}$. + +\vskip 0.2cm + +The unit of the reaction rates depends on the normalization applied to the flux. This normalization +takes place after the flux calculation, using the \moc{EVO:} module. Here is an example: + +\begin{verbatim} +INTEGER istep := 1 ; +REAL Tend := 0.0 ; +REAL Fuelpwr := 38.4 ; ! expressed in MW/tonne + +BURN MICROLIB := EVO: MICROLIB FLUX TRACKN :: + EDIT 0 + SAVE <<Tend>> DAY POWR <<Fuelpwr>> +; +\end{verbatim} + +\noindent where \moc{BURN} is the burnup object, \moc{MICROLIB} is the Microlib used to compute the flux, \moc{FLUX} is the flux +object and \moc{TRACKN} is the tracking object used to compute the flux. After this call, the record +{\tt 'FLUX-NORM'} in \moc{BURN} contains a unique real number, equal to the flux normalization factor. If \moc{MICROLIB} is +obtained using the \moc{LIB:} module, the \moc{DEPL} keyword with following data must be set (see \Sect{desclib}). +Unfortunately, the normalization factor is kept aside and is not applied to the flux present in object \moc{FLUX}. In +fact, only the advanced post-processing modules \moc{COMPO:} (see \Sect{COMPOData}) and \moc{SAP:} (see \Sect{SAPHYBData}) +are making use of this normalization factor. + +\eject |