\documentclass{elsarticle}
\usepackage{geometry}
\geometry{ a4paper, total={170mm,257mm}, left=20mm, top=20mm, }
\usepackage{layout}
\usepackage[english]{babel}
\usepackage{fancyhdr}
\usepackage{indentfirst}
\usepackage{hyperref}
\usepackage{subfigure}% in preamble
\usepackage{caption}
\usepackage{multicol}
\usepackage{wrapfig}
\usepackage{array}
\usepackage{lipsum}
\usepackage{multicol}
\begin{document}
\begin{frontmatter}
\title{In Vessel Corium Propagation Sensitivity Study Of Reactor Pressure Vessel Rupture Time With PROCOR Platform}
\author[mymainaddress,mysecondaryaddress]{E.Skrzypek}
\author[mymainaddress,mysecondaryaddress]{M.Skrzypek\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{maciej.skrzypek@ncbj.gov.pl}
\author[mythirdaddress]{Laurent Saas}
\author[mythirdaddress]{Romain Le Tellier}
\address[mymainaddress]{ National Center for Nuclear Research,7 Andrzeja Soltana Street, 05-400 Otwock, Poland}
\address[mysecondaryaddress]{Warsaw University of Technology, 21/25 Nowowiejska Street, 00-665 Warsaw, Poland}
\address[mythirdaddress]{CEA Cadarache, DEN/DTN/SMTA/LPMA, 13108 Saint-Paul-Lez-Durance, France}
\begin{abstract}
The problem of the corium propagation for PWRs in the Reactor Pressure Vessel (RPV) and the time of the RPV failure is one of main issues of study in area of severe accidents. The PROCOR numerical platform created by the CEA severe accident laboratory is modeling corium propagation for LWRs, its relocation to the Lower Plenum and RPV failure. The idea behind the platform was to provide the tool that will be sufficiently fast to be able to perform numerous calculations in reasonable time frame in order to perform statistical study. Therefore the work on the development of the models, describing in-vessel issues, is continuously performed through the simplified phenomena modeling, their verification and sensitivity studies. \\
The recent activities, in scope of PROCOR development, involved cooperation between French CEA experts and Polish PhD students, who were engaged in the topics of core support plate modeling and analysis of the phenomena of thin metallic layer on the top of the corium pool. Those issues were identified to strongly influence on the course of the severe accident and the timing of the RPV failure. In some sensitivity studies performed on a given generic high power Light Water Reactor with heavy reflector, two groups of RPV ruptures were distinguished related to the two issues, which has given the motivation for the further work on these topics. \\
The paper will present a sensitivity study of the corium propagation in order to identify the relevance of those two issues for the RPV time rupture.
\end{abstract}
\begin{keyword}
\texttt{Sensitivity study, PROCOR Platform, IVMR strategy}
\end{keyword}
\end{frontmatter}
\begin{multicols}{2}[ ]
\section{INTRODUCTION}
This work is related to the study of severe accidents in Light Water Reactors (LWR) for the improvement of their prevention and/or mitigation. In order to illustrate the importance of the two modelling topics that we are working on, the motivating study that we will present in this paper, is in the context of present Severe Accident Management Strategy.
The concept of the Severe Accident Management response is the In-Vessel Melt Retention (IVMR) strategy. This concept is being investigated under European Commission funded project from the Horizon 2020 - In-Vessel Melt Retention Severe Accident Management Strategy for Existing and Future NPPs \cite{IVMR-project}.
With this idea the melted core material can be kept inside of the RPV, by removing the risk of the vessel failure.
This is important, because of the fact that the RPV wall is one of the safety barriers for the nuclear power plant.
To ensure the ability of RPV integrity preservation, the study of the heat transfer at both sides of the vessel walls needs to be performed.
The IVMR strategy is a severe accident management strategy that incorporates the external vessel flooding to remove the heat from the in-vessel molten pool material. The heat is transferred from the molten pool to the external coolant through the vessel wall. This impacts highly the structure of the vessel due to high temperature and interaction between corium and steel walls (ablation).
%\vspace{0.1cm}
\includegraphics[width=0.9\linewidth]{figures/fe-ivmr.png}
\captionof{figure}{Heat transfer during IVMR strategy. Focusing effect.} \label{ivmr}
\vspace{0.1cm}
In Vessel Melt Retention strategy aims at containing the solid debris and liquid corium (relocated after the core degradation and melting) into the the lower plenum. For the existing reactor design the concept was considered feasible for the small power reactors. The strategy is already adopted for the VVER 440 type 213 based on through research work for the Finnish Loviisa NPP and Hungarian Paks NPP \cite{Sehgal_2012}. The concept is interesting from the safety point of view and there is a suggestion that it could be adopted for high power reactors with power of about 1000 MW or more. This is a challenge, because of the fact that the power density in such reactor types is higher and the feasibility of the method is not evident. The calculation and experiments, proving the efficiency of external vessel cooling for lower power reactors, were demonstrated by application of conservative assumptions. With such assumptions the heat removal at the vessel outer wall cannot be guaranteed, which indicates that best-estimate methods are needed to be applied \cite{Sangiorgi_2015}.\\
The one of most important issues, crucial for the evaluation of the IVMR strategy for high power reactors, is the study of the corium behaviour in the RPV. The study gives the indication of the areas, which are strongly influencing the RPV failure time and is determining possible failure modes.
This paper is focusing on the results from a sensitivity study on the corium propagation in the context of IVMR strategy and especially on the phenomena that are directly impacting vessel wall rupture - behaviour of the thin metallic layer on the top of the corium pool - "focusing effect" and the relocation of corium from the core to the lower plenum - particularly "core support plate failure mode", which will be described in more details.
\section{Tools}
To perform the calculations of the reactor corium pool propagation in core and the timing of the vessel rupture, the PROCOR platform (\cite{ERMSAR, Saas_ICAPP, Le.Tellier}) software was used, which uses URANIE software \cite{Gaudier_2010}, both developed in CEA. Before discussing the result of the computations, the tools will be briefly described in the following sections. \\
The PROCOR platform is a tool that is used to perform the sensitivity study of the corium propagation and all of the transient phenomena in the core region, as also later the vessel rupture. Features and capabilities of the platform are contained in packages, which later are gathered into different applications. The distributions of the PROCOR differ by the applications, which functionalities are specific to the reactor design. The main advantage and characteristics of PROCOR is its two part construction, which consists of set of simplified models and numerical tools, which are gathered as a library and a Monte-Carlo code launcher for the purposes of the sensitivity/uncertainty study \cite{ERMSAR}.
\subsection{Physical part - simplified modelling}
The physical part is composed of all simplified models to describe corium propagation in the core region and lower head with its behaviour. It contains also functionalities to deal with model and parameters. It takes the form of library, which is written in Java under object-oriented paradigm \cite{Skrien} and contains different packages. The important models, which are relevant for the study, are presented later on in this paper. In those one is for the corium pool thermal and stratification model, describing the corium pool behaviour in the core region and lower head. The other model is the debris bed model, treated as porous media, which is dealing with the coolability of the debris and its melting (upper and lower debris bed). In terms of the internal solid structures of the RPV, the steel structures ablation models are representing the vessel wall or core baffle/reflector as a 1D slabs. This are models dealing with the melting and melt-through of the heavy reflector in the core and RPV rupture in the lower head \cite{ERMSAR, Saas_ICAPP, Le.Tellier}.
\subsection{Statistics based on URANIE}
At the present time to perform sensitivity and uncertainty analysis Monte Carlo method is used. The statistical part of the PROCOR platform is consisting of the two parts. First part is a C++ executable based on URANIE, that provides the PROCOR dedicated coupling with the functionalities for parameter sampling and code launching. URANIE is a sensitivity and uncertainty analysis tool based on the ROOT framework \cite{ROOT}, it is a software developed at CEA \cite{Gaudier_2010} and it provides various tools for data analysis, sampling, statistical modelling, optimization, sensitivity analysis, uncertainty analysis and running code on high performance computers, etc. The second part is the set of CINT scripts for post-calculations uncertainty/sensitivity analysis. \cite{ERMSAR}
\section{Calculations}
To investigate the propagation of the corium pool in the Reactor Pressure Vessel and most important parameters influencing the RPV rupture time, a sensitivity study was performed. This study highlights how the uncertainty in the output of model, in terms of its distribution, is depending upon the uncertainty of some input parameters. This study is not a full statistical analysis of the IVMR strategy, but it aims at illustrating the importance of two modelling issues, which are part of work in the frame of authors Ph.D. theses.
\subsection{Input data}
The calculations are performed for a 1650 MW PWR type reactor. This is a generic reactor with specific feature of heavy reflector surrounding the core, which was used for the purpose of our study. The study is done for the Station BlackOut - SBO \cite{LOOP} scenario without safety injection. This accident sequence is not the fastest one, in comparison to the Large Break Loss of Coolant Accident - LBLOCA \cite{Bonelli_2012}, but it is an example of the scenario comparable to the Fukushima events, which led to core melt and probable RPV rupture. The PROCOR platform calculation starting point corresponds to the formation of the corium pool in the core and the degradation of the core is not computed itself by the code. This starting point is deduced from other integral type severe accident code - MAAP. For the analyses performed in the study MAAP4 calculations were used, what gave the initial core state with corium pool for the SBO sequence before the initiation of the core melt propagation. The sequence itself is the accident scenario, where the external and internal power sources needed for operation of the active cooling safety systems , are cut off and no portable power sources are available (Diesel Generators and Emergency Diesel Generators). This leads to the progressing core region dry-out and melting of the core structures. \\
\vspace{0.1cm}
\includegraphics[width=0.9\linewidth]{figures/vessel2.png}
\captionof{figure}{In-vessel core region initial configuration defined for the PROCOR platform.} \label{vessel}
\vspace{0.1cm}
In the Fig.\ref{vessel} the general view of the initial core and pool definition in the PROCOR code is shown and the translations based on the physical criteria of the corium state from integral type severe accident code into the platform. Later during the simulation starting from the point of the corium presence in the core \cite{Saas_ICAPP}, the corium pool with spherical and/or cylindrical shape is formed and results in the corium pool to be in contact with the peripheral core reflector and/or lower core support plate.
Table \ref{param.tab} is presenting the limited set of uncertain parameters and two changed manually (nb. 7 and 8), that were used in the PROCOR simulations to perform sensitivity analysis for the purpose of this study. All of them will be defined in the following sections. To have clear overview on the parameters that influence the RPV rupture mode and time, the ones concerning the corium pools creation inside the vessel were chosen, both in the core and lower plenum.
\end{multicols}
\begin{table}[htb]
\centering
\begin{footnotesize}
\caption{Parameters investigated during sensitivity study.}
\label{param.tab}
\begin{tabular}{|c|l| m{8.0cm}|}\hline
nb. & Model & \textbf{Sensitivity study parameter} \\ \hline \hline
1 & In-core thermochemistry kinetic 0D model & Uranium molecular diffusivity - $D_U$ \\
2 & Lower head thermochemistry kinetic 0D model & Uranium molecular diffusivity - $D_U$ \\
3 & Corium pool in lower head model & Boundary condition emissivity factor for debris - $f_{e,d}$ \\
4 & Lower debris bed in lower head model & Porosity - $\epsilon_{ld}$ \\
5 & Upper debris bed in lower head model & Porosity - $\epsilon_{ud}$ \\
6 & Corium pool & Corium expansion coefficient - $V_{exp}$ \\ \hline
7 & Main & Corium draining through core support plate model \\
8 & Vessel ablation model & Critical heat flux factor - $f_{\phi}$ \\ \hline
\end{tabular}
\end{footnotesize}
\end{table}
\begin{multicols}{2}[ ]
The one parameter that is highly influencing the phenomena of the vessel rupture is the Uranium diffusivity - $D_U$. The diffusivity is used in thermochamical model and determines the stratification of the corium pool into separate layers of top metallic, oxide and heavy metal layer using a simplified kinetic model \cite{Le.Tellier}.
$D_U$ influences mass transfer coefficient on the basis of heat-mass transfers analogy that relates the thickness of the mass transfer boundary layer $\delta_m$ to the thermal boundary layer $\delta_t$ and it is written as (\cite{Seiler}):
\begin{equation}
\frac{\delta_t}{\delta_m} = \frac{Sh}{Nu} = (Gr)^{1/12} \left(\frac{Sc}{Pr}\right)^{1/3} ,
\end{equation}
with the Sherwood number $Sh$ related to the mass transfer coefficient by $\frac{h_m H}{D_U}$.
The Uranium diffusion is present in the core and in the lower head of the RPV and the parameters of them are taken to the study with the same probability density function distribution. The nominal value is taken as equal to the Stokes-Einstein formula value \cite{Le.Tellier}:
\begin{equation}
D_U=\frac{k_B T}{6 \pi \eta r},
\label{diffusivity}
\end{equation} where $k_B$ - Boltzmann's constant, $T$ - absolute temperature, $\eta$ - dynamic viscosity and $r$ - radius of the spherical particle.
The emissivity factor for debris - $f_{e,d}$ - is used in top boundary condition of the corium pool (also in the lower head) and the equation for the radiative heat transfer evaluation (\ref{rad}). It is the dimensionless factor, which is applied to the upper layer emissivity in presence of debris. In this way the top boundary heat transfer is modified and the lower value of the factor will limit the top radiative heat transfer and increase the power transmitted laterally to the vessel wall. The formula of the heat flux:
\begin{equation}
\phi_{rad}=f_{e,d}\sigma(T_{surf}^4-T_\infty^4),
\label{rad}
\end{equation} where $\phi_{rad}$ - radiative heat flux, $\sigma$ - Stefan Boltzmann constant, $T_{surf}$ - body surface temperature and $T_\infty$ - surrounding temperature.
Another parameter investigated during this sensitivity study was the vessel rupture depending on the debris bed porosity - $\epsilon_{ld}$ and $\epsilon_{ud}$, as lower and upper, respectively.
This parameter does influence the position of the corium pool in the lower head. With its higher value, the corium pool is higher and can cause the core support plate melting. Apart from this, parameter influences the critical heat flux associated to the debris bed coolability due to residual water presence in the lower head for our study:
\begin{equation}
\phi^{crit}_{debris}=1.21\frac{\mathcal{H}_v}{((0.095+(\frac{\rho_w}{\rho_v})^{0.19}))^{2.63}}\sqrt{\frac{\epsilon^3 d \cdot g \Delta \rho \cdot \rho_v}{6(1-\epsilon)}},
\label{epsilon}
\end{equation} where $g$ is the gravity, $d$ - particle diameter, $\rho_w$ (resp. $\rho_v$) corresponds to the water density (resp. vapor density), $\mathcal{H}_v$ means the vaporization enthalpy \cite{Lindholm_2002}. When the critical heat flux is reached it will result in the melting of the debris. So while changing the porosity value - $\epsilon_{ld}$ and $\epsilon_{ud}$ the $\phi^{crit}_{debris}$ will increase with the porosity growth, the debris bed will be cooled easier with larger $\epsilon_{ld}$ and $\epsilon_{ud}$ value.
The parameter investigated during our study is corium pool expansion coefficient - $V_{exp}$, which for a spherical cap is determining the corium shape modification by the following relation:
\begin{equation}
\Delta h_{pool} = \alpha \Delta r_{pool}^+ + \beta
\end{equation}
$h_{pool}$ - pool height, $r_{pool}^+$ - top pool radius, $\alpha,\beta$ - expansion coefficients.\\
There are two possible choices for the expansion coefficients sets ($\alpha,\beta$)- "Ratio" and "Sum" option. For "Ratio" option, the ratio of the ablation velocity $v_{abl}$ on the top $z_{pool}^+$ and bottom $z_{pool}^-$ of the corium pool shape, where deformation is proportional to the local ablation speed:
\begin{eqnarray}
\alpha = \frac{v_{abl}(z_{pool}^+)}{v_{abl}(z_{pool}^-)}=\frac{\phi_{pool}^+}{\phi_{pool}^-} \nonumber \\
\beta = 0
\end{eqnarray}
$\phi_{pool}$ - corium heat flux at the top and bottom. \\
For the second choice - "Sum" option, the difference of ablation velocity of the lateral ablated component on the top and bottom of the associated corium pool shape:
\begin{eqnarray}
\alpha = 1 \nonumber \\
\beta = (v_{abl}(z_{pool}^+)- v_{abl}(z_{pool}^-))\Delta t= \frac{\Delta t(\phi_{pool}^+ - \phi_{pool}^-)}{\rho_c H_c (1-\epsilon_c)}
\end{eqnarray}
$\rho_c$ - density, $H_c$ - fusion enthalpy, $\epsilon_c$ - porosity. \cite{Saas_ICAPP}
%The radial propagation is slower for the "Sum" expansion coefficients and consequently the corium pool at the reflector melting is bigger, what causes the corium pool to look like a hemisphere. \cite{Saas_ICAPP}
The next two parameters - mode of corium draining to the lower head and critical heat flux factor - $f_{\phi}$ , were investigated during the study, but were not treated as random variables. They were changed for the sets of calculations as a constant values for the purpose of further analysis.\\
For the corium draining, the two cases regarding the behaviour of the core support plate and the possible axial transfer from the core to the lower head were considered. The "no axial draining" model through the core support plate, where the corium is slumping to the lower head only through the lateral direction. This approach is justified from a thermal-only analysis of the in-core corium pool interaction with the core support plate: indeed, thermal stationary computations show that the flux at the bottom of the corium pool in the core is low and consequently the crust on the bottom of the corium in the core becomes thick and does not break. The other case is "axial draining" model, in which the corium pool when entering into contact with the core support plate goes through the plate porosity or is causing the structure to fail, the assumption is that the crust surrounding the plate is not stable and directly breaks causing the corium transfer to the lower head. \\
The second parameter was the critical heat flux factor - $f_{\phi}$. The Critical Heat Flux (CHF) is computed with the ULPU correlation and is multiplied by $f_{\phi}=1.933$, so that the maximum CHF is about 3$\frac{MW}{m^2}$. This high value was selected in order to give more visible results of different vessel failure modes. The factor is indicating the heat flux that leads to the dryout of the vessel surface and consequently influences time of the vessel rupture. The use of the flux factor changes the wall critical heat flux value by the formula:
\begin{eqnarray}
\phi_{wall,i}^{crit}= \left\{ \begin{array}{l}
f_{\phi}\Phi^{crit}(\theta_i)\ if\ z_i \leq z_{water}\ and\ z_i \leq h_s \\
f_{\phi}\Phi^{crit}(0)\ if\ z_i \geq z_{water}\ and\ z_i \geq h_s\\
0\ otherwise\end{array}, \right.
\label{phi_crit}
\end{eqnarray}
where $i$ is the mesh of the vessel wall and the vessel wall is the spherical bottom and cylindrical part, $\theta_i$ is the local angle of the surface and $\Phi^{crit}$ is taken from the ULPU experiments \cite{Esmaili}. \\ The probability functions of parameters described above are presented in the Tab.\ref{tab:param}.
\end{multicols}
\begin{center}
\begin{table}[hbt]
\centering
\footnotesize {
\caption{Parameters taken to the statistical analysis}
\label{tab:param}
\begin{tabular}{| c | m{4.0cm} | m{4.0cm} | m{1.2cm} | m{1.2cm} | m{1.2cm} |m{1.5cm} |} \hline
& Parameters & Law & Min value& Nominal value & Max value & Standard deviation \\ \hline
1 & Uranium molecular diffusivity in core & Logtriangular & 1.81E-9 & 1.81E-8 & 1.81E-7 & - \\
2 & Uranium molecular diffusivity in lower head & Logtriangular & 1.81E-9 & 1.81E-8 & 1.81E-7 & - \\
3 & Emissivity factor for lower and upper debris in lower head & equiprobable (Bernoulli law $p=\frac{1}{3}$) & 0.0 & 0.25 & 0.5 & - \\
4 & Porosity for lower and upper debris bed in lower head model & Normal & 0.3 & 0.4 & 0.5 & 0.1 \\
5 & Volume anisotropic expansion option & equiprobable (Bernoulli law $p=\frac{1}{2}$, "Sum" and "Ratio" \cite{Saas_ICAPP}) & 0.0 & & 1.0 & - \\
\hline
\end{tabular}
}
\end{table}
\end{center}
\begin{multicols}{2}[ ]
\subsection{Results}
The Figures \ref{tvr} and \ref{tvr_m} show the results of the study for the reactor case, in which the core damage propagated until formation of the pool. The previous studies in \cite{Le.Tellier} and other studies have classified the possible accident propagations into three groups: early, late and no vessel failure cases. The parameters for our study differ in comparison to \cite{Le.Tellier} and the choice of parameters was done to maximize the number of early rupture mode in order to highlight the work on the thin metallic layer and core support plate.
With the "no axial draining" model, in most cases, focusing effect occurs quickly during the top steel layer formation due to structures ablation and leads to an early vessel rupture. In the "axial draining" case, there is a distinctive group of the "no failure of the RPV" cases, that indicates the corium pool stabilization and its cooldown (7\% of probability). It is related to massive addition of corium to lower head and very large steel layer, which presence results in no focusing effect (left Fig. \ref{tvr}, green group).
\vspace{0.1cm}
\includegraphics[width=0.9\linewidth]{figures/tvr_csp.png}
\captionof{figure}{Rupture and stabilization time groups for "no draining" and "massive draining" through the core support plate model, $tvr$ - vessel rupture time, $te$ - end of calculation time.} \label{tvr}
\vspace{0.1cm}
The earlier failure mode (first group of failure in red colour in the right Fig.\ref{tvr}) is directly connected to the early focusing effect appearance and heat transfer model in the thin metallic layer.
%, what translates to the values of less than 10 tons of metal and layer thickness lower than 10 cm.
This phenomena of the focusing effect is present in the top steel layer formed in the corium pool, while the first melting of the vessel and of the steel structures in the RPV.
The later ruptures (right Fig. \ref{tvr}, blue group) corresponds to the thermochemical effects, the mass transfer of steel during the achievement of stratification equilibrium, which is responsible of the decrease of the metallic layer thickness.
In our study (Fig. \ref{tvr}) the early rupture mode occur more often than the later rupture mode.
% This shows that the first focussing effect, while steel layer appearance is stronger than the thermochemical effects in the pool stratification process.
\vspace{0.1cm}
\includegraphics[width=1.0\linewidth]{figures/mass-tvr.png}
\captionof{figure}{Relation of the RPV time of rupture (tvr) and light metal, heavy metal and oxide layer in the pool mass - "no axial draining" model.} \label{tvr_m}
\vspace{0.1cm}
The high value of the heat flux to the walls is resulting in the failures with the lower masses of the formed pool ($\mathbf{mev_{hm}}$ - heavy metal mass, $\mathbf{mev_{ox}}$ - oxide mass) and especially molten metal ($\mathbf{mev_{lm}}$ - light metal mass) presented in the Fig. \ref{tvr_m}.
The model used in the calculations overestimates the lateral heat flux for very thin layer. In the PROCOR platform to define the heat fluxes the transient 0D energy conservation equation is solved with the following heat transfer correlations: top Globe and Dropkin \cite{G-D}, lateral Churchill and Chu \cite{Ch-Ch} or Chawla and Chan \cite{Cha-Ch} and bottom Bali \cite{Bali}. This correlations are questionable for layer thickness below 10 cm and do not take into account the time delay for the natural convection establishment.
This suggest the need for introduction of the new modelling enabling less conservative $\mathbf{tvr}$ estimation.
The studies planed for that issue will focus on the liquid phase of the metallic layer. Especially, they will include studies to investigate the heat transfer regimes in the metallic layer - the time delay of the convection establishment and description of the thermalhydraulics in the metallic layer and the goal is to propose the simplified realistic model, that could be incorporated to the PROCOR platform.
%The heat transfer for the upper boundary condition is influencing the time of the vessel rupture - Fig.\ref{tvr_fe}. The poorer the heat transfer on the top of the corium pool is, the earlier the time of the vessel rupture is and the larger lateral heat flux of the thin metallic layer becomes.
Another group of the accident course is the stabilization of the corium pool in the lower head. For the performed study this is an option with the "axial draining" model through the core support plate use.
The high impact, on the time of the vessel failure, of the core support plate modelling can be seen in the Fig. \ref{tvr_csp}. %The \textbf{trcsp} parameter on the Fig. \ref{tvr_csp} equal to -1 is indicating no failure of the core support plate cases in the study.
% These are the calculations in which the corium pool was not in contact with the plate.
From this figure the conclusion can be drawn, that the time of the vessel rupture - \textbf{tvr} is delayed for the cases where the contact with the core support plate was present and massive draining through the plate took place.
In the cases with the "axial draining" model through the core support plate, the way of the pool formation was identified to be influencing the possible \textbf{tvr}, which is presented in the Fig.\ref{tvr_csp}. The \textbf{$V_{exp}$} parameter is related to the geometrical modelling of the corium expansion \cite{Saas_ICAPP} in the RPV core region, when the value is above 0.5 ("Ratio" modelling option) the pool is hemispherical and larger. With "Sum" modelling option ($Vexp$ below the 0.5) we have earlier heavy reflector failure and consequently earlier appearance of the corium pool in the lower head. The result is that the rupture of the vessel occurs earlier than the core support plate rupture. The contact of the core support plate with the molten corium pool induce higher mass transfers of the molten materials to the lower head, which result in lower vessel walls thermal loads (lower lateral heat flux).
\vspace{0.1cm}
\includegraphics[width=1\linewidth]{figures/fe-tcsp.png}
\captionof{figure}{Relation of the RPV rupture time (tvr) and way of the core support modelling ($trcsp$ - time of core support plate rupture) - "axial draining" model $trcsp=0$ means no contact between core support plate and corium.} \label{tvr_csp}
\vspace{0.1cm}
The results with axial draining model in the Fig.\ref{tvr_csp} show we have less cases corresponding to RPV rupture when massive draining through the plate occurs. At present, the "axial' and "no axial" draining models in PROCOR are two extreme cases and we have to introduce a simplified thermal--mechanical model to have a realistic evaluation of the corium, that can drain through the plate. In this part the work will be done with the use of additional software - mechanical detailed code (Finite Element Code) i.e. ANSYS. The objective is to validate our model with ANSYS, based on detailed modelling - better thermomechanical coupling and using this modelling to build a set of reference cases that could be used for further validation or for introducing a better simplified model, for example response surface.
\section{Conclusions}
The results of the limited sensitivity analysis with PROCOR for SBO sequence have highlighted the further need for the improvement of the modelling of the two phenomena.
The first one related to the modelling the focusing effect responsible for the early vessel failures, more precisely, it deals with modelling of the natural convection for thin metallic layer. The work will be performed to find a simplified model for thin steel layer and perturbation analysis of the top boundary condition.
The second issue is related to the core support plate modelling, which influences the vessel failures. For this problem the actions are to develop an accurate thermomechanical modelling of the core support plate that is needed in upcoming PROCOR platform development.
These aspects improvements in the modelling will give help to have better understanding of the IVMR strategy utilization for nuclear reactors.
\section*{ACKNOWLEDGMENTS}
This work has been carried out within the framework of the PROCOR platform development funded by CEA, EDF and AREVA.
% Redefine the references label to be in uppercase
\renewcommand*{\refname}{\normalfont\bfseries\uppercase{REFERENCES}}
\renewcommand{\arraystretch}{0.1}
\setlength{\parsep}{0.1cm} \setlength{\itemsep}{0.1cm}
\bibliographystyle{unsrt}
\begin{small}
\begin{thebibliography}{99}
\bibitem{IVMR-project}
http://cordis.europa.eu
\bibitem{Sehgal_2012}
B. R. Sehgal, Nuclear Safety in Light Water Reactors: Severe Accident Phenomenology, Elsevier, 2012, pp. 550-551
\bibitem{Sangiorgi_2015}
M. Sangiorgi, In-Vessel Melt Retention (IVMR) Analysis of a VVER-1000 NPP, Bologna, 23-27 February 2015, 6th ASTEC user's club/ 2nd CESAM workshop
\bibitem{ERMSAR}
R. Le Tellier, L. Saas and F. Payot, Phenomenological analyses of corium propagation in LWRs: the PROCOR software platform
\bibitem{Saas_ICAPP}
L. Saas, R. Le Tellier, S. Bajard, A Simplified Geometrical Model for Transient Corium Propagation in Core for an LWR with Heavy Reflector, International Congress on Advances in Nuclear Power Plants, 2015
\bibitem{Le.Tellier}
R. Le Tellier, , L. Saas, S. Bajard, Transient stratification modelling of a corium pool in a LWR vessel lower head, Nuclear Engineering and Design, Volume 287, June 2015, Pages 68–77
\bibitem{Gaudier_2010}
F. Gaudier, URANIE : The CEA/DEN Uncertainty and Sensitivity platform , Procedia Social and Behavioral Sciences 2, (2010) 7660–7661, Published by Elsevier Ltd.
\bibitem{Skrien}
D. Skrien, Object-Oriented Design Using Java, Hardcover, January 22, 2008
\bibitem{ROOT}
www.root.cern.ch, User’s Guide, May 2014
\bibitem{LOOP}
P. Darnowski , E. Skrzypek, P. Mazgaj, K. Swirski, P. Gandrille, Total loss of AC power analysis for EPR reactor, Nuclear Engineering and Design
Volume 289, August 2015, Pages 8–18
\bibitem{Bonelli_2012}
A. Bonelli, O. Mazzantini, M. Sonnenkalb, Station Black-Out Analysis with MELCOR 1.8.6 Code for Atucha 2 Nuclear Power Plant, Science and Technology of Nuclear Installations Volume 2012 (2012)
%\bibitem{Meyer_2001}
%L. Meyer, M. Gargallo, Experiments on melt dispersion with lateral failure in the bottom head of the pressure vessel, Forschungszentrum Karlsruhe, Institut für Kern-und Energietechnik, Karlsruhe, Germany
\bibitem{Seiler}
J.M. Seiler et al., "Equations for solidification of corium without sparging gas - scaling criteria," in Proc. of OECD workshop on ex-vessel debris coolability. Karlsruhe, Germany, 1999.
\bibitem{Lindholm_2002}
I.Lindholm, A Review of Dryout Heat Fluxes and Coolabiliy of Particle Beds, VTT Energy, April 2002, Finland
\bibitem{Esmaili}
H. Esmaili, M. Khatib-Rahbar, Analysis of in-vessel retention and ex-vessel fuel coolant interaction for AP1000, Tech. Rep. NUREG/CR-6849 ERI/NRC04-201, U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research (2004)
\bibitem{G-D}
S. Globe, D. Dropkin, Natural convection heat transfer in liquids confined by two horizontal plates and heated from below, Journal of Heat Transfer, vol. 81, pp. 24–28, 1959.
\bibitem{Ch-Ch}
S.W. Churchill, H.H.S. Chu, Correlating equations of laminar rand turbulent free convection from a vertical plate, International Journal of Heat and Mass Transfer, vol. 18, pp. 1323–1329, 1975.
\bibitem{Cha-Ch}
T.C. Chawla, S.H. Chan, Heat transfer from vertical/inclined boundaries of heat-generating boiling pools, Journal of Heat Transfer, vol. 104, pp. 465–473, 1982.
\bibitem{Bali}
J.M. Bonnet and J.M. Seiler, "Thermohydraulic phenomena in corium pool: the Bali experiment," in Proc. of ICONE 7. Tokyo, Japan, 1999.
\end{thebibliography}
\end{small}
\end{multicols}
\end{document}