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\begin_body
\begin_layout Title
In Vessel Corium Propagation Sensitivity Study Of Reactor Pressure Vessel
Rupture Time With PROCOR Platform
\end_layout
\begin_layout Author
\noindent
Eleonora Skrzypek
\begin_inset Flex Authormark
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a,b
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\end_layout
\begin_layout Email
eleonora.skrzypek@ncbj.gov.pl
\end_layout
\begin_layout Author
\noindent
Maciej Skrzypek
\begin_inset Flex Authormark
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a,b
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\end_inset
\begin_inset Flex CorAuthormark
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cor1
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\end_layout
\begin_layout Email
maciej.skrzypek@ncbj.gov.pl
\end_layout
\begin_layout Author
\noindent
Laurent Saas
\begin_inset Flex Authormark
status open
\begin_layout Plain Layout
c
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\end_inset
\end_layout
\begin_layout Email
laurent.saas@cea.fr
\end_layout
\begin_layout Author
\noindent
Romain LeTellier
\begin_inset Flex Authormark
status open
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c
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\end_inset
\end_layout
\begin_layout Email
romain.le-tellier@cea.fr
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\begin_layout Address
\begin_inset Argument 1
status open
\begin_layout Plain Layout
a
\end_layout
\end_inset
National Center for Nuclear Research,7 Andrzeja Soltana Street, 05-400 Otwock,
Poland
\end_layout
\begin_layout Address
\noindent
\begin_inset Argument 1
status open
\begin_layout Plain Layout
b
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Warsaw University of Technology, 21/25 Nowowiejska Street, 00-665 Warsaw,
Poland
\end_layout
\begin_layout Address
\noindent
\begin_inset Argument 1
status open
\begin_layout Plain Layout
c
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CEA Cadarache, DEN/DTN/SMTA/LPMA, 13108 Saint-Paul-Lez-Durance, France
\end_layout
\begin_layout Corresponding author
Corresponding author
\begin_inset Argument 1
status collapsed
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cor1
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\end_layout
\begin_layout 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 distinguish
ed 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_layout
\begin_layout Keywords
Sensitivity study, PROCOR Platform, IVMR strategy
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Standard
\noindent
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
\begin_inset CommandInset citation
LatexCommand citep
key "IVMR-project"
\end_inset
.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.
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
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\begin_inset Graphics
filename fe-ivmr.png
lyxscale 10
width 6cm
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Heat transfer during IVMR strategy.
Focusing effect.
\begin_inset CommandInset label
LatexCommand label
name "IVMR"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
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).
\end_layout
\begin_layout Standard
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
\begin_inset CommandInset citation
LatexCommand citep
key "Sehgal_2012"
\end_inset
.
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 conservat
ive 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
\begin_inset CommandInset citation
LatexCommand citep
key "Sangiorgi_2015"
\end_inset
.
\end_layout
\begin_layout Standard
One of the 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.
\end_layout
\begin_layout Section
Tools
\end_layout
\begin_layout Standard
To perform the calculations of the reactor corium pool propagation in core
and the timing of the vessel rupture, the PROCOR platform
\begin_inset CommandInset citation
LatexCommand citep
key "ERMSAR,SAAS_ICAPP,Le.Tellier"
\end_inset
software was used, which uses URANIE software
\begin_inset CommandInset citation
LatexCommand citep
key "Gaudier_2010"
\end_inset
, both developed in CEA.
Before discussing the result of the computations, the tools will be briefly
described in the following sections.
\end_layout
\begin_layout Standard
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, whose functionalit
ies 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
\begin_inset CommandInset citation
LatexCommand citep
key "ERMSAR"
\end_inset
.
\end_layout
\begin_layout Subsection
Physical part- simplified modelling
\end_layout
\begin_layout Standard
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
\begin_inset CommandInset citation
LatexCommand citep
key "Skrien"
\end_inset
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, describin
g 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
\begin_inset CommandInset citation
LatexCommand citep
key "ERMSAR,SAAS_ICAPP,Le.Tellier"
\end_inset
.
The melt through the steel structure is possible due to the decay heat
presence, which is calculated by the separate model in the PROCOR platform.
This model is evaluating the power density in the single material or set
of materials and is associating it to the U or (U, Zr) elements according
to their mass fractions after reading the decay curve form the integrated
code calculations.
\end_layout
\begin_layout Subsection
Statistics based on URANIE
\end_layout
\begin_layout Standard
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
\begin_inset CommandInset citation
LatexCommand citep
key "ROOT"
\end_inset
, it is a software developed at CEA
\begin_inset CommandInset citation
LatexCommand citep
key "Gaudier_2010"
\end_inset
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/se
nsitivity analysis.
\begin_inset CommandInset citation
LatexCommand citep
key "ERMSAR"
\end_inset
\end_layout
\begin_layout Section
Calculations
\end_layout
\begin_layout Standard
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.
Two issues, the focusing effect and massive corium draining through the
core support plate phenomenon presence, identified as important for the
course of the severe accident for the high power PWR reactors are characterized
by different probability of occurrence during specific sequence of accident.
In the literature
\begin_inset CommandInset citation
LatexCommand citep
key "Loeffler_2004"
\end_inset
and related studies the probability of the reactor pressure vessel failure
ranges around 83-86% with associated probability of the corium retention
in the lower plenum of 10-13 depending on the methodology of the Probability
Safety Analysis.
What is shown in the later sensitivity study presented, the two phenomena
- focusing effect and core support plate failure, are representative accident
paths for the RPV failures and non-failures modes for the analysed reactor
sequence.
\begin_inset Float figure
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filename vessel2.png
lyxscale 10
width 7cm
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
In-vessel core region initial conguration dened for the PROCOR platform.
\begin_inset CommandInset label
LatexCommand label
name "vessel"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Calculations
\end_layout
\begin_layout Standard
The calculations are performed for a 1650 MW PWR type reactor.
This is a generic reactor with spe- cic feature of heavy re ector surrounding
the core, which was used for the purpose of our study.
The study is done for the Station BlackOut - SBO
\begin_inset CommandInset citation
LatexCommand citep
key "LOOP"
\end_inset
scenario without safety injection.
This accident se- quence is not the fastest one, in comparison to the Large
Break Loss of Coolant Accident - LBLOCA
\begin_inset CommandInset citation
LatexCommand citep
key "Bonelli_2012"
\end_inset
, but it is an example of the scenario comparable to the Fukushima events,
which led to core melt and prob- able RPV rupture.
The PROCOR platform calcula- tion 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.
\end_layout
\begin_layout Standard
In the Figure
\begin_inset CommandInset ref
LatexCommand ref
reference "vessel"
\end_inset
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
\begin_inset CommandInset citation
LatexCommand citep
key "SAAS_ICAPP"
\end_inset
}, 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.
\end_layout
\begin_layout Standard
Table
\begin_inset CommandInset ref
LatexCommand ref
reference "param.tab-1"
\end_inset
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_layout
\begin_layout Standard
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wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Parameters investigated during sensitivity study.
\begin_inset CommandInset label
LatexCommand label
name "param.tab-1"
\end_inset
\end_layout
\end_inset
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status open
\begin_layout Plain Layout
\backslash
scriptsize
\end_layout
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\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
nb
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Sensivity study parameter
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
In-core thermochemistry kinetic 0D model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Uranium molecular diffusivity -
\begin_inset Formula $D_{U}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Lower head thermochemistry kinetic 0D model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Uranium molecular diffusivity -
\begin_inset Formula $D_{U}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Corium pool in lower head model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Boundary condition emissivity factor for debris -
\begin_inset Formula $f_{e,d}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Lower debris bed in lower head model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Porosity -
\begin_inset Formula $\varepsilon_{ld}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Upper debris bed in lower head model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Porosity -
\begin_inset Formula $\varepsilon_{ud}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Corium pool
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Corium expansion coefficient -
\begin_inset Formula $V_{exp}$
\end_inset
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Main
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Corium draining through core support plate model
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Vessel ablation model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Critical heat flux factor -
\begin_inset Formula $f_{\phi}$
\end_inset
\end_layout
\end_inset
|
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
One parameter that is highly in uencing the phenomena of the vessel rupture
is the Uranium diusivity -
\begin_inset Formula $D_{U}$
\end_inset
.
The diusivity is used in thermochemical model and determines the stratication
of the corium pool into separate layers of top metallic, oxide and heavy
metal layer using a simplied kinetic model
\begin_inset CommandInset citation
LatexCommand citep
key "Le.Tellier"
\end_inset
.
\begin_inset Formula $D_{U}$
\end_inset
.
influences mass transfer coecient on the basis of heat- mass transfers
analogy that relates the thickness of the mass transfer boundary layer
\begin_inset Formula $\delta_{m}$
\end_inset
to the thermal boundary layer
\begin_inset Formula $\delta_{m}$
\end_inset
and it is written as (
\begin_inset CommandInset citation
LatexCommand citep
key "Seiler"
\end_inset
):
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\frac{\delta_{t}}{\delta_{m}}=\frac{Sh}{Nu}=(Gr)^{\nicefrac{1}{12}}\left(\frac{Sc}{Pr}\right)^{\nicefrac{1}{3}}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
with the Sherwood number
\begin_inset Formula $Sh$
\end_inset
related to the mass transfer coefficient by
\begin_inset Formula $\frac{h_{m}H}{D_{U}}$
\end_inset
.
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
\begin_inset CommandInset citation
LatexCommand citep
key "Le.Tellier"
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
D_{U}=\frac{k_{B}T}{6\pi\eta r},\label{diffusivity}
\end{equation}
\end_inset
where
\begin_inset Formula $k_{B}$
\end_inset
- Boltzmann's constant,
\begin_inset Formula $T$
\end_inset
- absolute temperature,
\begin_inset Formula $\eta$
\end_inset
- dynamic viscosity and
\begin_inset Formula $r$
\end_inset
- radius of the spherical particle.
\end_layout
\begin_layout Standard
The emissivity factor for debris -
\begin_inset Formula $f_{e,d}$
\end_inset
- is used in top boundary condition of the corium pool (also in the lower
head) and the equation for the radiative heat transfer evaluation (
\begin_inset Formula $\ref{rad}$
\end_inset
).
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:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
\phi_{rad}=f_{e,d}\sigma(T_{surf}^{4}-T_{\infty}^{4}),\label{rad}
\end{equation}
\end_inset
where
\begin_inset Formula $\phi_{rad}$
\end_inset
- radiative heat flux,
\begin_inset Formula $\sigma$
\end_inset
- Stefan Boltzmann constant,
\begin_inset Formula $T_{surf}$
\end_inset
- body surface temperature and
\begin_inset Formula $T_{\infty}$
\end_inset
- surrounding temperature.
\end_layout
\begin_layout Standard
Another parameter investigated during this sensitivity study was the vessel
rupture depending on the debris bed porosity -
\begin_inset Formula $\epsilon_{ld}$
\end_inset
and
\begin_inset Formula $\epsilon_{ud}$
\end_inset
, 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_inset Formula
\begin{equation}
\phi_{debris}^{crit}=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}
\end_inset
where
\begin_inset Formula $g$
\end_inset
is the gravity,
\begin_inset Formula $d$
\end_inset
- particle diameter,
\begin_inset Formula $\rho_{w}$
\end_inset
(resp.
\begin_inset Formula $\rho_{v}$
\end_inset
) corresponds to the water density (resp.
vapor density),
\begin_inset Formula $\mathcal{H}_{v}$
\end_inset
means the vaporization enthalpy
\begin_inset CommandInset citation
LatexCommand citep
key "Lindholm_2002"
\end_inset
.
When the critical heat flux is reached it will result in the melting of
the debris.
So while changing the porosity value -
\begin_inset Formula $\epsilon_{ld}$
\end_inset
and
\begin_inset Formula $\epsilon_{ud}$
\end_inset
the
\begin_inset Formula $\phi_{debris}^{crit}$
\end_inset
will increase with the porosity growth, the debris bed will be cooled easier
with larger
\begin_inset Formula $\epsilon_{ld}$
\end_inset
and
\begin_inset Formula $\epsilon_{ud}$
\end_inset
value.
\end_layout
\begin_layout Standard
The parameter investigated during our study is corium pool expansion coefficient
-
\begin_inset Formula $V_{exp}$
\end_inset
, which for a spherical cap is determining the corium shape modification
by the following relation:
\begin_inset Formula
\begin{equation}
\Delta h_{pool}=\alpha\Delta r_{pool}^{+}+\beta
\end{equation}
\end_inset
\begin_inset Formula $h_{pool}$
\end_inset
- pool height,
\begin_inset Formula $r_{pool}^{+}$
\end_inset
- top pool radius,
\begin_inset Formula $\alpha,\beta$
\end_inset
- expansion coefficients.
\end_layout
\begin_layout Standard
There are two possible choices for the expansion coefficients sets (
\begin_inset Formula $\alpha,\beta$
\end_inset
)- "Ratio" and "Sum" option.
For "Ratio" option, the ratio of the ablation velocity
\begin_inset Formula $v_{abl}$
\end_inset
on the top
\begin_inset Formula $z_{pool}^{+}$
\end_inset
and bottom
\begin_inset Formula $z_{pool}^{-}$
\end_inset
of the corium pool shape, where deformation is proportional to the local
ablation speed:
\begin_inset Formula
\begin{eqnarray}
\alpha=\frac{v_{abl}(z_{pool}^{+})}{v_{abl}(z_{pool}^{-})}=\frac{\phi_{pool}^{+}}{\phi_{pool}^{-}}\nonumber \\
\beta=0
\end{eqnarray}
\end_inset
\begin_inset Formula $\phi_{pool}$
\end_inset
- corium heat flux at the top and bottom.
\end_layout
\begin_layout Standard
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_inset Formula
\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}
\end_inset
\begin_inset Formula $\rho_{c}$
\end_inset
- density,
\begin_inset Formula $H_{c}$
\end_inset
- fusion enthalpy,
\begin_inset Formula $\epsilon_{c}$
\end_inset
- porosity.
\begin_inset CommandInset citation
LatexCommand citep
key "SAAS_ICAPP"
\end_inset
\end_layout
\begin_layout Standard
The next two parameters - mode of corium draining to the lower head and
critical heat flux factor -
\begin_inset Formula $f_{\phi}$
\end_inset
, were investigated during the study, but were not treated as random variables.
They were changed for the sets of calculations as constant values for the
purpose of further analysis.
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Standard
The second parameter was the critical heat flux factor -
\begin_inset Formula $f_{\phi}$
\end_inset
.
The Critical Heat Flux (CHF) is computed with the ULPU correlation and
is multiplied by
\begin_inset Formula $f_{\phi}=1.933$
\end_inset
, so that the maximum CHF is about 3
\begin_inset Formula $\frac{MW}{m^{2}}$
\end_inset
.
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:
\end_layout
\begin_layout Standard
\begin_inset 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}
\end_inset
where
\begin_inset Formula $i$
\end_inset
is the mesh of the vessel wall and the vessel wall is the spherical bottom
and cylindrical part,
\begin_inset Formula $\theta_{i}$
\end_inset
is the local angle of the surface and
\begin_inset Formula $\Phi^{crit}$
\end_inset
is taken from the ULPU experiments
\begin_inset CommandInset citation
LatexCommand citep
key "Esmaili"
\end_inset
.
\end_layout
\begin_layout Standard
The probability functions of parameters described above are presented in
the Tab.
\begin_inset CommandInset ref
LatexCommand ref
reference "tab.param"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Parameters taken to the statistical analysis.
\begin_inset CommandInset label
LatexCommand label
name "tab.param"
\end_inset
\end_layout
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
scriptsize
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Parameters
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Law
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Min value
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Nominal value
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Max value
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Standard deviation
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
1
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Uranium molecular diffusivity in core
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Logtriangular
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
-
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Uranium molecular diffusivity in lower head
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Logtriangular
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-9
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-8
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.81E-7
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
-
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Emissivity factor for lower and upper debris in lower head
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
equiprobable (Bernoulli law
\begin_inset Formula $p=\frac{1}{3}$
\end_inset
)
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.25
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
-
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Porosity for lower and upper debris bed in lower head model
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Normal
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.3
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.4
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.1
\end_layout
\end_inset
|
|
\begin_inset Text
\begin_layout Plain Layout
5
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Volume anisotropic expansion option
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
equiprobable (Bernoulli law
\begin_inset Formula $p=\frac{1}{2}$
\end_inset
, "Sum" and "Ratio"
\begin_inset CommandInset citation
LatexCommand citep
key "SAAS_ICAPP"
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
0.0
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
1.0
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
-
\end_layout
\end_inset
|
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Results
\end_layout
\begin_layout Standard
The Figures
\begin_inset Formula $\ref{tvr}$
\end_inset
and
\begin_inset Formula $\ref{tvr_m}$
\end_inset
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
\begin_inset CommandInset citation
LatexCommand citep
key "Le.Tellier"
\end_inset
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
\begin_inset CommandInset citation
LatexCommand citep
key "Le.Tellier"
\end_inset
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.
\end_layout
\begin_layout Standard
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 (upper Fig.
\begin_inset Formula $\ref{tvr}$
\end_inset
, blue group).
\end_layout
\begin_layout Standard
The earlier failure mode (first group of failure in red colour in the lower
Fig.
\begin_inset Formula $\ref{tvr}$
\end_inset
) is directly connected to the early focusing effect appearance and heat
transfer model in the thin metallic layer.
This phenomenon 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.
\end_layout
\begin_layout Standard
The first slumps of the molten material from the core region that are leading
to the RPV break are ranging from the t=23800s (6 h 36 min 40 s) with a
round of 30 000 kg of molten corium pool (heavy metal, oxide pool and light
metallic layer) created.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename tvr_csp.png
lyxscale 10
width 6cm
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Rupture and stabilization time groups for "no draining" and "massive draining"
through the core support plate model, te - end of calculation time.
\begin_inset CommandInset label
LatexCommand label
name "tvr"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The later ruptures (right Fig.
\begin_inset Formula $\ref{tvr}$
\end_inset
, 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.
\begin_inset Formula $\ref{tvr}$
\end_inset
) the early rupture mode occur more often than the later rupture mode.
\end_layout
\begin_layout Standard
The high value of the heat flux to the walls is resulting in the failures
with the lower masses of the formed pool (
\begin_inset Formula $\mathbf{mev_{hm}}$
\end_inset
- heavy metal mass,
\begin_inset Formula $\mathbf{mev_{ox}}$
\end_inset
- oxide mass) and especially molten metal (
\begin_inset Formula $\mathbf{mev_{lm}}$
\end_inset
- light metal mass) presented in the Fig.
\begin_inset Formula $\ref{tvr_m}$
\end_inset
.
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
\begin_inset CommandInset citation
LatexCommand citep
key "G-D"
\end_inset
, lateral Churchill and Chu
\begin_inset CommandInset citation
LatexCommand citep
key "Ch-Ch"
\end_inset
or Chawla and Chan
\begin_inset CommandInset citation
LatexCommand citep
key "Cha-Ch"
\end_inset
and bottom Bali
\begin_inset CommandInset citation
LatexCommand citep
key "Bali"
\end_inset
.
These correlations are questionable for layer thickness below 10 cm and
do not take into account the time delay for the natural convection establishmen
t.
This suggests the need for introduction of the new modelling enabling less
conservative
\begin_inset Formula $\mathbf{tvr}$
\end_inset
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.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename mass-tvr.png
lyxscale 10
width 6cm
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
Relation of the RPV time of rupture (tvr) and light metal, heavy metal and
oxide layer in the pool mass - "no axial draining" model.
\begin_inset CommandInset label
LatexCommand label
name "tvr_m"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
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.
\end_layout
\begin_layout Standard
In the simulations presented, for the SBO sequence as determined by the
PROCOR code, the core support plate failure occurs around 25000 s (6 h
56 min and 40 s).
The high impact, on the time of the vessel failure, of the core support
plate modelling can be seen in the Fig.
\begin_inset Formula $\ref{tvr_csp}$
\end_inset
.
\end_layout
\begin_layout Standard
From this figure the conclusion can be drawn, that the time of the vessel
rupture -
\begin_inset Formula $\textbf{tvr}$
\end_inset
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
\begin_inset Formula $\textbf{tvr}$
\end_inset
, which is presented in the Fig
\begin_inset Formula $.\ref{tvr_csp}$
\end_inset
.
The
\begin_inset Formula $\textbf{\ensuremath{V_{exp}}}$
\end_inset
parameter is related to the geometrical modelling of the corium expansion
\begin_inset CommandInset citation
LatexCommand citep
key "SAAS_ICAPP"
\end_inset
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 (
\begin_inset Formula $V_{exp}$
\end_inset
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).
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename fe-tcsp.png
lyxscale 10
width 6.8cm
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
Relation of the RPV rupture time (tvr) and way of the core support modelling
(
\begin_inset Formula $trcsp$
\end_inset
- time of core support plate rupture) - "axial draining" model
\begin_inset Formula $trcsp=0$
\end_inset
means no contact between core support plate and corium.
\begin_inset CommandInset label
LatexCommand label
name "tvr_csp"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The results with axial draining model in the Fig.
\begin_inset Formula $\ref{tvr_csp}$
\end_inset
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.
\end_layout
\begin_layout Section
Conclusion
\end_layout
\begin_layout Standard
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 of improvements in the modelling will give help to have better
understanding of the IVMR strategy utilization for nuclear reactors.
\end_layout
\begin_layout Section*
\noindent
Acknowledgments
\end_layout
\begin_layout Standard
This work has been carried out within the framework of the PROCOR platform
development funded by CEA, EDF and AREVA.
\end_layout
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "JPTbib"
options "elsarticle-num"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\begin_inset Graphics
filename elsarticle.layout
display false
\end_inset
\begin_inset Graphics
filename elsarticle.cls
display false
\end_inset
\begin_inset Graphics
filename elsarticleJPT.cls
display false
\end_inset
\begin_inset Graphics
filename LOGO.png
display false
\end_inset
\begin_inset Graphics
filename IHE.png
display false
\end_inset
\end_layout
\end_inset
\end_layout
\end_body
\end_document