Download Rayleigh-Bénard and Bénard-Marangoni convection in a thin

Rayleigh-Bénard and Bénard-Marangoni convection in a thin metallic layer on
top of corium pool
L. Saasa,∗, R. Le Tellier1 , E. Skrzypekb
a CEA/CAD/DEN/DTN/SMTA/LPMA,
b National
CEA Cadarache, 13115 Saint-Paul-lez-Durance, France
Center for Nuclear Research,7 A.Soltana Street, 05-400 Otwock, Poland
Abstract
To estimate the success of the In-Vessel Retention (IVR) strategy an accurate evaluation of the heat fluxes of the
corium pool in the lower head of the vessel is needed. Due to the ablation of the vessel wall and to thermochemical
effects, a thin metallic layer may be on top of the corium pool and may be responsible of the vessel failure. In order
to reduce uncertainties on the evaluation of the heat fluxes of this thin metallic layer, this paper present a thermalhydraulic study and especially the influence of the top free surface condition on the flow regime and on the heat
transfer. Non-dimensionalization of incompressible Navier-Stokes equations with Boussinesq approximation and
top free surface boundary condition is presented to introduce Rayleigh, Marangoni, Crispation and Galileo numbers. An analysis of the non-dimensionalized equations shows that Rayleigh-Bénard convection is the dominant
flow regime for large thicknesses, but for small thicknesses, Bénard-Marangoni effect takes place and may govern the convection flow regime. The Bénard-Marangoni effects could reinforce or decrease the Rayleigh-Bénard
convection: if surface tension decreases with temperature increase, the Bénard-Marangoni effects could enhance
the heat transfer on the top surface and may consequently diminish the lateral heat flux on the vessel. A simplified
stationary thermal model and a stability result based on Marangoni and Rayleigh numbers are used in order to
calculate a threshold thickness (about 3cm) below which Bénard- Marangoni effects have to be taken into account
in the thermal evaluation. The impact of Bénard-Marangoni effects on the Nusselt number and consequently on the
heat exchange is also quantified using correlations based on Marangoni number instead of Rayleigh number. Under
this threshold thickness, the top boundary condition with tension surface on the free surface has to be considered.
This condition must be used for detailled computations of the thin metallic layer. In this case, an evaluation of the
Galileo and Crispation numbers shows that the deformation of the top free surface may be neglected.
Keywords: Thermalhydraulics, Thin metallic layer, Rayleigh-Benard convection,
Marangoni-Benard convection
1. Introduction
In-Vessel Retention (IVR) [1], [2], [3], [4] is a severe accident management strategy which consists on avoiding
the vessel lower head rupture by external water cooling. The assessment of the IVR strategy is based on the evaluation of the thermal load of a stratified corium pool in the vessel lower head. This evaluation is still a challenge
for high power reactors due to large uncertainties regarding the transient thermochemical and thermalhydraulic
modelling of the stratified corium pool.
In the lower head of the vessel, on top of the stratified corium pool a thin metallic layer could be formed with
a thickness that may evolve during the transient from a centimetric to decimetric range [5] due to:
• the ablation of the vessel wall and of the internal structures that leads to steel addition in the thin metallic
layer at fusion temperature;
• the thermochemical phenomena [6], [7], that leads to mass exchange through the crust between the thin
metallic layer and the layers of the corium pool located below. The gaseous atmosphere (mixture of vapor,
fission products, and hydrogen) above the layer may also oxidize the thin metallic layer;
The thin metallic layer is heated by the corium pool from below and cooled at its lateral and top surfaces. If the
top cooling is not sufficient the lateral heat flux may be high because of small lateral surface of cooling and may
∗ Corresponding
author
Email address: [email protected] (L. Saas )
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
lead to the vessel rupture. This phenomenon is called focusing effect (a large thermal power has to be evacuated
on a small lateral surface).
In simplified analytical models [3],[4] or in severe accident scenario codes [8],[9],[10] that are commonly used
for IVR evaluation, heat exchange correlations associated to the thin metallic layer are steady state correlations
([12] for the lateral correlation in laminar regime, or [13] for the lateral correlation in laminar/turbulent regime,
and [11] for the upper correlation). They rely on simulant fluid experiments and correspond to unconfined cavities,
with large aspect ratio and rigid surfaces. In these correlations, only Rayleigh-Bénard convection is assumed to be
the dominant flow regime (only Prandtl and Rayleigh numbers are used for their evaluation). Simplified thermal
evaluations [14] of the thin metallic layer using the previous heat exchange correlations (as in the scenario codes
or in analytical models), shows that the lateral heat flux increases when the thickness diminishes (focusing effect).
Experimental results such as BALI-metal experiments (with water) [14], [15] emphasize that the thermal evaluation with this modelling could lead to overestimation of the stationary lateral heat flux and consequently to an
overestimation of the vessel rupture evaluation for IVR strategy.
In order to enhance the IVR evaluation, a part of the uncertainties and of the overestimation by the models
that are currently used on the lateral heat flux evaluation of thin metallic layer may be reduced by improving
the thermochemical and thermalhydraulic modelling of the thin metallic layer. This paper is focused on some
thermohydraulic phenomena that affects the behavior of this layer. The fine structure of the flow regime in the
thin metallic layer as observed in the BALI-metal experiments ([14], [15]) is not taken into account (prototypic
fluid and bounded cavity). The current correlations are not based on this observation. During the transient, before
natural convection is established, for a given height, heat fluxes are smaller than in steady state where they reach
their maximum level. The top boundary conditions of the thin metallic layer depend of the scenario of the severe
accident and could be not well defined (or its variation during transient). These boundary conditions directly
determine the upper heat flux (for example, the radiative heat exchange model between to parallel plate leads to an
under estimation of the upper heat flux [15] or the heat exchange in case of reflooding is not well defined (nucleate
boiling or film)). At present time the top surface characteristic is not accurately taken into account.
Figure 1: Thin metallic layer mass and heat transfer
Studies of these phenomena may provide some margin. Some of these different phenomena will introduce delay to
reach the maximum value of the lateral heat flux while the thickness of the layer may increase and the maximum
level of the lateral heat flux will decrease. In this case the layer may be sufficiently thick to avoid focusing effect.
Some of these different phenomena will directly decrease the level of the maximal lateral heat flux of the metallic
layer, for example the top boundary condition. Figure 1 illustrates the thin metallic layer with the different mass
and thermal exchanges.
In this paper, as a first step, we consider only thermal steady state of the metallic layer and we focus our study
on the top free surface boundary condition and its impact on the associated upper heat exchange. For thin layers
heated from below and with free surface on top, it has been shown that Marangoni effects could be the dominant
phenomenon in the flow regime [16], [17], , [18], [19], [20]. Because of Rayleigh-Bénard cells in natural convection, a thermal gradient at the top surface occurs. A surface flow may exists to compensate the variation of surface
tension with temperature and to ensure the force balance at the top free surface. Because of mass conservation, this
top surface flow will generate a circulating flow in the bulk. This flow is called Bénard-Marangoni effects, and is
associated with surface tension dependence on temperature. It is superimposed with the Rayleigh Bénard flow and
may reinforce or block it [21]. Modification of the flow due to Marangoni effects for small thickness will modify
the heat transfer of the metallic thin layer. Here, we want to evaluate this effect and its impact on the upper heat
transfer in the centimetric/decimetric range of layer thickness [5]. .
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
In all the document, we assume that no mass exchange occurs, so the thickness e of the thin metallic layer is
constant with a fixed composition. We assume that the thin metallic layer composition corresponds to the vessel
steel composition and that there is no crust laterally. Consequently the temperature at the lateral surface Tlat corresponds to the steel fusion temperature Tlat = T f usion . For simplicity, for the geometry of layer, we use a cylinder of
fixed radius R and fixed height e (equal to the thickness layer). The natural convection is assumed to be established
(no transient phenomena). A mechanically resistant crust separating the corium pool and the thin metallic layer is
present and the thin metallic layer is heated from below by the corium pool: the heat flux on the bottom surface
of the layer ϕdown coming from the corium pool is assumed to be fixed and constant (no transient for the corium
pool too). Physical properties of the metallic layer are assumed to be fixed except for the mass density (buoyancy force has to be considered for the Rayleigh-Bénard convection) and the surface tension (Marangoni effect).
Figure 1 present the different thermal boundary conditions and the assumption on the geometry with the associated notations: Tup is the upper surface temperature, T is the mean temperature of the layer, ϕlat designates the
lateral heat flux through the lateral surface Slat , and ϕup corresponds to the upper heat flux on the upper surface Sup .
The rest of the document is composed of three sections. In the first section, we present the different configurations of the thin metallic layer to introduce the configuration with a top free surface that we will study . Then,
in the second section, we present the Navier-Stockes equations with top free surface boundary conditions and we
perform the non-dimensionalization of theses equations in order to introduce the Rayleigh, the Marangoni, the
Crispation and the Galileo number. In the same section, we show that for small thicknesses Marangoni effects
can play a role in the flow regime. The third section is devoted to numerical evaluation of the Marangoni-Bénard
convection and to a comparison to the Rayleigh-Bénard and Bénard-Marangoni convection using stability results
[16] and an integral stationary thermal balance [14]. At the end, some conclusions on the modelling of the top free
surface is given with some perspectives.
2. Corium pool configurations with a top light metal layer: top boundary condition
The corium pool configurations in the lower head are the result of corium relocation from the core, of the thermochemical phenomena (stratification), of the thermal ablation of the vessel wall and the steel internal structures.
The configurations are dependent on the reactor type and the severe accident scenario. We are looking only on the
corium pool configurations where a metallic layer is on top of the corium pool and is separated from the corium
pool by a stable crust (see assumptions in the previous section).
In [14], when the top surface temperature Tup of the liquid layer is considered to be equal to the fusion temperature T f usion , there is no focusing effect because of an efficient cooling. Because we are interested in reducing
uncertainties on the heat transfer of the thin metallic layer, we will not study these configurations. They correspond to a cooling by nucleate boiling because of the presence of water on top of the thin metallic that is partially
solidified (Tup = T f usion ).
Anorther configuration corresponds to a thin metallic layer that is under a gaseous atmosphere with a upper
surface temperature which is above the fusion temperature of the thin metallic layer Tup > T f usion . This gas may
oxidize the thin metallic layer and an oxidic crust could be formed at the top surface. If the crust is sufficiently
thick and mechanically resistant there will be no free surface on top of the metallic layer (we can remark that this
oxidization process may also change the composition of the metallic layer). This configuration is also not studied
here but has to be dealt in another study where the oxidized crust thickness with its mechanical resistance has to
be evaluated.
The other configurations, that we are studying here, correspond to a thin metallic layer with a top free surface
with a upper surface temperature Tup which is greater than the fusion temperature Tup > T f usion (no crust). Some
gas and internal structures (degraded core for example) will be on top of the layer and thermal heat exchange
occurs between the layer and gas or the surrounding structures. In these configurations, we can assume that the
thermal boundary condition on the top surface corresponds to gas convection cooling, or radiative heat transfer to
4 −T4
the surrounding structures or both: ϕup = hgas (Tup − Tgas ) or ϕup = εσSB (Tup
structure ) where hgas is the heat exchange coefficient with the gas on top, Tgas the average gas temperature, ε corresponds to the global emissivity, σSB
corresponds to the Stefan-Boltzmann constant and Tstructure designates the average temperature of the surrounding
structures. The top boundary condition is simplified compared to the reactor case where slug-like or light-oxide
material in a very thin film may float on the layer may be present and will affecting both heat removal and mechanical behavior on the surface.
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
Now that we have defined the studied configuration of the thin metallic layer, we are going to evaluate the
existence and impact on heat transfer of Marangoni effects.
3. Thermal-hydraulic equations for the top light metal layer with free surface
In this section, we will study the behavior of the flow regime depending of the thickness of the layer. We
introduce Γ = Re which is called the geometrical aspect ratio. We assume that the flow regime can be modeled
by Navier-Stokes equations and that the approximation of Boussinesq is satisfied by the thin metallic layer. The
temperature variation of the density is given by:
ρ = ρ0 (1 + β (T − T0 ))
(1)
where ρ corresponds to the mass density, ρ0 is a reference density at T0 which is the reference temperature and T
the temperature of the thin metallic layer. β denotes the thermal expansion coefficient. We assume that the steel of
the layer is a Newtonian fluid. To take into account the free surface, as a first approximation, we assume that the
variation of the surface tension with the temperature is given by:
σ = σ0 − γ (T − T0 )
(2)
where σ is the surface tension, σ0 designates the surface tension at reference temperature T0 and γ is the surface
tension coefficient γ = − ∂∂ σT . When γ > 0, the Marangoni number is also positive [21] : the surface tension decreases with the increase of the temperature. In this case the Bénard Marangoni convection may reinforce the
natural Rayleigh-Bénard convection. When γ < 0, the Marangoni number is negative and the Marangoni effects
are opposed to the natural convection. In all the document, we assume that for the metallic layer γ > 0.
To simplify the presentation and the analysis in this section, we assume that the layer is a 2D slab of length
L = R and of height e. x denotes the coordinate of horizontal axis so that the layer is in [0, L] and the vessel
wall corresponds to x = 0 and x = L. z designates the coordinate of the vertical axis and the bottom of the layer
corresponds to z = 0. To take into account the top free surface and its possible deformation due to the flow in the
thin metallic layer, the top free surface is given by z(x,t) = e + η(x,t) (η corresponds to the surface deformation
at x at time t). The norm of the normal of the free surface at (x, z) is then:
s
2 s
∂z
∂ η(t, x) 2
N = 1+
= 1+
(3)
∂x
x
3.1. Incompressible Navier-Stokes Equations under Boussinesq approximation
We assume that the incompressible Navier-Stokes equations under the Boussinesq approximation are satisfied
for the thin metallic layer flow:
∇.~v = 0
∂~v
ρ0
+~v.∇~v
= −∇p + µ∆~v − ρ0 β (T − T0 )~g
∂t
∂T
+~v.∇T
= λ ∆T
ρ0C p
∂t
(4)
(5)
(6)
where ~v = (vx , vz ) is the velocity, t is the time, p is the pressure, µ corresponds to the dynamic viscosity, ~g the
gravity force, C p the heat capacity, λ the thermal conductivity. At the initial time, we assume that the metal of the
layer is at rest and at uniform temperature, so the associated initial conditions are given by: ~v = 0 and T = Tini ,
where Tini is the initial temperature of the layer. The thermal boundary conditions are:
• for the bottom surface, the heat flux is imposed by the outward heat flux of the corium pool below: −λ ∇T.~n =
ϕ pool , where ~n is the outer normal of the thin metallic layer;
• for the lateral surface the temperature corresponds to the fusion temperature: T = T f usion = Tlat . We recall
that we have previously assumed that the composition of the layer is the same as the vessel composition;
• for the upper surface, the upper heat flux is prescribed: −λ ∇T.n = ϕup . The different type of thermal
boundary for the upper surface has been explained in the previous section.
The mechanical boundary conditions are:
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
• a no slip condition for the bottom (crust between the layer and the corium pool) and lateral surfaces (vessel
wall) which corresponds to: ~v = 0
• for the upper surface at z = e + η(t, x), the surface is free and assuming that it could be deformed, the
equalization of the pressure, viscous, and surface tension forces at this surface give:
"
2
µ ∂ vz ∂ vz ∂ η
∂η
+
−
N2 ∂ z
∂x ∂x
∂x
"
!
∂η 2
∂ vx ∂ vz
∂η
µ
1−
+
+2
∂x
∂z
∂x
∂x
(p − pgas ) − 2
∂η
∂η
+ vx
= vz
∂t
∂x
#
∂ vx ∂ vz
σ ∂ 2η
+
= − 3 2
∂z
∂x
N ∂x
#
∂ vz ∂ vx
∂σ ∂η ∂σ
−
= N
+
∂z
∂x
∂x
∂x ∂z
(7)
(8)
(9)
where pgas is the pressure of the gas that is present above the upper surface of the layer.
3.2. Non-dimensionalization of the equation
To perform the non-dimensionalization of equations (4)-(9), we introduce the following scales and some nondimensionalized numbers:
2
• t = t0t ∗ with t0 the time scale. This scale is chosen to be the thermal diffusion time scale t0 = eα where
λ
is the thermal diffusivity. This scale is larger than the characteristic time of the flow because the
α = ρC
p
thermal diffusion is less effective than the natural convection;
• x = Rx∗ = Lx∗ = γ e x∗ with L the length scale and z = ez∗ ;
• vx = v0 v∗x and vz = v0 v∗z with v0 the velocity scale. We use the same scale for the velocity in each direction.
We fix it to be v0 = te0 . This scale corresponds to a Strouhal number equal to one St = t0ev0 = 1. With the
previous choices of scales, the Peclet number is also fixed to one Pe = vα0 e and consequently the Reynolds
1
number is given by Re = Pr
(Pr is the Prandtl number Pr = αν );
• T = Tre f + ∆T T ∗ with Tre f a reference temperature and ∆T the temperature scale. To simplify the calculus,
we take Tre f = T f usion and ∆T = T f usion − T0 ;
• p = pre f + p0 p∗ with p0 the pressure scale and pre f a reference pressure that we take equal to the pressure of
the gas above the thin metallic layer pre f = pgas . The choice for the pressure scale to the hydrostatic pressure
of the thin metallic layer p0 = ρ0 ge. This scale is chosen so that the equations are expressed in terms of the
3
Galileo number Ga = ge
(ν = µρ is the kinematic viscosity) and the Prandtl number instead of the Euler
ν2
number Eu (with the scale choices Eu = Ga Pr2 ) and to replace the Richardson number Ri by the Grashov
3
∆e3
number or the Rayleigh number (Ri = Gr Pr2 = Ra Pr2 with Gr = gβ ν∆Te
and Ra = gβαν
= Gr Pr where
2
β is the thermal expansion coefficient and g the gravity constant);
• the Marangoni number Ma =
γ∆Te
ρνα
and the Crispation number Cr =
ρνα
σe .
With these different scales and non-dimensionalized numbers introduced before, and using equation (1), the
equations (4)-(6) take the following forms:
∂ v∗x ∂ v∗z
+
∂ x∗ ∂ z∗
∂ v∗
∂ v∗
∂ v∗x
+ Γv∗x x∗ + v∗z ∗x
∂t
∂x
∂z
Γ
∂ v∗z
∂ v∗
∂ v∗
+ Γv∗x z∗ + v∗z ∗z
∂t
∂x
∂z
∂T∗
∂T∗
∂T∗
+ Γv∗x ∗ + v∗z ∗
∂t
∂x
∂z
= 0
= −Ga Pr2 Γ
(10)
∂ p∗
1 2 ∂ 2 v∗x
Γ
+
Γ
∂ x∗
Pr ∂ (x∗ )2
∂ p∗
1 ∂ 2 v∗z
= −Ga Pr2 ∗ + Ra PrT ∗ +
∂z
Pr ∂ (z∗ )2
2
∗
2
∗
∂ T
∂ T
=
Γ2
+
∗
2
∂ (x )
∂ (z∗ )2
Indicate here the SESSION name and the Paper Number (order as in the announcement)
(11)
(12)
(13)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
and, with the same scales and using equation (2), the equations (7)-(9) for the top boundary condition become:
∗
∂ z∗
∗ ∂z
+
Γv
= v∗z
(14)
x
∗
∂t ∗
∂ x#
"
∗
∂ v∗z
∂ v∗z
2
∂ z∗ 2
∂ z∗ ∂ v∗x
Γ2 1 ∂ 2 z∗
3 ∂ vz
Ga Pr p∗ − ∗ 2
+
Γ
−
Γ
+
Γ
=
−
(15)
(N ) ∂ z∗
∂ x∗ ∂ x∗
∂ x ∗ ∂ z∗
∂ x∗
(N ∗ )3 Cr ∂ (x∗ )2
"
∗ 2 ! ∗
#
∗
∂ v∗z
∂ vx
∂ z∗ ∂ v∗z
∂ v∗x
∂T
∂ z∗ ∂ T ∗
2 ∂z
∗
1−Γ
+
Γ
+
2Γ
−
Γ
=
ΓN
Ma
+
(16)
∂ x∗
∂ z∗
∂ x∗
∂ x ∂ z∗
∂ x∗
∂ x ∗ ∂ x ∗ ∂ z∗
r
where N ∗
1 + Γ2
=
∂ z∗
∂ x∗
2
is the non-dimensionalized norm of the top surface normal. The non-dimensionalized
thermal upper boundary condition is given by:
∂T∗
∂ z∗
=
ϕup e
λ ∆T .
3.3. Analysis
From the non-dimensionalized equations, we will perform an analysis of the flow regime and of the different
thermalhydraulic phenomena. The geometrical ratio Γ = Le is present in all the equations. In the conservation
equations, only Prandlt, Rayleigh and Galileo numbers appear. The top boundary conditions are written using
the Prandlt, the Galileo, the Marangoni and the Crispation numbers. The Prandtl number is associated with the
physical properties of the thin metallic layer and is fixed (for the metal Pr < 1, Pr ≈ 0, 09). The other numbers
are associated with the flow regime (∆T ). The Prandtl number is fixed (Pr ≈ 0, 09), because we assume that the
composition is fixed. First we will look at the balance equations (10)-(13). The lateral cooling (Tlat = T f usion )
is an unstable configuration for natural convection (horizontal thermal gradient) that induces a flow inside the
thin layer and consequently motion (velocity is not null). In the momentum equations (10)-(11), the velocity
terms are non-zero but bounded and consequently pressure and/or temperature gradients exist in the layer. If the
Rayleigh number is small compared to the Galileo number (Ra << Ga), the thermal gradient has to be taken into
account with the buoyancy forces and the Boussinesq approximation is satisfied. If the Rayleigh number is large
Ra >> 1 the flow regime corresponds to Rayleigh-Bénard convection (viscous forces are small compared to the
buoyancy forces). When the thickness of the thin metallic layer becomes small Γ = Re << 1, the variation of
vertical velocity and temperature are small. For the top boundary condition equations (14)-(16), if Γ << 1 (so that
1 ∂ 2 z∗
N ∗ ≈ 1), the right hand side terms are bounded and non-zero because of the fluid motion so the terms Cr
and
∂ (x∗ )2
∗
∂T
∂ z∗ ∂ T ∗
Ma ∂ x∗ + ∂ x∗ ∂ z∗ have to be large. The Crispation number Cr becomes also large when Γ << 1 (e.g. e << 1)
2 ∗
and the term ∂∂(x∗z)2 has to be large and it implies that the top free surface has to be deformed. The deformed
surface results in velocity gradient and fluid motion. The Marangoni number Ma becomes small when Γ << 1
(e.g. e << 1), so that the thermal gradient at the surface has to be large. The thermal gradient at the surface will
also lead to fluid motion. Marangoni effects will modify the motion at the free surface and play a role on the
flow regime for small thickness. They will modify the top heat exchange (velocity and thermal gradient will be
modified). If the deformation of the upper surface can be neglected η << 1 , we have:
∂ z∗ ∂ z∗
,
= 0 , and N ∗ = 1
∂t ∗ ∂ x∗
So that the mechanical top boundary conditions become:
v∗z = 0 , p∗ = 0 and
∂ v∗x
∂T∗
= ΓMa ∗
∗
∂z
∂x
(17)
If Marangoni effects could be neglected Ma << 1, the mechanical boundary conditions correspond to a no slip
condition as in the classical correlations:
v∗z = 0 , p∗ = 0 and v∗x = 0 (obtained using the mass conservation equation and the incompressibility of the fluid)
Table 1 gives the order of magnitude for the physical properties that appear in the non-dimensionalized numbers
for a ”steel” material taken from [15]. For steel the surface, tension and the surface tension coefficient are strongly
dependent of the composition and of the atmosphere [23], [24], [25] and are not given here : depending on these
conditions the surface coefficient can be positive or negative or change of sign with the variation of the temperature.
The surface tension may strongly change with the stainless steel composition [23] and also with oxydization [24].
Here, the ”steel” material is an approximative material because we do not know the range of variation of the
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
composition (different steels could be present in the vessel materials). For the surface tension, we use the pure
iron properties [22]. In Table 2, with physical properties of Table 1, we evaluate the non-dimensionalized numbers
for a thickness of e = 0, 1m and difference of temperature ∆T = 400K which corresponds to configuration of the
thin metallic layer that is observed in scenario code computations. During the transient the composition of the thin
metallic layer will change and its physical properties too. For the metallic layer, with the order of magnitude, we
can see that Ra << Ga, so that the Boussinesq approximation is justified. We have Ra >> 1 and consequently the
Rayleigh-Bénard convection takes place and Ma >> 1 and Cr << 1 so that Marangoni-Bénard effects modify the
flow. We have now to study which effects dominate the flow regime and the heat transfer.
Table 1: Physical properties order of magnitude for a ”steel”
Material
Steel
ρ
6720
Cp
674
λ
35
β
3 10−5
ν
6.8 10−7
σ0
1.92
γ
3.97 10−4
Table 2: Order of magnitude for the different nondimensionalized numbers for ”steel” (e = 0.1m and ∆T = 400K)
Material
Steel
Pr
8.8 10−2
Ga
2.12 1010
Ra
2.25 107
Ma
4.51 105
Cr
1.83 10−07
In general stability analysis with Bnard-Marangoni effects are done in a one dimensional frame or with adiabatic
boundary condition in a two dimensional frame [16], [17], [18], [19], [20]. In these studies lateral cooling, contrary
to the reactor case, is not taken into account and do not impact of the lateral thermal boundary condition. With
a stability analysis of a thin layer heated from below, D.A. Nield in [16] (one dimensional frame), shows that
a relation between the Rayleigh number Ra and Marangoni Ma can be used to determine which effect between
Rayleigh-Bénard convection and Marangoni- Bénard convection is the dominant flow regime. This relation is
based on critical value of the Rayleigh Rac and Marangoni Mac numbers that depend of the upper Biot number
h e
(Biup = up
λ if the upper heat flux is expressed as ϕup = hup (T − Tup )). This relation has been completed by Sarma
[17] and is given by:
Ra
Ma
+
= 1 + ε(Cr, Ga)
Rac Mac
(18)
where ε(Cr, Ga) depends on the Crispation number and Galileo number. Sarma has demonstrated that if Cr < 10−4
and Ga >> 1, the top free surface deformation can be neglected. In order to evaluate this relation and to know
at which thickness the Marangoni effects has to be taken into account in to the thermalhydraulic description of
the thin metallic layer. To use this relation, we have to evaluate the upper Biot number Biup and the difference
of temperature ∆ = T − Tup to compare the Rayleigh and Marangoni numbers to their critical values and also to
determine if the deformation of the free surface can be neglected. To complete the study of the flow regime, we can
also compute the Gr and Ra numbers to know if the flow regime is laminar or turbulent (the lateral flow is laminar
for Gr < 109 and the upper flow is laminar for Ra < 105 ). This will be done in the next section using a stationary
model.
4. Numerical evaluations for the top light metal layer
The stationary model that we use for the numerical evaluation is the same as in [14]. Similar models are used
in stationary analysis [3], [4] or in scenario code [5], [9], [10], [8].
4.1. Stationary thermal evaluation of the light metal layer
The geometry that is used for the evaluation is a cylinder of height e and radius R. The stationary thermal
balance that we consider to evaluate the upper Biot number and the difference of temperature of the thin metallic
layer is the following one:
ϕlat Slat + ϕup Sup = ϕdown Sdown
(19)
We consider that the temperature of the thin metallic layer is T . The upper heat exchange correlation [11] (Globe
& Dropkin) gives the heat exchange coefficient hup which is a function of the upper temperature difference ∆Tup =
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
h e
T − Tup and of the physical properties: ϕup = hup (T − Tup ). The Biot number is directly given by Bi = up
λ .
The lateral heat exchange correlation [12] (Churchill & Chu), or [13] (Chawla & Chan) gives the heat exchange
coefficient hlat which is a function of the lateral temperature difference ∆Tlat = T − Tlat and the physical properties:
ϕlat = hlat (T − Tlat . The upper boundary condition is assumed to be given by a radiative heat exchange as in [14]:
4 −T4
ϕup = εσSB (Tup
structure ).
4.2. Rayleigh-Benard vs Marangoni-Benard convection
In this subsection, we present the results obtain by the numerical evaluation with fixed physical properties
given in Table 1. The boundary conditions are: for the lateral surface the fusion temperature T f usion = 1658K, for
the bottom surface heat flux coming from the corium pool ϕdown = 0.6MW m2 , for the top surface the emissivity
ε = 0.6 and for the structure temperature Tstructure = 400K. The radius is fixed to R = 2m and thickness varies in
the range e = [0.005, 0.15]m. All the curves that we present correspond to variation depending on the thickness.
The Figure 2 present the variation of the heat flux concentration factor (ϕlat /ϕdown ) with respect with the thickness.
Figure 2: Evaluation of heat flux concentration factor ϕlat /ϕdowm depending on the thickness
In Figure 3, the evaluation of the Rayleigh Ra and Marangoni Ma numbers are compared to their critical values
(Rac , Mac ). We can see that the Marangoni effect has to be taken into account for thickness less than 0.05m and
that they dominates the flow for thickness around 0.03m.
Figure 3: Evaluation of the Marangoni Ma number and the Rayleigh Ra number variation depending on the thickness and comparison to their
critical values (critical Marangoni value Mac and critical Rayleigh value Rac )
As shown previously, the Bénard-Marangoni effects could dominate the flow for thickness below 0, 03m and must
be taken into account for thickness below 0, 04m. We have to evaluate what is the impact on the flow regime in
this case and especially on the upper heat transfer.
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
Figure 4: Evaluation of the Crispation number variation depending of the thickness
Figure 5: Evaluation of the Galileo number variation depending of the thickness
The figure 4 and 5 show that the Crispation number is less than Cr < 10−4 and that the Galileo number is large
Ga >> 1 so that the deformation of top surface of the thin metallic layer could be neglected. For thickness less than
the threshold thickness e = 0.03m, the boundary conditions that must be used are given by the non-dimensionalized
equations 17. The boundary condition are :
vz = 0 , p = 0 and
∂ vx
∂T
= Ma
∂z
∂x
Figure 6: Evaluation of the Grashof and Rayleigh number variation depending of the thickness. The red curve corresponds to the Rayleigh
number associated to the lateral surface, the black curve corresponds to the threshold for the Rayleigh number between laminar and turbulent
regime for the lateral flow. The blue curve corresponds to the Grashof number than must be sompared to the green cuvre wich corresponds to
the threshold between laminar and turbulente regim for the upper flow.
As shown in Figure 6, the flow close to the lateral wall is laminar until, the thickness reaches e = 0.025m.
Under the thickness threshold the flow regime is modified by the Marangoni effects. We have to determine if
the cooling will be more efficient or not with a Marangoni correlation: Bénard-Marangoni convection changes the
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
heat transfer on the upper surface which may affects the lateral heat exchange. Different heat exchange correlations
with Marangoni effect exists [30], [31], [28], [29] and can be compared to the Rayleigh correlation that are used
in general for IVR evaluation. These correlations are obtened using Computational Fluid Dynamics. A theoretical
bound of the Nusselt number depending on Marangoni number when Marangoni effects are dominant is also
established [32] (the upper bound for the Nusselt number with Marangoni effects is Nu = 0.838Ma2/7 ). To evaluate
the heat transfer with Marangoni correlation and compare it to heat transfer with Rayleigh correlation, we need to
compute the ∆up . We want to compare Rayleigh correlation of Globe and Dropkin [11] to the Marangoni Nusselt
correlation from [28] for low Prandtl number obtained for by using 3D CFD (Nu = 0.284Ma0.24 ). The difference
of temperature ∆Tup that is used is the one of the thermal problem defined in the previous subsection (with Rayleigh
correlations). If a thermal Marangoni correlation is used directly in the integral thermal balance in the range where
the Marangoni effect has to be taken into account, the thermal problem will be changed and also ∆Tup .
Figure 7: Comparison of Nusselt number for different correlations depending of the thickness : the black curve is the Nusselt number obtained
with the upper bound correlation with Ma, the red curve is the one obtained with the Globe & Dropkin correlation [11], the blue curve is the
one obtained with the Marangoni correlation [28]
Figure 7 shows that for small thickness the Marangoni effects will enhance the upper heat transfer because the
Nusselt number obtained with Marangoni correlation are larger than the one obtained with the Rayleigh correlation.
Figure 8 shows the modification of the heat flux concentration when Marangoni correlation is used for small
thickness in the thermal balance. For small thicknesses, the top cooling may be more efficient, but this result
does not take into account the lateral flow and the partition of power in the case of the thin metallic layer cooled
laterally and on the top: the strength of the lateral cooling flow has to be compared to the modified strength flow by
Marangoni effects. If the upper heat flux is larger the lateral heat flux diminishes and consequently some margin
on focusing effect will be gained. CFD (computational Fluid Dynamics) could be used to evaluate the impact of
the power balance when Marangoni effects dominate the flow.
Figure 8: Evaluation of heat flux concentration factor ϕlat /ϕdowm for the thin metallic layer with a modified correlation that takes into account
the Marangoni effects
5. Conclusion
In case of negative tension surface variation with temperature (γ > 0), this study has shown that for small
thickness less than 0, 04m for the thin metallic layer on top of a corium pool the Bénard-Marangoni convection has
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
to be considered. For thickness below 0, 03m it can dominate the flow regime. The top cooling could be reinforced
by these effects and results in decreasing the focusing effect, but the level of heat flux concentration is still large.
Other margins must be found on the focusing effect. In european H2020/IVMR project [35], some benchmarks
are done on CFD code on the thermalhydraulic of the stratified corium pool in order to evaluate the use of CFD
in this framework. It seems that CFD (Computations Fluid Dynamics) could be used to evaluate the impact of
the power balance when Marangoni effects dominate the flow [34]. All the results obtained in this paper assume
that surface tension diminish with the increase of temperature. Some materials (like stainless steel depending
of its composition) can have a more complex variation of surface tension with temperature and oxidization [23],
[24]. Some steel can have positive surface tension coefficient γ < 0 that leads to negative Marangoni [21] and that
will block the flow for small thickness. In reactor case, the composition of the thin metallic layer evolves due to
thermochemical effects and to ablation of the internal structures and of the vessel wall. At the top surface, slug-like
or light-oxide material in a very thin film may float on the layer may be present and will affecting both heat removal
and surface tension effect. For reactor computations, if the layer is assumed to be totally liquid, the variations with
temperature and composition of the physical properties of the thin metallic layer are needed in order to evaluate the
impact of Bénard-Marangoni convection. At the present time, in the scope of the H2020/IVMR european project
[35] the VITI [36] facility (CEA) is used to evaluate the variation of the surface tension with temperature and
composition of the thin metallic layer which may vary during the transient of the accident.
[1] T.G. Theofanous, C. Liu, S. Addition, S. Angelini, O. Kymalainen, T. Salmassi, In-Vessel Coolability And
Retention Of A Core Melt, Nuclear Engineering And Design, Vol. 169, 1997.
[2] H. Tuomisto, T.G. Theofanous, A consistent approach to severe accident management, Nuclear Engineering
and Design, Vol. 148, 1994.
[3] J.L. Rempe, D.L. Knudson, C.M. Allison, G.L. Thinnes, C.L. Atwood, M.J. Cebull, Potential for AP600
In-Vessel Retention through Ex-Vessel Flooding, Tech. Report., INNEL/EXT-97-00779, (1997).
[4] H. Esmali, M. Khatib-Rahbar Analysis of likelihood of lower head failure and ex-vessel fuel coolant interaction energetics for AP1000, Nuclear Engineering and Design, vol. 235, no. 1, p. 1583-1605 (2005).
[5] R. Le Tellier, L. Saas and S. Bajard, Transient stratification modelling of a corium pool in a LWR vessel
lower head, Nuclear Engineering and Design, Vol. 287, pp. 68-77, June 2015.
[6] M. Fischer, P. Levi, G. Langrock, A. A. Sulatsky, E. V. Krushinov, The impact of thermal chemical phenomena on the heat fluxes into the RPV during in-vessel melt retention, Proc. of Int. Congress on Advances in
Nuclear Power Plants, ICAPP 2011, Nice, France (2011).
[7] D. F. Tsurikov, V. F. Strizhov, S. V. Bechta, V. N. Zagriazkin, N. P. Kiselev, Main results of MASCA1 and 2
projects, Tech. rep., RRC Kurchatov Institute (2007).
[8] R.O. Gauntt, J.E. Cash, R.K. Cole, C.M. Erickson, L.L. Humphries, S.B. Rodriguez, M.F. Young, MELCOR
Computer Code Manual, NUREG/CR-6119, Rev 3 (2vols), (2005).
[9] Fauske & Associates Inc. , MAAP4 - modular accident analysis program for LWR power plants: Code
Structure and Theory, Vol. 2, Ch. 1, Electric Power Research Institute, (1994).
[10] J.-P. Van Dorsselaere, C. Seropian, P. Chatelard, F. Jacq, J. Fleurot, P. Giordano, N. Reinke, B. Schwinges,
H.-J. Allelein and W. Luther, The ASTEC integral code for severe accident simulation, Nuclear Technology,
vol. 165, no. 3 (2009).
[11] S. Globe, D. Dropkin, Natural-Convection Heat Transfer in Liquids Confined by Two Horizontal Plates and
Heated from Below, Transactions of the ASME, February 1959.
[12] S.W. Churchill and H.H.S. Chu, Correlating equations for laminar and turbulent free convection from a
vertical plate, Journal of Heat Mass Transfer, Vol. 18, pp. 1323-1329, 1975.
[13] 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, August 1982.
[14] J.M. Bonnet, Thermal hydraulic phenomena in corium pools: the BALI experiment, JAERI conference, 1999.
[15] C. Villermaux, Modlisation physique et numrique de la convection naturelle dans une couche de fluide de
faible rapport daspect dans le cadre des tudes daccidents graves de racteurs eau sous pression, Thse INPG,
2/02/1999.
Indicate here the SESSION name and the Paper Number (order as in the announcement)
The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017
Warsaw, Poland, 16-18 May 2017
[16] D.A. Nield, Surface tension and buoyancy effects in cellular convection, Journal of Fluid Mechanics, 1964,
Vol. 19, Issue 03.
[17] Sarma, G. S. R., Fourth International Conference on Physico-Chemical Hydrodynamics, New York, 1982.
[18] J.S. Vrentas, R. Narayanan, S.S. Agrawal, Free surface convection in a bounded cylindrical geometry, Journal
of Heat and Mass Transfer, Vol. 24, No. 9, pp. 1513-1529, 1981.
[19] J.A. Maroto, V. Pérez-Munuzuri, M.S. Romero-Cano, Introductory analysis of the Bénard-Marangoni convection, European Journal of Physics, v28 n2 p311-320 Mar 2007.
[20] L. Hadji, Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in RayleighBénard-Marangoni convection, Physical Review E, Vol. 53, No 6, June 1996, pp. 5982-5992
[21] G. Lebon, A. Cloot, Buoyancy and Surface-Tension Driven Instabilities in Presence of Negative Rayleigh
and Marangoni Numbers, Acta Mechanica 43, 141-158, 1982.
[22] Surface tension of nickel, copper, iron and their binary alloys, J. Brillo, I. Egry, Proceedings of the IV
International Conference/High Temperature Capillarity, Journal of Materials Science 40, pp.22132216, 2005.
[23] Determination of the surface tension of liquid stainless steel, Proceedings of the IV International Conference/High Temperature Capillarity Journal of Materials Science, Z. Li, K. Mukai, M. Zeze, K.C. Mills, Vol.
40, 2191 2195, 2005.
[24] Influence of Oxygen Partial Pressure on Surface Tension of Molten Type 304 and 316 Stainless Steels Measured by Oscillating Droplet Method Using Electromagnetic Levitation, S. Ozawa, K. Morohoshi, T. Hibiya,
J.
[25] Y. Su , Z. Li, K. C. Mills Equation to estimate the surface tensions of stainless steels, Proceedings of the IV
International conference/high temperature capillarity, ISIJ International, Vol. 54, No. 9, pp. 2097-2103, 2014.
Journal of Material Science 40, pp 2201-2205, 2005.
[26] C. Y. Paik, S. J. Lee, J. H. Scobel, Q. Zhou, W. Luangdilok, R.W. Reeves, R. E. Henry, In-Vessel Retention
Modelling Capabilities in MAAP5, M. Plys, NEA/CSNI/ (2012)2, 2012.
[27] F.B. Cheung, J. B. Novas, Heat Transfer in an optically thick horizontal fluid layer heated from below, Letters
in Heat and Mass transfer, Vol. 7, pp. 171-181, 1980.
[28] Bénard-Marangoni convection at large low Prandtl number, T. Boeck, A. Thess, Journal of Fluid Mechanic,
Vol. 399, pp. 251-275, 1999.
[29] Bénard-Marangoni convection at large Marangoni numbers: Results of numerical simulations, T. Boeck,
Advances in Space Research, Vol. 36, 2005.
[30] T. Boeck, A. Thess, Power-law scaling in Bénard-Marangoni convection at large Prandtl numbers, Physical
Review E, Vol. 64, 2001.
[31] T. Boeck, A. Thess, Turbulent Bénard-Marangoni Convection: Results of Two-Dimensional Simulations,
Physical Review Letters, Vol.80, No. 6, 9/02/1998.
[32] G. Hagstrom, C.R. Doering, Bounds on heat transport in Bénard-Marangoni convection, Physical Review E,
Vol. 81, 2010.
[33] J.M. Bonnet and J.M. Seiler, In-vessel corium pool thermal-hydraulics for bounding cases, Proc. of salt test
expert group meeting RASPLAV seminar, Munich, Germany (2000).
[34] C. Le Guennic, L. Saas, R. Le Tellier, Y. Wuc, M. Guingo, J. Laviville, Contribution of CFD studies to IVR
assessment, The 8th European Review on Severe Accident Research, Warsaw, Poland, 16-18 may 2017.
[35] In-Vessel Melt Retention Severe Accident Management Strategy for Existing and Future NPPs, European
project H2020 IVMR, https://gforge.irsn.fr/gf/project/ivmr.
[36] D. Grischenko, Recent progress in the gasfilm levitation as a method for thermophysical properties measurements: application to ZrO2 − Al2 O − 3, High Temp-High Pressures, 40(2), pp.127-149, 2011.
Indicate here the SESSION name and the Paper Number (order as in the announcement)