improvement of molten jet fragmentation modeling … paper...*corresponding author email address:...

* Corresponding author

Email address: [email protected] (A.Le Belguet)

Improvement of molten jet fragmentation modeling in MAAP

A.Le Belgueta*, E.Beuzeta, M.Torkhania

a EDF Lab, 7 boulevard Gaspard Monge, 91120 Palaiseau, France

Abstract

In the framework of severe accident analysis, Molten Fuel-Coolant Interaction (MFCI) is a major issue of

concern for the safety analysis regarding ex-vessel melt retention and coolability. As the melt is relocated in the

reactor pit, it may interact as a high-temperature jet with the remaining water. The fragmentation of a corium jet

falling into water is of particular interest since it influences the accident progression and consequences.

MAAP (Modular Accident Analysis Program) is an integral code used to simulate severe accident sequences.

The present work aims at improving the jet fragmentation modeling in order to get a more accurate and reliable

prediction of the system pressurization and melt fragmentation as a corium jet is falling into water. To that end,

some developments have been provided in the code using MAAP5.02a version.

The developments carried out mainly focus on heat transfer between melt and the surrounding environment

and on the evaluation of the size of debris resulting from jet fragmentation. For instance, the drag coefficient has

been determined depending on turbulence and on water boiling regime. Besides, the vapor production rate has

been evaluated by taking into account the film boiling heat transfer between corium and water. Finally, a new

correlation has been implemented to calculate the size of debris resulting from jet fragmentation, before secondary

fragmentation occurs.

FARO L-28 and L-31 tests have been simulated with MAAP in order to highlight the impact of these

developments. These MFCI experiments, conducted with prototypical melt composition and under realistic

accident conditions, provide consistent information on the underlying phenomena. Computed pressure and water

temperature are underestimated using the new modeling, but in a lesser extent when evaluating vapor properties

at film temperature instead of saturation conditions. Nonetheless, the developments provide a more physical

modeling of jet fragmentation. Further improvements have been identified to complement those carried out so far.

Keywords: Jet fragmentation, MAAP code, Molten Fuel-Coolant Interaction,

Reactor safety analysis.

1. Introduction

During a hypothetical severe accident in a Pressurized Water Reactor (PWR), molten corium resulting from

core degradation can relocate in the reactor pit and interact with the remaining water. As corium jet penetrates

water, it undergoes instabilities, resulting in jet break-up and droplet formation. Besides, intense heat transfer

between the high-temperature melt and the surrounding water occurs, leading to a significant steam production

and to system pressurization. Therefore, the description of Molten Fuel-Coolant Interaction (MFCI) is required to

evaluate its consequences.

In the last thirty years, many experimental and theoretical studies have been performed on MFCI, including

jet fragmentation, to provide models and correlations and predict the accident progression [1]. However, some

uncertainties remain regarding the interpretation of experimental data and their applicability to reactor-scale

analysis [2]. Test conditions are not always representative of severe accidents in terms of masses, temperatures,

pressure and simulant or prototypic corium composition. Besides, MFCI modeling encounters difficulties due to a

lack of knowledge but also to the complexity of physics.

MAAP (Modular Accident Analysis Program) is an integral code developed by Fauske & Associates, LLC

which simulates an overall severe accident sequence [3]. EDF adapted it to French nuclear power plants and

actively contributes to its development. Jet fragmentation mechanism is taken into account in MAAP on the basis

of entrainment similarity assumption and with simplified heat transfer modeling. User-defined parameters related

to correlations for jet fragmentation, to void fraction, to debris size or to final particle temperature, are used.

mailto:alix.le-belguet@

The 8th European Review Meeting on Severe Accident Research - ERMSAR-2017

Warsaw, Poland, 16-18 May 2017

Ex-vessel corium interactions and coolability, Paper N°208

The present study aims at improving jet fragmentation modeling in MAAP. Some developments in MAAP

are proposed, using MAAP5.02-EDF version. They mainly focus on heat transfer between corium and the

surrounding environment and are related to:

- drag coefficient, initially defined as a constant in MAAP, now evaluated according to turbulence and

water boiling regime,

- vapor production rate, evaluated by taking into account film boiling heat transfer between corium and

water,

- size of debris resulting from jet fragmentation, before secondary fragmentation occurs, calculated with

a new correlation.

After describing each development, simulation of FARO L-28 and L-31 tests with MAAP is presented to

highlight the impact of the new modeling of jet fragmentation. If still required, prospects of improvements are

proposed for MAAP.

2. Two-phase flow and particle sedimentation

2.1. Two-phase flow: ambient properties

As the corium jet falls into water, MFCI occurs and contributes to both heat up and vaporize water.

Vaporization is all the more intense than water subcooling is small. Therefore, void fraction (αV) increases locally

in the interaction zone.

Physical properties of a two-phase mixture are defined according to void fraction and to monophasic

properties of liquid (subscript L) and vapor (subscript V), evaluated respectively at coolant (TL) and saturation

(Tsat) temperatures. Ambient density (ρa) and dynamic viscosity (μa) can be defined as:

ρa = αVρV(Tsat) + (1-αV)ρL(TL),

μa = αVμV (Tsat) + (1-αV)μL(TL).

In MAAP code, ρa and μa used to describe molten jet fragmentation and corium debris sedimentation

correspond to water properties. Thus, ambient properties accounting for the void fraction have been implemented

in MAAP. They are involved in the evaluation of jet fragmentation, depending on the correlation used as described

in [4], as well as the size and the sedimentation of corium debris.

2.2. Particle sedimentation: a new evaluation of the drag coefficient

During sedimentation of high-temperature corium debris in water, the drag force FD tends to slow down the

fall of particles and promote their cooling. FD is defined as:

Dp2paD CAUρ

2

1F

with ρa the ambient flow density defined above, Up the particle velocity, Ap the particle cross-sectional area and

CD the drag coefficient of particles.

The drag coefficient varies depending on the shape and size of the element (here a sphere), on the fluid in

which this element falls (here liquid water and/or steam), and on the flow regime (laminar or turbulent). When the

fall velocity is high, drag is reduced in a turbulent flow. In case of high-temperature particles falling in water,

boiling regime also impacts the drag coefficient. CD can thus be evaluated depending on:

- flow turbulence: CD varies as a function of Reynolds number, Re,

- water boiling regime: CD is different whether there is a vapor film around particles, i.e. if the particle

temperature, Tp, is greater than the minimum film boiling temperature, TMFB.

In MAAP code, the drag coefficient is involved in the evaluation of debris sedimentation velocity, and

consequently of debris bed coolability. It is defined as a constant with a default value fixed at 1.0.

The 8th European Review Meeting on Severe Accident Research -ERMSAR-2017



CD as a function of Re

According to literature [5], variation of CD as a function of Re has been established as shown in Figure 1.

Different regimes are identified:

- CD decreases according to Stokes’law at low Re (Re << 1): in this laminar regime, viscosity is

predominant and drag coefficient is high.

- CD continues to decrease for 0.7 < Re < 800, without following a simple law.

- CD is constant from Re ≈ 103: CD ≈ 0.44.

As Re reaches 3 103, a sharp decrease of CD is observed. It corresponds to the laminar-turbulent transition of

the boundary layer. The drag coefficient comes to a minimum, around 0.07. At high Reynolds numbers (Re > 106),

CD tends to an almost constant value of about 0.2.

Figure 1 Drag coefficient as a function of Reynolds number for a solid sphere in steady state [5]

CD as a function of particle superheat

A recent study examined the effect of a stable vapor film around a high-temperature sphere falling in water

[6]. It showed a significant decrease of drag coefficient, up to 75%, due to water film boiling. Some experiments

were performed for 4 104 < Re < 106, using hydrophilic and superhydrophobic spheres at 200°C. Results are shown

in Figure 2.

Figure 2 Dependence of drag coefficient on Reynolds number for steel spheres (10 to 40 mm diameter) falling in 95°C

water. Blue solid squares: hydrophilic spheres at 95°C, with no vapor film. Red solid circles: superhydrophobic spheres at

200 °C with a vapor film. Red open circles: 200 °C superhydrophobic spheres (extrapolated values). Blue crosses: literature

values for smooth solid spheres. [6]




We observe that:

- For hydrophilic spheres with no vapor film, measures are close to literature values: the laminar-turbulent

transition at Re ≈ 3 105 leads to a decreases of CD from 0.5 to 0.2.

- For super hydrophobic spheres, film boiling causes a gradual decrease of CD for Re > 4 104. A smaller

decrease is obtained for Re > 6 105, with a drag coefficient that stabilizes around 0.06.

- For Re < 4 104, it was shown in a previous study [7] that CD value is identical with and without film

boiling.

Figure 3 summarizes the new evaluation of CD proposed and implemented in MAAP. MAAP reproduces

correctly experimental results, either in the film boiling regime or not, as long as Re is lower than 106. Beyond that

value, no data enable to predict CD.

Figure 3 Drag coefficient as a function of Reynolds number and the water boiling regime for the MAAP code

The minimum film boiling temperature TMFB determines the film boiling regime onset. It can be evaluated

according to Henry’s correlation as defined, for example, in the ASTEC code:

2/1

coriump

Lp

LHNHNMFBcλρ

cλρTTTKT

,

Where λ is the thermal conductivity, ρ the density, cp the heat capacity at constant pressure. THN is the

homogeneous nucleation temperature expressed as :

39252HN p108193.5p103907.2p10722.444.70567.459

9

5KT

with PaP105038.146.3203p 5 .

3. Evaluation of vapor production rate

During MFCI, vapor production rate around particles can be evaluated according to heat lost by corium

debris, Qc, and heat transferred from liquid-vapor interface to coolant, Qcl. In MAAP code, modeling of heat

transfer and steam generation is based on a simplified (but sufficient) approach. For a more physically consistent

modeling, some improvements are suggested.

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1E+01 1E+02 1E+03 1E+04 1E+05 1E+06 1E+07

Dra

g c

oef

fici

ent,

CD

Reynolds number, Re

MAAP - No vapor film MAAP - With a vapor film

[6] No vapor film (exp.) [6] With a vapor film (extrapolation)

[6] Hydrophilic spheres-No vapor film [6] Superhydrophobic spheres-With a vapor film

[8] Exp. Results-No vapor film




3.1. Heat transfer in film boiling regime

Heat transfer between corium and the surroundings depends on water boiling regime which is mainly in film

boiling, particularly for low subcooling and high superheat. In that case, both forced convection in film boiling

regime, Qconv, and radiative heat transfer, Qrad, can be considered at the sphere surface, as follows:

Qc = Qconv + J Qrad, with 8

7J [9].

In MAAP, evaluation of Qc is based on corium enthalpy change from the jet temperature, Tj, to the final

particle temperature, Tp. Tp is determined mechanistically considering only radiation heat transfer [4].

Forced convection heat transfer can be evaluated based on Epstein-Hauser correlation, used in MC3D [10]:

4/14

EH

EH

2

4EHEHEH

2/1L

EHD3MC,EHA

B

π

2

CA24

1

β

Re

2

3CNu

,

with CEH = 2,LVV

supV,p

EHhPr

TΔcA , 2/1

L

LVV

subV,p

V

LEHEH Pr

hPr

TΔc

λ

λβB ,

2/1

L

V

2/1

V

LEH

μ

μ

ρ

ρβ

,

L

ppLL

Lμ

DUUρRe

and

L

L,pL

Lλ

cμPr .

Heat flux lost by corium particles is then expressed as:

D3MC,EHsatpmpVpartconv NuTTDπλNQ ,

where Npart is the number of particles resulting from jet fragmentation, Tpm is the average temperature of particles,

depending on Npart and on corium debris cooling, and Dp the particle diameter.

Liquid and vapor properties are evaluated at TL and Tsat respectively, whereas they should be calculated at

mean temperature: TLm = (TL + Tsat)/2 and TVm = (TL + Tj)/2, with Tj the corium jet temperature. Although this is

expected to have a significant impact on convective heat transfer, steam tables in MAAP do not allow to reach

such temperatures.

In the new modeling, radiative heat transfer is determined considering the heated surface and the liquid-vapor

interface as two plane-parallel plate at respectively Tpm and Tsat. Assuming that corium is an opaque and gray body,

that steam is transparent and water is a black body, Qrad is expressed as follows:

4sat

4pmC

2ppartrad TTσεDπNQ .

3.2. Heat partition and vapor production

Part of the heat lost by corium particles is used to produce vapor (Qcv) while the remaining part is used to

heat up water (Qcl). Fcv characterizes the heat partition and is defined as:

c

cl

c

cvcv

Q

Q1

Q

QF ,

with Qc = Qcv + Qcl.

In MAAP code, Qcl and Qc are respectively expressed as:

LsatL,pLcl TTcmQ ,

pjj,ppc TTcmQ ,




where Lm (resp. pm ) is the water (resp. particle) mass flow rate.

Qcl is usually described in literature by a forced convection correlation around a solid sphere at the

temperature Tsat. The Whitaker’s correlation has been implemented in MAAP to evaluate Qcl [11]:

4/1

sat,L

L4.0L

3/2L

2/1LCF

μ

μPrRe06.0Re4.02Nu

.

Besides, 90% of the radiative heat transfer from corium particles are devoted to heat up water, while the

remaining 10% are absorbed at the liquid-vapor interface to produce vapor [12]. Qcl is thus expressed as:

radCFsubLppartcl Q9.0NuTΔλDπNQ .

3.3. Vapor condensation

Vapor produced around corium particle flows upward in water. It condensates in the surroundings, in

proportion that depends on water subcooling, vapor superheat and flow rate. In MAAP code, vapor condensation

is considered through the use of the Jakob number, Ja, such as:

Ja1FF cv*cv , with

fgg

wsatw,pw

hρ

TTcρJa

and hfg the latent heat of vaporization.

If Ja is greater than 1, all vapor condensates. It takes into account the temperature variation of entrained water,

with no phase change. It is not consistent with vapor condensation through convective heat transfer between

possibly superheated vapor and water. Besides, Fcv calculation involves void fraction, whose value is a user-

defined parameter.

Therefore, vapor bubble condensation has been evaluated considering forced convection heat transfer around

a solid sphere, using Whitaker’s correlation, as defined above. The heat flux between vapor bubble and the

surroundings is given by:

LsatCFLbbcond TTNuλDπNQ ,

where Nb is the number of generated bubbles, determined from the vapor mass flow rate, and Db is the bubble

diameter, assumed to be the corium particle diameter. Regarding vapor velocity needed to evaluate the Reynolds

number, it corresponds to the terminal velocity of a bubble rising in water.

Fcv can then be evaluated considering vapor condensation in water:

c

condcl

c

condcvcv

Q

QQ1

Q

QQF .

4. A new evaluation of the size of corium particles resulting from jet fragmentation

In MAAP code, the particle diameter, Dp, is assumed to be at least:

- the capillary length in case of gravity pour, Dp,g,

- the stable particle diameter, Dp,We, defined as a function of the critical Weber number in case of a

hydrodynamic fragmentation,

such that Dp is expressed as follows:

2jw

j

DpWe,p

2/1

j

j

Dpg,ppUρ

σWeFD,

ρg

σF2DminD




where FDp is a user-defined parameter to adjust particle diameter, σj the surface tension and We* the critical Weber

number. Classically, for a droplet in a liquid flow, We* is set at about 10-12 in vapor explosion codes and related

literature.

A new correlation has been implemented in MAAP to evaluate the size of corium particles resulting from

primary jet fragmentation. As molten jet penetrates water, corium particles are stripped from the jet. In the

Namiech’s model [13], attention has been focused on the erosion process where particles are part of the counter-

current vapor flow surrounding the jet. Particle diameter only accounts for primary fragmentation, without

considering secondary fragmentation. This can be indeed neglected in first approximation, but only for the

premixing phase, which is the scope of MAAP modeling, given the low coolant velocity.

Particle diameter can be evaluated from the derived correlation:

Dp

Dj,0

=33.272 (P

P0

)-0.958

(1-Tsat-TL

Tj-Tsat

)

0.484

(Uj

Uj,0

)

-1.485

(Uj

Um,0

)

1.591

(Um,0D

j,0

νV

)

-0.035

(σj

ρjDj,0Um,0

2)

0.065

with P0 = 1 bar, Uj,0 = 5 m.s-1, 0,j

V

L0,m gD

ρ

ρU . νV is the vapor cinematic viscosity. Let us mention that vapor

properties, ρV and νV, are evaluated at the jet temperature, Tj, as described in [13].

5. FARO tests simulation

5.1. FARO experimental program

The large-scale FARO experimental program was conducted to investigate MFCI under both in- and ex-

vessel severe accident conditions. The test facility is composed of a furnace where melt is produced by direct

electrical heating, intersection valves, a release vessel located above an interaction test section, and a venting

system that regulates pressure.

A FARO test consists in pouring by gravity a significative amount of corium melt into a pool of saturated or

subcooled water [14]. The main quantities measured during a test are pressures and temperatures in the gas region,

in the water pool and in the debris catcher bottom plate. Post-test debris analysis provides particle size distribution

and unfragmented melt mass.

Table 1 Main experimental conditions in FARO L-28 and L-31 tests

Unit L-28 L-31

Corium composition - 80 wt% UO2 + 20 wt% ZrO2

Melt temperature (Tj) K 3052 3003

Melt mass poured (mj) kg 175 92

Initial jet diameter (Dj,0) mm 50 50

Gas temperature (TV) K 465 300

Initial pressure (P0) bar 5.1 2.2

Melt fall height in gas phase (hg) m 0.89 0.77

Freeboard volume (VV) m3 3.53 3.49

Water mass (mL) kg 517 481

Water temperature (TL) K 423.7 290.8

Water subcooling (ΔTsub) K 2 105.9

Water depth (hL) m 1.44 1.45

Test section diameter (DL) m 0.71 0.71

Among FARO tests, L-28 and L-31 tests have been selected to evaluate the developments carried out on jet

fragmentation modeling in MAAP and quantify their impact on system pressurization in case of MFCI. These are

the only tests for which the jet break-up length has been measured. Simulation results are discussed in paragraph

5.2 and 5.3. Main experimental conditions are summarized in Table 1 [15]. It is worth mentioning that contrary to




L-28 test carried out in saturated conditions (ΔTSub = 2 K), L-31 test was performed with subcooled water (ΔTSub

= 105.9 K) and a lower corium mass (92 kg against 175 kg in L-28 test) (Table 1).

5.2. FARO L-28 test simulation

A first FARO L-28 simulation, regarded as the reference calculation, is performed without modification and

compared to experimental data. Pressure is 17% higher than the experimental value (Figure 4), resulting in an

overestimation of gas temperature of about 75 K (Figure 5) and of water temperature, but to a lesser extent (Figure

6). The computed final particle size (Figure 7) is in good agreement with the mean diameter of collected debris in

L-28 test. This is consistent with the fact that the user-defined parameter, FDp, has been adjusted to FARO

experiments.

When considering developments suggested above, pressure is largely underestimated, of about 44% (Figure

4), as well as water temperature (about 32 K, Figure 6). Besides, particle size, evaluated according to Namiech’s

correlation, is of the correct order of magnitude, although slightly overestimated. The particle mean diameter is

about 20% higher than in the experiment, which can be explained by secondary fragmentation that may occur in

the experiments, not accounted for in Namiech’s correlation. Taking secondary fragmentation into account may

lead to a better prediction of the debris size, as well as a higher and thus more consistent pressure level.

As mentionned in paragraph 3, vapor properties involved in heat transfer calculation are evaluated at

saturation temperature while vapor film temperature around corium particles is much higher. It can be

approximated by:

2

TTT

satpm

Vm

.

As vapor properties have a significant impact on convective heat transfer, some have been evaluated at TVm

by extrapolation, as steam tables in MAAP are not applicable at such temperature range: the specific enthalpy, hV,

and the density, ρV. The thermal conductivity, λV, and the specific heat capacity, cp,V, are also considered at TVm.

Computation results show that pressure and water temperature predictions are improved, although still

underestimated. This highlights the importance of having steam tables valid for a wider temperature range in

MAAP.

Figure 4 Pressure: comparison of L-28 test results to MAAP simulations with and without modifications

5,0E+05

8,0E+05

1,1E+06

1,4E+06

1,7E+06

2,0E+06

0 1 2 3 4 5 6 7 8

Pre

ssu

re (

Pa)

Time (s)

Exp. data

Reference

Modified

Modified+Vapor

Properties




Figure 5 Gas temperature: comparison of L-28 test results to MAAP simulations with and without modifications

Figure 6 Water temperature: comparison of L-28 test results to MAAP simulations with and without modifications

Figure 7 Particle diameter: comparison of L-28 test results to MAAP simulations with modifications

5.3. FARO L-31 test simulation

The same approach as for FARO L-28 test is followed for L-31 test. In the reference calculation, carried out

without considering modifications of jet fragmentation modeling, pressure is predicted within the correct order of

magnitude compared to experimental results (Figure 8), although slightly underestimated. Heat transfer from

corium to the surroundings is exclusively dedicated to heat up water and no steam is produced during melt

relocation. This is consistent with water subcooling degree which remains high on average all along the test

(ΔTSub,L-31 = 105.9 K, Table 1 and Figure 9). As obtained in L-28 test results, the particle diameter is well calculated

in MAAP thanks to a suitable choice of FDp parameter (Figure 10) [4].

When considering the developments suggested above, no change in pressure (Figure 8) nor in gas temperature

(Figure 9) is observed. However, prediction of water temperature is improved (Figure 10). Besides, Namiech’s

correlation slightly overestimates corium debris size (Figure 11). A sensitivity study using a multiplying factor in

Namiech correlation could be performed to evaluate the impact of debris size on system pressurisation.

400

450

500

550

600

0 1 2 3 4 5 6 7 8

Gas

tem

per

atu

re (

K)

Time (s)

Exp. data

Reference

Modified

Modified+Vapor

Properties

400

425

450

475

500

0 1 2 3 4 5 6 7 8

Wate

r te

mp

eratu

re

(K)

Time (s)

Exp. data

Reference

Modified

Modified+Vapor

properties

0,0E+00

1,0E-03

2,0E-03

3,0E-03

4,0E-03

5,0E-03

0 1 2 3 4 5 6 7 8

Part

icle

dia

met

er (

m)

Time (s)

Exp. data

(mean D)

Reference

Modified




Figure 8 Pressure: comparison of L-31 test results to MAAP simulations with and without modifications

Figure 9 Gas temperature: comparison of L-31 test results to MAAP simulations with and without modifications

Figure 10 Water temperature: comparison of L-31 test results to MAAP simulations with and without modifications

Figure 11 Particle diameter: comparison of L-31 test results to MAAP simulations with modifications

1,8E+05

2,0E+05

2,2E+05

2,4E+05

2,6E+05

2,8E+05

0 1 2 3 4 5 6

Pre

ssu

re (

Pa)

Time (s)

Exp. Data

Reference

Modified

280

320

360

400

440

480

0 1 2 3 4 5 6

Gas

tem

per

atu

re (

K)

Time (s)

Exp. Data

Reference

Modified

280

300

320

340

360

380

0 1 2 3 4 5 6

Wate

r te

mp

eratu

re

(K)

Time (s)

Exp. Data

Reference

Modified

0,0E+00

1,0E-03

2,0E-03

3,0E-03

4,0E-03

5,0E-03

0 1 2 3 4 5 6

Part

icle

dia

met

er (

m)

Time (s)

Exp. data

Reference

Modified




6. Conclusion

Developments have been carried out in MAAP5.02-EDF version to improve jet fragmentation modeling, and

more particularly:

- the sedimentation of corium debris in two-phase flow: ambient properties are considered and a drag

coefficient depending on turbulence and on water boiling regime is calculated,

- heat transfer between melt and the surroundings: vapor production rate is evaluated by taking into

account film boiling heat transfer between corium and water, as well as steam condensation with a more

coherent formulation of Fcv,

- debris size, with the implementation of Namiech’s correlation, that describes primary jet fragmentation

with no adjustment parameter nor arbitrary critical Weber number, thus giving more physical consistency

to MAAP modeling.

FARO L-28 and L-31 simulations with MAAP5.02-EDF version have shown the impact of these

developments. Even though computed pressure and water temperature are underestimated, the new modeling is

improved when considering steam properties at a more physical temperature, i.e. the vapor film temperature,

instead of saturation conditions. It then highlights the need for steam tables in MAAP to be valid for a wider

temperature range.

Nonetheless, these developments provide a more physical modeling of jet fragmentation while avoiding the

use of user-defined parameters. Further improvements have been identified to complement those carried out so far

such as extending steam tables validity in MAAP, or improving debris size and corium oxydation modeling.




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