Melt jet-breakup and fragmentation phenomena in nuclear reactors: A review of experimental works and solidification effects

Progress in Nuclear Energy 108 (2018) 188–203

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Progress in Nuclear Energy
journal homepage: www.elsevier.com/locate/pnucene

Review

Melt jet-breakup and fragmentation phenomena in nuclear reactors: A
review of experimental works and solidification effects

T

Yuzuru Iwasawaa,∗, Yutaka Abeb
a
b

Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8573, Japan
Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki, 305-8573, Japan

A R T I C LE I N FO

A B S T R A C T

Keywords:
Nuclear reactor
Severe accident
Fuel-coolant interaction
Jet-breakup
Fragmentation

Solidification effects

During severe accidents at Nuclear Power Plants (NPPs), fuel-coolant interaction (FCI) is a critical event in which
the melt released from the core region comes into contact with the coolant. The melt may eject in the form of a
melt jet and threaten the integrity of the NPP. Therefore, fragmentation of the melt jet and quenching of particulate fragments from the melt jet are invaluable from the viewpoint of safety assessment. To assess the integrity of an NPP, melt fragmentation phenomena that affects quenching and sustainable cooling of the debris
bed are important factors that must be predicted and evaluated precisely. The present review summarizes experimental works on the FCI phenomenon, especially, fragmentation of a melt jet during a severe accident in an
NPP. In addition, special attention is paid to solidification effects. Based on the literature survey, we discussed
the dominant factors governing the fragmentation mechanisms. Furthermore, we discuss the applicability of
various models for estimating these phenomena.

1. Introduction
For stabilization and termination of a severe accident in a Nuclear
Power Plants (NPP), investigating the risks and the progression of the
severe accident is important (Sehgal, 2006, 2012). During a severe
accident in an NPPs, fuel-coolant interaction (FCI), critical event in
which the melt released from the core region comes into contact with
the coolant, needs to be assessed for ensuring NPP integrity. The melt
may be injected in the form of a melt jet and threaten the integrity of
NPPs such as Light Water Reactors (LWRs) (Ma et al., 2016; Sehgal and
Bechta, 2016) and Sodium-cooled Fast Reactors (SFRs) (Suzuki et al.,
2014; Tobita et al., 2016). Therefore, fragmentation of the melt jet
(called as the jet-breakup), which means a coherent jet disappears in
this paper, and quenching of the particulate fragments from the melt jet
are invaluable from the viewpoint of safety assessment. For the safety
assessment, it is important to predict and evaluate precisely the characteristic values of the jet-breakup and the fragmentation phenomena
that affects the quenching and sustainable cooling of the debris bed
(Dinh et al., 1999).
If a melt jet directly hits the internal structures without jet-breakup,
it may threaten the integrity of a reactor vessel and, consequently,
threaten the integrity of a containment vessel. Hence, the jet-breakup

length, which refers to as the distance from the liquid (coolant) surface
to the location where a coherent melt jet disappears (Chu et al., 1995;

∗

Matsuo et al., 2008; Iwasawa et al., 2015a; Li et al., 2017), is important.
In addition, fine fragmentation of a melt jet may lead to vapor explosion, which threaten the integrity of the NPP. Even if vapor explosion
does not occur, molten fragments may threaten the integrity of the NPP,
when they directly hit the internal structures without quenching. In
addition, fine fragmentation of a melt affects debris bed formation and
decay heat removal. Therefore, it is important to estimate and evaluate
fragment size from the viewpoint of safety assessment.
FCI phenomena of a melt jet, such as jet-breakup and fragmentation
is known to be complex, mainly because of two interactions that occur
simultaneously: hydrodynamic (e.g., interfacial instability at two-phase
interface and liquid entrainment or stripping from interface), and
thermal (e.g., coolant boiling and solidification of a melt surface) (Chu
et al., 1995; Sugiyama et al., 1999; Nishimura et al., 2010; Manickam
et al., 2017). Many experiments have been carried out using various
combinations of melt and coolant. In addition to large-scale experiments using actual fuels, scoping experiments focusing on the fundamental processes of the FCI phenomena have also been carried out to
investigate each interaction and the dominant factors governing the FCI
phenomena.
The present review article summarizes experimental works on the
FCI phenomena, especially, jet-breakup and fragmentation of a melt jet
during a severe accident in NPPs. In addition, special attention is paid
to solidification effects. Based on the literature survey, this article

Corresponding author.
E-mail address: (Y. Iwasawa).

/>Received 6 August 2017; Received in revised form 21 March 2018; Accepted 11 May 2018

Available online 05 June 2018
0149-1970/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
( />

Progress in Nuclear Energy 108 (2018) 188–203

Y. Iwasawa, Y. Abe

Nomenclature

Greek symbol

d
D
Dj
E
E0
Fr
g
k
Lbrk
P
t
Ti
ΔTc
u
U
v

V
vj
vrel
x
y

δ
ε
γt
γy
ϕ
η
κ
λ
λn
λm
ρ
σ
ω

characteristic length
bending stiffness
jet diameter
Young's modulus
entrainment coefficient
Froude number
gravitational acceleration
wave number
jet-breakup length
pressure

time
initial interfacial temperature
coolant subcooling
fluctuating velocity in horizontal direction
uniform velocity in horizontal direction
fluctuating velocity in vertical direction
uniform velocity in vertical direction
jet velocity
relative velocity
horizontal direction
vertical direction

crust thickness
Poisson's ratio
temporal growth rate
spatial growth rate
velocity potential
displacement of interface
thermal conductivity
wavelength
neutral-stable wavelength
most-unstable wavelength
density
interfacial tension or surface tension
angular frequency

Subscript
0
j, 1
c, 2

s
w

initial
melt jet
coolant
sodium
water

2. Previous experiments on FCI phenomena

discusses dominant the factors governing jet-breakup and fragmentation. Furthermore, this article discusses the applicability of various
models for estimating these phenomena.
The remainder of this is organized as follows. In Chapter 2, previous experiments on FCI, including jet-breakup and fragmentation, are
reviewed and summarized. These review and summary are presented in
terms of melt and coolant composition. In Chapter 3, the dominant
factors governing the jet-breakup of a melt jet are discussed based on
the literature survey. In addition, existing models for estimating the jetbreakup length are presented based on their applicability. In Chapter 4,
the dominant factors governing the fragmentation of a melt jet are
discussed based on the literature survey. In addition, existing models
for estimating the fragment size are presented based on their applicability. In Chapter 5, the solidification effects of the FCI phenomena
are reviewed and summarized. A model for estimating the fragment size
considering the solidification effects is presented. Chapter 6 concludes
this article.

The following sections will summarize the previous experiments on
the FCI phenomena in terms of melt and coolant composition. Given
that the focus of this review article is on the jet-breakup and the fragmentation phenomena of a melt jet, the experiments considered herein
are those involving injected melts weighing several hundred grams to
several hundred kilograms (injected melt jets not melt droplets).

2.1. Oxide/sodium system
This section summarizes the previous experiments on the FCI phenomena involving oxide melt and sodium, which mainly target SFRs.
They are summarized in Table 1.
In the M-Series experiments conducted in the Argonne National
Laboratories (ANL) (Johnson et al., 1975; Sowa et al., 1979) and the
FLAG experiments conducted in the Sandia National Laboratories (SNL)
(Chu, 1982), uranium oxide melt was injected, with a focus on vapor
explosion. Zagorul'ko et al. (2008) conducted the experiment using the

Table 1
Previous experiments on FCI phenomena conducted using oxide/sodium system.
Organization
(Test facility or program)

Melt/Coolant

References

ANL
(M-Series)
SNL
(FLAG)
JRC
(BETULLA)

UO2-Mo, UO2-ZrO2-SS/
Sodium
Fe-Al2O3, UO2-ZrO2-SS/
Sodium

UO2, Al2O3/Sodium

Johnson et al. (1975)
Sowa et al. (1979)
Chu (1982)

JRC

UO2/Sodium

(FARO/TERMOS)
JAEA
(FR Tests)
IPPE
(Pluton)

Al2O3/Sodium
ZrO2- Fe/Sodium

189

Holtbecker et al. (1977)
Schins (1984)
Schins et al. (1984, 1986)
Schins and Gunnerson (1986)
Magallon et al. (1992)
Matsuba et al. (2012, 2015a,
2015b, 2016)
Zagorul'ko et al. (2008)

Progress in Nuclear Energy 108 (2018) 188–203

Y. Iwasawa, Y. Abe

installation at IPPE. The ZREX experiments conducted at ANL (Cho
et al., 1997, 1998) focused on hydrogen production upon the injection
of zirconia melt. The experiments conducted at the KROTOS facility in
JRC (Hohmann et al., 1995; Huhtiniemi et al., 1997a, 1997b;
Huhtiniemi and Magallon, 2001), those at FARO/TRERMOS and FARO/
FAT facility in JRC (Magallon and Hohmann, 1995; Magallon et al.,
1997, 1999; Magallon and Huhtiniemi, 2001), and the PREMIX experiments conducted in the FZK (Huber et al., 1996; Schütz et al., 1997;
Kaiser et al., 1997, 1999, 2001) involved injecting uranium oxide-zirconia (so-called corium) or alumina melt. The results of these experiments provided an important database and significant knowledge on
FCI phenomena in actual reactors. JAERI conducted a series of experiments called GPM (Moriyama et al., 2005), which involved injecting alumina-zirconia and stainless-carbon melts. Moriyama et al.
(2005) investigated a method for estimating the jet-breakup length and
fragment size. The MIRA experiments conducted at Royal Institute of
Technology (KTH) (Haraldsson and Sehgal, 1999; Haraldsson, 2000)
involved injecting various oxide melts. The DEFOR experiments
(Kudinov et al., 2008, 2010; 2013, 2015; Karbojian et al., 2009) were
also conducted at KTH. In this experiment, various oxide melts were
also injected to investigate the agglomeration of particulate fragments
from a melt jet. The French Alternative Energies and Atomic Energy
Commission (CEA) conducted experiments at the KROTOS facility, and
the TROI facility, which is in the Korea Atomic Energy Research Institute (KAIRI), under the OECD/NEA SERENA program. This program
focused on vapor explosion (Hong et al., 2013). Also, the Access to
Large Infrastructures for Severe Accidents (ALISA) project between
European and Chinese research institutions in the area of severe accident research is underway (Cassiaut-Louis et al., 2017). In this program,
the experiments for the study of FCI was conducted using the KROTOS
facility on the PULINIUS platform (Bouyer et al., 2015). In KAIRI, the
experiments at the TROI facility (Park et al., 2001, 2008, 2013; Song

et al., 2002a, 2002b, 2003a, 2003b, 2016, 2017; Kim et al., 2003, 2004,
2005, 2008, 2011; Song and Kim, 2005; Hong et al., 2013, 2015, 2016;
Na et al., 2014, 2016) involved injecting corium melt to investigate
vapor explosions. Recently, this research group also focused on InVessel Corium Retention External Reactor Cooling (IVR-ERVC). Na
et al. (2014, 2016), Hong et al. (2016), and Song et al. (2017) conducted experiments in which corium melt was injected without free fall.
Furthermore, experiments conducted at the MISTEE-jet and the JEBRA
facility in KTH involved injecting oxide and metallic melt (Manickam
et al., 2014, 2016, 2017). This research group discussed the difference
in fragmentation between oxide melt and metallic melt.

melt of thermite mixture in the Pluton test facility at the Institute for
Physics and Power Engineering (IPPE). In the experiments conducted at
the BETULLA facility in the Joint Research Centre (JRC) (Holtbecker
et al., 1977; Schins, 1984; Schins et al., 1984, 1986; Schins and
Gunnerson, 1986), uranium oxide and alumina melts were injected. The
JRC-based research group discussed differences in the fragmentation
phenomena between oxide melt and metallic melt into sodium. In the
large-scale experiments conducted at the FARO/TERMOS facility in
JRC (Magallon et al., 1992), 100 kg of uranium oxide was injected.
Because the conditions and the scale of these experiments were comparable to those of an actual SFR undergoing a severe accident, Suzuki
et al. (2014) referred to this experiment to evaluate safety assessment
procedures. Recently, Japan Atomic Energy Agency (JAEA) conducted
experiments involving the injection of alumina melt into sodium as FR
tests (Matsuba et al., 2012, 2015a, 2015b, 2016) to develop design
criteria for next-generation SFRs (Ichimiya et al., 2007; Kotake et al.,
2010; Aoto et al., 2011).
2.2. Metal/sodium system
This section summarizes the previous experiments on the FCI phenomena conducted using metallic melt and sodium, which mainly
target SFRs. They are listed in Table 2.
In the experiments conducted at the BETULLA facility in JRC (Benz

and Schins, 1982; Schins, 1984; Schins and Gunnerson, 1986; Schins
et al., 1986), stainless steel and copper melts were injected. The ANL
conducted the experiments in which they injected metallic melt into
sodium (Gabor et al., 1988). Gabor et al. (1988) visualized the fragments at the bottom of the test section using a radiograph. Central
Research Institute of Electric Power Industry (CRIEPI) conducted an
experiment involving metallic fuels (Nishimura et al., 2002, 2005,
2010) for SFRs. They focused on how the solidification effects influence
the FCI phenomena. JAEA conducted an experiment in which aluminum melt was injected into sodium (Matsuba et al., 2016). The melt
jet in sodium was visualized using X-rays.
2.3. Oxide/water system
This section summarizes the previous experiments on the FCI phenomena conducted using oxide melt and water, which mainly target
LWRs. They are listed in Table 3.
The FITS experiments conducted at SNL (Mitchell et al., 1981;
Corradini, 1981), the CCM experiments conducted at ANL (Spencer
et al., 1994), the MIXA experiments conducted at United Kingdom
Atomic Energy Authority (UKAEA) (Denham et al., 1994), the ALPHA
program at Japan Atomic Energy Research Institute (JAERI) (Yamano
et al., 1995), and the ECO experiment conducted at Forschungszentrum
Karlsruhe (FZK) (Cherdron et al., 2005) involved injecting oxide melt
by means of the thermite reaction. Zagorul'ko et al. (2008) conducted
the experiment using the melt of thermite mixture in the TVMT

2.4. Metal/water system
This section summarizes the previous experiments on the FCI phenomena conducted using metallic melt and water coolant, which
mainly target LWRs or SFRs. They are listed in Table 4.
These experiments are focused on fundamental processes of the FCI

Table 2
Previous experiments on FCI phenomena conducted using metal/sodium system.
Organization (Test facility or

program)

Melt/Coolant

References

JRC (BETULLA)

SS, Cu/Sodium

ANL
CRIEPI

U, U-Zr, U-Fe/
Sodium
Ag, Cu/Sodium

Benz and Schins (1982)
Schins (1984)
Schins and Gunnerson
(1986)
Schins et al. (1986)
Gabor et al. (1988)

JAEA

Al/Sodium

190

Nishimura et al. (2002,
2005, 2010)
Matsuba et al. (2016)


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Y. Iwasawa, Y. Abe

Table 3
Previous experiments on FCI phenomena conducted using oxide/water system.
Organization
(Test facility or program)

Melt/Coolant

References

SNL

Al2O3-Fe/Water
UO2-Mo/Water

Mitchell et al. (1981)
Corradini (1981)
Denham et al. (1994)

UO2-ZrO2-SS/Water

Spencer et al. (1994)

Al2O3-FeO,
Al2O3-Fe2O3/Water
ZrO2, Zr/Water

Yamano et al. (1995)

(FITS)
UKAEA
(MIXA)
ANL
(CCM)
JAERI
(ALPHA)
ANL
(ZREX)
JRC/CEA
(KROTOS)

JRC

Cho et al. (1997, 1998)

Al2O3, UO2-ZrO2/Water

Al2O3-Fe/Water

Hohmann et al. (1995)
Huhtiniemi et al. (1997a, 1997b)
Huhtiniemi and Magallon (2001)

Hong et al. (2013)
Bouyer et al. (2015)
Cassiaut-Louis et al. (2017)
Magallon and Hohmann (1995)
Magallon and Huhtiniemi (2001)
Magallon et al. (1997, 1999)
Magallon (2006)
Huber et al. (1996)
Schütz et al. (1997)
Kaiser et al. (1997, 1999, 2001)
Cherdron et al. (2005)

Al2O3-ZrO2, SS-C/Water

Moriyama et al. (2005)

CaO-B2O3, MnO2-TiO2,
WO3-CaO/Water
ZrO2- Fe/Sodium

Haraldsson and Sehgal (1999)
Haraldsson (2000)
Zagorul'ko et al. (2008)

CaO-B2O3, WO3-CaO,
MnO-TiO2, WO3-TiO2,
Bi2O3-TiO2, Bi2O3-CaO,
Bi2O3-WO3, WO3-ZrO2
/Water
UO2-ZrO2-Zr, UO2-ZrO2, ZrO2-Zr,

ZrO2, Al2O3
/Water

Kudinov et al. (2008, 2010, 2013, 2015)
Karbojian et al. (2009)

UO2-ZrO2/Water
(FARO/TERMOS, FAT)

FZK

Al2O3-Fe/Water
(PREMIX)

FZK
(ECO)
JAERI
(GPM)
KTH
(MIRA)
IPPE
(Pluton)
KTH
(DEFOR)

KAERI
(TROI)

KTH
(MISTEE-jet/JEBRA)

WO3-Bi2O3, WO3-ZrO2
/Water

Park et al. (2001, 2008, 2013)
Song et al. (2002a, 2002b, 2003a, 2003b,
2016, 2017)
Kim et al. (2003, 2004, 2005, 2008, 2011)
Kim et al. (2008, 2011)
Song and Kim (2005)
Hong et al. (2013, 2015, 2016)
Na et al. (2014, 2016)
Manickam et al. (2014, 2016, 2017)

thermocouples on the central axis of the nozzle. The experiment conducted at Chongqing University (CH) focused on vapor explosions (Lu
et al., 2016). The experiment conducted at Indira Gandhi Centre for
Atomic Research (IGCAR) (Mathai et al., 2015) and that conducted at
Indian Institute of Technology (IIT) (Pillai et al., 2016) focused on
agglomeration. Recently, an experiment was conducted at Tokyo Institute of Technology (TIT) using simulant metal to develop a method
for sealing NPPs (Takahashi et al., 2015; Secareanu et al., 2016). The
experiments conducted at University of Tokyo used simulant metal to
investigate the effects of internal structures such as Control Rod Guide
Tubes in Boiling Water Reactors (BWRs) on the jet-breakup and the
fragmentation phenomena (Wei et al., 2016).

phenomena (Spencer et al., 1986; Cho et al., 1991; Schins et al., 1992;
Hall and Fletcher, 1995; Dinh et al., 1999; Haraldsson, 2000). A few
research groups conducted experiments involving visualizing a melt jet.
Hall and Fletcher (1995) conducted the experiment of single nozzle and
multi nozzle geometry at Berkeley Technology Centre (BNL). Moreover,

the experiments conducted at the ANL (Gabor et al., 1992, 1994),
JAERI (Sugiyama et al., 1999; Sugiyama and Yamada, 2000; Sugiyama
and Iguchi, 2002), Korean Maritime University (KMU) (Bang et al.,
2003; Kim and Bang, 2016; Bang and Kim, 2017), University of Tsukuba (UT) (Abe et al., 2004, 2005; 2006; Matsuo et al., 2008; Iwasawa
et al., 2015a, 2015b), JAEA (Matsuba et al., 2013), KTH (Manickam
et al., 2014, 2017), and Shanghai Jiao Tong University (SJTU) (Li et al.,
2017) measured the fragment size and shape. Experiments conducted
from several viewpoints are summarized in this section. The experiment
conducted at Power Reactor and Nuclear Fuel Development Corporation (PNC) using the MELT-II facility (Kondo et al., 1995) and the experiment conducted at Pohang University of Science and Technology
(POSTEC) (Jung et al., 2016) focused on vapor generation around a
melt jet. Matsuba et al. (2013) conducted the experiments focused on
the fundamental processes using up to 400 kg melt. They measured
temperature distribution along a column of melt in water by installing

2.5. Experiment with other simulant materials
This section summarizes the previous experiments on the FCI phenomena using other simulant materials, mainly targeting LWRs or SFRs.
They are listed in Table 5.
The experiments conducted at PNC using the JET-I facility (Saito
et al., 1998) and the experiment conducted at ANL under the MFSBS
program (Schneider et al., 1992; Schneider, 1995) used volatile
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Y. Iwasawa, Y. Abe

Table 4
Previous experiments on FCI phenomena conducted using metal/water system.
Organization

(Test facility or program)

Melt/Coolant

References

ANL
IKE

Wood’ metal/Water
Wood’ metal/Water

JRC
ANL
BNL
PNC
(MELT-II)
KTH

Wood’ metal/Water
Al, Al-U/Water
Tin/Water
Wood’ metal/Water

Spencer et al. (1986)
Cho et al. (1991)
Berg et al. (1994)
Bürger et al. (1995)
Schins et al. (1992)
Gabor et al. (1992, 1994)

Hall and Fletcher (1995)
Kondo et al. (1995)

JAERI

Zn, Sn, Cu/Water

KMU
KMU
(COLDJET)
UT

Wood’ metal/Water
Wood’ metal/Water

JAEA
IGCAR
CH
IIT

Wood’ metal/Water
Wood’ metal/Water
Al, Pb, Bi/Water
Pb, Al/Water,
Glycerol
Wood’ metal, Ga/
Water
Wood’ metal/Water
Wood’ metal/Water

TIT
UT
MATE
(POSTEC)
KTH
(MISTEE-jet/JEBRA)
SJTU
(METRIC)

Wood’ metal/Water

Dinh et al. (1999)
Haraldsson (2000)
Sugiyama et al. (1999)
Sugiyama and Yamada (2000)
Sugiyama and Iguchi (2002)
Bang et al. (2003)
Kim and Bang (2016)
Bang and Kim (2017)
Abe et al. (2004, 2005, 2006)
Matsuo et al. (2008)
Iwasawa et al. (2015a, 2015b)
Matsuba et al. (2013)
Mathai et al. (2015)
Lu et al. (2016)
Pillai et al. (2016)

Wood’ metal/Water

Takahashi et al. (2015)

Secareanu et al. (2016)
Wei et al. (2016)
Jung et al. (2016)

Sn, Wood’ metal/
Water
Sn/Water

Manickam et al. (2014, 2017)
Li et al. (2017)

Table 5
Previous experiments on FCI phenomena using experiment other simulant materials.
Organization
(Test facility or program)

Melt/Coolant

References

PNC
(JET-I)
ANL
(MFSBS)
KTH

Water/Nitrogen, Freon

Saito et al. (1988)

Wood’ metal/Freon

Schneider et al. (1992)
Schneider (1995)
Dinh et al. (1999)

Water, Salt, Wood’ metal/
Paraffin oil, Salt

coolants such as nitrogen and freon. They investigated the effects of
vapor generation on the jet-breakup and the fragmentation phenomena.
Dinh et al. (1999) conducted an experiment to investigate the effects of
several variables such as physical properties and phase-change heat
transfer. Saito et al. (1988) proposed a semi-empirical correlation for
estimating the jet-breakup length. The details of the semi-empirical
correlation will be described in the next section.

and Fauske (2001) proposed these correlations, which are representative and widely used.
Based on experimental results, Saito et al. (1988) pointed out that the
important factors governing jet-breakup were the force balance among the
inertia forces of a melt jet, buoyancy force owing to density difference, and
the forces resulting from hydrodynamic and thermal interactions. Then,
they proposed the following semi-empirical correlation:

3. Jet-breakup phenomena

ρj 0.5
Lbrk
= 2.1 ⎜⎛ ⎟⎞ Fr 0.5
Dj

⎝ ρc ⎠

The following sections summarize the jet-breakup length obtained
from the previous experiments. The obtained values will be compared
with those obtained using existing methods, and the dominant factors
governing the jet-breakup phenomena will be discussed.

Fr =

(1)

vj2
gDj

(2)

where Lbrk denotes the jet-breakup length, Dj denotes the inlet diameter of
a melt jet, vj denotes the velocity of a melt jet, ρ denotes the density of
fluid, and Fr denotes the Froude number defined by Saito et al. (1988),
respectively. Subscripts j and c denote the jet and the coolant, respectively.
Epstein and Fauske (2001) proposed a semi-empirical correlation

3.1. Existing models and methods
This section briefly presents the existing methods for estimating the
jet-breakup length and the fragment size. Saito et al. (1988) and Epstein
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Y. Iwasawa, Y. Abe

Fig. 1. Normalized jet-breakup length obtained from previous experiments are organized by Froude number Fr. The values obtained using the correlations proposed
by Saito et al. (1998) and Epstein and Fauske (2001) are compared with the measured jet-breakup length.

based on previous works by Ricou and Splading (1961). Epstein and
Fauske (2001) introduced the tuning parameter E0 for adjusting the
difference between simple modeling and actual phenomena. Their
correlation is given by below:

that the correlation proposed by Saito et al. (1988) depends on vj. By
contrast, the correlation proposed by Epstein and Fauske (2001) is independent of vj.

0.5
Lbrk
1 ⎛ ρj ⎞
=
⎜
⎟
Dj
2E0 ⎝ ρc ⎠

3.2. Experimental data on jet-breakup length
(3)

Fig. 1 shows the jet-breakup length Lbrk obtained from previous
experiments organized by Froude number Fr. In Fig. 1, the vertical axis
represents the normalized Lbrk obtained using a jet diameter Dj and
density ratio of melt to coolant ρj/ρc, and the horizontal axis represents

where E0 denotes the tuning parameter called “the entrainment coefficient”, and its values range 0.05 and 0.1. This type of the correlation is
also called a “Taylor type” correlation (Taylor, 1963). We can recognize

Table 6
Previous experiments conducted under nearly saturated or saturated water conditions.
Organization
(Test facility or program)

Melt/Coolant

References

ANL
FZK
(PREMIX)
JAERI
(GPM)
MATE
(POSTEC)

Wood’ metal/Water
Al2O3-Fe/Water

Spencer et al. (1986)
Kaiser et al. (2001)

Al2O3-ZrO2, SS-C/Water

Moriyama et al. (2005)

Wood’ metal/Water

Jung et al. (2016)

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Y. Iwasawa, Y. Abe

Fr. The open plots indicate the Lbrk obtained in the experiments using
water, and the filled plots indicate the Lbrk obtained in the experiments
using sodium. Note that the legend in Fig. 1 shows the organization
where the experiments were carried out, the test facility or program,
melt-coolant composition, and related references. The dashed-dotted
line indicates the correlation for estimating Lbrk proposed by Saito et al.
(1988), and the dashed line indicates the correlation for estimating Lbrk
proposed by Epstein and Fauske (2001).
Several methods were used to measure Lbrk because of the limitations of the experiments. In the middle- or large-scale experiments,
thermocouples were set in the test facility to measure Lbrk from the
temperature history (e.g., KROTOS, FARO/TERMOS, FAT, PREMIX,
TROI). Alternatively, wired meshes were set up in the test facility to
detect the jet-breakup position (e.g., FARO/TERMOS, FR tests). In
several experiments, melt jets in the coolant were visualized using highspeed video cameras, and Lbrk was measured using the visualized
images (Abe et al., 2006; Matsuo et al., 2008; Iwasawa et al., 2015a; Li
et al., 2017). In previous experiments (Moriyama et al., 2005; Abe
et al., 2006; Matsuo et al., 2008; Iwasawa et al., 2015a; Li et al., 2017),
Lbrk was measured as the distance from liquid surface (coolant) where
the tip velocity of a melt jet decreases rapidly. This article summarizes

the experiments focused on vapor explosion, and the experiments
without jet-breakup in which a melt jet hit bottom of the test facility

Fig. 2. Effects of coolant subcooling on jet-breakup length obtained from previous experiments. The results obtained using the correlation proposed by
Epstein and Fauske (2001) are shown for reference.

Fig. 3. Normalized fragment sizes (MMDs) obtained from previous experiments are organized based on Weber number. The critical Weber number theory is
presented for reference.
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Y. Iwasawa, Y. Abe

the previous experiments in Table 6 in which the coolant temperature
was varied as a test condition. Then, we organized the Lbrk values in
Table 6 in terms of the coolant subcooling ΔTc in Fig. 2. In Fig. 2, the
vertical axis represents the normalized Lbrk, as in Fig. 1, and the horizontal axis represents ΔTc. The dashed line indicates the correlation for
estimating Lbrk proposed by Epstein and Fauske (2001) for reference.
In Fig. 2, high ΔTc means the coolant temperature is low, and low
ΔTc means the coolant temperature is nearly saturated or saturated.
Hence, we can interpret that at high ΔTc, coolant vapor may condense
easily, that is, a stable vapor film would be difficult to form.
From Fig. 2, we can recognize that as ΔTc increases, Lbrk tends to be
slightly short. However, owing to the lack of adequate number of the
experimental results, we cannot make a clear conclusion. Jung et al.
(2016) started an experiment focusing on the effects of vapor generation on the jet-breakup and the fragmentation phenomena. This program now is in progress. Therefore, they may provide further knowledge and clear conclusion in future works. Also, we need to verify not
only effects of coolant subcooling but also other integral effects including melt solidification/oxidation on the Lbrk. The authors consider
that this issue needs to be investigated. The research groups of

Manickam et al. (2017) may report in the future works.

directly (Magallon, 2006). In these experiments, the jet-breakup length
could not be measured. Therefore, we excluded these experimental
results from Fig. 1.
From Fig. 1, we can recognize that the Lbrk values obtained from the
previous experiments show two trends: Lbrk increases as Froude number
increases, and Lbrk remains almost constant in spite of some variations.
Moreover, we can see that the Lbrk values obtained from the previous
experiments targeting LWRs and SFRs have no significant differences,
regardless of the experimental scales. Almost all of the Lbrk values are
close to those obtained using the correlation proposed by Epstein and
Fauske (2001). However, the Lbrk values obtained from the experiments
using a volatile liquid such as freon and nitrogen and from those using
nearly saturated or saturated water as a coolant are close to the values
obtained using the correlation proposed by Saito et al. (1988).
In previous works, Moriyama et al. (2005) proposed criteria for
determining the applicability of the correlation for estimating Lbrk by
using the Bond number; Lbrk follows the correlation by Saito et al.
(1988) when the Bond number is small (< 50), and Lbrk follows the
correlation by Epstein and Fauske (2001) when the Bond number is
large (> 50). The criteria can be applied in the range of Bond number
from 10 to 100, and the range of Froude number from 100 to 500.
However, Jung et al. (2016) pointed out that the criteria cannot explain
the experimental results obtained by Abe et al. (2006) because the Lbrk
measured by them followed the correlation by Epstein and Fauske
(2001) even in the small Bond number. Abe et al. (2006) conducted the
experiments using low melting point metal in the subcooled condition.
Most of the experimental results conducted by Jung et al. (2016) followed the correlation by Saito et al. (1988) except the condition in a
highly subcooled water and a low melt superheat. Jung et al. (2016)

considered that the subcooled conditions result in a short jet-breakup
length by hindered the vapor generation and the results agreed with the
correlation by Epstein and Fauske (2001). Therefore, Jung et al. (2016)
focused the vapor generation on the jet-breakup and the fragmentation
phenomena, and conducted an experiment at the MATE facility to
verify the criteria. They focused on the effects of vapor generation on
the jet-breakup and the fragmentation phenomena. In the next section,
we will discuss the dominant factors governing Lbrk and the applicability of the correlations.

4. Fragmentation phenomena
The following sections summarize the fragment sizes obtained in the
previous experiments. The obtained values are compared with those
obtained using existing methods, and the dominant factors governing
the fragmentation phenomena are discussed.
4.1. Existing modeling schemes and methods
There are several classical theories for estimating fragment size such
as Kelvin-Helmholtz instability (KHI) and the critical Weber number
theory (CWT). These classical theories include only hydrodynamic interactions. Previous works (Dinh et al., 1999; Abe et al., 2006; Bang
et al., 2003; Matsuo et al., 2008; Iwasawa et al., 2015a, 2015b; Li et al.,
2017) have pointed out that these classical theories mostly pertain to
fragment size, although the actual interface phenomena involves complex and non-linear deformation.
The classical KHI in two-dimensions (JSME, 1995) is a linear stability theory that considers the balance between interfacial tension
force and pressure difference due to the velocity difference between the
two-phases. Then, we can obtain the characteristic wavelengths of the
KHI expressed as follows:

3.3. Dominant factors affecting jet-breakup length
In Table 6, we summarize the experiments in which nearly saturated
or saturated water was used as the coolant and where the Lbrk valued
obtained were close to those yielded by the correlation proposed by

Saito et al. (1988). In the experiments summarized in Tables 5 and 6, a
significant vapor film was generated and sustained around a melt jet.
Saito et al. (1988) pointed out that coolant vapor was generated at the
tip of a melt jet, and it immediately surrounded a melt jet as a vapor
film. They also pointed out that a vapor film disturbed the direct contact between a melt jet and the coolant, and promoted penetration of
the melt jet into the coolant. Similarly, Schneider et al. (1992) pointed
out that significant vapor generation around a melt jet tends to disturb
fragmentation owing to interfacial instability. From the theoretical
viewpoint, Epstein and Fauske (1985) confirmed that a vapor film
suppresses interfacial instability.
Therefore, we can consider that a melt jet tends to penetrate further
into the coolant when significant coolant vapor is formed around the
melt jet, which leads to suppression of fragmentation owing to interfacial instability. Consequently, the Lbrk values may agree with those
obtained using the correlation proposed by Saito et al. (1988). By
contrast, we can consider that intensive fragmentation occurs due to
interfacial instability under the conditions in which it is difficult for the
vapor film to be formed and sustained, which leads to the intensive jetbreakup. Consequently, the Lbrk values may agree with those obtained
using the correlation proposed by Epstein and Fauske (2001).
To investigate the effects of vapor generation on Lbrk, we focused on

λKHn =

λKHm =

2πσ (ρ1 + ρ2 )
2
ρ1 ρ2 vrel

(4)

3πσ (ρ1 + ρ2 )
2
ρ1 ρ2 vrel

(5)

where λKHn and λKHm denote the characteristic wavelengths called the
neutral-stable and the most-unstable wavelengths, respectively (Itoh
et al., 2004; Matsuo et al., 2008; Iwasawa et al., 2015b). Detailed descriptions of the physical meanings of these wavelengths will be presented in Chapter 5. Here, ρ denotes density, σ denotes interfacial
tension, and vrel denotes the relative velocity between the two-phases.
Previous works (Matsuo et al., 2008; Iwasawa et al., 2015a, 2015b)
employed vj of a melt jet as vrel under the assumption that the ambient
coolant was stationary.
The critical Weber number is used as the criteria for determining the
breakup of a liquid drop. The Weber number considers the hydrodynamic force that deforms a droplet and the interfacial force that helps
a droplet retains its shape, and it is expressed as follows:

We =
195

2
ρvrel
d
σ

(6)


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Y. Iwasawa, Y. Abe

out that the oxide and the metallic melt differed significantly in shape:
the oxide melt gained an almost angular shape unlike the metallic melt.
Recently, Manickam et al. (2017) reported the same results from their
experiments.
Schins and Gunnerson (1986) and Tyrpekl et al. (2014) mentioned
that differences in the fragmentation phenomena between the oxide
and the metallic melt. They mentioned that the effects of 1 (as follows)
are dominant in both oxide and metallic melt.

where d denotes the characteristic length. If the Weber number exceeds
the critical value, a droplet breaks up into smaller and more stable
droplets. As critical values, for example, 12 (Pilch and Erdman, 1987;
Uršič et al., 2010, Uršič and Leskovar, 2011) or 18 (Moriyama et al.,
2005; Matsuo et al., 2008; Iwasawa et al., 2015b; Manickam et al.,
2016, 2017) are used.
4.2. Experimental data on fragment size

1. Hydrodynamic fragmentation due to prompt boiling of the coolant.
This leads to prompt fragmentation and generates smooth spherical
fragments.

Fig. 3 shows the fragment sizes obtained in the previous experiments. In Fig. 3, the fragment sizes are organized based on Weber
number. Nishimura et al. (2010) suggested that the fragment sizes
could be correlated with the Weber number expressed by Eq. (6). As the
characteristic length d, the inlet diameter Dj of a melt jet is used instead
of droplet diameter.
In Fig. 3, the vertical axis represents the fragment sizes expressed by
the mass median diameter (MMD) normalized in terms of Dj and the

horizontal axis presents the Weber number. In many previous works,
MMD is used as the index of fragment size measured using sieves. The
open plots indicate the MMDs obtained in the experiments using water,
and the filled plots indicate the MMDs obtained in the experiments
using sodium. Note that the legend in Fig. 3 shows the organization
where the experiments were carried out, test facility or program, meltcoolant composition, and related references. Also, the CWT is shown
and compared with the MMDs for reference.
In Fig. 3, overall, the MMDs obtained in the previous experiments
decrease as the Weber number increases. The increase in vj, which induces fragmentation of a melt jet, may play an important role, although
the experimental conditions employed in the previous experiments
differ. We can see that the CWT, which is based only on the hydrodynamic interactions, can be used to estimate the MMDs when the
Weber number is low (almost of the order of 101-102 i.e. when the jet
diameter is of the order of a few critical diameters). At high Weber
numbers (almost of the order of 103-105), however, the MMDs obtained
in the previous experiments related to LWRs (i.e. using water) shows a
weak tendency against the Weber number. In addition, the MMDs obtained from in the previous experiments targeting SFRs (i.e. using sodium coolant) are smaller than those obtained in the previous experiments targeting LWRs. Fig. 3 includes the MMDs obtained in the
different melt superheat and coolant subcooling for water and sodium
coolant. Therefore, the variation of the MMDs indicates the influence of
melt superheat and coolant subcooling. Even if there is the variation of
the MMDs due to melt superheat and coolant subcooling, we can
identify the influence of coolant variation on the MMDs at high Weber
number.
The jet-breakup and the fragmentation phenomena involve not only
hydrodynamic interactions but also thermal interactions. Therefore, we
should verify whether the model (CWT) is applicable. In addition, we
need to consider and investigate the effects of the thermal interactions.

Schins and Gunnerson (1986) and Tyrpekl et al. (2014) also mentioned that the effects of 2 and 3 (as follows) are present in the oxide
melt.
2. Thermal fragmentation due to thermal stress acting on the crust of

the melt surface. This leads to shrinking and cracks the crust. This
occurs after hydrodynamic fragmentation.
3. Coolant ingression inside the shrunken and cracked crust plays an
important role in generating cracked and angular fragments.
On the other hand, in the metallic melt, Schins and Gunnerson
(1986) and Tyrpekl et al. (2014) also pointed out that thermoelasticity
of metal disturbs the cruck because the thermal stress does not exceed
the material strength of the crust. Therefore, hydrodynamic fragmentation becomes dominant in the metallic melt. Consequently, mainly
smooth spherical fragments were formed in the experiment involving
metallic melt.
Based on the previous works mentioned above, we can infer the two
reasons for the differences in the MMDs between the oxide/sodium and
the oxide/water system as follows:
1. In the case of oxide melt, the crust tends to be cracked owing to
the thermal stress caused by the temperature gradient in the crust
compared to the case of metallic melt (Schins and Gunnerson, 1986;
Tyrprekl et al., 2014).
2. Sodium has large effusivity than water. The effusivity determines
interfacial temperature between melt and coolant temperature to a
lower value for sodium coolant, which affects crust formation and
fragmentation phenomena (Schins and Gunnerson, 1986).
At this time, we need to note that the difference of opaque and semitransparent melt mentioned by Dombrovsky and Dinh (2008): in case of
opaque melt such as corium, radiative heat transfer from the melt
surface controls rapid crust formation from the melt surface. On the
other hand, in case of semi-transparent melt such as alumina, convective heat transfer controls crust formation although radiative heat
transfer from melt volume controls rapid solidification of the melt.
Then, we can suppose that the MMDs obtained from the previous
experiments for SFRs are smaller than those obtained from the previous
experiments for LWRs at high Weber number because of the difference
in the thermal stress that breaks the curst owing to the difference in the

effusivity of the coolant in addition to differences in the shrinkage and
the coolant ingression into the crust on the melt surface.
We can recognize that the solidification and the subsequent
cracking of the crust on the melt surface are important factors.
Investigation of the solidification effects on the jet-breakup and the
fragmentation phenomena in NPPs is important, but it is given the
complexity of the phenomena. Therefore, investigating the solidification effects separately is an effective approach.

4.3. Dominant factors affecting fragment size
In this section, we discuss the dominant factors governing the
fragmentation phenomena from the viewpoint of thermal interactions.
Schins and Gunnerson (1986) discussed the difference in the fragmentation phenomena based on an experiment in which oxide and metallic
was injected melt into sodium. They pointed out that the fragments
obtained from the experiments in which metallic melt was injected into
sodium coolant were smooth and round-shaped: the fragments obtained
in the experiments in which oxide melt was injected into sodium
coolant had a cracked shape and were brittle and fragile, with only a
few fragments being smooth and round-shaped. Moreover, Tyrprekl
et al. (2014) discussed the fragmentation phenomena based on morphology measurements conducted using metallographic, analytical, and
microscopic techniques for the fragments obtained in the experiment in
which oxide and metallic were injected melt into water. They pointed

5. Solidification effects on FCI phenomena
The following section will summarize the experimental works on the
FCI phenomena with a focus on the solidification effects. In SFRs and
LWRs, the influence of solidification effects on FCI is important.
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Table 7
Previous experiments on the FCI phenomena focused on solidification effects.
Organization

Melt/Coolant

Experiment

References

IKE

CRIEPI

Cu/Sodium

HU

Cu/Sodium

Droplet
Injection
Droplet
Injection
Jet
Injection
Droplet

Injection
Droplet/jet
Injection
Droplet
Injection

Bürger et al. (1985, 1986)

KTH

Wood’ metal/
Water
Wood’ metal/
Water
Sn, Tin/
Water
Pb-Bi/Water

NU
JAERI

In SFRs, stable film boiling could not be formed around a melt
(Kondo et al., 1995; Suzuki et al., 2014; Vanderhaegen and Belguet,
2014) because the melt and the coolant were partly in direct contact.
Fauske et al. (2002) pointed out that solidification of the melt surface
occurs during FCI in SFRs. Previous works (Bürger et al., 1985, 1986)
have pointed out that the solidification effects could suppress the
fragmentation phenomena. However, Magallon et al. (1992) reported
that intensive fragmentation occurred in the experiment at the FARO/
TERMOS facility, and the size of the fragments obtained from the experiment were of the order of 103∼101 μm. Therefore, the solidification

effects on the fragmentation phenomena are required to be investigated
from the viewpoint of debris coolability.
In LWRs, the solidification could also occur (Dombrovsky and Dinh,
2008; Uršič et al., 2012). The solidification effects are important,
especially with regard to the fragmentation phenomenon, which may
lead to vapor explosion (Bürger et al., 1985, 1986; Uršič et al., 2010,
2011; Uršič and Leskovar, 2011, 2012, 2015). Dombrovsky and Dinh
(2008), Uršič et al. (2010, 2011), Uršič and Leskovar, 2011 and Uršič
and Leskovar (2012) developed a model to calculate the temperature
involved in the solidification effects on a melt droplet, and Uršič et al.
(2015) reflected the model into an integrated code for evaluating vapor
explosion.

Yang and Bankoff (1987)
Sugiyama et al. (1999)
Li et al. (1998)
Haraldsson et al. (2001)
Nishimura et al. (2002, 2005, 2010)
Zhang et al. (2009)
Zhang and Sugiyama (2010, 2011, 2012)

In Table 7, Bürger et al. (1985, 1986) from IKE and Yang and
Bankoff (1987) from Northwestern University (NU) conducted experiments in which they observed the fragmentation phenomena. Moreover, they measured the shape and size of the fragments by injecting a
melt droplet into streaming water. Li et al. (1998) and Haraldsson et al.
(2001) from KTH, Nishimura et al. (2002) from CRIEPI, and Zhang
et al. (2009) and Zhang and Sugiyama (2010, 2011, 2012) from Hokkaido University (HU) conducted experiments in which they observed
the fragmentation phenomena and measured the shape and size of the
fragments by injecting melt droplets into static water. Then, Bürger
et al. (1985, 1986) pointed out that fragmentation and solidification
were competitive processes and classified various fragmentation modes

based on observation results and fragment shape. Li et al. (1998)
pointed out that the eutectic melt experience deeper fragmentation
than the non-eutectic melt. Also, Li et al. (1998) mentioned that noneutectic melt in the mushy zone (a mixture of liquid melt and fine solid
particles in the preceding cooling process) will become increasingly
viscous, and will prevents the fragmentation, especially when the melt
superheat is small. Yang and Bankoff (1987) and Haraldsson et al.
(2001) proposed the criteria under which a melt droplet breaks up
based on the work of Epstein (1977). Nishimura et al. (2002), Zhang
et al. (2009), and Zhang and Sugiyama (2010, 2011, 2012) pointed out
that fragment size increases owing to the solidification effects and
proposed an empirical correlation for estimating the fragment size.
Sugiyama et al. (1999) conducted an experiment in which they injected a simulant melt in the form of a melt jet and reported that the
sediments obtained in the experiment were cylindrical, which indicates
that the crust could be formed by solidifying the melt surface before the
jet-breakup. Moreover, the previous works (Nishimura et al., 2005,
2010; Iwasawa et al., 2015b) reported that sheet- and filament-shaped
fragments were formed in addition to the cylindrical sediments. The
previous works (Nishimura et al., 2010; Iwasawa et al., 2015a) also
reported that a melt jet breaks up under the condition that vj and
coolant temperature are relatively high, despite the solidification effects. At this time, Iwasawa et al. (2015a) concluded that the correlation proposed by Epstein and Fauske (2001) can be applied to the estimation of Lbrk. Moreover, Nishimura et al. (2010) proposed that an
empirical correlation for estimating fragment size under high Weber
number, which refers to a scenario in which hydrodynamic interactions
become dominant. In these works (Nishimura et al., 2005, 2010;
Iwasawa et al., 2015a, 2015b), the condition that solidification effects
become dominant is defined as the point when the initial interfacial
temperature Ti (Fauske, 1973) is lower than the melting point of a melt
in an initial condition. Under the condition that the solidification effects
become dominant, however, few works have investigated and constructed a mechanistic model for estimating the size of particulate
fragments from a melt jet. In the next section, an up-to-date model for
estimating fragment size, including the influence of the solidification

effects on a melt jet, will be presented.

5.1. Experimental works on solidification effects
This section summarizes the previous experiments on the FCI phenomena that focused on the solidification effects. They are listed in
Table 7. The previous experiments include in which several grams to
several hundred kilograms of melt were injected, that is, ranging from
melt droplets to melt jets. Although the experiments with the simulant
melt were summarized in Table 7, there are the study investigate the
solidification effects in the experiments with corium melt using simulation (Uršič et al., 2012; Uršič and Leskovar, 2012).
In the previous works, solidification of the melt surface has been referred to by specific terminology: surface freezing (Fauske et al., 2002) and
surface solidification (Yang and Bankoff, 1987; Cao et al., 2002; Iwasawa
et al., 2015a, 2015b). The crust formed on the melt surface leads to shrink
and crack due to thermal stress in the experiments conducted using oxide
melt (Schins and Gunnerson, 1986; Tyrpekl et al., 2014). Investigating the
influence of the solidification effects on the jet-breakup and the fragmentation phenomena in NPPs is difficult, and investigating the solidification effects separately is one of the effective approaches.
We have supplementarily mentioned that the cracking subsequent
to curst formation has been researched widely and extensively
(Cronenberg et al., 1974; Cronenberg and Fauske, 1974; Knapp and
Todreas, 1975; Ladisch, 1977; Corradini and Todreas, 1979; Cao et al.,
2002; Dombrovsky, 2009). Because details of these works are beyond
the scope of this article, we have omitted detailed description of these
significant works.
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∂v

∂v
1 ∂P
+V
=−
+g
∂t
∂y
ρ ∂y

(9)

where P denotes pressure, and g denotes gravitational acceleration. Eqs.
(7)–(9) are applied to the melt jet and the coolant. Note that the nonlinear term is neglected from the Euler equations because of the assumption that the interface fluctuation is small enough. This assumption also means that the interface displacement is small enough and can
be expressed as follows:

η (t , y ) = η0 ei (ωt − ky)

(10)

From Eq. (10), we can recognize that a constant-amplitude wave
with no decrease travels except if ω and k have positive imaginary
parts.
In the present model, the interface fluctuation induces the fluctuated velocities u and v. The relationship between the fluctuated velocity u and the interface displacement η is expressed as follows:

u=
Fig. 4. Schematic of linear stability model, including solidification effects.
Crust formation at the melt jet-coolant interface (Iwasawa et al., 2015b).

D
The following section presents a mechanistic model for estimating

fragment size, including the influence of the solidification effects on a
melt jet, proposed by Iwasawa et al. (2015b). The detailed description
and derivations are presented in the following sections. The present
model is based on previous works (Epstein, 1977; Haraldsson et al.,
2001).

∂ 2η
∂ 4η
− σ 2 = P1 − P2
∂y
∂y 4

(12)

where σ denotes interfacial tension, and D denotes bending stiffness. In
Eq. (12), the first term on the left-hand side denotes the elastic force
due to the mechanical strength of the crust, and the second term on the
left-hand side denotes the interfacial force acting on the crust. Note that
σ is defined as σ = σ1+σ2 (Epstein, 1977): σ1 refers to the interfacial
force between the melt jet and the crust, and σ2 refers to the interfacial
force between the coolant and the crust. Iwasawa et al. (2015b) used
the measured values in air as an alternative because estimation or
measurement of the exact values of σ1 and σ2 is difficult. The crust may
be formed on the melt jet-coolant interface during the jet-breakup and
the fragmentation phenomena. Therefore, the physical meaning and the
value of D may be different from those of D considered and measured in
solid material.
According to potential flow, the fluctuated velocities u, v are expressed using the velocity potential ϕ:

5.2.1. Modeling assumptions and governing equations

This section describes the assumptions and the governing equations
of the present model. The present model employs a linear stability
theory to calculate the growth of fluctuation at a melt jet-coolant interface based on previous works (Epstein, 1977; Haraldsson et al.,
2001) and assumes that the melt jet-coolant interface extends infinitely
along the vertical, as shown in Fig. 4, for simplification. The simplified
system shown in Fig. 4 considers interfacial instability with the crust,
which has a certain thickness δ. Subscripts 1 and 2 denote the melt jet
and the coolant, respectively. In Fig. 4, Cartesian coordinates are employed: the x-direction is along to the horizontal, and the y-direction is
along the vertical. Fluid 1 (region x < 0) with density ρ1 flows along
with the vertical with uniform velocity V1, and fluid 2 (region x < 0)
with density ρ2 flows along the vertical with uniform velocity V2. In
addition, following assumptions are applied:

u=−

∂ϕ
∂ϕ
, v=−
∂y
∂x

(13)

In addition, ϕ satisfies the following boundary conditions and the
continuity equation expressed as Eqs. (14) and (15).

∂ϕ1
∂ϕ2
(t , −∞, y ) = 0,
(t , ∞, y ) = 0

∂y
∂y

(1) Viscosity is assumed to be negligible compared to interfacial tension and crust stiffness.
(2) Crust is thin enough, so inertia is neglected. Moreover, interface
fluctuation is small enough, so the linear stability theory can be
applied.
(3) Melt jet and coolant are assumed to be incompressible and irrotational. Therefore, the present model assumes potential flow.
(4) The crust has no edge, that is, infinite size, and a constant thickness.
Also, thermal stress due to temperature gradient can be neglected.

∂ 2ϕ
∂x 2

+

∂ 2ϕ
∂y 2

=0

(14)

(15)

5.2.2. Derivation of interface growth rate
In this section, we derive the growth rate of the interface displacement to calculate the characteristic wavelengths based on the
linear stability theory. The temporal growth rate γt and the spatial
growth rate γy are employed. They are expressed as follows using Eq.
(10) (Itoh et al., 2004; Matsuo et al., 2008):

Based on the above assumptions, the present model employs the
continuity and the Euler equations in two dimensions as follows:

∂u
∂u
1 ∂P
+V
=−
∂t
∂y
ρ ∂x

(11)

To include the solidification effects, the modified Laplace law can be
employed and expressed as follows (Epstein, 1977; Haraldsson et al.,
2001):

5.2. Model for estimating fragment size (Iwasawa et al., 2015b)

∂u
∂v
+
=0
∂x
∂y

∂η
∂η

dη
+V
=
∂y
∂t
dt

(7)

(8)

1 dη ⎞
γt ≡ Re ⎜⎛
⎟ = Re(iω) = − ωi
⎝ η dt ⎠

(16)

1 dη ⎞
γy ≡ Re ⎜⎛
⎟ = Re(ik ) = − k i
⎝ η dy ⎠

(17)

From Eqs. (16) and (17), we derive the relationship between
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Y. Iwasawa, Y. Abe

Iwasawa et al. (2015b) assumed that only the temporal growth rate γt
could be considered, that is, the spatial growth rate γy could be ignored
when calculating the characteristic wavelengths.
According to the assumption, we can obtain γy = 0, that is, - ki = 0
from Eq. (17). This means that the k in the fluctuation has no imaginary
part; then, the first term of the left-hand side in Eq. (24) has real parts.
From Eq. (16), Eq. (24) needs to have an imaginary part to ensure
temporal growth of the fluctuation. Therefore, the second term of the
left side in Eq. (24) needs to be imaginary.
Consequently, we can obtain γt, expressed as follows, by using the
relationship k = 2/λ:

angular frequency ω and wave number k (dispersion relation) of the
fluctuation.
To derive the dispersion relation, Eq. (8) is rewritten as Eq. (18) by
integrating along the x-direction and replacing u and v using ϕ:

P=ρ

∂ϕ
∂ϕ
+ ρV
∂t
∂y

(18)

Now, we assume that ϕ is independent of the space x, y and the time
t, and it fluctuates periodically with time. In addition, we assume that
the interface fluctuation prevails only along the forward y-direction and
satisfies the boundary condition expressed by Eq. (14). Then, we can
rewrite ϕ as Eq. (19).

ϕ1 (t , x , y ) = ϕ01

e kx ei (ωt − ky),

ϕ2 (t , x , y ) = ϕ02

e−kx ei (ωt − ky)

1

ρ ρ (V1 − V2)2 2π 2
σ
2π 3
D
2π 5 2
⎛ ⎞ −
⎛ ⎞ −
⎛ ⎞⎤
γt = ⎡ 1 2
2
⎢ (ρ + ρ )
ρ1 + ρ2 ⎝ λ ⎠
ρ1 + ρ2 ⎝ λ ⎠ ⎥
⎝ λ ⎠

1
2
⎣
⎦

(19)

where ϕ01 and ϕ02 denote unknown constants.
We substitute Eqs. (10) and (19) into Eq. (18), and compute the
difference related to the P of each phase (melt jet and coolant) to obtain
Eq. (20).

(25)
On the right-hand side of Eq. (25), the first term refers to the destabilizing effects caused by a dynamic pressure drop owing to the
velocity deference between each phase (melt jet and coolant), second
term refers to the stabilizing effects of interfacial tension, and the third
term refers to the stabilizing effects of the elastic force due to mechanical strength of the crust. The interface becomes unstable when the
term in the square root of Eq. (25) has positive parts, which is the case
when first term is larger than the last two terms. Note that if the third
term is negligible (D = 0), we can easily introduce the KHI (Haraldsson
et al., 2001; Iwasawa et al., 2015b).

P1 − P2 = ϕ01 ρ1 (ω − kV1) ie kx ei (ωt − ky) − ϕ02 ρ2 (ω − kV2) ie−kx ei (ωt − ky)
(20)
Next, we substitute Eq. (10), Eq. (13), and Eq. (19) into Eq. (11) to
obtain Eq. (21).

ϕ01 ke kx = −η0 (ω − kV1) i,

ϕ02 ke−kx = η0 (ω − kV2) i

(21)

Furthermore, we obtain Eq. (22) by substituting Eq. (10) into Eq.
(12).

Dk 4 + σk 2η0 ei (ωt − kx ) = P1 − P2

5.2.3. Solidification effects on interfacial instability
This section presents the influence of the solidification effects on
interfacial instability. Fig. 5(a) shows the calculated relationship between γt and λ from Eq. (25) under the condition V1 - V2 = 1.0 m/s and
D = 0. Note that the previous work (Iwasawa et al., 2015b) employed
the values of physical properties from the previous experiments
(Iwasawa et al., 2015a). In the experiment, Iwasawa et al. (2015a)
employed low melting point alloy (Bi-Sn eutectic). In Fig. 5(a), the third
term on the left-hand side of Eq. (25) is zero, that is, D = 0. This means
that each phase is in direct-mechanical contact (no crust), and the KHI
can be calculated. From Fig. 5(a), if λ is large and of a certain length,
the temporal growth rate γt starts to increase abruptly from zero. This
indicates that the interface becomes unstable and starts to grow. The
value of λ at which the interface becomes unstable is called neutralstable wavelength λn (Itoh et al., 2004; Matsuo et al., 2008; Iwasawa
et al., 2015b).
Furthermore, from Fig. 5(a), if λ is lager than λn and a certain
length, γt is maximized. This indicates that the interface is as unstable

(22)

Finally, we obtain Eq. (23), which is the dispersion relation in the
present model, by substituting Eq. (20) into Eq. (22) and eliminating
e ± ky using Eq. (21). Note that Eq. (23) is arranged about ω.

(ρ1 + ρ2 ) ω2 − 2k (ρ1 V1 + ρ2 V2) ω − Dk 5 − σk 3 + (ρ1 V12 + ρ2 V22) k 2 = 0
(23)
This equation can be solved easily for ω because Eq. (23) is a
quadratic equation on ω. Therefore, we can obtain the dispersion relation in the present model as follows:

ω=

(ρ1 V1 + ρ2 V2) k
±
ρ1 + ρ2

ρ ρ (V1 − V2)2k 2
Dk 5
σk 3
+
− 1 2
ρ1 + ρ2
ρ1 + ρ2
(ρ1 + ρ2 )2

(24)

Iwasawa et al. (2015b) used MMD as an index of fragment size.
They pointed out that the spatial effects on the jet-breakup and the
fragmentation phenomena are averaged by using MMD. Therefore,

Fig. 5. Calculated temporal growth rate against wavelength: (a) Kelvin-Helmholtz instability, (b) solidification effects (Iwasawa et al., 2015b).
199

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Y. Iwasawa, Y. Abe

Fig. 6. Calculated variations in wavelength against relative velocity, and effects
of bending stiffness (Nm) (Iwasawa et al., 2015b).

as it can be. The wavelength λ at which the two-phase interface becomes as unstable as it can be is called the most-unstable wavelength λm
(Itoh et al., 2004; Matsuo et al., 2008; Iwasawa et al., 2015b). Note that
we can easily calculate λn and λm from γt = 0 and dγt/dt = 0, respectively. Hence, we confirm that the present model can estimate the
growth rate and the wavelength of the fluctuation.
Fig. 5(b) show the calculated relationship between γt and λ from Eq.
(25) under the condition of V1 - V2 = 1.0 m/s, D = 0, and D = 1.0 Nm.
From Fig. 5(b), we can recognize that γt and λ are smaller when D = 1.0
Nm than that when D = 0. According to the calculated results, we can
recognize that in the present model, the crust significantly shifts the
interfacial instability.
Fig. 6 shows the calculated results of the characteristic wavelengths,
λn, and λm, by varying D to investigate sensitivity. In Fig. 6, the values
of λn and λm calculated from the present model are compared with
those calculated from the KHI and the fragment diameter d calculated
based on the CWT. Note that the bending stiffness D is varied from 10−7
to 10−4 Nm. From Fig. 6, we can recognize that the λn and λm calculated using the present model are larger than those calculated without
the crust. In addition, we can recognize that as D increases, λn and λm
increase. According to Fig. 6, the present model significantly shifts the
interfacial instability as D increases, that is, the solidification effects.
5.2.4. Comparison with experimental data
This section presents the results of comparison between the characteristic wavelengths and the MMDs obtained in the previous experiment (Iwasawa et al., 2015b) and discuss the applicability of the present model.
In the previous work (Iwasawa et al., 2015b), spherical, filament-,

and sheet-shaped fragments were obtained in an experiment in which a
melt jet was injected; this experiment focused on solidification effects.
Iwasawa et al. (2015a, 2015b) used Ti to determine the initial conditions of the experiments. If Ti decreases to a value lower than the
melting point of the melt, the solidification effects could be dominant
(called surface solidification condition). By contrast, if Ti increases to be
higher than the melting point of the melt, the solidification effects could
be less dominant (called liquid-liquid contact condition).
Fig. 7(a) shows a comparison between the results of MMD measurement base only on the spherical fragments and the characteristic
wavelengths based on KHI, as well as fragment size based on the CWT.
In Fig. 7, the red-open symbols denote the MMDs obtained under the
liquid-liquid contact condition, and the blue-closed symbols denote the
MMDs obtained under the surface solidification. From Fig. 7(a), we can
recognize that the KHI and the CWT can be applied to estimate the
fragment size of spherical fragments. Fig. 7(b) shows a comparison

Fig. 7. Comparison of MMDs and the classical theories (KHI and CWT): (a)
MMDs of spherical fragments, (b) MMDs of filament- and sheet-shaped fragments (Iwasawa et al., 2015b).

between the results of MMD measurement based on filament- and sheetshaped fragments and the characteristic wavelengths based on the KHI,
as well as the fragment size based on CWT. From Fig. 7(b), by contrast,
the KHI and CWT cannot be applied for estimating the size of filament-,
and sheet-shaped fragments. Iwasawa et al. (2015b) pointed out that
filament-, and sheet-shaped fragments were generated under the condition that the solidification effects become dominant. Therefore, they
pointed out that a model considering the solidification effects was necessary to estimate the size of filament- and sheet-shaped fragments.
Fig. 8 shows a comparison between the results of MMD measurement based on filament-, and sheet-shaped fragments and the
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Y. Iwasawa, Y. Abe

of a melt jet during a severe accident in NPPs. In addition, we focused
on the solidification effects. Much progress has been made in terms of
understanding the jet-breakup and the fragmentation phenomena over
the past few decades. Based on a literature survey, in this article, we
presented the data obtained from the experimental works on jetbreakup length and fragment size. In addition, we discussed dominant
factors governing the jet-breakup and the fragmentation phenomena.
Furthermore, we discussed models for estimating the jet-breakup and
the fragmentation phenomena. The influence of the solidification effects on the FCI phenomena were summarized, and an up-to-date model
including the solidification effects for estimating the fragment size was
presented.
Acknowledgements
This work was supported by JSPS KAKENHI Grant Number 261960.
Y. I. is grateful to A. Kaneko (University of Tsukuba) and K. Koyama
(Mitsubishi FBR Systems, Inc.) for helpful discussions.
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6. Conclusions
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