MODELING OF MOLTEN FUEL COOLANT INTERACTIONS

M. TECH. PROJECT

Submitted in Partial Fulfillment of

the Requirements for the Degree of

MASTER OF TECHNOLOGY

in

CHEMICAL ENGINEERING

by

NIHARIKA SONI

(Roll no. 05302011)

DEPARTMENT OF CHEMICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY

BOMBAY 400 076

JULY 2007

i

ACCEPTANCE CERTIFICATE

Department of Chemical Engineering Indian Institute of Technology, Bombay

The project report entitled "Modeling of the Molten Fuel Coolant

Interactions" submitted by Ms. Niharika Soni (Roll No. 05302011) has

been corrected to my satisfaction and may be accepted for evaluation.

Date: July 3, 2007 Signature

Prof. Mahesh Tirumkudulu

ii

ACKNOWLEDGEMENT

I feel great pleasure in expressing my sincere gratitude towards my project

supervisor Prof. Mahesh S. Tirumkudulu for his encouragement and guidance

throughout the project work and also during preparation of this report. With his

enthusiasm and great efforts to explain things clearly and simply, I could

understand the concepts well. He has been considerate while dealing with my

mistakes and limitations. He always advised me patience during the difficult

period of this research work.

I extend my thanks to N. Kasinathan, J. Harvey and Jasmin Sudha from IGCAR,

for providing the orientation on the project. It gave us a valuable insight on the

background and significance of the project.

I also acknowledge my friends Sonia Pusha and Anant Gautam for their constant

moral support and encouragement. I am grateful to Nilesh Parmar for helping me

with FLUENT in the initial stage of project and Parag Malode for introducing me

to Linux. I would also like to thank my colleagues at Fluid Mechanics Laboratory

and my friends for their cooperation and support.

Finally, I am grateful to my parents, for their encouragement and unconditional

support during all my academic years.

Niharika Soni

Date : 3 July 2007 05302011

iii

ABSTRACT

An analytical study has been established to predict the behavior of Molten

Fuel Coolant Interactions. This study mainly focuses on the effect of surface

solidification on the growth of instability and fragmentation of a circular

cylindrical liquid jet falling in another fluid of lighter density. A linear stability

analysis of the interface between the two fluids with a thin crust, formed due to

solidification of melt jet, is performed. This leads to an explicit dispersion

relation, which includes the effect of bending resistance in the growth of

disturbance due to solidification, and the interfacial tension. The surface

solidification is observed to reduce the growth rate of the disturbances, and

therefore, stabilizes the interface. Furthermore, for a given We, a ‘critical crust

thickness’ is characterized at which no breakup is observed. The modified

Aeroelastic number (Ae*) at the critical crust thickness is obtained as a criterion

for the breakup of the melt jet, based on which a ‘stability curve’ is defined that

divides the regions of ‘breakup’ and ‘no breakup’ for a given jet velocity. The

effect of variation in the jet diameter and velocity on the stability of melt jet is

explained in terms of the crust thickness formed at the interface. To conclude, the

most probable fragment size calculated through the model is in well agreement

with the four different experimental studies.

Keywords: thermal fragmentation, surface solidification, temporal instability,

linear stability analysis, Molten Fuel Coolant Interactions, melt jet breakup

iv

Table of Contents

Certificate i

Acknowledgement ii

Abstract iii

List of Figures v

List of Tables vi

1. Introduction 1

2. Modeling 7

2.1 Theory 7

2.2 Mathematical model 8

3. Results and Discussions 17

4. Conclusion 21

Nomenclature 35

References 37

Appendix A 41

Appendix B 47

Appendix C 51

v

List of figures

Figure no. Title Page no.

1. Molten Fuel Coolant Interactions in the lower plenum of

a nuclear reactor core. 25

2. Schematic diagram of the molten jet in a coolant pool. 26

3. Flowchart representing the algorithm. 27

4. Effect of crust thickness on jet stability. 28

5. Weber no. vs. Aeroelastic no. at critical crust thickness plot

showing stability curve for Wood’s metal at 3 m/s. 29

6. Stability curves for Wood’s metal at different jet velocities. 30

7. Crust thickness calculated for the different experimental

studies. 31 8. Comparison of calculated fragment size for Wood’s metal

and water system with the experiments. 32

A.1 Schematic diagram of the molten core and coolant phases. 41

B.1 Variation in viscosity with distance in y direction. 50

B.2 Effect of variation in viscosity on the most unstable wave. 50

vi

List of tables

Table no. Title Page no.

1. Details of Wood’ metal and water experiments. 23

2. Properties of Wood’s metal used for calculations. 24

A.1 Comparison of calculated fragment size for Wood’s metal 46

and Water system ( non-boiling MFCI ) with experiment.

C.1 Comparison of calculated fragment size for Wood’s metal 53

and Water system ( non-boiling MFCI ) with experiment.

1

Chapter 1

Introduction

Severe accidents can occur if the supply of the coolant to the reactor core of a

fast breeder reactor decreases and there is no coolant available to remove the heat

from the reactor core. Such conditions can lead to significant heat release resulting in

melting of fuel rods and other materials in the reactor core. This situation can cause

the relocation of the molten core materials into the lower plenum (Figure 1). This

represents a serious challenge to the reactor vessel. In order to eliminate the

possibility of further accident progression and to collect this core melt in a cooled

state, the lower plenum of the reactor vessel is usually provided with an in-vessel core

catcher. The study of the interaction of the molten fuel with the coolant during its

passage to the core catcher is termed as Molten Fuel Coolant Interactions (MFCI).

The studies on such interactions envisage jets of superheated (above melting point)

molten core material that break up into droplets on the interaction with the coolant

due to interfacial instabilities inside the plenum while simultaneously being cooled by

the coolant. These interfacial instabilities can arise due to density differences,

interfacial tension and relative motion between the two phases. The droplet formation

from the surface of the jet, caused due to the Kelvin Helmholtz instability is assumed

to dominate the breakup process1. The resultant droplets are smaller than the jet

diameter that finally solidify and form debris.

The aim of any containment strategy should be to obtain a MFCI where the

resultant debris is fine enough to get completely solidified at the reactor core bottom.

To achieve this, the interaction should be optimized such that the growth of

2

hydrodynamic instabilities and fragmentation rate are higher than the solidification

rate. However, inefficient interactions can lead to the:

falling of the melt jet in the molten state through the pool and leading to cake

formation at the bottom of the plenum (low degree of fragmentation and cooling).

falling of the jet in a partially solidified form with no fragmentation (high degree

of cooling at the jet surface with almost no fragmentation).

falling of the jet as melt droplets which settle at the bottom and fuse together to

form a hot cake (fragmentation with low degree of cooling).

Hence, the success of this strategy will depend on the efficiency of the processes of

melt fragmentation and quenching in the deep pool of the sub-cooled coolant.

The instability and breakup of a liquid stream into droplets has been widely

studied starting with the first linear stability analysis of Rayleigh and Taylor2, to

further theoretical developments by Tomotika3, Meister and Scheele4, Lee and

Flumerfelt5, Kinoshita6 and Chacha7 and experiments by Meister and Scheele2,

Kitamura8, Saito9, Teng10, Arai11, Blasiot12, Chauhan13 and Cheong and Howes14 . All

of these studies investigated the jet breakup behavior based on the interfacial tension,

relative motion, density ratio (ratio of jet and ambient fluid density), jet fluid

viscosity and gravity and were limited to isothermal conditions. Hence, none of these

studies account for the effect of heat transfer on jet breakup.

As it can be expected, the Molten Fuel Coolant Interactions involve interfacial

instabilities in the presence of heat transfer and phase change. The high temperature

molten core jet behavior in the coolant during a severe accident would be largely

different from that under isothermal conditions. The simultaneous solidification of

melt and, in some cases, vaporization of coolant significantly influences the process.

Theofanus and Saito15 were the first to examine the liquid jet break up with heat

transfer and identified the effect of density of coolant phase in jet breakup and

mixing. Epstein and Fauske16 further examined the melt jet breakup including the

effect of film boiling. In the thick vapor film case, the vapor (coolant) density was

observed to determine the jet breakup whereas in the thin film case liquid coolant

density was found to be important. Schins and Gunnerson17 studied the boiling and

fragmentation behavior of the molten fuel in the liquid sodium experimentally. The

3

fuels used were copper and stainless steel, at initial temperature far above their

melting points; and uranium and alumina, initially at their melting points. They found

that the transition boiling is the dominating boiling mode for the tested fuels in sub-

cooled sodium. In case of oxide fuel both the fragmentation mechanisms, vapor

bubble formation and collapse and thermal stress shrinkage cracking were found to

prevail. This was evidenced by the presence of both smooth and fractured particulate.

In contrast, all metal fuel debris was smooth, suggesting fragmentation by the first

mechanism only. None of their test has shown evidence of an energetic MFCI. This

was followed by many other experimental studies using uranium oxide and alumina

in liquid sodium18, 19, 20 to visualize the mechanism of MFCI and measure jet breakup

length and debris/droplet size distribution. Corradini and Hohmann20 (1993) have

given a detailed analysis of the mechanism of fragmentation in the various stages of

fall of the melt jet in the coolant pool. They observed that, initially, because of the

relative velocity between the jet and the vapor, small melt droplets are stripped off the

jet surface. The leading edge of the jet is deformed into a "mushroom-like" shape as it

penetrates the coolant. The dynamic pressure and the shear force between the jet and

the coolant cause the melt flow from the front stagnation point to the rear of the

"mushroom-like" leading edge and in the process gets stripped off. This process is

called the boundary layer stripping and is confined to the leading edge only. As the jet

penetrates the coolant, the waves of short wavelength at the surface of the jet (upper

section above the leading edge) grow and become unstable due to the K-H instability.

Large relative velocity between the fluids results in small droplets (< 1 mm) and vice

versa. As the diameter of jet decreases due to the erosion from the surface, the jet

column near the leading edge would break-up into large discrete "lumps" due to

Rayleigh-Taylor�’s instability. Depending on the coolant volume and temperature and

the jet Weber number, the breakup into droplets will continue until a stable size

debris is formed. Kondo et al.21 investigated the possibility of energetic (highly

explosive vapor generation) MFCI in LMFBRs. They concluded that energetic

MFCIs are possible under conditions where stable film boiling occurs allowing a

large amount of the melt to penetrate and mix into coolant phase, and where the

interface temperature exceeds the spontaneous nucleation temperature of coolant to

4

initiate explosive boiling. In general such conditions in LMFBR are difficult to attain,

because both the minimum film boiling temperature and spontaneous nucleation

temperature of sodium are sufficiently high. Hence, no explosion is expected in

LMFBRs.

Since all the above mentioned experiments involved high vapor generation with

no explosion, the fragmentation phenomenon was complicated by vapor bubble

formation, collapse and mixing. Thus, for a better understanding of the hydrodynamic

instability and fragmentation aspect of the MFCI, experiments with molten Wood�’s

metal (melting point 70°C) and water as coolant have been chosen as the model

system by various groups22, 23, 24. The experiments at melt temperature below 100°C

exhibit non-boiling MFCI where fragmentation is achieved solely due to

hydrodynamic instabilities with heat transfer. Spencer et al22, investigated the breakup

of the molten jet in water with and without boiling by using Wood�’s metal and Cerro-

alloy. They found that the penetration of the jet leading edge was higher in non-

boiling case as compared to boiling case. This occurs due to further disintegration of

the interface by vapor collapse and mixing in latter case. Dinh et al.25 performed a

systematic study of jet breakup using various pairs of simulant liquids (water �–

paraffin, Cerrobend �– water, Cerrobend �– paraffin) to investigate the effects of jet

velocity, density ratio, melt coolant viscosities and heat transfer. They observed that

the density ratio affects the breakup behavior most significantly and no leading edge

breakup was observed when the density ratio was greater than 2.4. The effect of

viscosity of melt and coolant was found insignificant. Bang et al.26 investigated

molten jet breakup experimentally at various relative velocities for two different

diameters. Their studies demonstrated the dominance of Kelvin-Helmholtz instability

in the fragmentation process. Abe et al.27 observed the fragmentation behavior and

found that the size of the solidified fragment in static falling experiment is quite

larger from the jet ejection experiment. The increase in the effect of Kelvin-

Helmholtz instability caused smaller debris size in the latter case. Narayanan et al.28

analyzed the effect of coolant temperature, melt temperature (boiling and non-boiling

regimes) and mass of melt poured (amount of inventory) on fragmentation. As the

coolant temperature decreases larger mean fragment size was obtained due to higher

5

degree of heat transfer and solidification. Further, the fragments obtained from

boiling experiments were rough, more like flatten flakes with thin edges showing the

effect of vapor film formation. Higher inventory resulted in the formation of very

large connected globules, perhaps due to agglomeration of partially solidified droplets

on the target plate (collector for the debris).

Various theoretical studies investigated the mechanism of melt jet/stream

fragmentation and different models were proposed based on the above experimental

studies. These include numerical codes reported in literature such as PM-ALPHA29,

CHYMES30, IFCI31, SIMMER32, TEXAS33, 34, THIRMAL35, IKEJET36 etc are some

numerical codes available in literature. In most of these codes only hydrodynamic

instability models are employed except for the IKEJET, TEXAS and THIRMAL

codes that account for the effect of the vapor film collapse on the jet breakup. Kelvin

Helmholtz instability is considered as the major cause of breakup in all the above

mentioned instability analysis though the solidification of melt jet is neglected.

Cronenberg37 was the first to analyze the effect of surface solidification on the

fragmentation of a spherical melt drop of uranium oxide falling in coolant pool of

liquid sodium. The results of their parametric calculations indicate that the thermal

stresses induced in the thin outer shell and the pressurization of the inner molten core

are responsible for fragmentation. Followed by this work, Yang and Bankoff38 and Li

et al.39 performed experiments with melt drops of alumina and Pb-Bi alloy in water. It

was found that the fragmentation of melt drop can be classified into two regimes

namely; the freezing controlled regime and fragmentation controlled regime,

depending on the rate of crust formation. Haraldsson40 performed experiments for the

breakup of spherical drop of molten material (Pb-Bi alloy and Cerrobend-70 in water)

and compared the results with the predictions of the linear stability analysis of the

interface between two semi-infinite inviscid fluids with thin crust growing between

them. It was found that the modified Aeroelastic number which measures the ratio of

inertia forces to bending resistance can be employed to evaluate the drop breakup

behavior. Further, above a critical modified Aeroelastic number, the drop size is set

by the fragmentation controlled regime, where, the drop will breakup even though the

crust is formed at interface. Below the critical modified Aeroelastic number the

6

freezing controlled regime is obtained, where, no breakup of the droplets will occur.

Cao et al.41 presented a thermal fragmentation model for a spherical melt drop

including temperature and pressure induced stresses on the crust formed at the

interface. They developed a fragmentation rate correlation for the simulation codes.

The results show that in the case where the relative velocity is low and hydrodynamic

fragmentation is not dominant, the developed fragmentation rate correlation gives

much improved prediction of the front advancement of the stream of the melt droplets

in water as compared to the hydrodynamic fragmentation model.

The present study investigates the fragmentation of a circular cylindrical jet

moving in a coolant pool. Recall that all the previous theoretical studies concentrated

on the breakup of a melt drop rather than a melt jet. Further, the influence of the

solidification of the jet surface on the jet stability was not considered. Thus, in our

study, a linear stability analysis is performed on the cylindrical jet surface with thin

cylindrical crust, formed due to solidification of the melt at the interface. The bending

of the crust due to surface instability tends to compress the molten liquid part encased

within, which in turn generates an internal pressure on the crust. The bending stresses

therefore become an important component in the normal stress balance at the

interface. The analysis leads to a dispersion relation which includes the effect of

bending of solid crust along with the interfacial tension. Since we have assumed

cylindrical interface with interfacial tension, the effect of Rayleigh instability is also

accounted for in our model. This instability is important for break up of thin jets

where large wave instability dominates fragmentation.

The surface solidification is observed to reduce the growth rate of the

disturbances, and therefore, stabilizes the interface. Furthermore, for a given We, a

�‘critical crust thickness�’ is characterized at which no breakup is observed. The

modified Aeroelastic number (Ae*) at the critical crust thickness is obtained as a

criterion for the breakup of the melt jet, based on which a �‘stability curve�’ is defined

that divides the regions of �‘breakup�’ and �‘no breakup�’ for a given jet velocity. The

most probable fragment size calculated through the model is in good agreement with

the four different experimental studies.

.

7

Chapter 2

Modeling

2.1 Theory

The Molten Fuel Coolant Interaction shows two important physical

phenomena. First are the interfacial instabilities between two fluids due to fluid

-mechanical interactions and the second is the heat transfer from the superheated melt

to coolant resulting in solidification of the melt and vaporization of the coolant to

varying degrees. The breakup behavior of a circular cylindrical jet falling in an

another immiscible fluid has a varied range from axis-symmetric breakup at low

Weber numbers (ratio of inertial forces to surface tension) to surface stripping and

atomization regime at high Weber numbers. As the melt jets are enters the coolant

pool, the jet and coolant interface is subject to disturbances and these initial

disturbances will grow with time to break the jet. The two major mechanisms

responsible for breakup of jet are pinching of thin cylindrical jets at circular curvature

due to interfacial tension (Rayleigh Instability) and stripping of small droplets from

interface due to relative motion (Kelvin Helmholtz Instability). The analysis of such

phenomenon typically involves perturbing the governing equations and linearizing

them to predict the stability of a given state. If a state is unstable the linear stability

analysis predicts positive growth rates of the perturbation. The perturbation with the

maximum (positive) growth rate, also known as the most unstable wave, is assumed

to control the initial disintegration of the interface.

8

The heat transfer during interaction causes vaporization of coolant in various

extents depending upon the pressure conditions in that region and boiling

characteristics of coolant used. In present model, the initial temperature of the molten

fuel is assumed to be lower than the boiling temperature of the coolant so as to isolate

the key features of breakup phenomenon from the effect of vaporization.

The simultaneous solidification of melt significantly influences the jet

breakup behavior. Since the formation of a solid layer will oppose the bending of the

interface and will therefore dampen the perturbation. But being at a temperature near

to melting point it will be susceptible to cracking due to poor mechanical strength.

The MFCI analysis was started with proposing a simple model which

considered semi-infinite interface between molten core material and coolant phase

with no solid (see appendix A, B). The co-ordinate system being simpler, it was

easier to analyze and understand the process behavior. The effect of viscosity and its

variation with heat transfer were examined first. The viscosity of molten core phase is

observed to decrease the growth rate of surface wave, enhancing the interfacial

stability. The heat transfer between two phases caused increase in the viscosity of

molten core material but this variation is found to have no significant effect on

instability. The effect of surface solidification was then included (see appendix C)

and compared for viscous and inviscid molten phase. The predictions from both

analyses were comparable and it was found that once the surface solidification is

triggered, the effect of viscosity (fluid properties) is reduced. Finally, an advanced

model considering a circular cylindrical molten core material jet along with crust

shell is developed. The next section describes the complete model and derivation of

dispersion relation, followed by discussion of algorithm. Validation of model with

experiments and further results and discussion are presented in subsequent sections.

2.2 Mathematical model

Our model considers an axis-symmetric jet of superheated molten core

material, having density A, surface tension and radius R moving with velocity U

9

through an inviscid coolant of density B. A thin crust, formed due to solidification of

melt, having thickness is considered at interface. Figure 2 shows a schematic

diagram of the three phases and the coordinate system. The following assumptions

are made in formulating the equations for the model:

Uniform and steady state flow in molten core material phase; while the coolant

phase is initially stationary and of infinite expanse.

Both the phases are inviscid.

The flow in the coolant phase relative to fall of the melt jet is laminar. The theory

of linear stability analysis is valid.

The heat transfer between phases is because of forced convection from the melt

jet to coolant phase.

The crust layer is thin and the equations of bending theory of shells are applicable

with no edge effects. Thermal stresses are neglected.

The temperature of the molten jet is assumed to be below the boiling temperature

of coolant; hence vaporization in coolant phase is avoided.

Phases are immiscible so no mass transfer occurs across the interface.

Both the fluids are incompressible.

The effect of gravity is neglected.

Our model calculations are divided into two parts, namely, instability analysis

and heat transfer calculations. The heat transfer is decoupled from the momentum

transfer in that effect of heat transfer on instability is accounted through the formation

of crust but the effect of instability and surface growth on heat transfer is neglected.

Liquid jet phase velocity and pressure distribution

The governing equations of the liquid jet motion are conservation laws of

mass:

0r r zv v vr r z (1)

10

and momentum,

r r r A

A r zv v v p

v vt r z r , and (2)

z z z A

A r zv v v p

v vt r z z (3)

where, we implicitly assume that the flow is axisymmetric. Here, rv and zv are the

velocities in the radial and axial directions while Ap is the pressure in melt phase.

The growth of instability due to disturbances at the interface is studied in the

linearized form by considering a small disturbance (perturbation) at the interface,

0 ( )i zR e f t (4)

where, 0 is the initial amplitude of the disturbance that varies along the z direction,

is the wave number of the disturbance in z direction, and f(t) is a function in time.

Further, we assume f(t) to be exponential function of time,

f(t) = Ce t where, is the growth rate and C is an arbitrary constant. Accordingly, the velocities

and pressure in the two phases are perturbed about their base state,

, ( ) ,i z tr A Av W r e

, ,i z tz A Av U V r e

, ,i z tr B Bv W r e

, ,i z tz B Bv V r e

i z tA Ap P r e ,

i z tB Bp P r e .

Rewriting the governing equations (equations 1, 2 and 3) in terms of velocities

and pressure, the following linearized equations for the continuity and momentum

balance are obtained neglecting the non-linear terms we get,

11

A A

AdW WiVdr r , (5)

A

A AP

Ui Wr , and (6)

A A AUi V P i . (7)

On solving the above governing equations (equations 5, 6 and 7) simultaneously,

we find that the pressure satifies the Laplace equation,

21 0AA

dPd r Pr dr dr . (8)

Above is a typical example of the modified Bessel�’s equation of order 0, solution of

which is of the form,

0 0AP r AI r BK r (9)

where, A and B are constants.

Since the pressure is finite at the centre of jet (r = 0), B = 0. Substituting the above

equation in r and z momentum balance equations (equations 6 and 7), gives, 1

0A AV r i Ui AI r , and (10)

1

1A AW r Ui AI r . (11)

The above expressions for the velocities and pressure must satisfy the

kinematic and dynamic boundary conditions at the interface in the linearized form.

The kinematic boundary condition states that the velocity in r direction is equal to the

material derivative of interface in same direction,

, 0r A A r Rr

Dv W UiDt . (12)

Substituting the expression for AW r (equation 11) in above relation, expression for

constant A can be obtained as following, 2

0

1

A UiA

I r . (13)

12

The dynamic boundary condition is given by

, ,rr A rr B bendingp p (14)

where, rr is total normal stress in r direction, p is the pressure due to interfacial

tension between two phases and bendingp is the pressure on the solidified crust

(cylindrical shell) due to the pressure exerted by the inner molten material which

causes the bending of the crust layer.

Coolant phase velocity and pressure distribution

The governing equations in for phase B (coolant phase) are considered in

Cartesian coordinates. Using perturbed velocity component and pressure terms

defined for phase B and linearizing the equations lead to continuity,

BB

dWiVdr (15)

and momentum balance equations,

BB B

PW

r , and (16)

B B BV P i . (17)

The kinematic boundary condition applied to coolant phase at r = R in the linearized

form gives,

0B r RW . (18)

Note that the disturbance vanishes far away from the interface, i.e.

,r Bv 0, as r . (19)

The velocity profiles and pressure expressions are obtained by solving the

governing equations (equation 15, 16 and 17) subjected to the boundary conditions,

13

0

0

,

,

R rB

R rB

W e

V i e

20 .R rB BP e (20)

The dispersion relation for the inviscid liquid jet is derived using the normal

stress boundary condition (equation 14). The normal stress in the melt jet and coolant

phase is obtained from the corresponding relations for the pressure (equations 9 and

20 respectively) which are given by,

2

, 01

( ),A i zrr A A

Uip I R e f t

I R and (21)

20

, ( )R r i zrr B B Bp e e f t . (22)

The capillary pressure induced due to interfacial tension to the highest order in is

expressed as, 2

2 2

dpR dz . (23)

The pressure induced due to the bending of the shell42, 43 in terms of displacement, ,

and flexural modulus �‘D�’ is given by,

24

4 2 2

12 1bending

dp Ddz R , (24)

where, 3

212 1ED .

Here, E is the Young�’s modulus and is the Poisson�’s ratio for the solidified melt

crust. Substituting the expressions for the above stresses (equation 21, 22, 23 and 24)

in the normal stress boundary condition (equation 14) at r = R yields the following

dispersion relation,

and

14

2 22 2 4

0 2 2 21

12 11 1AB

UiI R D

I R R R .

The above expression can be written in the non dimensional form as the following, 2 2 2

40 2 2 22 * 2

1

1 11 12 1B

A

I RR R Ri R RU I R WeU Ae

(25)

where, We is the Weber number defined as 2 /ARU and Ae* is the modified

Aeroelastic number defined as40

3 32

* 22 212 1AR U RAe AeD

.

Ae in the above expression is Aeroelastic number 2 /AU E . Thus, two distinct

dimensionless numbers have been obtained. First is Weber number which

corresponds to instability and breakup and second is modified Aeroelastic number

which corresponds to degree of solidification and bending resistance at the interface.

Note that the above equation is quadratic in which is solved to calculate the

wave number for the highest growth rate, max. The diameter of droplet detached from

the interface is correlated to wave number of the most unstable wave1, 36, 44 as,

max

1.5d . (26)

The jet breakup length which is defined as the length up to which falling jet remains

as a continuous cylinder, is calculated from the time required for the perturbation

amplitude to increase to radius of the jet,

tbreakup = max 0

1 ln R, (27)

with the jet breakup length given by,

Lbreakup = U tbreakup , (28)

15

where, max is the growth rate of the most unstable wave. The prediction of the jet

breakup length becomes difficult because of the lack of knowledge of the magnitude

of initial disturbance, 0. Here, we follow the lead of previous authors1, 4, 45 by

assuming the amplitude of naturally occurring disturbances to be related to initial

radius of jet as 0 = Re-b where, b is a constant. Previous studies have proposed

different values of b ranging from 6�–18 obtained from their analysis.

As a check on our derivation, we note that for zero relative velocity and in

absence of solidification, the dispersion relation reduced to Rayleigh�’s relation for an

inviscid liquid jet in vacuum,

12 2

20

1

A

I RI RR .

The above analysis requires an estimate of the crust thickness, , to determine

the stability of the jet since it enters in the dispersion relation through bending stress

term in the dynamic boundary condition. This term can be significant and calculation

of becomes important. The basic physics of the problem suggests that for the

formation of solid crust thickness , the magnitude of sensible heat (due to

superheating) and the latent heat removed should equal the heat flux to the coolant.

The heat balance gives the thickness of crust layer formed per unit time as,

, , ,

''

A p A A fusion A fusion A

qt C T T H (29)

where, "q is the heat flux to the coolant , '' A Bq h T T .

The heat transfer from the jet to the coolant is assumed to occur by forced

convection. The Nusselt number based on radius of cylinder /RNu hR k can be

calculated using the Karman - Pohlhausen approximate method46 for the solution of

the boundary layer equation on a continuous cylinder. The analysis presents a general

solution for the heat transfer coefficient for any Prandtl number,

2 2 22 2

2 1 1 2 2 2 1 1Pr2 2 2 1 1

A KK

BdB A e A A e BK B K K BdA e AK A K K A

(30)

16

where, K = A for Pr 1, K = B for Pr 1 and K =A =B for Pr = 1. Here, B is the inverse

of the local Nusselt number based on the radius of the cylinder,1/ RNu , and A is a

dimensionless parameter based on a logarithmic velocity profile assumed in the

momentum boundary layer ,

1

R

R uA U R . (31)

A can be calculated using a curvature parameter, 2xX

UR from the following

relation, 1

12

1 2 !n n

n

nX An n (32)

where, is the kinematic viscosity of the ambient fluid flowing along the cylinder and

x is the position along the length. Equation (30) gives the local Nu and, the average

value of Nu can be calculated using the following integration,

0

1 X

R RL

Nu Nu dXX , (33)

where, 2LLX

UR and L is the length of the cylinder.

The above calculations require the knowledge of the length of the vertical

cylinder which in our case is equal to the jet breakup length. Further, the jet breakup

length depends upon the growth rate of the most unstable perturbation responsible for

disintegration of the interface. Hence we need to first assume a jet breakup length for

heat flux calculations, after which, it is compared with the jet breakup length

calculated from the growth rate of the most unstable wave obtained from the stability

calculations. The algorithm is summarized in figure 3. If the calculated length is

different from the assumed value, then a new value is chosen and the process is

repeated till it converges. Finally, the wave number for the most unstable wave is

used to calculated droplet diameter.

17

Chapter 3

Results and Discussions

The model analyzes the growth rate of a disturbance/perturbation introduced

at the interface of a molten core material jet and a coolant. The instability at the

interface corresponds to positive values of growth rate of the perturbation. Among

these, the fastest growing perturbation i.e., the wave with the maximum growth rate

controls the disintegration. The dispersion relation derived in the last section gives the

growth rate in terms of wave number. It is a quadratic equation with complex roots.

We are solving for the real part of growth rate which corresponds to a purely

temporal analysis. This approach does not consider traveling waves at the interface.

The effect of crust thickness on the stability of jet for a given jet diameter and

relative velocity is examined. Figure 4 presents the results, where it is observed that,

as the crust thickness is increased, the growth rate for disturbances of all the wave

numbers is decreased. This increase in stability occurs due to higher bending

resistance at the interface. The decrease in growth rate and wave number of most

unstable wave causes larger droplet size and jet breakup length. On further increase in

the crust thickness, the growth rate of most unstable wave becomes zero. This crust

thickness is termed as critical crust thickness. For any crust thickness higher than

critical crust thickness, no disturbance is found to have positive growth rate. Hence,

the jets are stable and they will fall as a partially solidified �‘rod�’ through the coolant

pool with no fragmentation.

With this background of having understood the effect of crust thickness on the

jet instability, we analyze the stability of Wood�’s metal and water system at various

jet diameters and velocities. The critical crust thickness is calculated for various

diameters at given velocity from the stability calculations. The modified Aeroelastic

18

number at the critical crust thickness is then plotted against Weber number (figure 5)

to obtain a �‘stability curve�’ for the system. This stability curve divides the regions of

�‘breakup�’ and �‘no breakup�’ on We vs. Ae* plot into two parts. The region above the

stability curve corresponds to the conditions where the crust thickness at interface is

small so that the disturbances can grow with time and lead to breakup. However, the

surface solidification dampens the growth rate of disturbances leading to larger

droplet size than those obtained in the absence of solidification. In other words,

instability dominates over the solidification phenomenon. Conversely, in the region

below the stability curve, the effect of surface solidification is dominating. Here, the

crust thicknesses are large and the growth of disturbance is completely suppressed.

No breakup is obtained in these conditions.

Also, the effect of increasing jet diameter at a given jet velocity on the critical

crust thickness is shown in figure 5. As the diameter increases, the jet becomes more

stable to Rayleigh stability and in fact at very large values of jet diameter, there is

almost no effect of jet�’s curvature and the interface behaves like a flat plane. Along

with this, the bending resistance at the interface decreases with increasing diameter

which makes it less stable. Thus, these two mechanisms act in opposition, although

the resultant is an overall decrease in the interfacial stability. In addition to this, with

the increase in the jet diameter the heat flux to the coolant also increases which

causes formation of thicker crust at the interface. Hence, at low Weber numbers, the

critical modified Aeroelastic number decreases (or critical crust thickness increases)

with increase in the Weber number. Furthermore at the neutral stability point (zero

growth rate) for all the critical crust thicknesses, the product of wave number and

radius ( R), is found to be a constant. Thus, for large values of radius the dispersion

relation reduces to,

2

40 2*

1

1 12 1I R RR RI R Ae

Consequently, the ratio /R at large radii is a constant, which essentially implies that

the modified Aeroelastic at critical crust thickness is also a constant. Thus, the

stability curve becomes constant at the high Weber numbers for a given jet velocity

(figure 5).

19

Furthermore, the relative position in the breakup region (figure 5) can be used

to estimate the relative droplet size and the jet breakup length obtained in various

conditions. For given We (given jet diameter), as Ae* increases (crust thickness

decreases), the droplet size and jet breakup length decreases (as explained in figure

4). Likewise, for given Ae*, as We increases (jet diameter increases), the wave

number of most unstable wave decreases but the growth rate increases (figure 6). The

decrease in wave number is because of the fact that the larger diameter jets require

large waves for breakup and consequently larger droplet are formed whereas the

growth rate is increased due to the increase in overall instability of the interface as

described earlier. This results in longer jet breakup length.

The stability curves obtained at various jet velocities are shown in figure 7.

The results show that with increase in relative velocity the stability curve shifts

towards lower values of the modified Aeroelastic number or higher values of the

critical crust thickness. Here, increase in the relative velocity intensifies Kelvin-

Helmholtz instability making the interface less stable (figure 8). Hence, a thicker

crust is required to stabilize the melt at higher jet velocity.

The overall effect of jet velocity on MFCI is twofold. First, with increase in

the velocity, the heat transfer coefficient increases which increases the heat flux to

coolant and degree of solidification of the melt jet. The higher crust thickness at

interface resists the growth of disturbance and results in longer breakup length. On

the other hand, higher relative motion between phases will further destabilize the

interface due to increase in Kelvin-Helmholtz type instability. Consequently the jet

will break sooner, without providing enough time for solidification. The crust

thicknesses calculated for experiments available in literature (details summarized in

table 1) are shown in figure 9. The values clearly show the competition of the above

mention effects of the jet velocity and diameter. For lower Weber numbers, the crust

thickness increases with We due to increase in heat transfer but at high We, it

decreases due to decrease in the interfacial stability.

The stability diagram in figure 5 and figure 6 also includes points that

correspond to the above mentioned experiments. These experiments were performed

under non-boiling conditions where the melt temperature was below the boiling point

20

of the coolant to avoid vaporization. The most probable droplet size predicted by the

model is compared with mean fragment / debris size observed in experiments (figure

10). In the calculations for the prediction of droplet size for these experimental

conditions, the leading edge advancement data provided by the authors is used to

estimate the breakup length and the average Nusselt number (calculated from

equations 30 �– 33). The crust thickness is then estimated using (29) where the time is

taken as that required for the jet to reach half its breakup length. This crust thickness

is taken as the average value and is used to calculate the droplet diameter. The model

predicts breakup in all the cases and is in good agreement with the observations.

21

Chapter 4

Conclusions

The molten fuel coolant interactions involve interfacial instabilities in the

presence of heat transfer and phase change. The simultaneous solidification of molten

core material, as it moves in the coolant pool, significantly influences the jet breakup

behavior. The study has analyzed the effect of surface solidification on fragmentation

and breakup of the melt jet. A linear stability analysis performed on the melt jet

(circular cylindrical) and coolant phases with a thin crust layer of core material

between them. A dispersion relation is obtained which gives the growth rate of any

disturbance at the interface in terms of its wave number.

The solid crust layer or shell formed at the interface hinders the growth of the

disturbance. The bending resistance of the solidified crust opposes the pressure

gradient responsible for the growth of the disturbance, and results in low growth rates

for all the waves. For a given Weber number, a �‘critical crust thickness�’ is observed at

which the growth rate of the most unstable wave becomes zero. It signifies the

condition at which no breakup is observed. The modified Aeroelastic number at the

critical crust thickness is obtained as a criterion for the breakup of the melt jet. A

stability curve is defined based on the modified Aeroelastic number calculated at the

critical crust thickness which divides the regions of breakup and no breakup of the

melt jet for a given jet velocity.

The overall effect of the jet diameter and velocity is explained in terms of the

crust layer thickness formed at the interface. The crust layer thickness increases with

increase in the jet diameter due to decrease in Rayleigh instability and higher heat

transfer. The velocity affects in two manners. Higher relative motion leads to increase

in instability (Kelvin-Helmholtz) and small breakup times which decreases the time

22

of heat transfer causing low degree of solidification. On other hand increase in heat

transfer coefficient at higher jet velocities increases the solidification at the interface.

The resultant of these three effects determines the crust layer thickness at the

interface which subsequently controls the breakup of melt jet. For low Weber

numbers the crust thickness increases with We due to increase in heat transfer but at

high We it decreases due to increase in the interfacial instability.

A complete model to calculate the most probable fragment size is developed.

The predictions are found in good agreement with four different experimental studies

on MFCI between Wood�’s metal and water (in non-boiling range). These results

indeed show that the solidification of the jet surface cannot be ignored in a molten

fuel coolant interaction and that the elastic nature of the solidifying crust plays an

important role in determining the breakup of the jet. The detailed experimental

analysis, however, is required to understand the behavior of jet breakup length with

temperature difference between the two phases and Weber number of the melt jet

falling.

23

Table 1. Details of Wood’s metal and Water system ( non - boiling MFCI )

experiments

Experiment Jet Dia. (m) Vel. (m/s) Melt Temp.(K) Coolant Temp.(K)

IKE23 0.004 16 363 353

ANL22

0.02 3 373 295

JRC Ispra24 0.05 2 373 298

Experiment at IGCAR28 0.008 0.8 371 300

24

Table 2. Properties of Wood’s metal used for calculations

Property

Density 9480 kg/m3

Interfacial Tension 1.2 N/m

Poisson�’s Ratio 0.33

Latent Heat of Fusion 41,237 J/kg

Specific Heat 184 J/kg/ºC

Temperature of Fusion 70 ºC

Young�’s Modulus* 0.002 GPa (at 60 ºC)

*The variation in Young�’s Modulus with temperature is obtained from a

correlation given by Dai et al.47. This study provides the temperature

dependence of the elastic modulus for Pb�–Bi (45 wt% Pb and 55 wt% Bi)

alloy. The similar trend of variation is assumed to be followed by Wood�’s

metal.

25

Figure 1. Molten Fuel Coolant Interactions in the lower plenum of a

nuclear reactor core1.

26

Figure 2. Schematic diagram of the molten jet in a coolant pool.

Molten Core Jet (Phase A)

Solidified Crust

z

r Coolant (Phase B)

2RU

Coolant (Phase B)

27

Figure 3. Flowchart representing the algorithm

28

Figure 4. Effect of crust thickness on maximum growth rate.

29

Figure 5. Weber no. vs. Aeroelastic no. at critical crust thickness

plot showing stability curve for Wood’s metal at 3 m/s.

30

Figure 6. Effect of jet diameter on most unstable wave.

31

Figure 7. Theoretically predicted stability curves for Wood are metal at

different jet velocities.

32

Figure 8. Effect of jet velocity on most unstable wave.

33

Figure 9. Crust thickness calculated for the different experimental

studies. A, B, C and D are points corresponding to IGCAR, JRC

Ispra, ANL and IKE experimental studies respectively.

34

Figure 10. Comparison of calculated fragment size for Wood’s metal

and water system (non-boiling MFCI) with the experiments listed in

table 1.

35

Nomenclature

Notation

D = Fractional modulus

E = Young�’s modulus

H = heat R = initial radius of melt jet

T = temperature

U = jet velocity

Cp = specific heat

Lj = jet breakup length

NuR = Nusselt number based on radius of cylinder

Pr = Prandtl number

I0 = modified Bessel function of first kind

I1 = modified Bessel function of second kind

f(t) = periodic function in time

f�’(t) = first time derivative of f(t)

d = droplet diameter/fragment size

h = heat transfer coefficient

i = iota

k = thermal conductivity

p = perturbed pressure

q" = heat flux to coolant

t = time

u = velocity in boundary layer around vertical cylinder

v = perturbed velocity

r, z = co-ordinates

36

Greek letters

= wave number

= crust thickness

= Poisson�’s ratio

= interfacial tension

= total stress

= density

= disturbance / perturbation introduced at interface

0 = initial amplitude of perturbation

= thermal diffusivity

= viscosity of molten core material phase

Subscripts

A = molten core material phase

B = coolant phase

= Interfacial tension

i = interface

37

References

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12. Blaisot, J. B., Adeline, S., Instabilities on a free falling jet under an internal flow breakup mode regime, Int. J. of Multiphase Flow 2003; 29: 629 �–653 13. Chauhan, A., Maldarelli, C., Rumschitzki, D. S., Papageorgiou, D. T., An experimental investigation of the convective instability of a jet, Chem. Engineering Science 2003; 58: 2421 �– 2432 14. Cheong B. S., Howes T., Capillary jet instability under the influence of gravity, Chem. Engineering Science 2004; 59: 2145 �– 2157 15. Theofanous, T. G., Saito, M., An assessment of Class-9 (Core-Melt) Accidents for PWR Dry-Containment Systems. Nuclear Engineering and Design, 1982; 66: 307-332. 16. Epstein, M., Fauske, H. K., Steam film instability and the mixing of core-melt jets and water. ANS Proceedings of the National Heat Transfer Conference, Denver, Colorado, USA 1985, .277�–284 17. Schins, H., Gunnerson, F. S., Boiling and fragmentation behavior during fuel-sodium interactions, Nuclear Engineering and Design 1986; 91: 221-235 18. Corradini, M. L., Kim, B. J., Vapor explosions in light water reactors: a review of theory and modeling, Prog. Nuclear Energy 1988; 22: 1�–117. 19. Chu, C. C., Corradini, M. L., One-dimensional transient fluid model for fuel: coolant interaction analysis. Nuclear Science Engineering 1989; 101: 48�–71 20. Corradini, M. L., Hohmann H., Multiphase flow aspects of fuel-coolant interactions in reactor safety research, Nuclear Engineering Design 1993; 145: 207-215 21. Kondo S. et al., Experimental Study on Simulated Molten Jet-Coolant Interactions, Nuclear Engineering and Design 1995, 155: 73-84 22. Spencer, B.W., Gabor, J.D., Cassulo, J.C., Effect of boiling regime on melt stream breakup in water, in T.N. Veziroglu (ed), Particulate Phenomena and Multiphase Transport. 1987; 3 23. Cho, S.H., Berg, E. V., Burger, M. and Schatz A., Experimental investigations with respect to the modeling of fragmentation in parallel shear flows of liquids, Proceedings Sprays and Aerosols Conference, Guildford, UK 1991; 165-169 24. Schins, H., Hohmann, H., Burger, M., Berg, E. V., Cho S. H., Breakup of melt jets in a water pool as a key process for analysis of lower PRV-head failure during core melt accidents in LWR, Jahrestaqung Kerntechnik "92, Kadsruhe, Germany, 1992

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25. Dinh T. N., Bui V. A., Nourgaliev R. R., Green J. A., Sehgal B. R., Experimental and analytical studies of melt jet-coolant interactions: a synthesis, Nuclear Engineering and Design 1999;189: 299�–327 26. Bang K. H., et al., Experimental study of melt jet breakup in water, Journal of Nuclear Science and Technology, 2003; 40: 807-813 27. Abe, Y., Kizu, T., Arai, T, Nariai, H, Chitose, K., Koyama,K., Study on thermal-hydraulic behavior during molten material and coolant interaction, Nuclear Engineering and Design 2004, 230: 277�–291 28. Narayanan K. S. et al., Assessment of thermal and hydrodynamic fragmentation in molten fuel coolant interaction with stimulant system, Proceedings of 14th

International Conference on Nuclear Engineering, 2006 29. Amarsooriya, W. H. and Theofanus, T. G., Premixing of steam explosions: A three fluid model, Nuclear Engineering and Design 1991; 126: 23-39 30. D.F. Fletcher and A. Thyagaraja, A mathematical model of premixing, ANS Proc. 25th National Heat Transfer Conference, Houston, TX, HTC-3, 1988; 184-190 31. Young M. F., FCI: An integrated code for calculation of all phases of fuel-coolant interaction, NUREG/CR-5084, SAND87-1048, 1987. 32. Bohl, W. R., An investigation of steam-explosion loading with SIMMER -2, Los Alamos National Laboratory Report, L.A. 1990 33. Chu, C. C., One dimensional transient fluid model for fuel-coolant interaction analysis, Ph. D. Thesis, University of Wisconsin, Madison, WI. 1986 34. Chu, C. C., Corradini, M. L., One-dimensional transient fluid model for fuel:coolant interaction analysis, Nuclear Science Engineering 1989; 101: 48�–71. 35. Chu, C. C., Stenicki, J. J., Spencer, B. W., The THIRMAL-1 melt-water interaction code, Proceedings of the 7th International meeting on Nuclear Reactor Thermal-Hydraulics (NURETH �– 7), Saratoga Springs, NY 1995; 2359-2389 36. Burger M., Cho S. H., Berg E. V., A. Schatz, Breakup of melt jets as pre-condition for premixing: Modeling and experimental verification, Nuclear Engineering and Design 1995; 155: 215-251 37. Cronenberg, A. W., Chawla, T. C., Fauske, H. K., A thermal stress mechanism for the fragmentation of molten UO2 upon contact with sodium coolant, Nuclear Engineering Design 1974; 30: 434-443

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38. Yang, J. W., Bankoff, S. G., Solidification effects on fragmentation of molten metal drops behind a pressure shock wave, J. Heat Transfer 1987; 104: 226-230 39. Li, H. X., Haraldsson, H. O., Dinh, T. N., Green, J. F., Sehgal, B. R., Fragmentation behavior of melt drop in coolant: Effect of melt solidification. Proceedings of the 3rd International Conference on Multiphase Flows, Lyon, France 1998 40. Haraldsson, H. O., Li, H. X., Yang, Z. L., Dinh. T. N., Sehgal, B. R., Effect of solidification on drop fragmentation in liquid-liquid media, Heat and Mass Transfer 2001; 37: 417-426 41. Cao, X., Tobita, Y., Kondo, S., A fragmentation model induced by surface solidification, J. Nuclear Science and Technology. 2002; 39: 628-636 42. Timoshenko, S., Goodier, J. N., Theory of elasticity, McGraw-Hill, New York 43. Donnell and Hamilton, L., Beams, plates and shells, New York, McGraw-Hill, 1976 44. Bradley, D., On the atomization of liquids by high-velocity gases, Part I, J. Phys. D: Applied Physics 1973; 6: I724-1736; Part II, J. Phys. D: Applied Physics 1973; 6: 2267-2272 45. McCarthy, M. L. and Molloy, N. A., Review of stability of liquid jets and the influence of nozzle design, Chem. Engineering Journal. 1974; 7: 1-20 46. Karnis J., and Pechoc, V., The thermal boundary layer on a continuous cylinder, Int. J. Heat Mass Transfer 1977; 21: 43-47 47. Dai, Y., Barbagallo, F., Groeschel, F., Compression properties of lead�–bismuth, Journal of Nuclear Materials 2003; 317: 252�–255 48. Akhmetzyanov K. G., et al., Measurement of viscosity of liquid Wood�’s metal, British Journal of Applied Physics 1989; 13: 527 49. Sergei Winitzki, A handy approximation of error function and its inverse, 2006; website: www.theorie.physik.uni-muenchen.de 50. Incropera, Frank P. and Dewitt, D. P., Fundamentals of heat and mass transfer, 5th Edition ISBN-10: 0471386502, New York, John Wiley and Sons, 1998

41

Appendix

Appendix A

Consider a semi-infinite system having one phase as superheated molten core

material and other phase as coolant. The molten fuel phase is moving with constant

velocity U having Newtonian viscosity . The coolant phase is considered inviscid

and stagnant. Figure A.1 shows the schematic diagram of molten core and coolant

phases.

Consider a small disturbance at interface, 0 exp( ) ( )i x f t .

Figure A.1: Schematic diagram of the molten core and coolant phases.

Accordingly, the velocities and pressure in the two phases will be,

1

1

2

( ) exp( ) ( )( )exp( ) ( )

( ) exp( ) ( )

x A

y A

x B

v U V y i x f tv W y i x f t

v V y i x f t

42

2

0

0

( ) exp( ) ( )

( ) exp( ) ( )

( ) exp( ) ( )

y B

A A

B B

v W y i x f t

p P P y i x f t

p P P y i x f t

The following assumptions are made in formulating the equations for the model:

The flow in the coolant phase relative to fall of the melt jet is laminar. The theory

of linear stability analysis is valid.

The heat transfer between phases is neglected.

Phases are immiscible so no mass transfer occurs across the interface.

Both the fluids are incompressible.

The effect of gravity is neglected.

Governing equations in phase A (in linearized form):

Continuity: AA

dWV idy

x - momentum balance:

22

2

'( ) 2( )

A AA A A A

d V dWf tUi V p i V i

f t dydy

y - momentum balance:

22

2

'( ) 2( )

A A AA A A

p d W dVf t Ui W i Wf t y dydy

Solving x and y momentum balance together to eliminate pressure terms and then

replacing VA by WA using continuity relation we get,

4 2 2'( )1 2( )A A A

f tD W D W Ui

f t2 4'( )1 0

( )A Af tW Uif t

(A-1.1)

43

The above expressions must satisfy the kinematic and dynamic boundary conditions

at the interface in linearized form. The kinematic boundary conditions states that

velocity in y direction is equal to material derivative at interface in same direction,

1 0y y

DvDt

00

'( )( )A y

f tW Uif t (A-1.2)

Using the above expression, eq. (A-1.1) can be re-written as,

0 04 2 2 2 4

0 0

1 12 0A Ay y

A A A A A

W WD W D W W

Redefining the variable WA as 0/AW , we can write the above expression as,

4 2 2 2 40 0

1 12 0A Ay yD D (A-1.3)

Consider the solution of the above fourth order ordinary differential equation in the

form,

exp( )C qy where, C is any arbitrary constant.

Substituting the above expression in eq. (A-1.2), the four roots of q can be solved as:

20Aq (say) and .

The disturbance in y direction should vanish very far away from the interface i.e.

infinity,

: ( ) 0Ay W y and

: ( ) 0Ay V y 0A

y

dWdy

44

Considering the above boundary conditions the solution of is calculated as,

1 2y yy A e A e where, A1 and A2 are constants (A-1.4)

The shear stresses at the interface should be zero, i.e,

0xy 11 0yx vv

x y 2

2 200

1y

y

ddy

Substituting the expression for (eq. A.1.4) in above boundary condition gives, 2

12

A

A . (A-1.5)

Governing equations in phase B (in linearized form):

Continuity: BB

dWV idy

x - momentum balance '( )( )B B B

f t V p if t

y - momentum balance '( )( )

BB B

pf t Wf t y

The kinematic boundary condition gives,

2 0y y

DvDt .

Also, the disturbance in y direction should vanish far away from the interface,

: ( ) 0By W y .

Solving the above governing equations and boundary conditions we get,

'( ) exp( )( )B

f tW h Ui yf t ,

45

'( ) exp( )( )B

f tV hi Ui yf t , and

'( ) '( ) exp( )( ) ( )B B

f t f thP Ui yf t f t . (A-1.6)

The dynamic boundary condition states that the normal stresses at the interface is

equal to the stress due to surface tension between two fluids,

2

00

( ) 2 AA B y

y

dWP P hdy , at 0 , (A.1.7)

where, stress due to interfacial tension is given by 2

2px .

The pressure in molten core, PA, can be obtained using x momentum balance for

phase A and profile for (eq. A-1.4). Similarly, coolant phase can be obtained from

the corresponding equation of pressure profile (eq A.1.6).

Substituting the pressure expressions for both the phases in eq (A.1.7) gives,

21 2 3 3 2

1 2 1 2 1 22 23 2 0A BA AA A A A A A Ui

(A-1.8)

The above equation (eq. A-1.8) can be solved for A2 for a given range of . The

growth rate then can be calculated using eq. (A-1.2). Once the wave number and

growth rate of most unstable wave are known, the droplet diameter and jet breakup

length can be predicted by using equations (26), (27) and (28).

The droplet size predicted by model is compared with Kelvin-Helmholtz formulation

for in-viscid fluids and the experiment performed at IGCAR28 (Table A.1).

46

Table A.1. Comparison of calculated fragment size for Wood’s metal and Water

system ( non-boiling MFCI ) with IGCAR experiment.

Droplet diameter

K-H theory (In-viscid fluids) 2.70 mm

Experimental Value28 4.75 mm

Simulation 3.34 mm

Discussion

As the table A.1 describes, the predictions with present model are much favorable

then basic Kelvin-Helmholtz instability formulation for inviscid fluids. Including

viscosity of molten core material phase has increased the droplet diameter. Because

as the surface wave grows, viscosity will cause shearing between streamlines and part

of kinetic energy of surface wave will be lost in viscous dissipation, hence, the

growth rate is decreased. In this way stability of interface has increased and larger

wave (higher wave length) is required for breakup which consequently leads to larger

droplet diameter.

47

Appendix B

The next step is to include the heat transfer between the superheated molten core

material and the coolant phase. As the two phases come in contact, transient heat

transfer across the interface begins. The moving molten core phase is infinite in the

direction of the flow hence, the heat convection in that direction is neglected. Thus,

the overall heat transfer is due to transient heat conduction in the direction

perpendicular to the interface.

Let, TA and TB are the temperatures of molten core phase and coolant respectively.

Applying the energy balance on the phases gives the following governing equations,

where, A and B are the thermal diffusivities and kA and kB are thermal

conductivities of the phases.

For Phase A (Molten core material phase),

2

2A A

A

T Tt y .

For Phase B (Coolant),

2

2B B

B

T Tt y .

The above differential equations are solved using two boundary conditions,

temperature continuity and heat continuity at the interface.

These boundary conditions can be written as,

At y = 0 TA = TB = Ti .

At y = TA = TA0 .

At y = - TB = TB0.

48

Solving for integration constants in the above equations using the given boundary

conditions, we get, the expression for interface temperature as,

0 0A BA B

A Bi

A B

A B

k kT T

Tk k ,

and the temperature profiles in the two phases as,

40 2

00

( , ) 21 exp 14 4 4

A

yt

A A

Ai A A A

T y t T y y yd erftT T t t ,

40 2

00

( , ) 21 exp 14 4 4

B

yt

B B

Bi B B B

T y t T y y yd erftT T t t .

Viscosity model:

The molten core phase is considered as Newtonian fluid near the melting point which

will increase as the phase cools down whereas, the coolant is assumed to be inviscid.

The viscosity of molten core material phase depends linearly on the temperature48.

Thus, the expression for viscosity can be written in terms of the temperature profile in

phase A as,

01

1

14i iya T T T erf b

t (A-2.1)

where, a and b are constants. We can simplify the above expression using the

following approximate expression for the error function49,

49

12

2

22

4

1 exp1

axerfx x

ax where, 0.14a .

Hence, the expression for viscosity (eq. A-2.1) can be rewritten as,

12 2

21

211

1

4 0.144

1 exp44 1 0.14

4

yty yerf

tt yt

.

Other derivatives of viscosity will be obtained by original relation (eq. A-2.1),

20

111

2 exp44 i

a yT Ty tt

2 2

013/ 22

11

2 2 exp44 i

ay yT Ty tt

Results and Discussion

The variation in viscosity due to the temperature gradient in the molten core material

phase is observed (figure B.1). The range of variation in viscosity was measured and

instability analysis (growth rate calculation) was performed at various points on

viscosity profile. The maximum growth rate and wave number calculated at each of

these points are found to overlap each other (figure B.2). Hence we can conclude that

the variation in viscosity due to heat transfer have no much effect on instability.

50

Figure B.1. Variation in viscosity with distance in y direction.

Figure B.2. Effect of variation in viscosity on the most unstable wave.

51

Appendix C

The last section proves that the variation in the fluid properties does not affect the

breakup in significant manner. Hence, in third development, effect of surface

solidification was included in the instability analysis. A solid crust of certain

thickness, calculated from heat transfer calculations, is assumed at the interface. This

crust resists the wave growth and a normal bending stress is developed at the

interface. The contribution of bending stress is accounted in normal stress balance

(dynamic boundary condition). The bending stress is at interface can be defined

as42,43, 4

4bendingdp Ddz

Now, the normal stress boundary condition (eq. A-1.8) can be re-written as,

21 2 3 3 2 41 2 1 2 1 22 23 2 0A BA A

A A A A A A Ui D

(A-3.1)

The crust calculation in this model requires the estimation of the heat transferred to

the coolant. Here, heat transfer is considered due to forced convection from an

infinite flat plate (the molten core material phase). The average Nusselt number for

the length of the plate is given by50,

0.5 0.3333

, 0.664Re Pravg L LNu .

52

The heat transfer coefficient can be calculated from Nusselt number as,

,avg Lavgkh NuL .

Now, the heat transferred to the coolant phase can be calculated as,

" avg A Bq h T T .

The crust thickness then calculated by using eq (29).Note that the calculation for

Nusselt number requires knowledge of length of flat plate. For the first calculation the

length is taken from experimental value of jet breakup length extracted from the melt

leading edge velocity vs. time plot as described earlier.

Results and Discussions

The table C.2 shows the comparison of the calculated most probable drop diameter

with the experimental value. The predicted size is much higher than the latter. This

difference could be due to the flat plate assumption taken, neglecting circular

curvature of interface. Typically, the heat transferred from flat plate is much higher

than cylinder. Hence, the crust thickness calculated at the interface comes out to be

greater in this case and leads to bigger drop size. The next advancement in model

should be inclusion of curvature and formulation in cylindrical co-ordinates. Similar

analysis was performed neglecting the viscosity of molten core material phase and

predicted droplet size is compared with the earlier (where viscosity is included). Both

the predictions are almost the same, which signifies that once the surface

solidification is triggered, the viscosity becomes less important. Therefore in the final

model viscosity is neglected for the simplicity.

53

Table C.1. Comparison of calculated fragment size for Wood’s metal and Water

system ( non-boiling MFCI ) with IGCAR experiment28.

Droplet diameter

Experimental Value (Narayanan et al., 2006)

4.75 mm

Simulation (Viscous molten core material phase)

9.90 mm

Simulation (In-viscid molten core material phase)

9.78 mm