An unsteady one-dimensional two-fluid model for fuel-coolant mixing in an LWR meltdown accident

Nuclear Engineering and Design 95 (1986) 285-295 North-Holland, Amsterdam

285

AN U N S T E A D Y O N E - D I M E N S I O N A L T W O - F L U I D M O D E L F O R F U E L - C O O L A N T M I X I N G IN A N L W R M E L T D O W N A C C I D E N T S.H. H A N * a n d S.G. B A N K O F F ** Chemical Engineering Department, Northwestern University, Evanston, 1L 60201, USA

The multiphase code PHOENICS is used to predict the mixing of molten core material with the lower plenum water pool in a severe light-water reactor accident. One-dimensionalbehavior is assumed, corresponding to catastrophic failure of the lower support plate, possibly due to a preliminary steam explosion. For the dangerous range of fuel drop sizes (10-100 mm) it is found that there is rapid level swell, with most of the drops being levitated upwards in a region of high steam volume fraction. This condition is thought to be unfavorable for an efficient explosion.

1, Introduction

In a LWR core melt accident, mixing of molten core material with an underlying water pool, either in-vessel or ex-vessel, or both, will probably occur. The fluid mechanics and energetics of this process is obviously of interest, both from the point of view of the possibility of a large-scale efficient steam explosion, and also the rate and amount of production of hydrogen. Several models have been proposed. Henry and Fauske [1] postulated that the fuel drops subdivided until the steam volumetric flux upwards exceeded the Kutateladze flooding limit [2], or equivalently the Zuber film boiling vapor flux [3]. The implication was that in the dangerous range of fuel drop diameters (say. 10 to 100 ram), the void fraction in the surrounding coolant mixture would be too large to permit an efficient explosion. Fuel particles much smaller than this range lose their heat to the coolant too rapidly during the premixing stage, while particles much larger than this range require too much viscous energy dissipation for interaction on explosive time scales (Cho, Fauske, and Grolmes [4]). The mechanism of subdivision in this one-dimensional model was not specified. Corradini [5] and Corradini and Moses [6] proposed a two-dimensional model, which allowed radial expansion of the falling fuel particle cloud, and hence was also time-dependent, in contrast to the Henry-Fauske model. However, the fuel

* Present address: Korea Power Engineering Company, Yeoeuido, Seoul, Korea. ** To whom communications should be addressed.

particle diameter, the particle volume fraction, and the local void fraction were all taken to be empirical functions of dimensionless time obtained from the data. Local relative velocities were not calculated, so that fluidization criteria could not be checked. Bankoff and Han [7] performed a quasi-steady one-dimensional calculation of the mixing, in which the drop size was held constant with time. This might correspond to a catastrophic failure of the lower support plate, possibly due to the combined effects of melting from above and a small surface explosion below. It was further assumed that the fuel particle concentration viewed from the falling front, changes only slightly with time. The validity of this assumption needs further investigation. In the present work the one-dimensional, unsteady mixing of fuel drops and coolant is studied numerically, using the PHOENICS code [8]. The growth of the steam volume fraction within the water and the penetration and mixing of fuel drops are estimated, using the mass, momentum, and energy conservation equation for fuel and coolant in a two-fluid formulation with generallyaccepted constitutive relations.

2. Statement of the model

As molten fuel drops fall into the water pool in the lower plenum at their terminal velocity and the mixing proceeds, the fuel drops would be in film boiling mode. A cylindrical coordinate system was introduced, with origin at the bottom of the pool, and the z-direction is upward. The geometry considered is shown in fig. 1. The following assumptions were made: (!) Fluid 1 is

0029-5493/86/$03:.50 ©~Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

S.H. Han, S.G. Bankoff / Modelfor fuel - coolant mtxing

286

where 4, stands for any of the dependem ~ariables. a; is the volume fraction, p, is the density. ~, i; the velocity vector. I~, is the exchange coefficient and .S~ is the source term for ~j. The continuity equation for the ith phase is

O

o

o

o

@

@

O *

o

u _

0

@

@

O(:,°t'~

• 0

0

0

b*:t/ V~

2m

°.

@

o

' 0

0 0

"~"°O 0

~,V2

t~

0

e

@ ©

°O

o

0

to

o

0

0

o

3

a-i(~,o,)

O

•

+ v . ( ~ , o , v , ) = 0;

i=

1,2,

(;)

where a] + a 2 = 1. The momentum equation for the ith phase is

~ fuel drop water-steam mixture

water

i = t,2,

=-aiVP+a,pig+FO(~-v,);

(

3m

)

where F,i is the interphase friction coefficient and given by

Fig. 1. Description of the system.

~ , = ~cD P ~ . i v , - v, I.

(5)

0~1('12

a homogeneous mixture of steam and water, and fluid 2 consists of the fuel drops. (2) The water is initially at the saturation temperature, and the temperature of the fuel drops remains constant. (3) The fuel drops are spherical and of equal diameter. (4) The ambient pressure remains constant throughout the mixing process. The one-dimensional mixing assumption corresponds to a complete failure of the lower support plate resulting from a preliminary explosion near the top of the pool. Slip between the water and steam has previously been shown to have little effect on the mixing calculations [7]. It is expected that the pool will be at saturation temperature if the pressure seals on the primary coolant system are lost, as implied by heat-up calculations by Denny [9]. The temperature change of the fuel particles 40-100 mm in diameter falling through the pool can be shown to be negligible. Assumptions (3) and (4) are made for convenience, and in the absence of better information. Only the effect of hydrostatic pressure was taken into consideration. The density of the homogeneous mixture depends only on its enthalpy; Pl = Pt( a + b h l ) -1;

To evaluate the drag coefficient, the fuel drop in the film boiling mode is considered to be a bubble with the drop diameter. One thus obtains [10] 1 C D = 0.45

The governing equations for the i th phase are in the general form given by

(6)

where the interphase heat transfer, Q.ij, is given by Q i j = 6 a z ( h , ~ + h~)(T2 - T l , ~ , t ) / d z "

(8)

The convective heat transfer coefficient for forcedconvection film boiling around a sphere is given by Witte [11] as

[ (1)

17.67(1 - or2) 9/7 ]2. 18.67 ( - - - - i - - ~2 )3/--~

+

Since the interphase heat transfer is dominant over viscous stress, interphase friction, gravity and other source terms in the energy equation, the energy equation for the ith phase becomes

a = 1 - plhl.sat/pghfg,

b = pl/pghfg

(4)

pgkgh'fgV

] 1/2

h~ = 2.98 d 2 ~ 2 7 ~ l , ~ a t )

(9)

where h;g = hfg + 0.68Cpg (T2 - Tl,sat)-

(10)

The radiation heat transfer coefficient is defined as = r h , j ¢ , + aiSq,,;

i = 1,2,

(2)

hr = oe2 ( T ; -- T ~ , s a t ) / ( T 2 -

TLsat ).

(ll)

S.H. Han, S.G. Bankoff / Model for fuel coolant mixing

original pool level, so that the fuel and coolant might be expected to disperse up the downcomer, and to interact further with the core. When the increase in the mass inflow due to the pool swell was taken into consideration, given by

3. Results The initial depth of the water pool was assumed to be 2 m and the diameter of the pool 3 m. The water pool was subdivided into ten uniform cells. The time interval was 0.05 s. The pool swell phenomenon due to rapid vaporization of water and fuel inflow was also taken into consideration. If no escape of fluids from the mixing region is assumed, the swell velocity of the pool can be estimated as

O2, ~ -/'~/2 = FJ"/2,0

dH/dt

(13)

U2,oo

where rh2,0 is the initial rate of fuel inflow and v2,~ is the terminal velocity of fuel drops, the pool swells more rapidly due to the increased mass inflow of the fuel drops. The overall behavior of fuel drops is, however, very similar, whether a constant or a variable fuel inflow is considered (figs. 4 and 5). Figs. 6 and 7 show the effects of ambient pressure on the void fraction within the coolant and the volume fraction of fuel drops, respectively. The homogeneous mixture of steam and water has a lower velocity at 1.0 MPa than at 0.48 MPa, as expected, and the void fraction of water near the top surface is about 80%. Most of the fuel drops are, however, still present in the region where a > 0.6 (70% of the fuel at 1.0 MPa, and 85% at 0.48 MPa). The effect of the fuel drop size is shown in figs. 8 to 11. The void fraction increases rapidly as the fuel drop diameter decreases. However, the fuel volume fraction curves are little changed for the drop sizes larger than 60 mm.

N

E 0jaz,

dH r~2 j 1 + - dt - Ap 2 pghfg

287

(12) '

where H is the height of the pool, A is the cross-sectional area of the pool, 0j is the volumetric heat source in the jth cell, and Azj is the height of the j t h cell. Figs. 2 and 3 show the penetration of fuel drops and growth of void fraction within the water, respectively, when a constant fuel inflow was assumed. It is seen that, due to the rapid increase in void fraction, most of the fuel drops are present in the region where the void fraction is greater than 60% in less than 0.5 s. Furthermore, the penetration of fuel drops is progressing very slowly, and the level swell (shown by the vertical dotted lines) places the high void fraction region well above the

0.20 0.25 sec

-0-~r 0.15

0.5 0.6

-<>- 0.7

q::l- 0.8 0.9 411- 1.0

g

7

0.10

I

I

/ ,,=

0.05

1

I 0 0.5

1.0

1.5

I 2.0

z

I

I

I

2.5

3.0

3.5

I

(m)

Fig. 2. Fuel volume fraction in the mixing zone (constant fuel mass flow rate): P =1.0 MPa, ~n2 = 9000 kg/s,

d 2=

I 4.0

50 mm.

288

S.H. Han, S.C. Bankoff / Model for fuel--coolant mixing 1.0

0.8

i

-~

0.25 sec 0.s

A

~

0.6

I

i

-O- 0.7

I

,

I

~ 0.9 0.8 0.6 -

~d~

-11- 1.0

~

d -~

0.4

/

1.0

1.5

I

i

I

I

i i

i i

/

i I

i

2.0

2.5

0

0.5

! !

! i J

V-// 7"/

0

~

z

a !

i !

! !

! J

z I

I

I

I

i

I !

I ,

i I

{

il

i I

!

I

!

3.0

"

i

i

3.5

4.0

(m)

Fig. 3. Coolant void fraction in the mixing zone (constant fuel mass flow rate): P = 1.0 MPa, rh 2 = 9000 kg/s,

d 2 =

50 mm.

the mixing zone instead of near the top surface after 1.6 s. Although accumulation of fuel drops at the bottom of the pool has not yet been achieved, there is the possibil-

The long-time behavior after a limited amount of fuel (2250 kg) is introduced is shown in fig. 12. It is seen that most of the fuel drops are present in the middle of

0.20 - 0 - 0.5 sec

4~ 0.8

0.15 z

o

'"" X = o>

0.10

0.05

0

P_.,---

v

1 2.0

1.0

z

3.0

(m)

Fig. 4. Fuel volume fraction in the mixing zone: P = 1.0 MPa, rh2.o = 9000 kg/s ( I • = for constant fuel mass flow rate).

4.0

S.H. Han, S.G. Bankoff / Model for fuel-coolant mixing

289

1.0 - 0 - 0.5 sec

O.8 0.8 z

0.6

Fz

0.4

0.2

0

0

1.0

I 3.0

2.0

z

4.0

(m)

Fig. 5. Coolant void fraction in the mixing zone: P =1.0 MPa, rh2, 0 = 9000 k g / s , d 2 = 50 m m (m • = for constant fuel mass flow rate). ter. T h e m a x i m u m p o i n t in t h e total v o l u m e f r a c t i o n o f fuel d r o p s m o v e s d o w n w a r d w i t h time. W h e n 4545 k g o f fuel h a s b e e n i n t r o d u c e d , fuel d r o p s b e g i n to a c c u -

ity o f f o r m a t i o n o f a stratified b o t t o m l a y e r at l o n g t i m e s , if p r e m a t u r e t r i g g e r i n g d o e s n o t occur. Fig. 13 s h o w s a s i m i l a r r e s u l t for fuel d r o p s o f 100 m m d i a m e 0.20

0.15

-

0.10

-

0.05

-

0

1.0 MPa

[]

0.48

z o

it.

ta.

0

0

1.0 z

I

I

2.0

3.0

4.0

(m)

Fig. 6. Fuel volume fraction in the mixing zone as a function of ambient pressure at t = 0.5

s:

t~/2, 0 =

9000 k g / s ,

d 2 =

50 mm.

S.H. Han, S.G. Bankof[ / Model for fuel coolant mixing

290 1.0

j.~E-] 1.0 MPa

jl~] - I "

0.8 F-..

0.6 ~Z

0.4

0.2

0 1.0

t

I

2.0

3.0

z

4.0

(m)

Fig. 7. Coolant void fraction in the mixing zone as a function of ambient pressure at t = 0.5 s: rh2, o = 9000 k g / s , d 2 = 50 m m

m u l a t e at t h e b o t t o m o f t h e p o o l a f t e r 3 s (fig. 14). Fig. 15 clearly s h o w s t h e f o r m a t i o n o f a b o t t o m l a y e r at t h e b o t t o m o f t h e p o o l for 6921 k g o f fuel a f t e r 5 s.

H o w e v e r , it s h o u l d b e n o t e d t h a t e v e n t h o u g h fuel d r o p s c a n p e n e t r a t e i n t o t h e b o t t o m o f t h e pool, t h e v o i d f r a c t i o n w i t h i n t h e w a t e r is still a b o v e 50% in m o s t

0.20 4cm

/

6

0.15

10

i 0.10 ~> ,=

0.05 0

0

1.0

2.0 z

i 3.0

(m)

Fig. 8. Fuel volume fraction in the mixing zone as a function of d 2 at t = 0.5 s: P = l . 0 MPa, vh2,o = 9000 k g / s .

4.0

S.H. Han, S.G. Bankoff / Modelfor fuel- coolant mixing

291

1.0 0

[]

0.8

~

4 cm

6 8

10

Z

0.6 ka.

Z

0.4

t~

0.2

0 0

I 2.0

1.0 z

Fig. 9. Coolant void fraction in the mixing zone as a function of

d 2 at

o f the m i x i n g zone. Triggering o f the e x p l o s i o n b y fuel h i t t i n g s u b m e r g e d structural steel a n d / o r i n s t r u m e n t p e n e t r a t i o n s m a y p r e v e n t f o r m a t i o n o f a b o t t o m layer.

4.0

3.0

(m) t = 0.5 s: P = 1.0 MPa, rh2,0 = 9000 kg/s.

W h e n these results are c o m p a r e d to the previous q u a s i - s t e a d y calculations [7], it is f o u n d that the void fraction w i t h i n w a t e r g r o w s e x p o n e n t i a l l y in b o t h cases.

0.20

0.15 -

O'

4 cm

A []

6 8

--~

I0

z

---or X =

0.10

= 0.05

2.0

1.0

z

3.0

(m)

Fig. 10. Fuel volume fraction in the mixing zone as a function of d 2 at t = 1.0 s: P = 1.0 MPa,

m2,o= 9000 kg/s.

4.0

S.H. Hen, S,G. Bankoff / Model for fuel coolant mixing

292 1.0

o.8

~

0.6

z~

0.4

LI-

0

4 cm

A

6

[]

8

--L ~

-

~ _ ~

-i

.......

u 0.2

0

J

0

I

1.0

I

2.0

3.0

4.0

z, (m) Fig. 11. C o o l a n t void fraction in the m i x i n g zone as a function of d 2 at l = 1.0 s: P = 1.0 MPa, vh2,o = 9000 k g / s .

1.0

0.10

~.41-j~

0.08

t = 0.6 sec /

0.06

~~~.,'"~

oo.

/B" /

..

/.b//

0.02

/

/

.~/"

/

.......

•

~ ,IP~

0.8

~

g

.

0.6

,'/ /,,/ IA// /

11/. I.~ I /

p/~_..~-/~

0.4

,.

~

Z

z

" 0.2

1.4

~"'J'/'/" f./.7 0.00

~ 0.5

A

"

L " 1.0

.0

/.¢" V"l 1.5

n

i

I

2.0

2.5

3.0

lr8 3.5

0.0 4.0

z (m) Fig. 12. Fuel v o l u m e fraction ( ) a n d c o o l a n t void fraction ( - . - ) in the m i x i n g zone: P = 1.0 MPa, rh2, o = 2250 k g / s , m 2 = 2250 kg, d 2 = 50 m m ( . . . . ) = fuel v o l u m e fraction for u n l i m i t e d fuel flow).

S.H. Han, S.G. Bankoff / Model for fuel-coolant mixing

293 .0

0.10

t

0.08

= I sec

0.8 O

z O ~=

Z -N

0.6

0.06

O

=, 0.4

0.04

z

,,= 0.2

0.02

0.00 0.0

I

I

I

I

I

.._i

0.5

1.0

1.5

2.0

2.5

3.0

'0.0 3.5

z (m) Fig. 13. Fuel v o l u m e f r a c t i o n ( - - )

a n d c o o l a n t v o i d f r a c t i o n ( . . . . ) in the m i x i n g zone: P = 1.0 M P a , rh 2,0 = 2250 k g / s , m 2 = 2250

kg, d 2 = 1 0 0 m m .

1.0

0.10

0.08

t = I sec

~..~ ~ : ~

-~

~o= = ~

0.8

-

C'b

Z

0.06

,..,i i ~"

0.6

-~ O

0.4

--q z

~.-/

0.02

0.00

0.2

I

" 0.0

//

0.5

1.0

0"//

I

I

I

I

1.5

2.0

2.5

3.0

z

Fig. 14. F u e l v o l u m e f r a c t i o n ( kg, d 2 = 100 m m .

0.0 3.5

(m)

) a n d c o o l a n t void f r a c t i o n ( . . . . ) in the m i x i n g zone: P = 1.0 M P a , rh2, 0 = 2 2 5 0 k g / s , m 2

=

4545

294

S.H. Han, S,G. Bankoff / Model for fuel coolant miring 0.10

1.0

0.8 -J

z o

7 * - Z- ' ~ / "

g

t I

0.06

---

--

_--~

-

%

4 0.6

z

L.~

= >

o--

-*

0.04

/

-" /

/

,'/ /~//"

-

i

0.4

.

/ "

/

//

//

-

I

/

~

4

1

0.2 / /

0.00 0.0

0.5

t 1.0

k 1.5

i 2.0

J 2.5

t 3.0

_1 0.0 3.5

z (m)

Fig. 15. Fuel volume fraction ( kg, d 2 = 100 ram.

) and coolant void fraction (. . . . ) in the mixing zone: P = 1.0 MPa, rh2,0 = 2250 kg/s, m 2

However, the volume fraction of fuel drops at the penetrating front is much less ( < 10%) than assumed in the quasi-steady calculation (50%), so that a direct comparison is difficult.

4. C o n c l u d i n g

discussion

It is found that the mixing is quite uneven, owing to the rapid steam generation, which increases the pool level and impedes the downwards motion of the fuel particles. Regions of high fuel concentration have mostly steam and little water in the coolant phase, while regions of high water content contain little fuel. Nevertheless, the theoretical maximum work, resulting from a constant-volume instantaneous temperature equilibration of the fuel and coolant, followed by an isentropic expansion, maintaining temperature equality of fuel and coolant, to the reactor volume, may still be sufficient to cause vessel failure. On the other hand, the rapid level swell will cause coolant to enter the core region very rapidly (time < 1 s). This could initiate a series of interactions, so that the energy release would be spread out over a longer time.

=

6921

Nomendature

A CD Cp d FO g h h c,h r h fg k ~h rhij P ~.ij qu S T t v z A zj

cross-sectional area drag coefficient specific heat diameter interphase friction coefficient gravitational acceleration enthalpy convective and radiational heat transfer coefficients, respectively latent heat of vaporization thermal conductivity mass flow rate interphase mass transfer rate pressure volumetric heat source volumetric heat source in the j t h cell any source term temperature time velocity vector vertical distance height of the j t h celt

S.H. Han, S.G. Bankoff / Model for fuel-coolant mixing Greek letters volume fraction exchange coefficient emissivity viscosity density Stefan-Boltzmann constant any dependent variable

Subscripts 0 1 2 g l sat

initial steam-water mixture fuel drops gas liquid saturated terminal

Acknowledgements This work was sponsored by the Electric Power Research Institute and by the Department of Energy through the Division of Educational Programs of Argonne National Laboratory. Dr. David Squarer is thanked for helpful comments. Prof. D.B. Spalding, Managing Director, C H A M , Ltd., London, is also thanked for use of the proprietary code P H O E N I C S .

295

References [1] R.E. Henry and H.K. Fauske, Required initial conditions for energetic steam explosions, Proc. Topical Mtg. Advances in Reactor Physics and Shielding, Sun Valley, Idaho (1980). [2] S.E. Kutateladze, Heat transfer in condensation and boiling, AEC-tr-3770, US Atomic Energy Commission (1952). [3] N. Zuber, Hydrodynamic aspects of boiling heat transfer. AECU-4439, US Atomic Energy Commission (1959). [4] D.H. Cho, H.K. Fauske and M.A. Grolmes, Some aspects of mixing in large-mass, energetic fuel-coolant interactions, Proc. Int. First Mtg. on Fast Reactor Safety and Related Physics, Chicago, IL (1976). [5] M.L. Corradihi, Molten fuel/coolant interactions: Recent analysis of experiments, Nucl. Sci. Eng. 86 (1984) 372-387. [6] M.L. Corradini and G.A. Moses, A dynamic model for fuel-coolant mixing, Proc. Int. First Mtg. on LWR Severe Accident Evaluation, Cambridge, MA (1983). [7] S.G. Bankoff and S.H. Han, Mixing of molten core material and water, Nucl. Sci. Eng. 85 (1983) 387-395. [8] D.B. Spalding, A general purpose computer program for multi-dimensional one- and two-phase flow, Math. Comp. in Simulation 13 (1981) 267-276. [9] D.B. Denny, The CORMLT code for the analysis of degraded core accidents, EPR! Report NP-3767 CCM (1984). [10] M. Ishii and N. Zuber, Drag coefficient and relative velocity in bubbly, droplet or particulate flows, AIChE J. 25 (1979) 843-855. [ll] L.C. Witte, Film boiling from a sphere, Ind. Eng. Chem. Fund. 7 (1968) 517-518.