Modelling of heat and mass transfer processes during core melt discharge from a reactor pressure vessel

N~e~

Engineenng a Design

Nuclear Engineering and Design 163 11996) 191 206

ELSEVIER

Modelling of heat and mass transfer processes during core melt discharge from a reactor pressure vesse! T.N. Dinh, V.A. Bui, R.R. Nourga!iev, T. Okkonen,

B.R. Sehgal

Royal In~thute of Teclmolog.r. Dirision ~!/".%ldear P,'~lce,"~'aji,t)'. Brim,/h'h~gen 60, 10044 Stock/iob~, Sweden Received ~ November 1995.

Abs~racq The objective of this paper is to study the heat and mass transfer process~ rdated ~o core melt discharge from a reactor vessel in a light water reactor severe accident. The phenomenology modell~c~ includes the convection in, and heat transfer from. the melt pool in contact with ~he vessel lower head wall, the fluid dynamics and heat transfer of the melt flow in the growing discharge hole and multi-dimensional heat conduction in the ablating lower head wall. A research programme is underway at the Royal Institute of Technology (Kungliga Tekniska H6gskolan, KTH) to (t) identify the dominant heat and mass transfer processes determining the characteristics of the lower head ablation process: (2) develop and vaEda~c c~dent anaiytieat/eomputational models for these processes; (3) apply models to assess the character of the melt discharge process in a reactor-scale situation; (4) determine the sensitivity of the melt discharge to structural differences and variations in the in-vesscl melt progression scenarios. The paper also presents a comparison with recent results of vessel hole ablation experiments conducted at KTH with a meR simulant. Keywo:ds: Modelling: Heat and mass transfer, Core melt discharge: Reactor pressure ves~l

1, Introduction and background in a light water reactor core meltdown accident (severe accident), the molten core material may cause Failure o f the lower head of the reactor pressure vessel (RPV) if sufficient internal or external cooling o f the vessel is not provided. Depending on the vessel design and accident sequence in question, the lower head integrity m a y be lost due to global or local creep rupture o f the lower head wall or, if the lower head contains penetrations, local penetration failure (Rempe et al., 1992). "11,.' initial failure site will enlarge rapidly due to heat transfer from the ejected melt

(corium) which is at a much higher temperature than the vessel melting point. Melt-induced loads on the containment a n d any further accident progression, involving interactions between the core melt and tbe coolant, structures and atmosphere in the reactor cavity of a pressurized water reactor (PWR), or in the pedestal (lower drywell) and suppression pool o f a boiling water reactor (BWR), will largely depend on the melt ejeclion characteristics. Previous work on melt interaclions with the vessel wall has been largely analytical (Rempe et al., 1992), except for the experiments conducted at Sandia National Laboratories ($NL) (for a rele-

0029-5493/96/S15.00 t~ 1996 Elsevier Science S.A. At[ righis reserv~i

S S D I 0029-5493195101167-6

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T.N. Dinh ~! al,[ Nuclear Eulz#aevr#ag and Desigu 163 (1996) 191-206

rant review, see Pilch 0993)). These experiments on hole ablation were performed with iron-alumina thermite and covered a limited hole enlargement range. Indeed, analytical relationships have shown that past experiments did not cover the range of characteristic ablation typical of either local penetration failures or circumferential vessel creep rupture (Pilch et al., 1993). The experimental data obtained have been used by Pilch 0993) to develop a one-dimensional model, equating the energy required to melt the wall material to the energy transferred from the molten fuel passing through the orifice. By comparison with former studies (Zion, 1981; Sienicki and Spencer, 1986), in order to be conservative, Pilch (1993) employed the difference between the temperature of the mollen fuel (T c} and the melting temperature of the wall (T,,.,~p), instead of the difference between the temperature of the molten fuel (Tr) and its melting point (Tr.~p). However, it is not clear how the assumptions on the physical mechanisms and correlations made in different models of the hole ablation process will hold for reactor-scale situa. tions. The lack of experimental evidence supporting, or contradicting, the chosen melt ejection and hole growth models was also emphasized in the recent DCH study for the Zion PWR (Pilch ctal., 1993). Melt ejection and lower head ablation experiments, using an o×idic melt material (Tr~ 10001500 K) discharged from a vessel with a low-melting-point metallic lower head (T~,.,~ ~ 600-900 K), are underway at the Royal Institute of Technology (Kungliga Tekniska H6gskolan, KTH), Stockholm. So far, we have employed the oxidie melt mixture PbO-B:O~ (80:20 (wt.%)). It has a melting point of about 900 K, and its melt phase viscosity of about 0.1 Pa s increases with solidification, a characteristic of the core melt as well, In the scoping tests on vessel ablation, pure lead whh a melting point of 600 K has been used as the tower head wall material. The integral scaling is based on the analysis by Pilch (1993). It was found that melt volumes of the order of 10-100 I are needed to model prototypic conditions, in which the initial lower head failure site flow rate is small compared with the vessel melt contents. In the scoping experiments performed so

far, with melt volumes of abom 3-'/ I. lhe melt has a substantial superheat and the lead plate thickneqs is varied in the range 2-4 cm (Sehgal et al., 1994). A detailed phennmenotogical model has been developed to support the experimental design and to analyse the results obtained, in the present paper, we describe this model. For reactor safety analyses and accident management considerations, the primary interests are the hole growth dynamics (Dhows(t)) and melt discharge flow parameters (melt flow rate, superheat. composition). The phenomenological considerations are built around three key elements: the thermal hydraulic behaviour of the core malt in the vessel lower head, the fluid dynamics and heat transfer of the mc|t flow in the ablating hole and the thermal and physical (phase change, mass transfer) response and feedback of the lower head wall (see Fig, I). During the core melt discharge, the com,,~,liv¢ heat fluxes (from melt flow to discharge hole boundaries) are the driving mechanisms for vessel wall ablation. Thus lhe heat transfer characteristics of a laminar entry region in experiments and of a turbulent entry region in prototypic situalions must b¢ analysed accurately. Based on a lentative identification, ranking and evaluation of related physical mechanisms, the most important phenomena are as follows: (I) crust formation and relocation dynamics; (2) temperature dep,.'udeuce of the melt properties; (3) multi-dimensional heat conduction and ablation front propagation in the vessel wall beneath the crust.

thickness- lSc~.

"~" '

Pcont

~,\\

~ - \ initial hale' D - (5~15) cm

Fig. I. The vessel hole ablation ,.+,cenario.

T.J~. Dinh el aL ~Nm'lear F~ag#~eerosgand Design I63 fl9961 191-206

The basic objective of model development is to study the scalability ot" experime,ual results, and the uncertainties inherent in such extrapolation due to modelling and data uncertainties. In order to ensure direct applicability of the data obtained. the prototypic nature of the experimental behaviour of the melt flow, heat transfer, wall behaviour and crust integrity must br established. Since this is hard to achieve, we must address the scaling distortions in our tests and their relevance to the reactor case. Analytical modelling helg's considerably in this task. Separate effects data are employed to validate analytical modelling. The models described below employ, whenever reasonable, only first-principle formulations or well-supported assumptions on physical mechanisms. Calculated results guide the applicability of available correlations for heat transfer and friction under prototypic and experimental conditions of interest. Furthermore, new or modified correlations can be introduced in the integrated model HAMISA {Hole Ablation Modelling In Severe Accidents) developed in this work.

2. Modelling of reel| discharge and vessel wall ablation The modelling efforts emphasize the integrated thermal hydraulics of the melt ejection and lower head ablation processes. A model named HAMISA.ID was developed to perform scaling analysis and to support the experimental facility design. The model considers two main cases: (ll hole ablation in, and core melt discharge from. the RPV in the protolypic case:, (2} experimental cases with metallic material as the lower head and an oxidic mixture as the corium simulant. The mathematical models of HAMISA.ID include transient mass and energy conservation in the melt pool, a set of transient, one-dimensional equations of mass. momentum and energy conservation of the melt flow in the growing discharge hole, as well as the closure relations required. These equations form a set of non-linear differential equations integrated, within a time loop, and along the hole. The fourth-order Runge-Kutta method is applied for numerical solution. The

193

discharge flow rate is determined frr ~1 the socalled P-SOLUTION algorithm. The ~essel wall melting calculations are performed in an axisymmetric geometry while tracking the ablation front. Thus the model provides the hole ablation rate, /)l,,,~¢lt,:), and the melt mass discharge rate, U , ~ tiofl(th as a function of time, which can then be compared with the data obtained in the experiments. The melt properties, e.g. viscosity and thermal conductivity, are varied as a function of temperature. The accuracy of the numerical methods employed has been tested against analytical solutions available for limiting cases. The HAMISA. I D model has been used to describe the SNL tests ¢Pilch, 1995), with reasonable agreement between prediction and data. This comparison was presented elsewhere (Sehgal et al., 1994). in general, the results obtained from the one-dimensional dynamic analysis are similar to the results obtained with the simplified models of Pilch (1993}. This ~_an be explained by the short time period of the discharge process analysed and the similar approach applied to define the ablation rate. For the SNL tests, the large melt superheats and small temperature differences between the melting points of the thermite melt and the vessel wall metal may have precluded crust formation. Calculations performed for reactor-specific situations demonstrate significant bifurcatiot~ of ablation dynamics, depending on whether or not a stable crust layer exists. The presence of the crust leads to much lower rules of vessel wall ablation in the initial phase, amplifying thereby the roles of pre-heating and heat conduction in the vessel wall and test plates. Due to such spatial and temporal complications, detailed multi-dimensional and dynamic modelling has been perlbrmed to support experimental design and interpretation, as well as the application of the measured data to proto. typic reactor accident conditions, Z I. Me# pool thermal hydraulics

The objectives of modelling the thermal hydraulics of the melt pool for the hole ablation process are to evaluate the thickness/compositlon of the crust lying on the lower head prior to the discharge process, and to model the forced con-

194

T.N. Dinla et .L[ Nuclear Engineering and Design 163 fl996) 191-206

veetion heat transfer during the melt discharge process from the RPV. For this study, a model was developed and_ employed to describe the twodimensional fluid flow and heat transfer (Dinh and Popov, 1993). The modified low-Reynoldsnumber turbulence model was applied to predict the heat transfer characteristics from the melt pool to its crusted boundaries (Dinh and Nourgalley, 1995). The heat fluxes obtained were then employed to calculate the quasi-static thickness of the crust between the melt pool and the vessel wall (in this section, the crust refers to solidified melt on the inside of the RPV lower head). During the melt discharge process, heat flux from the melt flow to the crust-vessel wall q~p can remelt the crust (solidified melt) layer. Calculations perfunned show thai, under both experimental and prototypic conditions, the melt flow in the melt pool during the discharge processes is mostly laminar. This is due to the low vessel overpressurization in our experiments and the large ratio between the melt pool radius and the discharge hole radius in reactor cases.

2.1.1. Heat transfer results The laminar flow model was used to analyse geometry and regime effects on heat transfer to the top surface of the vessel wall during the melt discharge process in small-scale experiments and in prototypic situations. The oxidic melt simulant and core melt were employed as respective working fluids. For the experimental conditions, the Nusselt number on the upper surface of the crust layer above the vessel wall, Nu.p~ was found to depend on the Reynolds number (Rehol~), the ratio between the pool radius and the discharge hole radius (rroo~frh~,c)and the dimensionless distance from the hole inlet (r*) as follows

A limited number of calculations were performed for r.'actor-specific conditions, where, in general, the Nusselt number NUupalso obeys Eq. (1). Thus this correlation (_+20°/o for near-bole regions) can be applied to assess the remelting process of crust overlying the top surface of the vessel wall and the experimental test plate.

2.1.2. Gas blowlhrough We are unaware of extensive computational efforts for predicting gas blo',vlhrough dynamics, and believe that it would be very difficult to develop a reliable computational scheme to describe this process. Several correlations have been obtained from previous studies, which have been derived mainly from dimensional az~alysis, and with fitting of the experimental data (see, for example, Smoglie and Reimann (t986)). Gluck et al. (1966a, 19661>) proposed a correlation for gas blowthrough onset in flat and hemispherical-bottomed cylinders H~l'gb =0.43D~ItanhfFr':zDh°'~ (21 Dhos~ Dhol~ ~, Dp~, ] Recently, Pilch and Griffith (1992) have colleered and examined a number of correlations describing gas blowthrough, with respect to their applicability to core melt discharge from RPV. They foo=,d that only the Gluck correlation embodies the effects of tank diameter, and the data base of this correlation s~ans the range of Froude number F r = Uc~li,,./~/(gDl~ole) and the Dpoo:/ Dbok values needed for reactor applications. Therefore the Gluck correlation was recommended for use in DCH analyses. However, it was found that there were no significant effects of the ratio Dpoo~/Dho,cfor the RPV hole ablation and melt discharge conditions (i.e. the ratio Dvool/Dh~~ is reduced from Eq. (2) for Dpool in the range 3.6-6A m and Dholein the range 0.05-0.5 m). in the present study, we are interested in the onset of gas blowthrough, rather than the annular gas-liquid discharge flow dynamics. For this putpose, a quasi-steady two-dimensional formulation is used 1o calculate the flow and pressure fields in the experimental crucible and RPV lower plenum, with melt discharge through holes of di~,'rent sizes. It can be shown that the boundary layer thickness in the hemispherical pool is so small that the whole flow field can be treated as potential flow. Nevertheless, the grid independence of numerical solutions was examined, based on th~ results obtained with computational meshes of different refinements. Calculations employing a low-Reynolds-number turbulence model indicate that shear-induced turbulence generation takes

T.N. Dhth l't al. Nudear Engmeeri:lg and D~,sign 163 (Iq96] I9t-206

Fig. 2+ The Ye|ocity field during the core melt discharge process: [.~,..,a,~m~ I0 m S *: Dladt:=O.~ m: D g p v = 3 . 6 m. T h e inler~'M between the velocity isolineS is I.~'4 m s I Tile

computaliona[ mesh for Ihe Io~¢r plenum section is 50 x 511 place only near the hole inlet and does not afl'ect the pressure field as a whole. The laminar model is thus applied to c"alculate the flow and pressure fields,

Fig. 2 describes the calculated isolines of the velocity field in a hemispherical lower head during the core melt discharge process through a hole of 0.5 m diameter. In this figure, the uppermost aizd lowermost isolines correspond to :,pproximately zero and approximately 12.4 m s ' ~ respectively. The calculated results of the dynamic pressure field were analysed to evaluate the potential "'crater formation" in the melt pool. First. dynamic pressure distributions at various heights from the lower plenum bottom are compared with the hydraulic heads of the respective melt columns (see Fig. 3). Our hypothesis is that. at the critical pool depth, the dynamic pressure variation above the discharge hole is equal to the corresponding hydraulic head. thereby inducing enough deformation of the free surface such that gas entrains in the discharge flow. Fig. 4 depicts the technique used to determine the blowthrough onset. A com. parison of the numerically determined critical pool depth with the experimental correlation of Gluck et al. (1966a, 1966b) for the hemisphericalbottomed cylinders is given in Fig. 5. It is seen that good agreement is achieved for the parameter ranges of interest.

195

It is worth noting here that the present work assumes at least partial depressurization of the reactor coolant system prior to the melt discharge, i.e. the discharge f~ow rates are in the intermediate range of 3 - [ 0 m s -t. The moment when a melt pool crater starts to dominate the flow patterns, in reactor cases, can be evaluated in the integrated HAMISA model using the Gluck correlation. In addition, the relative critical pool depth (k,~,,,, = Hr~,,,t.~,,,D,,.t,) decreases with increasing discharge hole diameter. It is found that annular-type discharge flow regimes can occur only at the very end of the ablation and discharge process. For an enltial melt ma~s of 100 tonnes, the fractions of melt mass discharged in the annular-type regime have been evaluated to be in the ranges 2%-5% and 4%~10% for cases with and without crust respectively (note that the crust affects the hole growth and hole size). 2.2. Discharge hole thermal hydraulics in this section, we present some results on the core melt ejection profess through a circular hole D h01e= 0.Sin, U e ~

~ t0n~$

0e+o0

a. m

s

N

~Oe+040.G

O,2

0,4

O,6

0,8

1,0

R(rn) Fig. 3. Calculat~l ~Pd a| d slribLa pus o f Ih¢ dynamic pressure in Ihc core m c h p o o l

196

~:N. Dinh e / a l l Nuclear R,tghwering amt D¢,~'ign Ib3 (I')96) 191 206 Thlt ~!~lll J hlo[e(liltnl~Ir rl_holl = 0,15m

!!

1. Experhm,,ml cmul#io,s Since the Reynolds numbers in the experiments are sufficiently low, laminar fl0w is the most probable experimental regime. This is why detailed analysis of thermal hydraulics within the discharge hole under experimental conditions must ix carried out in order to address the relevance of experiments to the prototypic reactor conditions. 2.2.1. I. Pre~,s,re drop. The apparent Fanning friction factor corresponds 1o the pressure drop in a certain flow length through the duct. Ir: a tube with a rricdonless infinite upstream section ( < : < 0}. the apparent Fanning friction factor is defined as follows

L~o(:. ) =

FIG. 4. Oelerrainalion o f gas blowthrough onsel from Ihu

dynamic prcssure and hydraulic head of the liquid column above the hole ( © corresponds to Ihe cfiti~.al pool depthl.

in the RPV lower head. A control-volume-based model was previously developed to solve the Navier-Stokes and energy equations in two-dimensional axisymmetric, narrowing channels with a constant wall temperature boundary condition (Dinh and Popov, 1993). while accounting for axial diffusion of both momentum and heat as well as viscous dissipation. Additionally, a 10WReynolds-number model of turbulence was e m ployed, due to the presence of laminar. transition-to-turbulence and turbulent regimes in the reactor-scale applications. By employing the first-principle modelling approaeit, it is possible to examine the effects of specific variations of viscosity and conductivity across the boundary layer due to solidification of the core melt near the wall, of velocity/temperature profiles at the hole inlet, of (narrowing) channel geometry and of the fluid Prandd number, For the limiting cases of interest~ the calculated results compare satisfactorily with previous boundary layer solutions and the measured data found in the literature.

P( - ~'~)- P(=') 2Reh,,k:.

(3)

The results presented in Table I assume that the viscosity varies strongly in the sublayer (0.95 < r/rh,,¢ < I). The maximum value of the viscosity at the wall is 25 times the bulk viscosity. It can be seen that. in the vicinity of the hole 1.6

IF -- II p r ~ wori(, 0 ¢~ole= 5~¢m : -- e)gJ.con'~a~on, O_ta~ ~ 5.¢m 1.4 A---~sse~worK, D_rm~,= TScm exp.com~ation, O_hole = 15cm t - - - e ~ese~ worg, D = 50¢m I i exp~om~aliOn, D_hole : 50cm

1.0

j

...,.II

/s

-~o

j f fa',~

~" 0.6 I

~"s'~"

O.4

"

! ,~/~f,~

o.o

25

s.o 7.5 Oiscffarge rate~, rnls

~0.o

Fig. :5. Comparison wRh the expcrimenlal correlation of' Gluck el al, (1966a, 1966b).

T.N. Dinh er al./.Vtrrh'ar Engim'¢.r#tg air# De.~igu 1'63 (1096) Iql. 206 "[fable 1 T h e a p p a r e n t F a n n i n g fri¢lion faclor

-'. = A:/(D,,,,~,Re+,,,,¢)

A p p a r e n t frict~ort factor Itl~ m vat

iI1~ m ¢on51

3.t38 ~ 10 ~ 1.256 × 10- a

464.3 170.9

222.8 11)-~.9

7.034 ~ 10-a

60.9

44.7

9 . I 6 6 × 10 J

19.4-

16.3

entrance, lhe viscosity variation leads to more than twofold higher v~!ues of the apparent Fanning friction factor, it should be noted that the thickness of the metallic plate, and thercb)- the hole length &c. in the experiments corresponds to z, in the range 10-4-10 "~+ It is seen that the friction factors are sensitive to the viscosity variations in this region.

Z Z I.Z The Nt+s~elt number. Table 2 presents the results of heat transfer calculations for four eases. The first ease is the standard case without property variation. The second case shows the effect of the above-described variation of viscosity across the sublayer on the heat transfer. The third and fourth cases include variations of both viscosity and conductivity across the sublayer. The conductivity is decreased parabolically near the wall. Values of the fluid conductivity in the wall-fluid interface are two and ten times smaller than that of the bulk conductivity for the third and fourth cases respectively. The Reynolds number is constant for the four cases, and equal to 500. In fact, the effects of the property variations on heat transfer are even greater than shown in Table 2+ since the discharge mass flow rates are also reduced due 1o increased wall friction (see Table 1 for the apparent Fanning friction factor). Similar catculations have also been performed for the reactor eases by employing the low-Reynold.~number turbulence model Most notably, there is also the development of a laminar boundary layer flow in the very short section of the discharge hole (:_ ~ 10-*+'k However, there are two differen+, factors determining the effects of the temperature, dependence of the fluid properties on the heat transfer rates. First. the

197

lower the value of : . + the more significant the effects of the fluid properties, due to the smaller thickness of the boundary layer with property variations. Second, the Reynolds numbers are higher under prototypic conditions, which limits tile property variations along and across the flow sections. In the present range of applications, we can conclude that the temperature dependence of the transport properties needs to be accounted for in an evaluation of the pressure drop and heat transfer coetticients inside the discharge hole. However. it is not known how the properties vary in the temperature range near the curium melt solidus point. We believe that experimental data, employing melt simnlants wilh temperatare-dependent viscosity, are required at relativdy high Reynolds numbers to reduce uncertainties in the assessments of convective heat fluxes and in the formulation of boundary conditions for the prototypic melt flow with a wide mushy region (ATm,~l~r=(Tliq.id~+T~,,,,~..)= 150 200 K). Analyses performed for converging ducts show dual effects of such geometry: narrowing channels cause a flow laminarization (for the same Reynolds number determined on the hole's local diameter and flow velocity), but can also subject channel wails to the hotter melt entering from the pool into the hole-wall boundary layer. Such aspects require further model development, e.g. coupling melt thermal hydraulics in the hole and the pool. Another analysis development issue is whether the core melt and its oxidic simulant behave as newtonian fluids, especially for cases with small melt superheats above the melt solidus point.

2.ZI.3. Discharge coefficient. In most ~,.aling studies and previous models, the velocities of the melt, ejected from the vessel, are calculated by means of Bernoulli's equation (Eq. (4)) with a discharge coeflieient CD in the range 0.6-I

v,j**,+,,°--c,,,/ where AP,,,r,.= #rgHr,,,,~ + P,~,~,,,, -- P¢ontairtrn,znt, The results obtained for melt ejected through

T,N. Do~h e! al. [ Noclear Engineering and Design 163 0996) 191-206

198

Table 2 The Nusselt number in laminar flows -'÷ =A-'/qDr~,-Peh,,l~)

Nusselt numbers

3.138 × 10-4 1.256 × I0 ~ 7.034 × I0 - ~ 9.166 × I0 -I

,u. ^" = const

p = var

p, ~,"= vat: x,,/tq, = 0.5

p, t¢ = var; h'.i h't, = 0A

22.21 11.56 5.82 3.65

t5.81 9.82 5.39 3.44

14.47 9.34 5.26 3.37

11,70 8.19 4.92 3.21

holes (in a two-dimensional axisymmetric steady state formulation) show the general applicability of Eq. (4) for the conditions of interest. Specifically, Eq. (4) applies to prototypie s;.tuations due to the low viscosity of the core melt (ttr ~ 0.005 Pa s) and the relatively small thickness of the vessel wall (L~,~,) compared with the discharge hole diameter (Dnok.} (L~jD,o~,
APeritif: ----

di*f d--~'- =

KI°~'PrUc~"~

(5)

2

dUf - mV~-~;

dV,, - p~--~

- ~,~r

-- ~ r - w('.. )pfUf{z )[O~ 2Dho|e

, ~--#rdV'-t2v, VfPko: 2prUf dDa°l~

Dhole dt

2Pr

dDhole

Dhot~

d.7

d~,,~-~

D~,,t~ d: _l

(6)

wilh APr.~.hot~=AP, olc-APoati~ as the given pressure drop in the hole. These equations, together with the mass conservation equation, comprise a method to calculate the melt ejection rates from a vessel which is more general than the conventional Bernoulli's equation, As already mentioned, this method is essential when analysing the oxidic simulant flow through a hole in the experimental test plates. Calculations performed for reactor-scale melt discharge processes show an increase in hole pressure drop due to the temperature dependence of the melt viscosity. However, as shown above (see Section 2.2.I), the fluid-wall friction inside the hole has a minor effect on the ejection rateS.

2.2.2. ProroO,pie couditions In typical accident scenarios, the core meh flows in the discharge hole at high Reynolds numbers (Re in the range 11)2-5 x 106) dependh~g, primarily, on the reactor system overpressurization and the size of the local failure site (hole diameter). The flow within the discharge hole may be characterized by the development of a laminar boundary layer at the entry region and its transition to turbulence. Nusselt numbers in the laminar flow region and turbulent region may be determined by Schliehting's correlation (Eq. (7)) and yon Kfirman's correlation (Eq. (8)) respectively (Schlichting, 1968) Nu = 0,332: = I''Pr- t,~

(7)

O.0288Rem= + t ~pr,5 Nu = 1 +0.85Re-l~s:=H°Pr-"t°~Pr - 1 + In[l + (5/6)iPr- I)]} (8)

T.N. Dinh et al. t Nm@ar Engitu'erh~l~ and Desig~ 163 (1996) 191-206

The Eqs. (7-8) were obtained, originally, for boundary layers over the flat plate, they are u~ed here wlth Nu numbers based on the channel diameter, These correlations are compared with the results of this study, employing numerical solutions for the fluid flow and heat transfer within the hole, in Figs. 6-8. We can see that the method of two-dimensional turbulent flow modelling employed in the present work can reproduce the beat transfer laws related to boundary layer development. The correlations of Dittus-Boeller and Petukhov et al. (lncropera and de Witt, 1990) for heat transfer in developed flows are also given for comparison. Despite the quantitative agreement achieved between the computational results and experimental correlations, it is worth noting that there arc significant difficulties in turbulence modelling of thermally and hydrodynamically developing flows. An attempt has been made to account for anisotropic effects in the thermally developing and developed f[ows (Bui and Dinh, ]995). However. the absence of experimental data for =+ < 10- ~ still renders

199

Oxid~.Jrr~t~[~ core melt. - 0.7 MPa ove,~r~sur;zatlo~ 5000 ~

~

2D I ~ m u ~

4000 ~;

----- - -/~"---~ ~

SchHc~i~g's ~ w ~ r kWsr m ~ l Van I¢llnn~'| turbuk~ layer mooel Diilus-8oel~r c o m ~ TH~P~IAL mcder I ~ i c ~

/

!

,

(prmn! work)

,

IQO0 -

0

oo

"[

............................ 0.1

o.2

0-3

0.4

0,5

0.6

0.7

0,8

0.9

1,0

Fig. 7. Forced convection hr.at Iransfer w i t h i n the discharge hole: Rch,,r ~.= 3 × 10~, Pr = 0.2. L is the hole ]eaglE.

Ox~iC.[fftelal[~ Oorl~ r1~11

5O0 rl~

~-I[ ~. 400 i ~

C-'--~ 2D simulation ( p r ~ n l work) -- -- - *~¢hliCNiflg'[i I~liltar layer model - - - - - Von Kannan's tudo~ent layer model :', - -- P,Oiffus-Boel~er correlate1

.i

100~

0

0.0

0.1

0.2

0,3

0.4 0.5 0.6 zJL

0.7

0.8 0.9

1.0

Fig. 6. Forc~ conv~..tionheat transfer ~ilhin the discharge hole; Reh~ = I0~. Pr =0,2` L is the hole length.

uncertain the prediction of heat transfer in the entry region of high-Reynolds-number flows (Re numbers up 1o 10: and .% in the range l0 v- 10-s). Thus further work on the validation and application of the model must focus on the effects of Re and fluid Pr numbers, as well as possible effects of the temperature dependence of the physical properties in the thermal boundary layer, Eqs, (7) and (8) have also been used in the THIRMAL code (Sienieki et al., 1993). However. the criterion for the laminar-to-turbulent transition was taken a s Re:.leans=pfU~jo=tio,Z=r.,ns/Pr=5 x 10~. This selection of Re:.tr=., is, perhaps, based on previous measurements on small-dlameter channels or flat plates, Sienicki ct aL (1993) reported that the minimum heat transfer coefficient will be either at the channel exit or at the location at which the laminar-to-turbulent transition occurs. As shown in Figs. 7 and 8, the THIRMAL model produces very sharp variations of the Nusselt numbers. We believe that this choice of Re:.,. . . . provides relatively small final hole sizes: determined by the local erosion rate.

T.N. Dinh et al.[ Ntu'lear ~zgh~eerh~gm~d Design 163 (1996) lg/~206

200

Our two-dimensional model for heat transl~r in the developing turbulent flow, however, indicates that the transition might occur at much smaller Re..,~,. Results obtained for Reynolds numbers ranging from i0 ~ up to 7.5 × 10¢' indicate Re: = 10~ as the most probable value of Re..~.... for the large-diameter (more than 5 cm inside diameter) circular channels, it was found that the fluid Prandfl number has a minor effect on Re;.~.... in the range Pr = 0.2-1. In the prototypie range of Re numbers (Re ~ 10o), the laminar-to-turbulent transition would take place near the hole entry. Furthermore, we should account for turbulence in the melt flow coming 1o the hote inlet, In such a case, the boundary layer may become turbulent from the inlet leading edge of the discharge hole. This may be seen in Fig. 9. which depicts a monotonic decrease in the Nusselt number for hydrodynamically developed flows (in the thermal entrance region) at low Re numbers (Re = 4 x 104). This is related to the fact that upstream turbulence is able to cause reductions in Re=:~,~. Presuming that forced convection heat transfer is the I~iflig gO~ melt, - 0.7 ~ i l ovlklp[ I,gstlnZIIl~n

8.000 i

O B l b u l ' H i q ' s d a ~ for Re-4,P.~..4. P¢=0,7 | ~ Oitlu|-Boeller r,:orrIlatiOn lot Re=4,E.O.. PtsO.7 | - - - - p t u C r i ~ d Pr_1-0.9 I ~ R~,,4,E,'.4. P~.O.7 - m o ~ l ~ d Pt_t [I I for Re=4.E+4, P¢~0.7

~

1 |

150

%,...

50t .... o

S

10

~S z/D_hole

20

25

30

Fig. 9. Local Nusselt number i n the thermal entrance region: R e h o ~ ¢ = 4 x I0~, Pr ~0.7.

governing mechanism, the presen! analysis indicates that the axial profile of the heat transfer coefficient, and therefore also of the wall ablation rate, will decrease monotonically towards the hole exit.

10000

-- - - -- - - L',~-L B - - -D ~

°F ............

2D simulation (pr~seot work} ,~hlichlkl g's I,~rnV~" layer model Yon Karman's tud:.ukl~t l a y s r r n ~ l Diltu so.Bael~ar can~14~on P ~ h o v correlation. THIRMAL model' t m n ~ k m

2.3. Crt¢st behaviour within the discharge hote

40¢~

O.e

0.1

0.2

0.8

0,4

D.5

0.6

0.7

0.$

0.9

LQ

Fig. 8. Forced conwction heat transfer within Ih¢ discharge hole: Rehol¢ = 3 x I06, Pr = 0.82. L is the hole length.

in the hole ablation process, the crust formation and dynamics play a very important role. The unsteady growth and decay of a solidified layer (crust) in a liquid flowing past a non-melting wall was studied by Epstein (1976), who developed an integral method employing second-order polynomials for the temperature profile. This method was later applied to calculate the process characteristics in a tube flow, with solidification in the liquid flow and melting in the initially solid wall (Yim et al., 1978}. So Far, no direct observations or measured data on the dynamic behavionr of the crust and molten wall layer within the discharge hole have been reported from simulant (e.g. Freon-ice wall system) or prototypic melt

T.N. Diuh er all Nudear En~#tevrit~gaml De.~ign 163 (1996) 191-206

Table 3 Crest formation and Temelling Parameter

Reaclor

Pc~,.,, tkg m - ~) /'/ca,,o. {Jkg - I ) q~., {MW m--')

8000 3 x 10~

~¢~r.,.,~,,~f~I(ram s-~ ,~,.~:.,~ {rnml t~r.urneL~Is) r¢~.~,o~his)

Experimenls (KTH) $000 3~ 10"

I

5

0.4

2 0.2 O.I 0.04

1

2.5 0.5

8000 3 x 10~

10 4 0A 0.02~ 0.01

material experiments, The crust formation and existence determine the rate o f vessel wall ablation, ff there is a stable crust, the melt superheat (e.g. 10-200 K in reactor cases) forms the driving temperature difference (ATra= Tr-Tf.m,), while without crust the watl ablation will be much more rapid, since the driving temperature difference is that between the melt temperature and the wall melting point (i.e. AT,~=Tf-T~,.~o=2700-1700~IO00 KL if the factor in the melt-wall heat transfer is about ten, as it may be between the limiting cases, the relative growth o f an initially small hole (proporI .'3. Pilch, 1993) may differ by a factor tional to T~f, of two or more. In order to assess the dynamics o f crust formation, growth and existence during the melt discharge process in both prototypic and experimental conditions, we have considered a set of phenomena, including conduction-controlled crust growth, crust remelting due to convective heat flux from the melt flow {q~,,,~), convection-induced crust sweep-out by melt flow and the falling film of the molten vessel wall beneath the crust. It was found that, although the remelting time periods ('Ccr.r~m~tl) are rather short for characteristic values o f the crust thickness, the remelting times are as much as 3 - 5 times larger than the characteristic times of conduction-controlled crust growth (r~,.vo~h) for the given values of the crust thickness ~ir.~,~,t (see Table 3). At the hole outlet. the heat transfer rates are relatively small, especially in the initial phase of the discharge process. Hence the crust growth and existence are dominant. Furthermore, the limiting mechanisms o f conveetion-indaced crust dynamics were consid-

81~ll 3× iOs

20 8 0.1)5 0.1J{16 0.0025

6000 2.5× l0 s

3

2.d 0.5 0.2 0,06

60oo 2.5 x I05 6 4.8

0.25 0.06 0.015

c~oo 2.5 x Ios 9 7.2 0.1 O.OIs

0.005

ered to evaluate typical values o f the crust thickhess. Order-of-magnitude assessments for crust-related parameters in reactor situations and in the KTH experiments are given in Table 4. It can be seen that the valtte o f the crust thickness is about 0.5 mm for both prototypic and experimental conditions. Such crust thicknesses are applied to the outlet of the discharge hole. The time characteristics related to the hole ablation process (ablation time, convection-controlled crust lifetime) for the prototypic and experimental conditions are similar. A number of models were developed to describe the crust formation and wall melting proo:sses. Originally. a separale layer model was proposed to model the thicknesses of the crust 6r.~,u~, and the molten wall layer J,,..,~. Previous studies (c.g~ Yim et al., I978) have employed a three.layer model with heat conduction and heat balance across the layers o f crust, mohen wall and solid wall to represent their transient behaviour. We believe that heat conduction is not the only operatire mechanism, as has been assumed in most previous studies (including the work o f Yim et aL, 1978) of the dynamics of the crust and vessel wall molten layer. There are other active mechanisms, e.g. the shear force from melt flow through the crust layer and the falling film o f t h e molten vessel wail. A simplified approach to the treatment of the crust within the discharge hole has been taken by assuming that heat conduction is the dominant process in the initial ablation phase, when the thickness o f the molten wall layer is less than a critical value (~iv,.ml < '~*.ml), e.g. 8"..ml= 1 mm for experimental conditions. The two equations o f heat conduction and phase change in the crust

202

T.N. Dinh et el./Nut'tear Eng#¢eeringand Design 163 (1996) t91-206

Table 4 Dynamics of the crust within the discharge hole Parameter

Reactor

Experiments(KTH)

Film driven crust dynamics Typical ablation ruto V.b (rams -~) Vessel:platethicknessL.~, (m) ~.~I (mr.)

3-10 (av. 6) 0.15 I

UI',Im :

0.9

I-6 (av. 3) 0,02-0.05 I 0.06-0.15 0.25

V~bL.~ll/~w,~l ( m S - - I )

Zlil....... : L~.lJ Ut~l~(s} Melt flow-drivencrusl dynamics U~t.,. (ms 1)

0.16

Reh.l¢

l0s 013164Re~o],~a 0.1-0.3 1.5-0.5 0.1 0,5

~[

I

3-10 10.7

~[

/J~, = U~.,.,),,,.,/'~ (m s -) ) rr.,.¢~.. = L... f Un~ (s) :r~,,=~= minlrt~im,~.... :n~t~,,.~) (s) (~r.=,..~Imm) (For r~r.B,o,~= r~,.~}

2.31 ~

Ti - T~.,~o (9)

q~,~

d~.,ml T i - T..m~ dt - - = ~ " ~ 6,,..m; -q~ona

(iO)

where

If the thickness o f the molten wall layer reaches its critical value, 6~,mt, the thickness o f the crust layer can be calculated from Eq. {1 I) and the heat flux imposed on the phase change interface, qlnt, is defined by Eq. (12) pf.,:ruslHfufionxru,~td~lrI'st

=

-q .....

+

"

"

_

,

(II)

(~,'f.,,,~,l~r ..... ) + (^'~,/~..a ) qim ------- ~w

~.mt*

0.3-1 990-3000 16/Reh~,= 0.02-0.1 1.5-0.2 0A5 0.6

~-- - - K'I~IC~IDA~.I ~f.Cflj.5| -

fl~Hfu~i°rt'w

0.5 0.12-0.3 0.10

assumptions need to be verified when more experimental observations and data become available.

and molten wall layers are applied d~r . . . . Pf.=ruslHfu~on.=ru~i ~ "

0.5 1.8 0"0g

( i 2)

in the present study, the calculated values o f qi.~ are employed as the boundary condition for the vessel wall heat conduclion and ablation analysis below (s¢¢ Eq. 03)). These crust-related

2.4. Vessel wall heat conduction and ablation

In this section, we examine the effects o f preheating and multi-dimensional heat conduction in the vessel wall and test plate, Preliminary assessments for experiments employing lead (a:; well as aluminium and tin) as simulants for the vessel wall show significant transients in the test plate's

temperature field prior and during melt discharge. As it takes time 0 5 - 3 0 s) for the melt discharge process to start after the first melt-wall contact, the thermal front may already penetrate through the relatively thin metallic layers (2-4 cm) o f the test plate due to the large values of the thermal diffusivity ( a = ( 2 0 - 4 0 ) x 10 -6 m 2 s - t ) o f the vessel materials (metals) used. The HAMISA.2D/ WALL model has been developed to describe the two-dimensional heat conduction whh phase change. An efficient numerical technique has been developed to solve the two-dimensional heat conduction equation with a moving phase change boundary. The idea o f the numerical method developed is to have a dominant direction o f boundary movement, i.e. the ablation front along the hole. The heat conduction in the other direction is taken into account in a semi-implicit man-

T.N, DOlh e~ al.' Nin'll,ar Engilu,i,riu,lf rim/Design I63 fl~)(ll 191 206

net, The moving rate of the phase change boundary of a given strip (horizontal layer) in the cylindrical coordinate system (r,:) is defined by the difference between the heat flux imposed on the interface (q~n~) and that taken away by heal conduction (q,,,,o) as follows (see also Fig. 1) I/jd:, t ) = x/r,>: + [2(q,., -

q=,.d)r.,At]/(p.~Hr~........ ) -

r,,

At

(13) where r,, is the position of the melting front at time to and At is the time step (At = t -- ,,,,)+ The calculated values of V~h are then used to track the phase change interface,

3. Analysis of the KTH hole ablation experiments A set of seeping experiments on hole ablation wa,q performed at KTH in 1994 (Sehgal ct aL, Immm

f

203

t994) in whicll a melt of PbO + B20 a oxidic mixt-re at approximately 1100 K was ejected through an initial hole of 10 mm diameter drilled in three lead plales of 20. 30 and 40 ram+ The melt volume employed in each case was approximatdy 5 1 (approximately 35 kg). The final sizes and shapes of the ablated holes are shown in Fig. 10, which depicts the two-dimensional character of the hole ablation process. It was also observed during the hole ablation process that the rate of hole ablation at ihe hole exit increased markedly after an initial period during which, perhaps, a certain portion of lhe plate in the vicinity of the ablating hole heated up to near the melting temperature. Aaalysis of these seeping experimems was performed using the HAMISA.2D/WALL model. Results of the calculations for cases with and without crust are presented respectively in Figs. II and 12 and Figs, 13 and 14. It can be seen from Figs. 12 and [4 that roughly similar diameters for the final hole were predicted for the two cases. Nevertheless, the transient lemperature fields in the lest plate dur-

'

i

A////J

0,04

I~lmm 0.03

i

/itili,,l,,']i

q;i I ~,' '

0.02

,t._. . . . .

I

0.01

0

I l,

la~m

Fig, 10. Final h o l e geometry in the KTH scoping hole ablation experiments: observations on three plates.

li~il//// 0

"0 20 30 40 50 60 70 60 90 I00 Hole ablalion front prooeiOla~km+mm

Fig. I I. HAMISA simulation of KTit hole ablation experiment: dynamics of hole enlargement process (ca~ I. T t Tr,m~,= 200 K: crust assumed: time step ur~cd In present

T.,V. Dh~hel al./ Nudeat E~gineering and Design 163 (1995) 191-206

204

rate,mm/s

2S.O

--

A v e r a g e d ablation Radius of the discharge hole exit, mm

20,0

/ /

-~ 15.0

I

{

/

g.O

.........

0 0 ~-~ ~ / ~

O.O

o ~

~'-~'--'--~'-~-~-~-'-~

5.0

10.0

15,0

20.0

Fig. 12. HAMISA simulation ol" KTH hole ablation experiment lease I ).

thermal and mechanical behaviour of the crust, which can limit the ablation in light water reactor sevet~ accidents. Other significant phenomena, e.g. hole flow entrance effects, multi-dimensional heat conduction phase change in the lower head wall and melt pool thermal hydraulics were also discussed in some detail. By taking advantage of these developments, we believe that it is possible to reduce the uncertainties in the quantification of the continued enlargement of the initial failure site in the RPV lower head. Calcalations, using the mechanistic models developed, have confirmed the effects of the core melt momentum and heat transport properties (p, re) and their temperature dependence. The current modelling has also highlighted the importane¢ of the walt thickness and therm,t properties in the scaling considerations for the hole ablation experiments. Further progress in model development and validation will rely on an analysis of further hole ablation experiments, The HAMISA model, when validated, will be valid for the prediction of hole ablation dynamics during the melt discharge process in prototypic reactor accidents.

difference between the melt temperature and the wall melting point. Under such conditions, the identification of the influence of the crust can be readily delineated. Analyses performed with the HAMISA.2D/WALL model help us to design the expedment~, e,g. define the initial thermal state of the test plate, set requirements for melt pouring times and design the lest plates.

4,

Conclusions

This paper describes the model development work performed at the Royal institute of Technology, Stockholm, to study the core melt discharge and RPV lower head ablation. The objective of the study was to provide an understanding of the physics of these complicated melt-structure interaction processes. Phenomenological and computational models were developed to treat together the dynamically coupled processes of melt flow, heat transfer to the yesset wall and hole ablation. The most difficult and most uncertain part of the modelling was the

003 •

i"/'

~ .f't/ / !/~~" : rj / /t

.

.~ ~. O.Ol~ 7 I

B

°

//

i ,'~ ' ~. ,' : ; .., e

i, -

,

i o.01; ,': . ~ •

r

0" ....................................................

0

10 20 30 40 50 60 70 80 ~) 100 HoEeabE~ionlmnl propagah0n, mm

Fig, 13. HAMISA simulation of KTH hole ablation expcri-

meat: dynamics of laole enlargement process (case 2, T~T~.,~p= 500 K: no crust assumed: time step used to present

T.N. Dhlh et ¢II. Nl~dear E~lg~lt,¢ti~lg o~ld De.fight 1~3 (1996) 191 ~206

~5

(qop(r* )Oh.,13/[~r(Tr- Tr.~,s,)]) 2s.o I i

--Avera~d ab,ati~r.to -m ~

P

;"1

-

P@h~d¢

Pr 20.0

r

// /

15.0

/ r*

/-"

Reh,,]= Re:

pressure (Pa) Peeler number (Peho~ = Re.,,~¢Prr) Prandtl number (Pr = pC~/Ic) radius or distance from the hole symmetry line Ira) dimensionless radius ( r * = ( r - r h , , l ¢ ) / Reynolds number based on Dh,,~ (Reh,,,~= prU~ti,,~,Dh,,iJttr) Reynolds namber based on Z (Re: =

PfU¢jcclk,nZ / lJr) g

10.0 ~..0

q

i

!

~

!

U v :,z

!

o%.o

1.-o

~o" slo

Time.

2o

slo

~io

71o

heat flux (W m -') time is} velocity in : direction (m s - ~} velocity in r direction (m s - i) : coordinate (m) dimensionless : for heat transfer data

(: + = Az/{DholcPehol¢)) 2,

dimensionless = for friction data (=. = Az/(Dh,,t~Rch,,t~))

Fig.. 14. HAMISA simulalion of KTH hole ablalion e~peri. ment ~cas¢ 2),

Greek k.ttcrs

Aekaowledgements

AP At A:

pressure difference (Pa) lime step (s) (~¢ Eq. 03)) distance from the discharge hole inlet Im) thickness o f layers (m)

K

heat conductivity ( W m - ~ K - ~) dynamic viscosity (Pa s) friction coefficient density (kg m--~) solidification time scale (s) remelting time scale (s) convection time scale (st residence time scale (s)

The authors wish to thank Dr. M.M. Pilch of Sandia National Laboratories for information on the Sandia experiments. Special thanks are due to Mr, J. Andersson for performing the seeping hole ablation experiments at KTH.

/1 P

Appendix A. Nomenclature CD

D

Jap~ Fr g h H /'/fu ~ion

L, L,~. Nu NUup

~'gro~,I11 Tremeh

discharge coefficient (see Eq. (4)1 diameter (m) apparent friction factor (see Eq. f3)) Froude number (Fr = U¢j~ti,,./

"tr¢~

gravitatiorJal acceleration (9.81 m s 2} heat transfer coefficient {W m - -~ K ~) melt pool I~,,el (m) heat o f fusion (J k g - i ) given thickness o f the RPV wall (m) Nusselt number (Nu = (hDh,,~l/K,.) Nusselt number (Nu,, o =

cond

x/(gD~,,O)

S~hscripts ab

cony cr, crust f fb] film

vessel wall ablation conduction convection crust of core melt or simulant material core melt flow boundary layer of discharge flow falling film characteristics o f the

206

T.N. Dinh er al./ Nuclear Engirwcring alrd Desiga I63 fl996) 191-206

fs gh hole int ml o pool

molten wall ]a:qcr simulant material of core melts gas blowthrough discharge hole interface molten layer reference value in-vessel melt pool

', transition

turbulent laminar-to-turbulent :r-:'~;uon

up

upper surface of the crust o:'erlying the vessel wall vessel lower head wall or its model

w References

V,A. Bul and T.N. Dinh. An approximation of lurbulen! Prandtl number in thermally developing flows. Proceedin~,s of Ire Second CFD CoHoqulom on Process Simulation, Helsinki Uni,~ersity of Technology, Espoo. Finland, June, 1995. pp. 149-163. T,N. Dinh and R.R. Nourgaliev. On turbulence modeling of large volumetric'arty healed liquid pools, Nucl. Eng. Des.+ in press. T.N. Dinh and A.A. Pop~v, Molten cori-m-reactor vessel interacdom scaling and other aspects, ASME [993 Winter Annual M~ting, New Orleans, Louisiana. USA. December 1993, Sect.ion 14C: Advanced Nuclear Power Plants. ASME Paper 93-WA/HT-81. 1993. M. Epstein, The growlh and ti~ay of a frozen layer in foxed flow, lnt. J. He.~l Mass Tranffer, 19 (1976) 1281-1288. D.F. Gluck, J.P. Gille, D.L Simkin and E.E. Zukowski, Distortion of the liquid surface under tow g conditions, in DJ. Simkin (~l.I. Aerospace Chemical Engineering, Vol. 62. No. 6L fIChE, New York, 1966a, p. 150. D.F. Gluck. J.P. Gille. E.E. Zoko,.vski and DJ. Simkin. Dislortion el" a Free suff~ee during tank clischarge, J, Space-

eralL 3 (1966b) 1691. F.P, iocropcra and D.P. de Witt, Fundamentals of H ~ t and Mass Transfer. Wiley. 3rd edn,. 1990. M,M. Pilch. Continued entargement of the initi'.q failure site in the reactor pressure vessel. Ap~ndix J. NUREG/CR.6075. SAND93-[535. $aw.dia National Laboratories. Albuquerque, NM. 1993. M.M. Pilch, personal communication. 1995. M.M. PLier and R.O. Griflith. Gas blowlhrough and flow quality correlations for use in the analyses of high pressure melt ejection (HPME) ~enls, SANDC)I-2322. Sandia National Laboratories. Albuquerque. NM. 1992. M.M. Pitch, H. Yan and T,G. Theoi~anous, The probability of containment failure by direct containment heating in Zion, NUREG/CR-6075, SAND93-1535, Sandia Nalionai Laboratories, Albuquerque, NM, 1993. J.L. Rempe et al., Light water r~clor lov,,er head failure analysis, ~IURIEG/CR~564E. USNRC, 1992. H. Schlichting. Boundary Layer Theory. McGraw-Hill. 6lh edn., 1968. BR. Schgal. J. Andersson. T.N. Dinh and T. Okkonan, Seeping experiments on vessel hole ablation during severe a,~idcnts, Proceedings of Ibe Workshop on S¢,~ereAccident Research in Japan. SARJ-4. Japan. Octob+'r-November. 1994, pp, 230-236, J.J, Sieoicki and B.W. Spencer, Superheat effects on localized vessel breach enlargement during cerium ejection. Trans. ANS, 52 (1986) ~22-524. .LJ. SLenicki.C.C. Chu, B.W. Spencer, W. Frid and G. Ldw~n. hidm. Ex-vessel melt-coolant interactions in deep water pool: studies and accident reanagcment for Swedish BWRs, Proeeed",rgs of the CSN! Specialist Meeting on F ~ I C ~ l a n t Interactions. Sanla Barb~.ra..lanuars- 5-8. 1993, NUREG/CP-0127. NEA/CSNIIR(93)8. C. $mogbe and L gcimann, Two-phase flow through small branch¢~ in a horizontal pipe with stratificd flow, Int. J. Mulliphas¢ Flow. 12 (4) (1986) 609-625. A. Yim, M. Epstein. S,G. Bunkoff+ G,A. La;mbert and G.M, Hauser. Freezing-me|ring heat transfer in a tube flow. Int. J. Heat Mass Transfer. 21 0.9781 1185-1196. Zion Probahilistic SaFely Study. Commonwealth Edison Co.. Chk,-,~go. |L, 19El.