Distribution Categor .LMFBR Safety (UC -79p)

ANL-76 -84

tRGONNE NATIONAL LABORATORY9700 South Cass AvenueArgonne, Illinois 60439

THE PARAMETRIC EFFECTSOF VARIED CHANNEL PRESSURE DROP

ON THE MOLTEN-CLADDING MOTION

by

W. L. Chen, M. Ishii,and M. A. Grolmes

Reactor Analysis and Safety Division

July 1976

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3

TABLE OF CONTENTS

Page

NOM ENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b

ABSTRACT.............. ...................... .... .7

I. INTRODUCTION. ........ ... .................... 7

II. DESCRIPTION OF THE MODEL . . . . . 10

A. Vapor-momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 11

B. Momentum Equation for Liquid-cladding Film . . . . . . . . . . . 12

C. Deflooding Process.. . . ....... 4. ... .. . . .0.0............16

D. Cladding-slumping Process..... .................. 17

III. METHOD OF SOLUTION.................... . . . . . . . . . 19

IV. SAMPLE CALCULATION AND DISCUSSION . . . . . . . . . . . . . . . 21

A. Effects of Varied Channel Pressure Drop on FFT F-typeSubassembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B. Effects of Sodium-vapor Chugging . . . . . . . . . . . . . . . . . . . 24

C. Comparison of Results for CRBR with Those for FFTF . . . . . 26

APPENDIX: Analysis of Upper Freezing . . . . . . . . . . . . . . . . . . . . 28

1. The Insula or Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2. The Reflector Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

REFERENCES......... ....................................... 31

4

LIST OF FIGURES

No. Title Page

1. Description of Cladding-motion Model. . . . . . . . . . . . . . . . . . . 8

2. Freezing in the Upper Plenum . . . . . . . . . . . . . . . . . . . . . . . . 9

3. Effect of Increased Total Pressure Drop Ap = 3pNaagCo on Clad-ding Motion for Test R5 Condition. . . . . . . . . . . . . . . . . . . . . . 19

4. Cladding Motion with the Standard Pressure Drop, Ap = pNag 4 o. . 22

5. Effect of Reduced Total Pressure Drop Ap =pNag o on Clad-ding Motion for Test R5 Condition. . . . . . . . . . . . . . . . . . . . . . 23

6. Effect of Increased Total Pressure Drop Ap = 2pNag#o on Clad-ding Motion for Test R5 Condition. . . . . . . . . . . . . . . . . . . . . . 23

7. Effect of the Total Channel Pressure Drop on the Extent of theUpper Penetration by Molten Cladding and Cladding Freezing . . . 24

8. Fluctuating Channel Pressure Drop. . . . . . . . . . . . . . . . . . . . . 24

9. Predicted Molten-cladding Motion for R5 Test Condition underChugging; Ap = PNagfo{1 + 0.189 sin[2r(t - tmo)/0.5]} . . . . . . . . 24

10. Molten-cladding Velocity and Sodium-vapor Velocity forR5 Test Condition under Chugging: tp = PNaEgo(1 +0.189 sin[Zr(t - tmo)/0.5]} . . . . . . . . .. . . .. . . . . . . . . . . . . 25

11. Predicted Molten-cladding Motion for R5 Test Condition under

Chugging; Ap = PNagio{l + 0.189 sin[2n(t - tmo - 0.25)/0.5]}.. . . 25

12. Molten-cladding Velocity and Sodium-vapor Velocity forRS Test Condition under Chugging; Ap P= P go{l +0.189 sin[27r(t - tmo - 0.25)/0.51 . . . . . . . . . . . . . . . . . . . . . . 25

13. Molten-cladding Velocity and Sodium-vapor Velocity for R5 TestCondition with Ap = pNaag o . . . . . . . . . . . . . . . . . . . . . . . . . . 25

14. Cladding Motion with Normal Steady-state Power and StandardPressure Drop, Ap = PNagto, in CRBR . . . . . . . . . . . . . . . . . . 26

15. Effect of Increased Power on Cladding Motion in CRBR . .. . . .. 27

TABLE

No. Title Page

I. Comparison of Events of Molten -cladding Motion for CRBR -typeDesigns with These for FFTF-type Designs under Normal PowerConditions . . . . . . . . . . . . . . . .. .. .. . . .. . . . . .. . . . . . . 26

5

NOMENCLATURE

6

NOMENCLATURE

A A parameter given by Eq. A.2

A' A parameter given by Eq. 18

Acnm Molten-cladding flow area per channel

Ag Vapor flow are i per channel

Ago Original flow area per channel

Cc Specific heat of cladding

D Hydraulic diameter for the flow:D 4(Ag +*Acm Pc

D Hydraulic diameter for the original channel:Do = SAgo ?-(LRc ' Dwr)

Dwr Shortest distance between fuel pins

fc Wall friction factor for the cladding

fs Friction factor for the vapor phase based on D

g Acceleration of gravity

kc Thermal conductivity of the cladding

Kfr Blockage -resistance coefficient

Lc L..tent heat of the cladding

Lch Hydraulic head corresponding to the amplitudeof chugging

'ins Length of the insultor

CF Length of heated section

Lo Upper level at the end of fuel pins4 ref Length of reflector

p Pressure

PC Wetted perimeter for the .low

Pg Wetted perimeter of the vapor phase

Rc Outer radius of the cladding region in a fuel

Rec Film Reynolds number, 4ocvcB, ic

RF Outer radius of the fuel region in a fuel pin

t Time

tbt Time when a complete blockage is formed bysolidifying of the slumping cladding

tir Time when the molten-cladding film reachesthe insulator sectio i

tmo Time when the film motion initiates

tpug Time to plug the total flow area at the upperplenum

tr Flow-reversal time

tref Time when the molten-cladding film reachesthe reflector section

trfd Time when the molten-cladding film drains totI," boundary between the reflector and theins lator

t, Starting time of slumping of molter claddinginto liquid sodium at the lower end of theheated section. at which Zdown(tst) - 0

Tc Cladding temperature

Tcp Cladding melting temperature

Ti Subcooled liiquid-sodium temperature at thebeginning of slumping of molten cladding

Tsat Se aum-saturated trnperature

v,. Cladding-film velocity

vg Vanur velocsy

vgi Inlet vapor velocity

Z Axial position

d own Lower end of the molten-cladding film

Zmo Position where the film motion starts

Z,.;p Upper end of the molten-cladding film

-Z Lower end of the melted-cladding section

Zz Upper end of the melted-cladding section

0 Void fraction if the vapor phase based on theflow area with stripped fuel pins o U Ag A.

rg Void fraction of the vapor phase based on thereal flow area (the flow area of the vapor andmolten cladding) ag f Ag (Ag - Acm)

og Ratio of flow area of intact channel to flowarea with stripped fuel pins a, Ago A

0 wr Ratio of wire to cladding mass

6 Thickness of molten-cladding film6c Original cladding thickness

6fr Thickness of the solidified layer

Opfr Pressure drop at blockage

Notch Phase-angle shift

t Deflooding parameter to account for thesmoothing of the film after flow reversal

Factor to account for various friction-factormodels

Kc Thermal diffusivity of the cladding

. Length of the melted-cladding section

hdr Level of drained molten cladding

Xfr L.ength of frozen layer

Afr(max) Maximum penetration of solidified layer

km Total length of the molten-cladding filmkm /up - /down

X84 Length of the bottom blockage

we Viscosity of molten cladding

wg Viscosity of sodium vapor

oc Cladding density

og Vapor density

DNa Liquid-sodium density

+c Liquid-cladding shear stress

'ch Period of chugging

Interfacial shear stress

X Martinelli parameter

7

TILE PARAMETRIC EFFECTSOF VARIED CHANNEL PRESSURE DROP

ON TILE MOLTEN-CLADDING MOTION

by

W. L. Chen, M. Ishii,

and M. A. Grolmes

ABSTRACT

The present cladding-relocation model was applied to

L- and R -series test conditions in which the pin bundles weresmall. The results were consistent with the experimental ob-

servation and with SAS calculations. One of the key assump-

tions of this model was that the total pressure drop over thevoided channel could be supplied as a boundary condition forthe vapor -momentum equation. The parametric study of the

total pressure drop was carefully performed for the FFTF-type subassembly, and the oscillatory pressure effect due to

the streaming sodium vapor generated by the chugging of thelower level of liquid sodium was also investigated.

There is an axial blanket region in CRBR or commercial -power -plant designs instead of a reflector in FFTF design above

the top of fuel. A significant difference exists in the thermalconductivities between the blanket and reflector, which could

delay the timings of all events of the cladding relocation for

the CRBR design case. The fuel may begin to lose its geome-try, and the molten fuel might spill out when the molten clad-ding is still around. Consequently, the vapor production dueto steel boiling might provide tne early vapor source for fueldispersal.

I. INTRODUCTION

A one -dimrensional cladding-relocation model based on a continuous filmflow with flooding and deflooding mechanisms has been developed.' The presentanalysis includes an upper film freezing in the insulator section, a plugging for-mation in the upper reflector section, end slumping of the molten cladding intoliquid sodium with eventual lower blockage formation.

The cladding-motion model considered here is shown in Fig. 1. It wasassumed that the coolant channel was voided before the initiation of molten-cladding motion. The average lower liquid level was taken at the bottom of the

Upper Level ofLiquid Sodium

- _ _ - Fuel

Solid Cladding

olten-Clod F-Imr

Sodium

Vapor

at}

Z zup

I r , c 1r down

Lower LeveLiquid Sod

Ialul

Re

R Coolant Channel

Wire Wrap

wr

Fig. 1. Description of Cladding-motionModel. ANL Neg. No. i00-75-7.

strongly dependent on the total

heated section, and the upper level at theend of the fuel pin. Then the total pressuredrop in the system, that through the length

to, was approximated by the liquid -sodiumhydro3tatic head, because plenty of bypass

area is available for liquid sodium. The's

pressure drop roughly corresponds to the

initial vapor velocity of 80 ms.

By neglecting the transverse inco-

herency in the power generation and in so-dium voiding within each subassembly

consisting of a few hur.red fuel pins, we

obtain the one-dimensional model by con-

sidering a typical coolant channel. Themodel is based on the liquid and vapor equa-

tions of motion, as well as the integral re-

lation between the film position and theliquid velocity. From these models, wehave developed a simple computer program

that includes the analysis of the upper andm lower blockage formations.

One of the key assumptions here wasthat the total pressure drop Ap over thevoided channel could be supplied as a bound-

ary condition for the vapor-momentum equa-

tion. Since the main driving force of the

initial upward motion of the molten claddingis the interfacial shear force due to thestreaming sodium vapor generated by the

chugging of the lower level of liquid sodium,

the magnitude of the vapor velocity is

pressure drop over the channel and the lengthof the voided section. Therefore the pressure-drop boundary condition is one

of the important paramters governing the upward motion of molten cladding,as well as the subsequent upper film freezing and plugging. The parametricstudy of the channel pressure drop on the molten-cladding motion is essential,including, for example:

1. Effects of varied channel pressure drop on an FFTF-type

subassembly.

? The sodium-vapor chugging effect.

The present study also focuses on the upper freezing and pluggingmechanisms during cladding relocation. There are two different sections ofpractical importance just above the fuel section. As shown in Fig. 2, the in-

sulator section stands next to the fuel section; the pellets in the insulator

8

9

section have a very low conductivity similar to that of the fuel pellets. Abovethe insulator section in the FFTF, there is a reflector that is made of solidInconel, whereas in CRBR or a commercial power-plant design, there will bean axial blanket region. In analyzing the freezing process at the upper plenum,we must make a distinction between the reflector and blanket sections, becauseof the significant difference in the thermal conductivities.

The results of the molten-cladding motion for CRBR -type reactor werecompared with those for FFTF-type reactor.

I''I1

IF I

FicSION GAS PLENUM

REFLECTOR

FUEL PELLETS

(a) UPPER PLENUM GEOMETRY

"J

' i

"/

{V [ "; /

1 _

E// Y i

L", . I

Fig. 2

Freezing in the Upper Plenum (FFTF-typefuel pins). ANL N&g. No. 900-75-70.

lb VARIOUS SOLIDIFIED LA;ER CONFIGURATIONS

I

SE IOSCI~

I oo

10

II. DESCRIPTION OF THE MODEL

A simple thermal-transient conduction model developed in Ref. 2 isused here for the thermal analysis of the fuel and cladding behavior with phasechange. By knowing the dryout time as a function of axial position for a fuelpin, we can easily determine the upper and lower ends and the length of themelted cladding section (Z2 , Z1 , and X = Zi - ZI).

If the film thickness is assumed to be uniform over the molten part. theconservation of the mass can be expressed as

6 = c-./gm' (1)

where we have neglected the curvature effect of the fuel pins. During the ini-tiation phase of loss-of-flow accidents considered in the present analysis, thevapor phase is assumed to be incompressible and no source of gas due to eitherfission-gas release or steel vaporization is included in the present analysis.Furthermore, the magnitude of the vapor velocity (20-80 ms) far exceeds thevelocities of melting boundaries, which are in the order of a few meters persecond under the standard power level. Consequently, the vapor continuityequation can be reduced to

a0vg = -vgi(t).

The one -dimensional equation of motions for the sodium vapor and themolten cladding are given by

bt( gvg) + (gv g) --gg - ?gg (2)at b p Z SPg(Ag + Acm.~

and

b b -~4g bp[(1 - Og)vcl + [(1 - g)vac -= - =

PgTg + Pcc(

+ C(Ag + Acmi)-

where Pg is the wetted perimeter of the vapor phase and PC is the wetted pe-rimneter for the flow. The shear stresses at the boundaries can be given as

af PqVgIvgIT g = 4 (1+ 300eb/D) 2

11

and

fc pcvclvcIc T 2

where f5 is the friction factor for the vapor phase based on Do (the hydraulic

diameter for the original channel) and the inlet vapor velocity vgi. The quantity

e is the deflooding parameter to account for the smoothing of the film after flow

reversal

A. Vapor-momentum Equation

The total pressure gradient along the surface of the fuel pin can be ob-

tained by summing up all four pressure gradients over the section of the meltedcladding, the section where the sodium vapor streams over the surface of the

intact cladding, the section where the molter film overlaps on the solid clad-ding, and the section where cladding freezes. By taking into account that the

inertia terms are small (being of the order of 1% of the hydrostatic head of theliquid sodium) and that the convective terms cancel out, and by assuming thatthe liquid-film thickness is relatively small compared with the hydraulic diam-

eter of the channel, we can easily approximate the result of integration of Eq. 2along the channel length by

A P gato jj-Pg vf g )zito 1 + 75--+ pfr' (4)

where Apfr is the pressure drop at the blockage formed by the freezing clad-ding in the upper plenum; it will be defined later in Eq. 7. Here we have ex-panded the cubic terms, and some simplifications have been made based on thefact that 1 - ot 1/4. The above approximation gives the frictional pressuredrop in the film section within 10% for X s0.9Xm-

Equation 4 is an integrated form of the vapor-momentum equation. Inthis formula, the overall pressure drop Ap is considered as input information.If the effect of the pressure-drop fluctuation due to the sodiurrm chugging can beneglected, Ap can be replaced by the liquid hydrostatic head. By knowing Apand X, we can calculate the velocity of the vapor phase from Eq. 4.

When the moltcn-cladding film reaches the unheated section as a resultof the upward motion induced by the sodium-vapor drag, liquid starts to freezeupon contact with the cold cladding surfaces. These solidified layers of clad-ding material over the origwal channel surface act as flow blockages for boththe molten-cladding flow and the streaming sodium vapor.

The motions of the sodium-vapor phase and molten-cladding film daringand after the upper freezing period can be analyzed by considering the additionalpressure drop caused by the blockage formation in the upper plenum. The

12

effect of the solidified layer on the motion of the two-phase gas -liquid flow is

analyzed by treating the partial blockage formed by solidified cladding as the

equivalent of a flow nozzle. By using the correlation derived by Murdock,'

we have

4 6. = fr i (1 + 1.26X)', (5)fr Do - 4 6fr Do - 4 6 fr 2

where X is the Martinelli parameter. Instead of calculating Y from the pres-sure drops for the liquid and vapor phases flowing alone through the nozzle,we use the standard Martinelli correlation4 for a pipe flow. In this c..se, thetwo-phase flow multiplier appearing in Eq. 5 can be approximated by a simplefunction of the vapor void fraction as

(1 + 1.26X)Z (). (6)\g/

Then the void fraction in Eq. 6 is approximated by the value at the inletof the upper plenum. Furthermore, if :t is assumed that the total film lengthdoes not change much during the freezing process, the pressure drop at the

blockage becomes

Apfr 'lfr g'2g91(7)

with

4 bfrDo I- CIO )5K1=- --Kf ~(Do - 47n

where m is the molten-cladding length at the beginning of freezing; i.e.,

ia m - m(tfr '

B. Momentum Equati-m for Liquid-cladding Film

Since the Reynolds number for the molten-cladding film, defined by

Rec -- 4PcVcb/w, is of the order of 1000 (at vc 60 cm/s and 5 = 0.0381 cm)or less, the use of the laminar wall friction factor" suffices for most cases.Thus we have

fc = 64/Rec, (8)

which is valid for high-shear flows. However, if the gravity effect on the shear-

stress distribution in the film is important, some modification of Eq. 8 may be

'3

appropriate. Based on the steady-state velocity profile for a thin, laminar-film flow, the modified friction factor becomes

f _ -( (c - (g9)g .)c P I ~' 4~ gj

PcV(: vcJ

For a slow transient process, the wail friction factor given by Eq. 9 ismore accurate, particularly for countercurrent flows. However, for a rapidly

accelerating upward film flow there are some uncertainties in applying Eq. 9,

because there may not be sufficient time to develop a downwardly convex para-

bolic velocity profile. In s.ch a case, the wall friction factor is expected to besomewhere between Eqs. 8 and 9.

By taking into account that 1 - a- 1/4 and substituting the wall friction

factor and the vapor velocity, we can approximate the result of integration ofthe liquid-film momentum equation (Eq. 3) along the continuous film length kmby

I d(vc%) 24CvclmSPc dt (~ c ~ g )g ~- - l Z.

Op - 'ggt0 1 + 75C(1 - Oo) + o(Am/X - 1) , (10)

(1. - c'o) o 1 + 75c(1 - /yv'o e + DoKfr/fs'to

where T = 1 for the standard wall friction factor, Eq. 8, and T = 2/3 for thegravity-modified friction factor. Eq. 9. Furthermore, in deriving the aboveequation, we have used the approximation 46cvo/Do -- I - oo

It is evident that the integral momentum equations for the vapor andliquid phases given above can cc applied to the prefreezing stage, since in thiscase the blockage resistance coefficient Kfr is zero i. Eqs. 4 and 10. The

vapor friction factor appearing in Eqs. 4 and 10 can be approximated by 0.02

for the present case, since the vapor flow is in the turbulent flow regime.

In the postfreezing stage, either a complete or a partial blockage formedby the solidified layer is left in the upper plenum region, and the molten-cladding film drains down the channel. For complete blockage formation, Kfrbecomes infinite; thus vgi = 0 and the interfacial shear-force term in Eq. 10drops out. For partial blockage, we assume that some molten-cladding filmwill be left on the blockage surface and that the two-phase resistance formulagiven by Eq. 7 can be extended to the postfreezing stage. This assumptionleads to a smaller gas velocity and a smaller upward interfacial force for thedraining molten cladding. Under these assumptions, the integral momentumequations for the vapor and liquid phase given by Eqs. 4 and 10 can be alsoapplied to the postfreezing stage of the cladding motion.

14

To solve the above equation for film motion (Eq. 10), it is necessary tospecify the relation between the molten-cladding length and the film velocity.By denoting the time when the cladding motion starts for this channel as tmo,we have

Zz(tmo) = Zi(tmo) = Zmo, (11)

where Zmo is the point at which the film motion initiates. To relate the filmmotion to the length of the film, the following postulate has been used. In gen-eral, we have

am = X(t) + thup(t) + 6Adown(t), (12)

where Xm = Zup - Zdown (the total length of the molten-cladding film) andX = Z2 - Z1 (the length of the melted -cladding section). For upward flow, inwhich vc(t) Z 0, we have:

If

t

tmo

v c (t)dt 2 ZZ(t) - Zmo,

then

t

Akup(t) = ttroo

vc(t)dt - [Z2 (t) - Zmol

otherwise,

axUP(t) =0.

For downward flow, in which vc < 0, thesection of the film is given by:

overlapped length in the lower

If

f vc(t)dt s Z1 (t) - Z(tr),Lr

then

t6adown v c(t)dt + [Z 1 (t) - Z(tr)

fr

(13)

(14)

(Contd.)

15

otherwise, (Contd.)

(14)

&Xdown = 0-

Here tr denotes the flow-reversal time from the upward to the downward di-rection. These two expressions for A)up and &&down are based on the relationbetween the integral motion of the center of mass of the molten cladding andthe motion of the melting boundary.

For the downward film motion, essentially three different ections needto be considered:

1. The film left on the unrmelted cladding surface at the upper section.

2. The film left on the exposed fuel surface.

3. The main draining film with the upper boundary moving down with

the center of mass of the film.

If we take into account that the amount of molten cladding in sections 1and 2 decreases rapidly because of the thinning effect due to draining and be-cause of the diminishing supply of molten cladding from the upper melt front,the main body of the film drains with the center of the mass of the film. Conse-

quently, we shall formulate the film motion based on section 3. Therefore, wehave, for tr t,1

Zup(t) = Zup(tr) + f vc(t)dttr

(15)and

AXup(t) = Zup(t) - Z?(t)

This parameter becomes negative when the film drained down beyond meltingboundary ZZ, thus exposing the fuel.

Note hers that, due to the existence of the uncovered fuel section, theintegral momentum equations for the vapor and the liquid phase derived abovedo not apply exactly to the case with A.up < 0. However, in view of the com-pensating effects of a widening hydraulic diameter at the bare-fuel section andthe increased frictional resistance at the lower overlapped section, the integralgas momentum balance, Eq. 4, is approximately valid for this case also.

Similarly, in the integral liquid momentum balance, Eq. 10, the masstransfer and the interfacial-friction terms for Aoup < 0 are only approximatelycorrect. However, by defining the total film length based on the main film body,i.e., film section 3 in the above discussion, we have included the effect of the

16

film thickness more accurately than by using the upper melting boundary ZZ.

Furthermore, if there is a considerable blockage formation in the upper insu-

lator or reflector section, the vapor velocity will be reduced so that the effect

of the interfacial-friction term on the film motion becomes insignificant. These

assumptions are justified also on the basis of the overall simplicity and con-

tinuity of the model.

C. Deflooding Process

Due to the increase in the interfacial resistance and possible formationof a flow blockage caused by freezing of molten cladding, the initial high gas

velocity is reduced significantly during a loss -of-flow accident. Therefore, it

is expected that the flow-reversal condition for the molten-cladding film will

be reached at a specific time. Since the transient process occurring in the

system is quite rapid, the flooding and flow-reversal conditions can be signifi-cantly different from those predicted from a quasi-steady-state condition. In

other words, the liquid inertia may have a considerable influence on the criticalgas velocity for the flow-reversal and deflooding process. In general, the ini-

tial film motion is upward; therefore, the liquid-inertia force a. ts in the upwarddirection just before the flow reversal, due to deceleration of the upward mo-

tion. This suggests that the deflooding of the molten-cladding film can be de-layed considerably and the critical vapor velocity for flow reversal can be

significantly below the steady-state flooding velocity.

Analyses of the flooding phenomena in rapid transient conditions are

almost nonexistent in the literature. Even under quasi-steady-state conditions,there are some uncertainties and conflicting results among existing floodingcorrelations when they are applied to liquid metals.6

Instead of imposing a condition leading to deflooding, we have assumedthat the rough interfacial-friction factor can be used up to the time of flow re-

versal. Thus the deflooding point is automatical predicted by solving the equa-tions of motion for the vapor and molten film. Therefore, we have

e = 1 for t < tr. (16)

After the flow reversal, the roughness of the interface decreases grad-

ually, and the momentum exchange between the phases is reduced. For t > trand c z 0, the deflooding parameter e can be calculated; therefore, we have(see Eq. 4)

e = pPgg - 1 KfrLo oo (17)

fsogvz (tr)to/2Do - fto)75(1 - o '

which applies for e z 0. If the above formula gives negative c, then the filmbecomes smooth and 4 should be taken as zero in Eqs. 4 and 10.

17

D. Cladding -slumping Process

Slumping of the molten-cladding film into the sodium two-phase mixtureor sodium liquid is a fairly complicated phenomenon. During the process, so-

dium boiling and molten-cladding quenching and solidification take place. The

real situation may be further complicated if the upper blockage from the so-lidified molten cladding is nonexistent or incomplete and the sodium chuggingstill exists in this stage of the accident.

When the molten cladding slumps into the liquid sodium at t = tbt at

which time Zdown(tbt) = 0, the wall conduction and convective -boiling heat

transfer removes the stored energy in the film, leading to the solidification.The previous analysis of the upper freezing shows that the time scale of com-plete blockage formation by the wall conduction alone is of the order of 0.2 s.

The solidification time can be further shortened by sodium convection at theouter side of the film. However, because of the very short time scale involvedin the bottom plugging process, conditions following the slumping will be in-sensitive to the details of the analysis of the blockage formation. From thispoint of view, a rather simple solidification model based on the conduction heat

transfer to the cold wall alone has been used. Furthermore, any convection

effect on the molten-cladding film caused by the liquid sodium was neglectedduring the freezing process.

By analogy to the upper freezing (see appendix), the thickness of the

bottom solidified layer of cladding material can be given by

Mfr 2A' x c(t - tst) for tst > t a tb , (18)

where

Ln[Cc(Tcp - Ti)/Lc] + 1.09

3.28

The time required to form a complete blockage by freezing of theslumped cladding is

~2tb- ts =its{ = Dllc

In other words, 6fr becomes Do/4 at t = tg and the channel is closed at thebottom.

If we simply extend the film-motion model described in the previoussections and neglect any effect caused by slumping and freezing. the length ofthe bottom blockage is given by

= - vc dt -Zdown(tbI), (0)Jtsl

18

where the mean film velocity vc is given by Eq. 10 with the film length Xmgiven by

(21)Xm = Zup(t);

the sodium vapor velocity is given by Eq. 4 for t < tst.-

After completion of the bottom blockage, we have

vgi = 0 for t-rtbt. (22)

Therefore, the liquid film simply drains down by gravity and accumulates atthe bottom of the heated section; the last term in Eq. 10 is eliminated.

Instead of making a detailed mass balance in order to obtain a correctexpression for the film thickness, we replace the total film length appearingin Eq. 10 by

km = Zup(t}t) for t > tbg. (Z3)

The film thickness for t > tbt, is given approximately by

6 (tbt) c Xfr fr + ZdownDo/4 (Z,1p(tbt)

Since for t > tbt the length Xfr of the upper solidified layer, its thickness 6 fr'and the bottom blockage length Zdown are fixed, 6 becomes constant. Thenthe level of drained molten cladding can be given by

I + a'wr c/5 6tXdr y D 0 /46 -1 Jtbt

1 + awroc/5 ft+dr Do/4ao0 - 1

tzt

-vc dt for kdr < Zi

-vc dt + Xdr(tZi) for kdr a Zi,

where tZ1 denotes the time when Xdr = Z1 -

and

(25)

(26)

(24)

19

III. METHOD OF SOLUTION

The necessary input data to the present model are the properties ofsodium and cladding materials. geometry of the fuel pins and flow channels,and the initial and boundary conditions.

The initial and boundary conditions consist of the information on the lo-cations of the upper and lower ends of the melted-cladding section and the total

pressure drop over the coolant channel. As explained in the previous section,the former information is supplied from the simple thermal-transient analysisof the fuel pin;Z therefore, we have Z1 = Z1 (t) and Z2 z Z2 (t), where Z1 (t) andZL(t) are given functions. This implies that the cladding motion starts at t =tmo, where Zi(tmo) = Zz(tmo). From the definition, we have %(t) = Zz - Z1 .These relations are shown in Fig. 3 together with the times at which the clad-ding dryout and cladding melting start.

Oertopping Flo Reversol9 rip I g iuppe, Bocage

Slort h pper Plug Completed

5 80 Top 0'F rue

S60 -l Moton Clod Fe

S40 2

20 Bottom ot Fuel Idown

Low Bockage Completed-20

-

25 30 35 40 45 509 16 0 s Time trom Boiling lnitielion.s

in Treat Time

Fig. 3. Effect of Increased Total Pressure Drop Ap 3pNagfo on 1laduingMotion for Test R5 Condition. ANL Neg. No. 900-75-354.

On the other hand, the total-pressure-drop boundary condition is givenby

Ap = ap(t).

The time -dependent pressure drop takes into account the effect of the sodiumchugging and associated pressure fluctuations, and the effect of the increasedtotal pressure drop due to the abnormal accident condition. An approximateform of Ap can be given as

Ap = cpNagto 1 + sin[Zn(t - tmo - Atch)/T ch] , (27)

where cpNag 4 ch is the amplitude of the pressure-drop oscillation, rch is itsperiod, Atch indicates the phase-angle shift, and c is the multiplying factor.

20

For the reference case of normal conditions, if we average out the effect ofchugging on Ap and recall that, in this stage of loss -of-flow accidents most ofthe liquid-sodium flow has been lost, an approximate form of Ap can be givenby the liquid-sodium hydrostatic head:

Ap =-PNatog'

Here we have also taken into account that plenty of bypass flow area is availablefor liquid sodium in the core region of reactors.

By knowing the pressure drop Ap, we can immediately calculate the so-dium vapor velocity vgi from the integral momentum balance for the vaporphase, Eq. 4. For the molten cladding, Eqs. 10 and 12 must be solved simui-taneously. As can be seen from these equations, the motion of the liquid filmis coupled with the length of the film, which is related to the time integral ofthe molten-cladding velocity through Eqs. 13 and 14. Gear's method7 is usedto solve the system of two ordinary differential equations simultaneously asan initial-value problem.

A simple computer code has been developed to solve the presentcladding-relocation model. In Sec. IV below, some sample calculations usingthis code will be given.

21

IV. SAMPLE CALCULATOR. AND DISCUSSION

The sample calculation considered here is the simulation of the

R-series seven-pin in-pile tests in the TREAT reactor.8 Tests R4 and R5

were run under basically the same loss-of-flow conditions with seven-pin test

sections. The difference between these two tests was the timing of the power

termination. The purpose of Test R4 was to obtain data for an integratedsequence of events up to and including complete meltdown, whereas in Test R5

the power was terminated before fuel melting in order to preserve evidenceof molten-cladding relocation.

The dimensions of the fuel pins are identical to those of the FFTR.

The test fuel power was kept constant at 29 kW/pin until the power termina-tion. A radially flat power profile was obtained by using different enrichmentfor the center pin, and the internal spacer wires and small-diameter wire

wraps on exterior pins gave the same power-to-flow ratio for interior andexterior subchannels.

The input data Z1 (t) and Zz(t) for the present cladding-motion modelhave been obtained from the simple thermal-transient analysis for fuel pins.3

The total pressure drop in the voided section was taken as the hydrostatic

head of the liquid sodium. The inlet temperature Ti was 327C, and the

liquid-sodium density was 0.873 g/cm3 . The fuel-pin geometry was that of

the FFTF: RF = 0.254 cm, Rc = 0.292 cm, to = 214.6 cm, LF = 91.44 cm,

tins = 2.03 cm, and fref = 14.5 cm. The channel dimensions of the seven-

pin tesL are given in Ref. 8.

In the present sample calculation, tI e center-channel geometry was

taken as a representative geometry; thus, the standard hydraulic diameterDo was 0.336 cm, and the area fraction of the intact cladding in the flow area

with stripped fuel pins, 1 - 00, was 0.2658. The sodium saturation tempera-

ture Tsat was 899 C, and the cladding melting temperature Tcp was 1427C.

During the molten-cladding motion, the steel properties corresponding

to the liquid state at melting were used: c = 0.065 g/cm-s and pc = 7.36 g/cm3. On the other hand, the sodium-vapor properties were calculated at thesaturation temperature; hence, Pg = 3.12 x 10-4 g/cm 3 and g = 1.98 x 104 g/

cm-s. Furthermore, in this study the wall friction factor with the gravity-

correction term, Eq. 9, has been used, therefore the factor 'I in the liquidmomentum equation should be 2/3.

The predicted cladding motion is shown in Fig. 4. In the experiment,

the cladding failed at 2.75 s which was 2.2 s after sodium-boiling initiation;in the sample calculation, the cladding motion started at 2.8 s. This model

indicates a small upper penetration of about 5.3-cm axial length, of which

2.5 cm is the complete blockage and an initial lower blockage of 4.4 cm, asshown in Fig. 4. However, at the time of the power termination, the film

1 Overlopping Start- Flow Reversal

Upper Plug Completed U

Moren - clad Fuil

r-Bottom of uei down

lower Blockage Coiv leted

5 T,, 3016s m

TREAT Time

35 "OTime from Boiling InictiotiOn, s

Fig. 4 Cladding Motion with the Standard Pre

p - pgQ. ANL Neg. No. 900-75

drained down completely andaccumulated 24.0 cm above the

pper Blockage top of the lower blockage. Since-.:.. the power was terminated before

the fuel meltdown occurred inTest R5, this accumulated moltencladding should have solidified inthe position. Posttest analysisshowed a smaller upper blockage(sl cm) than that calculated here.In addition to uncertainties in the

45 so test-bundle pressure drop at thetime of cladding melting, anyblockage formed in the test couldhave remelted due to axial con-

-s. D duction or radiant heat transfernot considered in this analysis.

The magnitude of the lower blockage is in excellent agreement with posttestresults, and the overall physical behavior of the cladding motion is quitesatisfactory.

A. Effects of Varied Channel Pressure Drop on FFTF-type Subassembly

Under the standard loss-of-flow accident condition, the sodium vaporis generated at the bottom of the heated section by chugging of the lower liquidlevel after the voiding of liquid sodium from the fuel section. Reentry of theupper liquid sodium into the hot core will not take place, since, in whole-coreloss-of-flow accidents, the rate of evaporation generally exceeds the rate ofcondensation in the channel as long as the power generation is not severelylowered. In this case, the total channel pressure drop fluctuates due to thesodium chugging. However, by averaging out tl'e effect of chugging on thepressure and recalling that, in this stage of loss .of-flow accidents, most ofthe liquid sodium flow has been lost, we can approximate Ap by the liquid-sodium hydrostatic head as in the previous sample calculation.

To tesb the sensitivity of the present cladding-motion model to theabove assumption, and to know the parametric effects of the varied channelpressure drop on the molten-cladding motion and the plugging formation, wehave made several sample calculations. Conditions were taken to be identicalto those of Test R5. However, the pressure-drop boundary condition waschanged as follows:

Case I: Ap = }pNa8 o

Case II: Ap = PNagto'

Case III: Ap = 2PNag o'

Case IV: Ap = 3PNagle-

22

120

S

e0

60

-40

20

0

.7n i v

"cv

23

The results of the predicted cladding motion and freezing are shownih Figs. 3-6 corresponding to Cases l-IV (not respectively). Comparison ofthese figures shows that the upper penetration by the molten cladding in-creases with increasing pressure drop in the channel. However, the freezingof molten cladding at the insulator and reflector sections increases the hy-draulic re sistance, and it may lead to a channel blockage, as in Cases II-IV.

S80

.60

S40

20

20 -25 Imo 30 35L 160s

in TREAT Time

40 45Time from Boiling Ivrliolion,s

Fig. 5

Effect of Reduced Total Pressure Drop

- }pNagfo on Cladding Motionfor Test R5 Condition. ANL Neg.No. 900-73-353 Rev. 2.

50 55 60

1V

120

Fig. tEffect of Increased Total Pressure Drop

Ap PNag'o on Cladding Motionfor Test R5 Condition. ANL Neg.No. ;U-75-357.

E

C

O

80

60

40

0

Overlapping- stal

Upper Plug Completed(Flow Reversal)

1,Upper Blockage

- Top ofFuel

-t Molten Ciod Film

tup

-122

bottom of Fuel

Lower Blockage Completed

2 5 30

L 160sin hveal Time

35 40Time from Boiling initiations

45 50

Since the growth of the frozen layer of cladding is iast, the moltencladding decelerates rapidly, leading to the flow reversal. In other words,the upper unheated region acts as a strong brake for the upward motion ofmolten cladding. This is clearly indicated in Fig. 7, where the uppermostpenetration by molten cladding is plotted against the total channel pressuredrop. For example, a twofold increase in tp from PNa gt0 results in further

penetration of molten cladding by only about 10 cm. Furtherrmore, in therange of expected Ap (= PNag4o), the upper penetration by steel into the coldregion is relatively minor (about 5 cm). This corresponds to an upper block-age formation of about 2.5-cm axial length, because the frozen cladding in theinsulator section forms a thin solid layer due to its small their ma.l sink and

flow Reversol

Overtopping Top of I etl

lb

1 Molten Clod fim

Down-Bottom of Fut

' - I Loiwt bockay Completed1 l 1 l__- r

.

-

Upprmost Penetration - T ----12G by Molten ClOd Gas Plenum

100-- - -- Reflector

80 IsltrTop of Molten Clad Film3 at Flow Reversal060-

I Fuela 40-

20k

only the molten cladding penetrated

into the reflector section can contrib-

ute to the blockage formation. Theseresults are related to an FTR-typefuel element with an Inconel reflector

Here freezing occurs in individualchannels.

B. Effects of Sodium-vapor Chugging

0L ___ L_ _ __.L J This study focuses on the

Oimensionless Channel Pressure Drop, Ap/P f9 chugging effect on the channel pre s -sure drop and the molten-cladding

Fig. 7. Effect of the Total Channel Pressure Drop on motion. To simulate the oscillationsthe Extent of the Upper Penetration by Molten of the lower and upper liquid-sodiumCladding and Cladding Freezing. ANL Ncg. levels and the fluctuating channel pres-No. 9Ct-75-355 Rev. 2.

sure drop due to sodium chugging, wehave used the pressure-drop Eq. 27 with c = 1 (see Fig. 8). Typical predic-tions for cladding motion with sodium chugging are shown ir. Figs. 9-12, whichare to be compared to the cases without chugging, given in Figs. 4 and 13. Inthese runs, we have used tch = 41 cm and Tch = 0.5 s. We found that boththe amplitude and the phase-angle shift are important in the initial stage ofthe cladding motion; however, after the upper plug has formed, the chugginghad no effect on the cladding motion, ab expected.

According to the values of 'ch' Tch, and 6tc, the predicted molten-cladding-film velocity before flow reversal can be significantly different fromthe case without chugging; however, the overall predicti^- of the cladding re-location is rather insensitive to the sodium chugging when tch to < 0.3.

7-

Pc 91c

a lch

E0KJU'---

I 4Ur

,5

Time,t1 160s

Fig. 8. Fluctuating Channel Pressure Drop,ANL Neg. No. 900-76-447.

F\ 0 Rrversol

Upper Plug Comp etedof ue Upper BloCbc,.

Lower loCboqe Complered -

lamo l 'j 35 40 45 50'#- Aurirrq Irhliaro {

.n 1Rf A' Tbw

Fig. D. Predicted Milten-cladding Motion for R5 TesCondition under Chugging. Ap PNag o1 '0.180 tii(21(t - t u /0.5]}. ANL Neg.No. 100-76-448.

24

Ka'f1mo

--

--

ON 1t 0,.04 (

r~ o

Slumping Sin? 6GL r Biocqe Comple

25 ' }3G C 40 45

Time from Boning mn-t, s

l100

0

'5

60

40

20

0

-20

2

Top of Fuel

Flow Reversal

Upper Plug CompletedUpper Blockage

-- - . - - - .LYLL/L LZULL / L

Mollfn-clod Film \

I 2\X\\\Lw Betom of Fuel C

Lower Blockage Completed

5 30 35 40 45 50

} Ti!60 s in IREAT Time

me from Boling Initiation, s

Fig. 10. Molten-cladding Velocity and Sodium-vapor Velocity for R5 Test Conditionunder Chugging; Ap - PNagfo{1 +0.189 sic[2(t - tmo)/0.51). ANL Neg.No. 900-76-449.

20

16

I2

08

04

1

04

08

I

2 5 tmo 30 35 40Time from Boilng rnitiaon, S

Fig. 11. Predicted Molten-cladding Motion for R5 TestCondition under Chugging; ,p = PNaggo{1 +0.189 sin[27r(t - tmo - 0.25)/0.5]}. ANL Neg.No. 900-76-450.

100

80

60

40

20 _

20 E

-40

.60

Eu

u

0

45

Fig. 12. Molten-cladding Velocity and Sodium-vaporVelocity for R5 Test Condition under Chugging;bC = pws'041 + 0.189 sin[2w(t - tmo - 0.2;)/0.5]}. ANL Neg. No. 900-76-451.

S6 Freezing Start 80

12 f- 1 Flow Reversol -0

08 tipper Plug Completed 40

04 ,20

-04 - 20

-0 8 - -40

Slumping Stort-1 2 - 60

Lower Blockage Completed6 -b-_-80

25 'mo 30 35 40 45Time frmm Boiling Initiation, s

E

Fig. 13. Molten-cladding Velocity and Sodium-vaporVelocity for R5 Test Conditio. with Ap =

PNag o. ANL Neg. No. 900-76-452.

Several reasons for thi3 can be considered. First, the upper plug can beformed in the early stage of the cladding motion; therefore, the time intervalduring which the chugging is important is limited. Second, since the positionof the molten-cladding film is predicted from the time integral of the filmvelocity, the velocity fluctuations caused by chugging had to be canceled.

From this study, it can be concluded that the present one-dimensionalcladding-relocation model is coherent in terms of the pressure-drop boundarycondition and is relatively insensitive to the fluctuations in the channel pressure

25

-

_.

E

Freezing Stotr -

Flog Peversal

Upper Hug Completed -

I -El

S umpn 51grr

I II I I

t

E

00

26

drop. This result justifies the present approach of treating Ap as a boundarycondition in analyzing the cladding motion.

C. Comparison of Results for CRBR with Those for FFTF

The result for a CRBR-type subassembly for the case of normal powerand standard pressure drop (Ap =

W. r, ,, - ,---*~Tr--OVERLAPPING FLOW UPPEP SOLIDIFIED

20- START REVERSAL LAYER(NO PLUGGiNG)

MOLTEN ELRrUELCLAD FILM ML MELEDL

LOWER BLOCKAGE COMPLETED -

0 0 20 30 40 50 60 70 80 90 IO N 12TIME FROM BOILING INITIATION ,S

Fig. 14. Cladding Motion with Normal Steady-state Power and Standard Pressure Drop,ep - PNaF' o, in CR.R. ANL Neg.No. ;-OO-75-227 Rev. 2.

pNL';o) is shown in Fig. 14. Table I com-pares these results to those for the FFTFcase. In general, for CRBR or commercial-power-plant design, no upper freezing plugcan be possibly formed. (A detailed analy-sis is presented in the appendix.) Conse-

quently, the smaller hydraulic resistancein the upper freezing section and higher

sodium-vapor shear resulted in a strongerupward cladding motion and a further upper

penetration of molten cladding into the axialblanket region. The timings of flow rever-sal, cladding slumping, and complete block-age formation due to freezing at the bottomof the channel were delayed considerablyfor the CRER-type designs with the normalpower condition, as indicated in Table I.Note also that the time of complete melt-

ing offuel at the hottest spot of the fuel pin is 8.20 s, and the cladding-slumping

time is 8.28 s, as shown in 2ig. 14. In other words, the fuel begins to lose itsgeometry, anO the molten fuel might spill out when the significant portion ofmolten cladding is still around the active fuel region.

TABLE I. Comparison of Events of Molten-cladding Motionfor CRBR-type Designs with These for FFTF-type

Designs under Normal Power Conditions

Event CRBR FFTF

Starting time, s 2.83 2.83

Cladding overlapping time, a 2.93 2.93

Flow-reversal time, s 5.52 3.37

Maximum penetration, cm 16.7 5.3

Cladding slumping time, s 8.28 4.12

Complete blockage time, s 8.35 4.18

Figure 15 shows the results of cladding motion in CRBR under theaccident condition with increased power (twice the steady-state power) follow-ing the sodium voiding. This case sirulates the reactivity effect due to voiding

-T 1 T F.L. N ---TOP FLOW REVER~SAL UPF 'E (SOL IDIFIEP LAYER

120 [OF FUEL -,. 'NPLUGGING)

BO MOLTEN

.~-S60F I I ; FUEL MELtED 1

4G r420 FUEL

L uMETNG o

BOT [ FFUkL- LOWER BLOCKAGE COMPLETED

0 0 20 30 40 50 60 70 80 90TIME FROM BOILING INITIATION . S

Fig. 15. Effect of Increased Power (TwiceFtcady-itate power) on Cladding

Motion in CRBR. ANL Neg.

No. 900-75-228.

complete melting of cladding and

on the cladding relocation and fuel melting.The complete melting of fuel begins at3.5 s from the initiation of boiling. This

timing is much earlier than that of the be-ginning of the cladding slumping into thebottom of the channel (6.64 s). In otherwords, both the cladding melting and fuelmelting are considerably accelerated, andthe initiation of motion for both materialsoccurs at much earlier stage than for thenormal-power case.

The most significant effect of thereactivity increase due to the sodium void-ing is that the time interval between the

that of fuel decreases as the power level in-creases, and under a significantly increased power level, even the order ofthese events can reverse. On the other hand, the motion of the molten clad-ding itself before the fuel melting is not significantly affected by the powerincrease, except for the effect of more rapid pr-pagation of the melting front.

This is easily explained, since the motion of the molten cladding is essentiallya dynamical phenomenon, whereas the melting of cladding and fuel materialsis a thermal phenomenon directly related to the heat generation in the fuel pins.

Although the motion of the molten cladding before fuel melting is notsignificantly affected by the power increase, the available time for the moltencladding to relocate inside the reactor core is rapidly diminishr:I due to theaccelerated fuel melting and the cladding motion initiated at the earlier stage.Consequently, the overall effect of the increased power is quickened timingsof melting with increasingly restricted cladding relocation, particularly inthe downward direction. It is evident, therefore, that the fuel-steel pool en-trainment can occur directly under these conditions as proposed by Fauske, 9

with vapor production due to steel boiling providing the early vapor sourcefor fuel dispersal and the subsequent shutdown of the reactor power.

27

28

APPENDIX

Analysis of Upper Freezing

There are two different sections of practical importance just abovethe fuel section. As shown in Fig. 2, the insulator section stands next to thefuel section: the insulator pellets have a very low conductivity similar to thatof fuel. Above the insulator section in the FFTF (Fast Flux Test Facility),

there is a reflector that is made of solid Inconel, whereas in CRBR orcommercial-power-plant designs, there will be an axial blanket region. Inanalyzing the freezing process at the upper plenum, we must distinguish be-

tween these two sections because of the significant difference in the thermalconductivities.

In the short time scale of our concern, the insulator pellets can be

considered as a completely insulating material such that the solid claddingand its thermal inertia. govern the solidification of the molten cladding in this

section. On the other hand, in the reflector section the entire cross sectionis essentially metal; thus the thermal inertia of the Inconel reflector shouldalso be included in the analyses. In what follows we shall study the growth of

the solidified layer in these two sections separately.

1. The Insulator Section

The temperature of the molten cladding that reaches the upper plenummay be approximated by the cladding melting temperature Tcp. since it has

flowed over the solid cladding surface having Tc Tcp. Furthermore, theenergy associated with the solidification is large so that the small differencein the initial superheat* is insignificant.

The temperature of the cladding in the upper plenum before contactingthe molten metal is estimated by the sodium saturation temperature Tsat, be-cause of the preexisting sodium boiling and the vapor streaming. By takingTc = Tsat in the unheated section, we neglect axial conduction from the fuelsection through the cladding.

The solution with a semiinfinite solid'0 in terms of the thickness of

the frozen layer is

8 fr = 2A c(t - tfr), (A.1)

where Kc is the therrral diffusivity of the cladding and parameter A is a

complicated implicit function of CC(Tcp - Tsat)/Lc. However, for Cc(Tcp -Teat)/Lc > 1.0, which is applicable to our case, it can be approximated by

*Here we define the liquid superheat of the cladding as the tcmper.ture above the solidiflation temperature.