Liquid Metal Cooled Systems. Sodium Boiling Dynamics

Chapter 4, Part 2 Liquid Metal Cooled Systems. Sodium Boiling Dynamics Michael A. Grolmes and Hans K. Fauske CONTENTS Page Nomenclature

261

1.0

Introduction

264

2.0

Fundamental Considerations

265

Sodium Superheat

266

2.1.1

Pressure-Temperature History

266

2.1.2

Inert Gas Effects

269

2.1

3.0

4.0

2.2

Flow Regime

274

2.3

Boiling Stability

276

2.4

Dryout Considerations

277

2.5

Boiling Incoherence Effects

281

Transient Sodium Voiding Models

284

3.1

Slug Models

285

3.2

The Current SAS Model

287 290

Applications 4.1

4.2

291

Unprotected Accidents 4.1.1

Loss-of-Flow Accidents

291

4.1.2

Heat Capacity

293

4.1.3

Transient Overpower Conditions

298 2

Protected Accidents

98

259

CONTENTS (Continued)

Page 4.2.1 LOPI

299

4.2.2 Decay Heat Removal Considerations

300 303

4.3 Boiling Behind Blockage 5.0 Concluding Remarks

306

Appendix

309

References

311

Problems

316

260

NOMENCLATURE

Upper Case

Description

A

flow area

C

heat capacity

D eq

hydraulic diameter

F

function

G

mass flow rate per unit cross section area

K

constant for flow regime stability, Equ. (4)

L

heated length

M

mass of component in heat capacity model

Mc

vapor momentum lost on condensation, Equ. (19)

P

pressure

Q

heat flux

R

radius

S

surface area

T

temperature

U

velocity

V

void position in heated zone

W

liquid film flow rate

Lower Case a 1 ,a 2

Description constants in Equ. (22)

b

film thickness

c

friction factor

f

function

g

gravity acceleration

h fg

latent heat of vaporization

j

superficial velocity

k

thermal conductivity 261

NOMENCLATURE (Continued)

Description

Lower Case q

linear heat rate, i.e., kw/m heat flux volume heat source term

s

distance normal to liquid surface area around vapor cavity

t

time

w

flow rate through channel

x*

dimensionless axial distance

z

axial distance

Superscript

Description refers to maximum pressure and temperature subcooling

*

dimensionless Description

Subscripts a

axial direction

avg

average

c

clad

ch

channel

of

effective

eq

equivalent

f

liquid

F

fuel

g

inert gas or vapor

h

heated surface

max

maximum

na

sodium

o

initial or inlet condition

p

pump

3

radial direction

sat

saturation condition 262

NOMENCLATURE (CONTINUED)

Subscripts sup

Description superheat

3

vapor

w

wall

Greek

Description

0-

surface tension

p

density

a

void fraction

D( )

change in ( )

0-

axial distance into liquid slugs

a

radial distance into liquid film

t

dummy time variable of integration

t

shear stress

y

ratio of clad surface area to cross section flow area

w

chopped cosine shape parameter

B

density temperature coefficient

263

CHAPTER 4, Part 2 LIQUID METAL COOLED SYSTEMS. SODIUM BOILING DYNAMICS

1.0 INTRODUCTION Coolant voiding ithin the nuclear core of a sodium-cooled liquid metal fast breeder reactor (LMFBR) is considered only in analysis of the consequences of low probability accidental events. Unlike current light water reactors, (LWRs) the primary coolant circuit in an LMFBR is a low-pressure (<_1 MPa) highly subcooled, nonboiling heat transport system. Because of the association with accident conditions, sodium voiding, with few exceptions, is a transient or dynamic condition. Sodium voiding can be postulated to result from power- and flow-transient conditions that overheat the coolant to its saturation temperature, leading to boiling, rapid fission gas release, fuel release, or in most whole-core accident conditions some combination of the above. Both rapid and slow fission gas discharges into the coolant channel have been analyzed (1) (2). It has generally been concluded that fission gas release, by itself, especially the more realistic slow gas release from a single fuel element, is an isolated event with no immediate severe consequences. Analysis and experimental data are well-documented in References (3) and (4). Likewise, sodium voiding resulting from the release of small quantities of molten fuel postulated to result from local fault conditions has been shown to be an isolated event (5). Sodium boiling has been postulated to occur as a result of either local blockages or whole-core loss-of-flow events. Local blockage will be discussed only to complete the treatment of boiling stability since the accumulation of either fuel or debris blockages in-core, large enough to cause boiling, is not considered a credible occurrence (6) (7). Furthermore, even should such conditions occur, sodium boiling would be confined to a wake region behind the blockage, being stabilized by the highly subcooled free stream flow (5). Inlet or exit blockages leading to coolant saturation are also not considered. Current core design practice is generally acknowledged sufficient to preclude such occurrences. Therefore, the principal emphasis of this material will be the development of the current status of sodium boiling in LMFBR geometry under whole core accident conditions. This emphasis generally focuses on the unprotected (failure to achieve neutronic shutdown by control rod actuation) loss-of-flow accident sequence as a reflection of worldwide efforts but in particular is related to loop type reactors of U.S. design.* We should note, however, that the possibility of avoiding sodium

*

See companion paper by Semeria et al., in this volume for a discussion of various reactor design features. 264

Liquid Metal Cooled Systems

265

boiling even for unprotected loss-of-flow accidents is not precluded. The French Super Phenix pool-type system design may have sufficient inherent negative reactivity feedback along with extended flow decay time sufficient to avoid boiling and achieve early termination of such accident sequences (8). Nevertheless, even were it possible to preclude sodium voiding by design, the need for understanding sodium boiling in LMFBR systems is and will be important in addressing safety and public risk questions. It is believed that almost two decades of scientific research has lead to an adequate understanding of sodium boiling behavior for these purposes. Current research efforts are generally directed to understanding specialized multidimensional effects of less concern to the overall assessment of public safety than to improved understanding of possible design margins. The historical significance of whole-core sodium voiding must be placed in the context of early attempts to provide a mechanistic source for reactivity transients leading to energetic whole-core disassembly. The calculations by MacFarlane (9) are typical in this regard. Considerations of the magnitude of sodium superheat in an LMFBR is also related to this problem. An initial sodium superheat of the order of 100° C [a value typical of experiment data in 1965 (10)] would lead to subassembly voiding on a time scale of the order of 0.2 sec. This time scale is about an order of magnitude more rapid than subassembly voiding for zero superheat at nominal power level. For a large core with a net positive sodium void reactivity coefficient in the range of $5 to $10, under assumptions of uniform core-wide voiding, $50/s to $75/s reactivity ramp rates can be estimated resulting in direct core disassembly from the initial voiding. More recently, however, with the establishment of negligible sodium superheat prior to boiling in an LMFBR and with the capability to account for core-wide incoherency effects and the concentrated safety analysis of smaller reactors like FFTF, SIR 300, and Phenix (small only in comparison to commercial-size reactors), the understanding of sodium boiling voiding in LMFBRs is no longer important as a direct source term for core disassembly ramp rates. Sodium voiding is, however, still important as an integral part of a sequence of phenomena eventually leading to core disassembly or in some special cases accident termination without disassembly. This paper will focus upon the current status of understanding sodium voiding due to boiling in an LMFBR geometry. Important phenomena required to understand current voiding models include: sodium superheat, two-phase flow regime stability, liquid film characteristics and parallel channel pressure drop-flow stability relationships. These will be discussed in Section 2.0. Current transient sodium boiling models will be discussed in Section 3.0, with specific model verification and applications discussed in Section 4.0. Combined effects of boiling, gas release and fuel release have only recently been incorporated into accident analysis. The significance of combined effects will be discussed in concluding remarks, Section 5.0, of this chapter. 2.0 FUNDAMENTAL CONSIDERATIONS All sodium boiling model developments have proceeded from most of the fundamental considerations developed in this section. These include: sodium liquid superheat prior to boiling inception, two-phase flow regime considerations, sodium boiling stability, heated surface dryout, and boiling incoherence effects within a subassembly. While it is currently understood that sodium liquid superheat prior to boiling initiation can be neglected, an extended discussion is provided because of the overall important significance of this result and because this result rests on a rather extensive body of experimental and theoretical work. Sodium boiling in LMFBR geometry is characterized by either the slug or the annular two-phase flow regime. The importance of the bubble flow regime is negligible in low-pressure

266

Heat Transfer and Fluid Flow in Nuclear Systems

systems. In a large number of applications, sodium boiling models are developed to describe an unstable transient event leading to subsequent clad and fuel melting. As such, the questions of liquid film dryout, boiling instability and boiling incoherency effects arise. These are treated in order in this section. Important considerations of the effect of heat capacity of the system are picked up later in Section 4.0. 2.1

Sodium Superheat

The subject of liquid-metal superheating was investigated in a number of laboratories in the 1960s and early 1970s, in view of its close connection to problems of fast reactor safety. In the early pool boiling experiments, superheats of the order of 0-500°C were observed for a variety of liquid metals, surface finishes, and other variables. The large spread in the data for incipient boiling superheat up to 1967 was graphically illustrated by Fauske (11) (Fig. la,b). The understanding of sodium superheat requires consideration of the basic mechanisms controlling vapor nucleation processes. More detailed studies were then carried out to determine the controlling variables in both static and circulating systems. It now appears that the incipient-boiling superheats in fast reactor systems will be very low, principally because of the effects of small quantities of inert gas in providing abundant vapor nucleation sources. The principal features of these studies are summarized below. 2.1.1 Pressure-Temperature History. In the absence of gas bubbles in the liquid, surface cavities previously not fully wetted and flooded by the liquid may be favored as the dominant source for vapor nucleation in many situations. The necessary conditions by which vapor or noncondensible gas can be trapped and remain inside the cavities on a heating surface has been discussed by Bankoff (12). Once gas is entrapped by a liquid phase, the curvature if the meniscus will determine whether or not the gas will remain trapped over a long period of time. However, it was pointed out by Bankoff (13) and independently by Deane and Rohsenow (14) that it is not necessary to consider the effects of contact angle if the cavity exhibits a reentrant geometry. As shown in Fig. 2, the minimum radius of curvature, upon penetration of liquid under prepressurization, will exist when the meniscus forms a hemisphere at the sharp corner at position 1. The deactivation or reflooding of such cavities is seen to depend only upon subcooling and system pressure, and not on contact angle. For a reentrant cavity of throat radius R, the condition for stability can be stated as follows:

f

_

~ R9

(RI

<

2s( T)

R

sat +

P

(1 )

where the primes refer to conditions within the cavity during the prepressurization period. This illustrates the importance of knowing the pressure-temperature history and the gas content of the surface prior to the transient. For a well-wetting liquid, and if reentrant cavities are available on the surface Fqu. (1) can be used to estimate the maximum size cavity available for nucleation. The incipient superheat can then be calculated from:

R

sat

-

f

=

2s(T)

R

(2)

Liquid Metal Cooled Systems

I000_

i

i

i

0

o 500

o

—

-

~~

~

i

i

i

I

q

0

o

8

267

i

i

~

I

_

0

~ F-

_

¢ w t rc

w a ~ u o w

O

50 —

rc

-

u ¢

w S

o w ~~

o-

10

—— -

D

5 —

D

-

I 600

I

I

0, REF. 3 (K) n, REF. 3 ( No) 0, REF. 2 ( No) n, REF. 5 ( Na) D, REF. 4 ( No) 1

I

1

1

1

700

1

800

Tsat.,

oo

1

I 900

Fi g. la Liquid Metal Superheat Data Presented at the 1970 AIC Conference. Ref. (30)

Heat Transfer and Fluid Flow in Nuclear Systems

268

I03

I

I

I

— ••

i _ •

~

ch G

•

-

q

—

-

q

0 O

° r O 0

9

O

D q

p

~ i~ ~DD

C OD ~

O

• CO • ~

q o o q

8

• ~ •

ii Ir ~ — • o _ w _ a rc

D

0

-•• • •

•

8

4

q

•

Q

00

O••••

s 0"

•

q q q

s

I 200

oo

° o 2 ~•

•

•

-

•

D

...

x

ca

u • •• • • _

i•• ~i D D

a

z

•

•

~~— _ _ • nn W ~ — n

W

• •

-

e•

.

u 1— H I O2 — .~

~

•

I

•

°

a w ~

•

•• •

~~

I

I

o

o • , REF. 11 (Na)

—

—

n, REFS.8 and 14(Na) 0, REF. 6 (Na) 0, REF. 7 (No) D REF. 9 ( Na) • , REF. 9 (K)

I400

I600

I800

Tsat.• oF

Fig. lb Liquid Metal Superheat Data Published Prior to the 1970 AIC Conference. Ref. (30)

Liquid Metal Cooled Systems

269

if no gas is present, and

P

sat +

R9

- Pf

_ 2s(T) R

(3)

if noncondensible gas is present. Equations (1) through (3) can be written in terms of advancing and receding contact angles and as such reflect the pressure-temperature history model of nucleation first advanced by Fabic (15) and Holtz (16), and later modified by Chen and Dwyer (17) (18).

LIQUID

Fig. 2 Penetration of Liquid into Reentrant Cavity. 2.1.2 Inert Gas Effects. The effect of inert gas in surface cavities was also considered and clearly demonstrated by Singer (19). The data discussed in Reference (19) and shown in Fig. 3 are particularly noteworthy since they represent incipient superheat data for which the relevant conditions were particularly wellcharacterized. A series of tests always started from frozen sodium that was melted and heated to a specific initial temperature and was brought to boiling repeatedly, starting from the same initial conditions in each sequence. Figure 3 shows two such test series, which were identical, except that the first series used a successively larger heat flux from test to test, while the second series involved the successive reduction of the heat flux. It is clear that the incipient-boiling superheat observed for the first test immediately after melting of the sodium was always much smaller (near zero) than that required for subsequent boiling tests in the same series. With repeated tests, the superheat is seen to increase, reaching a nominally unvarying value after three or four tests. This behavior can be attributed to a continuous loss of inert gas from the cavities due to repeated boiling. Ignoring this effect can lead to erroneous conclusions regarding the heat flux effects upon incipient superheat (19). The upper-limit value in Fig. 3 indicates that little or no inert gas is left in the active cavities when this value is reached. Furthermore, it is noted that this value can be predicted rather well from Equ. (2) (for conditions listed in Fig. 3, the calculated superheat value is 97°C). Also by accounting for the true incipient superheat location in forced convection system, Henry and Singer (20) (21) obtained good agreement between simple theory

270

Heat Transfer and Fluid Flow in Nuclear Systems

ioo

ca

z~~ JF

°4 m F Fa

_

50

t, 2

zw

j

o

Ts = 829°C

T1 = 502 °C

T~O = 432 C R~ = 1.36 b a r

z~~

L 0 =940mm R9= 0.14 bar

L h = 84 mm

SURFACE FINISH IOm m

00 50 Argon cover gas

0 50

100

1 50

200

250 300

WALL HEAT FLUX (q), w/cm2

Fig. 3 Variation of the Incipient-boiling Wall Superheat with Time and Heat Flux for Sodium. Ref. (19) and data.* The data were analyzed, assuming that a loss of history was experienced after each boiling event and that the deactivation conditions governing the next event were established by the operating procedures used between the runs. The system pressure was held constant at the boiling pressure; thus, the deactivations for these tests were accomplished by decreasing the temperature. Figures 4a and 4b show the comparison between predictions from Equ. (2) and data for three separate histories, and they are in good agreement over the velocity range investigated. Two results should be associated with the previous discussion. First, liquid metal superheat data behave in agreement with preboiling pressure-temperature deactivation considerations applied to wall cavities. Secondly, it can be concluded that inert gas effects also play a very powerful role in determining incipient-boiling superheat in alkali-metal loops. Although the physical laws involved (pre-boiling pressure-temperature relations) are relatively simple, the interactive effects may be complex and may mask other variables such as velocity, heat flux, temperature ramp, and the like. From a stability point of view, only reentrant or sharply necked cavities appear to be stable because of the low contact angles experienced with the sodium stainless-steel system. These cavities may be in equilibrium with gas bubbles in the coolant stream, and either one might act as nucleation sites. After long-term operation in reactor systems, the effects of surface corrosion on surface sites might become an important consideration. Experimental evidence was obtained by Hopenfeld (23) of corrosion by sodium on Type 304 stainless-steel surfaces under LMFBR conditions (sodium velocity, 6 m/s; heat flux, 568 W/cm 2°C; *As illustrated by Fauske (22), if the maximum superheat in the system is equated to the incipient nucleation, erroneous conclusions are reached relative to velocity, heat flux and ramp rate effects.

Liquid Metal Cooled Systems

t t / s ee D '2.2 ft/sec O —3.2 ft/sec r -4.5 ft/sec

271

o — I. 2

1 50

=

P

Pb, T '' 1000°F

1I0

0 0

50

~

o 3

7

5 (a)

BOILING

PRESSURE, Pb

9 ( P514)

a

Si

rc

Si

~ 200 N

J J

Q 3

O

1 50

O

R

12

psia, T

220° F

oO

O\

O

D i1 —m

\

100

P . I0psio, ( Alt

50

O 5

data

1300 I300°

nominally

at

F

3.2 ft/sec)

I

~~

~~

I

7

9

II

13

(b)

BOILING PRESSURE, Pb

1 15

( PSlA)

Fig. 4 Comparison of Pressure-Temperature History, Reentrant Cavity, Wall Nucleation Model with Convective Sodium Incipient Boiling Data. Ref. (20) (21)

272

Heat Transfer and Fluid Flow in Nuclear Systems

temperature, 642°C). By means of electronmicroscope techniques, it was shown that chromium and nickel are removed from the surface, leaving cavities of 1-2 micron diameter in the grain boundaries. In the absence of entrained gas bubbles, cavities in this size range would result in considerable superheating in a typical LMFBR system. A value of approximately 100-200°C is calculated from Equ. (3) if the partial pressure of inert gas in the cavity is set equal to the cover gas pressure in the reactor. However, as discussed in References (24) and (25), entrained gas bubbles are likely to be present in the reactor core. The gas bubbles may be due to mechanical entrainment at the free surface and/or gas precipitating from the liquid as it is cooled in the heat exchanger. The latter mechanism occurs because the solubility decreases with decreasing temperature, and the coolant is likely to be saturated with gas when it enters the heat exchanger. Likewise, dissolution of the gas bubbles will take place as they are flushed through the core because of the heating sodium. However, Thormeier (25) has demonstrated that the rate of dissolution is relatively low by accounting for transient mass diffusion of gas in the liquid. For the hypothetical case of a completely blocked subassembly and reactor conditions typical of the primary circuit of the SIR 300 'We liquid-metal fast-breeder reactor (26), the calculated lifetimes for both argon and helium gas bubbles are shown in Fig. 5. All the helium bubbles considered dissolved completely prior to reaching the saturation temperature. By contrast, argon bubbles above a certain size showed a different behavior. Because of the lower solubility and diffusion coefficient compared with helium, all argon bubbles of initial size greater than 10 microns still existed after the saturation temperature was reached.* In

5 \

~\

~

--

~I ~ I04

400

IF Sta rt of boiGnq

_Ar

_He

— He (DHe--=10D e---; I

i

600

i

800

1000

°

TEMPE RATURE, C

Fig. 5 Behavior of Bubbles Initially Filled with Gas During Heating of Sodium from 380°C to 945°C in 1.5 sec. Ref. (25) * This conclusion is supported by a more detailed study by France and Carlson (27).

Liquid Metal Cooled Systems

273

analyzing loss-of-flow accidents, it thus appeared that gas bubbles would be available as nucleation sites, at least if argon is used as cover gas, and would therefore prevent any significant superheating of the liquid sodium. To settle these questions definitively, France et al. (28) ran a series of sodium superheat measurements under carefully controlled conditions, using LMFBR simulation parameters, and measuring the actual location of boiling incipience. The heat transfer loop is shown in Fig. 6. The test section, which simulated a single fuel element, consisted of a 0.58 cm OD 45 kW heater with a 91 cm heated length surrounded by a 0.94 cm ID stainless steel tube. Twenty-two chromel-alumel thermocouples and 16 void sensor probes (voltage taps) were welded to the outer wall in the region -14.3 s L s 25 cm, where L = 0 represents the elevation of the top of the heated portion. Prior to boiling, the voltage signals were monitored at 0.016 s intervals. The plenum gas pressure was maintained at 1 atm and the sodium oxide level at ti15 ppm. The important point established in these runs was the need for sufficient time to saturate the sodium with argon gas in order to simulate steadystate reactor operating conditions. The loop operation time prior to the initiation of boiling by a simulated flow coastdown was varied in 26 runs with the bypass closed from 30.5 to 108 hr. The superheat at the point of incipient boiling in all these tests was zero. On the other hand, when the boiling was initiated only 0.8 to 1.5 hr after the time of filling the loop, maximum test section bulk superheats of 67-83°C were recorded. These recordings were not, however, incipient boiling superheats. For these runs vaporization occurred by flashing downstream from the heated zone where the incipient vaporization superheat was only 33°C. The

GAS SYSTEM

1200°F 100psio 5ogpm

I

G VACUUM PUMP ~~ I ~ i

_J ~ ~

VACUUM

~

CHAMBER

L-

lOgpm MAX

1000°F 500psia R,~ E EATER (1 /2in. SCH 40 TYPE 304 PIPE)

DUMP TANK

1

LMFBR HEAT TRANSFER SIMULATION LOOP

Fig. 6 Convective Sodium Experiment Loop of France. Ref. (28)

274

Heat Transfer and Fluid Flow in Nuclear Systems

controlled data of France convincingly demonstrated the overriding influence on incipient boiling superheat of gas bubbles, either circulating or trapped in wall cavities. The net result of these studies is that for most LMFBR primary heat transport systems, sodium boiling under abnormal conditions of power-flow mismatch, would occur at the local saturation temperature with negligible superheat. It is also significant that the numerous loss-of-flow type sodium boiling experiments carried out as integral tests, both in the laboratory and in-pile have produced bulk boiling with no evidence of measurable incipient liquid superheat (Reference 29 typical). It is also noteworthy that there has been no reported occurrence of significant liquid sodium superheat on the first approach to boiling — even in static tests. The occurrence of boiling on the first attempt, at the saturation temperature, is generally not reported in the discussion of subsequent superheating results (30). However, the first time is the significant event in reactor accident analysis. Thus, boiling under LMFBR conditions is generally accepted to initiate at the saturation temperature. 2.2

Flow Regime

For liquid-metal boiling near atmospheric pressure, typical of LMFBR loss-of-flow conditions, the slug and annular flow regimes prevail rather than the bubble flow regime common to steady boiling in high pressure LWRs. This is particularly true for the relative narrow coolant channels employed in an LMFBR and rapid bubble growth associated with low Prandtl number fluids. High liquid superheat, assumed in early model development efforts tended to emphasize the transient slug flow regime. Simulation experiments incorporating initial conditions of high liquid superheat confirmed the applicability of the slug flow regime. However, it was soon recognized (31) that even if the initial boiling development were slug-like at low superheat, vapor flow within the voided cavity must be treated according to assumptions typical of annular flow in order to correctly assess both the dryout time for the clad surface and the pressure acting on the liquid being driven from the channel in the assumed manner illustrated in Fig. 7. This flow regime was identified as bidirectional annular flow in Reference (31). The two-phase flow stability criterion suggested by Kutateladze (32) can be used to predict the flow regime transition from slug to annular flow

K

4

p

g

2

gJ

(4) 6

p

f

r

9

Note that K4 is the ratio of the dynamic pressure (tiP gj2) exerted on particles of the discontinuous phase to the weight of the heavier fluid. The bracketed term in the denominator of Equ. (4) is proportional to the maximum particle dimension based on mechanical stability criteria such as the Weber number of the Kelvin-Helmholtz wavelength. In other words, the Kutateladze criterion is simply that balance between dynamic pressure (or drag) and buoyancy forces that is consistent with fluid particle stability. Based upon experimental data K takes on a value of approximately 3 for the flow regime transition in question. For illustration purposes,

Liquid Metal Cooled Systems

275

LIQUID I VAPOR

a. SLUG FLOW, BOILING INLET FLOW RATE CONSTANT

VAPOR

LIQUID

b. ANNULAR FLOW, FLASHING

VAPOR

C. SLUG-ANNULAR FLOW, TRANSIENT FLOW CONDITIONS

Fig. 7 Typical Flow Regimes for Liquid Metal-vapor Flows the critical vapor flux, j, can be related to the voided length (in the heated zone), L, and heat flux according to L 9 p g hfg A

j

(5)

For LMFBR nominal power levels, Fig. 8 illustrates the early significance of the annular flow regime. This situation would be anticipated for an unprotected lossof-flow accident in an FFTF-like design. The single- or multi-slug bubble regime, which develops rapidly into bidirectional cocurrent annular flow (following flow reversal), typifies the flow patterns in single-channel systems or multichannel systems with uniform radial temperature profiles at boiling inception. Cocurrent annular flow (see Fig. 7b) characterizes the flow structure up to bulk flow reversal in multichannel (interconnected) systems with nonuniform radial temperature profiles at boiling inception. Following flow reversal, the flow pattern rapidly develops into bidirectional cocurrent annular flow with high void fraction. The latter combination of flow regimes (see Figs. 7b and 7c appears of particular interest in LMFBR subassemblies with wire-wrapped fuel pins where steep radial temperature profiles exist at the onset of boiling inception (see Section 2.5). However, for power conditions and axial temperature profiles associated with protected accidents such as loss-of-piping integrity and decay heat removal, the slug flow regime may indeed be more appropriate to describe the transient boiling process if a single channel or one-dimensional approach can be justified.

276

Heat Transfer and Fluid Flow in Nuclear Systems

1.0

Ro = 10 KW/ft 3 ft

ANNULAR-DISPERSED FLOW

0.1

0.01

SLUG FLOW

0.001

0

I

I

I

0.2

0.4

0.6

0.8

1.0

o

R/ R

Fig. 8 Sodium Boiling Flow Regime Map Related to LMFBR Channel Conditions. 2.3

Boiling Stability

For purpose of illustration we will use the unprotected loss-of-flow accident in an FFTF-like design. Because of the large radial power profile in-core, the onset of boiling is not uniform across the core. The first subassemblies to reach saturation may represent only about 10% of the total flow area. Furthermore, the onset of boiling in other subassemblies is distributed over many seconds in time, because of the relatively slow rate of flow decay. Under such conditions, boiling is reached in a reactor subassembly under conditions closely approximated by constant-pressure-drop, parallel channel system. Assuming for the moment that boiling is uniform within a subassembly, the highly unstable nature of sodium boiling for LMFBR loss-of-flow conditions is illustrated in Fig. 9. The S-curve noted in Fig. 9 can be constructed on the basis that

Liquid Metal Cooled Systems

277

Pressure head delivered by the pump A

a

a

0

Flow W

Fig. 9 Flow vs Pressure Drop Stability Considerations for Loss-of-Flow Conditions at Nominal Power for a FFTF Subassembly Design. annular flow prevails and applications of the corresponding standard conservation equation and relationship for momentum change and shear stress whose application to low pressure sodium has been demonstrated in (33) (34). Some of the key features are summarized in the Appendix. During the rundown of the pumps and as long as the coolant is subcooled, the flow decreases monotonically with decreasing pressure drop across the core (from A to B in Fig. 9). However, at the onset of boiling, point B of Fig. 9, the large liquid-to-vapor density ratio for sodium at low pressure causes a large volume change, rapid increase in local pressure gradient due to momentum change and increased friction, all requiring a rapid increase in total pressure drop if flow is to be maintained. Since this cannot occur because of the constant pressure drop parallel channel system, the flow in the boiling subassembly must rapidly decrease from point B to C. These are conditions which satisfy the requirements for the fundamental Ledinegg static boiling flow stability (35). Since the Ledinegg criterion

d(DR) dw

`

d(AP h) dw

(6)

is not satisfied, the boiling process is seen to be highly unstable, i.e., when point B is reached, the operating point moves to C. At low pressure and high power density typical of the unprotected loss-of-flow accident, this flow excursion requires that the coolant moving through the core must turn around, i.e., go from a positive to a negative velocity. This phenomenon is generally referred to as flow reversal. It is during this flow excursion that specific phenomena of interest to safety analysis such as voiding rate and dryout occur, and it is precisely this flow excursion that is described in current sodium boiling models for LMFBR geometry. 2.4

Dryout Considerations

During the transient boiling process associated with the flow excursion from B to C in Fig. 9, a liquid film is left behind on the heated wall. For low Prandtl

278

Heat Transfer and Fluid Flow in Nuclear Systems

number fluids like sodium, nucleation is suppressed in the film, and the heat transport is by liquid film evaporation. Since the liquid film is not replenished for the case illustrated in Fig. 9, the dryout time would be determined by the film thickness. Measurements by Grolmes and Fauske (36) on the liquid film thickness in slug flow for LMFBR conditions are shown in Fig. 10 and illustrate that the liquid film left behind is characterized by a liquid fraction of 0.15. This value is approximately the ratio of the maximum to average velocity for liquid in turbulent flow in round tubes and would imply that the slug bubble is traveling at a velocity characteristic of the centerline velocity in the liquid ahead as suggested in Reference (37).*

-

(1

io

a) =

Umax

I

I

I

Ua v

1_

(7)

turbulent flow

I

I

I

I

~

I

0 0.250 in • q

A O D

—

0.375 in AIR-WATER 0.500 in

®

— SINGLE BUBBLE SLUG EXPULSION

C I

I

I 2

—

UB =1.18OF

D/

O

~

6.00 mm F-11

annulus 9.00 mm 6.00mm

L

o

/ D~ ®f

l

I 4

I

I

I

6

I 8

I

IO

Average Liquid Velocity , UF , I/SEC

Fig. 10 Comparison of Bubble Velocity and Average Liquid Velocity for Single Bubble Slug Ejection. Liquid Film Thickness is Derived from Equ. (7) with U b = and U U . Umax f = avg * These results can be applied with confidence only to the situation where a light fluid, e.g., sodium vapor, is advancing against a heavy fluid, e.g., sodium liquid.

Liquid Metal Cooled Systems

279

If the liquid film remains undisturbed, the dryout time can be estimated from simple vaporization alone, as done in early slug bubble models (38) (40). However, as illustrated by Equ. (5) in Section 2, the flooding velocity is exceeded locally very early in the bubble growth process (a bubble length several centimeters would appear to be sufficient if all the generated power is removed by latent heat of evaporation). Because flooding is likely to occur within the bubble, the effect of axial liquid transport (W f,a ) as a result of induced upward and downward motion in the liquid film caused by vapor drag must be examined in addition to the radial component of liquid transport (W f,r) (see Fig. 11). As a result of the axial vapor velocity gradient at the time of flooding the induced motion in the liquid film is nonuniform and therefore will give rise to a net axial transport of mass locally in the liquid film. Neglecting entrainment, W f,r can be estimated from

QA h Wf,r

=

(8)

h fg

Based upon evaporation only, the estimated dryout time for a typical liquid film thickness (%0.15 mm) (36) would be approximately 0.2 to 0.3 s.

INCREASING VAPOR VELOCITY

G

N

~~

Fig. 11 Simplified Dryout Model Following Onset of Flooding An estimate of W f,a can be obtained by assuming quasi-steady incompressible flow, neglecting gravity, and specifying the appropriate constitutive equations of annular flow describing the increased wall and interface shear stresses following the onset of flooding. Combining Equations (A.1) through (A.4) (see Appendix) with a

280

Heat Transfer and Fluid Flow in Nuclear Systems

liquid continuity relation W f = R f(1 - a)p D2/4 leads to Wf rJ f,r

Figure 12 shows the ratio

_

r f(1 - a)

Wf,a/Wf,r

G L L

1 +75 (1 p fPg

1/2

a)

(9)

-

J

as a function of the liquid fraction (1 - a).

The initial liquid fraction prior to onset of flooding is taken as 0.15. As seen from Fig. 12, immediately following the onset of flooding, the liquid film will thin much more rapidly than in accordance with the rate calculated from evaporation alone. It is only after the initial film thickness has decreased by a factor of 10 or more that the axial and radial mass transports become of equal importance in determining the decrease in film thickness. These considerations show that calculation of the local dryout time based upon a stationary film will overestimate this time by as much as an order of magnitude. This conclusion appears to be in agreement with observations noted in Reference (41).

20

/r9=2.0c 103

~~ 10 E Q. 9

0 0.15

O.1O 0.05 LIQUID FRACTION, 1-a

O

Fig. 12 Illustration of the Relative Magnitude of Axial Liquid Film Transport Due to Vapor Shear Force in Comparison to Radial Transport Due to Evaporation at Nominal Power Levels. Curve is Derived from Equ. (9).

Liquid Metal Cooled Systems

281

Finally, it should be noted that an upper bound for dryout can be associated with an entrainment velocity which corresponds to a value of K in Equ. (4) of 3.7. 2.5

Boiling Incoherence Effects

G(t)/G(0), RELATIVE FLOW

In addition to the previously mentioned core-wide incoherence, a further incoherence in the boiling process is introduced by the noted large radial temperature profiles within a given subassembly (42). Calculations illustrated in Fig. 13

1.0 0.8 0.6 0.4 0.2 0.0

0

2

I

4

I

6

I

1

1

8

10

12

I

14

G

16

I

18

TIME , sec

UI~

w

J 1800

w

u-J 1600

i—

COBRA b = 0.02 INLET TEMPERATURE = 600°F AVERAGE LINEAR POWER = 10.58 KW/ft. FFTF AXIAL POWER DISTRIBUTION UNIFORM RADIAL POWER NOMINAL FFTF DIMENSIONS ~

h /

87°F/sec!

o

f

` 9 0E cec ll 9.0 f t / sec

~.~

1400 —

} _ 453°F

TIME = 10 sec 8.3 ft/sec

'

~

12oo ~ 1200

w

TIME = 0 sec, 24

ft/sec

1000

e—

800

F

—

I I I I I I 1 1 1 1 I I I 1'5 t+_

I 17 } 15 } 13 I 11 I 9} 7 f 5f 3 I 1} 31 5 f 71 9 I 11 G 13 } 15 G 17 f 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18

L—

I —

R

CHANNEL DESIGNATION Fig. 13 Radial Temperature Profile in FFTF Subassembly at Onset of Boiling for Unprotected Loss-of-Flow Conditions.

282

Heat Transfer and Fluid Flow in Nuclear Systems

show that a generally consistent temperature variation of 177°C to 215°C (max to min) might exist at the onset of boilinn. Thus, boiling at essentially zero superheat permits void growth in the central subassembly with net positive flow. The large momentum and frictional pressure drops noted above for the low pressure twophase flows, suggest flow diversion from boiling to nonboiling subchannels (interconnected channel effects). A local excursion of the classic Ledinegg instability will occur for each new subchannel that experiences boiling, and inlet flow reversal is not suggested to occur prior to reaching bulk coolant saturation.* (In connection with this conclusion, it is noted that local boiling within a subassembly due to a postulated planar blockage would be stable.) The magnitude of this incoherence is largely determined by the following two factors: (1) the edge-subchannel-to-total-area ratio (see Fig. 14) and (2) the axial temperature profile downstream of active fuel region (see Fig. 15). A fullsized subassembly typically has only 20% of its coolant flow goinq through the undercooled regions next to the hexcan wall; and in the case of an unprotected loss-of-flow accident, the temperature profile in the reflector-blanket-fission-gas

NUMBER OF FUEL PINS 7

19

37

2

3

4

61

91

127

169

217

5

6

7

8

9

~~0.Q ~ 0.7 w D

f 0.6 J

o w

0.4 rc

~~0.3 ~

~ 0. 2 rc w H

- 0.1

~

o

NUMBER OF RINGS

Fig. 14 Edge Subchannel-to-Total Flow Area Ratio as a Function of Bundle Size. * This implies that local dryout may also occur in a pin bundle with positive flow through the bundle.

Liquid Metal Cooled Systems

1000

1000 AXIAL TEMP. PROFILE

RADIAL TEMP. PROFILE

900 _ 800 _ z ~~ 700 _ w S 600 — w 500 —

~

400 0 20 40 60 80 100 PERCENT FLOW AREA

283

u

O

900

,y

800

~ w

700

~ ~

S

w

H

600

500 400

r

0

1

1

I

50 100 150 200 250 AXIAL DISTANCE (cm)

Fig. 15 Radial and Axial Temperature Profiles for Hypothetical FFTF Loss-of-Flow Transients. Solid Lines Refer to Unprotected Pump Coastdown. Dashed Lines Refer to Loss-of-Pipe Integrity with Scram. plenum regions at boiling inception or shortly thereafter coincide essentially with the corresponding saturation temperature profile (see Fig. 15). This implies that the flow diversion from boiling to nonboiling subchannels must take place outside a two-phase zone that occupies approximately half the length of the bundle and 80% of the flow cross section (see Fin. 16). The flow impedance is therefore very large, and requires an overall flow reduction by a factor of approximately 4 relative to flow conditions existing at boiling inception, i.e., a flow reduction well below the level required to satisfy boiling in the edge subchannels. This is equivalent to saying that flow diversion in a full-sized bundle is not important, and hence a one-dimensional approximation would appear (see Fig. 16a) adequate to describe the voiding history. The rapidity of achieving flow reversal then depends largely on the time to achieve saturated downstream conditions. This conclu* As will be sion is clearly supported by the recent work of Theofanous (43) (44). demonstrated in Section 4.0, the upstream voiding for the unprotected loss-of-flow accident is essentially determined by the fuel pin heat capacity in the active fuel region of the pin in the presence of saturated downstream conditions. On the other hand, for protected accidents, where fuel pin survivability rather than reactivity considerations is of prime concern, the downstream temperature profile is usually far removed from the saturation profile (see Fig. 15), and the extent of the initial two-phase zone is rather small (see Fig. 16b). For these cases substantial flow diversion is possible without significant decrease in the overall flow. Again the role of the heat capacity, in this case downstream of the active fuel region, dominates the problem. It follows that a one-dimensional approach may be adequate to represent first-order effects, and that a simple slug-flow regime (uniform bubble pressure) would appear appropriate to describe the transient boiling process. The combined effects of large heat sinks and low coolant boiling *

It would appear that the primary need for two-dimensional modeling may be found in interpretation of the experiments with limited bundle size as well as to what size bundle is adequate to simulate conditions in a full-size subassembly.

284

Heat Transfer and Fluid Flow in Nuclear Systems

a) LOF WITHOUT SCRAM I-D

2-D

I-D

TWO-PHASE ZONE

2-D

b) LOPI WITH SCRAM

f

W W

WW

INLET FLOW

INLET FLOW

Fig. 16 Comparison between One- and Two-Dimensional Boiling Geometries for Unprotected Loss-of-Flow Accident (a) and the Loss-of-Pipe Integrity with Scram (b). heat fluxes associated with protected accidents may only lead to temporary unstable boiling in the absence of dryout and return to all-liquid flow conditions, i.e., point C in Fig. 9 may never be reached under these conditions. These aspects are discussed further in Section 4.0.

3.0 TRANSIENT SODIUM VOIDING MODELS A review of the early attempts at transient sodium boiling model development for application to LMFBR accident analysis can be found in Reference (45). The annular flow model of MacFarlane (9) was handicapped by numerical instability problems. Attempts at overcoming these difficulties with solution techniques based upon the method of characteristics (46) were not universally successful. More recent attempts at drift-flux formulation of the problem achieved only limited success (47). However, the concept prevalent in the mid-1960s, that initiation of sodium boiling was likely to occur with attendant high superheat, was the principal reason for

Liquid Metal Cooled Systems

285

the widespread development of the single- and multi-bubble "slug type" voiding models. The emphasis on the development of the single- and multi-bubble slug ejection models is common to all but the French LMFBR development program. For reasons of system design, and also because the French program has consistently assumed that superheat prior to boiling in a reactor system would be negligible, their boiling model development has concentrated on the development of quasi-steady and transient homogeneous and annular flow patterns (34). 3.1

Slug Models

The model was originally proposed by Noyes (48), first implemented by Fauske and Cronenberg (40) and subsequently incorporated into the US SAS accident analysis model (49). Similar approaches emerged abroad including BLOW and NEIL (38) (39). The slug model is generally free of numerical stability problems because of its simplicity. The formation and growth of a typical slug bubble is illustrated in Fig. 17b. The presence of finite geometry and velocity field, nonuniform temperature, and continuous heat generation in the fuel-element make a spherical bubble growth calculation extremely difficult. A more convenient approach is to assume the bubble begins as an infinitesimally small strip of vapor occupying the entire channel with the exception of any film remaining on the wall (see Fig. 17): For this case the analysis of spherical and cylindrical growth and collapse becomes identical, and the uniform pressure within the bubble can be obtained by solving the NavierStokes equation describing the motion of the liquid slug on each side of the bubble together with the following simplified energy balance written for the vapor (see Fig. 17d): ph a g fg

, aDt a dT( x l t) = k f d x1 at

+ k

a

f

x1 =0

dT( x 2,t)

dx

(dT(s,t) fJ ~

(10)

+k

2

S

x2 =0

s=0

The temperature gradients at the upper and lower liquid-vapor interfaces are found by assuming plug flow in the upper and lower slugs of liquid. The heat equation for the upper slug may be written, neglecting radial gradients in the form aa 2 T( & 1,t) + Q(x1,t) _

ax

pfcf

2

a t( x 1,t)

(11)

at

1

subject to T( x 1,0) =

t(oo,t) < f

T(0,t) = f(t),

The temperature gradient at the liquid-vapor interface is found to be (50): t dT(x 1,t) d

_ - ~ [pa( t - t ) ]

d f( t) d t ' 2 d

f(0) ~p aT

o

x1 = 0 t

+

3 G

~[pa (t f

0 0

t)

-1 2

exp

4a ( t

xl

t

)

Q (x

1

, t) dt d S 1

(i2)

286

Heat Transfer and Fluid Flow in Nuclear Systems

T(x ,0)

+

2a 0

~ ~13 exp pat

[ 4a t

1

J

d xl

(13)

Note that f(T) is determined by the pressure within the vapor space, under the assumption of equilibrium between the vapor and the liquid surface at every instant.

SLUG FLOW APPROXIMATION

FORMATION OF BUBBLE SPHERICAL CYLINDRICAL GROWTH GROWTH

I

n A

P 0 R

- .

T Q

T

SLUG ELEMENT

~g1

D

A

P 0 D.

(N

A

~ a)

( b)

SLUG MODEL APPROXIMATION FOR SPHERICAL GROWTH V A P 0 R L

I

X i ,~l =0

DC

~ E =0 2

U

io )

I

(d) Fig. 17 Illustration of Slug Flow Pattern and Model Approximation.

Liquid Metal Cooled Systems

287

This temperature couples the energy and momentum equations written for the liquid slugs. For high thermal conductivity fluids (liquid metals), the temperature gradient at the liquid film-vapor interface can be approximated by dT( d, t) dd

Tw(z,t) - T 9(t) d

=0

d

(14 )

since the thermal boundary layer is equal to or larger than the liquid film thickness after a relatively short exposure time. For the l ow thermal conductivity fluids (nonmetallic fluids) the thermal boundary layer may remain small with respect to the liquid film thickness and varies both with time and distance (51). The validity of the spherical bubble growth approximation in finite geometry was verified by slug expulsion data obtained with superheated liquid freon (51). Figure 18 shows the comparison between experimental and predicted bubble growth for a superheat of 25.2°C. It is rather interesting that this model agrees well during the early stages of ejection, in view of the assumption of a disc of vapor filling the tube cross section. This early growth period accounts for an appreciable portion of the total ejection time. The ability of the model to predict the effective delay time before rapid expulsion commences is encouraging, and implies that the assumption of a slug geometry, even during the spherical growth period, approximates the surface-volume relationship of the vapor region satisfactorily, at all times. In the N E M I code (39) a detailed description of the spherical bubble growth is obtained by using the methods described in References (52) and (53). A similar approach is used in the BLOW-II model (38). As it became recognized that lower initial superheat was more realistic, the difficulties encountered because the single bubble was pushed downstream and collapsed were overcome by the multi-bubble model incorporated in the current SAS voiding model. The detailed equations and numerical procedures for the model are discussed in Reference (49). At l ow superheat, upon voiding initiation, many bubbles were calculated to be generated at the most favored location, and swept downstream to condense and disappear if sufficient heat sink were present. This process continued until flow reversal was brought about (49). The evolution and improvement of this model proceeded through successive stages of increasing sophistication in the treatment of vapor production and condensation within the vapor cavity. The motion of vapor within the cavity almost immediately reaches a level where its interaction with the residual liquid film demands an annular flow treatment, i.e., the assumption of uniform pressure within the vapor cavity is not realistic for reactor conditions corresponding to nominal power levels. Hence the term, bidirectional annular flow. In addition to the necessity for considering nonuniform pressure, the induced film motion is likely to lead to early dryout as compared to simple vaporization from the liquid film alone as illustrated in Section 2.4. 3.2

The Current SAS Model

Hoppner's (54) description of the current SAS model *

*

as illustrated in Fig. 19 will

Essentially the same feature, e.g., multi-bubble slug geometry including provision for bidirectional annular flow, is also incorporated in the German BLOW-3 voiding code (55).

Heat Transfer and Fluid Flow in Nuclear Systems

DISTANCE FROM BOTTOM OF CHANNEL, cm

288

70 60 50 40 30 20 SLUG THEORY 10

DTs = 25.2 °C

0 1.7

• • iR••ii~ii i i • 0.9

EXPERIMENTAL DATA

0.7 0

40 80 120 160 200 240 280 320 TIME, msec

Fig. 18 Comparison between Calculated Slug Model Voiding Rates and Pressures with Experimental Data for Freon 113 in a Circular Tube.

Liquid Metal Cooled Systems

289

Coolant Channel Upper Liquid Slug -Structure

Fuel Pin

Streaming Sodium Vapor

Moving Sodium Film

Lower Liquid Slug

Fig. 19 Bidirectional Tw -Phase Annular Flow Model for SAS Voiding. be used. Figure 19 illustrates a vertical coolant channel with uniform cross Section A and hydraulic diameter Deq bounded by fuel pin and structural surfaces. Liquid slugs separating the vapor regions in a uniform-area channel are governed by the following mass and momentum equations written for an Eulerian system: apf

aG f

at +

az

0

(15)

and Gf a

at + at

2 ( ~~

Pf

Gf ~f i i + dz + 4pf + c f 2rfDeg -

(16)

The conservation of energy in a liquid slug is provided in the following form which includes the continuity Equ. (15): QfCf at +

G Cf

f

az — YQ(z,t) + q(z,t)

(17)

At a given bubble length the calculation changes from the uniform-pressure model to the pressure-gradient model illustrated in Fig. 19.* The one-dimensional mass and momentum conservation equations for the vapor core are solved simultaneously for

* The critical bubble length can be determined by setting the average vapor velocity equal to the flooding velocity. For nominal power levels the critical bubble l ength is only a few centimeters. On the other hand, at decay heat power levels, the boiling process is better described by the uniform pressure model alone.

290

Heat Transfer and Fluid Flow in Nuclear Systems

all nodes in the bubble. They are written here in differential form: ar

aG

at ~

aG

a

_~+

at

az

~,' G 2 ' _9 Pg

ah t

(18)

q

fa

c

G,

G

_ dP M

g~ g ~ g 2p g0eq

(19)

c

dz

One-dimensional mass and momentum equations are written, as indicated in Fig. 19, for liquid-film slabs of thickness b moving with velocity U f:

at (bp f) + at

at

(bP U ) + f f

+

á

az

(b

rf~Gf) = Q/h fq c

(b0U2) = hfq Uf f f

hfg

U

g

(20)

- br g g-

dP

dt

b

r fUflufl p q (u~ - Uf )1(Uq - Uf ) - á

The vapor friction factor c q establishes the strong coupling between vapor velocities, axial pressure drop, and film velocities.* It is calculated according to a correlation of Wallis (56) for turbulent films:*

a

cg = a 1 Reg 2

1 + 300

b O

(22)

eq

Typical dynamic calculations obtained with the above slug-annular flow model will be illustrated in Section 4.1. The above discussion typifies current status of one-dimensional sodium voiding models. Examples of two-dimensional models can be found in References (43, 44, 57, 58). Only the work of Theofanous et al., represents a complete two-dimensional transient voiding calculation, which provides a comparison with one-dimensional calculations. This work concludes that one-dimensional calculations are adequate for accident analysis calculations. A somewhat simpler two-phase treatment as compared to the current SAS voiding model is also recommended. As will be discussed in Section 4, the fuel pin heat capacity controls the voiding rate and hence masks any effect from two-phase flow details. 4.0 APPLICATIONS Fundamental considerations of transient voiding model development have been discussed in previous sections of this report. These developments must now be considered in terms of experimental verification, evaluation of the results and applications to reactor accident analysis. This section will be developed within the framework of reactor accident applications. We will proceed from considerations of unprotected accidents such as the loss-of-flow scenario to protected

Note that Equ. 22 is equivalent to Equ. A.4.

Liquid Metal Cooled Systems

291

accidents where neutronic shutdown is assured such as the loss-of-piping integrity (LOPI) and the loss-of-heat sink (LOHS) accident. Finally, consideration will be given to local faults in which sodium boiling behavior within subassemblies must be summarized. In consideration of these applications, the role of heat capacity effects as they relate to the loss-of-flow accident scenario will be illustrated. The purpose of such illustrations is to show in a fundamental way how one can evaluate and assess heat capacity effects in subassembly voiding.

4.1

Unprotected Accidents

We now consider unprotected accidents that have received substantial attention in accident analysis of fast reactors around the world (59) (65). Particular examples here will be related to the US-FFTF and CRBR reactors.

4.1.1 Loss-of-Flow Accidents. The loss-of-flow accident scenario is characterizec as a loss of pumping power with failure to actuate neutronic shutdown of the reactor core. The accident proceeds with continually reduced inlet flow to the core, overheating of the sodium coolant and reactor structure, until finally sodium boiling is initiated within high-powered subassemblies in the reactor core. Early sodium boiling experiments related to these conditions were carried out in either single pin out-of-pile test facilities or in-pile experiments with nonprototypic short, heated length fuel samples. Experimental results indicated two-dimensional effects and other effects uncharacteristic of the fundamental considerations presented earlier. In other words, such experiments were difficult to analyze witr voiding models, which have been characterized as current state-of-the art. This provided an incentive for both understanding of these differences and difficulties and creating more prototypic experimental conditions. Among the first prototypic sodium boiling experiments were the R-series, in-pile TREAT experiments, and the OPERA out-of-pile electrically heated-pin sodium boiling experiments (29) (66). Both experiment series were executed in 7-pin test bundles with full length fuel elements or full length electrically heated fuel element simulators. Prototypic conditions here refer to geometric dimensions characteristic of the US/FFTF reactor. Approximately 5 loss-of-flow experiments were executed in the R-series in-pile apparatus with 7-pin bundles and one 7-pin loss-of-flow experiment was executed in the OPERA test facility. Experimental results were very similar and comparable in all tests. Figure 20 shows a characteristic voiding envelope for the R-series loss-of-flow test R -4 and comparison with SAS voiding calculations. It was noticed that in most of these 7-pin experiments that the onset of boiling in the experiment occurred approximately 1 to 1½ seconds prior to the best estimate SAS calculations. While geometry in these 7-pin experiments was designed to minimize two-dimensional effects, the time delay between the onset of boiling in the experiment and the onset of boiling in the SAS calculations could still be attributed to the two-dimensional radial temperature profile which did exist in these 7-pin bundles at the onset of boiling. Nonetheless, the significant point to note in Fig. 20 is that after inlet flow reversal which occurred with minimal upstream voiding in the experiment, there was essentially complete agreement between experimental results and SAS calculations. This, in fact, is viewed as very excellent confirmation that the overall one-dimensional modeling features of the SAS code are adequate to describe onedimensional sodium boiling processes. Other significant results in these boiling experiments were the continuing observation of essentially zero liquid superheat at the onset of sodium boiling. Results relative to clad melting, the role of fission gas and subsequent fuel melting from these integral experiments are discussed here (67).

292

Heat Transfer and Fluid Flow in Nuclear Systems

7-PIN R-4 EXPERIMENT

I

I

I

1

1

I

I

I

I

I

VOID PROFILE AND

~ 250

GAS RELEASE

SAS 3A COMPARI SON -•-•-

w ~

200

0

J

START VOID

J I50

FISSION GAS PLENUM

)-

Z

START SAS VOID

III EXPERIMENT FLOW REVERSAL

13

I

1 14

I

I 15

I

I 16

FUEL ZONE

I

I 17

I

I8

TIME, SEC

Fig. 20 Comparison of Sodium Voiding in 7-pin R-4 Test with SAS Analytical Model. With the good agreement between 7-pin experiments and analysis, the question of prototypicality was carried one further step. That is, how applicable are 7-pin test results to 217 pin or larger size subassemblies (68). Analysis would indicate that for large pin bundle size, the time difference between the onset of boiling and flow reversal would decrease (68) (69). These considerations were based upon relative size effects, magnitude of the radial temperature gradients and their effect on suppressing radial void growth in large size bundles. The results indicated that between 19 and 61 pins that differences between experiment and onedimensional analysis should rapidly diminish. Large sample size experiments followed in SLSF loss-of-flow tests (70) (71). Test P-2, a 19-pin bundle, also simulating an unprotected loss-of-flow condition in an FFTF reactor was executed. However, in this experiment, inlet flow reversal following boiling inception was delayed by as much as 4 seconds compared to the 12 seconds delay for the earlier 7-pin experiments (70). This result was contrary to expectation, and renewed interest in voiding coherency and questioned the validity of one-dimensional approach to boiling and voiding in a subassembly under loss-of-flow conditions. However, subsequent examination of this test indicated that flow tube failure contributed to excessive heat loss in the bundle. When these additional heat losses were taken into account, the longer time between the onset of boiling and inlet flow reversal could be explained.

Liquid Metal Cooled Systems

293

Subsequent to test P-2, tests R-3a and R-3 were executed (71). Test P-3a and P-3 contained a 37-pin bundle, again simulating an FFTF loss-of-flow condition. These test results were unencumbered by excessive heat loss conditions during the transient and the results were particularly satisfying, reestablishing trends indicated by earlier analysis. This is particularly well illustrated in Fig. 21 where the comparison between the SAS model and the inlet flow meter of the R-3a tests are illustrated. The time difference between the onset of boiling and the onset of inlet flow reversal between experiment and analysis is less than 1 second. Similar comparison was obtained for P-3. This, all things considered, could be taken as excellent agreement between experiment and analysis and reinforces the notion that voiding in larger size subassemblies becomes increasingly more one-dimensional in these loss-of-flow conditions. 4.1.2 Heat Capacity Consideration in Loss-of-Flow Accidents. While the overall comparison between experiment and one-dimensional analysis indeed seems encouraging, the experiments to date do not provide information relative to details of the liquid film motion within the voided region, dryout, rewetting, evaporation and condensation. All of these features are incorporated within the SAS code. It has been noticed for some time that variation of these internal details did indeed effect such local conditions as the time of first dryout but did not have a noticeable effect upon the overall voiding envelope, and thus the overall comparison with experimental data as shown in the previous figures. This lack of sensitivity requires some additional consideration, as it is attributed to the fundamental role of heat capacity effects in determining the overall voiding rate.

i

i

I

I

i

BOILING INCEPTION

R3A (37-PIN)

w ~~ —

~ \N SAS 3D

~

INLET FLOW REVERSAL

i

I

Is TIME

Fig. 21 Comparison of Inlet Flow Rate during Voiding for OPERA 7-pin Test, R3A 19-pin Test and SAS 3D Prediction of R3A Test.

294

Heat Transfer and Fluid Flow in Nuclear Systems

The importance of heat capacity effects on the fuel and clad can be illustrated by the following analysis (72). Figure 22 illustrates the axial power distribution, coolant, clad and fuel temperature profiles, both at steady state and the onset of boiling. If flow reversal occurs abruptly as experiments seem to indicate, further convective heat transfer can be ignored within the bundle. Under these conditions the bundle containing fuel, clad and coolant continues to overheat at the assumed constant power. One can therefore evaluate the time required for the cladding to reach the temperature equal to the sodium saturation temperature based upon the axial power profile and the axial temperature profile at the onset of boiling (see Fig. 22). The significance of the clad overheating to the sodium boiling temperature is simply related to an assumption that sodium voiding within the heated zone, under such conditions, will progress only as fast as the clad can overheat to the sodium saturation temperature. The details of the analysis which follows are quite simple and straightforward, but also quite illustrative in terms of the important effects which can be calculated. The axial power distribution and the axial coolant temperature distribution are given by Equ. (23) and Equ. (24), respectively, q

- cos(wx*)

(23)

qmax

and T c*-

To

Te_To

_I 2

1 +

sin( w x*) sin(w)

(24)

1200 CLAD TEMPERATURE YIHICH CORRESPONDS TO SODIUM SATURATION

i000 900

1.0ó 800— Q

700 L

u

'

.

~ 6 ~~ 600 — sI 4

500~

~ .2

n rc °0

i

~ I

o

400— 300

/

,

1

i

i

AXIAL POWER

PROFILE `~Pmox=12 kW/ft

C r max

V

/

CLAD AND COOLANT ‚N TEMPERATURE PROFILE AT BOILING FOR SLOW LIE

' /

STEADY STATE COOLANT TEMPERATURE PROFILE 20

40

60

80

10 0

DISTANCE, cm

Fig. 22 Illustration of Conditions at the Onset of Boiling for an FFTF Type Fuel Pin/Coolant Channel During Unprotected Loss-of-Flow Simulation.

295

Liquid Metal Cooled Systems In Equations (23) and (24), x* is the dimensionless axial coordinate, x* = 2x/L

,

where x varies

and L is the length of the heated zone, and w is the shape parameter for the cosine power distribution defined by q

mdx _

w

(25)

sin(w)

qdVq

The significance of this shape parameter as it relates to the maximum to average power distribution characteristic of the fuel bundle is illustrated in Fig. 23. It is assumed that the fuel clad and sodium increase in temperature at the same rate, and if one neglects initial time delays, then this rate is given by Equ. (26). q

dT( x*) _ dt

cos(w x*)

(26)

max

M

c

f + Mf

C

C

ha c + Mf

C

na

From Equ. (26) and the initial temperature profile at the onset of boiling, one can estimate a time for the clad to reach a temperature equivalent to the sodium

1.2 1.1 1.0 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

w

Fig. 23 Shape Parameter for Chopped Cosine Distribution on a Function of max-to-avg. Ratio.

296

Heat Transfer and Fluid Flow in Nuclear Systems

saturation temperature for each axial location, x* , according to Equ. (27) sat - T(x*)

T

ts(x*) _

(27)

dT(x*) dt

Combining Equations (25), (26) and (27), this time is represented by Equ. 28 as

{1

(T sa t - T o ) ts (c*) _

q max

-

2

(

1

+

sin(w)*)~}

(28)

cos(wx*)

C

ef In Equ. (28) the term C ef is the effective system heat capacity, which is a short form of the denominator of Equ. (26). Equation (28) now represents a relation for the time at each axial location for the clad to reach the sodium saturation temperature. If this time is plotted in a manner comparable to the customary results of the SAS voiding model, Fig. 24 results. It is noted that both the shape and the location of the curve generated by Equ. (28) are in agreement with the upstream voiding calculation of the detailed SAS transient analysis. One can even extend this calculation further to the onset of clad melting by appropriately substituting the temperature for clad melting in place of the temperature for clad reaching the sodium saturation temperature in Equ. (28). A similar curve is generated as illustrated in Fig. 25. Again, both the shape and the location in time are in good agreement with the detailed SAS transient analysis. The two curves in Fig. 25 for clad melting represent temperatures corresponding to the clad solidus temperature, and an effective temperature corresponding to the latent heat of clad melting. One can also do the same for the onset in fuel melting under the assumptions that the power profile and the power generation rate remain constant in time. Thus, it is demonstrated that heat capacity effects are of the first order importance in l oss-of-flow transients following flow reversal. Assuming flow reversal occurs with the onset of boiling, an effective voiding rate, dV(x*) dt

, can be determined by differentiating Equ. (28): qmax ~

dV(x*) _

dt

C

w

1

sin(wx*) [cos(wx*)]2

(T ef sat

1

-

(1 +

T

o

)

sin(wx*)

sin(w)

11 J ]

_

1 2sin(w)

(29)

Equation (29) shows in a fundamental way that the vo'ding rate varies directly with the power density and inversely with the heat capacity of the system and the inlet subcooling. It has been found that the above simple approximation reproduces most SAS voiding profiles for slow loss-of-flow transients at constant power, thus providing a convenient estimate amenable to hand calculations. Again, the good agreement further suggests that the one-dimensional models in SAS are inherently numerically accurate. That is to say that while all of the details included in the one-dimensional voiding models relative to film motion dynamics are physically

Liquid Metal Cooled Systems

297

SAS CALCULATION OF VOIDING FOR PUMP FAILURE 200 LIQUID

I80 I60 I40 I20

AXIAL LOCATION

ill 80

CLAD MELTING

60 40

CLAD TEMPERATURE

20

EQUAL TO SODIUM BOILING

O

LIQUID

DRYOUT

VOID

0

mmmmmm II1II ~ 1' iiIiIiii i iiiiiIiii I

2

3

TIME FROM START OF VOIDING, SEC

Fig. 24 Comparison of SAS Calculation of Voiding and Clad Melting for Loss-of-Flow at Constant Power with Simplified Heat Capacity Model.

i

298

Heat Transfer and Fluid Flow in Nuclear Systems

100 90 80 70 E

60 50

u 40 ö 30 20 10 0

20

40

60

80

100

LENGTH z, cm

Fig. 25 Voidinq Rate into Heated Zone as Calculated by Equ. (29). realistic, they appear to be only of second order importance in influencing the voiding rate under these conditions. This is consistent with Theofanous' conclusions that a simple homogeneous model provides good agreement with the SAS Code, which uses two-fluid equations. The voiding rate determined from Equ. (29) is plotted in Fig. 25. This shows, as is illustrated in previous figures, that the voiding rate varies in time, being most rapid early in time, and decreasing as voiding over the length of the subassembly progresses to completion. 4.1.3 Transient Overpower Conditions. The unprotected transient overpower accident also represents an accident in which sodium boiling occurs. However, in this accident much less attention has been paid to sodium boiling dynamics than in the loss-of-flow accident. The reason for this is that in the transient overpower accident, the onset of voiding, clad and fuel melting are nearly simultaneous. So, in this accident, the significant concern is not so much the voiding rate per se, but the motion of the fuel immediately after failure. This is a case where combined effects of gas release at fuel failure, fuel motion, and sodium boiling are present almost simultaneously. Early fuel motion away from the core midplane can lead to a reduction in power such that the sodium flow can be restored and the accident sequence terminated. However, relative to sodium boiling, the onset of flow reversal again should be present at the onset of boiling. Parallel channel, flow instability considerations are equally applicable and were it not for other factors being of overriding importance, heat capacity considerations would indicate a more rapid upstream voiding because of the higher power levels. 4.2

Protected Accidents

Two classes of protected accidents may involve considerations of sodium boiling within some core subassemblies. They are (1) a postulated pipe break condition referred to as loss-of-pipe integrity (LOPI) and (2) loss of primary heat sink (CONS) at decay heat power levels. The potential for sodium boiling in both

Liquid Metal Cooled Systems

299

considerations is highly system dependent. We will mention some of these system dependencies and then focus on the generic features of boiling in each case. 4.2.1 LOPI. The loss-of-pipe integrity conditions obviously refers to a condition of concern only in a loop type LMFBR. Even in this system, the occurrence of boiling is dependent upon the number of primary flow loops, the plant power level, and other thermal hydraulic conditions. It is generally acknowledged that the guillotine break in a low pressure LMFBR piping system is less likely than for a high pressure piping system. For an LMFBR, leak or pipe crack, detection before break is considered likely. However, analysis of such conditions have proceeded for US FFTF and CRBR loop type reactors. The course of the accident is characterized by both rapid loss-of-flow and rapid neutronic shutdown as indicated in Fig. 26. The residual flow (from auxiliary pumping power) is sufficient for long term heat removal as shown in Fig. 27. The time scale of importance here is measured in seconds. It is correspondingly the question of removal of the stored energy in the fuel pin, which amounts to about 2 to 3 equivalent full-power seconds, without loss-of-flow stability and clad melting. It was generally shown that most low and intermediate power subassemblies within the FTR and CRBR cores when subjected to a LOPI protected transient did not reach boiling. However, some high power subassemblies reached the sodium saturation temperature. SLSF tests P-1 (73) and W-1 (74) were executed in the context of LOPI flow and power transient. The P-1 test was run according to a pre-programmed FTR pipe rupture transient while tests W-1 represented a series of flow and power transients to boiling, starting from various power levels. Figures 28 and 29 show the P-1 test transient data and a SAS post test analysis of the limited boiling associated with this case. The benign and limited nature of the sodium boiling process as indicated by the SAS calculation, Fig. 29, would appear consistent with the equally benign nature of the flow meter trace in Fig. 28. In fact, the only indication of the occurrence of sodium boiling is the flattened exit temperature

a

I

1.0

‚-~

FO

H

..

I

1 .0

I

FFTF Pipe Rupture Power Decay

.8

H . ~~ D a Ll

.6

~~

Fa

o

a .4

.. .4 F-

°

rc

°

.2

O

.-~-i

a

Z

FFTF Pipe Rupture Flow Decay

.2

—

0

o

I

5

I

I

10

15

TIME, SEC

I

20

5

I 10

I 15

TIME, SEC

26a, b Flow Rate and Reactor Power vs Time for Simulated Loss-of-Pipe Integrity Accident with Scram.

20

300

Heat Transfer and Fluid Flow in Nuclear Systems

5—

o

N

FFTF Pipe Rupture Characteristics With Boiling Suppressed

04

I—

w

3 ~~ J

w

2'

92 a

o

1

O ~

I

O

0 ~~ a

I

1

2

3

TIME, SEC

Fig. 27 Transient Power to Coolant Flow Ratio for Loss-ofPipe Integrity Accident with Scram. in Fig. 28, which is of the same order of time duration as the SAS calculation of sodium boiling. It is important to recognize that such analyses, exploring the limits of stable flow conditions, push the limits of analytical capabilities to describe the actual boundary. The actual limits are in fact influenced by details of the fuel pin model, voiding model, pressure drop in the vapor cavity and two-dimensional bypass effects. However, in a more fundamental matter, the controlling physics is determined by (1) the timing of the flow and power transient and (2) the stored energy in the pin and the downstream heat sink of the fission gas plenum. 4.2.2 Decay Heat Removal Considerations. At the other extreme of the time scale is the loss of heat sink conditions. Here one is concerned with the ability of the core to survive boiling for an extended period of time without dryout so as to prevent clad and potential fuel melting on a long time scale. SAS analysis by Dunn (75) for the FFTF reactor indicate the potential for removal of up to 4% of nominal power for a protected period of time. Typical boiling traces are shown in Fig. 30. The reasonableness of the result can be related to the following fundamental considerations. It is a good assumption that liquid film drainage inside the voided region can occur so long as the vapor velocity at the end of the heated zone does not exceed the flooding velocity or the velocity required for the slug to annular flow regime transition. This velocity is effectively given by Equ. (4). For Equ. (4) with K = 3.0,

r

u

= 0.72 g/cm 3, P g

=

5.2 x 10-4 g/cm 3,

s=

150 dyne/cm and

-

g = 980 cm/sec , the critical velocity jq = 24 m/sec. Now if one considers only latent heat transport, Equ. (5) determines the minimum heat that can be removed by latent heat transport process only, ignoring other convective effects. The flow area of an FTR subassembly is 43 cm 2 and for sodium h fg = 3800 j/g.

Liquid Metal Cooled Systems

160

I I Reactor Power

T

I

301

1800

I Sodium Exit Temperature

.- 120

Test Flow 120 —

1400 —80

~~ S

O

1000

LL 80 Sodium

—40

40

~

~~ H

rce

O N

Inlet Temperature

w

F-

S ~

° D

600

l

I

2

I

I

4

i

I

I

6

i 8

i

200 10

TIME, SEC Fig. 28 Test Data for SLSF P-1 Loss-of-Pipe Integrity Flow Transient. The maximum power from Equ. (5) for these conditions turns out to be 200 Kw. A high power subassembly in the FFTF core is worth about 5.5 Mw. The ratio of 200 Kw to 5.5 Mw represents about 3.7% heat removal due to latent heat transport. This is consistent with Dunn's SAS analysis and recent boiling experiments in pin bundle geometry simulating decay heat power levels (76, 77, 78). The important consideration is that at low power, or low vapor velocity, liquid film drainage can occur. Other estimates such as Reference (79), which ignore these flow regime stability considerations and assume an annular flow friction factor between vapor and liquid film, reach a conclusion that the limit of heat removal with boiling is an order of magnitude less. However, the above estimates are more physically sound. Also, in the context of heat removal at decay power levels without loss of heat sink, the concern for intermittent sodium boiling arises because of potential loss of temperature driving potential for natural circulation. Again, the previous estimate of heat removal would assure no permanent loss of coolant due to clad melting. However, so long as some source of heat removal were in place, which maintained an overall core inlet temperature well below saturation, single-phase, natural convection flow would be the more likely mode of heat removal. This can be illustrated from the following simple relationship from natural convective flow driven by buoyant effects. The ratio of heat removal at some reduced natural convective flow, to heat removal at full flow is given by Q __ p U(Tout Ro o To U A Q~

T

in )

nat-cony

(30)

Heat Transfer and Fluid Flow in Nuclear Systems

302

O

O

0.0

0.4

0.8

1.2

TIME SINCE START OF VAPOR FORMATION (SEC)

Fig. 29 SAS Voiding Calculation for R-1 Test Transient Illustrates Intermittent Boiling with Flow Recovery. The driving potential for flow across the core is DR nat-conv = g

o ~ (T

T

out

in

(31)

)

i f it is assumed that the velocity U across the core is approximately given by U nI(R , and that the ratio of the friction factors is approximately unity then one can write

Q ti Qo

out - Tin

T

DR

-

°1-(Tout - Tin )

0

° P

(32)

o

The following property values are relevant to provide a numerical estimate: T

920° C

R

= 300°C

b = 238 x 10-3

DT o

= 150°C

DR o = 10 6

out =

Lo

= 200 cm

4

dyne

cm i1ith the above, Q/Qo

ti

0.3. This illustrates the large margin for natural convec-

tive heat removal of the LMFBR that exists between the normal inlet temperature and the normal sodium saturation temperature. This margin would in fact be corrected downward when proper account is taken for the flow resistance of the total

Liquid Metal Cooled Systems

303

o M

O

u

-

9 F0' '0

O O

110.0

150.0

1 90.0

230

TIME (S)

Fig. 30 SAS Voiding Calculation for FFTF at Decay Heat Power Level. coolant circuit and property variations. * However, it is also consistent with Equ. (32) that a T of %100°C is sufficient to remove decay heat at a 2% out - T in power level. Correspondingly, for any subcooling which provides a temperature difference in excess of ü100°C, single-phase natural convection heat transfer would be more likely than stable boiling.

4.3

Boiling Behind Blockage

Localized boiling in pin bundle geometry due to postulated coolant channel blockages has received worldwide attention. Initially, it was thought possible that local boiling could occur in the wake of a relatively small blockage, involving 6-10 subchannels, and result in rapid boiling propagation across the subassembly, the concern here being the inability to detect the event in time to preclude subassembly flow starvation and meltdown. However, recent excellent progress in understanding fluid mechanics and heat transfer in the wake behind blockages (80) (81) rules out the concern for developing dangerous internal blockages during operation prior to limited fuel pin failures that can be detected by fission product detectors. Detailed computer codes such as SABRE (82) and WAKE (83) have been constructed to describe wake flows. * See Semeria et al., for a detailed discussion of natural convection flow modeling in LMFBRs.

304

Heat Transfer and Fluid Flow in Nuclear Systems

Experimental data and analysis show that in order to produce coolant boiling, a thin planar blockage involving more than 200 subchannels appears to be required. This is due to convective mass exchange from recirculation in the wake of the blockage which was ignored in early analysis. The case of flow through a porous planar blockage has also been considered with the WAKE code (83). A small seepage rate can actually increase the maximum temperature in the recirculation zone by reducing the level of recirculating flow and at the same time not greatly decreasing the residence time. However, the WAKE code results would indicate that this effect is of no great significance, since the effect is very small below 15% seepage rate and washed out completely above 20% seepage rate. Planar blockages greater than 50% of the subassembly cross-sectional area can therefore be detected by thermocouples before they lead to coolant boiling. Furthermore, analysis show that if coolant boiling is postulated behind such a blockage, it will not lead to flow instability and rapid subassembly voiding (84, 85, 87). The flow blockage model used by Fauske (87) is illustrated in Fig. 31. The axial temperature profile in the wake region is assumed to be linear, and the maximum temperature behind the blockage is given by the specified incipient-boiling superheat. The choice of wake dimension as well as temperature profile is considered reasonable on the basis of experiments (80). The boiling process is described by the single bubble concept (84) (40), along with the following assumptions: (a) inertia-controlled growth based on the work of Theofanous et al. (52), (b) isothermal collapse since the heat capacity of the vapor content is small compared to the cladding material, (c) neglect of the heat generation in the fuel rods since the bubble lifetimes are relatively short, (d) uniform pressure and temperature in the bubble, and (e) liquid film adhering to the fuel pins in thermodynamic equilibrium with the vapor. Based on the above assumptions, the vapor temperature in the growing bubble is given by

Ti

T

_ 1 =

T sup

sup -

2

N

T

2D

~B

(33)

and in the collapsing bubble by T1

1

R

0 - 2

( ~sur

R1

D

EB,max =

constant

CloV kage

TEMPERATURE

[AT

Assumed Profile

D

~·

Real Profile

~~--2D `\ Normal Conditions

—., Flow

DISTANCE

Fig. 31 Illustration of Flow Blockage Model.

(34)

Liquid Metal Cooled Systems

305

Assuming equilibrium within the bubble, the vapor pressure in the bubble is given by R n = R(T n )

(35)

The coupling between Equations (33) and (34) and the standard momentum equations written for the liquid slugs yielded the bubble history as a function of incipient superheat (see Fig. 32). It is seen that the lifetime of bubbles is small due to the large subcooling (the flow blockage in the example is 50% of the total cross section of an FFTF subassembly) and the waiting time between bubbles whose growth characteristics can result in sizable overall flow disturbances (large incipient superheat) is considerably longer than the hydraulic time constant of a typical LMFBR subassembly. Based upon planar blockage experiments and analyses performed by investigators from the Netherlands, West Germany, Italy, Japan, England, and the United States, the general consensus is that planar blockage of substantial size cannot lead to immediate or rapid subassembly disruption (88) (92). Most impressive are the experiments carried out at the Karlsruhe Nuclear Center (89). The tests involved an electrically heated bundle of 169 pins, with a centrally located blockage extending over 49% of the flow area.

BUBBLE LEHG

The main conclusion reached is that even under drastically reduced flow conditions, cooling in the region behind a planar blockage is preserved even with intensive local boiling, in agreement with early theoretical predictions by Gast (85) and Fauske (87). Flow instability, bulk-coolant boiling, and gross cladding melting resulting from a thin planar blockage appears therefore likely only if the blockage is sufficiently large to lead to bulk temperatures approaching the saturation temperature of the coolant. This implies that a thin, planar internal blockage

0.020

0.040

0.060

0.080

TIME, SEC

Fig. 32 Predicted bubble history as a function of incipient superheat..

306

Heat Transfer and Fluid Flow in Nuclear Systems

must almost approach the same dimension as required by an inlet blockage (>90% of the flow area) in order to cause failures or bulk boiling. Formation of internal planar blockage of this size (even with grid spacers) without prior pin failures must be considered an incredible event.

5.0 CONCLUDING REMARKS

These discussions of sodium boiling in fast breeder reactor geometry have tended to emphasize the results of the U.S. LMFBR development program. However, adequate reference has been made to important contributions from other nations so that the interested reader may pursue these sources in further depth. Nonetheless, these discussions and the numerous references (by no means exhaustive) do indicate a very mature technology area. In summarizing the current state of maturity, we choose four marks of comparison: (1) fundamental scientific-engineering understanding of principles, (2) understanding and applications to safety and consequence assessment in LMFBR accident analysis, (3) codification of analytical models and (4) overall experiment verification. An almost universal consensus seems to exist with respect to a fundamental scientific engineering understanding of principles. Sodium liquid superheat is treated fundamentally as a nucleation phenomenon. Pressure-temperature history considerations seem to adequately describe wall nucleation processes for both static and flowing systems and can be reproduced in laboratory experiments. At the same time, not all factors which effect surface chemistry, wetting, cavity size, and the like are understood from first principles. However, when present, inert gas solubility effects can override pressure-temperature history effects, leading to zero liquid superheat in a heat transport circuit. These effects are dominant in a reactor system and are reproducible in laboratory experiments, implying negligible liquid superheat prior to sodium boiling in LMFBR geometry. Sodium boiling in the low-pressure LMFBR coolant system is characterized by the slug and annular flow regime. Considerations of flooding criteria for co-current vapor liquid flow indicate the dominance of the annular, and bi-directional annular flow regime for boiling at nominal power levels. At decay heat power levels, the criteria for annular flow is unlikely to be satisfied, indicating a uniform pressure slug flow pattern in which liquid film drainage can occur. For both flow regimes, two-phase hydrodynamic consideration of pressure drop, void fraction and phase change for boiling sodium is well understood. Parallel-channel, Ledinegg type instability consideration can occur both core-wide and within subassemblies. Because of the low pressure and high-power density features of the system, boiling tends to be unstable under most conditions with early dryout of the heated surface. For local blockages, the assumed boiling in the wake region, however, is stabilized by the subcooling and recirculation flow in the wake. For whole core loss-of-flow conditions, non-uniform radial and axial temperature distributions give rise to non-coherence effects at boiling initiation. The conditions contributing to both the presence of these effects and their nominal significance in unprotected accidents are also widely recognized. The current understanding of fundamental considerations is quite satisfactory for applications purposes. In fact, over the last five to ten years, most efforts have been directed to improving applications rather than seeking additional fundamental developments.

Liquid Metal Cooled Systems

307

With respect to understanding sodium boiling applications for safety and consequence assessment in LMFBR accident analysis, we find the following to be most significant: Local Faults: It is generally accepted that an impervious planar blockage within an LMFBR subassembly cannot accumulate to a size sufficient to cause local boiling. Even were such boiling to occur, it would not lead to flow instability and loss of cooling within the subassembly. In fact, boiling from all local faults does not appear to have overall safety implications, assuming detection features are present in the design. Whole-Core Accidents: With negligible liquid superheat prior to boiling, sodium voiding by itself is no longer a direct source of large reactivity insertions leading to core disassembly. In whole-core loss-of-flow accidents in which sodium boiling occurs at near nominal power levels, boiling is unstable and leads to lossof-coolable geometry. Sodium voiding, however, is only one component of the reactivity effects which must be accounted for to understand accident consequences. Since such sequences lead to clad and fuel melting, it is the temporal aspects of radial progression (core-wide incoherences) and details of combined fuel and coolant motion that are of overriding importance. In whole-core loss-of-flow condition with neutronic shutdown, sodium boiling is likely to be only of a temporary nature, with long-term cooling provided by single-phase convection at flow to power ratios sufficient to preclude boiling. In whole-core accidents, sodium boiling by itself is no longer a direct contributor to accident consequences. However, certain fundamental considerations of sodium boiling relative to its stability are important to understanding accident sequence progression. For some previously postulated accident conditions such as local faults and loss-of-pipe integrity, system designs should be adequate to remove sodium boiling from safety concerns.* Therefore, we conclude that current understanding of sodium boiling is adequate for addressing safety consequences of LMFBR system designs. Much progress, worldwide, has been made in the development of working sodium boiling analysis codes for LMFBR geometry. For loss-of-flow conditions, we have indicated again the fundamental role of system heat capacity and boiling stability considerations. After flow reversal, voiding and overheating of the core is more determined by the heat capacity of the fuel and clad than two-phase sodium boiling dynamics. Understanding of these effects adds to the confidence that can be attributed to the boiling modules of highly developed accident analysis codes such as SAS and BLOW-3A. Current code developments have tended to emphasize core-wide incoherency effects, and within-subassembly incoherency effects. Progress has been made in both areas. Current versions of the SAS code can provide up to 33,channel descriptions of the reactor core. While many attempts have been made to formally code multidimensional boiling effects within a subassembly, the porous media approach of Theofgnous is results currently the most successful. It is also significant that Theofgnous indicate minimal significance of multidimensional boiling within a subassembly, thus reinforcing the conclusion that current one-dimensional models, accounting for core-wide incoherency, are sufficient for accident analysis. It would appear that some utility can be attributed to a multidimensional boiling model for the purposes of analysis and interpretation of small scale experiments. *

From a safety point of view, designs which potentially preclude boiling under unprotected loss-of-flow conditions would also be extremely attractive provided such designs were otherwise economic, safe and practical.

308

Heat Transfer and Fluid Flow in Nuclear Systems

Finally we conclude that the overall status of experiment verification is adequate for fundamental consideration and whole-core unprotected accident applications. Verification of the limits of stable boiling of sodium at decay heat power and natural circulation flow conditions appears to be one of the few areas requiring additional experiments. Work is currently ongoing to provide this information.

APPENDIX

TWO-PHASE PRESSURE DROP

In order to solve the standard conservation equations written for two-phase slip flow, relationships for the void fraction and wall shear stress are required. By using incompressible flow theory for the liquid and vapor phases, 4~~ Momentum for both phases:

dz

+

Dw

= 0

(A.1)

= 0

(A.2)

40. Momentum for vapor phase: dZ +

and the following relationships for the shear stresses, w

~i =

1 2 cfpfUf =Z

(A.3)

c g[1 + 75(1 - a)] p g Uq

(A.4)

Wallis (56) derived the following expression for void fraction in annular flow,

(1 - a) 2[1 + 75(1 - a)]_ a

2. 5

c

2

(A.5)

where for turbulent-turbulent flow c 2 becomes (35)

~ c tt

1 xx

~0.9 ~ p f ~ 0.5

G úg ~0.1

(A.6)

Experimental void fraction values in the annular f ow regime for flashing liquid sodium were obtained by Fauske (33) and are compared with Equ. (A.5) in Fig. A.1. The above expression for the void fraction can be used together with the well-known expression for the two-phase friction multiplier for annular flow F2 = 1 f 1 - a 309

DR

DR

TR fo

(A.7)

310

Heat Transfer and Fluid Flow in Nuclear Systems

to obtain an estimate for the two-phase frictional pressure drop. Comparison of Equ. (A.7) to data is illustrated in Fig. A.2. This general agreement has also been noted in Reference (93).

0.0 I

0.1

1.0 10 LOCKHART - MARTINELLI PARAMETER, Ctt

I00

Fig. A.1 Experiment Liquid Volume Fraction Comparison for Potassium and Sodium.

10- I

10°

io

LOCKHART MARTINELLI PARAMETER, C tt

Fig. A.2 Experiment Two-Phase Friction Pressure Drop Comparison for Potassium and Sodium.

Liquid Metal Cooled Systems

311

REFERENCES 1. 2. 3.

4. 5. 6. 7. 8.

9. 10.

11. 12. 13.

14. 15. 16. 17. 18. 19. 20.

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PROBLEMS (See Chapter 4, Part I and the Appendix for Thermal Properties of Sodium.) Problem 1. A sodium boiling experiment is being conducted in a 1 liter vessel filled with 400 gm of sodium. Prior to heating, the vessel is pressurized to three atmospheres with nitrogen while the sodium is maintained at a temperature of 600°C. The pressure is then reduced to 1 atmosphere and maintained at that level while heating slowly to incipient boiling. What is the expected wall superheat prior to boiling? After reaching boiling the apparatus is again pressurized with nitrogen to three atmospheres and allowed to cool to 350°C. The pressure is reduced to 0.4 atmosphere and maintained at that level while heat is applied. What is the expected superheat prior to boiling? Problem 2. A typical LMFBR fuel pin is 7 mm in diameter with an associated sodium coolant volume fraction of 0.4. Boiling occurs at reduced flow and full power corresponding to 300 w/cm average. Under normal operating conditions the inlet temperature is 290°C, the pin length is 100 cm and the sodium velocity is 8 m/sec. What is the normal outlet temperature? What reduction in flow rate is required to cause sodium boiling at 2 atm pressure? Referring to Equations 4 and 5 in the text, discuss the likely two phase flow regime after boiling. What is the length of the boiling zone required to establish this flow regime? Problem 3. Assuming an axial power shape factor of 1.4 max to average calculate the time to void the heated length (100 cm) of the fuel pin in Problem 2, using the simplified heat capacity approach. Change the shape factor to unity and increase the power to 500 w/cm. Recalculate the voiding time over the heated length. Problem 4. With the aid of Equ. 4 and 5 estimate the dry-out heat What percent of normal power does this represent? Compare this to vection heat removal potential assuming the core inlet temperature the exist temperature is limited by the saturation temperature for

flux for sodium. the natural conis 360°C and sodium at 2 atm.

Problem 5. Given a subassembly, whose charactertics are as follows: Coolant flow area is 50 cm2, normal sodium flow rate is 600 g/(cm2 -sec), normal pressure drop at 8 atm, normal temperature increase is 150°C. Calculate the flow reduction and temperature increase for a 10% area blockage at the inlet to the subassembly. (Assume a thin planar blockage.) Repeat for a 50% area blockage. What area blockage is require to cause boiling at the exit of the subassembly?