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Incoherency effects in clad relocation dynamics for LMFBR CDA analyses
Nuclear Engineering and Design 36 (1976) 59-67 © North-Holland Publishing Company
INCOHERENCY EFFECTS IN CLAD RELOCATION DYNAMICS FOR LMFBR CDA ANALYSES T.G. THEOFANOUS, M. DiMONTE and P.D. PATEL Nuclear Engineering Department, Purdue University, West Lafayette, Indiana 4 7907, USA
Received 27 September 1975
Reactivity feedbacks owing to clad relocation during a core disruptive accident (CDA) are of primary importance in LMFBR safety. A calculation of these feedbacks would require a knowledge of clad dynamics. An experimental and theoretical investigation of the flooding phenomenon (which is relevant to clad dynamics) was carried out. The experimental data indicate that flooding velocities agree well with Wallis's correlation. An extension of this correlation to LMFBR conditions gave rise to a set of flow regime maps, which allow a first order estimate of cladding velocities and could be utilized for evaluating reactivity feedbacks.
phase, vertical, gas-liquid flow systems, and may occur in a variety of physical and geometric conditions. The regime of primary interest for LMFBR clad relocation dynamics consists (at least initially) of an'essentially 'annular' flow pattern: a molten cladding film adhering to the fuel pin and an upward sodium vapor flow due to the pressure gradient imposed across the core. The majority of previous work in this regime has been performed using vertical tube sections with a variety of lengths, diameters, fluids and flow conditions [ 2 - 4 ] . A number of correlations exist and sizeable discrepancies amongst data and/or correlations are not uncommon. In usual applications the liquid and gas flow rates are specified such that, taking into account the geometry of the system, the flooding limit is avoided. Knowledge of the so-called 'flooding velocity' of the gas as a function of the liquid flow rate is sufficient for such purposes. On the other hand, for the applications of interest here, the total available pressure gradient is specified instead, and the resulting flow must be predicted. From the single-phase pressure drop, an upper limit of the gas velocity in such a system ma~; be calculated. In the absence of gas-flow bypass (absence of dry channels in parallel) this velocity may be compared to the critical flooding to determine the flooding limit. However, in the presence of bypass, a detailed determination of the entire pressure field would be required prior to the estimation of the velocity fields and hence flooding boundaries. Further-
1. Introduction The experience accumulated from FFTF-related studies indicates that, for small-size LMFBRs, a LOFA initiating-phase calculation is quite insensitive to the details of the contributing physical processes. Calculations performed within the SAS framework invariably lead to a 'transition phase' characterized by a boiling steel-fuel pool within a plugged up core region. It is possible, for such behavior, to 'conservatively bound' the expected range of events and their consequences, provided a judicious choice of assumptions is made and sensitivity calculations are performed. For a large LMFBR, such as the CRBRP, the problem becomes increasingly more complex. The major goal again is to estimate reactivity consequences throughout the initiating phase to determine the potential for: (i) early disassembly and shutdown; (ii) transition phase; and (iii) termination without extensive loss of geometry. However, due to an increased magnitude of possible reactivity feedbacks due to sodium voiding and clad motion, the behavior of a large LMFBR requires a closer examination. The discussions in ref. [ I ] illustrate this point well. An investigation of the flooding phenomena in the subassembly, taking clad melting incoherencies into account, could lead to a better assessment of the reactivity feedbacks possible. Flooding phenomena, in general, refer to the limits of steady-state operability of counter-current, two59
60
T.G. Theofanous et al. / Clad relocation dynamics
more, in both cases, mere knowledge of the flooding velocity does not yield a description of the flow process following flooding inception. The gist of the above remarks is that, for the purposes of assessing cladding relocation, the knowledge of flooding velocity is of limited value. It appears preferable to deal in terms of the post-flooding pressure gradient and to approach the flooding limit from cocurrent upward annular flow rather than from the traditional counter-current regime [3]. In this manner the 'hazy' region between flooding inception and complete cocurrent (all liquid moving up) climbing film flow can also be defined. For a film continually supplied with liquid, flooding inception appears to correspond to interfacial instabilities which, by increasing the interfacial friction, give rise to a flow transient with continually increasing liquid holdup, leading, in turn, to a chaotic phase distribution. A different transient wo~.!d be expected for a fixed liquid film, as is the case for the molten cladding. For small hydraulic diameter channels these effects are further accentuated by the more readily obtained flow regime transitions. On the other hand, for cocurrent upflow, an annular flow regime would be expected. In experimental work, therefore, it would be beneficial to avoid such unrepresentative flow transitions by approaching the limits of upflow and incident flooding from upward annular flow. The purpose of this paper is to employ the above considerations, together with a well-known correlation of two-phase frictional pressure drop, to predict the amount of bypass and define the boundaries for upward cladding motion for different degrees of axial and radial cladding melting incoherencies and different amounts of total available pressure drop across the subassembly. Some experimental data obtained in an alcohol-air system,.with and without air bypass, are presented and their consistency with the theoretical predictions is demonstrated.
2.Experimental Flooding experiments were performed in the flow loop shown in fig. 1. The large inlet and outlet plenums provided sufficient isolation of the test section from outside pressure disturbances. Experiments were run at room temperature and the pressure level in the
PLENUM ~
EXHAUST
UQUID FEED ~
T
E
FILM
ST SEGTION Y-TRANSDUC~ X-Y PLOTTER~ X-TRANSDUCER
IFIGE METER
""~~i
PLENUM
GAS INPUT
Fig. 1. Flow apparatus. system was very nearly atmospheric. Instantaneous flow and test section pressure drop measurements were obtained by means of a Pace Engineering model Pg00 and a Validyne model DP45 differential pressure transducers (0-1 in. H20 range). The test section, shown in fig. 2, was machined out of acrylic sheets. In the bypass configuration a three subchannel arrangement
[ []
PRESSURETAP
I
WITHOUT BY-PASS :-:,.1 (b)
ij to
l
(a)
T.~. ', .~..~ .'J (C]
WiTH BY-PASS PRESSURETAP NOTE: DIMENSIONSIN GM.
~SHIM STOGK
Fig. 2. Test section.
61
Z G. Theofanous et ai. / Clad relocation dynamics
was produced by the permanently fixed runners shown in fig. 2(c). The no-bypass configuration was produced by inserting acrylic runners in the two outer subchannels as shown in fig. 2(b). The liquid fflrn was, in most runs, produced on the lower face of the central subchanriel. The wick structure shown at the top of fig. 2(a) was found adequate for the controlled injection of a uniform liquid film. A rotameter was utilized for the liquid flow rate determination. Ethyl alcohol was chosen as the working liquid and air as the gas. Alcohol wets the acrylic material of the test section extremely well and is, therefore, convenient for sequential runs. Another desirable property is that at the end of each run the channel, at times being wet throughout from two-phase mixing, could be easily dried by simply passing an air current through. Prior to attaching the top plate of the test section, precise liquid £flm thickness measurements were made over the width of the central subchannel for the whole range of anticipated liquid flow rates. The good wetting properties of the system now turned into a disadvantage, as the film tended to 'climb' up the sides of the runners, producing very strong thickness variations normal to the flow direction. This problem was alleviated by the use of the 0.012 in. brass shim-stock runners shown in fig. 2. The alcohol f'tlm could not climb over the edge of the shim and a very nearly uniform film thickness was achieved. Also, except for the high liquid rates producing turbulent films, the laminar film thickness predictions was found accurate. The dry test section flow-rate/pressure-drop curves were obtained prior to each run. These dry tests agreed with smooth wall predictions, within 5% for the bypass test section and within 15% for the no-bypass one. Two-phase runs were performed for a range of liquid flow rates (film thickness) in the two test sections under both quasi steady state and highly transient conditions. For both, the liquid film with no air flow was established first. Subsequently, for the quasi steady-state runs, the gas flow rate was increased gradually while the film flow phenomena was visually observed and the test section pressure drop against gas flow rate was recorded using a Houston Instrument Omnigraphic 2000 X - Y recorder. For the transient runs the gas control valve was left at a position corresponding to the particular transitions of interest (as determined from the quasi steady-state
runs), and the gas flow was shut off by another on-off valve. Subsequent to the formation of the liquid film this latter valve was turned on while the transient pressure/flow record was obtained and visual observations, in the form of high-speed movies, were made. These transient records agree well with the quasi steady-state ones, indicating that any possible alcohol vaporization had an insignificant effect on the results. A typical set of experimental results is presented in figs. 3 - 6 . At least three major transitions were visually identified, 'departure', 'stopped' and 'all up'. Due to the high frequency pressure oscillations, the locus o f uncertainty for each transition is marked by the shaded regions on these figures. The runs with bypass, as may be seen, were always smoother than the no-bypass ones. The first striking observation is that not one of the three transitions appears to touch-off a major transient in pressure drop of the type observed in recent experimental work [3]. Differences in the twophase flow structure are probably responsible for this discrepancy. A modification of the present experiment (allowing liquid film on both sides of the channel). showed a sharp jump in pressure drop (at flooding) to be associated with a chaotic two-phase flow structure. Upon 'departure' a large-wave pattern rapidly appeared. In addition to their large amplitude and wavelength these waves are distinctive also in their random velocities and spatial distribution, often leading to coalescence. They also appear to possess a random oscillatory lateral velocity component. The name 'departure' was given to characterize this first visible transiI.O
O~
~~--~STOPPED~ ALLUP~ WITFIHLM
-- 0.6 1
I
o.olsr TEST
o.o~25
o.o27z
SECTION ~LOW RATE eFT')SE¢)
Fig. 3. Experimental test section pressure drop versus gas flow rate.
62
Z G. Theofanous et aL / Clad relocation dynamics I.O
ALL UP STATIONARY
0.6
oI
°t
LL
sToP
WITHFILM
FILMTHICKNESS__2,7X _ I(~CM NOBY-PASS '
0.0157 ' Q02~ TEST SEGTIONFLOWRATE(FT~/SEG)
O~)i
Fig. 4. Experimental test section pressure drop versus glas flow rate.
TESTSE~n0" FLOW ~ATE t~SEC) Fig. 6. Experimental test section pressure drop versus gas flow rate.
tion. These waves appear similar to the disturbance waves mentioned by Wallis [4]. The associated transition was the only one resulting in a visible pressure effect in the form of high frequency oscillations. These oscillations diminished below and beyond this point. The test section could be operated indefinitely at points significanti~y beyond 'departure'. However, it is doubtful whether this would be possible in a test section with a significantly smaller hydraulic diameter. At points marked 'stopped', the wave pattern appeared stationary. However, some liquid was still dripping out of the bottom of the test section. Finally, the 'all up' transition marks the threshold for no
liquid flow out of the bottom of the test section. Of the whole range of flow regimes and phenomena the states 'in' and 'around' the 'stopped' transition appear the most chaotic. The film tends to accumulate, splash, bridge, etc. Occasionally bridging led to liquid spreading into the dry subchannels. Sometimes it gave rise to liquid seepage into the pressure tap(s) and led to erroneous dips in pressure drop of the type shown in fig. 4. Significantly all these effects were absent in the transient runs operating at condition 'in' or 'beyond' the 'all up' transition. In this case the liquid feed flow was diverted away from the test section entrance, and the film present within the test section moved upwards, simultaneously thinning, in an orderly fashion. As this happened the pressure drop reading moved, gradually, from the 'all up' region, vertically down to the corresponding dry test section value. The 'average' gas velocity in the test section, corresponding to the three transitions observed, is plotted against liquid film thickness in the manner of Grolmes et al. [3], in fig. 7. The correlation (solid line) of Grolmes et al. is also shown for comparison. It appears that this correlation best describes the inception of the first transition ('departure'). In fact, considering the variability common in such experiments, the results for both bypass and no-bypass are amazingly close. It is also seen that more than a doubling of the average gas velocity at departure is required to attain the 'all up' transition. From the measured pressure drop, the amount of gas bypass in the dry portion of the test section was
I'0f
/ 02 ~
ALL U
/
P
~
~,M THICKNESSz.2xlgcM BY-RLSS
0.0494 0D696 TEST SEOTIONFLOWRATE{FT~SEC)
Fig. 5. Experimental test section pressure drop versus gas flow rate.
T.G. Theofanous et aL / Clad relocation dynamics • "ALL UP" POINT, BYPASS 0 "STOP" POINT, NO BYPASS & "ALL UP" POINT, NO BYPASS • " DEPARTURE" POINT, BYRLSS • "STOP" POINT, BYPASS GROLMES e l (31. (3) D "DEPARTURE' POINT, NO BYPASS 1
S &
&
e
~r
"-"5
o
O
tl ti
o
o
3 '
1.16
'
210 ' 21.4 2 2!8 FiLM THIGKNESS ~ (~ I ~ l
31.2
'
'
Fig. 7. Gas velocity for the various transitions versus liquid film thickness. calculated. By subtracting this flow from the measured total flow in the test section the true gas flow rate in the wet subchannel could be deduced. The results are plotted in fig. 8, in the coordinates ]g,/i, suggested by WaUis's flooding correlation [4]. With the exception of one, most likely erroneous, point the 'stopping' transition appears in agreement with the generally accepted value of C = 1. The 'all up' transitions lie just above the C = 1.1 line and agree remarkably well with the generally accepted upflow transition Oflg ~ I [4]. The three Wood's metal 'simulation' experiments of Henry [5,2] are also shown by the arrows labeled (a) and (c). These were constant pressure drop experiments, obtained at two different initial argon velocities GROt.MES et al. (3} |-O.OE CM + 11-0006 CM I ~ d - 0 . 5 GM 2 ~ d - 1 . 5 CM 3~Cl.2.5 GM HENRY et OI ( 7 ) --I(0),(b),{c)
nO
~
LO
+j
• A • 0 • a •
P R E S E N T WORK "ALL UP' BYPASS 'ALL UP" NO BYPASS "STOP" BYPASS 'STOP" NO BYPASS "DEPARTURE" BYPASS "OI[PARTURE" NO BYPASS TWO FILM TESTS
•
(34 and 61 ft/sec). Run (c) attempted to flood but no cladding levitation was observed and in run (a) upward cocurrent flow was observed. For this run, the true gas velocity would be reduced from its initial value (due to increased friction) and unfortunately was not measured. Nevertheless, these two runs are in agreement with the correlation. Arrow (b) is the flooding point prediction of these experiments based on the Grolmes et al. correlation. Interestingly, this point also agrees with Wallis's correlation. Since, at melting inception, the Wood's metal was stationary, the value ofj~ in plotting these experimental points was taken as zero. To further explore the relationship between these two correlations, predictions of the Grolmes et al. correlation, for several randomly selected combinations of tube diameters and film thickness (or equivalently, liquid flow rates) are also shown in fig. 8. It appears that the flooding lines defined by points with the same film thickness are parallel to each other and the constant C depends on the tube diameter. Similarly, if Wallis's correlation (with C = 1) was to be plotted on Grolmes's coordinates of fig. 7, the resulting line would depend on the particular tube diameter chosen. Additional work is needed to further explore this 'diameter effect'. However, the two correlations are similar in origin (balancing gravity to frictional forces) and the differences depend on the method chosen to represent friction. Furthermore, for the range of hydraulic diameters of interest here (less than 0.4 cm) all correlations and data are in agreement. The results of the modified experiment previously mentioned (which more closely matched the wetted perimeter conditions of ref. [3] ) are also shown in fig. 8. These results also agree well with Wallis's correlation. The pressure drop under cocurrent upflow conditions is a sensitive function of the void fraction. The dimensionless expression
.i*l/~ .)q/~
t
Jg,Jr .c }tl
'
Ap* = 10-2];2 (1 + 75(1 -- or) }
(1)
"1-1
0.6 a
0"0
63
'
0
o~,
Jf Fig. 8. Wallis's plot for the various transitions.
d,
correlates a wide range of experimental data in waterair and alcohol-air systems [4]. Setting ]~ = 1 in this equation we can calculate the pressure gradient required for incipient levitation at any particular void fraction. These results compare favorably with our measured pressure gradient at the point of stopping.
64
T.G. Theofanous et al. / Clad relocation dynamics
Beyond this point the comparison is not as good, due most likely to small changes in void fraction in the channel. Because of the importance of this correlation in the analysis of the next section, work in this area is continuing. Henry's [5] reported pressure gradient included the pressure drop in the dry portions of his channel. In lieu of pertinent dimensions (omitted in his published test section figure) the reported pressure drop is imposed over the 8 in. length of the wet channel to calculate: Ap* _ 0.18 for the 34 ft/sec run and Ap* = 0.54 for the 61 ft/sec run. For his void fraction of about 0.80 the above equation yields a minimum levitation pressure drop of AP* = 0.21. Comparing these figures it is clear why the low velocity run is just at the incipient levitation point, whereas the second run represents an 'all up' case. In view of the lack of experimental verification of eq. (11 with liquidmetal systems this limited comparison appears encouraging. However, additional data are required in this area. In conclusion it appears that a j~ value in excess of one is required for cladding levitation and the minimum levitation pressure drop is given as a function of ct by Ap*N10-2{ 1+75(1-c0} --
~-572-
"
(2)
3. Analytical
FLOW AREA AW I'WET'I ~FLOW AREA AD I'DRY"I
~
PORTION OF SUBASSEMBLY WITH MOLTEN CLAD t"WET'I
L'r SUBASSEMBLY WITH SOLID CLAD I'DRY'I
Io SUBASSEMBLY
L
l
TOTAL FLOW AREA A
SODIUM VAPOR
Fig. 9. Subassemblygeometryand cladmeltingincoherency definition. termine/3, ~, and/i and hence evaluate upflow potential, as functions of the zXPT (or m/, and degree of axial and radial incoherency (L w and Aw). The void fraction for the clad-vapor system at the instant of melting is a = 0.76. From conservation of mass considerations we have /3 = ( A d / a ) 7 + ( A w / A )5 --¢id"), +.Ziw5 .
Consider the cladding melting incoherency characterized by L w and A w, as illustrated in fig. 9. Also consider a total available pressure drop represented by a factor m greater than the minimum corresponding to the static sodium head AXPT = m P s g L T .
The single phase pressure drop over sections 1D and 2D may be written (AP)I D = f pg ~ 1
(3)
Let Vg0 be the velocity through an entirely dry subassembly under a pressure gradient of Psg, i.e. Vg0 "~ 240 ft/sec. The velocity under a pressure gradient m P s g is (rn) 1/2 Vg0. Due to the increased resistance from the two-phase friction, the velocity through the dry portion ID will decrease by a factor/3 03< 11 to VgtD =/3(m)1/2 Vg0. Similarly the velocities through the regions 2D and 2W will change by factors 7 and respectively, to Vg2D = ?(m) 1/2 Vg0 and Vg2w = 8(m)I/2 Vg0. The object of this formulation is to de-
(4)
(LT - L w)=/32mPsg(L T - LW) , 1
2 .... f VI2D(zar)2W =-~ Pg ~ Z~W= 7 2 m P s g L w .
(5) (6)
In combination with eq. (31 the total pressure drop requirement is satisfied if
(1 - ( L w / L T ) ) / 3 2 + ( L w / L T ) 7 2 = 1 or
(1 -/~W)/3 2 +/~W',/2 = 1.
(7)
65
Z G. Theofanous et al. / Clad relocation dynamics
The remaining equation is obtained from the equality of the pressure drop across the parallel interconnected sections 2D and 2W: AP* = m3 , 2 (p s /Pc l) =
m('y2/10)
o8 f
] =o6
(8) ~"
0.8 0.9
0.4
with
:?
{
AP*= 10-2/'~ 2 1 + 7 5 ( 1 - a ) /
(9)
o.,
0.4
0.8
and
.,
Fig. 11. Flow regime map for the 'all up' transition, for a =
aVg2W
Ig - ( g O H P , ) l / 2
0.80.
_ aVgo(m)l/2~
(gOHP,)l/2 ~" 3~5(m)1/2.
(10)
Eqs. (4), (7)-(10) form a complete algebraic system. The solution of this system is presented in graphical form in figs. 10-15 for the range of void l;ractions of interest. The value o f m affects only the magnitude of ]g, as in eq. (10). It cancels out when eqs. (8) and (10) are substituted into eq. (9). By normalizing, therefore, ]g by (m) 1/2 as was done in figs. 10-15, the solution for any value o f m may be read immediately. Thus, for any combination of/~w, "4w and a, the value of/g/(m) 1/2 may be determined. Conversely, for any a and m the range of/~w - AW combination resulting in levitation (]~ > 1.1)may be determined. For example, for (x = 0.76 and m = 1 (only hydrostatic head) upflow is predicted for any 0 < L w < 0.1 and 0.7 < A w < 1.0 combination. For the same void fraction and a pressure drop four times the hydrostatic head (m = 4) upflow is predicted for any combination of/~w and.4w. Since this value ofm represents
only a modest available pressure drop, as compared to SAS-calculated values [6], it may be concluded that significant upflow of cladding should be expected independently of the degree of incoherency in melting. The rate of upflow is also sensitive to the value of m. As the void fraction increases all contours move in the direction of the upper left corner of these plots indicating stability of the upflow regime, i.e. as levitation proceeds, void fraction increases and levitation becomes even easier. For/g < 0.5 draining is predicted. However, due to regime changes under such conditions, care should be exercised in interpreting the draining regime. After the flow regime has been determined, the rate of upward cladding motion can also be determined. The two-phase pressure drop can also be related to the superficial liquid velocity ]~ [4]
AP*= 10-2 (./'~2/(1 - 002) + (1 - o0 .
(11)
0.7
=0.5 0.8
0.8 f
-~.
).6
Z. 0.4
0.0 o.o
0.7 08 0.9 1.0 I.I 0.4
Z.
~
I ~
5
0.4
0.0 0.0
0.8
oe
~9" 1.0 I.I 12
0.4~ A.
0.8
Fig. 10. Flow r e , m e map for the 'aLl up' transition, for e =
Fig. 12. Flow regime m a p for the 'all u p ' transition, for a =
0.76.
0.84.
66
T. G. Theofanous et al. / Clad relocation dynamics
~'=1.0
"L., 0.8
LI 1.2 1.3 1.4 1,5 1.6
0.4
0.0 0.0
0.4
0.0
0,8
0.4
0.8
X.,
Fig. 13. Flow regime map for the 'all up' transition, for a = 0.90.
Fig. 15. Flow regime map for the 'all up' transition, for a = 0.98.
This, upon combination with eq. (9), yields
rect experimental support in liquid-metal systems with appropriate ranges of void fraction and hydraulic diameter values especially with reference to the twophase flow regimes is needed. Extension of these re. suits to complete transient predictions including film thinning and role of exit blockages and wirewrap is also feasible.
/
j~ = /;2(1 _ a) 2 1 + 75(1 - a) a5/2
1
_ 102( 1 _ a) 3
/
(12) For void fractions of interest in LMFBRs, and for conditions where Ig > 2, eq. (12) can be reduced to
•
/~=Ig(1--a)
(
! + 7c:/2 5(1-a)
)
1/2 .
(13)
From eq. (13) we can, assuming a p* ~ 1000, obtain Vf
= 3.16 X 10-2a
(1 + 75(1 _ or) }1/2 a5/2 .
(14)
Finally, for a particular combination o f / f w andAw, the value of/~ may be read off the appropriate figure and If can be calculated, for any void fraction of interest, from eq. (12). This approach is expected to yield reasonable initial cladding velocity predictions. However, additional di-
0.8
O4
O.O OD
~ 0.4
d '
=1.3
4. Conclusions For pressure drops currently estimated using SAS, cocurrent upward clad flow would be expected under any degree of clad melting incoherency. The clad velocity can be determined for any degree of incoherency and pressure driving force from the flow maps presented here. There has been no verification of the SAS calculated lower plenum pressurization due to voiding, and the effect of core.wide incoherencies on this process is not known. Due to the large reactivity effects of clad motion in large LMFBRs, the results of this work indicate that the degree of lower plenum pressurization should be thoroughly assessed. Furthermore, the importance of two-phase flow regimes and the need of their experimental assessment under more prototypic conditions is noted.
1.4
Acknowledgement 1.6 1.7 1.8 t.9 2.O 0.8
Fig. 14. Flow regime map for the 'all up' transition, for a = 0.94.
This work was performed under the auspices of the Nuclear Regulatory Commission.
Nomenclature A, AW, L T, L L -- defined in fig. 9 A w =Aw/A = radial melting incoherency
T.G. Theofanous et al. / Clad relocation dynamics
Lw =Lw/Lr DI_t or d
[
g
Jg /f
= axial melting incoherency = hydraulic diameter o f subchannel = friction factor = acceleration due to gravity = superficial vapor velocity = superficial liquid velocity = dimensionless gas flux
(= ]g /( gDH P*)1/2) ]f
Vg Vf Ps Pcl P* = Oel/Pg APT A~P*
= dimensionless liquid flux (= jf/(gDH) 1/2) = gas velocity = liquid velocity = void fraction = liquid sodium density = cladding density = density ratio = total available pressure drop = dimensionless pressure gradient (= (dP/dz)/Pclg).
67
References [1] T.P. Speis, C.L. Allen, R.E. Alcouffe, R.P. Denise, W.E. Kastenberg and T.G. Theofanous, Studies of core disruptive accidents and licensing aspects of fast breeder reactors, The European Nuclear Conference, ANS Trans. 20 (1975) 547. ]2] H.K. Fauske, M.A. Grolmes and R.E. Henry, An assessment of voiding dynamics in sodium-cooled fast reactors, ANL/ RAS 74-20, Aug. (1974). 131 M.A. Grolmes, G.A. Lambert and H.K. Fauske, Flooding in vertical tubes, to be published in I. Chem. Eng. Symp. Series-Multiphase Flow Systems (Apr. 1974) Univ. of Strathclyde, Glasgow, U.K. [41 G.B. Wallis,One Dimensional Two-Phase Flow, McGrawHill (1969). 15] R.E. Henry et al., Cladding relocation experiments, ANS Trans. 18 (1974) 209. [6] R. Alcouffe, NRC, Private communication, Mar. 1975.
INCOHERENCY EFFECTS IN CLAD RELOCATION DYNAMICS FOR LMFBR CDA ANALYSES T.G. THEOFANOUS, M. DiMONTE and P.D. PATEL Nuclear Engineering Department, Purdue University, West Lafayette, Indiana 4 7907, USA
Received 27 September 1975
Reactivity feedbacks owing to clad relocation during a core disruptive accident (CDA) are of primary importance in LMFBR safety. A calculation of these feedbacks would require a knowledge of clad dynamics. An experimental and theoretical investigation of the flooding phenomenon (which is relevant to clad dynamics) was carried out. The experimental data indicate that flooding velocities agree well with Wallis's correlation. An extension of this correlation to LMFBR conditions gave rise to a set of flow regime maps, which allow a first order estimate of cladding velocities and could be utilized for evaluating reactivity feedbacks.
phase, vertical, gas-liquid flow systems, and may occur in a variety of physical and geometric conditions. The regime of primary interest for LMFBR clad relocation dynamics consists (at least initially) of an'essentially 'annular' flow pattern: a molten cladding film adhering to the fuel pin and an upward sodium vapor flow due to the pressure gradient imposed across the core. The majority of previous work in this regime has been performed using vertical tube sections with a variety of lengths, diameters, fluids and flow conditions [ 2 - 4 ] . A number of correlations exist and sizeable discrepancies amongst data and/or correlations are not uncommon. In usual applications the liquid and gas flow rates are specified such that, taking into account the geometry of the system, the flooding limit is avoided. Knowledge of the so-called 'flooding velocity' of the gas as a function of the liquid flow rate is sufficient for such purposes. On the other hand, for the applications of interest here, the total available pressure gradient is specified instead, and the resulting flow must be predicted. From the single-phase pressure drop, an upper limit of the gas velocity in such a system ma~; be calculated. In the absence of gas-flow bypass (absence of dry channels in parallel) this velocity may be compared to the critical flooding to determine the flooding limit. However, in the presence of bypass, a detailed determination of the entire pressure field would be required prior to the estimation of the velocity fields and hence flooding boundaries. Further-
1. Introduction The experience accumulated from FFTF-related studies indicates that, for small-size LMFBRs, a LOFA initiating-phase calculation is quite insensitive to the details of the contributing physical processes. Calculations performed within the SAS framework invariably lead to a 'transition phase' characterized by a boiling steel-fuel pool within a plugged up core region. It is possible, for such behavior, to 'conservatively bound' the expected range of events and their consequences, provided a judicious choice of assumptions is made and sensitivity calculations are performed. For a large LMFBR, such as the CRBRP, the problem becomes increasingly more complex. The major goal again is to estimate reactivity consequences throughout the initiating phase to determine the potential for: (i) early disassembly and shutdown; (ii) transition phase; and (iii) termination without extensive loss of geometry. However, due to an increased magnitude of possible reactivity feedbacks due to sodium voiding and clad motion, the behavior of a large LMFBR requires a closer examination. The discussions in ref. [ I ] illustrate this point well. An investigation of the flooding phenomena in the subassembly, taking clad melting incoherencies into account, could lead to a better assessment of the reactivity feedbacks possible. Flooding phenomena, in general, refer to the limits of steady-state operability of counter-current, two59
60
T.G. Theofanous et al. / Clad relocation dynamics
more, in both cases, mere knowledge of the flooding velocity does not yield a description of the flow process following flooding inception. The gist of the above remarks is that, for the purposes of assessing cladding relocation, the knowledge of flooding velocity is of limited value. It appears preferable to deal in terms of the post-flooding pressure gradient and to approach the flooding limit from cocurrent upward annular flow rather than from the traditional counter-current regime [3]. In this manner the 'hazy' region between flooding inception and complete cocurrent (all liquid moving up) climbing film flow can also be defined. For a film continually supplied with liquid, flooding inception appears to correspond to interfacial instabilities which, by increasing the interfacial friction, give rise to a flow transient with continually increasing liquid holdup, leading, in turn, to a chaotic phase distribution. A different transient wo~.!d be expected for a fixed liquid film, as is the case for the molten cladding. For small hydraulic diameter channels these effects are further accentuated by the more readily obtained flow regime transitions. On the other hand, for cocurrent upflow, an annular flow regime would be expected. In experimental work, therefore, it would be beneficial to avoid such unrepresentative flow transitions by approaching the limits of upflow and incident flooding from upward annular flow. The purpose of this paper is to employ the above considerations, together with a well-known correlation of two-phase frictional pressure drop, to predict the amount of bypass and define the boundaries for upward cladding motion for different degrees of axial and radial cladding melting incoherencies and different amounts of total available pressure drop across the subassembly. Some experimental data obtained in an alcohol-air system,.with and without air bypass, are presented and their consistency with the theoretical predictions is demonstrated.
2.Experimental Flooding experiments were performed in the flow loop shown in fig. 1. The large inlet and outlet plenums provided sufficient isolation of the test section from outside pressure disturbances. Experiments were run at room temperature and the pressure level in the
PLENUM ~
EXHAUST
UQUID FEED ~
T
E
FILM
ST SEGTION Y-TRANSDUC~ X-Y PLOTTER~ X-TRANSDUCER
IFIGE METER
""~~i
PLENUM
GAS INPUT
Fig. 1. Flow apparatus. system was very nearly atmospheric. Instantaneous flow and test section pressure drop measurements were obtained by means of a Pace Engineering model Pg00 and a Validyne model DP45 differential pressure transducers (0-1 in. H20 range). The test section, shown in fig. 2, was machined out of acrylic sheets. In the bypass configuration a three subchannel arrangement
[ []
PRESSURETAP
I
WITHOUT BY-PASS :-:,.1 (b)
ij to
l
(a)
T.~. ', .~..~ .'J (C]
WiTH BY-PASS PRESSURETAP NOTE: DIMENSIONSIN GM.
~SHIM STOGK
Fig. 2. Test section.
61
Z G. Theofanous et ai. / Clad relocation dynamics
was produced by the permanently fixed runners shown in fig. 2(c). The no-bypass configuration was produced by inserting acrylic runners in the two outer subchannels as shown in fig. 2(b). The liquid fflrn was, in most runs, produced on the lower face of the central subchanriel. The wick structure shown at the top of fig. 2(a) was found adequate for the controlled injection of a uniform liquid film. A rotameter was utilized for the liquid flow rate determination. Ethyl alcohol was chosen as the working liquid and air as the gas. Alcohol wets the acrylic material of the test section extremely well and is, therefore, convenient for sequential runs. Another desirable property is that at the end of each run the channel, at times being wet throughout from two-phase mixing, could be easily dried by simply passing an air current through. Prior to attaching the top plate of the test section, precise liquid £flm thickness measurements were made over the width of the central subchannel for the whole range of anticipated liquid flow rates. The good wetting properties of the system now turned into a disadvantage, as the film tended to 'climb' up the sides of the runners, producing very strong thickness variations normal to the flow direction. This problem was alleviated by the use of the 0.012 in. brass shim-stock runners shown in fig. 2. The alcohol f'tlm could not climb over the edge of the shim and a very nearly uniform film thickness was achieved. Also, except for the high liquid rates producing turbulent films, the laminar film thickness predictions was found accurate. The dry test section flow-rate/pressure-drop curves were obtained prior to each run. These dry tests agreed with smooth wall predictions, within 5% for the bypass test section and within 15% for the no-bypass one. Two-phase runs were performed for a range of liquid flow rates (film thickness) in the two test sections under both quasi steady state and highly transient conditions. For both, the liquid film with no air flow was established first. Subsequently, for the quasi steady-state runs, the gas flow rate was increased gradually while the film flow phenomena was visually observed and the test section pressure drop against gas flow rate was recorded using a Houston Instrument Omnigraphic 2000 X - Y recorder. For the transient runs the gas control valve was left at a position corresponding to the particular transitions of interest (as determined from the quasi steady-state
runs), and the gas flow was shut off by another on-off valve. Subsequent to the formation of the liquid film this latter valve was turned on while the transient pressure/flow record was obtained and visual observations, in the form of high-speed movies, were made. These transient records agree well with the quasi steady-state ones, indicating that any possible alcohol vaporization had an insignificant effect on the results. A typical set of experimental results is presented in figs. 3 - 6 . At least three major transitions were visually identified, 'departure', 'stopped' and 'all up'. Due to the high frequency pressure oscillations, the locus o f uncertainty for each transition is marked by the shaded regions on these figures. The runs with bypass, as may be seen, were always smoother than the no-bypass ones. The first striking observation is that not one of the three transitions appears to touch-off a major transient in pressure drop of the type observed in recent experimental work [3]. Differences in the twophase flow structure are probably responsible for this discrepancy. A modification of the present experiment (allowing liquid film on both sides of the channel). showed a sharp jump in pressure drop (at flooding) to be associated with a chaotic two-phase flow structure. Upon 'departure' a large-wave pattern rapidly appeared. In addition to their large amplitude and wavelength these waves are distinctive also in their random velocities and spatial distribution, often leading to coalescence. They also appear to possess a random oscillatory lateral velocity component. The name 'departure' was given to characterize this first visible transiI.O
O~
~~--~STOPPED~ ALLUP~ WITFIHLM
-- 0.6 1
I
o.olsr TEST
o.o~25
o.o27z
SECTION ~LOW RATE eFT')SE¢)
Fig. 3. Experimental test section pressure drop versus gas flow rate.
62
Z G. Theofanous et aL / Clad relocation dynamics I.O
ALL UP STATIONARY
0.6
oI
°t
LL
sToP
WITHFILM
FILMTHICKNESS__2,7X _ I(~CM NOBY-PASS '
0.0157 ' Q02~ TEST SEGTIONFLOWRATE(FT~/SEG)
O~)i
Fig. 4. Experimental test section pressure drop versus glas flow rate.
TESTSE~n0" FLOW ~ATE t~SEC) Fig. 6. Experimental test section pressure drop versus gas flow rate.
tion. These waves appear similar to the disturbance waves mentioned by Wallis [4]. The associated transition was the only one resulting in a visible pressure effect in the form of high frequency oscillations. These oscillations diminished below and beyond this point. The test section could be operated indefinitely at points significanti~y beyond 'departure'. However, it is doubtful whether this would be possible in a test section with a significantly smaller hydraulic diameter. At points marked 'stopped', the wave pattern appeared stationary. However, some liquid was still dripping out of the bottom of the test section. Finally, the 'all up' transition marks the threshold for no
liquid flow out of the bottom of the test section. Of the whole range of flow regimes and phenomena the states 'in' and 'around' the 'stopped' transition appear the most chaotic. The film tends to accumulate, splash, bridge, etc. Occasionally bridging led to liquid spreading into the dry subchannels. Sometimes it gave rise to liquid seepage into the pressure tap(s) and led to erroneous dips in pressure drop of the type shown in fig. 4. Significantly all these effects were absent in the transient runs operating at condition 'in' or 'beyond' the 'all up' transition. In this case the liquid feed flow was diverted away from the test section entrance, and the film present within the test section moved upwards, simultaneously thinning, in an orderly fashion. As this happened the pressure drop reading moved, gradually, from the 'all up' region, vertically down to the corresponding dry test section value. The 'average' gas velocity in the test section, corresponding to the three transitions observed, is plotted against liquid film thickness in the manner of Grolmes et al. [3], in fig. 7. The correlation (solid line) of Grolmes et al. is also shown for comparison. It appears that this correlation best describes the inception of the first transition ('departure'). In fact, considering the variability common in such experiments, the results for both bypass and no-bypass are amazingly close. It is also seen that more than a doubling of the average gas velocity at departure is required to attain the 'all up' transition. From the measured pressure drop, the amount of gas bypass in the dry portion of the test section was
I'0f
/ 02 ~
ALL U
/
P
~
~,M THICKNESSz.2xlgcM BY-RLSS
0.0494 0D696 TEST SEOTIONFLOWRATE{FT~SEC)
Fig. 5. Experimental test section pressure drop versus gas flow rate.
T.G. Theofanous et aL / Clad relocation dynamics • "ALL UP" POINT, BYPASS 0 "STOP" POINT, NO BYPASS & "ALL UP" POINT, NO BYPASS • " DEPARTURE" POINT, BYRLSS • "STOP" POINT, BYPASS GROLMES e l (31. (3) D "DEPARTURE' POINT, NO BYPASS 1
S &
&
e
~r
"-"5
o
O
tl ti
o
o
3 '
1.16
'
210 ' 21.4 2 2!8 FiLM THIGKNESS ~ (~ I ~ l
31.2
'
'
Fig. 7. Gas velocity for the various transitions versus liquid film thickness. calculated. By subtracting this flow from the measured total flow in the test section the true gas flow rate in the wet subchannel could be deduced. The results are plotted in fig. 8, in the coordinates ]g,/i, suggested by WaUis's flooding correlation [4]. With the exception of one, most likely erroneous, point the 'stopping' transition appears in agreement with the generally accepted value of C = 1. The 'all up' transitions lie just above the C = 1.1 line and agree remarkably well with the generally accepted upflow transition Oflg ~ I [4]. The three Wood's metal 'simulation' experiments of Henry [5,2] are also shown by the arrows labeled (a) and (c). These were constant pressure drop experiments, obtained at two different initial argon velocities GROt.MES et al. (3} |-O.OE CM + 11-0006 CM I ~ d - 0 . 5 GM 2 ~ d - 1 . 5 CM 3~Cl.2.5 GM HENRY et OI ( 7 ) --I(0),(b),{c)
nO
~
LO
+j
• A • 0 • a •
P R E S E N T WORK "ALL UP' BYPASS 'ALL UP" NO BYPASS "STOP" BYPASS 'STOP" NO BYPASS "DEPARTURE" BYPASS "OI[PARTURE" NO BYPASS TWO FILM TESTS
•
(34 and 61 ft/sec). Run (c) attempted to flood but no cladding levitation was observed and in run (a) upward cocurrent flow was observed. For this run, the true gas velocity would be reduced from its initial value (due to increased friction) and unfortunately was not measured. Nevertheless, these two runs are in agreement with the correlation. Arrow (b) is the flooding point prediction of these experiments based on the Grolmes et al. correlation. Interestingly, this point also agrees with Wallis's correlation. Since, at melting inception, the Wood's metal was stationary, the value ofj~ in plotting these experimental points was taken as zero. To further explore the relationship between these two correlations, predictions of the Grolmes et al. correlation, for several randomly selected combinations of tube diameters and film thickness (or equivalently, liquid flow rates) are also shown in fig. 8. It appears that the flooding lines defined by points with the same film thickness are parallel to each other and the constant C depends on the tube diameter. Similarly, if Wallis's correlation (with C = 1) was to be plotted on Grolmes's coordinates of fig. 7, the resulting line would depend on the particular tube diameter chosen. Additional work is needed to further explore this 'diameter effect'. However, the two correlations are similar in origin (balancing gravity to frictional forces) and the differences depend on the method chosen to represent friction. Furthermore, for the range of hydraulic diameters of interest here (less than 0.4 cm) all correlations and data are in agreement. The results of the modified experiment previously mentioned (which more closely matched the wetted perimeter conditions of ref. [3] ) are also shown in fig. 8. These results also agree well with Wallis's correlation. The pressure drop under cocurrent upflow conditions is a sensitive function of the void fraction. The dimensionless expression
.i*l/~ .)q/~
t
Jg,Jr .c }tl
'
Ap* = 10-2];2 (1 + 75(1 -- or) }
(1)
"1-1
0.6 a
0"0
63
'
0
o~,
Jf Fig. 8. Wallis's plot for the various transitions.
d,
correlates a wide range of experimental data in waterair and alcohol-air systems [4]. Setting ]~ = 1 in this equation we can calculate the pressure gradient required for incipient levitation at any particular void fraction. These results compare favorably with our measured pressure gradient at the point of stopping.
64
T.G. Theofanous et al. / Clad relocation dynamics
Beyond this point the comparison is not as good, due most likely to small changes in void fraction in the channel. Because of the importance of this correlation in the analysis of the next section, work in this area is continuing. Henry's [5] reported pressure gradient included the pressure drop in the dry portions of his channel. In lieu of pertinent dimensions (omitted in his published test section figure) the reported pressure drop is imposed over the 8 in. length of the wet channel to calculate: Ap* _ 0.18 for the 34 ft/sec run and Ap* = 0.54 for the 61 ft/sec run. For his void fraction of about 0.80 the above equation yields a minimum levitation pressure drop of AP* = 0.21. Comparing these figures it is clear why the low velocity run is just at the incipient levitation point, whereas the second run represents an 'all up' case. In view of the lack of experimental verification of eq. (11 with liquidmetal systems this limited comparison appears encouraging. However, additional data are required in this area. In conclusion it appears that a j~ value in excess of one is required for cladding levitation and the minimum levitation pressure drop is given as a function of ct by Ap*N10-2{ 1+75(1-c0} --
~-572-
"
(2)
3. Analytical
FLOW AREA AW I'WET'I ~FLOW AREA AD I'DRY"I
~
PORTION OF SUBASSEMBLY WITH MOLTEN CLAD t"WET'I
L'r SUBASSEMBLY WITH SOLID CLAD I'DRY'I
Io SUBASSEMBLY
L
l
TOTAL FLOW AREA A
SODIUM VAPOR
Fig. 9. Subassemblygeometryand cladmeltingincoherency definition. termine/3, ~, and/i and hence evaluate upflow potential, as functions of the zXPT (or m/, and degree of axial and radial incoherency (L w and Aw). The void fraction for the clad-vapor system at the instant of melting is a = 0.76. From conservation of mass considerations we have /3 = ( A d / a ) 7 + ( A w / A )5 --¢id"), +.Ziw5 .
Consider the cladding melting incoherency characterized by L w and A w, as illustrated in fig. 9. Also consider a total available pressure drop represented by a factor m greater than the minimum corresponding to the static sodium head AXPT = m P s g L T .
The single phase pressure drop over sections 1D and 2D may be written (AP)I D = f pg ~ 1
(3)
Let Vg0 be the velocity through an entirely dry subassembly under a pressure gradient of Psg, i.e. Vg0 "~ 240 ft/sec. The velocity under a pressure gradient m P s g is (rn) 1/2 Vg0. Due to the increased resistance from the two-phase friction, the velocity through the dry portion ID will decrease by a factor/3 03< 11 to VgtD =/3(m)1/2 Vg0. Similarly the velocities through the regions 2D and 2W will change by factors 7 and respectively, to Vg2D = ?(m) 1/2 Vg0 and Vg2w = 8(m)I/2 Vg0. The object of this formulation is to de-
(4)
(LT - L w)=/32mPsg(L T - LW) , 1
2 .... f VI2D(zar)2W =-~ Pg ~ Z~W= 7 2 m P s g L w .
(5) (6)
In combination with eq. (31 the total pressure drop requirement is satisfied if
(1 - ( L w / L T ) ) / 3 2 + ( L w / L T ) 7 2 = 1 or
(1 -/~W)/3 2 +/~W',/2 = 1.
(7)
65
Z G. Theofanous et al. / Clad relocation dynamics
The remaining equation is obtained from the equality of the pressure drop across the parallel interconnected sections 2D and 2W: AP* = m3 , 2 (p s /Pc l) =
m('y2/10)
o8 f
] =o6
(8) ~"
0.8 0.9
0.4
with
:?
{
AP*= 10-2/'~ 2 1 + 7 5 ( 1 - a ) /
(9)
o.,
0.4
0.8
and
.,
Fig. 11. Flow regime map for the 'all up' transition, for a =
aVg2W
Ig - ( g O H P , ) l / 2
0.80.
_ aVgo(m)l/2~
(gOHP,)l/2 ~" 3~5(m)1/2.
(10)
Eqs. (4), (7)-(10) form a complete algebraic system. The solution of this system is presented in graphical form in figs. 10-15 for the range of void l;ractions of interest. The value o f m affects only the magnitude of ]g, as in eq. (10). It cancels out when eqs. (8) and (10) are substituted into eq. (9). By normalizing, therefore, ]g by (m) 1/2 as was done in figs. 10-15, the solution for any value o f m may be read immediately. Thus, for any combination of/~w, "4w and a, the value of/g/(m) 1/2 may be determined. Conversely, for any a and m the range of/~w - AW combination resulting in levitation (]~ > 1.1)may be determined. For example, for (x = 0.76 and m = 1 (only hydrostatic head) upflow is predicted for any 0 < L w < 0.1 and 0.7 < A w < 1.0 combination. For the same void fraction and a pressure drop four times the hydrostatic head (m = 4) upflow is predicted for any combination of/~w and.4w. Since this value ofm represents
only a modest available pressure drop, as compared to SAS-calculated values [6], it may be concluded that significant upflow of cladding should be expected independently of the degree of incoherency in melting. The rate of upflow is also sensitive to the value of m. As the void fraction increases all contours move in the direction of the upper left corner of these plots indicating stability of the upflow regime, i.e. as levitation proceeds, void fraction increases and levitation becomes even easier. For/g < 0.5 draining is predicted. However, due to regime changes under such conditions, care should be exercised in interpreting the draining regime. After the flow regime has been determined, the rate of upward cladding motion can also be determined. The two-phase pressure drop can also be related to the superficial liquid velocity ]~ [4]
AP*= 10-2 (./'~2/(1 - 002) + (1 - o0 .
(11)
0.7
=0.5 0.8
0.8 f
-~.
).6
Z. 0.4
0.0 o.o
0.7 08 0.9 1.0 I.I 0.4
Z.
~
I ~
5
0.4
0.0 0.0
0.8
oe
~9" 1.0 I.I 12
0.4~ A.
0.8
Fig. 10. Flow r e , m e map for the 'aLl up' transition, for e =
Fig. 12. Flow regime m a p for the 'all u p ' transition, for a =
0.76.
0.84.
66
T. G. Theofanous et al. / Clad relocation dynamics
~'=1.0
"L., 0.8
LI 1.2 1.3 1.4 1,5 1.6
0.4
0.0 0.0
0.4
0.0
0,8
0.4
0.8
X.,
Fig. 13. Flow regime map for the 'all up' transition, for a = 0.90.
Fig. 15. Flow regime map for the 'all up' transition, for a = 0.98.
This, upon combination with eq. (9), yields
rect experimental support in liquid-metal systems with appropriate ranges of void fraction and hydraulic diameter values especially with reference to the twophase flow regimes is needed. Extension of these re. suits to complete transient predictions including film thinning and role of exit blockages and wirewrap is also feasible.
/
j~ = /;2(1 _ a) 2 1 + 75(1 - a) a5/2
1
_ 102( 1 _ a) 3
/
(12) For void fractions of interest in LMFBRs, and for conditions where Ig > 2, eq. (12) can be reduced to
•
/~=Ig(1--a)
(
! + 7c:/2 5(1-a)
)
1/2 .
(13)
From eq. (13) we can, assuming a p* ~ 1000, obtain Vf
= 3.16 X 10-2a
(1 + 75(1 _ or) }1/2 a5/2 .
(14)
Finally, for a particular combination o f / f w andAw, the value of/~ may be read off the appropriate figure and If can be calculated, for any void fraction of interest, from eq. (12). This approach is expected to yield reasonable initial cladding velocity predictions. However, additional di-
0.8
O4
O.O OD
~ 0.4
d '
=1.3
4. Conclusions For pressure drops currently estimated using SAS, cocurrent upward clad flow would be expected under any degree of clad melting incoherency. The clad velocity can be determined for any degree of incoherency and pressure driving force from the flow maps presented here. There has been no verification of the SAS calculated lower plenum pressurization due to voiding, and the effect of core.wide incoherencies on this process is not known. Due to the large reactivity effects of clad motion in large LMFBRs, the results of this work indicate that the degree of lower plenum pressurization should be thoroughly assessed. Furthermore, the importance of two-phase flow regimes and the need of their experimental assessment under more prototypic conditions is noted.
1.4
Acknowledgement 1.6 1.7 1.8 t.9 2.O 0.8
Fig. 14. Flow regime map for the 'all up' transition, for a = 0.94.
This work was performed under the auspices of the Nuclear Regulatory Commission.
Nomenclature A, AW, L T, L L -- defined in fig. 9 A w =Aw/A = radial melting incoherency
T.G. Theofanous et al. / Clad relocation dynamics
Lw =Lw/Lr DI_t or d
[
g
Jg /f
= axial melting incoherency = hydraulic diameter o f subchannel = friction factor = acceleration due to gravity = superficial vapor velocity = superficial liquid velocity = dimensionless gas flux
(= ]g /( gDH P*)1/2) ]f
Vg Vf Ps Pcl P* = Oel/Pg APT A~P*
= dimensionless liquid flux (= jf/(gDH) 1/2) = gas velocity = liquid velocity = void fraction = liquid sodium density = cladding density = density ratio = total available pressure drop = dimensionless pressure gradient (= (dP/dz)/Pclg).
67
References [1] T.P. Speis, C.L. Allen, R.E. Alcouffe, R.P. Denise, W.E. Kastenberg and T.G. Theofanous, Studies of core disruptive accidents and licensing aspects of fast breeder reactors, The European Nuclear Conference, ANS Trans. 20 (1975) 547. ]2] H.K. Fauske, M.A. Grolmes and R.E. Henry, An assessment of voiding dynamics in sodium-cooled fast reactors, ANL/ RAS 74-20, Aug. (1974). 131 M.A. Grolmes, G.A. Lambert and H.K. Fauske, Flooding in vertical tubes, to be published in I. Chem. Eng. Symp. Series-Multiphase Flow Systems (Apr. 1974) Univ. of Strathclyde, Glasgow, U.K. [41 G.B. Wallis,One Dimensional Two-Phase Flow, McGrawHill (1969). 15] R.E. Henry et al., Cladding relocation experiments, ANS Trans. 18 (1974) 209. [6] R. Alcouffe, NRC, Private communication, Mar. 1975.