Helmholtz phenomenon

Nuclear Engineering and Design 210 (2001) 53 – 77 www.elsevier.com/locate/nucengdes

Liquid entrainment by an expanding core disruptive accident bubble—a Kelvin/Helmholtz phenomenon Michael Epstein a,*, Hans K. Fauske a, Shigenobu Kubo b, Toshio Nakamura c, Kazuya Koyama d a

Fauske and Associates, Inc., 16W 070 West 83rd Street, Burr Ridge, IL60521, USA b The Japan Atomic Power Company, Tokyo, Japan c Nagasaki R&D Center, Mitsubishi Hea6y Industries, Ltd., Nagasaki, Japan d Ad6anced Reactor Technology Co., Ltd., Tokyo, Japan

Received 18 April 2001; received in revised form 25 June 2001; accepted 9 August 2001

Abstract The final stage of a postulated energetic core disruptive accident (CDA) in a liquid metal fast breeder reactor is believed to involve the expansion of a high-pressure core-material bubble against the overlying pool of sodium. Some of the sodium will be entrained by the CDA bubble which may influence the mechanical energy available for damage to the reactor vessel. The following considerations of liquid surface instability indicate that the Kelvin– Helmholtz (K–H) mechanism is primarily responsible for liquid entrainment by the expanding CDA bubble. First, an instability analysis is presented which shows that the K – H mechanism is faster than the Taylor acceleration mechanism of entrainment at the high fluid velocities expected within the interior of the expanding CDA bubble. Secondly, a new model of liquid entrainment by the CDA bubble is introduced which is based on spherical-core-vortex motion and entrainment via the K–H instability along the bubble surface. The model is in agreement with new experimental results presented here on the reduction of nitrogen-gas-simulant CDA bubble work potential. Finally, a one-dimensional air-over-water parallel flow experiment was undertaken which demonstrates that the K – H instability results in sufficiently rapid and fine liquid atomization to account for observed CDA gas-bubble work reductions. An important byproduct of the theoretical and experimental work is that the liquid entrainment rate is well described by the Ricou–Spalding entrainment law. © 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Following a postulated fast reactor excursion involving all or part of the reactor core, a bubble containing liquid fuel and its vapor and possibly * Corresponding author. Tel.: + 1-630-887-5210; fax: +1630-986-5481. E-mail address: [email protected] (M. Epstein).

molten steel and its vapor will emerge from the core and enter the upper plenum ‘pool’ of cold sodium. This two-phase bubble, often referred to as the core disruptive accident (CDA) bubble, has a high internal pressure and forces the sodium coolant before it as it expands. Impact of the sodium pool (slug) with the reactor head raises the possibility of serious consequences if a break should occur.

0029-5493/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S0029-5493(01)00436-8

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M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

A key phenomena of interest for determining the damage potential of the impacting sodium slug is the entrainment of sodium by the expanding fuel bubble. If very large amounts of sodium are entrained, fuel vapor condensation will result and the bubble driving pressure and slug impact force are reduced. Of course if the amount entrained is small, it has an insignificant effect on the slug impact force. Since the entrained sodium is volatile there is an intermediate range of sodium entrained mass for which sodium vaporization occurs within the fuel bubble leading to an enhanced driving pressure and slug impact force (Cho et al., 1974). Two different physical processes that might lead to liquid sodium entrainment have been identified clearly in the literature and are illustrated schematically in Fig. 1. One mechanism [(a) in Fig. 1] emerges from the well-known fact that wind blowing over a liquid surface generates surface waves. In regions where the fuel vapor flow is parallel to the liquid sodium the fuel vapor is expected to generate surface waves on the liquid sodium. Sodium droplet formation will occur when the ratio of amplitude to wavelength for these waves becomes too large for the crests to remain an integral part of the liquid. This wind-

Fig. 1. Illustration of: (a) Kelvin Helmholtz; and (b) Taylor — instability liquid sodium entrainment mechanisms during postdisassembly fuel expansion.

wave entrainment mechanism is usually referred to as the Kelvin–Helmholtz (K–H) mechanism, in honor of their independent pioneering theoretical work on the generation of water waves by wind (see Lamb, 1945). Entrainment rates predicted by K– H theory depend on the local fuel vapor velocity and on the fuel vapor-to-liquid sodium density ratio. The second entrainment mechanism is the wellknown Taylor (1950) ‘instability’. When a fluid is accelerated either by gravity or by inertia in a direction perpendicular to an interface across which there is a discontinuity in density, waves grow on the surface provided that the acceleration is directed from the less dense fluid to the more dense fluid. For example, during the late one-dimensional acceleration stage of the sodium pool the upper portion of the bubble interface [(b) in Fig. 1] may be subject to breakup by means of this type of instability. The resulting entrainment rate depends on the instantaneous acceleration of the interface. Surface tension may also play a role in the Taylor entrainment process. Experiments performed by Cagliostro and Florence (1972) (see also Cagliostro et al., 1974) were the first to demonstrate a reduction in work done by an expanding, initially high-pressure, high-temperature, noncondensible gas in the presence of a relatively cold liquid (water). In their experiments, a piston was initially supported in a vertical cylinder, a short distance below an explosive charge. In most of the experiments water was placed on top of the piston. Upon firing the explosive charge, a gas pressure of approximately 200 atm was produced essentially instantaneously above the liquid surface. By this technique, the water column/piston composite was accelerated downward by the hot product gas at accelerations of up to 104 g. The reduction in gas work due to the presence of water was clearly revealed by the gas pressure and volume (P–V work) measurements. Epstein (1973a,b) proposed that the role of water in reducing the gas expansion work is to present a large surface area generated by Taylor instability to effect sufficient heat transfer from the hot expanding gas. He assumed that all the entrained liquid is converted into droplets of diameter of the order of the most probable Taylor-wavelength

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

and that the volume rate at which droplets are entrained is proportional to the square-root of the product of the liquid surface acceleration and the most probable wavelength, as suggested by the pioneering Taylor instability experiments of Lewis (1950). Good agreement between the theory and the experimental P– V measurements of Cagliostro and Florence (1972) was obtained for a reasonable choice of the proportionality coefficient. A modified version of Epstein’s entrainment model was used by Corradini et al. (1980) to assess the effects of sodium entrainment and heat transfer on CDA bubble expansion work, which is a measure of the disruptive mechanical energy of the core disruptive accident. The question of interest is whether the Taylor instability is primarily responsible for sodium entrainment within the CDA bubble which experiences both planar and spherical expansions as well as source-fluid jet-induced vortexing. It should be recognized that the Cagliostro and Florence (1972) experiments modeled by Epstein (1973a,b) involved planar expansions and the only operative entrainment mechanism was the Taylor instability. A number of experiments were conducted in an attempt to provide information on the dynamics of expanding bubbles similar to the CDA bubble illustrated schematically in Fig. 1. Three experimental programs carried out by Tobin and Cagliostro (1980) (see also Kufner et al., 1982), Simpson et al. (1981) (see also Simpson, 1981) and Kaguchi et al. (2000) simulated the post-disassembly-bubble expansion in scaled models of a fast breeder reactor (FBR) and involved the transient development of upward gas discharges and saturated liquid discharges into liquid pools that varied between 1/30 and 1/7 scale. The laboratory containment vessels used in the work of Tobin and Cagliostro and Simpson et al. were made of transparent acrylic cylinders and high-speed movies were taken of the motions of the bubble and the liquid pool. A notable feature of the photographic records is the formation of large vortices within the expanding bubble. The bubble volume history was determined by using the bubble profile captured on the high-speed movies and assuming a surface of revolution. Entrainment of pool liquid into the bubble was calculated from

55

the difference between the estimated bubble volume and the measured slug displacement volume. Referring to the nitrogen gas discharge experiments, Tobin and Cagliostro reported that about 25% of the bubble volume was occupied by entrained liquid when the liquid impacted the vessel cover while Simpson et al. reported about 7% liquid entrainment at slug impact. This difference may be due to the large differences in nitrogen gas source pressure and/or scale model size, namely about 10 MPa and 1/30 scale in the experiments of Tobin and Cagliostro and about 1.5 MPa and 1/7 scale in the experiments of Simpson et al. Tobin and Cagliostro (1980) and Simpson et al. (1981) estimated the peak kinetic energy of the liquid slug (at impact) by assuming that the whole liquid pool moves with the velocity of the water surface as determined from the movie data. The slug kinetic energy was found to be less than the ideal bubble work based on an expansion from the source pressure and volume to the final bubble volume at slug impact. The energy conversion efficiencies were typically of the order of 50% for nitrogen gas discharges and of the order of 10% for saturated liquid discharges. The CDA bubble experiments performed in Japan fall into two categories. One is concerned with the FBR vessel structural response to the simulant CDA bubble (Kaguchi et al., 2000), the other with the energy loss during the CDA bubble expansion. The CDA bubble energy loss measurements are presented for the first time in this paper. It is argued in this paper that the K– H instability is primarily responsible for liquid entrainment by the expanding CDA bubble. An analysis of combined K–H and Taylor instability is presented which shows that the K–H mechanism is dominant for conditions existing in the CDA bubble. Experiments were carried out for measuring the pressure–volume (P –V) relationships of a nitrogen-gas simulant CDA bubble. The results establish an unambiguous bubble energy loss (gas work reduction) relative to the P– V work produced by an ideal bubble expansion. A new model of liquid entrainment by the CDA bubble is introduced based on the K–H mechanism and a bubble-core vortex flow. Comparison of the model

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M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

with the experimental results on CDA nitrogenbubble gas work reduction proves favorable. Finally a one-dimensional air-over-water parallel flow experiment was performed in which the only possible mechanism of entrainment is the K–H instability. The experiments demonstrate that the K –H instability is capable of rapidly injecting a quantity of atomized liquid into the gas stream to effect a significant gas work reduction.

2. Taylor instability versus Kelvin – Helmholtz instability During the expansion of the CDA bubble there is likely to be considerable gas circulation (vortex motion) within the bubble so that the liquid at the bubble surface is subject to a strong lateral flow of gas. The upward motion of the bubble alone is sufficient to generate a Hill’s vortex (MilneThomson, 1960) within the bubble of gas circulation strength equal to the upward (axial) velocity of the bubble surface (see Section 4). The constant injection of gas from below will act to significantly strengthen the vortex. The gas, then, is expected to generate waves of the Kelvin– Helmholtz type over the entire bubble surface. Some segments of the bubble surface will also experience acceleration during the expansion process. Thus waves of both the K– H and Taylor types may be produced on these surface segments. For a fixed surface acceleration the nature of the waves on the surface, K– H or Taylor, depends markedly on the magnitude of the local gas velocity parallel to the bubble surface. The characteristic time ~ for the ‘linear growth’ of a two-dimensional surface wave due to both Kelvin–Helmholtz and Taylor instabilities is (from Lamb, 1945) 1 a(zf −zg) z z u2 | = k + f g 2k 2 − k 3, 2 ~ zf + zg (zf +zg) zf +zg

sional liquid entrainment mechanism is assumed. The most rapidly growing ridge wave, which is found when 1/~ 2 is a function of k exhibits a maximum (i.e. when ~ is a minimum), is detached as a ridge of liquid of diameter comparable with u/2 and width W. The mass rate of liquid entrainment per unit area of liquid surface is then m; ¦en 

where k is the wave number (inverse of wavelength u, i.e. k =2y/u), | is the liquid surface tension, u is the velocity of the lateral gas flow, a is the acceleration normal to the surface, and zg and zf are the densities of bubble gas and liquid, respectively. The following idealized two-dimen-

(2)

where ~min and kmin are the values of ~ and k for the most rapidly growing wave. The quantity kmin is determined by differentiating Eq. (1) with respect to k and setting the result equal to zero. Substituting the expression for kmin into Eq. (1) gives the equation for ~min. Finally, inserting the functions for kmin and ~min into Eq. (2) gives the desired result for m; ¦en ; namely m; ¦en 

1

3

(zgzf )1/2u

'

6b + 1, 1+ (1+ 3b)1/2

(3)

where b=

|azf . z 2gu 4

(4)

In deriving Eq. (3) the gas density zg has been neglected in comparison with the liquid density zf. It is clear from Eq. (3) that the mechanism of entrainment is controlled by the magnitude of the parameter b. If b 1 then the expression for entrainment by the K–H mechanism is recovered; namely m; ¦en 

1

(zgzf )1/2u.

(5)

3 On the other hand, if b 1 the Taylor-instabilitycontrolled entrainment rate asymptote is obtained: m; en 

(1)

zf y(umin/4)2Wzf  , ~min(uminW) ~minkmin

2 (z z )1/2 ub 1/4 = 31/4 g f

'  2 3|a 3 zf

1/4

z f.

(6)

Note that the numerical coefficients in the asymptotic forms given by Eq. (5) and Eq. (6) do not correspond to those that have been inferred from entrainment measurements. For example, experimental work (Ricou and Spalding, 1961) shows that the numerical coefficient in Eq. (5) is of order 0.1 (see Section 4.2). The reason for the dis-

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77 Table 1 Parameter b as a function of gas velocity u b

u (m s−1)

2.5×103 1.6×102 4.0 0.25 0.016

1.0 2.0 5.0 10.0 20.0

crepancy is that Eq. (3) is only valid for the early, linear stage of the instability and, presumably, the empirical coefficients correct for the late, nonlinear stage of wave growth. Nevertheless, it is reasonable to assume that Eq. (3) for the early behavior is a good indicator as to which instability will ultimately dominate the entrainment process. Consider the magnitude of the parameter b for a typical simulated CDA gas bubble expansion (Simpson et al., 1981; Kaguchi et al., 2000). The bubble pressure during the expansion is roughly 2.0 MPa and, therefore, the bubble gas density is approximately zg =22.0 kg m − 3. The acceleration of the bubble surface is estimated to be a= 2× 104 m s − 2. By inserting these estimates into Eq. (4), together with the values | = 0.06 kg s − 2 and zf =103 kg m − 3 for water, the magnitude of b as a function of the gas velocity u parallel to the liquid surface is obtained. The results are tabulated in Table 1. Clearly, only a modest transverse gas velocity (Y 15 m s − 1) adjacent to the bubble surface is required to ‘wipe away’ the effects of the Taylor instability and thereby render the entrainment process largely K –H controlled. The gas enters the bubble with velocities in excess of 300 m s − 1. While the vortexing velocities will be less the source velocity, they are likely to be high enough to ensure that the K –H mechanism is primarily responsible for liquid entrainment.

3. CDA bubble residual energy measurements A series of experiments simulating the post-disassembly expansion phase in 1/20 and 1/10 scale

57

models of an FBR was performed (Kaguchi et al., 2000). Several of these experiments were aimed at determining the amount of work done by the expanding simulant CDA bubble. The data obtained from two 1/10-scale bubble-work experiments, namely PC3-1 and PC3-4, provided sufficient information to allow conclusions to be drawn with respect to the liquid entrainment mechanism (see Section 4). The experimental system is similar in form to that used by previous investigators (Tobin and Cagliostro, 1980 Simpson et al., 1981) and is represented schematically in Fig. 2. A more detailed drawing of the apparatus is given in Kaguchi et al. (2000). Briefly, high-pressure nitrogen gas was initially stored in a discharge reservoir (or simulated reactor core and core barrel of total volume 1.7 m3). A glass rupture disk separated the nitrogen gas from an overlying liquid pool of water (0.61-m deep) contained in a cylindrical rigid vessel (1.04 m wide by 0.78 m high). The water simulates the pool of liquid sodium in the FBR upper plenum while the high pressure nitrogen gas simulates the CDA bubble. A cylindrical bar (0.286 m diameter by 0.628 m high) supported from the top of the vessel and located

Fig. 2. Proposed streamlines in CDA bubble. For clarity only one of the two vortex patterns is shown. The symbol P denotes locations of two key pressure measurements in Tests PC3-1 and PC3-4.

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M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Fig. 3. Measured pressure history of cover gas.

along the axial center of the water pool represented FBR upper internal structure (UIS, see Fig. 2). The experiment was initiated by bursting the rupture disk, thereby allowing the nitrogen gas to enter the water pool through an 0.286-m diameter opening (simulated core barrel) at the bottom of the test vessel. Pressures were recorded in the liquid pool, in the bubble, and in the vessel cover gas. The measured vessel cover gas pressures versus time are shown in Fig. 3. Note that the initial cover gas pressure was 0.19 MPa. The ‘stagnation’ pressure histories recorded at the center of the bottom surface of the UIS are shown in Fig. 4. The locations of the pressure transducers that provided the data in Fig. 3 and Fig. 4 are indicated in Fig. 2. The displacement of the surface of the liquid pool (i.e. water pool/cover gas interface) was measured with a level meter based on an electrical conductivity principle and the results are shown in Fig. 5. The velocity history of the surface of the pool was inferred from the surface displacement measurements (Fig. 6). From the measured pool surface displacement and the known cross-sectional area of the cover gas (0.785 m2), the cover gas volume history was calculated.

The functional relationship between the measured cover gas pressure and volume (obtained by eliminating the intermediate variable, time) was found to follow a near-isentropic path. This was an important finding in terms of making an accurate determination of the liquid pool (or slug) kinetic energy. In Tests PC3-1 and PC3-4 the initial cover gas gap was 17 cm. By way of comparison, the cover gas gaps in the experiments of Tobin and Cagliostro and Simpson et al. were 2.5 and 4.5 cm, respectively. The large gap in the tests reported here prevented the slug from impacting the vessel cover. Instead the cover gas was compressed in a near-isentropic manner to a minimum volume and then expanded. Thus an accurate measure of the slug kinetic energy was obtained by equating it with the compression work done on the cover gas, rather than by assuming that all of the pool liquid moves with the speed of the liquid surface. The actual expansion work W done by the nitrogen-gas simulant CDA bubble was determined in two different ways. Both methods started with the bubble work integral W= Pb dVb where Pb is the bubble pressure and dVb is

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

the incremental increase in the bubble volume. In the first method Pb was identified with the stagnation pressure measured at the base of the UIS and shown in Fig. 4. In the second method the effective bubble pressure history was calculated by assuming that the nitrogen gas flow from the top of the discharge reservoir to the bubble is wellrepresented by the compressible orifice flow equation with an orifice coefficient of 0.8. By equating the calculated orifice flow with the measured rate of change of bubble volume, the effective bubblepressure time history was determined. The two methods of calculating W are within 2% of one another. The cover gas compression work (or slug kinetic energy) and the actual bubble work are listed in Table 2. The cover gas compression work is less than the actual expansion work done by the nitrogen-gas simulant CDA bubble, indicating that there are significant energy losses associated with the CDA bubble expansion. The difference between the CDA bubble expansion work and the cover gas compression work is defined as the residual energy. In the next section it is shown that the K– H entrainment mechanism, together with a vortex model of the flow within the CDA

59

bubble, is capable of rationalizing the measured residual energies.

4. A Hill’s vortex model of Kelvin– Helmholtz-governed entrainment in the FBR/CDA bubble

4.1. Vortex model Visual observations by Simpson et al. (1981) and Tobin and Cagliostro (1980) indicate that the CDA bubble shape is near spherical and vortexing occurs within the bubble. The major assumption of the analysis presented here is that the flow in the bubble interior is an inviscid Hill’s spherical vortex whose strength is determined by the source velocity U of the gas flow that feeds the bubble. The bubble and vortex are illustrated in Fig. 2. In many of the simulant CDA bubble experiments, including Tests PC3-1 and PC3-4, a cylindrical bar supported from the top of the test vessel was employed to represent the upper internal structure (UIS) of a real reactor. The cylinder is shown in Fig. 2.

Fig. 4. Pressure (stagnation) history at base of upper internal structure.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

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Fig. 5. Measured displacement of liquid surface versus time.

The gas flows out of the core barrel, impinges against the lower boundary of the UIS and then flows parallel to this boundary and the core barrel opening until it enters the bubble in a tangential manner with average velocity U (see Fig. 2). It follows that the motion of the gas within the bubble in response to this tangential inlet flow is toroidal. The tangential inlet flow is regarded as a stagnation-type flow which drives the vortex motion in the spherical domain of the CDA bubble. Thus the motion within the bubble is assumed to have much in common with the high Reynolds number Hill’s vortex flow induced within a spherical droplet or gas bubble moving at velocity U (see, e.g. Chao, 1962 or Harper and Moore, 1968). It should be noted that in the absence of the UIS the vortex motion within the bubble will be the reverse of that shown in Fig. 2, that is clockwise flow rather than counter-clockwise flow. The components of the velocity vector for the inviscid axisymmetric vortex are 3U (2r 2 −R 2) sin q, 2R 2

(7)

3U 2 (R −r 2) cos q, 2R 2

(8)

uq = ur =

where the variables and parameters are defined in Fig. 2. Obviously in writing Eq. (7) and Eq. (8) we have ignored the region within the bubble occupied by the UIS. It is also assumed that the strength of the vortex is not influenced by liquid that enters the bubble via entrainment. This assumption is consistent with the density-independent form of the equations for a purely inviscid vortex.

4.2. Ricou–Spalding form of K–H entrainment equation Ricou and Spalding (1961) performed experiments on the entrainment of ambient air by gas jets of vastly different jet and ambient densities, z and z , respectively. They were able to correlate their data with the following equation 6en = Eo

  z z

1/2

u,

(9)

where 6en is the mean inflow (entrainment) velocity across the edge of a turbulent flow (jet), u is the mean velocity over the cross-section of the turbulent flow, and Eo is an entrainment coefficient of order 0.1. In the years since the work of

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Ricou and Spalding, the use of Eq. (9) has been successfully extended to problems involving the entrainment of liquid by a submerged high-speed gas jet (Weimer et al., 1973; Carreau et al., 1985; Loth and Faeth, 1989, 1990; Fauske and Grolmes, 1992). In Section 5 the power of the Ricou –Spalding equation is further revealed by its application to liquid entrainment from an initially stratified two-phase region. In the two-phase gas/liquid parallel flow application, Eq. (9) takes the form 6en = Eo

 zg zf

1/2

u,

(10)

where u is the gas velocity relative to the liquid velocity. The functional form of Eq. (10) may be derived from K– H instability theory (see Eq. (5) with m; ¦en replaced by zf6en). Thus the Ricou– Spalding entrainment law is a manifestation of the K–H instability.

4.3. Liquid entrainment by CDA bubble 6ortex An increment of bubble surface area dA in the spherical polar coordinate system shown in Fig. 2 is

dA= 2 yR 2 sin q dq.

61

(11)

The increment of liquid mass entrained per unit time dm; en across dA via the Kelvin–Helmholtz mechanism is dm; en = Eo(zgzf )1/2uq (R, q) dA,

(12)

where uq (R, q) is the gas velocity ‘parallel’ to the spherical bubble surface (i.e. Eq. (7) evaluated at r= R). In writing Eq. (12) it is assumed that the free stream flow given by Eq. (7) and Eq. (8) extends all the way to the eroding surface of the bubble. Strictly speaking zg and u in Eq. (10) should be identified with the local density and velocity of the mixture of fine scale droplets and gas within the ‘boundary layer’ adjacent to the bubble wall. A more sophisticated analysis which includes the presence of the two-phase fluid boundary layer in a uniform potential flow, but which is too lengthy to present here, shows that the simple modeling represented by Eq. (12) results in about a factor of two overestimation of the entrainment rate. This penalty, however, is outweighed by several advantages, including simple first-order differential equations for complex entrainment flow geometries and surprisingly

Fig. 6. Measured velocity of liquid surface versus time.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

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Table 2 Experiment work energy results Test

Reservoir pressure (MPa)

Bubble work (kJ)

Cover gas compression work (kJ)

Residual energy (kJ)

PC3-1 PC3-4

1.04 2.57

104.0 251.0

88.0 161.0

16.0 90.0

good agreement with experiments. Combining Eq. (7), Eq. (11), and Eq. (12) and integrating the result over the bubble surface yields m; en =3yEoUR 2(zgzf)1/2

&

y

sin2 q dq

0

=

3y 2 E (z z )1/2UR 2. 2 o g f

(13)

Now Eq. (13) expresses the instantaneous rate at which liquid is being entrained by the bubble gas when the bubble radius is R and the inlet flow velocity is U. By virtue of the assumption of a spherical CDA bubble R is related to the instantaneous bubble volume Vb by R=

  3 4y

1/2

V 1/3 b .

(14)

and U may be expressed in terms of the instantaneous mass rate of flow of nitrogen gas m; g from the test section gas reservoir (simulated core region) to the bubble via the mass conservation equation U=

m; g

,

AUIS zg

(15)

where AUIS is the cylindrical segment flow area at the radial outer edge of the region between the UIS lower boundary and the core barrel orifice. Calculations of the instantaneous m; g were carried out using the compressible orifice flow equation with an orifice coefficient of 0.8. To estimate the total liquid mass men entrained at the end of the bubble expansion process, Eq. (13) is converted into a ordinary differential equation with the independent variable being the instantaneous bubble volume Vb by writing m; en =

dmen dVb dmen dm = =ACG6s(Vb) en. dt dt dVb dVb

(16)

In Eq. (16) the derivative of bubble volume with respect to time (dVb/dt) has been replaced by the cross-sectional area ACG of the cover gas multiplied by the upward-directed velocity 6s of the water pool/cover gas interface (see Fig. 2). The relationship 6s(Vb) was obtained from the measurements of the pool surface displacement and velocity histories shown in Fig. 5 and Fig. 6. Eliminating m; en between Eq. (13) and Eq. (16) gives

 

dmen 3y 2 3 = dVb 2 4y

2/3

Eom; g(zf/zg)1/2V 2/3 b . AUISACG6s(Vb)

(17)

The bubble gas density zg in Eq. (17) is related to the bubble volume by the bubble-gas mass balance equation dVb d d (V z )= m; g, (Vb zg)= dt dVb b g dt

(18)

where the volume of the bubble occupied by the entrained liquid has been neglected (see below). Expanding the derivative and solving for dzg/dVb yields



n

m; g 1 dzg = − zg . dVb Vb ACG6s(Vb)

(19)

Thus by integrating two coupled ordinary differential equations, namely Eq. (17) and Eq. (19), the entrained liquid mass men as a function of bubble volume is obtained. Two such integrations were performed for the geometric parameter values ACG = 0.785 m2, AUIS = 0.136 m2 pertinent to Tests PC3-1 and PC3-4 and for zf = 103 kg m − 3, and Eo = 0.1. The predicted total liquid masses entrained as well as the percents of the bubble volume occupied by entrained liquid are listed in Table 3. The blowdown conditions of the 1/7th scale nitrogen discharge experiments of Simpson et al. (1981) were similar to the blowdown conditions of

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77 Table 3 Predicted entrained mass at end of bubble expansion Test

Reservoir (core) pressure, MPa

men (kg)

KE =

1.04 2.57

8.0 10.8

153y z2ƒ R 3U 2, 210

6.9 7.2

459 (m + mg)U 2 840 en

= 0.546 (men + mg) Tests PC3-1 and PC3-4. Indeed, the values given in Table 3 are in reasonable agreement with the measured percentage entrainment values reported by Simpson et al. which varied between 6 and 12%.

4.4. Kinetic energy of two-phase CDA bubble at end of expansion An increment of bubble volume dVb in the spherical polar coordinate system in Fig. 2 is dVb =2yr 2 sin q dr dq.

(20)

Assuming homogeneous two-phase vortex flow within the bubble, the bubble kinetic energy KE is given by the integral KE = 12z2ƒ 2y

&& y

0

R

(u 2q +u 2r )r 2sin q dr dq,

(21)

0

where z2ƒ, the two-phase bubble density, is the sum of the mass men of the entrained liquid and the mass mg of the bubble gas divided by the bubble volume at the end of the bubble expansion; that is mg + men = 43yR 3z2ƒ.

(23)

or, from Eq. (22)

% Entrainment

KE = PC3-1 PC3-4

63

(22)

Substituting Eq. (7) and Eq. (8) into Eq. (21) and carrying out the indicated integrations gives, after considerable algebra,





m; g 2 . AUISzg

(24)

The last equality in the above equation follows from Eq. (15). The entrained liquid mass men is given in Table 3. All the other predicted end-of-expansion parameters that appear in Eq. (24), together with the measured residual energy, are listed in Table 4 for Tests PC3-1 and PC3-4. Recall that the residual energy is the difference between the actual CDA bubble expansion work and the cover gas compression energy. It is worth noting again that during the cover gas compression in Tests PC3-1 or PC3-4 the measured relation between cover gas pressure P and volume V was PV k = constant, indicating an adiabatic process. The agreement between the predicted bubble kinetic energy and the test residual energy inferences (measurements) is very good considering both theoretical and experimental inaccuracies. One could argue that this agreement is somewhat fortuitous and simply due to a lucky choice of the vortex strength 3U/(2R), where U was taken to be the velocity of the horizontal flow between the UIS and the core barrel opening. The characteristic vortex strength could have just as well been based on the vertical velocity of the core-barrel orifice flow. This choice would undoubtedly give somewhat different numerical results; however, the predicted bubble kinetic energy versus core barrel pressure trend remains the same. The important point to be made here is that the theory can be made to coincide with the residual energy

Table 4 Predicted two-phase bubble kinetic energy at end of expansion Test

Core barrel mass flux m; g (kg s−1)

Bubble gas density zg (kg m−3)

Bubble gas mass mg (kg)

Bubble kinetic energy KE (KJ)

Measured residual energy (KJ)

PC3-1 PC3-4

91.0 292.0

9.65 18.86

1.13 2.40

24.0 93.4

16.0 90.0

64

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Fig. 7. Schematic diagram of experimental apparatus for study of K – H instability entrainment and reduction of gas expansion work. All dimensions are in centimeters.

measurements by assigning a reasonable value to the vortex strength. This agreement suggests that the energy expended in bringing the Kelvin– Helmholtz-governed entrained liquid up to the gas velocity is the most likely explanation of the bubble-expansion-energy loss (i.e. residual energy). An attempt was made to explain the residual energy on the basis of Taylor-instability entrained liquid (see Eq. (6)). The theory could not be made to fit both of the residual energy data points listed in Table 4 with a fixed numerical value of the entrainment coefficient (proportionality constant in Eq. (6)).

5. Experiments on Kelvin– Helmholtz instability governed entrainment and reduction of gas expansion work

5.1. Experimental apparatus The apparatus is illustrated in Fig. 7. The figure shows only one of several entrainment section designs that were used in the study (see Section 5.4). The entrainment section (channel) shown in

the figure was constructed of transparent plastic; it had a 10× 10 cm2 cross section, and a 50-cm long and 5.0-cm deep water reservoir was fastened to the floor of the entrainment channel. The reservoir occupied the full 10-cm width of the channel and when full contained 2.5 kg of water and the water surface was flush with the floor of the channel. A 5.08-cm diameter blowdown orifice supplied gas (air) at a high flow rate to one end of the entrainment channel. An 11.3-cm diameterhorizontal-transparent-Plexiglas tube containing a moveable plastic piston was attached to the other end of the entrainment channel. The piston itself was also constructed of plastic; it was 11.16 cm in diameter, 15.8 cm in length, and weighed 1.82 kg. The piston had two Teflon O-rings which fit tightly in rectangular slots milled around the outside of the piston. The upstream O-ring formed a seal between the high pressure gas in the entrainment section and the atmosphere, while the downstream O-ring guided the piston along the cylinder axis. Upon initiating the gas blowdown by bursting the rupture disk (diaphragm), a high pressure was produced rapidly in the liquid entrainment chan-

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

nel. By this technique, the piston was accelerated horizontally away from the reservoir and simultaneously a high velocity flow of gas was established above the water reservoir surface. The gas flow destabilized the plane liquid surface resulting in the formation, growth and entrainment of waves. During the course of the transient, the gas pressure was monitored by pressure transducers mounted along the walls of the reservoir channel and the piston acceleration tube. The horizontal motion of the piston was determined from the pressure transducer records. The reduction of gas work due to the presence of entrained water was revealed by performing experiments with and without water in the reservoir channel. Five purely entrainment experiments without the piston and 13 gas work reduction experiments with the piston were carried out to determine entrainment behavior and the work reduction potential of the entrained liquid.

5.2. Entrainment test results Fig. 8 shows the pressure histories measured in the entrainment channel upstream of the reservoir

65

during Test 1 of the entrainment-only series of tests. Pressure transducer P1 was located midway between the blowdown orifice and the liquid reservoir and P2 was located at the upstream end of the liquid reservoir (see Fig. 7). The pressure versus time curves in Fig. 8 are similar to those recorded during all the entrainment-only tests. A sequence of narrow pressure spikes ( ms) was immediately followed by a relatively broad-based ( 0.1 s) triangular-shaped pressure pulse that was detected practically simultaneously by both of the pressure transducers. The short precursor pressure spikes were undoubtedly due to the bursting of the relief diaphragm. The broad triangular pulse is believed to be a direct result of the entrainment process simply because the triangular pulse did not appear in tests which were conducted with an empty reservoir. The triangular pulse did not begin until about 17.0 ms after the appearance of the precursor shock pulses. The time delay between the bursting of the diaphragm and the quasi-steady flow of gas from the gas supply tank and the onset of liquid entrainment is identified with the round trip time of a shock wave in the 297-cm long blowdown

Fig. 8. Pressure –time histories at pressure transducer locations P1 and P2 during channel flow entrainment Test 1; fit of linear segments to triangular pressure pulse.

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M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

pipe, or 17.0 ms. Indeed the supply tank pressure did not begin to decrease until about 17.0 ms after the bursting of the diaphragm. The gas supply tank thermocouples and pressure transducers indicated an isothermal and linear supply tank gas depressurization over the time duration of interest. The essentially steady mass flow rate of gas from the supply tank into the entrainment channel during the entrainment event was determined from the slope (negative) of the supply tank depressurization curve. A video camera was used to provide a feeling for the liquid surface morphology and rapidity of the entrainment process. Image contrast was enhanced by mixing an aqueous dye (food coloring) with the reservoir water. The video tape revealed trains of large waves of several centimeters in length. Above the turbulent liquid surface a spray suddenly appeared (within one video frame of about 40 ms) across the whole depth of flowing gas. Presumably the spray developed from much smaller capillary waves superimposed on the larger waves. The reservoir appeared to empty in a time span between two and three video frames. This observation is consistent with the duration of the triangular pulse in Fig. 8. From the K – H entrainment concept, Eq. (9), the instantaneous water mass men lifted from the reservoir by the blowdown gas is given by the differential equation: dmen = Arevzfwen =ArevEo (z2ƒzf)1/2u2ƒ, dt

(25)

where z2ƒ and u2ƒ are, respectively, the instantaneous density and flow area-averaged velocity of the homogeneous (assumed), two-phase entrained liquid/gas mixture above the liquid reservoir and Arev is the geometric surface area of the water in the reservoir cavity (Arev =0.05 m2). Eq. (25) may be regarded as a more rigorous application of the entrainment concept than in the previous problem of liquid entrainment by the CDA bubble in that the presence of the liquid fragments in the flow above the liquid surface is accounted for. The two-phase quantities z2ƒ and u2ƒ may be related to the corresponding instantaneous all-gas quantities zg, ug upstream of the reservoir, say at the

location of pressure transducer P1, via the quasisteady momentum equation (z2ƒ u2ƒAch)u2ƒ = (zg ugAch)ug = m; g ug,

(26)

where Ach is the flow cross-sectional area of the entrainment section (Ach = 0.01 m2) and m; g is the steady state gas flow rate from the supply tank. Using the left-hand equality in the above equation; Eq. (25) becomes dmen = Arev Eo (zgzf)1/2ug, dt

(27)

which is the form that one would obtain directly by assuming that liquid is entrained by the all-gas flow. Using the ideal gas law for isothermal flow in the entrainment section, an assumption justified by thermocouple measurements, zg may be related to the upstream pressure P1 by zg =

zg,oP1 , Po

(28)

where subscript ‘o’ refers to the initial (atmospheric) condition in the entrainment section. Combining Eqs. (26)–(28) yields



n

zf P o dmen ArevEom; g = dt Ach zg,o P1(t)

1/2

.

(29)

Integrating Eq. (29) from the incipient entrainment condition at time t= 0 to the end of the entrainment process at time t= ten and solving the result for the entrainment coefficient Eo gives Eo =

  !&  n "

Ach men zg,o Arev m; g zf

1/2

ten

0

Po P1(t)

1/2

dt

−1

. (30)

As already mentioned, it seems reasonable to presume that the measured broad triangular shaped pressure pulses (see, e.g. Fig. 8) are the fluid mechanical signatures of the entrainment events. Thus for a specific entrainment experiment, ten in the above equation is identified with the width tp of the triangular pressure pulse. To integrate Eq. (30) in closed form the following piecewise continuous function is used to represent the triangular segment of P1(t):

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

67

Table 5 Estimated gas flow rates, triangular pulse fitting parameters, and entrainment coefficient inferences from Eq. (32) Test

Supply tank pressure, MPa

m; g kg s−1

Pmax/Po

tp, s

Hen, m

Eo

1 2 3 4 5

0.69 0.45 0.77 0.76 0.76

1.70 1.00 1.95 1.89 1.85

3.04 2.0 3.38 1.75 3.18

0.093 0.115 0.092 0.36 0.14

0.048 0.045 0.05 0.05 0.05

0.14 0.16 0.14 0.03 0.09

P1(t) Á t ÃPo +(Pmax −Po)t max =Í ÃP +(P −P ) t −tp max o Ä o tmax −tp





0 5t 5 tmax tmax 5t5 ten (31)

In Eq. (31) tmax is the time at which the peak pressure Pmax is achieved. Fig. 8 shows Eq. (31) fitted to the measured triangular pulse of Test 1. Substituting Eq. (31) into Eq. (30) and carrying out the indicated integral with ten =tp results in the desired expression for Eo: Eo =



n

AchHen(zfzg,o)1/2 Pmax/Po −1 . (Pmax/Po)1/2 −1 2m; gtp

(32)

In the above equation the mass entrained was eliminated in favor of the depth Hen of the liquid that was removed from the reservoir cavity by using the obvious relationship men =Arev Hen zf.

(33)

Also note that tmax does not appear in Eq. (32) as it cancels out during the algebraic manipulations. The experiment parameters and the numerical inferences of Eo for the entrainment tests are summarized in Table 5. All the estimated numerical values of Eo are of the expected order ( 0.1), except perhaps for Test 4. In Test 4 the entrainment section was turned around so that the reservoir cavity was close to the blowdown orifice. Therefore, most of the liquid surface was submerged in the expansion zone caused by the sudden enlargement just downstream of the orifice. It is clear from the low value of Eo that the expanding gas does not entrain liquid very well if at all.

The expansion zone covers an axially length of about five channel diameters (Idel’chik, 1960), or about 50 cm of the entrainment section. In Test 4 then parallel gas flow only occurred over approximately 25% of the liquid surface, in the downwind end of the reservoir. If parallel gas flow is a requirement for entrainment, the factor of three to four lower estimated value of Eo for Test 4 compared with the Eo values for the other entrainment tests is explained. Indeed the laboratory study of Trabold et al. (1987) on turbulent air jets discharged into ambient air showed that entrainment is restricted unless the streamlines of the ambient flow are nearly parallel to the jet axis. We conclude this section by noting that the satisfactory interpretation of the experimental entrainment results using the Ricou–Spalding entrainment law and the assumption of homogeneous flow indicates that K–H induced water entrainment, droplet breakup and acceleration up to the gas phase velocity occurs on a time scale less than the time it takes a parcel of gas to traverse the axial length of the water reservoir ( 10 ms).

5.3. Work reduction tests: theoretical considerations Each work reduction measurement involved the performance of two piston acceleration tests with identical gas supply and geometry conditions. One test was conducted without water in the entrainment section reservoir and in the other test the reservoir was filled with water. The gas volume in the entrainment section of the no-water test was kept at the same value as in the water test by inserting solid ‘filler’ materials into the reservoir

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

68

cavity. Since the piston O-ring friction was negligible (see Section 5.4), the maximum work done by the expanding gas was determined by the kinetic energy (velocity) of the piston just as it exited the piston acceleration tube. The piston mass is denoted by the symbol mp. The piston exit velocity without and with water entrainment are denoted by the symbols up,ne and up,e, respectively. Then the piston exit kinetic energy (or gas work) without entrainment is KEne = 12mpu 2p,ne,

(34)

and the piston exit kinetic energy (or gas work) with entrainment is 1 2

2 p,e

KEe = mp u .

(35)

The reduction in gas work Wred due to entrainment is then 1 2

Wred =KEne −KEe = mp u

2 p,ne

1 2

2 p,e

− mp u .

  n

(36)

The work reduction ratio is (in percent) up,e Wred = 100 1 − KEne up,ne

2

.

(37)

Assuming that the entrained liquid mass men is accelerated to the piston velocity up,e and that this process of acceleration is entirely responsible for the gas work reduction Wred, then Wred = 12men u 2p,e.

(38)

Eliminating Wred between Eq. (37) and Eq. (38) results in the following expression for the entrained mass as a function of mp and the ratio up,ne/up,e: men = mp

  n up,ne up,e

2

−1 .

(39)

Finally, it is of interest to inquire whether the predicted entrained mass given by Eq. (39) is consistent with the K– H mechanism of entrainment as represented by the differential form of the Ricou –Spalding entrainment law, Eq. (27), with zg given by Eq. (28),

 n

dmen P1(t) = ArevEo(zg,ozf)1/2 dt Po

1/2

ug .

(40)

Neglecting gas compressibility effects that occur early in the piston acceleration transient, it is

proposed that the gas velocity is dictated by the piston velocity up and well-represented by the quasi-steady mass balance ugAch = upAp,

(41)

where Ach and Ap are, respectively, the cross-sectional areas for gas flow in the entrainment and piston sections of the apparatus. Substituting ug in Eq. (41) into Eq. (40) and solving the result for Eo gives Eo =

menAch , ArevAp(zg,ozf)1/2 I

& n

(42)

where I is defined by the integral ~

I=

0

P1(t) Po

1/2

up(t) dt.

(43)

The upper limit ~ of the integral in Eq. (43) is the time it takes for the piston to traverse the length of the piston acceleration tube. The pressure P1(t) is the measured pressure history at the upstream end of the entrainment section. The piston velocity history up(t) is determined by the derivative of a polynomial fit to the piston displacement measurements or by solving the equation of piston motion (see Section 5.4).

5.4. Work reduction tests: experimental program The experimental program is summarized in Table 6 and the various entrainment/piston section composite configurations and designs that were used in the study are shown in Fig. 9. As can be seen from the figure, three of the configurations involved the square entrainment section with its water trough located near or far from the blowdown orifice. These configurations are labeled A, B, C. Configuration D refers to a cylindrical entrainment section in which water occupied the entire length of the section. Before we describe the detailed experimental results, it would appear worthwhile to explain briefly the reasons for working with these different configurations. Configuration A shown in Fig. 9 was used in Tests 1–4. In this configuration the leading edge of the water reservoir was only 18 cm (1.8 channel diameters) downstream of the blowdown orifice.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Tests 1 and 2 were performed with an initial supply tank pressure of 0.45 MPa. The tests show no detectable gas work reduction. The initial supply tank pressure was increased to 0.78 MPa in Tests 3 and 4. Again no detectable gas work reduction was observed. Test 3 which was performed without water represented the benchmark test for all the subsequent entrainment tests with the square entrainment section (Tests 4, 5, 6), since all the subsequent tests were initiated with a 0.78 MPa rupture disc. Test 5 was conducted with Configuration B (see Fig. 9). In this configuration the water reservoir is far enough downstream to ensure a fully expanded parallel gas flow above the liquid surface. Nevertheless, no measurable gas work reduction was obtained. From the video tapes of Tests 1–5 it becomes quite clear that, except for early in the transient, the gas flow above the liquid surface was controlled by the piston motion and that the gas velocities behind the piston and above the liquid reservoir were too low ( 55 m s − 1) to entrain enough liquid to influence the piston motion and were probably significantly lower than the local gas velocities in the model scale CDA tests PC3-1 and PC3-4 (Y 100 m s − 1). It appeared that a more appropriate experiment could be performed by decreasing the flow area above the liquid reservoir. Accordingly, a set of results

69

was obtained in Tests 6 and 7 with reduced gas flow areas as described below. In order to quickly determine the gas work reduction potential at higher gas velocities without the need for structural modifications to the apparatus, a water-filled balloon was placed in the downstream end of the reservoir in Test 6 (Configuration B in Fig. 9). Additional water was added to the reservoir so that the water inventory was essentially the same as in the previous tests. The balloon burst early in the transient and exposed the released water to an effectively higher gas velocity. A large gas work reduction was achieved in Test 6. Encouraged by this result an apparatus modification was made in preparation for Test 7. The test section appears in Fig. 9 as Configuration C. The shaded structure that hangs from the ceiling of the entrainment section was the new feature of the test section. This piece was inserted for the purpose of increasing the gas velocity above the liquid reservoir by about a factor of two, by reducing the cross-sectional area for gas flow from 100 to 50 cm2. Significant gas work reduction was measured in Test 7. Unfortunately, the square plastic entrainment section failed just before the piston left its acceleration tube. While the experimental results from this test could still be used, the entrainment section was beyond repair.

Table 6 Test matrix Test

Entrainment/piston section configuration (see Fig. 9)

Liquid in reservoir (kg)

Supply pressure (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13

A A A A B B C Db Db D D D D

No (filler) 2.5 No (filler) 2.5 2.5 2.5a 2.5 No (filler) 2.5 2.5 No (filler) 3.6 No (filler)

0.45 0.45 0.78 0.78 0.77 0.76 0.80 0.78 0.78 0.69 0.69 0.69 0.69

a b

Most of liquid inventory in water-filled balloon. Steel-tube entrainment section.

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M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Fig. 9. Entrainment section/piston acceleration tube configurations used in work reduction tests.

It was decided to replace the square entrainment channel with cylindrical (or tube-like) entrainment sections. One tube was constructed of steel and had an internal diameter of 10.16 cm and a length of 66 cm. The other tube was made of transparent plastic, 11.3 cm in diameter, and 71.0 cm in length. The diameter of the plastic, tubular entrainment section was the same as that of the piston section. Unlike the square entrainment section, the circular entrainment section did not have a trough-type reservoir. Instead the water to be entrained occupied a semi-cylindrical cavity bounded on one axial

end by the inner surface of the gas-supply-side flange and on the other axial end by the inner surface of the piston. Test 8 and 9 were conducted with the steel-tube entrainment section and Tests 10–13 were carried out with the plastic-tube entrainment section (Configuration D in Fig. 9). Tests 8 and 9 and Tests 12 and 13 showed reduction in gas work, but Tests 10 and 11 did not. Apparently the gas expansion over the relatively shallow water layer employed in Test 10 limited the quantity of entrained liquid to an amount that was too small to affect the piston kinetic energy.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

5.5. Work reduction tests: experimental results and interpretation The measured pressures at P1, P2, and P3 during Test 1 (no water) are shown in Fig. 10. Note that all the pressures are practically equal to one another indicating a uniform pressure throughout the entrainment section. Pressure P1 is slightly lower than pressures P2 and P3 due to flow expansion in the upstream region and pressure recovery in the downstream region of the entrainment section. The pressure in the entrainment section increases at first, since initially the entrainment section is a sealed enclosure due to the presence of the piston. As the piston accelerates along the cylinder more volume is made available for gas expansion and ultimately the gas flow from the supply tank can not keep pace with the rate of volume increase. Consequently the pressure in the entrainment section peaks and begins to decrease. Shortly after the piston leaves the acceleration tube, following a time delay approximately equal to the 5.9 ms sonic wave round-trip in the acceleration tube, the pressure rapidly decreases in the entrainment section. It turns out that the pressure in the acceleration tube follows the pressure history in the entrain-

71

ment section. This behavior can be illustrated by comparing any one of the pressure transducer records within the acceleration tube with either P2 or P3. Fig. 11 shows the pressures P3 and P7 as functions of time. Transducer P7 was located about midway along the piston acceleration tube (see Fig. 7). Note that as soon as the piston passes the location P7 (at 7.381 s) the pressure P7(t) instantaneously jumps to pressure P3 and then follows the P3 versus time curve until the piston exits the acceleration tube. There is another feature of Fig. 11 that is worthy of discussion at this stage. The pressure at P7 is observed to execute two jumps as the piston passes. It first jumps to a plateau value of about 0.068 MPa and remains at this value for about 4 ms. It is believed that this pressure plateau represents the pressure of the gas that ‘leaks’ into the space between the two piston rings. Indeed it takes about 4 ms for the piston (actually the distance between rings) to pass location P7. Therefore the second pressure jump, from 0.068 MPa to P3, is used to determine the displacement as a function of time of the high-pressure side of the piston. Many of the pressure transducer records for the piston acceleration tube exhibited this type of behavior, namely a precursor pressure

Fig. 10. Pressure –time histories at pressure transducer locations P1, P2 and P3 during Test 1.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

72

Fig. 11. Pressure – time histories at pressure transducer locations P3 and P7 during Test 1.

rise of duration that correlates inversely with the piston velocity followed by a second, usually more abrupt pressure rise. In all cases the arrival of the high-pressure side of the piston was identified with the second pressure rise. The piston displacement versus time curve for Test 1, as determined from the pressure transducers P4 to P11, is shown as the open circles in Fig. 12. Time zero in this figure corresponds to the time the diaphragm bursts. There is a semi-theoretical way of determining the piston displacement history which can be used to check the measured displacement history. Writing Newton’s second law of accelerated motion of the piston, gives mp

dup =Ap [P(t) − P ], dt

(44)

where P(t) is the pressure imposed on the piston’s high-pressure side and up is the instantaneous piston velocity which is related to the distance X traveled by the piston via dX =up. dt

(45)

Based on the discussion given in the foregoing, either P2 or P3 can be used to represent P(t) in

Eq. (44). The solid curve in Fig. 12 was constructed by substituting a polynomial fit to the measured series of points P3(t) into Eq. (44) and integrating the result. As can be seen from Fig. 12 the agreement between the measured and calculated piston displacement-versus time curves is very good. This agreement indicates that the motion of the plastic piston inside the acceleration tube may be regarded as frictionless. The calculated piston velocity at location P11 is up =45.5 m s − 1. The measured value, as determined from the slope of the line that passes through the last two data points in Fig. 12, is up = 44.0 m s − 1. A purely theoretical model of the piston motion can be developed by writing the equation for the conservation of the mass of the gas in the entrainment section plus the mass of the gas in the piston acceleration tube: d (V z + Ach X zg)= m; g. dt sq g

(46)

In the above equation Vsq is the volume of the square reservoir section. Since the gas flow rate m; g is approximately constant during the period of piston motion, we may integrate Eq. (46) to get Vsqzg + AchXzg = m; gt+ Vsqzg,o.

(47)

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Now zg is related to the pressure P (or P1) by Eq. (28) which when substituted into the above equation and solving for P gives P m; gt +Vsq zg,o = . Po Ach zg,o X +Vsq zg,o

(48)

For a given gas flow rate m; g, Eq. (44), Eq. (45), and Eq. (48) are sufficient to predict P, X and up as functions of time. This system of equations is hereafter referred to as the piston motion model, or the ‘model’. The model calculations were carried out with m; g = 1.07 kg s − 1, determined from the supply tank pressure decay curves for Tests 1 and 2. The model predicts a piston velocity up =46 m s − 1 at the exit transducer. This is nearly equal to the velocity up = 45.5 m s − 1 calculated by combining Eq. (44) with the measured pressure history and is close to the velocity up =44 m s − 1 determined from the slope of the measured piston displacement-versus time curve. In Test 2 the liquid reservoir was filled with 2.5 kg of water. The initial gas supply pressure was Po =0.45 MPa, the same as that in Test 1. The measured pressure versus time curves and the measured piston displacement versus time curve for Test 2 were essentially identical to those for

73

Test 1. The video tape of this tests revealed a highly roughened water surface and an optically thick two-phase region in the wake of the piston. Nevertheless, the pressure history data indicated that the entrainment process had no measurable effect on the piston motion. As mentioned previously, significant reductions in gas work were measured after the square entrainment section was modified or replaced in order to reduce the area for gas flow and thereby increase the gas velocity over the liquid surface. For example, test pair 8 and 9 employed a cylindrical entrainment section of reduced gas-flow area relative to that of the original square entrainment section. In Test 8 a 2500 cm3 semi-cylindrical, solid block of wood filled the lower half of the steel entrainment tube. In Test 9, 2500 cm3 of water occupied the lower half of the steel entrainment section (see Configuration D in Fig. 9). The liquid surface coincided with the bottom rim of the blowdown orifice. The supply pressure in both tests was 0.78 MPa. The pressure versus time records at locations P1 and P2 in the non-entrainment Test 8 are shown in Fig. 13. The pressure at P1 is somewhat less than the pressure at P2. This difference is due to the gas expansion from the blowdown orifice in the vicinity of P1

Fig. 12. Measured and calculated piston displacement vs. time. Time relative to disc rupture.

74

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Fig. 13. Measured pressure versus time curves at P1 and P2 during Test 8; time relative to disc rupture.

and the recovery of pressure in the vicinity of P2. In contrast with Test 8, the measured Test 9 pressure difference P1 −P2 is large (see Fig. 14). Clearly liquid entrainment was responsible for the pressure gradient across the liquid pool in Test 9. The dramatic increase in pressure at P2 after the piston left the test section during Test 9 (see Fig. 14) was probably caused by the rapid increase in gas velocity above the liquid surface and accompanying increased flow of entrained liquid into the piston acceleration tube. The piston displacement versus time curves for Tests 8 and 9, as obtained from the pressure transducers P4 to P11, gave the piston exit velocities up,ne = 56.4 m s − 1 (from Test 8) and up,e = 45.4 m s − 1 (from Test 9). It is of interest to determine whether the K–H mechanism of entrainment, as represented by Eq. (10), is consistent with the data obtained from Tests 8 and 9. This is accomplished by using the methodology already discussed in Section 5.3 (see Eq. (42) and Eq. (43)). The integral I in Eq. (43) is evaluated using the pressure history P1(t) at the upstream transducer and up(t) calculated from the piston motion model. The result is I = 2.05 m. The piston-velocity-equilibrated entrained liquid mass is, from Eq. (39) and the measured ratio

up,ne/up,e = 1.24, men = 0.98 kg, which is 6.4% of the available volume (entrainment section plus piston section). Interestingly enough, this compares well with values inferred from our CDA bubble residual energy measurements (see Table 3) and with the percentage entrainments found by Simpson et al. (1981) in their CDA nitrogen gas bubble tests. The work reduction ratio is, from Eq. (37), Wred/KEne = 35%. The appropriate geometric parameters corresponding to 2500 cm3 of liquid in the 10.16-cm diameter, 66.0-cm long steel circular entrainment section are Ach = 43.2 cm2, Arev = 670 cm2, and Ap = 100 cm2. Substituting the above parameter estimates into Eq. (42) gives Eo = 0.09, a result which is in near-perfect accord with the Ricou and Spalding (1961) form of the K–H entrainment mechanism. All the gas work reduction Wred and entrainment coefficient Eo inferences are presented in Table 7. Two test numbers comprise each entry in the first column of the table. The first numbers of the column entries identify the ‘benchmark’ tests that were conducted without water. The second numbers of the pairs of numbers identify the tests with water entrainment but otherwise identical gas supply and geometry conditions as their benchmark tests. As should be apparent by now

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

the results of two tests from a specific pair of tests were compared with one another to determine the extent of gas work reduction due to entrainment. The second column of Table 7 gives the velocity up,ne of the piston the instant it leaves the acceleration tube, as determined from the piston displacement measurements in the tests performed with no water. The third column of Table 7 gives the measured velocity up,e of the piston at the exit of the acceleration tube in the tests with water entrainment. The maximum gas velocity above the liquid surface, ug,max, appears in the fourth column of the table; it is assumed to be proportional to up,e and is calculated from Eq. (41) with up = up,e. The fifth column of Table 7 shows the gas work reduction Wred normalized by the piston kinetic energy (gas work) in the absence of entrainment (see Eq. (37)). The entrainment coefficient values Eo inferred from the gas work reduction experiments are listed in the last column of the table. To within the accuracy of the piston displacement measurements, roughly 2.0 m s − 1 out of about 60 m s − 1, the piston velocities up,e in water entrainment Tests 2, 4, 5, and 10 were the same as those measured in the respective non-entrainment Tests 1, 3, and 11. Thus in Table 7, Wred = 0 and

75

Eo is listed as indeterminate for Tests 2, 4, 5 and 10. These entries are not meant to imply that there was no water entrained in these tests but, instead, that the amount entrained was too small to result in a measurable and unambiguous piston velocity difference up,ne − up,e. The low and ineffective entrainment rate in these tests was probably due to a low ug,max and/or a gas expansion region that covered a significant fraction of the liquid surface. By comparing the measurements taken during Test 6 with Test 3, the relevant benchmark experiment without water, it is clear that an entrainment-induced gas work reduction was achieved in Test 6. However, this test was performed with a water-filled balloon which presented an unknown, initial water surface area to the gas flow. Thus Eo could not be estimated for Test 6. In Tests 7, 9, and 12 the condition for significant gas work reduction by liquid entrainment was met and based on the findings of this study this condition appears to be a high gas velocity (ug,max Y 90 m s − 1) parallel to the liquid surface. In Tests 7, 9, and 12 the initial liquid reservoir geometry was well defined and a non-negligible Wred was determined from the data; therefore, the effective entrainment coefficient could be esti-

Fig. 14. Measured pressure versus time curves at P1 and P2 during Test 9; time relative to disc rupture.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

76

Table 7 Gas work reduction and entrainment coefficient values inferred from piston displacement measurements Test pair (no water, water)

up,ne (m s−1)

up,e (m s−1)

ug,max (m s−1)

Wred/KEne (%)

Eo

1,2 3,4 3,5 3,6 a ,7 8,9 11,10 13,12

44.0 58.7 58.7 58.7 57.6 56.4 53.0 55.6

44.0 55.0 55.0 44.0 45.0 45.4 51.0 43.9

44.0 55.0 55.0 88.0 90.0 90.8 78.7 87.8

0 0 0 44 39 35 0 38

– – – – 0.15 0.09 – 0.1

a

up,ne estimated with piston motion model.

mated and the Eo values are given in Table 7. A comparison of these values with the expected value of approximately 0.1 is an indirect confirmation of the Ricou– Spalding entrainment equation, since the entrained mass is not directly measured in the experiments but estimated from the gas work reduction measurements. An extra large wave was observed during Test 7 which may have been responsible for the somewhat enhanced value Eo =0.15 inferred from this test. The videotape information suggested that extra large waves were more likely to form in the square channel than in the circular entrainment section.

6. Conclusions It was proposed that the Kelvin– Helmholtz (K–H) instability is primarily responsible for liquid entrainment by the expanding CDA bubble. The work reported in this paper that tends to support this proposal may be summarized as follows: 1. An analysis of entrainment due to combined K –H and Taylor instabilities was presented and the results show that only modest CDA bubble-gas cross flows parallel to the bubble surface are required to suppress the Taylor instability in favor of the K– H instability. 2. New experimental data were presented on CDA bubble energy losses. The data were obtained from an existing experimental facility that simulates the CDA bubble expansion in a

1/10 scale model of the FBR upper plenum (Kaguchi et al., 2000). 3. A new model of liquid entrainment by the expanding CDA bubble was introduced. The flow field within the bubble is assumed to be a spherical core vortex of strength dictated by the source gas flow and entrainment occurs via the K –H instability along the bubble surface where strong relative flows persist. The model is capable of rationalizing the measured bubble energy losses mentioned in Item 2 above, based on the idea that the work potential of the expanding bubble is reduced by an amount equal to the kinetic energy required to bring the entrained liquid mass from rest up to the gas velocity within the bubble at the end of the expansion. 4. A one-dimensional air-over-water parallel flow experiment was undertaken that demonstrates: (i) that entrainment of the K–H type can occur on the time scales of CDA bubble expansions; and (ii) that the liquid mass entrained by the K–H mechanism is of sufficient quantity and atomized into a sufficiently large number of small droplets to reduce the bubble expansion work and thereby account for the bubble work reductions observed during the laboratory-scale CDA bubble experiments. It is also apparent from this work that the K –H instability/entrainment phenomena can readily be included in future models of CDA vapor bubble expansions by adopting the Ricou and Spalding (1961) entrainment equation.

M. Epstein et al. / Nuclear Engineering and Design 210 (2001) 53–77

Acknowledgements The experiment described in Section 5 was conducted as a part of the research and development program for fast breeder reactors under sponsorship of the nine Japanese electric power companies, Electric Power Development Co., Ltd., and the Japan Atomic Power Company.

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