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Condensation dynamics of submerged steam jet in subcooled water
International Journal of Multiphase Flow 39 (2012) 66–77
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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Condensation dynamics of submerged steam jet in subcooled water Soon Joon Hong a,⇑, Goon Cherl Park b, Seok Cho c, Chul-Hwa Song c a
FNC Tech. Co., Building 135-308, Seoul National University (SNU), San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea Department of Nuclear Engineering, Seoul National University (SNU), San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea c Korea Atomic Energy Research Institute (KAERI), 150, Dukjin-Dong, Yusong-Gu, Taejon 305-353, South Korea b
a r t i c l e
i n f o
Article history: Received 25 July 2011 Received in revised form 14 October 2011 Accepted 17 October 2011 Available online 26 October 2011 Keywords: Condensation oscillation Dominant frequency Submerged steam jet Mechanistic model Steam mass flux Pool temperature
a b s t r a c t Condensation oscillation of submerged steam jet in water pool was investigated. From the experiments it was found that the dominant frequency of condensation oscillation was proportional to steam mass flux for steam mass flux under 300 kg/m2 s and inversely proportional for over 300 kg/m2 s. The frequency was always inversely proportional to pool temperature. For the high steam mass flux region (over 300 kg/m2 s), one-dimensional mechanistic model was developed based on the balance of the kinetic energy that the steam jet gives and the pool water receives, adopting the submerged turbulent jet theory. The proposed model excellently predicted the dominant frequencies for the steam mass flux 300–900 kg/m2 s and water temperature 35–75 °C. For the higher water temperature, the developed model also could predict the dominant frequencies by adjusting the ratio of jet expansion coefficients of vapor dominant region and liquid dominant region. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The phenomena of a submerged condensable gas jet discharging into a liquid pool have attracted attention from many investigators with a number of interests. This process involves condensation of a pure component (steam into water), dissolution of a gas into a bulk liquid phase (ammonia into water), or a more complicated reactive situation. In particular, recent advanced nuclear power plants use a submerged steam jet condensation in safety systems for the safe quenching of discharged steam. In each of these systems, it has been of interest to know the jet condensation length (penetration length of steam jet), heat transfer, condensation regime map, and condensation oscillation as a function of the operating conditions. Traditionally the penetration length of steam jet (or steam jet condensation length) has been measured by visualization (taking a picture). Kerney et al. (1972), Weimer et al. (1973), Kudo (1974), DelTin et al. (1983), Kim (1996, 2001) and Chun et al. (1996) proposed empirical correlations for steam jet penetration length to nozzle diameter (X/d0) as a function of steam mass flux and pool temperature (or subcooling) for a wide range of steam mass flux (0–2000 kg/m2 s) and pool temperatures (13–95 °C). The length to diameter (X/d0) was proportional to steam mass flux and pool temperature. Several studies on heat transfer of submerged steam jet condensation were experimentally conducted. ⇑ Corresponding author. Tel.: +82 2 872 6084; fax: +82 2 872 6089. E-mail address: [email protected] (S.J. Hong). 0301-9322/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2011.10.007
Cumo et al. (1978), Simpson and Chan (1982), Kim (1996, 2001) and Chun et al. (1996) found that the heat transfer coefficient is a function of nozzle diameter, steam mass flux and pool temperature. And Chun (1983), Sonin (1984) and Sonin et al. (1986) induced the liquid turbulent intensity for the explanation of heat transfer. Since the steam jet condensation shows different characteristics according to steam mass flux and pool temperature, several condensation regime maps or stability maps have been proposed. Cho et al. (1998) suggested a condensation regime map based on visual observation and acoustics, which are applicable for a wide range of steam mass flux (up to 450 kg/m2 s) and pool temperature (20–95 °C), and can cover most of the previous ones, as shown in Fig. 1. This experiment was carried out for several nozzle diameters (5–20 mm) and the pool was open to atmosphere. At low steam mass flux, ‘chugging (denoted by ‘C’ in Fig. 1)’ occurs. As the steam mass flux increases, a ‘transient region from chugging to condensation oscillation (TC)’ is found, where subcooled water does not enter the nozzle any more. Instead, a cloud of tiny steam bubble is formed near the nozzle exit. Further increase of the steam mass flux results in the ‘condensation oscillation (CO)’. In this region, steam condenses outside the nozzle and the steam/water interface oscillates violently. ‘Bubbling condensation oscillation (BCO)’ occurs within the same steam mass flux region of CO, but at higher liquid temperature. The steam plumes are detached from injector and relatively large steam bubbles of irregular shape are condensed. When the steam mass flux is relatively high, ‘stable condensation (SC)’ occurs. As the pool temperature increased, the
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
o
Pool Temperature [ C]
100
Not Condensate in Water BCO
IOC
80
TC
60
CO SC
40 C 20 0
100 200 300 2 Steam Mass Flux [kg/m -s]
400
Fig. 1. Condensation regime map by Cho et al. (1998).
67
interface of steam jet becomes unstable. This region is ‘interfacial oscillation condensation (IOC)’. Oscillation of steam jet condensation was also an area of interest. First of all, chugging was experimentally studied by several investigators (Anderson et al., 1978; Sonin, 1981, 1984; Chun, 1983) because of the large hydrodynamic load. Intensive theoretical analyses were conducted by Okazaki (1979), Aya et al. (1980, 1983), Pitts (1980), Chen and Dhir (1982), and Utamura et al. (1984). For CO region in the regime map, Chan (1978) measured the oscillation frequency and found that the frequency is inversely proportional to nozzle diameter. He explained it by the aid of Rayleigh bubble equation. But, in 1982, he together with Simpson and Chan, 1982 again suggested an experimental frequency correlation using Strouhal number for frequency (St), steam Reynolds number (Re) and pool Jacob number (Ja). The frequency is proportional to steam mass flux and pool subcooling, but inversely proportional to nozzle diameter. More intensive experimental study was con-
(a)
(b)
Fig. 2. Schematic diagram and bird’s eye view of GIRLS: (a) Schematic diagram and (b) Bird’s eye view.
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
ducted by Damasio et al. (1985) with many test sets up to maximum steam mass flux 250 kg/m2 s. They also suggested an experimental frequency correlation similar to Simpson’s, but added Weber number (We) to increase prediction capability. Nariai and Aya, 1986 analytically studied chugging oscillation and the condensation oscillation. However, it should be noted that all of these studies were limited to the low steam mass flux (<300 kg/m2 s). Thus, for higher steam mass flux (SC and IOC in condensation regime map), more studies are necessary. The objective of this study is to investigate the dynamic characteristics of steam jet condensation frequency. The range of steam mass flux was 200–900 kg/m2 s, and that of pool temperature was 35–95 °C (subcooling is 65–5 K). In particular, the high steam mass flux region (>300 kg/m2 s) is intensively discussed experimentally and theoretically. 2. Experiment 2.1. Test facility and measurement instruments An experimental apparatus, General Investigation Rig for Liquid/Steam Jet Condensation (GIRLS), was constructed as shown in Fig. 2. All the water was demineralized and degassed. Saturated steam from a steam generator was provided through a main steam line to cylinder pool, whose diameter is 1.8 m, height 1.5 m, water level 1.3 m. A sparger, a steam quenching device, was especially manufactured as a dual pipe in order to minimize heat loss in submerged part, as shown in Fig. 3. The steam was discharged horizontally into subcooled water in a cylinder pool through a single hole of diameter 10 mm. In the cylinder pool, a cooling system was installed to easily control the pool temperature. Summary of the measurement instrumentations is presented in Table 1, together with their uncertainties. For reliable measurement for the dynamic pressure transducers (DPTs), noise level tests were conducted, and the maximum noise level was turned out to be about 1.6 Pa (Hong, 2001). Important test conditions are summarized in Table 2.
Fig. 3. Design of sparger.
2.2. Development of measurement data analysis method The dynamic characteristics of steam jet condensation are known to be extremely complicated and to have strong randomness. Thus, it is not easy to find the dominant frequency. Fourier transform (or Fast Fourier Transform (FFT)) is a powerful tool to find the frequency by transforming the time domain to frequency domain. Based on this FFT, an advanced method to analyze the frequency was developed. The idea is based on the averaging
method. At first, the total samplings are divided into several subsets, and each set is processed by FFT, which produces the set of pair of frequency and complex number. The complex number is again expressed as polar form composed of amplitude and phase. For the same frequency of each set, the corresponding amplitudes are averaged throughout the subsets. Fig. 4 shows the diagram of this method. These processes were programmed with C++ computer language, and named GIRLS Analysis Program based on FFT
Table 1 Measurement Instruments. Instrument
Model
Descriptions
Uncertainties
Thermocouple, K-type (TC) Pressure transmitter (PT) Flow transmitter, vortex type (FT)
Omega Rosemount 3051P Rose mount 8800A
Pool and steam temperature Static pressure of steam Steam flowrate
DPT (Piezoelectric type) Steam table
Kistler 7061B
Pressure oscillation (Dynamic pressure)
Thomas and Peter (1984)
Steam properties
0.6 °C 0.0005 MPa (0.5 kPa) 1.35% for reading value (4 kg/h for 300 kg/h) Negligible delay of response (natural frequency: 15 kHz) 0.05%
Table 2 Important Test Conditions. Hole size (mm)
Steam discharge
Water level (m)
Submergence (m)
Pool temperature (°C)
Steam mass flux (kg/m2 s)
10
Horizontal
1.3
1.1
35–95
200–900
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 4. Diagram of GAPF.
300 Processed by Only FFT
Pressure(Pa)
(a)
200
100
0 0
100
200
300
400
Frequency(Hz) 80 Processed by GAPF
(b)
Pressure(Pa)
60
40
20
0 0
100
200
300
400
Frequency(Hz) Fig. 5. Performance of GAPF. (a) Result processed by only FFT. (b) Result processed by GAPF.
(GAPF). The process using GAPF has the effect of several tests by just one test, and the randomness greatly decreases. Thus, the GAPF makes the curves smooth and makes it easy to find the dominant frequency as shown in Fig. 5. More detailed description, benchmarking, and performance testing can be found in Hong (2001). 2.3. Results and discussions Fig. 6 shows a sequence of photographs taken by high speed camera of 1000 frames/s in four typical condensation regimes (CO, BCO, SC, and IOC). The structure of steam jet and the variation of steam jet penetration length can be clearly seen. The steam jet consists of vapor cone, two phase mixture, and main region as investigated by several previous workers (DelTin et al., 1983; Kim, 2001). The flow oscillation varies arises as follows. Initially injected vapor cone encounters the cold pool water, and near the end of the cone the local water temperature increases. The increase of pool temperature decreases the condensing power of water. Thus, in order to condense all the steam provided, the cone become inevitably prolonged and obtains the larger heat transfer area. The
69
prolonged cone again encounters new fresh cold water, and the condensing power of water becomes higher than the provided steam. Then, the vapor cone reduces in size. However, the warm water soon accumulates near the end of the cone, and condensing power of water decreases. These processes repeat and an instability results. As a result, the steam jet condensation length varies, and such a variation of the length induces the pressure oscillation. Fig. 7 is summarized test results of dominant frequency processed by GAPF. The dominant frequencies spread broadly according to test conditions, however the results show some consistent trends. The frequencies at the steam mass flux 200 kg/m2 s are very similar to those of Damasio et al. (1985). As a whole, the frequencies tend to increase as the steam mass flux increases when the steam mass flux is lower than around 300 kg/m2 s, even though some exceptions exist. However, when the steam mass flux is higher than around 300 kg/m2 s, the frequencies tend to monotonously decrease as the steam mass flux increases. Such a different trend is related with condensation regimes in Fig. 1. Condensation regimes are distinguished at around 300 kg/m2 s, that is, CO&BCO vs. SC&IOC. The vertical dot line in Fig. 7 means the regime transition criteria for steam mass flux. As investigated by previous workers (Cho et al., 1998), there are clear differences between CO and SC, or between BCO and IOC. The most noticeable difference is the motion of the interface near the circumference of the jet. In CO the circumference oscillates. Thus, the oscillation frequency increases as the steam mass flux increases, since the higher steam mass flux makes the circumference to reach the maximum expansion more rapidly. However, in SC the oscillation of only the jet end is remarkable. Thus, as the jet condensation length is longer the more time is required for the cycling of the jet oscillation. Noting that the jet condensation length is prolonged as the steam mass flux increases, it is reasonable that the frequency becomes lower. Similarly, as the pool temperature increases, the jet condensation length is prolonged and the frequency decreases. For a pool temperature over 80 °C the steam jet is enormously prolonged, and the oscillation of both circumference and jet length itself becomes important. Thus, the above explanation does not thoroughly cover this region, and another regimes were defined as shown in Fig. 1 (BCO and IOC). This slightly different phenomenon will be discussed latter once more in modeling section. Consequently, the frequency trend for higher steam mass flux was experimentally found, and the quantitative data or frequency data for regime transition between CO and SC was proposed by this study.
3. One-dimensional mechanistic model for SC and IOC 3.1. Derivation of governing equation The derivation of governing equation is based on the balance of kinetic energy (KE) that the steam jet gives and the ambient water receives when the steam jet grows. On this basic idea, the fundamental theory of submerged turbulent jet was adopted. Through the observation of steam jet in Fig. 6, Fig. 8 was suggested as a simplified structure of submerged steam jet. It is mainly composed of vapor dominant region (VR) and liquid dominant region (LR). Followings are principal assumptions for the derivation of balance equation. (1) The effective width of LR is proportional to the distance from the exit of hole, and the conceptual diameter is k2x, where k2 is jet expansion coefficient for LR. Such an assumption is based on the theory of submerged turbulent jet. With similar reason, the effective width of VR at X is proportional to the
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distance from the exit of hole, and the conceptual diameter is k1X, where k1 is jet expansion coefficient for VR. (2) A mean velocity, u(x), can represent the liquid velocity in LR. (3) The entrained water does not affect the total kinetic energy (KE) in VR and LR. This assumption is justified by the fact that the entrainment is also caused by the kinetic energy transfer from VR and LR to ambient liquid. The mass conservation of liquid at the point X and x(>X) gives
p
dX p ¼ ðk2 xÞ2 uðxÞ; ðk1 XÞ dt 4 4 2
ð1Þ
where u(x) is an area-averaged velocity at x, and t is time. Rearrange yields
uðxÞ ¼
k1 X k2 x
2
dX dt
ð2Þ
Throughout the LR, X < x < 1, the liquid gains the kinetic energy from the motion of the VR. Using Eq. (2) and the relation dV ¼ p4 ðk2 xÞ2 dx, following kinetic energy can be obtained
KE ¼
Z
1
X
! 1 p dX 2 k41 3 2 q u ðxÞdV ¼ ql X 2 2 l dt 8 k2
ð3Þ
ql is density of LR. The net work, Wl, done against the LR as the VR grows from x = 0 to X is given by Wl ¼
Z
X
Ps 0
p 4
ðk1 xÞ2 dx P1
1p p 2 3 ðk1 XÞ2 X ¼ k X ðPs P1 Þ 34 12 1
ð4Þ
Ps is the pressure of VR, and P1 is the pressure of LR. In the first line of this equation, the second term subtracts work done against the LR to accommodate the volume change of the VR. Equating the Eq. (3) and Eq. (4), arranging, and differentiating with respect to X gives
Fig. 6. Six sequent photos of high speed camera for steam jet condensation. (a) at pool temperature 53 °C and steam mass flux 210 kg/m2 s (Frequency = 327 Hz, Period = 0.0031 s, X/d0 2.7, CO region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (b) at pool temperature 85 °C and steam mass flux 209 kg/m2 s (Frequency = 113 Hz, Period = 0.0088 s, X/d0 5.8, BCO region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (c) at pool temperature 47 °C and steam mass flux 625 kg/m2 s (Frequency = 228 Hz, Period = 0.0035 s, X/d0 3.7, SC region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (d) at pool temperature 85 °C and steam mass flux 611 kg/m2 s (Frequency = 97 Hz, Period = 0.0103 s, X/d0 8.4, IOC region in Fig. 1) X/d0 is based on Kim (2001). See Table 3.
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
71
Fig. 6 (continued)
2
X
d X dt
2
þ
2 2 3 dX 1 k2 ðPs P1 Þ ¼ 0 2 dt ql k1
ð5Þ
This equation is referred to as the ‘jet equation’’. The form of Eq. (5) is the form of second order one dimensional non-linear ordinary differential equation. The form of Eq. (5) is very similar to Rayleigh bubble equation. This means that the spherical bubble is characterized by the radius, and the steam jet can be characterized by its length. However, two expansion coefficients k1 and k2 appear in jet equation, reflecting the turbulent jet expansion. 3.2. Solution for the derived governing equation Perturbation solution method was used to solve the jet equation (Moody, 1990). The following initial conditions were considered
t ¼ 0;
X ¼ X E ð1 þ eÞ;
dX ¼0 dt
ð6Þ
The equilibrium jet condensation length is disturbed by the amount eX E in Eq. (6). A series expansion of XðtÞ is assumed in the form of
XðtÞ ¼ X 0 ðtÞ þ eX 1 ðtÞ þ
ð7Þ
The problem can be linearized by keeping terms up to the order of e. For the pressure of VR, it was assumed that the VR undergoes the polytropic process, and given by
n 3n 3n VE XE XE ¼ P1 ¼ P1 V X X 0 ð1 þ eX 1 =X 0 þ0 Þ 3n XE X1 ¼ P1 1 3ne þ ; X0 X0
Ps ¼ P1
ð8Þ
where n is a coefficient of polytropic process, and VE is equilibrium volume corresponding jet condensation length, XE. The other terms in jet equation and initial condition also can also be expanded in a similar manner. Now, the expanded terms in the jet equation can be rearranged according to the order of e. The 0th order of governing equation (GE) and initial conditions (ICs) are as followings; 0th order: 2
GE : X 0
d X0 dt
2
# 2 " 3n 2 3 dX 0 1 k2 XE þ P 1 ¼0 2 dt ql k1 1 X 0
ð9aÞ
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 6 (continued)
ICs : X 0 ð0Þ ¼ X E ;
X_ 0 ð0Þ ¼ 0
ð9bÞ
The solution of the above set can be directly obtained without complicated method, since the initial undisturbed state is the solution. The solution is given by
X 0 ðtÞ ¼ X E
ð10Þ
The 1st order of equation and initial condition are as followings 2
GE :
d X1 dt
2
This solution shows that the frequency is inversely proportional to steam jet length. This is similar to the explanation on bubble oscillation by Moody (1990). He explained the bubble oscillation using the bubble diameter by the aid of Rayleigh equation. He showed that the larger the bubble size, the lower the frequency.
þ
2 k2 X1 P ð3nÞ 2 ¼ 0 ql k1 1 X0
ð11aÞ
X_ 1 ð0Þ ¼ 0
ð11bÞ
1
ICs : X 1 ð0Þ ¼ X E ;
The solution is easily obtained as
1 k2 X 1 ðtÞ ¼ X E cos X E k1
sffiffiffiffiffiffiffiffiffiffiffiffi ! 3nP1 t
ql
ð12Þ
Thus, the frequency, f, the target of this analysis, is given by
sffiffiffiffiffiffiffi pffiffiffiffiffiffi 3n k2 1 P1 f ¼ 2p k1 X E ql
ð13Þ
4. Comparison of experiments and predictions 4.1. For the pool temperature 35–75 °C In order to get substantial frequency, all of the terms in Eq. (13) have to be specified. At first, for the equilibrium jet condensation length, XE, experimental correlations of Kerney’s type, which considers both steam mass flux, G0, and pool conditions, T1, were used.
X ¼ c1 Bc2 ðG0 =Gm Þc3 d0 B ¼ cp ðT s T 1 Þ=hfg Gm ¼ 275 kg=m2 s
ð14Þ
cp is specific heat of pool, Ts is saturated temperature and hfg is latent heat at pool pressure. Three correlations are suggested in Table 3. It was revealed that the Kerney et al.’s length is the longest, Kim,
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 6 (continued)
700
Dominant Frequency (Hz)
o
Pool Temperature o o ~40 C ~45 C o o ~60 C ~65 C o o ~80 C ~85 C
600
500
CO&BCO
~35 C o ~55 C o ~75 C o ~95 C
o
~50 C o ~70 C o ~90 C
SC&IOC
400
300
200
100
0 100
200
300
400
500
600
700
800
900
1000
2
Steam Mass Flux (kg/m s) Fig. 7. Test results – frequency vs. steam mass flux.
Fig. 8. Modeling of submerged steam jet.
73
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 9. Comparison of experiments and predictions.
Y.S.’s is the shortest, and the Kim, H.Y.’s lies between the two. The ambient pressure or water pressure P1 was assumed to be atmospheric pressure, 101,325 Pa, and the density of water was obtained from steam tables for each pool temperature. An adiabatic process was assumed, i.e. n = c = 1.32. The ratio of jet expansion coefficients for VR and LR is assumed to be k2/k1 = 3.26 from trial and error. This ratio of coefficients means that the expansion ration in LR is larger than that in VR by 3.26 times.
Fig. 9 shows the comparison of predicted frequencies to experimentally measured ones of Fig. 7. It shows good agreement between experiment and prediction except for pool temperature over 80 °C and steam mass flux under 300 kg/m2 s, as shown in Fig. 10. The prediction using Kim, H.Y. was bounded within just 15% error for steam mass flux 300–900 kg/m2 s and pool temperature 35–75 °C. In Figs. 10 and 11 the average and the root mean square (RMS) are defined as follows
75
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77 Table 3 Correlations for steam jet condensation length. Author
Nozzle (mm)
Steam mass flux (kg/m2 s)
Pool temperature (°C)
Correlation
Kerney et al. (1972)
0.4–9.5
378–2044
28–85
X d0
¼ 0:3583B8311 ðG0 =Gm Þ0:6446
Kim (1996)
1.35–10.85
286–1487
13–84
X d0
¼ 0:5923B0:66 ðG0 =Gm Þ0:3444
Kim (2001)
5–20
250–1188
35–80
X d0
¼ 0:51B0:70 ðG0 =Gm Þ0:47
Av erage Dev iation ¼ RMS ¼
N 1X ðfpred fexp Þi N i
ð15Þ
rffiffiffiffi N 1X ðfpred fexp Þ2i N i
ð16Þ
It should be noted that from the condensation regime map in Fig. 1 the pool temperature 80 °C and the steam mass flux 300 kg/m2 s are transition boundaries. Thus, it is necessary to exclude the boundary data, and to consider only the data in SC region since phenomena near transition conditions are usually unstable. This analysis shows more accurate prediction, only 10% error, as shown in Fig. 11.
500
Using the Correlatio n of Kerney (1972)
Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
4.2. For the higher pool temperature (80–95 °C)
+25%
300 -25%
200 Average Deviation = -24.9 Hz Root Mean Square = 32.1 Hz
100
0 0
100
200
300
400
500
Experiment (Hz) 500
Using the Correlation of Kim, Y.S. (1996)
Prediction (Hz)
400
Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
+20%
300
As discussed in the previous section, the developed model does not relatively work well for higher pool temperatures (over 80 °C) using k2/k1 = 3.26. However, it is noticeable that the overall trend is very similar. This fact means that if k2/k1 are suitably adjusted, good prediction is obtainable. For adjusted k2/k1 Table 4 was used together with Kim, H.Y.’s correlation for steam jet condensation length, and this shows better prediction as shown in Fig. 12. However, the predictability at pool temperature 95 °C still is not so good. This means that the limitation of the prediction model because of the difference condensation regime from SC (in Fig. 1 this region is described as ‘Not Condensate in Water’). It may be noted that the value k2/k1 decreases with pool temperature. This implies that the expansion coefficient s of LR and VR become similar. For the IOC region, using modified ratio of jet expansion coefficient (k2/k1) for each pool temperature gave an improved predictions. However, consistent coefficient ratio which is independent of pool temperature was not achievable.
-20%
500 Using the Correlation of Kim, H.Y. (2001)
200
400
Average Deviation = +7.1 Hz Root Mean Square = 17.1 Hz
100
0 0
100
200
300
400
500
Experiment (Hz)
Prediction (Hz)
Prediction (Hz)
400
Pool Temperature 35~75 C Steam Mass Flux 400~900 kg/m^2s
+10% -10%
300
200
500
100
Using the Correlation of Kim, H.Y (2001)
Prediction (Hz)
400 Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
+15%
Average Deviation = -5.3 Hz Root Mean Square = 11.6 Hz
0 0
100
-15%
300
200
300
400
500
Experiment (Hz) Fig. 11. Prediction accuracies according to jet length correlations and conditions for steam mass flux 400–900 kg/m2 s and pool temperature 35–75 °C.
200
100
Average Deviation = -3.0 Hz Root Mean Square = 14.1 Hz
Table 4 k2/k1 For higher pool temperature than 80 °C.
0 0
100
200
300
400
500
Experiment (Hz) Fig. 10. Prediction accuracies according to jet length correlations and conditions for steam mass flux 300–900 kg/m2 s and pool temperature 35–75 °C.
Pool temperature (°C)
Steam mass flux (kg/m2 s)
k2/k1
80 85 90 95
400–900 400–900 400–900 400–900
2.82 2.61 2.20 1.08
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77 120
150 Using the Correlation of Kim, H.Y. (2001)
+5%
o
Pool Temperature 85 C 2 Steam Mass Flux 400~900kg/m s
o
Pool Temperature 80 C 2 Steam Mass Flux 400~900kg/m s
-5%
90
Prediction (Hz)
Prediction (Hz)
120
Using the Correlation of Kim, H.Y. (2001)
+5%
90
60
-5%
60
30 30
0
0 0
30
60
90
120
150
0
30
60
Experiment (Hz) 80
120
40 Using the Correlation of Kim, H.Y. (2001)
+10%
Using the Correlation of Kim, H.Y. (2001)
o
o
Pool Temperature 90 C 2 Steam Mass Flux 400~900kg/m s
Pool Temperature 95 C 2 Steam Mass Flux 400~900kg/m s
30
-10% Prediction (Hz)
60
Prediction (Hz)
90
Experiment (Hz)
40
20
+35% 20
-35% 10
0
0 0
20
40
60
80
0
Experiment (Hz)
10
20
30
40
Experiment (Hz)
Fig. 12. Prediction for the pool temperature 80–95 °C and steam mass flux 400–900 kg/m2 s.
5. Concluding remarks
References
This study presents the experimental results and one-dimensional mechanistic model for submerged steam jet condensation oscillation. Photos taken by high speed camera showed the detailed structure and the motion of steam jet condensation. Experiment showed a different trend of dominant frequency around the transition condition of steam mass flux 300 kg/m2 s. When the steam mass flux was under 300 kg/m2 s, the frequency increased as the steam mass flux increased. However, when the steam mass flux was over 300 kg/m2 s, the frequency decreased as the steam mass flux increased. The frequency decreased as the pool temperature increased regardless of steam mass flux. Using these experimental results, condensation regime map in Fig. 1 can be extended to the steam mass flux 900 kg/m2 s. For the high steam mass flux region, a one-dimensional mechanistic model was developed based on the balance of kinetic energy that the steam jet gives and the water receives, adopting the theory of submerged turbulent jet. The proposed model could predict the experiments with an error of 10% for steam mass fluxes between 400 and 900 kg/m2 s and pool temperature between 35 and 75 °C (SC region). For the IOC region, adjustment of k2/k1 was required for the better prediction. And, at a pool temperature 95 °C, good prediction was not achievable in spite of the adjustment of k2/k1.
Anderson, W.G., Huber, P.W., Sonin, A.A., 1978. Small Scale Modeling of Hydrodynamic Forces in Pressure Suppression System, NUREG/CR-0003. Aya, I., Nariai, H., Kobayashi, M., 1980. Pressure and fluid oscillations in vent system due to steam condensation, (I) – Experimental results and analysis model for chugging. J. Nucl. Sci. Technol. 17 (7), 499–515. Aya, I., Nariai, H., Kobayashi, M., 1983. Pressure and fluid oscillations in vent system due to steam condensation (II) – High-frequency component of pressure oscillations in vent tubes under at chugging and condensation oscillation. J. Nucl. Sci. Technol. 20 (3), 213–227. Chan, C.K., 1978. Dynamic pressure pulse in steam jet condensation. In: 6th International Heat Transfer Conference, pp. 395–399. Chen, C.L., Dhir, V.K., 1982. Hydrodynamics of a bubble formed at the vent pipe exit. Int. J. Multiph. Flow 8, 147–163. Cho, S., Song, C.-H., Park, C.K., Yang, S.K., Chung, M.K., 1998, Experimental study on dynamic pressure pulse in direct contact condensation of steam discharging into subcooled water. In: First Korea-Japan Symposium on Nuclear Thermal Hydraulics and Safety, NTHAS98, Pusan, Korea. Chun, J.H., 1983. Scaling Laws and Rate correlations for Steam Condensation on Turbulent Water, PhD. Dissertation, MIT. Chun, M.H., Kim, Y.S., Park, J.W., 1996. An investigation of direct condensation of steam jet in subcooled water. Int. Commun. Heat Mass Transfer 23, 947–958. Cumo, M., Farello, G.E., Ferrari, G., 1978. Direct heat transfer in pressuresuppression systems. In: Proceedings of 6th International Heat Transfer Conference Toronto, vol. 5, pp. 101-106. Damasio, C., Del Tin, G., Fiegna, G., Malandrone, M., 1985. Experimental study on the unstable direct contact condensation regimes. In: Proceedings of the 3rd International Topical Meeting on Reactor Thermal Hydraulics, pp. 6.C1–6.C-8. DelTin, G. et al., 1983. Thermal and fluid-dynamic features of vapor condensing jets. Heat and Technology 1, 13–35.
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77 Hong, S.J., 2001. Dynamic Characteristics of Steam Jet Condensation in Sparger, PhD. Dissertation, Seoul National University (SNU). Kerney, P.J., Faeth, G.M., Olson, D.R., 1972. Penetration characteristics of submerged jet. AIChE J. 18 (3), 548–553. Kim, Y.S., 1996. An Investigation of Direct Condensation of Steam Jet in Subcooled Water, PhD. Dissertation, Korea Advanced Institute of Science and Technology (KAIST). Kim, H.Y., 2001. A Study on the Characteristics of Direct Contact Condensation of a Steam Jet Discharging into a Quenching Tank through a Single Horizontal Pipe, PhD. Dissertation, Korea Advanced Institute of Science and Technology (KAIST). Kudo, A., 1974. Basic study on vapor suppression. In: 5th International Heat Transfer Conference, Tyokyo, vol. 3, pp. 221–225. Moody, F.J., 1990. Introduction to Unsteady Thermofluid Mechanics. John Wiley & Sons. Nariai, H., Aya, I., 1986. Fluid and pressure oscillations occurring at direct contact condensation of steam flow with cold water. Nucl. Eng. Des. 95, 435–445. Okazaki, M., 1979. Analysis for pressure oscillation phenomena induced by steam condensation in containment with pressure suppression system, (I)-model and analysis for experimental conditions of Marviken reactor. J. Nucl. Sci. Technol. 16 (1), 30–42.
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Pitts, J.H., 1980. Steam chugging in a boiling water reactor pressure-suppression system. Int. J. Multiph. Flow 6, 329–344. Simpson, M.E., Chan, C.K., 1982. Hydrodynamics of a subsonic vapor jet in subcooled liquid. J. Heat Transfer 104, 271–278. Sonin, A.A., 1981. Scaling laws for small-scale modeling of steam relief into water pools. Nucl. Eng. Des. 65, 17–21. Sonin, A.A., 1984. Suppression Pool Dynamics Research at MIT, NUREG/CP-0048, pp. 400–421. Sonin, A.A., Ahimko, M.A., Chun, J.H., 1986. Vapor Condensation onto a turbulent Liquid (I)-The steady condensation rate as a function of liquid-side turbulent. International Journal of Heat Mass Transfer 29 (9), 1319–1332. Thomas Jr., F. Irvine, Peter, E. Liley, 1984. Steam and Gas Tables with Computer Equations. Academic Press, Inc.. Utamura, M., Moriya, K., Uozumi, H., 1984. Numerical analysis on pressure propagation in pressure suppression system due to steam bubble collapse. Journal of Nuclear Science and Technology 21, 279–287. Weimer, J.C., Faeth, G.M., Olson, D.R., 1973. Penetration of vapor jet in subcooled liquid. AIChE Journal 19 (3), 552–558.
Contents lists available at SciVerse ScienceDirect
International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Condensation dynamics of submerged steam jet in subcooled water Soon Joon Hong a,⇑, Goon Cherl Park b, Seok Cho c, Chul-Hwa Song c a
FNC Tech. Co., Building 135-308, Seoul National University (SNU), San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea Department of Nuclear Engineering, Seoul National University (SNU), San 56-1, Shillim-Dong, Kwanak-Gu, Seoul 151-742, South Korea c Korea Atomic Energy Research Institute (KAERI), 150, Dukjin-Dong, Yusong-Gu, Taejon 305-353, South Korea b
a r t i c l e
i n f o
Article history: Received 25 July 2011 Received in revised form 14 October 2011 Accepted 17 October 2011 Available online 26 October 2011 Keywords: Condensation oscillation Dominant frequency Submerged steam jet Mechanistic model Steam mass flux Pool temperature
a b s t r a c t Condensation oscillation of submerged steam jet in water pool was investigated. From the experiments it was found that the dominant frequency of condensation oscillation was proportional to steam mass flux for steam mass flux under 300 kg/m2 s and inversely proportional for over 300 kg/m2 s. The frequency was always inversely proportional to pool temperature. For the high steam mass flux region (over 300 kg/m2 s), one-dimensional mechanistic model was developed based on the balance of the kinetic energy that the steam jet gives and the pool water receives, adopting the submerged turbulent jet theory. The proposed model excellently predicted the dominant frequencies for the steam mass flux 300–900 kg/m2 s and water temperature 35–75 °C. For the higher water temperature, the developed model also could predict the dominant frequencies by adjusting the ratio of jet expansion coefficients of vapor dominant region and liquid dominant region. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The phenomena of a submerged condensable gas jet discharging into a liquid pool have attracted attention from many investigators with a number of interests. This process involves condensation of a pure component (steam into water), dissolution of a gas into a bulk liquid phase (ammonia into water), or a more complicated reactive situation. In particular, recent advanced nuclear power plants use a submerged steam jet condensation in safety systems for the safe quenching of discharged steam. In each of these systems, it has been of interest to know the jet condensation length (penetration length of steam jet), heat transfer, condensation regime map, and condensation oscillation as a function of the operating conditions. Traditionally the penetration length of steam jet (or steam jet condensation length) has been measured by visualization (taking a picture). Kerney et al. (1972), Weimer et al. (1973), Kudo (1974), DelTin et al. (1983), Kim (1996, 2001) and Chun et al. (1996) proposed empirical correlations for steam jet penetration length to nozzle diameter (X/d0) as a function of steam mass flux and pool temperature (or subcooling) for a wide range of steam mass flux (0–2000 kg/m2 s) and pool temperatures (13–95 °C). The length to diameter (X/d0) was proportional to steam mass flux and pool temperature. Several studies on heat transfer of submerged steam jet condensation were experimentally conducted. ⇑ Corresponding author. Tel.: +82 2 872 6084; fax: +82 2 872 6089. E-mail address: [email protected] (S.J. Hong). 0301-9322/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmultiphaseflow.2011.10.007
Cumo et al. (1978), Simpson and Chan (1982), Kim (1996, 2001) and Chun et al. (1996) found that the heat transfer coefficient is a function of nozzle diameter, steam mass flux and pool temperature. And Chun (1983), Sonin (1984) and Sonin et al. (1986) induced the liquid turbulent intensity for the explanation of heat transfer. Since the steam jet condensation shows different characteristics according to steam mass flux and pool temperature, several condensation regime maps or stability maps have been proposed. Cho et al. (1998) suggested a condensation regime map based on visual observation and acoustics, which are applicable for a wide range of steam mass flux (up to 450 kg/m2 s) and pool temperature (20–95 °C), and can cover most of the previous ones, as shown in Fig. 1. This experiment was carried out for several nozzle diameters (5–20 mm) and the pool was open to atmosphere. At low steam mass flux, ‘chugging (denoted by ‘C’ in Fig. 1)’ occurs. As the steam mass flux increases, a ‘transient region from chugging to condensation oscillation (TC)’ is found, where subcooled water does not enter the nozzle any more. Instead, a cloud of tiny steam bubble is formed near the nozzle exit. Further increase of the steam mass flux results in the ‘condensation oscillation (CO)’. In this region, steam condenses outside the nozzle and the steam/water interface oscillates violently. ‘Bubbling condensation oscillation (BCO)’ occurs within the same steam mass flux region of CO, but at higher liquid temperature. The steam plumes are detached from injector and relatively large steam bubbles of irregular shape are condensed. When the steam mass flux is relatively high, ‘stable condensation (SC)’ occurs. As the pool temperature increased, the
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
o
Pool Temperature [ C]
100
Not Condensate in Water BCO
IOC
80
TC
60
CO SC
40 C 20 0
100 200 300 2 Steam Mass Flux [kg/m -s]
400
Fig. 1. Condensation regime map by Cho et al. (1998).
67
interface of steam jet becomes unstable. This region is ‘interfacial oscillation condensation (IOC)’. Oscillation of steam jet condensation was also an area of interest. First of all, chugging was experimentally studied by several investigators (Anderson et al., 1978; Sonin, 1981, 1984; Chun, 1983) because of the large hydrodynamic load. Intensive theoretical analyses were conducted by Okazaki (1979), Aya et al. (1980, 1983), Pitts (1980), Chen and Dhir (1982), and Utamura et al. (1984). For CO region in the regime map, Chan (1978) measured the oscillation frequency and found that the frequency is inversely proportional to nozzle diameter. He explained it by the aid of Rayleigh bubble equation. But, in 1982, he together with Simpson and Chan, 1982 again suggested an experimental frequency correlation using Strouhal number for frequency (St), steam Reynolds number (Re) and pool Jacob number (Ja). The frequency is proportional to steam mass flux and pool subcooling, but inversely proportional to nozzle diameter. More intensive experimental study was con-
(a)
(b)
Fig. 2. Schematic diagram and bird’s eye view of GIRLS: (a) Schematic diagram and (b) Bird’s eye view.
68
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
ducted by Damasio et al. (1985) with many test sets up to maximum steam mass flux 250 kg/m2 s. They also suggested an experimental frequency correlation similar to Simpson’s, but added Weber number (We) to increase prediction capability. Nariai and Aya, 1986 analytically studied chugging oscillation and the condensation oscillation. However, it should be noted that all of these studies were limited to the low steam mass flux (<300 kg/m2 s). Thus, for higher steam mass flux (SC and IOC in condensation regime map), more studies are necessary. The objective of this study is to investigate the dynamic characteristics of steam jet condensation frequency. The range of steam mass flux was 200–900 kg/m2 s, and that of pool temperature was 35–95 °C (subcooling is 65–5 K). In particular, the high steam mass flux region (>300 kg/m2 s) is intensively discussed experimentally and theoretically. 2. Experiment 2.1. Test facility and measurement instruments An experimental apparatus, General Investigation Rig for Liquid/Steam Jet Condensation (GIRLS), was constructed as shown in Fig. 2. All the water was demineralized and degassed. Saturated steam from a steam generator was provided through a main steam line to cylinder pool, whose diameter is 1.8 m, height 1.5 m, water level 1.3 m. A sparger, a steam quenching device, was especially manufactured as a dual pipe in order to minimize heat loss in submerged part, as shown in Fig. 3. The steam was discharged horizontally into subcooled water in a cylinder pool through a single hole of diameter 10 mm. In the cylinder pool, a cooling system was installed to easily control the pool temperature. Summary of the measurement instrumentations is presented in Table 1, together with their uncertainties. For reliable measurement for the dynamic pressure transducers (DPTs), noise level tests were conducted, and the maximum noise level was turned out to be about 1.6 Pa (Hong, 2001). Important test conditions are summarized in Table 2.
Fig. 3. Design of sparger.
2.2. Development of measurement data analysis method The dynamic characteristics of steam jet condensation are known to be extremely complicated and to have strong randomness. Thus, it is not easy to find the dominant frequency. Fourier transform (or Fast Fourier Transform (FFT)) is a powerful tool to find the frequency by transforming the time domain to frequency domain. Based on this FFT, an advanced method to analyze the frequency was developed. The idea is based on the averaging
method. At first, the total samplings are divided into several subsets, and each set is processed by FFT, which produces the set of pair of frequency and complex number. The complex number is again expressed as polar form composed of amplitude and phase. For the same frequency of each set, the corresponding amplitudes are averaged throughout the subsets. Fig. 4 shows the diagram of this method. These processes were programmed with C++ computer language, and named GIRLS Analysis Program based on FFT
Table 1 Measurement Instruments. Instrument
Model
Descriptions
Uncertainties
Thermocouple, K-type (TC) Pressure transmitter (PT) Flow transmitter, vortex type (FT)
Omega Rosemount 3051P Rose mount 8800A
Pool and steam temperature Static pressure of steam Steam flowrate
DPT (Piezoelectric type) Steam table
Kistler 7061B
Pressure oscillation (Dynamic pressure)
Thomas and Peter (1984)
Steam properties
0.6 °C 0.0005 MPa (0.5 kPa) 1.35% for reading value (4 kg/h for 300 kg/h) Negligible delay of response (natural frequency: 15 kHz) 0.05%
Table 2 Important Test Conditions. Hole size (mm)
Steam discharge
Water level (m)
Submergence (m)
Pool temperature (°C)
Steam mass flux (kg/m2 s)
10
Horizontal
1.3
1.1
35–95
200–900
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 4. Diagram of GAPF.
300 Processed by Only FFT
Pressure(Pa)
(a)
200
100
0 0
100
200
300
400
Frequency(Hz) 80 Processed by GAPF
(b)
Pressure(Pa)
60
40
20
0 0
100
200
300
400
Frequency(Hz) Fig. 5. Performance of GAPF. (a) Result processed by only FFT. (b) Result processed by GAPF.
(GAPF). The process using GAPF has the effect of several tests by just one test, and the randomness greatly decreases. Thus, the GAPF makes the curves smooth and makes it easy to find the dominant frequency as shown in Fig. 5. More detailed description, benchmarking, and performance testing can be found in Hong (2001). 2.3. Results and discussions Fig. 6 shows a sequence of photographs taken by high speed camera of 1000 frames/s in four typical condensation regimes (CO, BCO, SC, and IOC). The structure of steam jet and the variation of steam jet penetration length can be clearly seen. The steam jet consists of vapor cone, two phase mixture, and main region as investigated by several previous workers (DelTin et al., 1983; Kim, 2001). The flow oscillation varies arises as follows. Initially injected vapor cone encounters the cold pool water, and near the end of the cone the local water temperature increases. The increase of pool temperature decreases the condensing power of water. Thus, in order to condense all the steam provided, the cone become inevitably prolonged and obtains the larger heat transfer area. The
69
prolonged cone again encounters new fresh cold water, and the condensing power of water becomes higher than the provided steam. Then, the vapor cone reduces in size. However, the warm water soon accumulates near the end of the cone, and condensing power of water decreases. These processes repeat and an instability results. As a result, the steam jet condensation length varies, and such a variation of the length induces the pressure oscillation. Fig. 7 is summarized test results of dominant frequency processed by GAPF. The dominant frequencies spread broadly according to test conditions, however the results show some consistent trends. The frequencies at the steam mass flux 200 kg/m2 s are very similar to those of Damasio et al. (1985). As a whole, the frequencies tend to increase as the steam mass flux increases when the steam mass flux is lower than around 300 kg/m2 s, even though some exceptions exist. However, when the steam mass flux is higher than around 300 kg/m2 s, the frequencies tend to monotonously decrease as the steam mass flux increases. Such a different trend is related with condensation regimes in Fig. 1. Condensation regimes are distinguished at around 300 kg/m2 s, that is, CO&BCO vs. SC&IOC. The vertical dot line in Fig. 7 means the regime transition criteria for steam mass flux. As investigated by previous workers (Cho et al., 1998), there are clear differences between CO and SC, or between BCO and IOC. The most noticeable difference is the motion of the interface near the circumference of the jet. In CO the circumference oscillates. Thus, the oscillation frequency increases as the steam mass flux increases, since the higher steam mass flux makes the circumference to reach the maximum expansion more rapidly. However, in SC the oscillation of only the jet end is remarkable. Thus, as the jet condensation length is longer the more time is required for the cycling of the jet oscillation. Noting that the jet condensation length is prolonged as the steam mass flux increases, it is reasonable that the frequency becomes lower. Similarly, as the pool temperature increases, the jet condensation length is prolonged and the frequency decreases. For a pool temperature over 80 °C the steam jet is enormously prolonged, and the oscillation of both circumference and jet length itself becomes important. Thus, the above explanation does not thoroughly cover this region, and another regimes were defined as shown in Fig. 1 (BCO and IOC). This slightly different phenomenon will be discussed latter once more in modeling section. Consequently, the frequency trend for higher steam mass flux was experimentally found, and the quantitative data or frequency data for regime transition between CO and SC was proposed by this study.
3. One-dimensional mechanistic model for SC and IOC 3.1. Derivation of governing equation The derivation of governing equation is based on the balance of kinetic energy (KE) that the steam jet gives and the ambient water receives when the steam jet grows. On this basic idea, the fundamental theory of submerged turbulent jet was adopted. Through the observation of steam jet in Fig. 6, Fig. 8 was suggested as a simplified structure of submerged steam jet. It is mainly composed of vapor dominant region (VR) and liquid dominant region (LR). Followings are principal assumptions for the derivation of balance equation. (1) The effective width of LR is proportional to the distance from the exit of hole, and the conceptual diameter is k2x, where k2 is jet expansion coefficient for LR. Such an assumption is based on the theory of submerged turbulent jet. With similar reason, the effective width of VR at X is proportional to the
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
distance from the exit of hole, and the conceptual diameter is k1X, where k1 is jet expansion coefficient for VR. (2) A mean velocity, u(x), can represent the liquid velocity in LR. (3) The entrained water does not affect the total kinetic energy (KE) in VR and LR. This assumption is justified by the fact that the entrainment is also caused by the kinetic energy transfer from VR and LR to ambient liquid. The mass conservation of liquid at the point X and x(>X) gives
p
dX p ¼ ðk2 xÞ2 uðxÞ; ðk1 XÞ dt 4 4 2
ð1Þ
where u(x) is an area-averaged velocity at x, and t is time. Rearrange yields
uðxÞ ¼
k1 X k2 x
2
dX dt
ð2Þ
Throughout the LR, X < x < 1, the liquid gains the kinetic energy from the motion of the VR. Using Eq. (2) and the relation dV ¼ p4 ðk2 xÞ2 dx, following kinetic energy can be obtained
KE ¼
Z
1
X
! 1 p dX 2 k41 3 2 q u ðxÞdV ¼ ql X 2 2 l dt 8 k2
ð3Þ
ql is density of LR. The net work, Wl, done against the LR as the VR grows from x = 0 to X is given by Wl ¼
Z
X
Ps 0
p 4
ðk1 xÞ2 dx P1
1p p 2 3 ðk1 XÞ2 X ¼ k X ðPs P1 Þ 34 12 1
ð4Þ
Ps is the pressure of VR, and P1 is the pressure of LR. In the first line of this equation, the second term subtracts work done against the LR to accommodate the volume change of the VR. Equating the Eq. (3) and Eq. (4), arranging, and differentiating with respect to X gives
Fig. 6. Six sequent photos of high speed camera for steam jet condensation. (a) at pool temperature 53 °C and steam mass flux 210 kg/m2 s (Frequency = 327 Hz, Period = 0.0031 s, X/d0 2.7, CO region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (b) at pool temperature 85 °C and steam mass flux 209 kg/m2 s (Frequency = 113 Hz, Period = 0.0088 s, X/d0 5.8, BCO region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (c) at pool temperature 47 °C and steam mass flux 625 kg/m2 s (Frequency = 228 Hz, Period = 0.0035 s, X/d0 3.7, SC region in Fig. 1) X/d0 is based on Kim (2001). See Table 3. (d) at pool temperature 85 °C and steam mass flux 611 kg/m2 s (Frequency = 97 Hz, Period = 0.0103 s, X/d0 8.4, IOC region in Fig. 1) X/d0 is based on Kim (2001). See Table 3.
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
71
Fig. 6 (continued)
2
X
d X dt
2
þ
2 2 3 dX 1 k2 ðPs P1 Þ ¼ 0 2 dt ql k1
ð5Þ
This equation is referred to as the ‘jet equation’’. The form of Eq. (5) is the form of second order one dimensional non-linear ordinary differential equation. The form of Eq. (5) is very similar to Rayleigh bubble equation. This means that the spherical bubble is characterized by the radius, and the steam jet can be characterized by its length. However, two expansion coefficients k1 and k2 appear in jet equation, reflecting the turbulent jet expansion. 3.2. Solution for the derived governing equation Perturbation solution method was used to solve the jet equation (Moody, 1990). The following initial conditions were considered
t ¼ 0;
X ¼ X E ð1 þ eÞ;
dX ¼0 dt
ð6Þ
The equilibrium jet condensation length is disturbed by the amount eX E in Eq. (6). A series expansion of XðtÞ is assumed in the form of
XðtÞ ¼ X 0 ðtÞ þ eX 1 ðtÞ þ
ð7Þ
The problem can be linearized by keeping terms up to the order of e. For the pressure of VR, it was assumed that the VR undergoes the polytropic process, and given by
n 3n 3n VE XE XE ¼ P1 ¼ P1 V X X 0 ð1 þ eX 1 =X 0 þ0 Þ 3n XE X1 ¼ P1 1 3ne þ ; X0 X0
Ps ¼ P1
ð8Þ
where n is a coefficient of polytropic process, and VE is equilibrium volume corresponding jet condensation length, XE. The other terms in jet equation and initial condition also can also be expanded in a similar manner. Now, the expanded terms in the jet equation can be rearranged according to the order of e. The 0th order of governing equation (GE) and initial conditions (ICs) are as followings; 0th order: 2
GE : X 0
d X0 dt
2
# 2 " 3n 2 3 dX 0 1 k2 XE þ P 1 ¼0 2 dt ql k1 1 X 0
ð9aÞ
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 6 (continued)
ICs : X 0 ð0Þ ¼ X E ;
X_ 0 ð0Þ ¼ 0
ð9bÞ
The solution of the above set can be directly obtained without complicated method, since the initial undisturbed state is the solution. The solution is given by
X 0 ðtÞ ¼ X E
ð10Þ
The 1st order of equation and initial condition are as followings 2
GE :
d X1 dt
2
This solution shows that the frequency is inversely proportional to steam jet length. This is similar to the explanation on bubble oscillation by Moody (1990). He explained the bubble oscillation using the bubble diameter by the aid of Rayleigh equation. He showed that the larger the bubble size, the lower the frequency.
þ
2 k2 X1 P ð3nÞ 2 ¼ 0 ql k1 1 X0
ð11aÞ
X_ 1 ð0Þ ¼ 0
ð11bÞ
1
ICs : X 1 ð0Þ ¼ X E ;
The solution is easily obtained as
1 k2 X 1 ðtÞ ¼ X E cos X E k1
sffiffiffiffiffiffiffiffiffiffiffiffi ! 3nP1 t
ql
ð12Þ
Thus, the frequency, f, the target of this analysis, is given by
sffiffiffiffiffiffiffi pffiffiffiffiffiffi 3n k2 1 P1 f ¼ 2p k1 X E ql
ð13Þ
4. Comparison of experiments and predictions 4.1. For the pool temperature 35–75 °C In order to get substantial frequency, all of the terms in Eq. (13) have to be specified. At first, for the equilibrium jet condensation length, XE, experimental correlations of Kerney’s type, which considers both steam mass flux, G0, and pool conditions, T1, were used.
X ¼ c1 Bc2 ðG0 =Gm Þc3 d0 B ¼ cp ðT s T 1 Þ=hfg Gm ¼ 275 kg=m2 s
ð14Þ
cp is specific heat of pool, Ts is saturated temperature and hfg is latent heat at pool pressure. Three correlations are suggested in Table 3. It was revealed that the Kerney et al.’s length is the longest, Kim,
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 6 (continued)
700
Dominant Frequency (Hz)
o
Pool Temperature o o ~40 C ~45 C o o ~60 C ~65 C o o ~80 C ~85 C
600
500
CO&BCO
~35 C o ~55 C o ~75 C o ~95 C
o
~50 C o ~70 C o ~90 C
SC&IOC
400
300
200
100
0 100
200
300
400
500
600
700
800
900
1000
2
Steam Mass Flux (kg/m s) Fig. 7. Test results – frequency vs. steam mass flux.
Fig. 8. Modeling of submerged steam jet.
73
74
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77
Fig. 9. Comparison of experiments and predictions.
Y.S.’s is the shortest, and the Kim, H.Y.’s lies between the two. The ambient pressure or water pressure P1 was assumed to be atmospheric pressure, 101,325 Pa, and the density of water was obtained from steam tables for each pool temperature. An adiabatic process was assumed, i.e. n = c = 1.32. The ratio of jet expansion coefficients for VR and LR is assumed to be k2/k1 = 3.26 from trial and error. This ratio of coefficients means that the expansion ration in LR is larger than that in VR by 3.26 times.
Fig. 9 shows the comparison of predicted frequencies to experimentally measured ones of Fig. 7. It shows good agreement between experiment and prediction except for pool temperature over 80 °C and steam mass flux under 300 kg/m2 s, as shown in Fig. 10. The prediction using Kim, H.Y. was bounded within just 15% error for steam mass flux 300–900 kg/m2 s and pool temperature 35–75 °C. In Figs. 10 and 11 the average and the root mean square (RMS) are defined as follows
75
S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77 Table 3 Correlations for steam jet condensation length. Author
Nozzle (mm)
Steam mass flux (kg/m2 s)
Pool temperature (°C)
Correlation
Kerney et al. (1972)
0.4–9.5
378–2044
28–85
X d0
¼ 0:3583B8311 ðG0 =Gm Þ0:6446
Kim (1996)
1.35–10.85
286–1487
13–84
X d0
¼ 0:5923B0:66 ðG0 =Gm Þ0:3444
Kim (2001)
5–20
250–1188
35–80
X d0
¼ 0:51B0:70 ðG0 =Gm Þ0:47
Av erage Dev iation ¼ RMS ¼
N 1X ðfpred fexp Þi N i
ð15Þ
rffiffiffiffi N 1X ðfpred fexp Þ2i N i
ð16Þ
It should be noted that from the condensation regime map in Fig. 1 the pool temperature 80 °C and the steam mass flux 300 kg/m2 s are transition boundaries. Thus, it is necessary to exclude the boundary data, and to consider only the data in SC region since phenomena near transition conditions are usually unstable. This analysis shows more accurate prediction, only 10% error, as shown in Fig. 11.
500
Using the Correlatio n of Kerney (1972)
Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
4.2. For the higher pool temperature (80–95 °C)
+25%
300 -25%
200 Average Deviation = -24.9 Hz Root Mean Square = 32.1 Hz
100
0 0
100
200
300
400
500
Experiment (Hz) 500
Using the Correlation of Kim, Y.S. (1996)
Prediction (Hz)
400
Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
+20%
300
As discussed in the previous section, the developed model does not relatively work well for higher pool temperatures (over 80 °C) using k2/k1 = 3.26. However, it is noticeable that the overall trend is very similar. This fact means that if k2/k1 are suitably adjusted, good prediction is obtainable. For adjusted k2/k1 Table 4 was used together with Kim, H.Y.’s correlation for steam jet condensation length, and this shows better prediction as shown in Fig. 12. However, the predictability at pool temperature 95 °C still is not so good. This means that the limitation of the prediction model because of the difference condensation regime from SC (in Fig. 1 this region is described as ‘Not Condensate in Water’). It may be noted that the value k2/k1 decreases with pool temperature. This implies that the expansion coefficient s of LR and VR become similar. For the IOC region, using modified ratio of jet expansion coefficient (k2/k1) for each pool temperature gave an improved predictions. However, consistent coefficient ratio which is independent of pool temperature was not achievable.
-20%
500 Using the Correlation of Kim, H.Y. (2001)
200
400
Average Deviation = +7.1 Hz Root Mean Square = 17.1 Hz
100
0 0
100
200
300
400
500
Experiment (Hz)
Prediction (Hz)
Prediction (Hz)
400
Pool Temperature 35~75 C Steam Mass Flux 400~900 kg/m^2s
+10% -10%
300
200
500
100
Using the Correlation of Kim, H.Y (2001)
Prediction (Hz)
400 Pool Temperature 35~75 C Steam Mass Flux 300~900 kg/m^2s
+15%
Average Deviation = -5.3 Hz Root Mean Square = 11.6 Hz
0 0
100
-15%
300
200
300
400
500
Experiment (Hz) Fig. 11. Prediction accuracies according to jet length correlations and conditions for steam mass flux 400–900 kg/m2 s and pool temperature 35–75 °C.
200
100
Average Deviation = -3.0 Hz Root Mean Square = 14.1 Hz
Table 4 k2/k1 For higher pool temperature than 80 °C.
0 0
100
200
300
400
500
Experiment (Hz) Fig. 10. Prediction accuracies according to jet length correlations and conditions for steam mass flux 300–900 kg/m2 s and pool temperature 35–75 °C.
Pool temperature (°C)
Steam mass flux (kg/m2 s)
k2/k1
80 85 90 95
400–900 400–900 400–900 400–900
2.82 2.61 2.20 1.08
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S.J. Hong et al. / International Journal of Multiphase Flow 39 (2012) 66–77 120
150 Using the Correlation of Kim, H.Y. (2001)
+5%
o
Pool Temperature 85 C 2 Steam Mass Flux 400~900kg/m s
o
Pool Temperature 80 C 2 Steam Mass Flux 400~900kg/m s
-5%
90
Prediction (Hz)
Prediction (Hz)
120
Using the Correlation of Kim, H.Y. (2001)
+5%
90
60
-5%
60
30 30
0
0 0
30
60
90
120
150
0
30
60
Experiment (Hz) 80
120
40 Using the Correlation of Kim, H.Y. (2001)
+10%
Using the Correlation of Kim, H.Y. (2001)
o
o
Pool Temperature 90 C 2 Steam Mass Flux 400~900kg/m s
Pool Temperature 95 C 2 Steam Mass Flux 400~900kg/m s
30
-10% Prediction (Hz)
60
Prediction (Hz)
90
Experiment (Hz)
40
20
+35% 20
-35% 10
0
0 0
20
40
60
80
0
Experiment (Hz)
10
20
30
40
Experiment (Hz)
Fig. 12. Prediction for the pool temperature 80–95 °C and steam mass flux 400–900 kg/m2 s.
5. Concluding remarks
References
This study presents the experimental results and one-dimensional mechanistic model for submerged steam jet condensation oscillation. Photos taken by high speed camera showed the detailed structure and the motion of steam jet condensation. Experiment showed a different trend of dominant frequency around the transition condition of steam mass flux 300 kg/m2 s. When the steam mass flux was under 300 kg/m2 s, the frequency increased as the steam mass flux increased. However, when the steam mass flux was over 300 kg/m2 s, the frequency decreased as the steam mass flux increased. The frequency decreased as the pool temperature increased regardless of steam mass flux. Using these experimental results, condensation regime map in Fig. 1 can be extended to the steam mass flux 900 kg/m2 s. For the high steam mass flux region, a one-dimensional mechanistic model was developed based on the balance of kinetic energy that the steam jet gives and the water receives, adopting the theory of submerged turbulent jet. The proposed model could predict the experiments with an error of 10% for steam mass fluxes between 400 and 900 kg/m2 s and pool temperature between 35 and 75 °C (SC region). For the IOC region, adjustment of k2/k1 was required for the better prediction. And, at a pool temperature 95 °C, good prediction was not achievable in spite of the adjustment of k2/k1.
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