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Experimental study of water film falling and spreading on a large vertical plate
Progress in Nuclear Energy 54 (2012) 22e28
Contents lists available at SciVerse ScienceDirect
Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Experimental study of water film falling and spreading on a large vertical plate Y.Q. Yu*, S.J. Wei, Y.H. Yang, X. Cheng School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 May 2011 Received in revised form 23 September 2011 Accepted 23 September 2011
In this paper, experimental studies on water film falling and spreading on a large vertical flat plate are carried out. The experiments aim at investigating the flow characteristics of a falling film. The Reynolds number of falling film ranges from 50 to 900. Capacitance probes and high speed camera are chosen to measure the water film thickness and wave velocity. Plates with different coatings are used to study the effect of the surface condition on the coverage rate. Based on the comparison with theoretical solutions and other experimental data in open literature, the reliability of the experimental data is proven. Experimental results on film thickness, its probability density function, as well as coverage rate and wave velocity are presented and discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Water film Film thickness Surface wave
1. Introduction Film flow is widely applied in industrial fields due to its high thermal efficiency and small flow flux. In the PCCS (Passive Containment Cooling System) of the Generation III nuclear power plant AP1000, the film flow on the surface of the containment is an important measure. During accidents water in the tank on the top of the containment is driven by gravity and forms water film on the surface of the containment, which keeps the containment the final heat sink in accident conditions. The hydrodynamics of thin film flow has been extensively studied for several decades. For all that, the mechanism of some mass transfer and break-up of the film flow still lack knowledge. Therefore, investigation on the hydrodynamics of film flow is still attracting significant attention. With the development of the measurement technology, extensive experimental studies on film flow have been carried out by many researchers. Most experiments were performed on the wall of tubes; only modest attention has been devoted to the film flow on a flat plate. Park and Nosoko (2003) experimentally observed the evolution of solitary waves into three-dimensional waves on a flat plate at Re ¼ 10 w 100. They pointed out the transition of flow pattern and mass transfer, described the hydrodynamic features of falling film. The measurements also show that the surface wave greatly enhances mass transfer. Nosoko et al. (1996) obtained the relation between film velocity, wave amplitude, wave length, fluid property and Reynolds number (Re ¼ 14 w 90) of falling film on flat plates. Moran et al. * Corresponding author. E-mail address: [email protected] (Y.Q. Yu). 0149-1970/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2011.09.007
(2002) measured the thickness, velocity distribution and wall shear stress of falling film on inclined plate at Re ¼ 11 w 220, the results show that: the Nusselt theory (1916) underestimates the film thickness and overestimates the film velocity. In addition, it was found that the peaks in wall shear stress precede the film thickness peaks. The study suggested that the film experience acceleration in the wave region. Ambrosini et al. (2002) studied the statistical characteristics of falling film under different film temperature and plate inclination. The experiment captured the film thickness distribution and wave velocity at Re ¼ 140 w 3200. The effects of plate inclination and film temperature on wave velocity are studied. The presence of two different types of transition of flow pattern is indicated. Making use of a particular definition of the dimensionless film thickness, it is possible to observe a qualitative similarity between the mean film thickness of free falling film and of gas sheared annular flow film. Non-Newtonian liquid spreading on inclined plate was studied experimentally by Sutalo et al. (2006). Film shape, film width and time dependent velocity profiles for falling film have been reported. They agree well with the theoretical value and CFD calculations. The containment of AP1000 is over 60 m. The range of Reynolds number of water film is up to 3700. The current experimental studies on falling film on flat plate have limitation on size of the plate and Reynolds number. It is hard to fully reflect the hydrodynamics of falling film on the containment surface. In order to have a better understanding on the film behavior, this paper describes an experimental study of falling film on a large scale vertical plate at relatively large Reynolds number. The statistical characteristics of the falling film are reported and analyzed.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
23
Table 1 Experimental parameters of WABREC. Parameter
Range
Parameter
Range
Water temperature Plate size Measurement frequency
25 C 25m 1000 Hz
Flow rate Reynolds number Measurement time
100 w 1500 l/h 50 w 900 30 s
Fig. 3. Schematic diagram of film wave.
Fig. 1. Sketch of the experimental loop.
2. Experimental apparatus The test rig named WABREC (WAter Behaviour in REctangle Channel) consists of a vertical stainless plate which is 2 m wide and 5 m long. The schematic diagram of the loop is shown in Fig. 1. The surface of the plate is painted with both organic and inorganic zinc coating (AP1000 containment coating). The water pumped from the main container to the water distribution tank spreads on the plate when the water overflows the distribution tank. Water temperature is kept constant at 25 C with temperature control system. The flow rate is measured by flow meter. The wave pattern and the wave speed are monitored and measured by high speed camera at three longitudinal distances. (0.06 m, 2.74 m, 3.81 m from the inlet). Capacitance probes are adopted in this paper to measure the film thickness, as illustrated in Fig. 2(a). According to the capacitance principle, the voltage V0 signal is proportional to the film thickness h, as presented as below:
Cs Vs V0 ¼ h ε0 S where ε0 is the dielectric constant, S the area of the probe, Cs and Vs constants. The calibration of the capacitance probes is carried out in
static condition. The calibration results are reported in Fig. 2(b). The measurement error is less than 5% within the test range (film thickness <1200 mm).Given the full development of the falling film, the probe is placed at 60 cm from the entrance). The probe can be displaced horizontally to determine the film thickness distribution across the plate width. This signal is received by the data acquisition system and sampled with a scan rate of 1000 Hz. Experiments were run at a temperature of 25 C and at 22 different volumetric flow rates, covering a Reynolds number range of 50 w 850. The Reynolds number of water film is defined as follow:
Re ¼
4G
m
Where G is mass flow rate (kg/m∙s),m dynamic viscosity (Pa∙s). For each run, film thickness data were obtained at equally spaced time intervals to compute time averaged parameters over the sampling period of 30 s. The detailed experimental parameters are shown in Table 1. 3. Results and discussions 3.1. Statistical characteristics of film thickness Most researchers point out that the wave shape is shown in Fig. 3. The wave consists of film substrate large solitary wave and the capillary wave in the front of the wave. The samples of film thickness time series obtain in this research are shown in Fig. 4. It
Fig. 2. Diagram of capacitance probe and calibration data points and best fit line.
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Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
Fig. 4. Samples of film thickness time series.
basically accords with the common description in the open literature. The capillary wave is very obvious at high flow rate. Chu and Dukler (1975) suggested defining the substrate thickness with film thickness PDF (Probability Density Function). The PDF of film thickness under different Reynolds number is reported in Fig. 5. There appears only one peak which can be considered as film substrate thickness. The width of the PDF which indicates the wave amplitude increases with the Reynolds number increasing. At larger Reynolds number, there appear additional peaks at greater film thickness values which correspond to the small and large amplitude waves. Fig. 6 shows the film thickness corresponding to the PDF peak value, the average film thickness and the minimum film thickness. The definition of average film thickness and film thickness deviation are shown as follow:
PN average film thickness ¼
i¼1 hi
standard film thickness deviation N sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 6. Film Thickness corresponding to peak value of PDF and average film thickness PN 2 under different Reynolds number. i¼1 ðhi hÞ ¼ N1
Obviously, the film thickness corresponding to the PDF peak value is close to the minimum film thickness under low Reynolds number while the average film thickness is much larger. In other word, the minimum film thickness can be regarded as substrate thickness approximately under low Reynolds number. The maximum, minimum, average and standard deviation of film thickness is displayed in Fig. 7. Conclusively, the maximum and
Fig. 5. PDF of film thickness under different Reynolds number.
the minimum film thickness increase with the increase in Reynolds number, especially the maximum value. That means the amplitude of the wave is increasing. When Reynolds number is small, the maximum film thickness increases sharply (Fig. 7). It suggests first transition appears at Reynolds number about 110. The second transition appears at Reynolds number about 300. The growth rate of the standard deviation is small in the range between two transition points. Ambrosini et al. (2002) also observed two transitions in their experiments. The present authors argue that both
Fig. 7. Maximum, minimum, average film thickness and film thickness standard deviation.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
25
Fig. 8. Typical wave pattern with different Reynolds number.
Fig. 9. Wave velocity under different Reynolds number.
transitions of statistical characteristics are coupled with the change of film wave pattern. The second transition is captured by the high speed camera during experiments. Fig. 8 shows the typical wave pattern with different Reynolds number. The pattern and shape of the solitary wave at Re ¼ 220 is relatively regular and smooth. At Re ¼ 620 (Fig. 8(b)), there appear large amplitude waves, which seem to have complex structure. We calls it heap wave, which does not appears at low Reynolds number, however, occurs frequently at high Reynolds number conditions.
3.2. Wave velocity Wave velocity is measured at three different longitudinal distances (3.81 m, 2.74 m, 0.06 m from the entrance). Fig. 9 shows the wave velocity under different Reynolds number in WABREC and other experiments. The wave velocity is basically the same at locations beyond 2.74 m from the entrance. It indicates that the
wave is fully developed at that location. For large Reynolds number, the flow is fully developed after a short distance, e.g. 0.06 m, from the entrance. Compared with the wave velocity reported by other authors, this experiment shows higher wave velocity. This is caused by the large size of the plate, which enables a strong interaction between solitary waves. 3.3. Time averaged film thickness Nusselt (1916) put forward the laminar model for liquid film in 1916. The model assumes that film surface is flat and flow is laminar. Additional shear stress due to gas flow is ignored. He proposed the following equation to describe the film thickness:
Table 2 Time averaged film thickness relation based on experiment for tubes (Yan et al., 2005). Author
Re range
Nusselt
0 < Re < 1000 w 2000
Relation h ¼ 0.909Re1/3(y2/g)1/3
Brauer
h ¼ 0.208Re8/15(y2/g)1/3
Ganchev Gimbutis
h ¼ 0.137Re0.58(y2/g)1/3 h ¼ 0.136Re0.538(y2/g)1/3
Takahama
h ¼ 0.228Re0.526(y2/g)1/3
Karapantsios Härkönen
509 < Re < 13,090
h ¼ 0.214Re0.538(y2/g)1/3 h ¼ 0.218Re0.53(y2/g)1/3
Jiang
400 < Re < 5000
h ¼ 0.295Re0.498(y2/g)1/3
Fig. 10. Time averaged film thickness under different Reynolds number.
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Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
Fig. 11. Comparison of measured non-dimensional film thickness with correlations and test data by other authors. Table 3 Non-dimensional film thickness relation and their application range. Author
Application range
Relation
Nusselt (1916)
No shear stress laminar falling film (0 < Re < 1000 w 2000) High Reynolds number falling film (Re < 11,020) Annular film under gas shear stress Annular film under gas shear stress Falling film on the tube (400 < Re < 5000)
hþ ¼ 0.707Re0.5
Takahama and Kato (1980) Kosky (1971) Asali et al. (1985) Jiang (Yan et al., 2005)
hþ ¼ 0.089Re0.789 hþ ¼ 0.0512Re0.875 hþ ¼ 0.34Re0.6 hþ ¼ 0.1307Re0.747
h ¼
2 1=3 1=3 n 3 Re1=3 g 4
Later, some other authors proposed similar equations based on various film experiments in tubes. Some equations are summarized in Table 2. The relation between time averaged film thickness and Reynolds number obtained in the present study is shown in Fig. 10. The two-dimensional CFD results obtained by the present authors are also shown in Fig. 10. It is seen that the CFD results agree well with Nusselt theory. The experimental data is a little lower than the value predicted by Nusselt theory. The experimental data have a good agreement with the equation reported by Jiang. All the equations considered have larger discrepancies in the low Reynolds number region and better agreement at high Reynolds number conditions. Normalizing the Nusselt theory, non-dimensional film thickness is obtained. Table 2 summarizes some non-dimensional film thickness equations with the following definition:
hþ ¼
Fig. 12. Dimensionless film thickness from various experiments versus Re=Fi w1=11
hw ; w ¼
n
rffiffiffiffi
sc 2 2 ; s ¼ sw þ si r c 3 3
Where n and r are the liquid kinematic viscosity and density, sc characteristic shear stress depending on wall shear stress sw and interfacial shear stress si. In the case of free falling film it is si ¼ 0 and sw is calculated by force balance. Fig. 11 reports non-dimensional film thickness collected at WABREC test facility and some data collected by other authors. Most experimental data gained in WABRE appear around the relation listed in Table 3 In order to verify the reliability of the
Fig. 13. Coverage rate on wet plate.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
27
Fig. 14. Film thickness and Ap as function of volume flow rate.
present experimental data; the data are compared with those from other authors (Fig. 9(b)). It is seen that the present test data are in good agreement with those of Ambrosini and show larger deviation to the data of Moran. This is due to similar experimental conditions of WABREC tests and tests of Ambrosini. Both tests are performed on a vertical plate and use water as fluid, while the test of Moran is performed on an inclined plate and uses silicon as fluid. Alekseenko et al. (1994) propose the application of the dimensionless quantity,Fi w which depends on fluid property and plate inclination, and if defined as below:
Fi w ¼
s3 r3 gcos qn4
Dimensionless film thickness as a function of Re=Fi w1=11 is shown in Fig. 12. As seen, all test data are close to each other.
3.4. Coverage rate The coverage rate of water film is significant to the heat transfer of PCCS. The wet perimeter on the plate is measured by rulers at three longitudinal distances, i.e. 4 m, 2 m, and 0.2 m from the entrance, respectively. In order to estimate the coverage rate, we define two dimensionless numbers as follow:
Cp ¼
Lwet Awet Ap ¼ Lall Aall
Where Lwet is the length of the wet line, Lwet the width of the plate, Awet the area of the wet surface, Aall the area of the plate. The coverage rate is measured at two different surface conditions (wet plate and dry plate) and two coatings (organic coating and inorganic coating). Fig. 13 shows the coverage rate on wet surface. With the development of the film flow, the width of the film shrinks a little. The plate surface with either inorganic zinc or organic zinc coating achieves high coverage rate (Cp, Ap > 90%) under wet surface condition, even at low Reynolds conditions. The coverage rate decreases quickly when the Reynolds number is lower than about 50.
Fig. 15. Change in dimension of a film with liquid flow rate (Q1 < Q2 < Q3).
With the change of the flow rate, the relation between the film thickness and Ap is reported in Fig. 14(a). According to the change rate of this two parameters, the flow process can be divided into three region (AeC). In region A, the film thickness increases slowly, while AP increases sharply. In region B, the film thickness and Ap show synchronous growth. In region C, Ap is kept almost constant, whereas film thickness increases rapidly. These phenomena reveal existence of the critical film thickness. In order to further verify the critical film thickness, the research also measure the width of the stream which is formed by V shaped groove (Fig. 14(b)). Three region is still obvious for stream flow. Doniec (1984) suggests the idea of the critical film thickness. The change in dimension of a film is shown in Fig. 15. For large scale plate, when the film thickness reaches the critical value, the film thickness keeps at this critical value even with the increase in flow rate, which leads to the widening of the film. The coverage rate increases sharply (region A). When the film covers the whole plate, the increase in flow rate results in film thickening (region C). Referring to the minimal energy principle (Doniec, 1984), the critical film thickness is:
hmax ¼
1=5 45 1=5 m2 slv 1=5 ð1 cos qÞ 7 r3 g 2
where hmax is the critical film thickness, m the kinematic viscosity, slv the surface free energy difference between liquid and gas, q the contact angle. The critical film thickness varies from 100 to134 for whole plate condition and varies from 222 to 269 for stream condition while the theoretical values are 103 and 229 respectively. Given the effect of the advanced contact angle and the receding contact angle (Fig. 16), the theoretical values agree well with the experimental data.
Fig. 16. Schematic diagram of the dynamic contact angle for a droplet flowing on an inclined plate.
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Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
complex structure and carry huge amount of mass are formed at large Reynolds number. The substrate film thickness is much smaller than the average film thickness while the minimum film thickness can be considered as substrate film thickness at low Reynolds number. Empirical correlation for film thickness as a function of Reynolds numbers proposed by Jiang is recommended for prediction of film thickness on large flat plate. Nusselt theory over predicts the mean film thickness. The wet plate with either organic or inorganic coating has high coverage rate, also at low Reynolds number. Decreasing the critical film thickness leads to an increase in the spreading of water film. This paper gives a good agreement between the theoretical critical film thickness with the experimental data.
Acknowledgements Fig. 17. Coverage rate on wet and dry plate.
Fig. 17 shows the coverage rate on wet and dry plate. Because the contact angle of water film on wet plate is much smaller than that on dry plate, the dry plate needs much higher flow rate to reach high coverage rate than that for wet plate. Therefore, the critical film thickness for wet plate is much thinner. The film spreads slowly before it reaches the critical point on the dry plate.
4. Conclusions This paper reports the experimental study of water film falling on large vertical plate. The statistical characteristics of the falling film such as average film thickness, maximum film thickness, film thickness standard deviation, film thickness probability density function and wave velocity are obtained and analyzed. The coverage rate and the critical film thickness are measured and discussed. The results show: The data collected in this experiment are close to the data from other experiment after proper data processing, which proves the reliability of the experimental data. Two transition points which also appear in other experiments are found in this experiment. The wave pattern transition is captured by high speed camera. The heap waves which have
A support from National Key Projects 2010ZX06002-005 is gratefully acknowledged. References Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G., 1994. In: Fukano, T. (Ed.), Wave Flow of Liquid Films. Begell House. Ambrosini, W., Forgione, N., Oriolo, F., 2002. Statistical characteristics of a water film falling down a flat plate at different inclinations and temperatures. Int. J. Multiphase. Flow. 28, 521. Asali, J.C., Hanratty, T.J., Andreussi, P., 1985. Interfacial drag and film height for vertical annular flow. AIChE. J. 31, 895e902. Chu, K.J., Dukler, A.E., 1975. Statistical characteristics of thin, wavy films III: structure of the large waves and their resistance to gas flow. AIChE. J. 21, 583e593. Doniec, A., 1984. Laminar flow of a liquid down a vertical solid surface: maximum thickness of liquid rivulet. PhysicoChem. Hydrodyn. 5 (2), 143e152. Kosky, G., 1971. Thin liquid films under simultaneous shear and gravity forces. Int. J. Heat. Mass. Transfer. 14, 1120e1224. Moran, K., Inumaru, J., Kawaji, M., 2002. Instantaneous hydrodynamics of a laminar wavy liquid film. Int. J. Multiphase. Flow. 28, 731e755. Nosoko, T., Yoshimura, P.N., Nagata, T., Oyakawa, K., 1996. Characteristics of twodimensional waves on a falling liquid film. Chem. Eng. Sci. 51, 725. Nusselt, N., 1916. berflachenkondensation des Wasserdampfes. Zeit. Ver. D. Ing. 60, 541e569. Park, C.D., Nosoko, T., 2003. Three-dimensional wave dynamics on a falling film and associated mass transfer. AIChE. J. 49, 2715. Sutalo, I.D., Bui, A., Rudman, M., 2006. The flow of non-Newtonian fluids down inclines. J. Non-Newton. Fluid. Mech. 136, 64e75. Takahama, H., Kato, S., 1980. Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow. Int. J. Multiphase. Flow. 6, 203e215. Yan, W.P., Ye, X.M., Li, H.T., 2005. Flow and heat transfer characteristics of liquid thin film flow. J. North. China. Electric. Power. Univ. 32 (1), 59e65.
Contents lists available at SciVerse ScienceDirect
Progress in Nuclear Energy journal homepage: www.elsevier.com/locate/pnucene
Experimental study of water film falling and spreading on a large vertical plate Y.Q. Yu*, S.J. Wei, Y.H. Yang, X. Cheng School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 May 2011 Received in revised form 23 September 2011 Accepted 23 September 2011
In this paper, experimental studies on water film falling and spreading on a large vertical flat plate are carried out. The experiments aim at investigating the flow characteristics of a falling film. The Reynolds number of falling film ranges from 50 to 900. Capacitance probes and high speed camera are chosen to measure the water film thickness and wave velocity. Plates with different coatings are used to study the effect of the surface condition on the coverage rate. Based on the comparison with theoretical solutions and other experimental data in open literature, the reliability of the experimental data is proven. Experimental results on film thickness, its probability density function, as well as coverage rate and wave velocity are presented and discussed. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Water film Film thickness Surface wave
1. Introduction Film flow is widely applied in industrial fields due to its high thermal efficiency and small flow flux. In the PCCS (Passive Containment Cooling System) of the Generation III nuclear power plant AP1000, the film flow on the surface of the containment is an important measure. During accidents water in the tank on the top of the containment is driven by gravity and forms water film on the surface of the containment, which keeps the containment the final heat sink in accident conditions. The hydrodynamics of thin film flow has been extensively studied for several decades. For all that, the mechanism of some mass transfer and break-up of the film flow still lack knowledge. Therefore, investigation on the hydrodynamics of film flow is still attracting significant attention. With the development of the measurement technology, extensive experimental studies on film flow have been carried out by many researchers. Most experiments were performed on the wall of tubes; only modest attention has been devoted to the film flow on a flat plate. Park and Nosoko (2003) experimentally observed the evolution of solitary waves into three-dimensional waves on a flat plate at Re ¼ 10 w 100. They pointed out the transition of flow pattern and mass transfer, described the hydrodynamic features of falling film. The measurements also show that the surface wave greatly enhances mass transfer. Nosoko et al. (1996) obtained the relation between film velocity, wave amplitude, wave length, fluid property and Reynolds number (Re ¼ 14 w 90) of falling film on flat plates. Moran et al. * Corresponding author. E-mail address: [email protected] (Y.Q. Yu). 0149-1970/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.pnucene.2011.09.007
(2002) measured the thickness, velocity distribution and wall shear stress of falling film on inclined plate at Re ¼ 11 w 220, the results show that: the Nusselt theory (1916) underestimates the film thickness and overestimates the film velocity. In addition, it was found that the peaks in wall shear stress precede the film thickness peaks. The study suggested that the film experience acceleration in the wave region. Ambrosini et al. (2002) studied the statistical characteristics of falling film under different film temperature and plate inclination. The experiment captured the film thickness distribution and wave velocity at Re ¼ 140 w 3200. The effects of plate inclination and film temperature on wave velocity are studied. The presence of two different types of transition of flow pattern is indicated. Making use of a particular definition of the dimensionless film thickness, it is possible to observe a qualitative similarity between the mean film thickness of free falling film and of gas sheared annular flow film. Non-Newtonian liquid spreading on inclined plate was studied experimentally by Sutalo et al. (2006). Film shape, film width and time dependent velocity profiles for falling film have been reported. They agree well with the theoretical value and CFD calculations. The containment of AP1000 is over 60 m. The range of Reynolds number of water film is up to 3700. The current experimental studies on falling film on flat plate have limitation on size of the plate and Reynolds number. It is hard to fully reflect the hydrodynamics of falling film on the containment surface. In order to have a better understanding on the film behavior, this paper describes an experimental study of falling film on a large scale vertical plate at relatively large Reynolds number. The statistical characteristics of the falling film are reported and analyzed.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
23
Table 1 Experimental parameters of WABREC. Parameter
Range
Parameter
Range
Water temperature Plate size Measurement frequency
25 C 25m 1000 Hz
Flow rate Reynolds number Measurement time
100 w 1500 l/h 50 w 900 30 s
Fig. 3. Schematic diagram of film wave.
Fig. 1. Sketch of the experimental loop.
2. Experimental apparatus The test rig named WABREC (WAter Behaviour in REctangle Channel) consists of a vertical stainless plate which is 2 m wide and 5 m long. The schematic diagram of the loop is shown in Fig. 1. The surface of the plate is painted with both organic and inorganic zinc coating (AP1000 containment coating). The water pumped from the main container to the water distribution tank spreads on the plate when the water overflows the distribution tank. Water temperature is kept constant at 25 C with temperature control system. The flow rate is measured by flow meter. The wave pattern and the wave speed are monitored and measured by high speed camera at three longitudinal distances. (0.06 m, 2.74 m, 3.81 m from the inlet). Capacitance probes are adopted in this paper to measure the film thickness, as illustrated in Fig. 2(a). According to the capacitance principle, the voltage V0 signal is proportional to the film thickness h, as presented as below:
Cs Vs V0 ¼ h ε0 S where ε0 is the dielectric constant, S the area of the probe, Cs and Vs constants. The calibration of the capacitance probes is carried out in
static condition. The calibration results are reported in Fig. 2(b). The measurement error is less than 5% within the test range (film thickness <1200 mm).Given the full development of the falling film, the probe is placed at 60 cm from the entrance). The probe can be displaced horizontally to determine the film thickness distribution across the plate width. This signal is received by the data acquisition system and sampled with a scan rate of 1000 Hz. Experiments were run at a temperature of 25 C and at 22 different volumetric flow rates, covering a Reynolds number range of 50 w 850. The Reynolds number of water film is defined as follow:
Re ¼
4G
m
Where G is mass flow rate (kg/m∙s),m dynamic viscosity (Pa∙s). For each run, film thickness data were obtained at equally spaced time intervals to compute time averaged parameters over the sampling period of 30 s. The detailed experimental parameters are shown in Table 1. 3. Results and discussions 3.1. Statistical characteristics of film thickness Most researchers point out that the wave shape is shown in Fig. 3. The wave consists of film substrate large solitary wave and the capillary wave in the front of the wave. The samples of film thickness time series obtain in this research are shown in Fig. 4. It
Fig. 2. Diagram of capacitance probe and calibration data points and best fit line.
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Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
Fig. 4. Samples of film thickness time series.
basically accords with the common description in the open literature. The capillary wave is very obvious at high flow rate. Chu and Dukler (1975) suggested defining the substrate thickness with film thickness PDF (Probability Density Function). The PDF of film thickness under different Reynolds number is reported in Fig. 5. There appears only one peak which can be considered as film substrate thickness. The width of the PDF which indicates the wave amplitude increases with the Reynolds number increasing. At larger Reynolds number, there appear additional peaks at greater film thickness values which correspond to the small and large amplitude waves. Fig. 6 shows the film thickness corresponding to the PDF peak value, the average film thickness and the minimum film thickness. The definition of average film thickness and film thickness deviation are shown as follow:
PN average film thickness ¼
i¼1 hi
standard film thickness deviation N sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 6. Film Thickness corresponding to peak value of PDF and average film thickness PN 2 under different Reynolds number. i¼1 ðhi hÞ ¼ N1
Obviously, the film thickness corresponding to the PDF peak value is close to the minimum film thickness under low Reynolds number while the average film thickness is much larger. In other word, the minimum film thickness can be regarded as substrate thickness approximately under low Reynolds number. The maximum, minimum, average and standard deviation of film thickness is displayed in Fig. 7. Conclusively, the maximum and
Fig. 5. PDF of film thickness under different Reynolds number.
the minimum film thickness increase with the increase in Reynolds number, especially the maximum value. That means the amplitude of the wave is increasing. When Reynolds number is small, the maximum film thickness increases sharply (Fig. 7). It suggests first transition appears at Reynolds number about 110. The second transition appears at Reynolds number about 300. The growth rate of the standard deviation is small in the range between two transition points. Ambrosini et al. (2002) also observed two transitions in their experiments. The present authors argue that both
Fig. 7. Maximum, minimum, average film thickness and film thickness standard deviation.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
25
Fig. 8. Typical wave pattern with different Reynolds number.
Fig. 9. Wave velocity under different Reynolds number.
transitions of statistical characteristics are coupled with the change of film wave pattern. The second transition is captured by the high speed camera during experiments. Fig. 8 shows the typical wave pattern with different Reynolds number. The pattern and shape of the solitary wave at Re ¼ 220 is relatively regular and smooth. At Re ¼ 620 (Fig. 8(b)), there appear large amplitude waves, which seem to have complex structure. We calls it heap wave, which does not appears at low Reynolds number, however, occurs frequently at high Reynolds number conditions.
3.2. Wave velocity Wave velocity is measured at three different longitudinal distances (3.81 m, 2.74 m, 0.06 m from the entrance). Fig. 9 shows the wave velocity under different Reynolds number in WABREC and other experiments. The wave velocity is basically the same at locations beyond 2.74 m from the entrance. It indicates that the
wave is fully developed at that location. For large Reynolds number, the flow is fully developed after a short distance, e.g. 0.06 m, from the entrance. Compared with the wave velocity reported by other authors, this experiment shows higher wave velocity. This is caused by the large size of the plate, which enables a strong interaction between solitary waves. 3.3. Time averaged film thickness Nusselt (1916) put forward the laminar model for liquid film in 1916. The model assumes that film surface is flat and flow is laminar. Additional shear stress due to gas flow is ignored. He proposed the following equation to describe the film thickness:
Table 2 Time averaged film thickness relation based on experiment for tubes (Yan et al., 2005). Author
Re range
Nusselt
0 < Re < 1000 w 2000
Relation h ¼ 0.909Re1/3(y2/g)1/3
Brauer
h ¼ 0.208Re8/15(y2/g)1/3
Ganchev Gimbutis
h ¼ 0.137Re0.58(y2/g)1/3 h ¼ 0.136Re0.538(y2/g)1/3
Takahama
h ¼ 0.228Re0.526(y2/g)1/3
Karapantsios Härkönen
509 < Re < 13,090
h ¼ 0.214Re0.538(y2/g)1/3 h ¼ 0.218Re0.53(y2/g)1/3
Jiang
400 < Re < 5000
h ¼ 0.295Re0.498(y2/g)1/3
Fig. 10. Time averaged film thickness under different Reynolds number.
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Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
Fig. 11. Comparison of measured non-dimensional film thickness with correlations and test data by other authors. Table 3 Non-dimensional film thickness relation and their application range. Author
Application range
Relation
Nusselt (1916)
No shear stress laminar falling film (0 < Re < 1000 w 2000) High Reynolds number falling film (Re < 11,020) Annular film under gas shear stress Annular film under gas shear stress Falling film on the tube (400 < Re < 5000)
hþ ¼ 0.707Re0.5
Takahama and Kato (1980) Kosky (1971) Asali et al. (1985) Jiang (Yan et al., 2005)
hþ ¼ 0.089Re0.789 hþ ¼ 0.0512Re0.875 hþ ¼ 0.34Re0.6 hþ ¼ 0.1307Re0.747
h ¼
2 1=3 1=3 n 3 Re1=3 g 4
Later, some other authors proposed similar equations based on various film experiments in tubes. Some equations are summarized in Table 2. The relation between time averaged film thickness and Reynolds number obtained in the present study is shown in Fig. 10. The two-dimensional CFD results obtained by the present authors are also shown in Fig. 10. It is seen that the CFD results agree well with Nusselt theory. The experimental data is a little lower than the value predicted by Nusselt theory. The experimental data have a good agreement with the equation reported by Jiang. All the equations considered have larger discrepancies in the low Reynolds number region and better agreement at high Reynolds number conditions. Normalizing the Nusselt theory, non-dimensional film thickness is obtained. Table 2 summarizes some non-dimensional film thickness equations with the following definition:
hþ ¼
Fig. 12. Dimensionless film thickness from various experiments versus Re=Fi w1=11
hw ; w ¼
n
rffiffiffiffi
sc 2 2 ; s ¼ sw þ si r c 3 3
Where n and r are the liquid kinematic viscosity and density, sc characteristic shear stress depending on wall shear stress sw and interfacial shear stress si. In the case of free falling film it is si ¼ 0 and sw is calculated by force balance. Fig. 11 reports non-dimensional film thickness collected at WABREC test facility and some data collected by other authors. Most experimental data gained in WABRE appear around the relation listed in Table 3 In order to verify the reliability of the
Fig. 13. Coverage rate on wet plate.
Y.Q. Yu et al. / Progress in Nuclear Energy 54 (2012) 22e28
27
Fig. 14. Film thickness and Ap as function of volume flow rate.
present experimental data; the data are compared with those from other authors (Fig. 9(b)). It is seen that the present test data are in good agreement with those of Ambrosini and show larger deviation to the data of Moran. This is due to similar experimental conditions of WABREC tests and tests of Ambrosini. Both tests are performed on a vertical plate and use water as fluid, while the test of Moran is performed on an inclined plate and uses silicon as fluid. Alekseenko et al. (1994) propose the application of the dimensionless quantity,Fi w which depends on fluid property and plate inclination, and if defined as below:
Fi w ¼
s3 r3 gcos qn4
Dimensionless film thickness as a function of Re=Fi w1=11 is shown in Fig. 12. As seen, all test data are close to each other.
3.4. Coverage rate The coverage rate of water film is significant to the heat transfer of PCCS. The wet perimeter on the plate is measured by rulers at three longitudinal distances, i.e. 4 m, 2 m, and 0.2 m from the entrance, respectively. In order to estimate the coverage rate, we define two dimensionless numbers as follow:
Cp ¼
Lwet Awet Ap ¼ Lall Aall
Where Lwet is the length of the wet line, Lwet the width of the plate, Awet the area of the wet surface, Aall the area of the plate. The coverage rate is measured at two different surface conditions (wet plate and dry plate) and two coatings (organic coating and inorganic coating). Fig. 13 shows the coverage rate on wet surface. With the development of the film flow, the width of the film shrinks a little. The plate surface with either inorganic zinc or organic zinc coating achieves high coverage rate (Cp, Ap > 90%) under wet surface condition, even at low Reynolds conditions. The coverage rate decreases quickly when the Reynolds number is lower than about 50.
Fig. 15. Change in dimension of a film with liquid flow rate (Q1 < Q2 < Q3).
With the change of the flow rate, the relation between the film thickness and Ap is reported in Fig. 14(a). According to the change rate of this two parameters, the flow process can be divided into three region (AeC). In region A, the film thickness increases slowly, while AP increases sharply. In region B, the film thickness and Ap show synchronous growth. In region C, Ap is kept almost constant, whereas film thickness increases rapidly. These phenomena reveal existence of the critical film thickness. In order to further verify the critical film thickness, the research also measure the width of the stream which is formed by V shaped groove (Fig. 14(b)). Three region is still obvious for stream flow. Doniec (1984) suggests the idea of the critical film thickness. The change in dimension of a film is shown in Fig. 15. For large scale plate, when the film thickness reaches the critical value, the film thickness keeps at this critical value even with the increase in flow rate, which leads to the widening of the film. The coverage rate increases sharply (region A). When the film covers the whole plate, the increase in flow rate results in film thickening (region C). Referring to the minimal energy principle (Doniec, 1984), the critical film thickness is:
hmax ¼
1=5 45 1=5 m2 slv 1=5 ð1 cos qÞ 7 r3 g 2
where hmax is the critical film thickness, m the kinematic viscosity, slv the surface free energy difference between liquid and gas, q the contact angle. The critical film thickness varies from 100 to134 for whole plate condition and varies from 222 to 269 for stream condition while the theoretical values are 103 and 229 respectively. Given the effect of the advanced contact angle and the receding contact angle (Fig. 16), the theoretical values agree well with the experimental data.
Fig. 16. Schematic diagram of the dynamic contact angle for a droplet flowing on an inclined plate.
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complex structure and carry huge amount of mass are formed at large Reynolds number. The substrate film thickness is much smaller than the average film thickness while the minimum film thickness can be considered as substrate film thickness at low Reynolds number. Empirical correlation for film thickness as a function of Reynolds numbers proposed by Jiang is recommended for prediction of film thickness on large flat plate. Nusselt theory over predicts the mean film thickness. The wet plate with either organic or inorganic coating has high coverage rate, also at low Reynolds number. Decreasing the critical film thickness leads to an increase in the spreading of water film. This paper gives a good agreement between the theoretical critical film thickness with the experimental data.
Acknowledgements Fig. 17. Coverage rate on wet and dry plate.
Fig. 17 shows the coverage rate on wet and dry plate. Because the contact angle of water film on wet plate is much smaller than that on dry plate, the dry plate needs much higher flow rate to reach high coverage rate than that for wet plate. Therefore, the critical film thickness for wet plate is much thinner. The film spreads slowly before it reaches the critical point on the dry plate.
4. Conclusions This paper reports the experimental study of water film falling on large vertical plate. The statistical characteristics of the falling film such as average film thickness, maximum film thickness, film thickness standard deviation, film thickness probability density function and wave velocity are obtained and analyzed. The coverage rate and the critical film thickness are measured and discussed. The results show: The data collected in this experiment are close to the data from other experiment after proper data processing, which proves the reliability of the experimental data. Two transition points which also appear in other experiments are found in this experiment. The wave pattern transition is captured by high speed camera. The heap waves which have
A support from National Key Projects 2010ZX06002-005 is gratefully acknowledged. References Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G., 1994. In: Fukano, T. (Ed.), Wave Flow of Liquid Films. Begell House. Ambrosini, W., Forgione, N., Oriolo, F., 2002. Statistical characteristics of a water film falling down a flat plate at different inclinations and temperatures. Int. J. Multiphase. Flow. 28, 521. Asali, J.C., Hanratty, T.J., Andreussi, P., 1985. Interfacial drag and film height for vertical annular flow. AIChE. J. 31, 895e902. Chu, K.J., Dukler, A.E., 1975. Statistical characteristics of thin, wavy films III: structure of the large waves and their resistance to gas flow. AIChE. J. 21, 583e593. Doniec, A., 1984. Laminar flow of a liquid down a vertical solid surface: maximum thickness of liquid rivulet. PhysicoChem. Hydrodyn. 5 (2), 143e152. Kosky, G., 1971. Thin liquid films under simultaneous shear and gravity forces. Int. J. Heat. Mass. Transfer. 14, 1120e1224. Moran, K., Inumaru, J., Kawaji, M., 2002. Instantaneous hydrodynamics of a laminar wavy liquid film. Int. J. Multiphase. Flow. 28, 731e755. Nosoko, T., Yoshimura, P.N., Nagata, T., Oyakawa, K., 1996. Characteristics of twodimensional waves on a falling liquid film. Chem. Eng. Sci. 51, 725. Nusselt, N., 1916. berflachenkondensation des Wasserdampfes. Zeit. Ver. D. Ing. 60, 541e569. Park, C.D., Nosoko, T., 2003. Three-dimensional wave dynamics on a falling film and associated mass transfer. AIChE. J. 49, 2715. Sutalo, I.D., Bui, A., Rudman, M., 2006. The flow of non-Newtonian fluids down inclines. J. Non-Newton. Fluid. Mech. 136, 64e75. Takahama, H., Kato, S., 1980. Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow. Int. J. Multiphase. Flow. 6, 203e215. Yan, W.P., Ye, X.M., Li, H.T., 2005. Flow and heat transfer characteristics of liquid thin film flow. J. North. China. Electric. Power. Univ. 32 (1), 59e65.