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Experiment study on film width and thickness of free falling water film on a large inclined plate
Nuclear Engineering and Design 358 (2020) 110445
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Experiment study on film width and thickness of free falling water film on a large inclined plate
T
Po Hu , Xingguan Huang, Kai Bao, Guixue Zhu ⁎
School of Nuclear Science and Engineering, Shanghai Jiao Tong University, 800 Dongchuan Rd, Shanghai, China
ARTICLE INFO
ABSTRACT
Keywords: Free falling film Film thickness Film width Film wettability
An experiment was setup to study film width and film thickness of a free falling water film on a large inclined steel plate with no heat transfer. Based on results a semi-empirical correlation was developed to predict the film width and film thickness for partially film-covered condition, and the prediction of the film width was in agreement, to within ± 20%, with the experiment results. And different film thickness correlations were examined with experiment data under fully film-covered condition, and it showed that at lower Reynolds the Nusselt theory predicted the average film thickness well and at higher Reynolds the Lel correlation predicted better. The transition region of Reynolds between these two was related to the measuring location relative to the liquid inlet.
1. Introduction
=
Free falling liquid film has been used widely in different industrial applications, such as tube evaporator, wetted-wall absorber and passive cooling containment system (PCCS) in nuclear power plant. As shown in Fig. 1, PCCS is one of the passive safety systems adopted by Westinghouse in their AP series nuclear reactor designs (Sha et al., 2004). It uses the counter-current natural circulating air flow and gravity-driven water film to cool down the containment steel wall during accidents. To uniformly distribute the water film, a weir with V-shaped inlets was used. The weir design was based on experiment to assure the film coverage above the design limit. In order to better design equipment using falling film, a thorough study on heat transfer and hydrodynamic behavior of liquid film is needed, which may include the film thickness development, film wettability, and heat transfer capability under a wide range of Reynolds number. The current paper studied water film width and film thickness under partially covered and fully covered conditions on a plate with different inclination angles. Nusselt (1916) has studied free falling film analytically based on a 2D, shear-free interface model, and he predicted the falling film thickness and film velocity as following,
u(y) =
⁎
gsin ( y µ
y2 ) 2
(1)
3µm L 2 gsin
1/3
(2)
or in dimensionless form
= (3Re )1/3 = (
Re =
m Lµ
2 g sin
µ2
)1/3
(3) (4)
As shown in Fig. 2, the u(y) is the velocity profile in the film and um is average velocity over the film thickness, and y is the distance away from the wall surface vertically. δ is the thickness of the film, ρ is the density, μ is the viscosity, g is the gravitational acceleration, θ is the inclination angle from the horizontal, m is the film mass flow rate, and L is the film width or wetted perimeter. Selected free falling film experiments and reported film thickness correlations are shown in the Table 1. As shown in the Fig. 11, for film thickness tests in tube, the Takahama and Karapantsios correlations predict dimensionless thickness with up to 40% difference comparing to Nusselt theory for Reynolds less than 600. For film thickness tests on plate, Lel correlation predicts the thickness within 10% difference comparing to Nusselt theory for Reynolds less than 600, and it deviates distinctively from Nusselt theory for Reynolds larger than 600. Huang correlation is quite close to the Nusselt theory up to Reynolds equal to 2000, and the difference is less than 10%. Therefore it seems that the average film thickness on a fully covered plate is predicted well with
Corresponding author. E-mail address: [email protected] (P. Hu).
https://doi.org/10.1016/j.nucengdes.2019.110445 Received 27 May 2019; Received in revised form 14 November 2019; Accepted 14 November 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
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kinetic viscosity (m2/s) polar angle (°)
Nomenclature g L m Re x y u
acceleration of gravity (m/s2) film width (m) mass flow rate (kg/s) m Reynolds number Re = Lµ coordinate along the surface (m) coordinate perpendicular to the surface (m) or thickness ratio equal to MLFT / PCLFT film velocity (m/s)
Subscript a b m mean min
Greek symbols ρ Δ
c
µ
the MWR point the beginning of the constant film thickness region, corresponding to the lowest flow rate with constant thickness average value average value minimum
Abbreviation
density (kg/m3) film thickness (m) dimensionless minimum liquid film thickness (m) surface tension (N/m) inclination angle (°) contact angle (°) dynamic viscosity (kg/s-m)
MLFT MWR PCCS PCLFT RSD
minimum liquid film thickness minimum wetting rate passive containment cooling system partially covered liquid film thickness Relative standard deviation
Nusselt theory at lower Reynolds less than 600, and at higher Reynolds, the thickness will deviate from Nusselt theory and approach Karapantsios and Lel correlations. The current study examined the phenomenon with experiment for Reynolds more than 1000. All the tests above were conducted with test section fully covered by the fluid film, when the flow rate starts to decrease, at beginning the water film thickness will decrease, then at certain flow rate, the water film width will start to decrease and the surface will be partially covered by a continuous film, as shown by Yu et al. (2012). But few studies in the literature studied film behavior under this partially film-covered condition, one of these is the study of Lan et al. (2010), which studied free falling liquid film thickness and width with experiment, and the data was compared to the CFD calculation, and the test section dimension was limited as 0.10 m × 0.26 m. When the flow rate decreases further, a specific minimum wetting rate (MWR) will be reached, which is the lowest flow rate needed to ensure that surface remains covered by a continuous liquid film. The MWR attracted a lot of attention from different branches of scientific community, and various theories have been proposed such as force balance and minimum total energy
Fig. 2. Schematic of a 2D liquid film flowing down a vertical wall.
criterions. El-Genk and Saber (2001) compared four different analytical expressions for MWR and minimum liquid film thickness (MLFT), and proposed a set of new equations for vertical surface as following, and c is the contact angle
Fig. 1. Schematic of AP1000 passive containment cooling system (image courtesy of Westinghouse Electric Corporation). 2
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Table 1 Selected free falling film experiments and film thickness correlations. Reference
Reported film thickness correlations
Takahama and Kato (1980)
= 0.473Re 0.526
Karapantsios and Karabelas (1995)
= 0.451Re 0.538
Lel et al. (2005)
= 1 + 0.615Re0.47
Huang et al. (2014)
= 1.666Re 0.322
Comments Tests in Tube Measuring location 1.7 m from inlet Tests in Tube Measuring location 1.7–2.6 m from inlet Tests on plate Measuring location 1.2 m from inlet Tests on plate Measuring locations 1–4.5 m from inlet
Fig. 3. Schematic of experiment apparatus, measuring equipment, and weir plate with V-shaped inlets.
m=L
µ g
3 0.2
15µ2 = ( 3 2 )0.2 g
(0.67
2.83 min
+ 0.26
9.51 min)
2. Experiment (5)
min
(6)
cos c )0.22
(7)
2.1. Experiment apparatus Experiment apparatus schematic is shown in Fig. 3. A steel plane plate (5 × 1.23 × 0.04 m) is carefully manufactured to maintain a good planarity, and then painted with nonorganic zinc paint (Carbozinc 11 HSN). The plate is mounted on a tilting frame which can stay at any inclination angle from horizontal to vertical position. Therefore the water film behavior on the plate with different inclination angles can be used to study the similar phenomena of different parts on the containment dome area. The water delivery system consists of a supply tank, flow meter and valves, distribution box and drain. The distribution box is adjustable to test different shape and arrangement of inlets. A plexiglass baffle plate is mounted parallel to the steel plate, and the gap between them is 30 cm. Water temperature was measured with T type thermocouple at the inlet of the water distribution box, and water flow rate was measured with electromagnetic flow meters as shown in Fig. 3. The water film thickness was measured with confocal sensors (Precitec Messkopf 12 mm) at 1000 Hz sampling rate, which was mounted on a rail and can measure film thickness at any position on the reference line above the
where min
= (1
And the film width can be derived as
L=
2 (1 1
cos 2 cos
c) c
0.5
(8)
The current paper studied the film width and film thickness on an inclined plate under partially covered condition with experiment. And a semi-empirical model was proposed to predict the film width and film thickness with a single V-shaped fluid inlet, therefore for AP1000 type PCCS containment in which a series of V-shaped inlets on weir were used, this new model can be used to estimate the total film width on the containment surface. This paper mainly focused on the hydrodynamic behavior of water film, while its heat transfer behavior have been addressed in other paper (Hu et al., 2018). 3
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Table 2 Test parameters.
Flow direction
Number of tests
0.5m
1m
Water flow rate (kg/s)
Plate inclination angle(°)
Water film temperature (℃)
(a) Test parameters under partially covered condition 18 0.02–1.5 15, 45, 90
8–20
(b) Test parameters under fully covered condition 17 0.01–1.5 15, 45, 90
8–20
confocal sensor was calibrated with a set of standard optical flats and the relative error was shown in Fig. 5.
5m
2.2. Experiment procedure
1m
A test started with cleaning and drying out the plate surface, then the plate was adjusted to the designated inclination angle, after that the water flow rate was turned to designated value and the water film began to cover the plate. In the current study, two different water inlets were used, a single V-shaped inlet was used to study film behavior under the partially covered condition, and three adjacent V-shaped inlets were used for fully covered condition. After the water film coverage on the plate had been stabilized after 20 min from first appearance of flow on plate, the film width at six reference lines, and the film thickness at 6 positions on two reference lines as shown in Fig. 4 were measured, if the wetted width was decreased and the designated positions were uncovered due to the lower flow rate, three new evenly distributed positions would be used in the film covered zone on the same reference line. The water film temperatures were ambient temperatures and closely monitored. The test plan is shown in Table 2.
0.5m 0.5m 0.5m 1.23m Film thickness measuring location Film width measuring location Fig. 4. Measuring locations of film thickness and film width on the experiment plate.
3. Results and discussion
plate, while in the fully covered condition the thicknesses at 3 positions (0.3 m, 0.6 m, 0.9 m from the side of plate) on two reference lines were measured. The water film width, or wetted length was measured with laser guided ruler at six reference lines as shown in Fig. 4. All the measuring instruments were calibrated and the errors referring to actual readings were shown in the following table, and the
3.1. Water film width profile under partially covered condition Film width was measured at six reference lines as shown in Fig. 4, and the film width was gradually changing along the flow direction under partially covered condition. The width profiles in different inclination angles with various flow rates were shown in Fig. 6. Because
Fig. 5. Relative error of confocal sensor.
4
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Fig. 6. Water film width developments under various flow rates for three different inclination angles.
the impingement from the momentum component perpendicular to the plate of the outflow from V-shape inlet had a large impact on the film width formation, and in the lower inclination angle this momentum component was larger, a prematurely developed film width would form at the entry section, and then the film width in these cases decreased downstream the prematurely developed zone and then increased to a stable value. In most cases, the widths were stabilized only at a distance of 1 m to the flow outlet, which may indicate the free falling film need a distance around 4 m to reach a fully developed film width, therefore in the following analysis only the average film width and film thickness at this developed zone (1 m to the outlet) were used.
3.2. Water film width and film thickness in developed zone under partially covered condition The averaged water film thickness and film width at the developed zone at various flow rates for three different inclination angles are shown in Fig. 7. The flow rates in all three inclination angles started from small values, under which the film covered 20% or above of the plate, and increased to the values under which the film fully covered plate. In the tests with same inclination angle, the film thickness changed quite slowly with changing flow rate, the corresponding relative standard deviations (RSD) of three inclination angle cases were less than 10%. The film width increased linearly when the flow rates
Fig. 6. (continued)
5
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Fig. 6. (continued)
Fig. 7. Film thickness and film width at different inlet mass flow rates at partially covered condition for three different inclination angles.
increased for all three inclination angle cases. If a constant water film thickness can be assumed for the partialcovered film on the inclined plate, the slope of the linear curve of film width versus mass flow rate as shown in Fig. 7 can be estimated by ratio of added film width versus added mass flow rate as illustrated in Fig. 8(b)–(d). When the flow rate increases from Fig. 8(b) condition, the added mass flow rate can be assumed only to expand the film laterally and the film thickness remains the same. If a simple velocity profile is assumed as Eq. (1), then the ratio of added film width versus added
mass flow rate can be expressed as
L = m u
1 PCLFT
(9)
From Eq. (1) we can have
u = Then
6
gsin 3µ
2
(10)
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Fig. 7. (continued)
Fig. 7. (continued)
3µ L = 3 m gsin 2 PCLFT
to other inclination angle θ by replacing gravitation acceleration g with gsin(θ). And if MLFT is assumed as the constant film thickness when the film was expanding as proposed by El-Genk and Saber (2001) and Doniec (1991), then the predicted MLFT and corresponding film width, and derived slope can be compared with the experiment results as shown in following table.
(11)
A average film thickness is needed to estimate the ratio of added film width versus added flow rate in Eq. (11). As proposed by El-Genk, the MWR, MLFT and corresponding film width can be estimated with Eqs. (5), (6), (8) for 90° inclination angle using 49° contact angle for the experiment surface, and these equations were assumed to be applicable
7
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Fig. 8. Schematic of water film width expanding model and a measured water film profile. Table 3 Comparison of predicted MLFT, film width and width-flowrate slope based on El-Genk correlation and experiment results. (a) Correlation prediction Inclination angle (°)
Predicted MWR Eq. (5) (kg/s)
Predicted film width from Eq. (8) (m)
Predicted MLFT from Eq. (6)(um)
Predicted slope from Eq. (11) (m-s/kg)
15 45 90
0.000192 0.000102 0.0000696
0.00277 0.00182 0.00142
630 415 323
6.420 7.921 9.117
(b) Experiment results Inclination angle (°)
Film width from Fig. 7 experiment correlations with MWR predicted in (Table 3a) (m)
Measured film thickness (um)
Slope from Fig. 7 experiment correlations (m-s/ kg)
15 45 90
0.144 0.302 0.332
644 663 707
5.4113 1.8273 0.7389
(c) Comparison of Correlation prediction and experiment results Inclination angle (°)
Relative difference in film width
Relative difference in thickness
Relative difference in slope
15 45 90
−98.1% −99.4% −99.6%
−2.17% −37.41% −54.31%
18.65% 333.55% 1133.69%
The predicted slope, MLFT and film width were not in agreement with corresponding values from experiment except the MLFT for inclination angle at 15°. The reason that El-Genk’s correlation was not applicable here may be the uncertainty in its MWR prediction (Lu et al.,
2017), but more importantly was that the experiment results in Fig. 7 imply that even with infinitesimal flow rate, the film width will be much larger than the predicted value at MWR, therefore this correlation can’t extrapolate to the condition where the flow rate is larger than the 8
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Fig. 9. Comparison of predicted film widths and experiment values under partially covered condition.
MWR, and a new model is needed. Therefore we propose that when the flow rate increases from MWR, both the water film thickness and film width will increase until the film thickness reaches a Partially Covered Liquid Film Thickness (PCLFT), as illustrated from Fig. 8(a) to Fig. 8(b). If the flow rate increases further, the film thickness will stay constant as PCLFT, and the film width will expand linearly relative to the increasing flow rate, as illustrated from Fig. 8(b) to Fig. 8(c) and Fig. 8(d). And the film thickness will increase again when the film reaches the side walls or the test section is fully covered, and film behavior under this condition will be discussed in the next section. If we assume that the film thickness profile b ( , rb) of the water film at Fig. 8(b) can be correlated to the thickness profile a ( , ra ) of Fig. 8(a) as a( b(
, ra ) r sin( ) = a , rb) rbsin( )
MLFT PCLFT
Fig. 10. (continued)
(12)
Fig. 10. (continued)
Therefore the film thickness and film width from Fig. 8(a) and Fig. 8(b) can be correlated as
Fig. 10. Comparison of film thickness from experiment measurements and Nusselt theory, Lel correlation predictions for three different inclination angles under fully covered condition.
a (0, ra) b (0,rb)
as 9
=
La = Lb
MLFT PCLFT
(13)
And the mass flow rate from Fig. 8(a) and Fig. 8(b) can be correlated
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Fig. 11. Comparison of nondimensional film thickness from current and literature experiments and Nusselt theory predicts and four other correlations under fully covered condition.
Fig. 12. Comparison of nondimensional film thickness from current and literature experiments and Nusselt theory predicts under fully covered condition.
2 ma = mb 2
0
La 2
0
Lb 2
And for growing thickness region from Fig. 8(a) to Fig. 8(b), with the assumption of Eq. (12), Eq. (15) can be used to predict the film width as following:
a (x ) uadx b (x ) ubdx
(14)
By introducing Eq. (10), the Eq. (14) can be rewritten as
ma = mb
0 0
La 2 Lb 2
2 a (x ) a dx
2 2 2 a 0 ra sin (
)sin( ) d
2 b (x ) b dx
2 2 2 b 0 rb sin (
)sin( ) d
=
4 MLFT 4 PCLFT
L = LMWR 4
Lb =
L (m m
mb)
(17)
The parameters in the prediction Eq. (16) can be determined if is known, and it can be estimated with an empirical method as following. Firstly an average film thickness from all the thickness measurements is used to estimate the PCLFT , then the corresponding mass flow rate mtemp and film width Ltemp were calculated with Eq. (15) and Eq. (17). For only thickness measurements in constant thickness region should be used, the PCLFT is estimated again with the thickness measurements whose mass flow rate larger than mtemp , the iteration continues until estimated PCLFT is not changed. The current measurements for all inclination angles were with mass flow rate larger than corresponding mtemp , and this is the reason of the linear relation
(15)
PCLFT
For La and ma are film width and MWR estimated from Eqs. (5)–(8), with Eq. (11), Eq. (13) and Eq. (15), a linear correlation can be formulated to predict the film width under partial coverage condition for constant thickness ( PCLFT ) region from Fig. 8(b) to Fig. 8(d) as following:
L
m mMWR
(16) 10
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Fig. 13. Comparison of Karapantsios film thickness data measured at 1.72 m and 2.45 m with Nusselt theory prediction under fully covered condition.
between mass flow rate and film width for all test data in Fig. 7. Then a thickness ratio is defined as y = MLFT / PCLFT and the ratio values can be estimated based on the data in Table 3 for three different inclination angles. Therefore a correlation can be formulated as following.
y=
than 550 were larger than the Nusselt theory predictions and in agreement with Lel correlation. But current study showed film thickness measurements were in agreement with Nusselt theory predictions for Reynolds up to 900, and current study measurements for Reynolds larger than 900 were in agreement to Lel correlation for Reynolds up to 1350. One possible reason for the shifting of Reynolds value from 550 to 900 as a criteria for choosing the appropriate predicting methods between Nusselt theory and Lel correlation, is that the difference of measuring positions relative to the inlet, this indicates that the film is not well developed at near entry location. As shown in Fig. 12. Lel’s data were measured at 1.2 m downstream from the inlet, but the data in current study were measured at locations over 4 m downstream from the inlet. The similar shifting pattern can be found in tube tests as shown in Fig. 13 in which thickness measured at 1.72 m deviated from Nusselt theory prediction at Re = 450, and thickness measured at 2.45 m deviated from Nusselt theory prediction at Re = 620. This suggest that when Reynolds increases the film flow will deviate from a laminar pattern as Nusselt theory indicated, and the upstream of the film will deviate earlier than the downstream ones due to the entry affect. Based on the discussion, a film thickness prediction strategy on a fully covered plate can be recommended as following: As shown in Fig. 12, the average film thickness can be predicted with Nusselt theory closely (around 10% deviation) at lower Reynolds, and can be predicted with Lel correlation at higher Reynolds (up to Reynolds equal to 1400), while the transition Reynolds is related to the measuring location downstream relative to the inlet, for measuring location around 1.5 m, the transition Reynolds is around 550, and for measuring location around 4 m, the transition Reynolds is about 900.
(18)
0.277ln( ) + 0.6071
and PCLFT
=
MLFT (
0.277ln( ) + 0.6071)(0 <
90°)
(19)
Then a plot of predicted film width versus increasing flow rate is shown in Fig. 9. This prediction method with Eq. (16) and Eq. (19) can estimate the film width under partial coverage condition with less than 20% error. 3.3. Water film thickness under fully covered condition In the current study, water film thickness was measured at locations 4 m downstream from the inlet, and the average thickness was increased with increasing flow rate under full coverage condition as shown in Fig. 10. Comparing to the Nusselt theory prediction Eq. (2) and Lel correlation prediction as shown in Table 1, the film thickness with Reynolds less than 900 were in better agreement with Nusselt theory, but thickness with Reynolds larger than 900 were in better agreement with Lel correlation. For Lel correlation was designed to accommodate the larger film thickness better than Nusselt theory when Reynolds was larger than 600. The current experiment results were also compared with similar experiment data and correlations in the literature as shown in Fig. 11. In order to compare experiment film thickness data from different fluids and different inclination angles, a dimensionless film thickness was used as defined in Eq. (3). The film thickness data from Karapantsios, and predictions from Karapantsios correlation and Takahama correlation were based on tube tests, and they were slightly different from data from plate tests, and the thickness results from tube tests were smaller than that of plate tests for Reynolds less than 300, and larger than that of plate tests for Reynolds larger than 550. All film thickness data from experiments on plate, such as Takamasa and Hazuku (2000) (measuring location 0.366 m downstream to the inlet), Ambrosini et al. (2002); Lel et al. and current study, were in agreement with predictions from Nusselt theory, Lel correlation, and Huang correlation for Reynolds less than 550. Lel et al.’s film thickness measurements for Reynolds larger
4. Conclusion The free falling film width and film average thickness were examined in the current study under partially film-covered condition and fully covered condition on a large inclined plate. As the flow rate increases from minimum wetting rate (MWR) under partially covered condition, the film width will expand, and the film thickness will increase to a partially covered liquid film thickness (PCLFT) instead of staying at minimum liquid film thickness (MLFT) as proposed in the literature. Then the film width will expand linearly relative to the increasing flow rate until the film fully covers the plate. A corresponding 11
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semi-empirical correlation was developed to predict the film width and film thickness, and the width prediction is in good agreement with experiment measurements in the developed zone, which is 4 m downstream to the fluid inlet. Once the film covered the plate fully, the film thickness will increase with increasing flow rate as Nusselt theory predicts at low Reynolds, at higher Reynolds the film will deviate from laminar pattern as Nusselt theory assumed and film thickness grows larger than the Nusselt theory predicts and can be predicted by Lel correlation. The transition Reynolds from the Nusselt theory to the Lel correlation is related to the measuring location relative to the fluid inlet, and the upstream film deviates earlier than the downstream one.
References Ambrosini, W., Forgione, N., Oriolo, F., 2002. Statistical characteristics of a water film falling down a flat plate at a different inclination and temperatures. Int. J. Multiph. Flow 28, 1521–1540. Doniec, A., 1991. Laminar flow of a liquid rivulet down a vertical solid surface. Can. J. Chem. Eng. 69 (2), 198–202. El-Genk, M.S., Saber, H.H., 2001. Minimum Thickness of a flowing down liquid film on a vertical surface. Int. J. Heat Mass Transfer 44, 2809–2825. Hu, P., Du, K.S., Zhai, S.W., Yang, Y.H., 2018. Experiment study of water film/air countercurrent flow heat transfer on a vertical plate for passive containment cooling system. Nucl. Eng. Des. 328 (3), 73–79. Huang, X.G., Yang, Y.H., Hu, P., Bao, K., 2014. Experimental study of water–air countercurrent flow characteristics in large scale rectangular channel. Ann. Nucl. Energy 69, 125–133. Karapantsios, T.D., Karabelas, A.J., 1995. Longitudinal characteristic of wavy falling films. Int. J. Multiph. Flow 21 (1), 119–127. Lan, H., Wegener, J.L., Armaly, B.F., Drallmeier, J.A., 2010. Developing laminar gravitydriven thin liquid film flow down an inclined plane. J. Fluids Eng. 132 (8) 08130101-081301-8. Lel, V.V., Al-Sibai, F., Leefken, A., Renz, U., 2005. Local thickness and wave velocity measurement of wavy films with a chromatic confocal imaging method and a fluorescence intensity technique. Exp. Fluids 39, 856–864. Lu, Y., Stehmann, F., Yuan, S., Scholl, S., 2017. Falling film on a vertical flat plate – Influence of liquid distribution and fluid properties one wetting behavior. Appl. Therm. Eng. 123, 1386–1395. Nusselt, W.K. 1916. Die oberflachnek ondensation des wasserdamphes. VDI-Zs 60,541. Sha, W., Chien, T.H., Sun, J.G., Chao, B.T., 2004. Analysis of large-scale test for AP-600 passive containment cooling system. Nucl. Eng. Des. 232, 197–216. Takahama, H., Kato, S., 1980. Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow. Int. J. Multiph. Flow 6, 203–215. Takamasa, T., Hazuku, T., 2000. Measuring interfacial waves on film flowing down a vertical plate wall in the entry region using laser focus displacement meters. Int. J. Heat Mass Transfer 43, 2807–2819. Yu, Y.Q., Wei, S.J., Yang, Y.H., Cheng, X., 2012. Experimental study of water film falling and spreading on a large vertical plate. Prog. Nucl. Energy 54, 22–28.
CRediT authorship contribution statement Po Hu: Conceptualization, Methodology, Writing - review & editing, Project administration. Xingguan Huang: Data curation. Kai Bao: Data curation. Guixue Zhu: Data curation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The Test facility was supported by State Key Science and Technology Projects (Project number: 2011ZX06002-005).
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Experiment study on film width and thickness of free falling water film on a large inclined plate
T
Po Hu , Xingguan Huang, Kai Bao, Guixue Zhu ⁎
School of Nuclear Science and Engineering, Shanghai Jiao Tong University, 800 Dongchuan Rd, Shanghai, China
ARTICLE INFO
ABSTRACT
Keywords: Free falling film Film thickness Film width Film wettability
An experiment was setup to study film width and film thickness of a free falling water film on a large inclined steel plate with no heat transfer. Based on results a semi-empirical correlation was developed to predict the film width and film thickness for partially film-covered condition, and the prediction of the film width was in agreement, to within ± 20%, with the experiment results. And different film thickness correlations were examined with experiment data under fully film-covered condition, and it showed that at lower Reynolds the Nusselt theory predicted the average film thickness well and at higher Reynolds the Lel correlation predicted better. The transition region of Reynolds between these two was related to the measuring location relative to the liquid inlet.
1. Introduction
=
Free falling liquid film has been used widely in different industrial applications, such as tube evaporator, wetted-wall absorber and passive cooling containment system (PCCS) in nuclear power plant. As shown in Fig. 1, PCCS is one of the passive safety systems adopted by Westinghouse in their AP series nuclear reactor designs (Sha et al., 2004). It uses the counter-current natural circulating air flow and gravity-driven water film to cool down the containment steel wall during accidents. To uniformly distribute the water film, a weir with V-shaped inlets was used. The weir design was based on experiment to assure the film coverage above the design limit. In order to better design equipment using falling film, a thorough study on heat transfer and hydrodynamic behavior of liquid film is needed, which may include the film thickness development, film wettability, and heat transfer capability under a wide range of Reynolds number. The current paper studied water film width and film thickness under partially covered and fully covered conditions on a plate with different inclination angles. Nusselt (1916) has studied free falling film analytically based on a 2D, shear-free interface model, and he predicted the falling film thickness and film velocity as following,
u(y) =
⁎
gsin ( y µ
y2 ) 2
(1)
3µm L 2 gsin
1/3
(2)
or in dimensionless form
= (3Re )1/3 = (
Re =
m Lµ
2 g sin
µ2
)1/3
(3) (4)
As shown in Fig. 2, the u(y) is the velocity profile in the film and um is average velocity over the film thickness, and y is the distance away from the wall surface vertically. δ is the thickness of the film, ρ is the density, μ is the viscosity, g is the gravitational acceleration, θ is the inclination angle from the horizontal, m is the film mass flow rate, and L is the film width or wetted perimeter. Selected free falling film experiments and reported film thickness correlations are shown in the Table 1. As shown in the Fig. 11, for film thickness tests in tube, the Takahama and Karapantsios correlations predict dimensionless thickness with up to 40% difference comparing to Nusselt theory for Reynolds less than 600. For film thickness tests on plate, Lel correlation predicts the thickness within 10% difference comparing to Nusselt theory for Reynolds less than 600, and it deviates distinctively from Nusselt theory for Reynolds larger than 600. Huang correlation is quite close to the Nusselt theory up to Reynolds equal to 2000, and the difference is less than 10%. Therefore it seems that the average film thickness on a fully covered plate is predicted well with
Corresponding author. E-mail address: [email protected] (P. Hu).
https://doi.org/10.1016/j.nucengdes.2019.110445 Received 27 May 2019; Received in revised form 14 November 2019; Accepted 14 November 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
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kinetic viscosity (m2/s) polar angle (°)
Nomenclature g L m Re x y u
acceleration of gravity (m/s2) film width (m) mass flow rate (kg/s) m Reynolds number Re = Lµ coordinate along the surface (m) coordinate perpendicular to the surface (m) or thickness ratio equal to MLFT / PCLFT film velocity (m/s)
Subscript a b m mean min
Greek symbols ρ Δ
c
µ
the MWR point the beginning of the constant film thickness region, corresponding to the lowest flow rate with constant thickness average value average value minimum
Abbreviation
density (kg/m3) film thickness (m) dimensionless minimum liquid film thickness (m) surface tension (N/m) inclination angle (°) contact angle (°) dynamic viscosity (kg/s-m)
MLFT MWR PCCS PCLFT RSD
minimum liquid film thickness minimum wetting rate passive containment cooling system partially covered liquid film thickness Relative standard deviation
Nusselt theory at lower Reynolds less than 600, and at higher Reynolds, the thickness will deviate from Nusselt theory and approach Karapantsios and Lel correlations. The current study examined the phenomenon with experiment for Reynolds more than 1000. All the tests above were conducted with test section fully covered by the fluid film, when the flow rate starts to decrease, at beginning the water film thickness will decrease, then at certain flow rate, the water film width will start to decrease and the surface will be partially covered by a continuous film, as shown by Yu et al. (2012). But few studies in the literature studied film behavior under this partially film-covered condition, one of these is the study of Lan et al. (2010), which studied free falling liquid film thickness and width with experiment, and the data was compared to the CFD calculation, and the test section dimension was limited as 0.10 m × 0.26 m. When the flow rate decreases further, a specific minimum wetting rate (MWR) will be reached, which is the lowest flow rate needed to ensure that surface remains covered by a continuous liquid film. The MWR attracted a lot of attention from different branches of scientific community, and various theories have been proposed such as force balance and minimum total energy
Fig. 2. Schematic of a 2D liquid film flowing down a vertical wall.
criterions. El-Genk and Saber (2001) compared four different analytical expressions for MWR and minimum liquid film thickness (MLFT), and proposed a set of new equations for vertical surface as following, and c is the contact angle
Fig. 1. Schematic of AP1000 passive containment cooling system (image courtesy of Westinghouse Electric Corporation). 2
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Table 1 Selected free falling film experiments and film thickness correlations. Reference
Reported film thickness correlations
Takahama and Kato (1980)
= 0.473Re 0.526
Karapantsios and Karabelas (1995)
= 0.451Re 0.538
Lel et al. (2005)
= 1 + 0.615Re0.47
Huang et al. (2014)
= 1.666Re 0.322
Comments Tests in Tube Measuring location 1.7 m from inlet Tests in Tube Measuring location 1.7–2.6 m from inlet Tests on plate Measuring location 1.2 m from inlet Tests on plate Measuring locations 1–4.5 m from inlet
Fig. 3. Schematic of experiment apparatus, measuring equipment, and weir plate with V-shaped inlets.
m=L
µ g
3 0.2
15µ2 = ( 3 2 )0.2 g
(0.67
2.83 min
+ 0.26
9.51 min)
2. Experiment (5)
min
(6)
cos c )0.22
(7)
2.1. Experiment apparatus Experiment apparatus schematic is shown in Fig. 3. A steel plane plate (5 × 1.23 × 0.04 m) is carefully manufactured to maintain a good planarity, and then painted with nonorganic zinc paint (Carbozinc 11 HSN). The plate is mounted on a tilting frame which can stay at any inclination angle from horizontal to vertical position. Therefore the water film behavior on the plate with different inclination angles can be used to study the similar phenomena of different parts on the containment dome area. The water delivery system consists of a supply tank, flow meter and valves, distribution box and drain. The distribution box is adjustable to test different shape and arrangement of inlets. A plexiglass baffle plate is mounted parallel to the steel plate, and the gap between them is 30 cm. Water temperature was measured with T type thermocouple at the inlet of the water distribution box, and water flow rate was measured with electromagnetic flow meters as shown in Fig. 3. The water film thickness was measured with confocal sensors (Precitec Messkopf 12 mm) at 1000 Hz sampling rate, which was mounted on a rail and can measure film thickness at any position on the reference line above the
where min
= (1
And the film width can be derived as
L=
2 (1 1
cos 2 cos
c) c
0.5
(8)
The current paper studied the film width and film thickness on an inclined plate under partially covered condition with experiment. And a semi-empirical model was proposed to predict the film width and film thickness with a single V-shaped fluid inlet, therefore for AP1000 type PCCS containment in which a series of V-shaped inlets on weir were used, this new model can be used to estimate the total film width on the containment surface. This paper mainly focused on the hydrodynamic behavior of water film, while its heat transfer behavior have been addressed in other paper (Hu et al., 2018). 3
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Table 2 Test parameters.
Flow direction
Number of tests
0.5m
1m
Water flow rate (kg/s)
Plate inclination angle(°)
Water film temperature (℃)
(a) Test parameters under partially covered condition 18 0.02–1.5 15, 45, 90
8–20
(b) Test parameters under fully covered condition 17 0.01–1.5 15, 45, 90
8–20
confocal sensor was calibrated with a set of standard optical flats and the relative error was shown in Fig. 5.
5m
2.2. Experiment procedure
1m
A test started with cleaning and drying out the plate surface, then the plate was adjusted to the designated inclination angle, after that the water flow rate was turned to designated value and the water film began to cover the plate. In the current study, two different water inlets were used, a single V-shaped inlet was used to study film behavior under the partially covered condition, and three adjacent V-shaped inlets were used for fully covered condition. After the water film coverage on the plate had been stabilized after 20 min from first appearance of flow on plate, the film width at six reference lines, and the film thickness at 6 positions on two reference lines as shown in Fig. 4 were measured, if the wetted width was decreased and the designated positions were uncovered due to the lower flow rate, three new evenly distributed positions would be used in the film covered zone on the same reference line. The water film temperatures were ambient temperatures and closely monitored. The test plan is shown in Table 2.
0.5m 0.5m 0.5m 1.23m Film thickness measuring location Film width measuring location Fig. 4. Measuring locations of film thickness and film width on the experiment plate.
3. Results and discussion
plate, while in the fully covered condition the thicknesses at 3 positions (0.3 m, 0.6 m, 0.9 m from the side of plate) on two reference lines were measured. The water film width, or wetted length was measured with laser guided ruler at six reference lines as shown in Fig. 4. All the measuring instruments were calibrated and the errors referring to actual readings were shown in the following table, and the
3.1. Water film width profile under partially covered condition Film width was measured at six reference lines as shown in Fig. 4, and the film width was gradually changing along the flow direction under partially covered condition. The width profiles in different inclination angles with various flow rates were shown in Fig. 6. Because
Fig. 5. Relative error of confocal sensor.
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Fig. 6. Water film width developments under various flow rates for three different inclination angles.
the impingement from the momentum component perpendicular to the plate of the outflow from V-shape inlet had a large impact on the film width formation, and in the lower inclination angle this momentum component was larger, a prematurely developed film width would form at the entry section, and then the film width in these cases decreased downstream the prematurely developed zone and then increased to a stable value. In most cases, the widths were stabilized only at a distance of 1 m to the flow outlet, which may indicate the free falling film need a distance around 4 m to reach a fully developed film width, therefore in the following analysis only the average film width and film thickness at this developed zone (1 m to the outlet) were used.
3.2. Water film width and film thickness in developed zone under partially covered condition The averaged water film thickness and film width at the developed zone at various flow rates for three different inclination angles are shown in Fig. 7. The flow rates in all three inclination angles started from small values, under which the film covered 20% or above of the plate, and increased to the values under which the film fully covered plate. In the tests with same inclination angle, the film thickness changed quite slowly with changing flow rate, the corresponding relative standard deviations (RSD) of three inclination angle cases were less than 10%. The film width increased linearly when the flow rates
Fig. 6. (continued)
5
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Fig. 6. (continued)
Fig. 7. Film thickness and film width at different inlet mass flow rates at partially covered condition for three different inclination angles.
increased for all three inclination angle cases. If a constant water film thickness can be assumed for the partialcovered film on the inclined plate, the slope of the linear curve of film width versus mass flow rate as shown in Fig. 7 can be estimated by ratio of added film width versus added mass flow rate as illustrated in Fig. 8(b)–(d). When the flow rate increases from Fig. 8(b) condition, the added mass flow rate can be assumed only to expand the film laterally and the film thickness remains the same. If a simple velocity profile is assumed as Eq. (1), then the ratio of added film width versus added
mass flow rate can be expressed as
L = m u
1 PCLFT
(9)
From Eq. (1) we can have
u = Then
6
gsin 3µ
2
(10)
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Fig. 7. (continued)
Fig. 7. (continued)
3µ L = 3 m gsin 2 PCLFT
to other inclination angle θ by replacing gravitation acceleration g with gsin(θ). And if MLFT is assumed as the constant film thickness when the film was expanding as proposed by El-Genk and Saber (2001) and Doniec (1991), then the predicted MLFT and corresponding film width, and derived slope can be compared with the experiment results as shown in following table.
(11)
A average film thickness is needed to estimate the ratio of added film width versus added flow rate in Eq. (11). As proposed by El-Genk, the MWR, MLFT and corresponding film width can be estimated with Eqs. (5), (6), (8) for 90° inclination angle using 49° contact angle for the experiment surface, and these equations were assumed to be applicable
7
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Fig. 8. Schematic of water film width expanding model and a measured water film profile. Table 3 Comparison of predicted MLFT, film width and width-flowrate slope based on El-Genk correlation and experiment results. (a) Correlation prediction Inclination angle (°)
Predicted MWR Eq. (5) (kg/s)
Predicted film width from Eq. (8) (m)
Predicted MLFT from Eq. (6)(um)
Predicted slope from Eq. (11) (m-s/kg)
15 45 90
0.000192 0.000102 0.0000696
0.00277 0.00182 0.00142
630 415 323
6.420 7.921 9.117
(b) Experiment results Inclination angle (°)
Film width from Fig. 7 experiment correlations with MWR predicted in (Table 3a) (m)
Measured film thickness (um)
Slope from Fig. 7 experiment correlations (m-s/ kg)
15 45 90
0.144 0.302 0.332
644 663 707
5.4113 1.8273 0.7389
(c) Comparison of Correlation prediction and experiment results Inclination angle (°)
Relative difference in film width
Relative difference in thickness
Relative difference in slope
15 45 90
−98.1% −99.4% −99.6%
−2.17% −37.41% −54.31%
18.65% 333.55% 1133.69%
The predicted slope, MLFT and film width were not in agreement with corresponding values from experiment except the MLFT for inclination angle at 15°. The reason that El-Genk’s correlation was not applicable here may be the uncertainty in its MWR prediction (Lu et al.,
2017), but more importantly was that the experiment results in Fig. 7 imply that even with infinitesimal flow rate, the film width will be much larger than the predicted value at MWR, therefore this correlation can’t extrapolate to the condition where the flow rate is larger than the 8
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Fig. 9. Comparison of predicted film widths and experiment values under partially covered condition.
MWR, and a new model is needed. Therefore we propose that when the flow rate increases from MWR, both the water film thickness and film width will increase until the film thickness reaches a Partially Covered Liquid Film Thickness (PCLFT), as illustrated from Fig. 8(a) to Fig. 8(b). If the flow rate increases further, the film thickness will stay constant as PCLFT, and the film width will expand linearly relative to the increasing flow rate, as illustrated from Fig. 8(b) to Fig. 8(c) and Fig. 8(d). And the film thickness will increase again when the film reaches the side walls or the test section is fully covered, and film behavior under this condition will be discussed in the next section. If we assume that the film thickness profile b ( , rb) of the water film at Fig. 8(b) can be correlated to the thickness profile a ( , ra ) of Fig. 8(a) as a( b(
, ra ) r sin( ) = a , rb) rbsin( )
MLFT PCLFT
Fig. 10. (continued)
(12)
Fig. 10. (continued)
Therefore the film thickness and film width from Fig. 8(a) and Fig. 8(b) can be correlated as
Fig. 10. Comparison of film thickness from experiment measurements and Nusselt theory, Lel correlation predictions for three different inclination angles under fully covered condition.
a (0, ra) b (0,rb)
as 9
=
La = Lb
MLFT PCLFT
(13)
And the mass flow rate from Fig. 8(a) and Fig. 8(b) can be correlated
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Fig. 11. Comparison of nondimensional film thickness from current and literature experiments and Nusselt theory predicts and four other correlations under fully covered condition.
Fig. 12. Comparison of nondimensional film thickness from current and literature experiments and Nusselt theory predicts under fully covered condition.
2 ma = mb 2
0
La 2
0
Lb 2
And for growing thickness region from Fig. 8(a) to Fig. 8(b), with the assumption of Eq. (12), Eq. (15) can be used to predict the film width as following:
a (x ) uadx b (x ) ubdx
(14)
By introducing Eq. (10), the Eq. (14) can be rewritten as
ma = mb
0 0
La 2 Lb 2
2 a (x ) a dx
2 2 2 a 0 ra sin (
)sin( ) d
2 b (x ) b dx
2 2 2 b 0 rb sin (
)sin( ) d
=
4 MLFT 4 PCLFT
L = LMWR 4
Lb =
L (m m
mb)
(17)
The parameters in the prediction Eq. (16) can be determined if is known, and it can be estimated with an empirical method as following. Firstly an average film thickness from all the thickness measurements is used to estimate the PCLFT , then the corresponding mass flow rate mtemp and film width Ltemp were calculated with Eq. (15) and Eq. (17). For only thickness measurements in constant thickness region should be used, the PCLFT is estimated again with the thickness measurements whose mass flow rate larger than mtemp , the iteration continues until estimated PCLFT is not changed. The current measurements for all inclination angles were with mass flow rate larger than corresponding mtemp , and this is the reason of the linear relation
(15)
PCLFT
For La and ma are film width and MWR estimated from Eqs. (5)–(8), with Eq. (11), Eq. (13) and Eq. (15), a linear correlation can be formulated to predict the film width under partial coverage condition for constant thickness ( PCLFT ) region from Fig. 8(b) to Fig. 8(d) as following:
L
m mMWR
(16) 10
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Fig. 13. Comparison of Karapantsios film thickness data measured at 1.72 m and 2.45 m with Nusselt theory prediction under fully covered condition.
between mass flow rate and film width for all test data in Fig. 7. Then a thickness ratio is defined as y = MLFT / PCLFT and the ratio values can be estimated based on the data in Table 3 for three different inclination angles. Therefore a correlation can be formulated as following.
y=
than 550 were larger than the Nusselt theory predictions and in agreement with Lel correlation. But current study showed film thickness measurements were in agreement with Nusselt theory predictions for Reynolds up to 900, and current study measurements for Reynolds larger than 900 were in agreement to Lel correlation for Reynolds up to 1350. One possible reason for the shifting of Reynolds value from 550 to 900 as a criteria for choosing the appropriate predicting methods between Nusselt theory and Lel correlation, is that the difference of measuring positions relative to the inlet, this indicates that the film is not well developed at near entry location. As shown in Fig. 12. Lel’s data were measured at 1.2 m downstream from the inlet, but the data in current study were measured at locations over 4 m downstream from the inlet. The similar shifting pattern can be found in tube tests as shown in Fig. 13 in which thickness measured at 1.72 m deviated from Nusselt theory prediction at Re = 450, and thickness measured at 2.45 m deviated from Nusselt theory prediction at Re = 620. This suggest that when Reynolds increases the film flow will deviate from a laminar pattern as Nusselt theory indicated, and the upstream of the film will deviate earlier than the downstream ones due to the entry affect. Based on the discussion, a film thickness prediction strategy on a fully covered plate can be recommended as following: As shown in Fig. 12, the average film thickness can be predicted with Nusselt theory closely (around 10% deviation) at lower Reynolds, and can be predicted with Lel correlation at higher Reynolds (up to Reynolds equal to 1400), while the transition Reynolds is related to the measuring location downstream relative to the inlet, for measuring location around 1.5 m, the transition Reynolds is around 550, and for measuring location around 4 m, the transition Reynolds is about 900.
(18)
0.277ln( ) + 0.6071
and PCLFT
=
MLFT (
0.277ln( ) + 0.6071)(0 <
90°)
(19)
Then a plot of predicted film width versus increasing flow rate is shown in Fig. 9. This prediction method with Eq. (16) and Eq. (19) can estimate the film width under partial coverage condition with less than 20% error. 3.3. Water film thickness under fully covered condition In the current study, water film thickness was measured at locations 4 m downstream from the inlet, and the average thickness was increased with increasing flow rate under full coverage condition as shown in Fig. 10. Comparing to the Nusselt theory prediction Eq. (2) and Lel correlation prediction as shown in Table 1, the film thickness with Reynolds less than 900 were in better agreement with Nusselt theory, but thickness with Reynolds larger than 900 were in better agreement with Lel correlation. For Lel correlation was designed to accommodate the larger film thickness better than Nusselt theory when Reynolds was larger than 600. The current experiment results were also compared with similar experiment data and correlations in the literature as shown in Fig. 11. In order to compare experiment film thickness data from different fluids and different inclination angles, a dimensionless film thickness was used as defined in Eq. (3). The film thickness data from Karapantsios, and predictions from Karapantsios correlation and Takahama correlation were based on tube tests, and they were slightly different from data from plate tests, and the thickness results from tube tests were smaller than that of plate tests for Reynolds less than 300, and larger than that of plate tests for Reynolds larger than 550. All film thickness data from experiments on plate, such as Takamasa and Hazuku (2000) (measuring location 0.366 m downstream to the inlet), Ambrosini et al. (2002); Lel et al. and current study, were in agreement with predictions from Nusselt theory, Lel correlation, and Huang correlation for Reynolds less than 550. Lel et al.’s film thickness measurements for Reynolds larger
4. Conclusion The free falling film width and film average thickness were examined in the current study under partially film-covered condition and fully covered condition on a large inclined plate. As the flow rate increases from minimum wetting rate (MWR) under partially covered condition, the film width will expand, and the film thickness will increase to a partially covered liquid film thickness (PCLFT) instead of staying at minimum liquid film thickness (MLFT) as proposed in the literature. Then the film width will expand linearly relative to the increasing flow rate until the film fully covers the plate. A corresponding 11
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semi-empirical correlation was developed to predict the film width and film thickness, and the width prediction is in good agreement with experiment measurements in the developed zone, which is 4 m downstream to the fluid inlet. Once the film covered the plate fully, the film thickness will increase with increasing flow rate as Nusselt theory predicts at low Reynolds, at higher Reynolds the film will deviate from laminar pattern as Nusselt theory assumed and film thickness grows larger than the Nusselt theory predicts and can be predicted by Lel correlation. The transition Reynolds from the Nusselt theory to the Lel correlation is related to the measuring location relative to the fluid inlet, and the upstream film deviates earlier than the downstream one.
References Ambrosini, W., Forgione, N., Oriolo, F., 2002. Statistical characteristics of a water film falling down a flat plate at a different inclination and temperatures. Int. J. Multiph. Flow 28, 1521–1540. Doniec, A., 1991. Laminar flow of a liquid rivulet down a vertical solid surface. Can. J. Chem. Eng. 69 (2), 198–202. El-Genk, M.S., Saber, H.H., 2001. Minimum Thickness of a flowing down liquid film on a vertical surface. Int. J. Heat Mass Transfer 44, 2809–2825. Hu, P., Du, K.S., Zhai, S.W., Yang, Y.H., 2018. Experiment study of water film/air countercurrent flow heat transfer on a vertical plate for passive containment cooling system. Nucl. Eng. Des. 328 (3), 73–79. Huang, X.G., Yang, Y.H., Hu, P., Bao, K., 2014. Experimental study of water–air countercurrent flow characteristics in large scale rectangular channel. Ann. Nucl. Energy 69, 125–133. Karapantsios, T.D., Karabelas, A.J., 1995. Longitudinal characteristic of wavy falling films. Int. J. Multiph. Flow 21 (1), 119–127. Lan, H., Wegener, J.L., Armaly, B.F., Drallmeier, J.A., 2010. Developing laminar gravitydriven thin liquid film flow down an inclined plane. J. Fluids Eng. 132 (8) 08130101-081301-8. Lel, V.V., Al-Sibai, F., Leefken, A., Renz, U., 2005. Local thickness and wave velocity measurement of wavy films with a chromatic confocal imaging method and a fluorescence intensity technique. Exp. Fluids 39, 856–864. Lu, Y., Stehmann, F., Yuan, S., Scholl, S., 2017. Falling film on a vertical flat plate – Influence of liquid distribution and fluid properties one wetting behavior. Appl. Therm. Eng. 123, 1386–1395. Nusselt, W.K. 1916. Die oberflachnek ondensation des wasserdamphes. VDI-Zs 60,541. Sha, W., Chien, T.H., Sun, J.G., Chao, B.T., 2004. Analysis of large-scale test for AP-600 passive containment cooling system. Nucl. Eng. Des. 232, 197–216. Takahama, H., Kato, S., 1980. Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow. Int. J. Multiph. Flow 6, 203–215. Takamasa, T., Hazuku, T., 2000. Measuring interfacial waves on film flowing down a vertical plate wall in the entry region using laser focus displacement meters. Int. J. Heat Mass Transfer 43, 2807–2819. Yu, Y.Q., Wei, S.J., Yang, Y.H., Cheng, X., 2012. Experimental study of water film falling and spreading on a large vertical plate. Prog. Nucl. Energy 54, 22–28.
CRediT authorship contribution statement Po Hu: Conceptualization, Methodology, Writing - review & editing, Project administration. Xingguan Huang: Data curation. Kai Bao: Data curation. Guixue Zhu: Data curation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The Test facility was supported by State Key Science and Technology Projects (Project number: 2011ZX06002-005).
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