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Analysis of fluctuation and breakdown characteristics of liquid film on corrugated plate wall
Annals of Nuclear Energy 135 (2020) 106946
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Analysis of fluctuation and breakdown characteristics of liquid film on corrugated plate wall Bo Wang, Bowen Chen, Ru Li, Ruifeng Tian ⇑ Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 17 May 2019 Received in revised form 16 July 2019 Accepted 19 July 2019
Keywords: PLIF Fluctuation characteristics Liquid film thickness Steam water separation Corrugated plate
a b s t r a c t Corrugated plate dryer is a very vital separation equipment for nuclear engineering and it is meaningful to study the flow, fluctuation and breakdown characteristics of the liquid film on corrugated plate wall. Liquid film thickness of steady flow is measured based on plane laser induced fluorescence (PLIF) technique and time series of liquid film thickness are obtained. Besides, amplitude and frequency domain information of liquid film fluctuations are analyzed by probability density function (PDF), power spectral density (PSD) and wavelet method. A two-dimensional liquid film breakdown model is established. Critical airflow velocity when the water film breakdown is measured experimentally. Based on theoretical results, equations for calculating critical airflow velocity of the corrugated plate under different liquid film thicknesses are fitted in the form of nike function. Fitting equations for calculating critical airflow velocity in the form of nike function agree well with experimental results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Internal flow in the corrugated plate is a typical multiphase flow problem due to the presence of gas, liquid film, and corrugated wall surfaces. Research on the method of improving the separation efficiency of corrugated plates is a hot research topic. The secondary carrying phenomenon means that when the inlet gas velocity of the corrugated plate rises to a certain extent, the separation efficiency does not increase but decreases. The internal secondary carrying phenomenon is a key factor affecting the separation efficiency of the corrugated plate. This phenomenon occurs because after the steam is separated once, large droplets are trapped by the wall surface and some of the droplets that have been captured may re-enter the flow channel under the action of the airflow to continue moving. The research on the characteristics of the liquid film on the wall under the action of no airflow is the research basis of the separation mechanism of the corrugated plate separator under the action of airflow. Therefore, it is of great value to study the flow and breakdown characteristics of the liquid film on the corrugated plate wall. In the experiment, the water film breakdown phenomenon is commonly observed at the corner of corrugated plate. Thus the corrugated plate consisting of many plates is used instead of using only one flat plate in the research. For the aspect for free falling film research, in 1916, Nusselt firstly ⇑ Corresponding author. E-mail address: [email protected] (R. Tian). https://doi.org/10.1016/j.anucene.2019.106946 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
studied the flow characteristics of the free falling liquid film on plate surface and established the smooth laminar flow model. In 1957, Beniamin used the power series expansion technology to solve the N-S equation (Beniamin, 1957). In 1955, Yih et al. carried out the research on the stability of free falling film fluctuations and analyzed the stability of isothermal falling film (Yih, 1963). To analyze the flow characteristics of the free falling film, Qian carried out the long-wave equation into the autonomous equation of the dynamic system and analyzed the nonlinear stability on the film flow state (Qian et al., 2002). In 2006, Wang et al. calculated the performance parameters of the corrugated plate dryer by establishing a two-phase field model (Wang and Huang, 2006). And it reveals that the trend of pressure loss in simulation results is the same as the experimental result. For the aspect for liquid film breakdown research, previous researches have focused on the breakdown of liquid film on the wall. However, there are few studies on the breakdown phenomenon at the corrugated plate corner and most of them focus on experimental research. In 2007, Li et al. carried out an experiment on a corrugated plate dryer to obtain the separation efficiency of the dryer, which explains why the double-hook corrugated plate dryer has a greater separation efficiency. In 2008, Wegener et al. studied on the breakdown phenomenon of the liquid film at the corner of the horizontal wall under the airflow shear stress (Wegener et al., 2008). It was observed that the liquid film exhibits a complete strip shape as it is breakdown. Results reveal that while the corner angle of the corrugated plate
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Nomenclature u
v
p x y d u v t p R d L R Rel Reg f1 f2 ug ML uc n i hi ðt Þ d uy t0 p0
Water film flow velocity in x-axis direction Water film flow velocity in y-axis direction Total pressure the water film is subjected to Dimensionless length in x-axis direction Dimensionless length in y-axis direction Dimensionless water film thickness Dimensionless flow velocity in x-axis direction Dimensionless flow velocity in y-axis direction Dimensionless time interval Dimensionless pressure Dimensionless radius Linear distance between two walls Liquid film wetted perimeter Curvature radius Reynolds number of water Reynolds number of air Axial force the thin film is subjected to Radial resultant force the thin film is subjected to Critical airflow velocity Mass flow rate Inlet airflow velocity Number of samples to be extracted Number of the sample to be extracted Thickness of the No. i liquid film sample at time t Water film boundary layer thickness Water film velocity in y-axis direction Characteristic time interval Characteristic pressure
wall is bigger than 60 degrees, the influence on the liquid film breakdown can be ignored if the angle is increased again. In 2011, Alamu et al. analyzed an experimental study on streamlined corrugated plates and broken-line corrugated plates. It proves that the streamlined type is better controlled than the broken-line type (Alamu and Azzopardi, 2011). Due to the complexity of the liquid film breakdown process, the empirical formulas obtained from experiments have not been unified. In this paper, the free falling film thickness is measured by Planar Laser Induced Fluorescence (PLIF) method and figures of liquid film thickness under different Reynolds number are obtained. The probability density distribution (PDF), power spectral density (PSD) and wavelet method of liquid film thickness time series are calculated by MATLAB program. Furthermore, the amplitude and frequency domain information of liquid-film fluctuations are analyzed. In addition, the critical air flow velocity when the liquid film ruptures is measured. Two-dimensional liquid film breakdown model is established and the critical airflow velocity is derived and verified by the dimensionless method. Ultimately, combined with theoretical results of this paper, the empirical correlations for the critical airflow velocity are fitted. 2. Experimental research 2.1. Experimental system and PLIF methods A schematic diagram of the free liquid film and liquid film breakdown research test bench of this paper is shown in Fig. 1. Devices are as follows. 1: Data acquisition system, 2: Liquid storage tank No. 1, 3: Steady-pressure tank, 4: Liquid film generator, 5: Rectifier, 6: Blower, 7: Pump, 8: Liquid storage tank No. 2, 9: Liquid storage tank No. 3, 10: Laser with wavelength of 532 nm, 11: Four measuring position point, 12: Anemometer, 13: Flow meter, 14:
cf Rx ðsÞ xT ðt Þ T Fs Sx ðxÞ
Non-dimensional damping coefficient Auto correlation function Stationary random signal Sampling period Sampling frequency Self-power spectral density of the signal
Greek symbols h The flexural angle of the CP corner lal Liquid phase dynamic viscosity lg Dynamic viscosity of the air d Water film thickness ql Liquid density qg Air density sls Friction force the water film is subjected to on the wall sal Airflow shear stress the water film is subjected to r1 Surface tension at air-liquid interface r2 Surface tension at liquid-solid interface Acronysms CP Corrugated Plate CCD Charge Coupled Device MIS Metal-Insulator-Semiconductor PLIF Planar Laser Induced Fluorescence N-S Navier-Stokes PDF Probability Density Distribution PSD Power Spectral Density
Electronically controlled valve, 15: Corrugated plate, and 16: High speed camera. A 6 mm wide slit is formed at the junction of the apparatus No. 4 and the apparatus No. 15 to form a liquid film. The structure of apparatus No. 4 is shown in reference (Wang and Tian, 2019) and (Wang and Tian, 2019). In this paper, the liquid film thickness is measured by the PLIF method (Borisenko et al., 2003); (Drosos et al., 2004). Therefore, it is necessary to previously add Rhodamine B staining agent having a maximum absorption wavelength of 555 nm to the apparatus No. 2, and to uniformly agitate the liquid in the apparatus No. 2. The basic principle of the PLIF method is that the vertical irradiation of the liquid film on the wall by the laser causes the Rhodamine B particles to transition from the ground state to the excited state, and the excited state unstable particles move back to cause the fluorescent agent to emit light, which makes the image of the liquid film high-speed camera captured. The electronically controlled valve should be first opened before the experiment so that the liquid in the apparatus No. 2 flows to the wall surface of the apparatus No. 15 so that the wall surface is sufficiently wetted. This experiment needs to be carried out in the dark state, as shown in Fig. 2. Water and air are used as the liquid phase and the gas phase, respectively, for cold state experiments. In the experiment, the liquid in the apparatus No. 2 flows into the apparatus No. 3. The liquid level on the left side of the apparatus No. 3 does not change, which stabilizes the liquid film flowing to the wall of the apparatus No. 15. This portion of the liquid flows into the apparatus No. 9 and afterwards flows into the apparatus No. 8. The liquid on the right side of the apparatus No. 3 flows directly into the apparatus No. 8 and flows into the apparatus No. 2 under the action of a pump, which creates a circulatory system that can generate the steady liquid film flow. In the experiment, the liquid film images are captured by the apparatus No. 16 and the liquid film Reynolds number
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
3
Fig. 1. Schematic diagram of experimental system.
under the direct irradiation by the laser, the discrepancy in color between the liquid film image and the surrounding test environment exists. Thus the program can be written to acquire the gray value at each position in the Fig. 3a and the contour and boundary of the liquid film as shown in Fig. 3b can be obtained based on the boundary differential principle. In the experiment, an image of an object of a known size is photographed by a high-speed camera (Tokuhiro and Kimura, 1993; Jayanti et al., 1993; Bankoff, 1994). And the size of the scale can be obtained, thereby obtaining the true thickness of the liquid film. 2.3. Analysis of error and uncertainty in measuring liquid film thickness by PLIF method
Fig. 2. Water film thickness measurement by PLIF technology.
is adjusted by the electronically controlled valve which is the apparatus No. 14 in Fig. 1 to change the experimental conditions. The liquid film Reynolds number can be calculated according to the liquid film wetted perimeter L, the mass flow rate M L measured by the apparatus No. 13, and the liquid phase dynamic viscosity lal . The specific calculation method is as follows in Eq. (1).
Re ¼
4M L Llal
ð1Þ
2.2. Liquid film thickness image processing methods The liquid film image captured by the high-speed camera is shown in Fig. 3a, in which the distribution of the liquid film, airflow direction, wall surface, and corrugated plate corner is pointed out, respectively. Since the liquid film image is generated by the light issued by the particles in Rhodamine B fluorescent agent
Fig. 3. Water film image processing.
In the analysis and processing of the image, some errors will inevitably be introduced, which will cause certain difficulties and deviations in the study of liquid film thickness. For image analysis of liquid film thickness, some scholars have proposed using the differential image algorithm to extract liquid film thickness information. The essence of the difference image algorithm is to subtract a grayscale value from an already calibrated image and a real-time image, and the size and position of two images are exactly the same. The image after the differential image algorithm is eliminated from the fixed background and the geometric information of the measurement object can be obtained. Fig. 4 shows the principle of the differential image algorithm. In this experiment, the background image with fixed position and
Fig. 4. Schematic diagram of the differential image algorithm.
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size is the image of the corrugated plate wall, and the measurement object that changes in real time is the liquid film surface, as shown in Fig. 5. However, it is difficult to meet the two requirements of the differential image algorithm in the experiment, that is, the size and position between the calibrated object and the measurement object are exactly the same. This is because the experimental system must have minute vibrations due to the flow of the water circulation. Therefore, in view of this shortcoming of the differential image algorithm, a direct measurement method that can simultaneously track the position information of the corrugated plate wall and the liquid film surface is proposed in this paper. In this method, the corrugated plate wall in contact with the liquid film is used as a calibration object together with the liquid film to avoid interference caused by vibration of the experimental system. Specifically, on the wall of the corrugated plate that is in contact with the liquid film, pixel points are respectively selected on the left and right sides of the liquid film, and the connecting lines of two pixel points are used as the base of the liquid film. The comparison of measurement results of the liquid film thickness data by the difference image algorithm and the direct method is shown in Fig. 6. Results show that there is a significant constant frequency volatility in the liquid film thickness of the differential image algorithm, and the liquid film thickness measured by the direct method has no obvious periodicity and is more random. Since the amplitude of this oscillation is small (about 20 lm, as can be seen from Fig. 6), it is difficult to be observed by the naked eye in actual experiments. The measure precision of the airflow velocity measure instrument in Fig. 1 for the experiment is 0.01 m/s. Since the liquid film thickness is not obtained by direct measurement, an error will inevitably occur, and the error transmission process is given below. The average thickness of the liquid film at a certain time is assumed to be as follows.
Pn
hðt Þ ¼
i¼1 hi ðt Þ
ð2Þ
n
Fig. 5. Liquid film grayscale image.
where n is the number of samples to be extracted. i is the serial number of the sample to be extracted. hi ðt Þ is the thickness of the No. i liquid film sample at time t. Images taken by high-speed cameras do not actually reflect the specific size of the object, thus a standard size needs to be introduced into the image. If the relative position of the camera and the measuring object does not change within a certain period of time, the following equation can be obtained.
hi ðtÞ ¼
a hix ðtÞ ax
ð3Þ
where a is the true length value of the standard size, ax is the number of pixels occupied by the standard size in the image, and hix ðtÞ is the number of pixels corresponding to the No. i liquid film thickness at time t. Therefore, the following equation can be obtained from Eqs. (2) and (3).
hðt Þ ¼
a n
Pn
i¼1 hix ðt Þ
ax
ð4Þ
Thus according to the transmission of the absolute error, the following equation can be obtained.
n X @hðt Þ @hðt Þ DhðtÞ ¼ Dhix ðtÞ þ Dax ð Þ t @h @a x ix i¼1 n X a Dhix ðtÞ þ a hix ðtÞ Dax ¼ n a 2 n a x x i¼1 n a 1 X jDhix ðtÞj þ hix ðt Þ jDax j ¼ n ax i¼1 ax
ð5Þ
In the experiment, 55 pixels are used to represent a distance of 3 mm, then a ¼ 3 and ax ¼ 55. Assume that the resolution of the object during image processing is 1 pixel, and the measurement object is 1 mm. In addition, considering the small vibration generated by the sway of the gantry during the system cycle in the experiment, the absolute error of the PLIF method in measuring the thickness of the liquid film in this experiment is about 0.073 mm. Based on Eq. (5), the following method for improving the measurement accuracy can be obtained. (1) Use a high-resolution camera with a higher resolution (the camera resolution of this document is 10241024) for image capture. (2) Apply liquid film thickness treatment to software with stronger liquid film boundary recognition ability. The level of edge detection technology of digital images directly affects the accuracy of liquid film thickness measurement. In this paper, the optimal step edge detection (Step Edge) is used to detect the Canny operator to extract the liquid film boundary. (3) Shorten the distance between the camera and the measurement object to reduce the scale, that is, use more pixels to simulate the same actual distance. 2.4. Time series of liquid film thickness
Fig. 6. Comparison of two image processing methods.
The measurement points are taken at positions 5 cm, 10 cm, 15 cm and 18 cm below the entrance of the liquid film. The inflection angle is 30 degrees while the height of the wall is 20 cm. The length of flange is 50 mm and the spacing of plates is 17.1 mm. Experiments were carried out on corrugated plates with a wall roughness of 0.025 mm. Additionally, the sampling frequency and time were 500 Hz and 1 s, respectively (as shown in Fig. 7).
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Fig. 7. Curve of free liquid film thickness under different Reynolds numbers.
3. Research on liquid film volatility Liquid film volatility can be analyzed by power spectral density (PSD), probability density function (PDF), and wavelet method. The basic principle of power spectral density is described in Chapter 3.1.
where xT ðt Þis sampled and converted to X ðK Þ by sampling frequency Fs, and the discrete sequence Aðx; T Þ is sampled by sampling frequency Fs and converted into AðkÞ. And X ðK Þ and AðkÞ are one-toone correspondence. Thus
AðkÞ ¼
3.1. Principle of power spectral density In 1898, Schuster first proposed the theory of power spectral density estimation based on Fourier transform periodic graph method. He directly performed discrete Fourier transform (DFT) on the discrete data and then squared the amplitude to obtain the power spectral density of the signal. In 1965, Cooly Tukey proposed the Fast Fourier Transform (FFT), which solved the practical problem of too much computation. This method has been widely used. From the Wiener-Sinqin correlations, the auto correlation function of the stationary random signal is Fourier transformed and the self-power spectral density of the signal is obtained as shown in Eq. (6).
Z
Sx ðxÞ ¼
1
1
jwt
Rx ðsÞe
ds
ð6Þ
where Rx ðsÞ is the auto correlation function. For a stationary random signal xT ðt Þ, its Fourier transform is as follows in Eq. (7).
Z
Aðx; T Þ ¼
1 1
xT ðt Þejwt dt
ð7Þ
Thus
1 Sx ðxÞ ¼ limT!1 jAðx; T Þj2 T
ð8Þ
1 X ðkÞ Fs
ð9Þ
The sampling period can be calculated by the following equation.
T¼
N Fs
ð10Þ
Thus the power spectrum estimation value of the continuous random signal is as follows in Eq. (11). K
S ð xÞ ¼ x
1 2 jX ðK Þj N Fs
ð11Þ
The power spectral density of the signal xT ðt Þ is as follows. K
S ðk Þ ¼ x
1 2 jX ðkÞj N
ð12Þ
Then the estimated function of the power spectral density is shown in Fig. (13).
Gx ðwÞ ¼ 2Sx ðwÞ ¼
2 K S ðkÞ Fs x
ð13Þ
Therefore, according to the above calculation process, the steps of estimating the power spectrum of MATLAB in actual calculation are as follows. First of all, seven parameters necessary for calculating the random signal power spectral density estimation function PSD, that is, the random signal, the Fourier transform point, the sampling
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frequency, the window function vector, an inter-segment overlap number, and the credibility are determined. Furthermore, the signal data is segmented and the power spectral density correction factor is calculated, respectively. Ultimately, the data of each segment is preprocessed first, and afterwards the Fourier transform is applied to finding the variables of each segment, thereby realizing the output, which is the core part of the power spectral density calculation. 3.2. PSD analysis The PSD curve is utilized to obtain frequency domain information of liquid film thickness fluctuations because the characteristic parameters of the PSD curve can reflect the volatility characteristics of the liquid film. Because the peak height of the PSD curve represents the square of the amplitude of the liquid film thickness signal. It represents the energy of the signal in that frequency region. The frequency value corresponding to the peak in the PSD curve represents the inherent volatility frequency of the liquid film. Width of the wave in the PSD curve represents the frequency structure of the liquid film thickness signal. For example, the narrow and steep shape of the wave indicates that the liquid film has a characteristic frequency. Otherwise, the liquid film has a multi-frequency characteristic. The area enclosed by PSD curve and coordinate axis is equal to the mean square value of liquid film thickness signal, that is, the total energy of the liquid film thickness signal. The liquid film volatility energy is not only related to the amplitude, but also related to the frequency. Based on the classical periodogram and the Welch mean period graph theory, MATLAB was used to prepare a prediction program for the power spectral density of liquid film thickness fluctuations. The power spectral density curve of the liquid film thickness volatility signal at positions of 5 cm, 10 cm,
15 cm and 18 cm below the liquid film inlet under different Reynolds numbers is obtained in Fig. 8. Fig. 8 shows the liquid film volatility power spectrum at different Reynolds numbers and at different heights. It can be seen from the figure that under the same Reynolds number, as the measurement height increases, the PSD curve of the liquid film as a whole shows a small upward trend. Because the area enclosed by the PSD curve and the coordinate axis reflects the average value of the volatility of the liquid film time series. As the height of the measurement increases, the volatility of the liquid film is remarkably enhanced. The superposition of different liquid films on the wall exists as the height of the measurement increases, which increases the volatility of the liquid film. This conclusion is in complete agreement with the phenomenon of liquid film accumulation at the bottom of corrugated plate dryers that are common in the industry. Similarly, at the same measurement height position, as the Reynolds number increases, the liquid film increases the flow instability under the joint action of the wall friction and its own gravity, which also leads to an increase in the volatility of the liquid film. What’s more, it can be seen from the Fig. 8 that the thickness of the liquid film is distributed throughout the entire frequency band, and the PSD curves of the liquid film thickness at different Reynolds numbers are substantially the same, so that the liquid film thickness has no characteristic frequency, that is, the liquid film thickness is of significant multi-frequency features. The energy of liquid film fluctuations is mostly concentrated in the low frequency region, and the appearance of solitary waves causes the volatility of the liquid film to increase sharply, which makes the liquid film volatility instability enhanced. The PSD curve does not exhibit significant or obvious peak characteristics, thus the liquid film does not have a characteristic frequency.
Fig. 8. Curve of power spectral density of liquid film thickness.
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
3.3. PDF analysis and wavelet analysis The evolution of the surface wave of liquid film is carried out in time and space. The PSD characteristics (power spectrum characteristics) of surface waves of different wave characteristics are also significantly different, which can be used as one of the criteria for judging the volatility characteristics of liquid film (as shown in Fig. 9). The liquid film thickness value corresponding to the peak of the PDF curve represents the average value of the liquid film thickness to some extent. The numerical value can also reflect the degree of volatility of the liquid film. At the same measurement height, as the Reynolds number increases, the liquid film thickness gradually increases in the PDF curve, that is, the liquid film volatility gradually increases. At the same time, under the same Reynolds number, the higher the measurement height, the larger the liquid film thickness corresponding to the peak value of the PDF curve. The PDF curve has no distinct bimodal characteristics, so the multi-frequency of the liquid film thickness can be demonstrated. In the wavelet method research (as shown in Fig. 10), whether the liquid film Reynolds number increases or the measurement height increases, the wavelet-time distribution in the time–frequency image is wider, which is consistent with the conclusions obtained from the PSD curve. 4. Breakdown model at corrugated plate corner 4.1. Dimensionalization of two-dimensional governing equations The above studies can show that when there is no air flow and the Reynolds number is not very large, gravity does not cause the
7
liquid film itself to breakdown, and there is no periodic volatility and characteristic frequency of the liquid film thickness. Therefore, gravity can be neglected to establish a two-dimensional model of liquid film breakdown under laminar flow conditions. The tangential direction of each point on the wall surface of the corrugated plate is the x-axis and the normal direction is the y-axis, thus a two-dimensional orthogonal coordinate system is established, as shown in Fig. 11. Fig. 11 shows a schematic diagram of the distribution of the boundary layer portion, the non-boundary layer portion, the air flow and the wall surface of the liquid film. The lateral and longitudinal velocities at point A in the boundary layer of the liquid film are u and v, respectively, and l is the distance from point A to the center of curvature. R is the radius of curvature. Thus the governing equations in the coordinate system are as follows.
@u l y @u @u uv y l 1 @p þ u þv þ ¼ @t l @x @y l q @x l " 2 ðl y Þ @ 2 u @ 2 u 1 @u u 2ðl yÞ @ v þ þm þ þ 2 2 @x2 @y2 l @y l2 @x l l y l dðl yÞ yðl yÞ dðl yÞ @u vþ 3 þ 3 dx dx @x l l @v l y @v @ v u2 y l 1 @p u þ þv þ ¼ l l q @x @t @x @y l " 2 2 2 ðl y Þ @ v @ v 1 @ v v 2ðy lÞ @u þm þ þ þ 2 2 @x @x2 @y2 l @y l2 l l l y dðl yÞ yðl yÞ dðl yÞ @ v þ 3 uþ 3 @x dx dx l l
Fig. 9. Curve of probability density of liquid film thickness.
ð14Þ
ð15Þ
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
Fig. 10. Curve of liquid film wavelet timing methods.
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
R @u @ v v þ þ ¼0 y þR @x @y y þR Order 11 1 1 1e Simplified equations can be obtained in Eqs. (18, 19, 20) when neglecting the higher order terms.
Fig. 11. Schematic diagram of two-dimensional boundary layer flow at CP corner in curvilinear coordinate system.
R @u @ v v þ þ ¼0 y þ R @x @y y þ R
ð17Þ
Order of each term in the above equations is as follows.
@u R @u @u u v þ u þv þ @t y þR @x @y y þR " R @p 1 R2 @ 2 u @ 2 u ¼ þ þ þ 2 y þR @x Re ðy þRÞ @x2 @y2
e
11
e 1 1 e2 2
1
1 @u u 2R @v R dR þ v y þR @y ðy þRÞ2 ðy þRÞ2 @x ðy þRÞ3 dx # y R dR @u þ ðy þRÞ3 dx @x Order 11 1e
1 1 12 12
e
1 13
1e
e
3
1
e 1 1 e e 1 11 1 1 e2 1 e2
e 1 12 12
1
1 13
@u @ v þ ¼0 @x @y
ð20Þ
11
The model of the liquid film with radius r and width dr is established as shown in Fig. 12 and basic assumptions are as follows. First of all, flow velocity only changes with ordinate. Besides, flow is assumed as a circular motion since the corner is arc-shaped. Finally, the thickness of the non-boundary layer can be ignored as liquid film thickness is very small. The velocity is considered to be 99% of the velocity of the mainstream. Besides, the flow velocity at the liquid–solid interface is considered to be zero. Thus, velocity in y-axis direction is as follows (Dadvand et al., 2009; Tan et al., 2009; Ghiaasiaan et al., 1997; Kim et al., 2001; Ju et al., 2018; Owen and Ryley, 1985; Alekseenko et al., 1985).
u¼
0:99U y d
Radial force at the position of radius r + dr and r are f + df and f, respectively. Therefore, radial resultant force and centrifugal force can be obtained in Eqs. (22 and 23).
f 1 ¼ ðf þ df Þðr þ dr Þda f rda ¼ da½dðf r Þ þ dfdr da dðf r Þ
ð22Þ
u2 rdrda r
ð23Þ
f 2 ¼ ql
As f 1 is equal to f 2 , thus
Z
Rl
"
Rs
# Z Rl Z Rl u2y ql rdrda dðf rÞda ¼ da ql u2y dr dðf r Þ ¼ 0 r Rs Rs
1
e1 1 e 13
ð21Þ
ð24Þ
e2
1 @v v 2R @u R dR u þ y þR @y ðy þRÞ2 ðy þRÞ2 @x ðy þRÞ3 dx # y R dR @ v þ ðy þRÞ3 dx @x Order 11 1
ð19Þ
11
@v R @v @v u2 þ u þv þ @t y þR @x @y y þR " R @p 1 R2 @2v @2v þ þ ¼ þ 2 y þR @y Re ðy þRÞ @x2 @y2 Order
@p ¼0 @y
4.2. Rupture model
d d ¼ f ðf < 1Þ L
Order 1 1 1 1
ð18Þ
ð16Þ
Eqs. (14, 15, and 16) are processed in a dimensionless method, that is, the term of the equation higher than the one order is ignored and afterwards the dimension is restored, which will result in a simplified governing equation. Take the dimensionless lengthd as the a term of f order. Thus
1 1e e 1
@u @u @u 1 @p @2u þu þv ¼ þm 2 @t @x @y @y q @x
R
f rjRls ¼ f ðRl Þ Rl f ðRs Þ Rs ¼
r1 R1
R1
r2 R2
R 2 ¼ Dr
ð25Þ
The surface tension at air–liquid interface and liquid–solid interface are r1 and r2 , respectively. Shear force and friction force are shown in Eqs. (26 and 27).
Fig. 12. Geometry and schematic diagram of CP wall, water film and air at the CP corner.
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1 2
sal ¼ cf qg u2g sls ¼ ll
ð26Þ
@u 0:99U ¼ ll @y d
ð27Þ
Non-dimensional damping coefficient cf under laminar flow condition is shown in Eq. (28).
cf ¼
1:328
No.
H/ mm
h/ degree
d/ mm
L/ mm
1# 2# 3#
200 200 200
30.0 45.2 40.4
17.1 19.3 14.6
50 50 50
ð28Þ
Re0:5 g
As sls is equal to Eq. (29).
ug ¼ 2:4
Table 1 Parameters of the corrugated plate.
sal , critical airflow velocity can be obtained in
dcosh ðr1 r2 Þl2al
ql qg lg
!13
1 h
ð29Þ
According to Eq. (29), critical airflow velocity is positively correlated to d, h and it is negatively correlated to surface tension, density of gas and liquid phase. It is in complete agreement with the industrial corrugated plate dryer. 5. Results and discussion Parameters of the three kinds of corrugated plate are as follows in Table 1. Therefore, according to theoretical results in Eq. (29), the calculation equation of the critical airflow velocities of the three different parametric corrugated plates are as follows.
ug ¼
8:617 h
ð30Þ
ug ¼
8:415 h
ð31Þ
ug ¼
7:829 h
ð32Þ
The units of ug and h are m/s and mm, respectively. Liquid film thickness distribution in this experiment ranges from 0.2 to 1.65 mm. The airflow velocity is negatively correlated with the thickness of the liquid film. When the thickness of the liquid film is small, the liquid film flow is stable and gravity has little effect on the liquid film flow. Therefore, the mainstream of the liquid film is difficult to be breakdown and when the liquid film thickness is less than 0.4 mm, the value of critical airflow velocity is even above 20 m/s. As the Reynolds number increases, the coupling of gravity and airflow makes the liquid film more susceptible to be breakdown, thus the critical airflow velocity is reduced. It can be seen from Fig. 13 that the difference between the experimental value of the critical airflow velocity and the theoretical value gradually decreases as the Reynolds number increases. For the corrugated plate dryer with the same structural parameters, theoretical results of the critical airflow velocity are greater than experimental results. And there are two factors contributing to this result. First of all, the theoretical analysis is based on the two-dimensional laminar flow condition by ignoring the gravity. The coupling of gravity and airflow in the actual flow condition makes the liquid film more susceptible to be breakdown. In addition, the phenomenon of liquid film reflow that is common in corrugated plate dryers is observed at the bottom of the corrugated plate. The reason for this phenomenon is that when the airflow is blown over the surface of the liquid film mainstream, it will be subjected to the resistance and thus forming a cyclone below it, which makes the liquid film more susceptible to be breakdown (Karimi and Kawaji, 1999; Wang and Tian, 2019; Choi and Sang,
Fig. 13. Results of critical airflow velocity.
1996; Chung et al., 2014; Bashter et al., 1996; Fang et al., 2020). In summary, the theoretical value of the critical airflow velocity is slightly larger than experimental results. The distribution of experimental results in Fig. 13 is not uniform, thus it is difficult to describe them with a single function. The thickness of the liquid film can be artificially divided into three sections, namely 0–0.68 mm, 0.68–1.11 mm and 1.11–1.6 mm. The relationship between the liquid film thickness and the critical airflow velocity in actual conditions is not a simple inverse proportional correlation derived from the simplified two-dimensional model. And the distribution shape of experimental results is similar to the common nike function in mathematics and the equation is shown in Eq. (33).
f ð xÞ ¼
k1 þ k2 þ k3 x x
ð33Þ
Therefore, experimental results are fitted in the form of nike function shown in Eq. (34). And according to the structural parameters of the corrugated plate and derivation results of the model, the fitting equations for the calculation of the critical airflow velocity of three corrugated plates with different structural parameters are revealed in Eqs. (35, 36, 37, 38, 39, 40, 41, 42 and 43).
ug ¼
k1 þ k2 þ k3 h h
ð34Þ
When the liquid film thickness is in the range of 0–0.68 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
4:378 þ ð2:89Þ þ 2:60h h
ð35Þ
2#ug ¼
9:137 þ ð17:444Þ þ 21:341h h
ð36Þ
3#ug ¼
11:511 þ ð27:701Þ þ 30:558h h
ð37Þ
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
When the liquid film thickness is in the range of 0.68–1.11 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
12:833 þ ð13:247Þ þ 7:914h h
ð38Þ
2#ug ¼
11:236 þ ð9:803Þ þ 5:567h h
ð39Þ
3#ug ¼
7:988 þ ð2:922Þ þ 1:734h h
ð40Þ
When the liquid film thickness is in the range of 1.11–1.65 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
14:683 þ ð9:093Þ þ 0:556h h
ð41Þ
2#ug ¼
12:943 þ ð4:505Þ þ 0:760h h
ð42Þ
3#ug ¼
19:140 þ ð25:578Þ þ 7:517h h
ð43Þ
Fig. 13 reveals that above fitting results agree well with experimental results. Nevertheless, it is unable to obtain a formula for calculating the critical airflow velocity that can be applicable to all the corrugated plate dryers. Authors of this paper are still trying to find out the correlation between these empirical equations and the structural parameters of the corrugated plate to achieve the unification of the calculation equation, which will be the prospective research direction. 6. Conclusions and prospects The free falling film thickness is measured by the PLIF method, and the liquid film thickness curve is obtained. The probability density distribution (PDF), power spectral density (PSD) and wavelet method of liquid film thickness time series are calculated by MATLAB program. Besides, the amplitude and frequency domain information of liquid-film fluctuations are analyzed. The critical airflow velocity of three kinds of corrugated plates is measured experimentally. Ultimately, combined with theoretical results of this paper, a semi-empirical correlation for the critical airflow velocity is obtained. The specific conclusions are as follows. (1) The evolution of the surface wave of liquid film is carried out in time and space. The PDF features (probability distribution characteristics) and PSD characteristics (power spectrum characteristics) of surface waves with different wave characteristics also have significant differences, which can be used as the judgment benchmark of characteristics of liquid film volatility. The PDF curve has no obvious bimodal characteristics. And the PSD curve and the wavelet curve have no significant peak characteristics. Therefore, the spectrum has no characteristic frequency, that is, the liquid film has multifrequency characteristics. (2) The energy of the liquid film volatility is mostly concentrated in the low frequency region and the appearance of the solitary wave causes the volatility of the liquid film to increase sharply, thus the volatility of the liquid film volatility is enhanced. (3) Factors affecting the development of liquid film volatility instability mainly include liquid film thickness, wave frequency and longitudinal component of gravity. The increase of the thickness of the liquid film accelerates the longitudinal evolution of the liquid film volatility, and the same
11
longitudinal position exhibits different volatility patterns due to the difference in liquid film thickness. The fluctuating frequency plays a substantial role in the spatiotemporal evolution of the surface wave characteristics of the liquid film. The longitudinal component of gravity can accelerate the evolution of surface waves and is an unstable factor. Therefore, the impact of the above three factors on the analysis of liquid film fluctuations is the prospective research trend. In this paper, the first two influencing factors are analyzed, but the influence of gravity on the liquid film is not considered enough, which will be improved in future research. (4) Both experimental and theoretical results reveal that the critical airflow velocity is negatively correlated with the liquid film Reynolds number. Comparing results with theoretical results of the critical airflow velocity as derived in this paper, the fitting equations for the critical airflow velocity are obtained. And the equations fit well with experimental results. Nevertheless, theoretical results of the critical airflow velocity in this paper are based on many hypothetical preconditions and it is derived in two-dimensional coordinate system under laminar flow conditions. However, the liquid film flow in the corrugated plate under the action of the air flow is a typical strong turbulence problem, thus the derivation of the liquid film breakdown turbulence model under the three-dimensional condition considering gravity will be the prospective research direction. Besides, Eqs. (35, 36, 37, 38, 39, 40, 41, 42 and 43) are not universal because they are just fit for the certain structure of the corrugated plate. According to nine equations, it is different to conclude a universal equation. The research of a universal to calculate the critical airflow velocity is the prospective study. Furthermore, the flow inside the corrugated plate is a strong turbulence phenomenon, but many calculations in the laminar flow state are used in the derivation. However, the liquid film Reynolds number of the experiments performed in this paper is relatively small. Therefore, within a certain range, the formula of laminar flow is available. For experimental and theoretical derivation in the case of strong turbulence, they will be studied in the future.
Acknowledgement Authors would like to acknowledge the National Natural Science Foundation of China (No. 51676052) for funding provided. Wang. B would like to give great thank to Fundamental Science on Nuclear Safety and Simulation Technology Laboratory of Harbin Engineering University for his studentship and help. References Alamu, M.B., Azzopardi, B.J., 2011. Wave and Drop Periodicity in Transient Annular Flow. Nucl. Eng. Des. 241, 579–592. Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G., 1985. Wave Formation on Vertical Falling Liquid Films. Int J Multiphase Flow 11 (5), 607–627. Bankoff, S.G., 1994. Significant questions in thin film heat transfer. J. Heat Transfer 116, 11–16. Bashter, I.I., Abdo, E.S., Makarious, A.S., 1996. A comparative study of the attenuation of reactor thermal neutrons in different types of concrete. Ann. Nucl. Energy 23 (17), 1189–1195. Beniamin, T.B., 1957. Wave Formation in Laminar Flow down an Inclined Plane. J. Fluid Mech. 2, 554–574. Borisenko, V., Lougheed, T., Hesse, J., Füreder-Kitzmüller, E., Fertig, N., Behrends, J.C., Woolley, G.A., Schütz, G.J., 2003. Simultaneous Optical and Electrical Recording of Single Gramicidin Channels. Biophys. J. 84 (1), 612–622. Choi, C.J., Sang, Y.L., 1996. Transient analysis of a condensation experiment in the noncondensable gas-filled closed loop using an inverted U-tube. Ann. Nucl. Energy 23 (14), 1179–1188.
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Chung, Y.J., Bae, K.H., Kim, K.K., Lee, W.J., Lee, W.J., 2014. Boiling heat transfer and dry-out in helically coiled tubes under different pressure conditions. Ann. Nucl. Energy 71, 298–303. Dadvand, A., Khoo, B., Shervani-Tabar, M.T., 2009. A collapsing bubble-induced micro-injector: an experimental study. Exp. Fluids 46 (3), 419–434. Drosos, E.I.P., Paras, S.V., Karabelas, A.J., 2004. Characteristics of developing free falling films at intermediate Reynolds and high Kapitza numbers. Int. J. Multiphase Flow 30 (7), 853–876. Fang, D., Wang, M.-J., Duan, Y.-G., et al., 2020. Full-scale numerical study on the flow characteristics and mal-distribution phenomenon in SG steam-water separation system of an advanced PWR. Prog. Nucl. Energy 118, 103075. Ghiaasiaan, S.M., Wu, X., Sadowski, D.L., et al., 1997. Hydrodynamic Characteristics of Countercurrent Two-Phase Flow in Vertical and Inclined Channels: Effect of Liquid Properties. Int. J. Multiphase Flow 23, 1063–1083. Jayanti, S., Hewitt, G.F., Low, D.E.F., Hervieu, E., 1993. Observation of flooding in the taylor bubble of co-current upwards slug flow. Int. J. Multiphase Flow 19 (3), 531–534. Ju, P., Liu, Y., Ishii, M., et al., 2018. Prediction of rod film thickness of vertical upward co-current adiabatic flow in rod bundle. Ann. Nucl. Energy 121, 1–10. Karimi, G., Kawaji, M., 1999. Flow Characteristics and Circulatory Motion in Wavy Falling Films with and without Counter-current Gas Flow. Int. J. Multiph. Flow 25 (12), 1305–1319. Kim, J.H., Kim, T.W., Lee, S.M., Park, G.C., 2001. Study on the natural circulation characteristics of the integral type reactor for vertical and inclined conditions. Nucl. Eng. Des. 207 (1), 21–31.
Owen, I., Ryley, D.J., 1985. The Flow of Thin Liquid Films around Corners. Int. J. Multiphase Flow 11 (1), 51–62. Qian, H.-Q., Sun, H.-D., Liu, L., 2002. Stability Analysis of Long Wave Equation for the Free Falling Flow of Liquid Film. J. Xi’an Jiaotong Univ. 36 (11), 1121–1124. Tan, S.C., Su, G.H., Gao, P.Z., 2009. Heat transfer model of single-phase natural circulation flow under a rolling motion condition. Nucl. Eng. Des. 239 (10), 2212–2216. Tokuhiro, A., Kimura, N., 1993. An experimental investigation on thermal striping: Mixing phenomena of a vertical non-buoyant jet with two adjacent jets as measured by ultrasound Doppler velocimetry. Nucl. Eng. Des. 188 (1), 49–73. Wang, X.-M., Huang, S.-Y., 2006. The Behavior Analysis of Droplets in the Steamwater separator. J. Eng. Thermophys. 27 (S1), 181–184. Wang, B., Tian, R.-F., 2019. Judgement of critical state of water film rupture on corrugated plate wall based on SIFT feature selection algorithm and SVM classification method. Nucl. Eng. Des. 347, 132–139. Wang, B., Tian, R.-F., 2019. Investigation on flow and breakdown characteristics of water film on vertical corrugated plate wall. Ann. Nucl. Energy 127, 120–129. Wang, B., Tian, R.-F., 2019. Study on characteristics of water film breakdown on the corrugated plate wall under the horizontal shear of airflow. Nucl. Eng. Des. 343, 76–84. Wegener, J.L., Friedrich, M.A., Lan, H., Drallmeier, J.A., Armaly, B.F., 2008. A Criterion for Predicting Shear-driven Film Separation at An Expanding Corner with Experimental Validation. ILASS Americas. In: 21th Annual Conference on Liquid Atomization and Spray System. Orlando, FL, May 2008. Yih, C.S., 1963. Stability of Liquid Flow down An Inclined Plane. Phys. Fluids 6 (3), 321–334.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Analysis of fluctuation and breakdown characteristics of liquid film on corrugated plate wall Bo Wang, Bowen Chen, Ru Li, Ruifeng Tian ⇑ Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 17 May 2019 Received in revised form 16 July 2019 Accepted 19 July 2019
Keywords: PLIF Fluctuation characteristics Liquid film thickness Steam water separation Corrugated plate
a b s t r a c t Corrugated plate dryer is a very vital separation equipment for nuclear engineering and it is meaningful to study the flow, fluctuation and breakdown characteristics of the liquid film on corrugated plate wall. Liquid film thickness of steady flow is measured based on plane laser induced fluorescence (PLIF) technique and time series of liquid film thickness are obtained. Besides, amplitude and frequency domain information of liquid film fluctuations are analyzed by probability density function (PDF), power spectral density (PSD) and wavelet method. A two-dimensional liquid film breakdown model is established. Critical airflow velocity when the water film breakdown is measured experimentally. Based on theoretical results, equations for calculating critical airflow velocity of the corrugated plate under different liquid film thicknesses are fitted in the form of nike function. Fitting equations for calculating critical airflow velocity in the form of nike function agree well with experimental results. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Internal flow in the corrugated plate is a typical multiphase flow problem due to the presence of gas, liquid film, and corrugated wall surfaces. Research on the method of improving the separation efficiency of corrugated plates is a hot research topic. The secondary carrying phenomenon means that when the inlet gas velocity of the corrugated plate rises to a certain extent, the separation efficiency does not increase but decreases. The internal secondary carrying phenomenon is a key factor affecting the separation efficiency of the corrugated plate. This phenomenon occurs because after the steam is separated once, large droplets are trapped by the wall surface and some of the droplets that have been captured may re-enter the flow channel under the action of the airflow to continue moving. The research on the characteristics of the liquid film on the wall under the action of no airflow is the research basis of the separation mechanism of the corrugated plate separator under the action of airflow. Therefore, it is of great value to study the flow and breakdown characteristics of the liquid film on the corrugated plate wall. In the experiment, the water film breakdown phenomenon is commonly observed at the corner of corrugated plate. Thus the corrugated plate consisting of many plates is used instead of using only one flat plate in the research. For the aspect for free falling film research, in 1916, Nusselt firstly ⇑ Corresponding author. E-mail address: [email protected] (R. Tian). https://doi.org/10.1016/j.anucene.2019.106946 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
studied the flow characteristics of the free falling liquid film on plate surface and established the smooth laminar flow model. In 1957, Beniamin used the power series expansion technology to solve the N-S equation (Beniamin, 1957). In 1955, Yih et al. carried out the research on the stability of free falling film fluctuations and analyzed the stability of isothermal falling film (Yih, 1963). To analyze the flow characteristics of the free falling film, Qian carried out the long-wave equation into the autonomous equation of the dynamic system and analyzed the nonlinear stability on the film flow state (Qian et al., 2002). In 2006, Wang et al. calculated the performance parameters of the corrugated plate dryer by establishing a two-phase field model (Wang and Huang, 2006). And it reveals that the trend of pressure loss in simulation results is the same as the experimental result. For the aspect for liquid film breakdown research, previous researches have focused on the breakdown of liquid film on the wall. However, there are few studies on the breakdown phenomenon at the corrugated plate corner and most of them focus on experimental research. In 2007, Li et al. carried out an experiment on a corrugated plate dryer to obtain the separation efficiency of the dryer, which explains why the double-hook corrugated plate dryer has a greater separation efficiency. In 2008, Wegener et al. studied on the breakdown phenomenon of the liquid film at the corner of the horizontal wall under the airflow shear stress (Wegener et al., 2008). It was observed that the liquid film exhibits a complete strip shape as it is breakdown. Results reveal that while the corner angle of the corrugated plate
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
Nomenclature u
v
p x y d u v t p R d L R Rel Reg f1 f2 ug ML uc n i hi ðt Þ d uy t0 p0
Water film flow velocity in x-axis direction Water film flow velocity in y-axis direction Total pressure the water film is subjected to Dimensionless length in x-axis direction Dimensionless length in y-axis direction Dimensionless water film thickness Dimensionless flow velocity in x-axis direction Dimensionless flow velocity in y-axis direction Dimensionless time interval Dimensionless pressure Dimensionless radius Linear distance between two walls Liquid film wetted perimeter Curvature radius Reynolds number of water Reynolds number of air Axial force the thin film is subjected to Radial resultant force the thin film is subjected to Critical airflow velocity Mass flow rate Inlet airflow velocity Number of samples to be extracted Number of the sample to be extracted Thickness of the No. i liquid film sample at time t Water film boundary layer thickness Water film velocity in y-axis direction Characteristic time interval Characteristic pressure
wall is bigger than 60 degrees, the influence on the liquid film breakdown can be ignored if the angle is increased again. In 2011, Alamu et al. analyzed an experimental study on streamlined corrugated plates and broken-line corrugated plates. It proves that the streamlined type is better controlled than the broken-line type (Alamu and Azzopardi, 2011). Due to the complexity of the liquid film breakdown process, the empirical formulas obtained from experiments have not been unified. In this paper, the free falling film thickness is measured by Planar Laser Induced Fluorescence (PLIF) method and figures of liquid film thickness under different Reynolds number are obtained. The probability density distribution (PDF), power spectral density (PSD) and wavelet method of liquid film thickness time series are calculated by MATLAB program. Furthermore, the amplitude and frequency domain information of liquid-film fluctuations are analyzed. In addition, the critical air flow velocity when the liquid film ruptures is measured. Two-dimensional liquid film breakdown model is established and the critical airflow velocity is derived and verified by the dimensionless method. Ultimately, combined with theoretical results of this paper, the empirical correlations for the critical airflow velocity are fitted. 2. Experimental research 2.1. Experimental system and PLIF methods A schematic diagram of the free liquid film and liquid film breakdown research test bench of this paper is shown in Fig. 1. Devices are as follows. 1: Data acquisition system, 2: Liquid storage tank No. 1, 3: Steady-pressure tank, 4: Liquid film generator, 5: Rectifier, 6: Blower, 7: Pump, 8: Liquid storage tank No. 2, 9: Liquid storage tank No. 3, 10: Laser with wavelength of 532 nm, 11: Four measuring position point, 12: Anemometer, 13: Flow meter, 14:
cf Rx ðsÞ xT ðt Þ T Fs Sx ðxÞ
Non-dimensional damping coefficient Auto correlation function Stationary random signal Sampling period Sampling frequency Self-power spectral density of the signal
Greek symbols h The flexural angle of the CP corner lal Liquid phase dynamic viscosity lg Dynamic viscosity of the air d Water film thickness ql Liquid density qg Air density sls Friction force the water film is subjected to on the wall sal Airflow shear stress the water film is subjected to r1 Surface tension at air-liquid interface r2 Surface tension at liquid-solid interface Acronysms CP Corrugated Plate CCD Charge Coupled Device MIS Metal-Insulator-Semiconductor PLIF Planar Laser Induced Fluorescence N-S Navier-Stokes PDF Probability Density Distribution PSD Power Spectral Density
Electronically controlled valve, 15: Corrugated plate, and 16: High speed camera. A 6 mm wide slit is formed at the junction of the apparatus No. 4 and the apparatus No. 15 to form a liquid film. The structure of apparatus No. 4 is shown in reference (Wang and Tian, 2019) and (Wang and Tian, 2019). In this paper, the liquid film thickness is measured by the PLIF method (Borisenko et al., 2003); (Drosos et al., 2004). Therefore, it is necessary to previously add Rhodamine B staining agent having a maximum absorption wavelength of 555 nm to the apparatus No. 2, and to uniformly agitate the liquid in the apparatus No. 2. The basic principle of the PLIF method is that the vertical irradiation of the liquid film on the wall by the laser causes the Rhodamine B particles to transition from the ground state to the excited state, and the excited state unstable particles move back to cause the fluorescent agent to emit light, which makes the image of the liquid film high-speed camera captured. The electronically controlled valve should be first opened before the experiment so that the liquid in the apparatus No. 2 flows to the wall surface of the apparatus No. 15 so that the wall surface is sufficiently wetted. This experiment needs to be carried out in the dark state, as shown in Fig. 2. Water and air are used as the liquid phase and the gas phase, respectively, for cold state experiments. In the experiment, the liquid in the apparatus No. 2 flows into the apparatus No. 3. The liquid level on the left side of the apparatus No. 3 does not change, which stabilizes the liquid film flowing to the wall of the apparatus No. 15. This portion of the liquid flows into the apparatus No. 9 and afterwards flows into the apparatus No. 8. The liquid on the right side of the apparatus No. 3 flows directly into the apparatus No. 8 and flows into the apparatus No. 2 under the action of a pump, which creates a circulatory system that can generate the steady liquid film flow. In the experiment, the liquid film images are captured by the apparatus No. 16 and the liquid film Reynolds number
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
3
Fig. 1. Schematic diagram of experimental system.
under the direct irradiation by the laser, the discrepancy in color between the liquid film image and the surrounding test environment exists. Thus the program can be written to acquire the gray value at each position in the Fig. 3a and the contour and boundary of the liquid film as shown in Fig. 3b can be obtained based on the boundary differential principle. In the experiment, an image of an object of a known size is photographed by a high-speed camera (Tokuhiro and Kimura, 1993; Jayanti et al., 1993; Bankoff, 1994). And the size of the scale can be obtained, thereby obtaining the true thickness of the liquid film. 2.3. Analysis of error and uncertainty in measuring liquid film thickness by PLIF method
Fig. 2. Water film thickness measurement by PLIF technology.
is adjusted by the electronically controlled valve which is the apparatus No. 14 in Fig. 1 to change the experimental conditions. The liquid film Reynolds number can be calculated according to the liquid film wetted perimeter L, the mass flow rate M L measured by the apparatus No. 13, and the liquid phase dynamic viscosity lal . The specific calculation method is as follows in Eq. (1).
Re ¼
4M L Llal
ð1Þ
2.2. Liquid film thickness image processing methods The liquid film image captured by the high-speed camera is shown in Fig. 3a, in which the distribution of the liquid film, airflow direction, wall surface, and corrugated plate corner is pointed out, respectively. Since the liquid film image is generated by the light issued by the particles in Rhodamine B fluorescent agent
Fig. 3. Water film image processing.
In the analysis and processing of the image, some errors will inevitably be introduced, which will cause certain difficulties and deviations in the study of liquid film thickness. For image analysis of liquid film thickness, some scholars have proposed using the differential image algorithm to extract liquid film thickness information. The essence of the difference image algorithm is to subtract a grayscale value from an already calibrated image and a real-time image, and the size and position of two images are exactly the same. The image after the differential image algorithm is eliminated from the fixed background and the geometric information of the measurement object can be obtained. Fig. 4 shows the principle of the differential image algorithm. In this experiment, the background image with fixed position and
Fig. 4. Schematic diagram of the differential image algorithm.
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
size is the image of the corrugated plate wall, and the measurement object that changes in real time is the liquid film surface, as shown in Fig. 5. However, it is difficult to meet the two requirements of the differential image algorithm in the experiment, that is, the size and position between the calibrated object and the measurement object are exactly the same. This is because the experimental system must have minute vibrations due to the flow of the water circulation. Therefore, in view of this shortcoming of the differential image algorithm, a direct measurement method that can simultaneously track the position information of the corrugated plate wall and the liquid film surface is proposed in this paper. In this method, the corrugated plate wall in contact with the liquid film is used as a calibration object together with the liquid film to avoid interference caused by vibration of the experimental system. Specifically, on the wall of the corrugated plate that is in contact with the liquid film, pixel points are respectively selected on the left and right sides of the liquid film, and the connecting lines of two pixel points are used as the base of the liquid film. The comparison of measurement results of the liquid film thickness data by the difference image algorithm and the direct method is shown in Fig. 6. Results show that there is a significant constant frequency volatility in the liquid film thickness of the differential image algorithm, and the liquid film thickness measured by the direct method has no obvious periodicity and is more random. Since the amplitude of this oscillation is small (about 20 lm, as can be seen from Fig. 6), it is difficult to be observed by the naked eye in actual experiments. The measure precision of the airflow velocity measure instrument in Fig. 1 for the experiment is 0.01 m/s. Since the liquid film thickness is not obtained by direct measurement, an error will inevitably occur, and the error transmission process is given below. The average thickness of the liquid film at a certain time is assumed to be as follows.
Pn
hðt Þ ¼
i¼1 hi ðt Þ
ð2Þ
n
Fig. 5. Liquid film grayscale image.
where n is the number of samples to be extracted. i is the serial number of the sample to be extracted. hi ðt Þ is the thickness of the No. i liquid film sample at time t. Images taken by high-speed cameras do not actually reflect the specific size of the object, thus a standard size needs to be introduced into the image. If the relative position of the camera and the measuring object does not change within a certain period of time, the following equation can be obtained.
hi ðtÞ ¼
a hix ðtÞ ax
ð3Þ
where a is the true length value of the standard size, ax is the number of pixels occupied by the standard size in the image, and hix ðtÞ is the number of pixels corresponding to the No. i liquid film thickness at time t. Therefore, the following equation can be obtained from Eqs. (2) and (3).
hðt Þ ¼
a n
Pn
i¼1 hix ðt Þ
ax
ð4Þ
Thus according to the transmission of the absolute error, the following equation can be obtained.
n X @hðt Þ @hðt Þ DhðtÞ ¼ Dhix ðtÞ þ Dax ð Þ t @h @a x ix i¼1 n X a Dhix ðtÞ þ a hix ðtÞ Dax ¼ n a 2 n a x x i¼1 n a 1 X jDhix ðtÞj þ hix ðt Þ jDax j ¼ n ax i¼1 ax
ð5Þ
In the experiment, 55 pixels are used to represent a distance of 3 mm, then a ¼ 3 and ax ¼ 55. Assume that the resolution of the object during image processing is 1 pixel, and the measurement object is 1 mm. In addition, considering the small vibration generated by the sway of the gantry during the system cycle in the experiment, the absolute error of the PLIF method in measuring the thickness of the liquid film in this experiment is about 0.073 mm. Based on Eq. (5), the following method for improving the measurement accuracy can be obtained. (1) Use a high-resolution camera with a higher resolution (the camera resolution of this document is 10241024) for image capture. (2) Apply liquid film thickness treatment to software with stronger liquid film boundary recognition ability. The level of edge detection technology of digital images directly affects the accuracy of liquid film thickness measurement. In this paper, the optimal step edge detection (Step Edge) is used to detect the Canny operator to extract the liquid film boundary. (3) Shorten the distance between the camera and the measurement object to reduce the scale, that is, use more pixels to simulate the same actual distance. 2.4. Time series of liquid film thickness
Fig. 6. Comparison of two image processing methods.
The measurement points are taken at positions 5 cm, 10 cm, 15 cm and 18 cm below the entrance of the liquid film. The inflection angle is 30 degrees while the height of the wall is 20 cm. The length of flange is 50 mm and the spacing of plates is 17.1 mm. Experiments were carried out on corrugated plates with a wall roughness of 0.025 mm. Additionally, the sampling frequency and time were 500 Hz and 1 s, respectively (as shown in Fig. 7).
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
Fig. 7. Curve of free liquid film thickness under different Reynolds numbers.
3. Research on liquid film volatility Liquid film volatility can be analyzed by power spectral density (PSD), probability density function (PDF), and wavelet method. The basic principle of power spectral density is described in Chapter 3.1.
where xT ðt Þis sampled and converted to X ðK Þ by sampling frequency Fs, and the discrete sequence Aðx; T Þ is sampled by sampling frequency Fs and converted into AðkÞ. And X ðK Þ and AðkÞ are one-toone correspondence. Thus
AðkÞ ¼
3.1. Principle of power spectral density In 1898, Schuster first proposed the theory of power spectral density estimation based on Fourier transform periodic graph method. He directly performed discrete Fourier transform (DFT) on the discrete data and then squared the amplitude to obtain the power spectral density of the signal. In 1965, Cooly Tukey proposed the Fast Fourier Transform (FFT), which solved the practical problem of too much computation. This method has been widely used. From the Wiener-Sinqin correlations, the auto correlation function of the stationary random signal is Fourier transformed and the self-power spectral density of the signal is obtained as shown in Eq. (6).
Z
Sx ðxÞ ¼
1
1
jwt
Rx ðsÞe
ds
ð6Þ
where Rx ðsÞ is the auto correlation function. For a stationary random signal xT ðt Þ, its Fourier transform is as follows in Eq. (7).
Z
Aðx; T Þ ¼
1 1
xT ðt Þejwt dt
ð7Þ
Thus
1 Sx ðxÞ ¼ limT!1 jAðx; T Þj2 T
ð8Þ
1 X ðkÞ Fs
ð9Þ
The sampling period can be calculated by the following equation.
T¼
N Fs
ð10Þ
Thus the power spectrum estimation value of the continuous random signal is as follows in Eq. (11). K
S ð xÞ ¼ x
1 2 jX ðK Þj N Fs
ð11Þ
The power spectral density of the signal xT ðt Þ is as follows. K
S ðk Þ ¼ x
1 2 jX ðkÞj N
ð12Þ
Then the estimated function of the power spectral density is shown in Fig. (13).
Gx ðwÞ ¼ 2Sx ðwÞ ¼
2 K S ðkÞ Fs x
ð13Þ
Therefore, according to the above calculation process, the steps of estimating the power spectrum of MATLAB in actual calculation are as follows. First of all, seven parameters necessary for calculating the random signal power spectral density estimation function PSD, that is, the random signal, the Fourier transform point, the sampling
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
frequency, the window function vector, an inter-segment overlap number, and the credibility are determined. Furthermore, the signal data is segmented and the power spectral density correction factor is calculated, respectively. Ultimately, the data of each segment is preprocessed first, and afterwards the Fourier transform is applied to finding the variables of each segment, thereby realizing the output, which is the core part of the power spectral density calculation. 3.2. PSD analysis The PSD curve is utilized to obtain frequency domain information of liquid film thickness fluctuations because the characteristic parameters of the PSD curve can reflect the volatility characteristics of the liquid film. Because the peak height of the PSD curve represents the square of the amplitude of the liquid film thickness signal. It represents the energy of the signal in that frequency region. The frequency value corresponding to the peak in the PSD curve represents the inherent volatility frequency of the liquid film. Width of the wave in the PSD curve represents the frequency structure of the liquid film thickness signal. For example, the narrow and steep shape of the wave indicates that the liquid film has a characteristic frequency. Otherwise, the liquid film has a multi-frequency characteristic. The area enclosed by PSD curve and coordinate axis is equal to the mean square value of liquid film thickness signal, that is, the total energy of the liquid film thickness signal. The liquid film volatility energy is not only related to the amplitude, but also related to the frequency. Based on the classical periodogram and the Welch mean period graph theory, MATLAB was used to prepare a prediction program for the power spectral density of liquid film thickness fluctuations. The power spectral density curve of the liquid film thickness volatility signal at positions of 5 cm, 10 cm,
15 cm and 18 cm below the liquid film inlet under different Reynolds numbers is obtained in Fig. 8. Fig. 8 shows the liquid film volatility power spectrum at different Reynolds numbers and at different heights. It can be seen from the figure that under the same Reynolds number, as the measurement height increases, the PSD curve of the liquid film as a whole shows a small upward trend. Because the area enclosed by the PSD curve and the coordinate axis reflects the average value of the volatility of the liquid film time series. As the height of the measurement increases, the volatility of the liquid film is remarkably enhanced. The superposition of different liquid films on the wall exists as the height of the measurement increases, which increases the volatility of the liquid film. This conclusion is in complete agreement with the phenomenon of liquid film accumulation at the bottom of corrugated plate dryers that are common in the industry. Similarly, at the same measurement height position, as the Reynolds number increases, the liquid film increases the flow instability under the joint action of the wall friction and its own gravity, which also leads to an increase in the volatility of the liquid film. What’s more, it can be seen from the Fig. 8 that the thickness of the liquid film is distributed throughout the entire frequency band, and the PSD curves of the liquid film thickness at different Reynolds numbers are substantially the same, so that the liquid film thickness has no characteristic frequency, that is, the liquid film thickness is of significant multi-frequency features. The energy of liquid film fluctuations is mostly concentrated in the low frequency region, and the appearance of solitary waves causes the volatility of the liquid film to increase sharply, which makes the liquid film volatility instability enhanced. The PSD curve does not exhibit significant or obvious peak characteristics, thus the liquid film does not have a characteristic frequency.
Fig. 8. Curve of power spectral density of liquid film thickness.
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
3.3. PDF analysis and wavelet analysis The evolution of the surface wave of liquid film is carried out in time and space. The PSD characteristics (power spectrum characteristics) of surface waves of different wave characteristics are also significantly different, which can be used as one of the criteria for judging the volatility characteristics of liquid film (as shown in Fig. 9). The liquid film thickness value corresponding to the peak of the PDF curve represents the average value of the liquid film thickness to some extent. The numerical value can also reflect the degree of volatility of the liquid film. At the same measurement height, as the Reynolds number increases, the liquid film thickness gradually increases in the PDF curve, that is, the liquid film volatility gradually increases. At the same time, under the same Reynolds number, the higher the measurement height, the larger the liquid film thickness corresponding to the peak value of the PDF curve. The PDF curve has no distinct bimodal characteristics, so the multi-frequency of the liquid film thickness can be demonstrated. In the wavelet method research (as shown in Fig. 10), whether the liquid film Reynolds number increases or the measurement height increases, the wavelet-time distribution in the time–frequency image is wider, which is consistent with the conclusions obtained from the PSD curve. 4. Breakdown model at corrugated plate corner 4.1. Dimensionalization of two-dimensional governing equations The above studies can show that when there is no air flow and the Reynolds number is not very large, gravity does not cause the
7
liquid film itself to breakdown, and there is no periodic volatility and characteristic frequency of the liquid film thickness. Therefore, gravity can be neglected to establish a two-dimensional model of liquid film breakdown under laminar flow conditions. The tangential direction of each point on the wall surface of the corrugated plate is the x-axis and the normal direction is the y-axis, thus a two-dimensional orthogonal coordinate system is established, as shown in Fig. 11. Fig. 11 shows a schematic diagram of the distribution of the boundary layer portion, the non-boundary layer portion, the air flow and the wall surface of the liquid film. The lateral and longitudinal velocities at point A in the boundary layer of the liquid film are u and v, respectively, and l is the distance from point A to the center of curvature. R is the radius of curvature. Thus the governing equations in the coordinate system are as follows.
@u l y @u @u uv y l 1 @p þ u þv þ ¼ @t l @x @y l q @x l " 2 ðl y Þ @ 2 u @ 2 u 1 @u u 2ðl yÞ @ v þ þm þ þ 2 2 @x2 @y2 l @y l2 @x l l y l dðl yÞ yðl yÞ dðl yÞ @u vþ 3 þ 3 dx dx @x l l @v l y @v @ v u2 y l 1 @p u þ þv þ ¼ l l q @x @t @x @y l " 2 2 2 ðl y Þ @ v @ v 1 @ v v 2ðy lÞ @u þm þ þ þ 2 2 @x @x2 @y2 l @y l2 l l l y dðl yÞ yðl yÞ dðl yÞ @ v þ 3 uþ 3 @x dx dx l l
Fig. 9. Curve of probability density of liquid film thickness.
ð14Þ
ð15Þ
8
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
Fig. 10. Curve of liquid film wavelet timing methods.
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
R @u @ v v þ þ ¼0 y þR @x @y y þR Order 11 1 1 1e Simplified equations can be obtained in Eqs. (18, 19, 20) when neglecting the higher order terms.
Fig. 11. Schematic diagram of two-dimensional boundary layer flow at CP corner in curvilinear coordinate system.
R @u @ v v þ þ ¼0 y þ R @x @y y þ R
ð17Þ
Order of each term in the above equations is as follows.
@u R @u @u u v þ u þv þ @t y þR @x @y y þR " R @p 1 R2 @ 2 u @ 2 u ¼ þ þ þ 2 y þR @x Re ðy þRÞ @x2 @y2
e
11
e 1 1 e2 2
1
1 @u u 2R @v R dR þ v y þR @y ðy þRÞ2 ðy þRÞ2 @x ðy þRÞ3 dx # y R dR @u þ ðy þRÞ3 dx @x Order 11 1e
1 1 12 12
e
1 13
1e
e
3
1
e 1 1 e e 1 11 1 1 e2 1 e2
e 1 12 12
1
1 13
@u @ v þ ¼0 @x @y
ð20Þ
11
The model of the liquid film with radius r and width dr is established as shown in Fig. 12 and basic assumptions are as follows. First of all, flow velocity only changes with ordinate. Besides, flow is assumed as a circular motion since the corner is arc-shaped. Finally, the thickness of the non-boundary layer can be ignored as liquid film thickness is very small. The velocity is considered to be 99% of the velocity of the mainstream. Besides, the flow velocity at the liquid–solid interface is considered to be zero. Thus, velocity in y-axis direction is as follows (Dadvand et al., 2009; Tan et al., 2009; Ghiaasiaan et al., 1997; Kim et al., 2001; Ju et al., 2018; Owen and Ryley, 1985; Alekseenko et al., 1985).
u¼
0:99U y d
Radial force at the position of radius r + dr and r are f + df and f, respectively. Therefore, radial resultant force and centrifugal force can be obtained in Eqs. (22 and 23).
f 1 ¼ ðf þ df Þðr þ dr Þda f rda ¼ da½dðf r Þ þ dfdr da dðf r Þ
ð22Þ
u2 rdrda r
ð23Þ
f 2 ¼ ql
As f 1 is equal to f 2 , thus
Z
Rl
"
Rs
# Z Rl Z Rl u2y ql rdrda dðf rÞda ¼ da ql u2y dr dðf r Þ ¼ 0 r Rs Rs
1
e1 1 e 13
ð21Þ
ð24Þ
e2
1 @v v 2R @u R dR u þ y þR @y ðy þRÞ2 ðy þRÞ2 @x ðy þRÞ3 dx # y R dR @ v þ ðy þRÞ3 dx @x Order 11 1
ð19Þ
11
@v R @v @v u2 þ u þv þ @t y þR @x @y y þR " R @p 1 R2 @2v @2v þ þ ¼ þ 2 y þR @y Re ðy þRÞ @x2 @y2 Order
@p ¼0 @y
4.2. Rupture model
d d ¼ f ðf < 1Þ L
Order 1 1 1 1
ð18Þ
ð16Þ
Eqs. (14, 15, and 16) are processed in a dimensionless method, that is, the term of the equation higher than the one order is ignored and afterwards the dimension is restored, which will result in a simplified governing equation. Take the dimensionless lengthd as the a term of f order. Thus
1 1e e 1
@u @u @u 1 @p @2u þu þv ¼ þm 2 @t @x @y @y q @x
R
f rjRls ¼ f ðRl Þ Rl f ðRs Þ Rs ¼
r1 R1
R1
r2 R2
R 2 ¼ Dr
ð25Þ
The surface tension at air–liquid interface and liquid–solid interface are r1 and r2 , respectively. Shear force and friction force are shown in Eqs. (26 and 27).
Fig. 12. Geometry and schematic diagram of CP wall, water film and air at the CP corner.
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B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
1 2
sal ¼ cf qg u2g sls ¼ ll
ð26Þ
@u 0:99U ¼ ll @y d
ð27Þ
Non-dimensional damping coefficient cf under laminar flow condition is shown in Eq. (28).
cf ¼
1:328
No.
H/ mm
h/ degree
d/ mm
L/ mm
1# 2# 3#
200 200 200
30.0 45.2 40.4
17.1 19.3 14.6
50 50 50
ð28Þ
Re0:5 g
As sls is equal to Eq. (29).
ug ¼ 2:4
Table 1 Parameters of the corrugated plate.
sal , critical airflow velocity can be obtained in
dcosh ðr1 r2 Þl2al
ql qg lg
!13
1 h
ð29Þ
According to Eq. (29), critical airflow velocity is positively correlated to d, h and it is negatively correlated to surface tension, density of gas and liquid phase. It is in complete agreement with the industrial corrugated plate dryer. 5. Results and discussion Parameters of the three kinds of corrugated plate are as follows in Table 1. Therefore, according to theoretical results in Eq. (29), the calculation equation of the critical airflow velocities of the three different parametric corrugated plates are as follows.
ug ¼
8:617 h
ð30Þ
ug ¼
8:415 h
ð31Þ
ug ¼
7:829 h
ð32Þ
The units of ug and h are m/s and mm, respectively. Liquid film thickness distribution in this experiment ranges from 0.2 to 1.65 mm. The airflow velocity is negatively correlated with the thickness of the liquid film. When the thickness of the liquid film is small, the liquid film flow is stable and gravity has little effect on the liquid film flow. Therefore, the mainstream of the liquid film is difficult to be breakdown and when the liquid film thickness is less than 0.4 mm, the value of critical airflow velocity is even above 20 m/s. As the Reynolds number increases, the coupling of gravity and airflow makes the liquid film more susceptible to be breakdown, thus the critical airflow velocity is reduced. It can be seen from Fig. 13 that the difference between the experimental value of the critical airflow velocity and the theoretical value gradually decreases as the Reynolds number increases. For the corrugated plate dryer with the same structural parameters, theoretical results of the critical airflow velocity are greater than experimental results. And there are two factors contributing to this result. First of all, the theoretical analysis is based on the two-dimensional laminar flow condition by ignoring the gravity. The coupling of gravity and airflow in the actual flow condition makes the liquid film more susceptible to be breakdown. In addition, the phenomenon of liquid film reflow that is common in corrugated plate dryers is observed at the bottom of the corrugated plate. The reason for this phenomenon is that when the airflow is blown over the surface of the liquid film mainstream, it will be subjected to the resistance and thus forming a cyclone below it, which makes the liquid film more susceptible to be breakdown (Karimi and Kawaji, 1999; Wang and Tian, 2019; Choi and Sang,
Fig. 13. Results of critical airflow velocity.
1996; Chung et al., 2014; Bashter et al., 1996; Fang et al., 2020). In summary, the theoretical value of the critical airflow velocity is slightly larger than experimental results. The distribution of experimental results in Fig. 13 is not uniform, thus it is difficult to describe them with a single function. The thickness of the liquid film can be artificially divided into three sections, namely 0–0.68 mm, 0.68–1.11 mm and 1.11–1.6 mm. The relationship between the liquid film thickness and the critical airflow velocity in actual conditions is not a simple inverse proportional correlation derived from the simplified two-dimensional model. And the distribution shape of experimental results is similar to the common nike function in mathematics and the equation is shown in Eq. (33).
f ð xÞ ¼
k1 þ k2 þ k3 x x
ð33Þ
Therefore, experimental results are fitted in the form of nike function shown in Eq. (34). And according to the structural parameters of the corrugated plate and derivation results of the model, the fitting equations for the calculation of the critical airflow velocity of three corrugated plates with different structural parameters are revealed in Eqs. (35, 36, 37, 38, 39, 40, 41, 42 and 43).
ug ¼
k1 þ k2 þ k3 h h
ð34Þ
When the liquid film thickness is in the range of 0–0.68 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
4:378 þ ð2:89Þ þ 2:60h h
ð35Þ
2#ug ¼
9:137 þ ð17:444Þ þ 21:341h h
ð36Þ
3#ug ¼
11:511 þ ð27:701Þ þ 30:558h h
ð37Þ
B. Wang et al. / Annals of Nuclear Energy 135 (2020) 106946
When the liquid film thickness is in the range of 0.68–1.11 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
12:833 þ ð13:247Þ þ 7:914h h
ð38Þ
2#ug ¼
11:236 þ ð9:803Þ þ 5:567h h
ð39Þ
3#ug ¼
7:988 þ ð2:922Þ þ 1:734h h
ð40Þ
When the liquid film thickness is in the range of 1.11–1.65 mm, the fitting equation of critical airflow velocity of the 1#, 2#, 3# corrugated plates are as follows.
1#ug ¼
14:683 þ ð9:093Þ þ 0:556h h
ð41Þ
2#ug ¼
12:943 þ ð4:505Þ þ 0:760h h
ð42Þ
3#ug ¼
19:140 þ ð25:578Þ þ 7:517h h
ð43Þ
Fig. 13 reveals that above fitting results agree well with experimental results. Nevertheless, it is unable to obtain a formula for calculating the critical airflow velocity that can be applicable to all the corrugated plate dryers. Authors of this paper are still trying to find out the correlation between these empirical equations and the structural parameters of the corrugated plate to achieve the unification of the calculation equation, which will be the prospective research direction. 6. Conclusions and prospects The free falling film thickness is measured by the PLIF method, and the liquid film thickness curve is obtained. The probability density distribution (PDF), power spectral density (PSD) and wavelet method of liquid film thickness time series are calculated by MATLAB program. Besides, the amplitude and frequency domain information of liquid-film fluctuations are analyzed. The critical airflow velocity of three kinds of corrugated plates is measured experimentally. Ultimately, combined with theoretical results of this paper, a semi-empirical correlation for the critical airflow velocity is obtained. The specific conclusions are as follows. (1) The evolution of the surface wave of liquid film is carried out in time and space. The PDF features (probability distribution characteristics) and PSD characteristics (power spectrum characteristics) of surface waves with different wave characteristics also have significant differences, which can be used as the judgment benchmark of characteristics of liquid film volatility. The PDF curve has no obvious bimodal characteristics. And the PSD curve and the wavelet curve have no significant peak characteristics. Therefore, the spectrum has no characteristic frequency, that is, the liquid film has multifrequency characteristics. (2) The energy of the liquid film volatility is mostly concentrated in the low frequency region and the appearance of the solitary wave causes the volatility of the liquid film to increase sharply, thus the volatility of the liquid film volatility is enhanced. (3) Factors affecting the development of liquid film volatility instability mainly include liquid film thickness, wave frequency and longitudinal component of gravity. The increase of the thickness of the liquid film accelerates the longitudinal evolution of the liquid film volatility, and the same
11
longitudinal position exhibits different volatility patterns due to the difference in liquid film thickness. The fluctuating frequency plays a substantial role in the spatiotemporal evolution of the surface wave characteristics of the liquid film. The longitudinal component of gravity can accelerate the evolution of surface waves and is an unstable factor. Therefore, the impact of the above three factors on the analysis of liquid film fluctuations is the prospective research trend. In this paper, the first two influencing factors are analyzed, but the influence of gravity on the liquid film is not considered enough, which will be improved in future research. (4) Both experimental and theoretical results reveal that the critical airflow velocity is negatively correlated with the liquid film Reynolds number. Comparing results with theoretical results of the critical airflow velocity as derived in this paper, the fitting equations for the critical airflow velocity are obtained. And the equations fit well with experimental results. Nevertheless, theoretical results of the critical airflow velocity in this paper are based on many hypothetical preconditions and it is derived in two-dimensional coordinate system under laminar flow conditions. However, the liquid film flow in the corrugated plate under the action of the air flow is a typical strong turbulence problem, thus the derivation of the liquid film breakdown turbulence model under the three-dimensional condition considering gravity will be the prospective research direction. Besides, Eqs. (35, 36, 37, 38, 39, 40, 41, 42 and 43) are not universal because they are just fit for the certain structure of the corrugated plate. According to nine equations, it is different to conclude a universal equation. The research of a universal to calculate the critical airflow velocity is the prospective study. Furthermore, the flow inside the corrugated plate is a strong turbulence phenomenon, but many calculations in the laminar flow state are used in the derivation. However, the liquid film Reynolds number of the experiments performed in this paper is relatively small. Therefore, within a certain range, the formula of laminar flow is available. For experimental and theoretical derivation in the case of strong turbulence, they will be studied in the future.
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