Wave characteristics of the falling liquid film in the development region at high Reynolds numbers

Journal Pre-proofs Wave characteristics of the falling liquid film in the development region at high Reynolds numbers Zongyao Wei, Yifei Wang, Ziwei Wu, Xin Peng, Guangsuo Yu PII: DOI: Reference:

S0009-2509(19)30944-3 https://doi.org/10.1016/j.ces.2019.115454 CES 115454

To appear in:

Chemical Engineering Science

Received Date: Revised Date: Accepted Date:

15 June 2019 24 October 2019 21 December 2019

Please cite this article as: Z. Wei, Y. Wang, Z. Wu, X. Peng, G. Yu, Wave characteristics of the falling liquid film in the development region at high Reynolds numbers, Chemical Engineering Science (2019), doi: https:// doi.org/10.1016/j.ces.2019.115454

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Wave characteristics of the falling liquid film in the development region at high Reynolds numbers Zongyao Wei, Yifei Wang*, Ziwei Wu, Xin Peng, Guangsuo Yu Shanghai Engineering Research Center of Coal Gasification, Institute of Clean Coal Technology, East China University of Science and Technology, Shanghai 200237, China

Abstract: The wave characteristics of the turbulent falling liquid film in the development region along the vertical Perspex plate were studied by using an Ultrasonic Doppler Velocimetry (a non-intrusive technique). The instantaneous thickness of the falling film at the center of the plate with axial distance ranges from 50 mm to 1300 mm was measured and the Rel （the liquid film Reynolds number）ranges from 2.28×103 to 1.43×104. The results show that the mean thickness, wave frequency and amplitude increase while the growth rate decreases with the increase of Rel. The Probability Density Function of the liquid film instantaneous thickness tends to an approximately normal distribution with increasing Rel. Furthermore, the Plateau-Rayleigh instability can cause the breakup of the liquid film surface and resist the increase of the liquid film thickness, which results in the fluctuation range of the liquid film approximate to a fixed value when the Rel＞1.20×104. Keywords: liquid falling film, film thickness, statistical property, breakup

1. Introduction The falling film is an efficient heat transfer method with the advantages of small flow rate, small temperature difference, high heat-mass transfer coefficient, and simple structure. Hence, the falling film system is widely used in many fields, such as chemical engineering, thermal engineering, food industry, and nuclear industry (Commenge, 2017; Lin, 2018; Zhang, 2018; Du, 2018.). Many studies have been carried out on the characteristics of the liquid film flow. The experimental studies of the liquid film flow show that the liquid film is always divided into laminar flow, fluctuating laminar flow, turbulent flow, and fluctuating turbulent flow (Dai et al, 2005). Nusselt (1916) proposed a laminar-flow models for the liquid film. The liquid film was assumed to be a thin film with infinite width, smooth interface, constant physical parameters, and no heat transfer in the flow direction. However, the surface of the liquid film still fluctuated even when the Re was small. According to a study of Telles and Dukler (1970), the laminar flow would turn into the turbulent flow when Re≥1000 and the mean film thickness was larger than Nusselt’s prediction in the turbulent film flow. Besides, Ambrosini et al. (2002) studied the process from laminar flow to turbulent and illuminated the flow state near the transition as well as the influence factors of the threshold. Guzanov et al. (2018) studied the wave evolution on the surface of the isothermal liquid film falling down a vertical plate and found three typical scenarios of the

evolution in the range of low and moderate liquid flow rates. The transverse modulation of the 2D waves was the main evolution mode at the low liquid flow rate. The 2D waves broke up into the 3D waves when the Rel ＞40 (this value depended on the physical properties of the working fluid). Then, when the Rel＞400, the fully developed steady 3D waves were observed. Even at a low Rel, a small disturbance can cause a large amplitude fluctuation on the liquid film surface (Aktershev et al, 2013). Initial quiescent smooth film flow, twodimensional regular solitary wave pattern riding on film flow, and three-dimensional irregular wave patterns were recorded by Lu et al. (2016) via shadowgraphy with a moderate Rel conditions. And Lu et al. found that the wave evolution on the film flow at moderate Rel is controlled by the gravity, drag, and the Rayleigh-Taylor instability. Meanwhile, the viscous force of the fluid plays a critical role in the laminar flow, as it can increase the stability of the film flow due to the damping. According to the research of Mudawar et al. (1993), there are many obvious differences between high viscous fluids and low viscous fluids. Especially, in turbulent flows, the fluctuation of the liquid film is determined by the inertial force resulting from the change of instantaneous velocity of the liquid film. The vortex is one of the main flow models of the fluid, which exists universally in the liquid film flow. The change of the vortex inside liquid films influences the fluctuation of the liquid film surface. Doro (2013) identified that the change of the streamwise pressure gradient at the teardrop-shaped and capillary wavefront was the

main reason for capillary wave and vortex backflow. Meanwhile, Alekseenko et al. (2007) clarified that the increase of the wave amplitude will slightly change the location of the vortex center. Furthermore, with the increase of inclination angle and the corresponding growth of longitudinal velocity, the vortex motion becomes less intensive and the vortex center shifts toward the interface. Considerable research efforts have been devoted to the prediction of film flow. A long-wave equation was derived by Sheintuch and Dukler (1989), including the viscidity, surface tension, and interfacial shear effects for film thickness. Then, Rob and Boersma (1995) applied the Orr-Sommerfeld equation to study the stability of thin liquid films under the shear of the downstream gas. Kalhadasis and Demekhin (2003) established the integral-boundary-layer equations for a viscous fluid flowing down a uniformly heated planar wall. Aktershev and Alekseenko (2005) derived the two-wave equation for film thickness with phase transition and dispersion formulas. Furthermore, there are many excellent literature dealing with the mechanism of liquid film flow by CFD simulation (Szulczewska et al, 2003; Valluri, 2005; Helbig, 2005; Trifonov, 2017; Landel, 2015), but their Reynolds number range is all less than 1000. Similar to the above work, much work so far has focused on the liquid film flow with a low Reynolds number condition (Re<1.0×104) or the liquid film distribution in stable region. However, little attention has been paid to the wave characteristic of the liquid film flow with high Reynolds number and development region. In fact, turbulence and fluctuation of the falling film are very drastic in some industrial devices

(such as Scrubbing-Cooling Chamber (Yan et al, 2017)) and the main heat transfer area of those devices is in the development region. Thus, it is important to study the fluctuation of the liquid film in the development region under high Reynolds number for industrial application. Kapitza has pointed out earlier that the reduction in film effective thickness is the main reason for the enhancement in heat transfer. Jayanti et al. (1997) proposed that even in the presence of surface waves, the heat conduction process of wall rather than the backflow zone determines the heat transfer coefficient of the free falling film. Moreover, they also observed the heat transfer coefficient increases as the film effective thickness decreases due to the surface waves. In the Opposed Multi-Burner (OMB) gasifiers, the high-temperature syngas from the upper combustion chamber enters the scrubbing-cooling chamber through the scrubbing-cooling tube. The syngas is directly contacted with the liquid falling film flowing along the inner wall of scrubbing-cooling tube. In this process, the liquid film will cool, humidify and scrub the high-temperature syngas, and collect the slag particles. During the cooling process, the heat-mass transfer coefficient will be reduced, and the cooling efficiency will be deteriorated due to the large film thickness. The film break and drywall can be observed when the liquid film is too thin (this critical film thickness is related to the Minimum wetting rate) (Yu et al., 2014), which may make the hightemperature syngas directly contact with the tube wall and cause the deformation of the downcomer and endanger production.

The falling film cooling system in the scrubbing-cooling chamber of the OMB gasifier was investigated in this study with the establishment of the single-channel scrubbing cooling tube model. In addition, in order to provide the theoretical basis of steady operation and enlargement for the cooling system of the OMB gasifier, the distribution of the turbulent falling film thickness in the development region was measured by Ultrasonic Doppler Velocimetry (UDV).

2. Experiments system 2.1. Experiment setup The experiments were carried out in a Perspex square tube with the dimensions H=1500 mm in vertical height and 80 mm×80 mm in inner cross section. Water was pumped from the water tank and then transported to the square tube inlet. The flow rate of the water was controlled by the flow meter. A buffer tank was installed before the plate inlet. The liquid films were formed by a slot distributor with 3 mm×80mm in inner cross section, and flowed along a vertical plate on one side of the square tube. Finally, the film flowed out of the bottom of the square tube into the water storage tank for recycling. The instantaneous thickness of the liquid film at the center of the plate with the axial distance ranging from 50 mm to 1300 mm was measured by UDV. The sketch of the experimental setup is shown in Fig.1. The coordinate system is shown in Fig.2, and the main experimental conditions are shown in Table 1. Moreover, the liquid film Reynolds number Rel is defined as: Rel 

4



(1)

where Γ is the mass flow rate per unit circumferential length of the film and μ is the

dynamic viscosity. Table 1 Experiment conditions Temperature (K)

Pressure (atm)

Ul (m/s)

Rel

287±2.5

1.0

0.46~2.89

2.28×103~1.43×104

2.2. Ultrasonic Doppler Velocimetry (UDV) The Ultrasonic Doppler Velocimetry (UDV) method based on ultrasonic technology was used and the principle of UDV method is shown in Fig.3. The sensor emits ultrasonic waves with a certain frequency to the experimental object, and then the sound waves propagate in the fluid after passing through the propagation medium and the pipe wall. Since there is a certain angle (Doppler angle, less than 90°) between the ultrasonic wave and fluid flow, the sound waves will reflect when it encounters the tiny particles such as solid particles and bubbles in the fluid, which changes the frequency of the sound waves. This phenomenon is named the Doppler Effect. The reflected wave time of particles at different positions received by the sensor is different. After the dynamic spectrum analysis and signal calculation, the particle velocity of the flowing fluid at different positions on the ultrasonic propagation path is obtained and the particle velocity can be regarded as the fluid velocity at the same position (Fischer et al, 2008; Jaafar et al, 2009). The UDV was applied to measure the instantaneous thickness of the liquid film. Ultrasonic waves generate a strong reflected wave when passing through the wall-water and water-air interfaces. The time and intensity of the reflected waves were obtained by the transducer. Then, according to the time difference between the two receiving

reflected waves and the propagation speed of the ultrasonic waves in the medium, the distance between the two interfaces as the thickness of the liquid films can be calculated. The ultrasonic technique was used by Li et al. (2004), Azevedo et al. (2017), Jayakumar et al. (2019) to measure the thickness of the liquid film with different flow conditions. And the results show that the measurement of liquid film thickness via ultrasonic probe is accurate.

5 6

7

8

2

10 1

3

4

11

9

Fig. 1. Schematic diagram of the experimental apparatus. (1- water pump; 2- flowmeter; 3- Pump; 4- recycle tank; 5- slot distributor; 6- water inlet; 7- single channel scrubbing cooling tube; 8- transducer; 9-water tank; 10-UDV; 11- computer） The frequency of the sound wave was 4 MHz and the diameter of the transducer detection area was 5 mm. The spatial resolution of the film thickness measurement was 0.12 mm, and the time resolution was 10ms. In the experiments, the sample rate was 100 records per second and the Doppler angle was 85°.

2.3 Error analysis In this paper, the physical properties of the water were obtained by the

Thermophysical Properties of Fluid Systems of the National Institute of Standard and Technology (NIST). The main errors in this work were analyzed as follows. 2.3.1 The Error of the sound velocity The parameters of the sound velocity used by the UDV were obtained via NIST and adjusted in real-time according to the fluctuation of liquid film temperature. In our previous work, the error of the sound velocity in the water has been calculated. The sound velocity obtained via NIST was recorded as VN, and the sound velocity measured by UDV was recorded as VM. The error of the sound velocity in the water was defined as:

E

VN  VM VM

100%

(2)

The maximum error of the sound velocity in this paper is 0.49%. y Transducer

Large wave

Transmitting media

δ

Small wave

Scattering volumes

Flow

Mean film thickness Substrate Velocity profile

Amplitude x

A

Fig. 2. Schematic diagram of the falling film.

Fig. 3. Principle of UDV measurement.

2.3.2 The accuracy of thickness measurement Fig.4 shows the experimental setup for the accuracy of thickness measurement.

The HD images of the still water were taken by the digital camera, and the true height of still water was obtained by image processing software ImageJ. The true height was recorded as hI, and the height measured by UDV was recorded as hM. Therefore, the accuracy of thickness measurement was calculated as follows:

E

hM  hI hI

100%

(3)

In this paper, the accuracy of the thickness measurement is 0.53%.

Fig. 4. Experimental setup for the accuracy of thickness measurement 2.3.3 The precision of thickness measurement The mean thickness of the falling film at a fixed position was measured 20 times repeatedly. Then, the Relative Standard Deviation (RSD) of the mean thickness obtained by repeatability measurement was expressed as the precision of thickness measurement, and the RSD was defined as follows:

RSD 

 δ

i

δ



(4)

δ

Where 𝛿𝑖 is the per measurement value. As a result, the maximum RSD is less than

3.00%. 2.3.4 Flow rate pulsations of the film formation The flow rate of the falling film at the flow exit was measured repeatedly. The result shows that the maximum RSD for the mean flow rate of the falling film at the flow exit is 2.93%. Thus, the flow rate pulsation of the film formation was small. The main errors of the experimental system were summarized as follows:

Table 2 Main errors of the experimental system Measuring apparatus Temperature Volume flow rate

Raytek-raynger ST60

±1%（or ±1℃）

Rotor flowmeter LZB40

Slot distributor

Vernier caliper

Sound velocity

NIST

Film thickness

UDV

Errors ±2% 150mm±0.02mm 0.50%

Accuracy

0.53%

precision

3.00%

2.4 Sidewall effects For the turbulent falling film flowing along with a finite width plate, the sidewall effects were prominent. Fig. 5 shows the uniformity of falling film thickness in the central region (within 20 mm on each side of the centerline) of the plate. The RSD of the liquid film thickness at different horizontal positions is used to characterize the uniformity of the liquid film in the horizontal direction. The area from the flow exit to the 1/4 pipe length is the main region of heat exchange, which syngas can achieved most temperature drop (Peng et al, 2010). As shown in Fig. 5, at x=50 mm and x=300 mm, the RSD is close to 0.05. The results show

that the horizontal distribution of the liquid film in the main heat exchange area, which is the focus of research, is relatively uniform. Then, the sidewall effect becomes obvious as the flow distance increases. As can be seen from Fig.5, the RSD at x=750 mm, increases with the increase of Rel and is significantly greater than the other height positions. However, the research shows the heat exchange in this region is slight (Peng et al, 2010). Then, the flow state of the liquid film tends to be stable and the distribution of the liquid film also becomes uniform. The flow behavior of the turbulent falling film is a complex three-dimensional flow problem, and it is difficult to comprehensively analyze this process. Furthermore, it is inaccurate to illustrate the axial distribution of the liquid film by using the mean value of the liquid film thickness at different horizontal positions. The distribution rule of the falling film thickness at the centerline can give a good reflection on the development of the falling film at the top and bottom of the plate, and also can explain the change of the falling film in the middle of the plate. Therefore, we mainly focused on the wave characteristics of the falling film at the centerline. Besides, a further study on the three-dimensional flow problem of the turbulent falling film could be carried out based on this.

0.5

50mm 300mm 750mm 1200mm

0.4

RSD

0.3

0.2

0.1

0.0

6000

8000

10000

12000

14000

Rel

Fig. 5. Uniformity of the falling film thickness in the central region.

3. Results and discussions 3.1. Liquid film thickness distribution 3.1.1. Axial distribution The experimental section of this paper is 1.5m in length, which can be considered that the liquid film has been fully developed and the accurate statistics such as maximum, minimum and mean film thickness or turbulent characteristics can be obtained (Karapantsios, 1995; Takahama et al., 1980; Portalski et al., 1972). The axial distribution of the liquid film time mean thickness is shown in Fig. 6. From Fig.6, the mean thickness of the liquid film is divided into three regions as the axial distance increases: the inlet region I, the development region (the film thickness increase region II and the film thickness decrease region III) and the stable region Ⅳ. With the increase of the axial distance, the mean thickness of the liquid film in the region Ⅰ rapidly decreases, and increases at first and then decrease in the development region. For the liquid film in the stable region, the mean thickness no longer changes

significantly. However, the distribution of the falling film thickness on the whole plate which under the different Rel has a large difference. When Rel＜7.50×103, as shown in Fig. 6 (a), the region Ⅰ and the region Ⅳ are evident. The mean thickness of the liquid film from the exit of the slot distributor to the x=200mm decreases rapidly, due to the velocity increase of the liquid film. For the turbulent vertical falling film, gravity and wall shear stress are the main forces (Mascarenhas, 2013). The direction of gravity is vertically downward, which accelerates the liquid film; and the direction of wall shear stress is vertical upward, which is the main resistance in the film flow. The wall shear stress will cause a velocity gradient in the liquid film and slow the film. At the inlet of the plate, the gravity effect of the liquid film is greater than the resistance caused by the wall shear stress, thus the liquid film accelerates. As a result, the thickness of the liquid film is reduced to maintain the flow cross-sectional area. In the development region, the mean thickness of the liquid film presents a small fluctuation, and the boundary between region Ⅱ and Ⅲ are vague. When x ≥ 950mm, the mean thickness of the liquid film no longer changes significantly with the increase of the axial distance, indicating that the development of the liquid film reaches a steady state. The reason is that the liquid film at the end of the plate is thinner and faster, the gravity and wall shear stress of the liquid film is balanced macroscopically. With the increase of Rel, as shown in Fig. 6 (b), the length of the region Ⅰ shortened greatly, and the region Ⅱ and Ⅲ can be seen clearly in the development

region, and the origination of the region Ⅳ moves backward. In the development region, the mean thickness of the liquid film increases at first and then decreases. There are three reasons for the increase of the film thickness in the region Ⅱ. Firstly, the increase of the liquid film velocity caused by gravity is slight because the liquid film has a large initial velocity. Therefore, the decrease of the flow cross-sectional area caused by the increase of velocity is unobvious. Then, the formation of the liquid film is a process in which the liquid flows to a semi-infinite space from slit, similar to the jet. The liquid film has a velocity component in y direction which increases with the increase of Rel, and adds the film thickness. Finally, the liquid film has a horizontal velocity component, probably caused by the jet effects, turbulence effects or other factors. And the baffling of the sidewall causes the upstream transverse flow converge in the downstream central area and increases the thickness of the liquid film. However, the film thickness cannot increase continuously because of the weakening of baffling, the flow area of liquid film extends to the sidewall, and the breakup of surface waves. As shown in Fig.6 (b), at x=500mm, the mean thickness of the liquid film reaches the maximum, and then the mean thickness of the liquid film monotonously decreases until stable.

5

Ⅱ+Ⅲ

3

2

Rel=6.85×103 Rel=7.43×103 Rel=8.00×103

4

/mm

/mm

4

Ⅰ

5

Rel=2.28×103 Rel=2.86×103 Rel=3.43×103 Ⅳ

3

2

Ⅱ 1

Ⅳ

1 0

200

400

600

800 x/mm

1000 1200 1400

0

(a)

200

400

600

800 x/mm

1000 1200 1400

(b)

5

Rel=1.31×104 Rel=1.37×104 Rel=1.43×104

4

/mm

Ⅲ

3

Ⅲ

Ⅱ

Ⅳ

2

Fig. 6 Axial distribution of the liquid film mean thickness at different Rel.

1 0

200

400

600

800 x/mm

1000 1200 1400

(c) When Rel increases further, the region Ⅰ disappears, and the critical Rel≈7.50×103. Besides, as for the highly turbulent film (Rel ≥1.0×104) shown in Fig.6 (c), the film mean thickness in the region Ⅱ increases first and then stabilizes. Because the liquid film thickness cannot increase due to the breakup of the liquid film surface, this phenomenon will be discussed in detail in the section 3.3. Generally, the increase of Rel will reduce the influence of the region Ⅰ, but the region Ⅱ, Ⅲ and Ⅳ will be clearer. When Rel ＞7.50×103, the inflection point between region Ⅱ, Ⅲ and Ⅳ are fixed to x=500mm and x=1200mm. 3.1.2. Influence of the liquid film Reynolds number

In order to obtain the universal distribution of the liquid film thickness, the nondimensional film thickness β is defined (Mascarenhas, 2013): 2

 δu *  3  β   ν  

u* 

τw  δg ρ

(5)

(6)

where δ is the liquid film thickness, u* is the friction velocity, τw is the wall shear stress. And τw can be calculated by the force balance of the liquid film，as for the vertical falling film τw=ρgδ. The distribution of the non-dimension mean thickness of the liquid film at different axial positions with Rel is shown in Fig. 7. On the whole plate, the liquid film thickness increases with the increase of Rel. However, results in Fig. 7 (a) and (b) show that the liquid film thickness increases with the increase of Rel, while the growth rate decreases gradually. Because the film thickness decreases rapidly after leaving the slit at low Rel, the influence of Rel increase is obvious for the thin film. As the continuous increase of Rel, the film fluctuates greater, as well as the wave amplitude. However, the liquid film cannot maintain the complete waveform due to the breakup of the liquid film surface, hence the growth rate of the liquid film thickness reduce gradually. On the bottom part, as shown in Fig. 7 (c) and (d), the development of the liquid film tends to be stable. With the increase of Rel, the thickness of the liquid film increases monotonously, but the growth rate of the liquid film thickness reduces with the increase of the axial distance.

100

100

x=50mm x=200mm

x=300mm x=450mm

60

60



80



80

40

40

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0

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10000

8000

12000

0

14000

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10000

12000

14000

10000

12000

14000

Rel

(a)

(b)

100

100

x=750mm x=900mm

x=1200mm x=1300mm

60

60



80



80

40

40

20

20

0

4000

6000

8000

10000

12000

14000

0

4000

Rel

6000

8000

Rel

(c) (d) Fig. 7. Distribution of the non-dimensional mean film thickness at different axial positions with the effect of Rel

3.2. Statistical property The main characteristic of turbulence is randomness, and its physical quantity changes randomly with spatial and temporal. It is extremely difficult to describe the turbulence fully in general. However, according to the turbulent statistical theory, there is accurate statistical characteristics in turbulence, although its behavior is disordered. 3.2.1. Liquid film instantaneous thickness distribution The distribution of the liquid film instantaneous thickness at different Rel is shown in Fig. 8. During the experiment, the liquid film is in a high turbulent state and the liquid film instantaneous thickness distribution at different heights has a large difference.

In the region Ⅰ, when Rel is low (less than 7.5×103), there are obvious solitary waves on the liquid film surface, and the relatively smooth liquid film dominated by capillary can still be seen between two large waves. With the increases of Rel, the fluctuation of the liquid film in the region Ⅰ is intensified, and the smooth part disappears gradually. As for the liquid film in the development region, the wave amplitude increases distinctly. As the Rel increases, the maximum thickness of the liquid film in the region Ⅳ changes rarely, However, the substrate thickness has an obviously increase. 8

8

x=50mm

6 4

4

2

2

8

8

x=300mm

6 4

4

2

2

8

x=750mm

6 4

x=300mm

6

/mm

/mm

x=50mm

6

8

x=750mm

6 4

2

2

8

x=1300mm

6

8 4

2

2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x=1300mm

6

4

0.0

t/s

0.5

1.0

1.5

2.0

2.5

3.0

t/s

(a) Rel= 5.71×103 (b) Rel= 1.43×104 Fig. 8. Liquid film instantaneous thickness distribution. ·3.2.2 Frequency statistic It is generally accepted that the falling film comprise mainly of (a) a thin liquid substrate; (b) large waves, which are surface undulations with height larger than the mean film thickness; and (c) ripples, which are surface undulations with height smaller than the mean film thickness (Kostoglou et.al, 2010). Fig. 9 (a) ~ (d) shows the frequency of surface waves with the effect of Rel. In the region Ⅰ and Ⅱ, the ripple frequency increases first, then decreases and finally tends

to be smoothly with the increasing Rel, and the inflection points are Rel≈7.00×103 and Rel≈1.00×104. In the region Ⅲ, the ripple frequency decreases slowly. Furthermore, the frequency of the ripple in the region Ⅳ changes little. And the profiles of the large wave frequency with Rel are opposite to ripples’. Moreover, it can be seen from Fig. 9 (e) that the ratio of the large wave frequency and ripple frequency tends to a constant with increasing Rel. 100

100

Large waves Ripples

Large waves Ripples

60

60

f, Hz

80

f, Hz

80

40

40

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20

0

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0

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Rel

(a) x=50mm

(b) x=300mm

100

100

Large waves Ripples

Large waves Ripples

60

60

f, Hz

80

f, Hz

80

40

40

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Rel

(c) x=750mm

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(d) x=1200mm

12000

14000

x=50mm x=300mm x=750mm x=1200mm

fsmall wave /flarge wave

4

Fig. 9 (a) ~ (d) Frequency of surface 2

waves and (e) The ratio of the large 0

waves frequency to the ripple frequency. 4000

6000

8000

10000

12000

14000

Rel

(e) Fig. 9 also shows that, for turbulent falling liquid film, the effect of Rel on the surface wave frequency decreases with the increase of flow distance. Meanwhile, the evolution between large waves and ripples mainly occurs in the upper part of the plate. Besides, the increase of the liquid film fluctuation is starting with the increase of the ripple frequency. The frequency of large waves begins to increase when Rel is greater than a critical value. This is owing to the phenomenon that the interaction between ripples can promote the formation of large waves. Adomeit and Renz (2000) found that the interaction between waves and substrate dominates the surface waves of liquid films at low Reynolds number when they studied the flow mechanism of 3D waves in fluctuation laminar flow; however, at a high Reynolds number, the collision between waves that usually occur in ripples is the critical. In the initial stage of wave formation, ripples collide continuously, and the rate of the resonance wave generating during the fusion process is more than 20% faster than the ordinary ripples (Song et al, 2012). Then, the resonance waves will catch up with the front waves and collide with them. There are two consequences of the collision:

one is resonance waves interact with the front waves which thinner and slower, then slow down and disappear gradually; the other is resonance waves collide with ripples repeatedly, and as a result, the mass accumulates and forms a large wave. This is the main reason why the ripple frequency increases first and then decreases with increasing Rel in the region Ⅰ and Ⅱ. Moreover, large waves will change local Reynolds number around it, causing the liquid film to produce a significant strain structure (Adomeit and Renz, 2000) and promote the ripples appear. It is generally accepted that the large wave can be regarded as a liquid block that slides on the liquid film. The wave with a large mass and high speed, as described previously, will decelerate and frequently cause the local turbulence because the direct interaction with the slower substrate. Meanwhile, collisions between large waves will also change the local flow structure, produce a local high shear stress and induce turbulent vortexes. This kind of the liquid film strain structure caused by large waves can promote the formation of ripples. And this is the main factor in the decrease of the ripple frequency reduction amplitude when Rel≥1.00×104 in Fig. 9 (a) (b). 3.2.2. Probability density function (PDF) In order to analyze the wave characteristics of the falling film flowing along the plate, the Ksdensity function in Matlab is used to fit the Probability density function (PDF) of the liquid film instantaneous thickness distribution. Fig. 10 shows the PDFs of the liquid film instantaneous thickness distribution at different heights with Rel. The PDFs of the liquid film instantaneous thickness at x=50mm in the region Ⅰ

are shown in Fig. 10 (a). With the increase of Rel, the PDFs peak shifts to the right, the peak height appears an “increase- stabilization-decrease” change, and the PDFs width decreases at first and then increases. Overall, the PDFs develops from “sharp” to “flat”, and the PDFs in region Ⅰ are more sharp than other positions. In the region Ⅱ, as shown in Fig. 10 (b), with the increase of Rel, the PDFs peak shifts to the right, the peak height increases first and then decreases, while the PDFs width decreases first and then increases. Compared with other locations, the curves more flat and the change amplitude is larger. This means that the film in region Ⅱ is more sensitive to Rel. The film in this region has a bigger thickness distribution range indicating that the film fluctuates most. Similarly, with the increase of Rel, the PDFs of the liquid film instantaneous thickness in the region Ⅲ shift to the right, the height decreases, and the width increases, but the change amplitude is small. In the region Ⅳ, the PDFs of the liquid film instantaneous thickness at different Rel have a considerable overlap area witch indicate that the effect of Rel is small. 2.0

1.0

Rel=3.43×10

3

Rel=4.57×10

3

Rel=5.71×10

3

Rel=7.43×10

3

Rel=9.14×10

3

Rel=1.14×10

4

Rel=1.43×10

4

0.5

Rel=2.28×103 Rel=3.43×103 Rel=4.57×103

1.5

Rel=5.71×103 Rel=7.43×103

PDF

1.5

PDF

2.0

Rel=2.28×103

Rel=9.14×103

1.0

Rel=1.14×104 Rel=1.43×104

0.5

0

2

4

  mm

(a) x=50mm

6

8

0

2

4

  mm

(b) x=300mm

6

8

2.0

1.0

Rel=3.43×10

3

Rel=4.57×10

3

Rel=5.71×10

3

Rel=7.43×10

3

Rel=9.14×10

3

Rel=1.14×10

4

Rel=1.43×10

4

Rel=2.28×103 Rel=3.43×103 Rel=4.57×103

1.5

Rel=5.71×103 Rel=7.43×103

PDF

1.5

PDF

2.0

Rel=2.28×103

Rel=9.14×103

1.0

Rel=1.14×104 Rel=1.43×104

0.5

0.5

0

2

4

6

  mm

8

0

2

4

6

  mm

8

(c) x=750mm (d) X=1300mm Fig. 10. Probability density function of the liquid film thickness In order to describe the change of PDFs of the liquid film instantaneous thickness more accurately, the Skewness Sδ and Kurtosis Kδ are calculated in this paper. The Skewness Sδ and Kurtosis Kδ are defined as follows: δ i δ  n   Sδ   (n  1)(n  2)  SD 

3

(7)

4

δ i δ  3(n  2) 2 n(n  1)    Kδ   (n  1)(n  2)(n  3)  SD  (n  2)(n  3)

(8)

where the SD is the Standard Deviation of the liquid film instantaneous thickness:

SD 

δ

i

δ



2

(9)

n 1

As for the normal distribution, the S and K are both 0. 70

x=50mm x=200mm x=300mm x=450mm x=750mm x=900mm x=1200mm x=1300mm

6 5

S

4 3 2

x=50mm x=200mm x=300mm x=450mm x=750mm x=900mm x=1200mm x=1300mm

60 50 40

K

7

30 20

1

10 0

0 -1

4000

6000

8000

10000

Rel

12000

14000

4000

6000

8000

10000

Rel

12000

14000

(a) Skewness (b) Kurtosis Fig. 11. Skewness Sδ and Kurtosis Kδ Fig. 11 shows the Skewness Sδ and Kurtosis Kδ distribution of the PDFs of the liquid film instantaneous thickness at different axial positions. As shown in Fig. 11 (a), the Sδ of the PDFs is greater than 0. This indicates that the PDFs of the liquid film instantaneous thickness are positive skewness in most cases, which means the right side of the PDFs are longer than the left side, in other words, the data that δ＜𝛿 is majority and the fluctuation of the liquid film is dominated by ripples. In the region Ⅰ, such as x=50mm and x=200mm, the Sδ of the PDFs increases at first and then decreases. This is because the increase of the liquid film fluctuation is starting with the increase of the ripple frequency. The Skewness distribution of the PDFs in the region Ⅱ is similar to that in the region Ⅰ, but the transition point advances. Besides, the Sδ is negative at Rel ≈1.40×104, which means the number of large waves is larger than ripples’ and the liquid film in the region Ⅱ fluctuates most. In the region Ⅲ and Ⅳ, as the flow distance increases, the effects of Rel on the PDF Sδ gradually weakened. And the Sδ fluctuates around 1.51 at the end of the plate. The main reason for this is that the evolution of the ripple and the large wave on the film surface approaches a dynamic equilibrium with the increase of flow distance. The Kδ of the PDFs of the liquid film instantaneous thickness at different axial positions is shown in Fig. 11 (b). In most cases, the Kδ of the PDFs is greater than 0. The positive value indicates that the distribution of the liquid film thickness is steeper than the normal distribution, and the film thickness distribution has a mode.

When Rel≤4.0×103, the Kδ of the liquid film instantaneous thickness PDFs in the region Ⅰ and Ⅱ is larger than that in other cases. At low Rel, the liquid film has a long inlet region and the fluctuation of the liquid film is dominated by ripples. Meanwhile, the film is smooth relatively and the thickness distribution is concentrated near the mean value, therefore the Kδ is larger than other regions. Subsequently, as the increase of Rel, the Kδ of the liquid film instantaneous thickness PDFs in the region Ⅰ and Ⅱ decreases quickly at first, and then slowly. It shows that the distribution range of the liquid film thickness increases, and the mean film thickness cannot represent the film thickness very well. Furthermore, the PDFs Kδ close to a constant when Rel＞7.50×103. And the fixed Kδ means the range of the liquid film thickness has a maximum value. In the region Ⅲ and Ⅳ, the Kδ is small, and changes slightly with the increase of Rel, which indicates that the PDFs tend to an approximately normal distribution. That is the thickness distribution of the liquid film in the stable region is distributed in the maximum range. 3.2.3. Standard Deviation and Relative standard deviation The Standard Deviation (SD) and Relative standard deviation (RSD) of the liquid film instantaneous thickness at different axial positions are shown in Fig.12. Moreover, the RSD is defined as follows： RSD 

SD δ

(10)

The Standard Deviation evaluates the absolute deviation of the liquid film instantaneous thickness relative to the mean value, and reflecting the degree of the

liquid film fluctuation. As can be seen from Fig. 12 (a), in the region Ⅰ, the SD increases first and then tends to be a plateau. This fact shows that the increase of Rel can cause an increase in the surface wave amplitude, but this effect is gradually weakened. That is the amplification of the surface wave amplitude via Rel effects has limits. Increasing Rel will intensify the film fluctuation. And the drastic fluctuation will stimulate the collision of ripples, which promotes the generation of large waves. However, excessive disturbance will make the large wave unable to maintain the complete waveform and break into small droplets. Hence, the large wave amplitude has a maximum value. In the region Ⅱ, the SD increases as the increasing Rel, but the growth rate gradually reduces. Then, in the region Ⅲ, the SD changes little. As for the region Ⅳ, the SD decreases gradually. The main reason is the fluctuation range at the end of plate is certain but the mean film thickness increases with increasing Rel. 1.5

1.0

1.2

0.8

0.9

x=750mm x=900mm x=1200mm x=1300mm

RSD

SD

0.6

0.6

0.4

x=50mm x=200mm x=300mm x=450mm

0.3

0.0

x=50mm x=200mm x=300mm x=450mm

4000

6000

8000

10000

Rel

x=750mm x=900mm x=1200mm x=1300mm 12000

14000

0.2

0.0

4000

6000

8000

10000

12000

14000

Rel

(a) Standard Deviation (b) Relative standard deviation Fig. 12. Standard Deviation and Relative standard deviation The Relative Standard Deviation evaluates the relative deviation of the liquid film instantaneous thickness relative to the mean value. In Fig. 12 (b), with the increase of

Rel, except for the condition that in the upper part of the plate (x=50mm 、x=200mm、 x=300mm 、 x=450mm) and Rel ＜ 4.00×103 the RSD increases slightly, the RSD decreases at first and then tends to be flat. However, this does not mean that the fluctuation of the liquid film decreases with the increase of Rel. Because of the breakup of the liquid film, the instantaneous thickness fluctuation has a maximum value. This means that, for the turbulent falling film, the increasing Rel increases more the wave frequency but wave amplitude. 3.2.4. Extreme value distribution The extremum value distribution is used to describe effectively the development of the liquid film. However, owing to the randomness of the turbulent liquid film, it is difficult to illuminate accurately the fluctuation of the liquid film only by the maximum and minimum. In order to reduce the influence of the random number, the statistic maximum 𝛿′𝑚𝑎𝑥 and the statistic minimum 𝛿′𝑚𝑖𝑛 are adopted in this paper. The data of the liquid film instantaneous thickness are arranged in the ascending order, and the 𝛿′𝑚𝑖𝑛 is the mean of the first 100 data, the 𝛿′𝑚𝑎𝑥 is the mean of the last 100 data. The distribution of non-dimensional statistic maximum 𝛽′𝑚𝑎𝑥 is shown in Fig. 13 (a). In the region Ⅰ, the 𝛽′𝑚𝑎𝑥 increases rapidly first and then tends to be flat when Rel ＞7.50×103. Because the influence of the increase of Rel for the liquid film in the region Ⅰ is significant at low Rel. Then, the film will breakup to maintain the 𝛽′𝑚𝑎𝑥 stability when Rel ＞7.50×103. The 𝛽′𝑚𝑎𝑥 of the liquid film in the region Ⅱ increases with the increase of Rel, but the growth rate reduces gradually. The fluctuation of the liquid film

in the region Ⅱ is the most, and the wave collision probability is bigger than other axial positions. And the drastic fluctuation facilitates the accumulation of wave mass and forms the large amplitude wave. Moreover, the wave in the region Ⅱ can maintain a bigger amplitude because it is thicker than other positions. Therefore, the 𝛽′𝑚𝑎𝑥 can increase continuously even if Rel is large. The 𝛽′𝑚𝑎𝑥 of the film in the region Ⅲ increase with the increase of Rel, but the growth rate is lower than others. As for the region Ⅳ, the 𝛽′𝑚𝑎𝑥 changes little. 140

70

120

60

40

60 40

x=50mm x=200mm x=300mm x=450mm

20 0

x=750mm x=900mm x=1200mm x=1300mm

50

80

’min

’max

100

x=50mm x=200mm x=300mm x=450mm

4000

6000

8000

10000

Rel

x=750mm x=900mm x=1200mm x=1300mm 12000

14000

30 20 10 0

4000

6000

8000

10000

12000

14000

Rel

(a) Statistic maximum (b) Statistic minimum Fig. 13. Non-dimensional statistic maximum and minimum The falling film is usually used in the cooling system and accompanied intense heat and mass transfer, and the phase change. Moreover, the liquid film will evaporate and disappear if the film is too thin, which will result in the drywall and endanger production. Thus, the minimum thickness is important for the liquid film. Chu and Dukler (1975) used the crest of the PDF to define the substrate of the liquid film. However, the fluctuation of the turbulent liquid film is drastic and there is no distinct smooth part, the thickness of the PDF crest cannot be used to describe the liquid film substrate. Hence, the statistic minimum thickness is used to represent the liquid film

substrate in this paper. The distribution of 𝛽′𝑚𝑖𝑛 is shown in Fig. 13 (b). Overall, the 𝛽′𝑚𝑖𝑛 increases with the increase of Rel, but the growth rate is different. The growth rate of the 𝛽′𝑚𝑖𝑛 of the liquid film in the region Ⅰ and Ⅱ decreases gradually, indicating that the 𝛽′𝑚𝑖𝑛 of the film on the upper part of the plate increases first and then tends to be stable. In the region Ⅲ, the film substrate approaches a linear increase. Besides, the growth rate of the 𝛽′𝑚𝑖𝑛 of the liquid film in the region Ⅳ increases at first and then comes close to a constant, and the thickness of the liquid film substrate is thinner than other positions. 1.0

4

x=50mm x=200mm x=300mm x=450mm

x=750mm x=900mm x=1200mm x=1300mm

0.8

3 max/ 

min/ 

0.6 0.4

2

x=50mm x=200mm x=300mm x=450mm

0.2 0.0

1

4000

6000

8000 10000 12000 14000 Rel

4000

6000

x=750mm x=900mm x=1200mm x=1300mm

8000 10000 12000 14000 Rel

(a)

(b) Fig. 14. Non-dimensional statistic maximum and minimum calculated by liquid film mean thickness In order to quantify better the extremum value distribution of the liquid film, the non-dimensional statistic maximum and minimum calculated by the liquid film mean thickness 𝛿′𝑚𝑎𝑥/𝛿 and 𝛿′𝑚𝑖𝑛/𝛿 are adopted in this study, and the results are shown in Fig.14. From Fig.14 (a), the 𝛿′𝑚𝑎𝑥/𝛿 of the liquid film in the region Ⅰ and Ⅱ increases first and then reduces, the 𝛿′𝑚𝑎𝑥/𝛿 of the liquid film in the region Ⅲ and Ⅳ reduces

continuously, and all tend to be flat at last. As a result, with the increase of Rel, the maximum thickness of the liquid film in the inlet region and development region close to 1.5 𝛿, the maximum thickness of the liquid film in the stable region close to 1.7 𝛿 because the thinner mean thickness. As shown in Fig.14 (b), the 𝛿′𝑚𝑖𝑛/𝛿 of the liquid film in the inlet region and development region increases first and then tends to be flat. But the 𝛿′𝑚𝑖𝑛/𝛿 of the liquid film in the stable region increases continuously. In the stable region, due to the effects of the liquid film breakup, the growth rate of 𝛿′𝑚𝑎𝑥 is much less than the growth rate of 𝛿′𝑚𝑖𝑛. This means the growth rate of 𝛿′𝑚𝑎𝑥 < the growth rate of 𝛿 < the growth rate of 𝛿′𝑚𝑖𝑛. Hence, the 𝛿′𝑚𝑖𝑛/𝛿 of the liquid film in the stable region increases with the increasing Rel. Finally, the consequence is that the minimum thickness of the liquid film close to 0.7 𝛿 with the increase of Rel.

3.3. Liquid film surface breakup As for the turbulent falling film, the liquid that waves on the film surface will break into small droplets when the local disturbance is too large (Mascarenhas, 2013; Li et al, 2019). In this study, the breakup of the liquid film surface is the most important influence factor except Rel. For the liquid film flowing along the vertical wall, the mass accumulation of the large wave during the traveling is the direct caused of the liquid film surface breakup. Mudawar and Houpt (1993) pointed out that the large waves can carry 40-70% of the total mass flow and play an important role in the transport of mass in the film. Adomeit (2000) and Song (2012) pointed out that the small amplitude wave with high frequency

will accumulate mass in the process of collision and absorption, and promoting the formation of the large amplitude wave. Meanwhile, Stokes pointed out that the velocity of Stokes waves depends not only on the water depth and wavenumber, but also on the amplitude (Fu, 2015). This mean that the large wave is faster, and it is easy to catch up the front wave and absorb the front wave to further increase its mass, which leads to the breakup of the liquid film surface. The breakup of the liquid film surface in the region Ⅱ is particularly evident. In this region, the interaction between waves-waves and waves-substrate is enhanced by gravity, sidewall effects and transverse development (Lu, 2016; Patnaik et al, 1996). As a result, the process of acceleration, collision, coalesce and deceleration of waves was promoted (Adomeit, 2000). Furthermore, the interaction of the large wave with the slower substrate and the breakup of the wave can cause the significant deformation of the liquid film, and this process will enhance the local Rel and the fluctuation of the liquid film. Plateau–Rayleigh instability is used to describe the perturbation phenomenon in the flow of the water column, which measured first by Plateau (1873) and illuminated by Rayleigh (1878) via mathematical model. The Plateau–Rayleigh instability can be used to explain the breakup of the liquid film surface (Eggers, 1997; Hunter, 2012). With the increase of Rel, the frequency and velocity of the large wave increase, which is conducive to the greater mass accumulation of the large wave. The fluctuation of the liquid film will present as a small threadlike water column when the large wave deviates too far from the equilibrium position and the perturbation will enhance or reduce the

radius in some parts of the water column. Meanwhile, the local Reynolds number will increase because of the acutely interaction between the large wave and substrate, causing a strong disturbance. As a result, the pressure in the smaller radius parts of the water column is bigger than others. And then, this pressure will cut the water column and form droplets and result in the breakup of the liquid film surface.

4. Conclusions In this study, the instantaneous thickness of the liquid film falling down a vertical plate is measured by UDV, and the wave characteristics of the turbulent falling liquid film in the development region are discussed. The conclusions can be drawn as follows: (1) As the axial distance increases, the mean thickness distribution of the liquid film is divided into three regions: the inlet region I, the development region (the film thickness increase region II and the film thickness decrease region III), and the stable region Ⅳ, and the region I can be ignored when the Rel＞7.50×103. The mean thickness of the liquid film in different axis positions increases with the increase of Rel, but the growth rate is different. The growth rate in the region I and II decrease gradually, and the mean thickness of the liquid film in the region III and Ⅳ increase steadily. (2) The PDFs of the liquid film instantaneous thickness shift to the right with the increase of Rel, and develops from “sharp” to “flat”. The PDFs of the liquid film instantaneous thickness are positive skewness (Sδ＞0) in most cases, which means the fluctuation of the liquid film is dominated by the ripples. With the increase of Rel, the skewness decreases and the large wave frequency increases. In most cases, the Kδ of the PDFs is greater than 0, which indicates that the distribution of the liquid film

thickness is steeper than the normal distribution. Besides, the PDFs tend to an approximately normal distribution with the increase of the Rel. (3) Plateau-Rayleigh instability can cause the breakup of the liquid film and resist the increase of the liquid film thickness. Therefore, the fluctuation range of the liquid film instantaneous thickness will approach fix when the Rel ＞ 1.20×104. As the Rel increases, the maximum thickness of the liquid film in the region I, II and III becomes close to 1.5 𝛿; the maximum thickness of the liquid film in the region Ⅳ is close to 1.7 𝛿 , and the substrate close to 0.7 𝛿 .

Acknowledgements This work was supported by the National Key R&D Program of China (2017YFB0602601), the Shanghai Engineering Research Center of Coal Gasification (18DZ2283900).

Nomenclature E

Error

x

Axial distance, m

H

Height, m

y

Coordinate perpendicular to the wall, m

h

Liquid level, m

Greek symbols

i

Direction index

β

Non-dimensional film thickness, m

Kδ

Kurtosis

𝛽′𝑚𝑎𝑥

Non-dimensional statistic maximum film thickness, m

n

Data number

𝛽′𝑚𝑖𝑛

Non-dimensional statistic minimum film thickness, m

RSD

Relative standard deviation

Γ

Mass flow rate per unit, Kg•m-1•s

SD

Standard deviation

δ

Film thickness, m

Sδ

Skewness

𝛿′𝑚𝑎𝑥

Statistic maximum film thickness, m

T

Time, s

𝛿′𝑚𝑖𝑛

Statistic minimum film thickness, m

u*

Friction, m/s

V

Mean flow rate, m/s

μ

Dynamic viscosity, m2•s-1

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Highlights: ► The mean thickness of the turbulent liquid film falling down a vertical plat can be divided into three regions. ► Relationships between the liquid film instantaneous thickness and Rel were discussed. ► The distribution of the liquid film thickness tends to an approximately normal distribution with increasing Rel. ► The fluctuation range of the liquid film instantaneous thickness will approach a fixed value as the Rel increases.