Experimental characterization of stable liquid rivulets on inclined surfaces: Influence of surface tension, viscosity and inclination angle on the interfacial area [original]

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Bonart, H., Marek, A., & Repke, J.-U. (2017). Experimental characterization of stable liquid rivulets on

inclined surfaces: Influence of surface tension, viscosity and inclination angle on the interfacial area.

Chemical Engineering Research and Design, 125, 226–232. https://doi.org/10.1016/j.cherd.2017.07.022

Henning Bonart, André Marek, Jens-Uwe Repke

Experimental characterization of stable

liquid rivulets on inclined surfaces:

Influence of surface tension, viscosity and

inclination an

le on the interfacial area

Accepted manuscript (Postprint)Journal article |

Experimental Characterization of Stable Liquid Rivulets

on Inclined Surfaces: Influence of Surface Tension,

Viscosity and Inclination Angle on the Interfacial Area

Henning Bonarta, Andr´e Marekb, Jens-Uwe Repkea

aTechnische Universit¨at Berlin, Process Dynamics and Operations Group,

Str. des 17. Juni 135, 10623 Berlin, Germany

bTU Bergakademie Freiberg, Leipziger Str. 28, 09599 Freiberg, Germany

Abstract

In this work, liquid rivulets on inclined, smooth surfaces were examined

experimentally using light-induced fluorescence. The influence of viscosity,

surface tension and inclination angle was studied in terms of the Reynolds and

Kapitza numbers. Detailed results on the interfacial area of the rivulets were

obtained. Based on the experimental results, a correlation of the interfacial

area in dependence on the Reynolds and Kapitza numbers is proposed. It is

found, that the correlation can reproduce the experiments very well.

Keywords: Rivulet flow, Light-induced fluorescence, Interfacial area

1. Introduction1

Liquid film flow over solid surfaces are of great importance for chemical2

engineering applications demanding high heat and mass transfer rates. Ex-3

amples include distillation and absorption processes as well as falling film4

reactors and reboilers. Affected by various parameters like mass flow rates,5

surface structures and physical properties like viscosity, the gravity driven6

flow down inclined plates can exhibit a range of different flow patterns. At7

low flow rates the liquid film tends to split into several rivulets and droplets.8

A rivulet is a narrow film which spreads freely on a solid surface [1]. This9

leads to a smaller interfacial area between the liquid flow and an overflowing10

Email address: [email protected] (Jens-Uwe Repke)

Preprint submitted to Chemical Engineering Research and Design May 11, 2017

gas phase than widespread film flows [2]. As a consequence, the effective area11

for mass and heat transfer is greatly reduced.12

To accurately predict the mass and heat transfer performance of rivulets,13

an understanding of the fluid dynamics is essential. Accordingly, a number14

of experimental and numerical studies have been conducted on the hydrody-15

namics of rivulets. First detailed experiments and calculations on straight16

rivulets on inclined plates were conducted by [3], in which the influence of the17

liquid flow rate on the rivulet’s width was examined. Later on, calculations18

and experiments were performed for example to describe the velocity distri-19

bution in rivulets [4], the stability of rivulets [5] and the influence of curved20

surfaces [6]. In [7], light-induced fluorescence measurements were used to de-21

scribe shape and tip speed of rivulets. The wetting behavior and rivulet for-22

mation were researched by [2] using numerical simulations and experiments.23

Detailed measurements of the width of rivulets on different plate materials24

were reported by [8] and numerical simulations and experiments on the width25

and thickness of rivulets were performed by [9]. In [10], the film thickness26

and velocity distribution of rivulets and droplets were measured using light-27

induced fluorescence. Numerical simulations to characterize the influence of28

surface textures on the wetting of plates and the formation of rivulets were29

conducted by [11]. Calculations on the shape of the wavy surface of rivulets30

were compared with existing experimental measurements in [12]. Recently,31

comprehensive numerical simulations were performed in [13] to calculate the32

interfacial area of rivulets under the variation of liquid properties, surface33

properties and inclination angles. However, little attention has been paid to34

the direct experimental characterisation of the rivulet’s interfacial area.35

In the present paper a new correlation for the interfacial area of rivulets36

based on the Reynolds number and the Kapitza number is proposed. A total37

of 108 distinct measurements of the interfacial area for four different fluid38

mixtures on a smooth, flat plate were conducted. The experiments were39

performed using the light-induced fluorescence method for a wide range of40

fluid properties and flow conditions. In this way, the influence of surface41

tension, viscosity, inclination angle and mass flow on the interfacial area was42

identified.43

2. Experiments44

2.1. Setup45

For this contribution a new apparatus was designed to investigate the46

interfacial area of liquid rivulets. The measurements were conducted using47

light-induced fluorescence, see [7, 10, 14] for details. The experimental setup48

is shown in Fig. 1. The flat plate on which the rivulet was formed had a length

Figure 1: Experimental setup.

of 300 mm and a width of 150 mm. The width of the measurement cell was50

large enough to prevent contact of the rivulet with the outer walls. The plate51

was made of stainless steel with a surface roughness of 37 µm. The liquid52

inlet was constructed as a point distributor acting like an overflow weir. The53

measurement cell as well as the calibration cell were illuminated uniformly by54

a continuously operated single light source (manufacturer: igb-tech GmbH,55

type: FRDA202) with wavelengths from 340 nm-440 nm and peak wavelength56

at 395 nm. A long pass filter with 420 nm cutoff wavelength was added in57

front of the lens to block reflected light from other sources then the fluorescent58

dyes. The emitted light in both measurement and in-situ calibration cell was59

captured with a single 5.5 MP sCMOS camera (manufacturer: PCO AG,60

type: pco.edge) with an exposure time of 25 ms. In Fig. 2 a photo of the61

plate is displayed with a rivulet in the center and the calibration cell in62

the upper right corner. The light source and the camera were positioned63

Figure 2: Photo of the light-induced fluorescence of the plate with the rivulet and the

in-situ calibration cell (upper right corner). Brightness of the image is enhanced for better

visibility.

orthogonal to both the measurement cell and the calibration cell. The height64

inside the calibration cell increased linear from 0.3 mm to 2.0 mm to represent65

several liquid film thicknesses, see Fig. 3. The whole setup was installed into

inflow outflow

0.3 mm

2.0 mm

Figure 3: Side view of the in-situ calibration cell.

a single frame with all components fixed at defined positions. The frame67

with all components was inclined to the horizontal in its entirety. In this68

way, accurate measurements for different inclination angles were ensured.69

2.2. Performed Experiments70

The used liquids and dyes are listed together with their measured physical71

properties in Tab. 1. Triton X-102 was used as the surfactant. The contact72

angles of the fluids were measured using the sessile drop method. Note, that73

it is not easily possible to vary the surface tension of a liquid without varying74

Identifier Dye Density ρ

(kg/m3)

Surf.

tension σ

(mN/m)

Dyn.

viscosity η

(mPa s)

Contact

angle θ

(◦)

Kaa

DC10bCoumarin 152 940 18 10.419 27 36

DC5bCoumarin 152 920 18 5.073 30 91

Water/SurfactantbFluorescein 998 29 1.114 53 1150

WatercFluorescein 998 72 1.178 79 2070

aKapitza number with inclination angle α= 90◦, see Eqn. 3 for details.

bDow Corning 200 with different viscosities

cDeionized water

Table 1: Physical properties of the examined liquids measured at 25 ◦C.

its contact angle using the same surface. As a consequence, the liquids and75

the surface were chosen such that the contact angles were at least in a narrow76

range over all experiments. Tab. 2 lists the examined Kapitza and Reynolds77

numbers used in the performed experiments. The measurements were per-78

formed with the four different mixtures each at nine different flow rates and79

three different inclination angles (60◦, 75◦, 90◦). In total, 108 distinct con-80

figurations were probed. For the present experiments, the gas phase above81

the rivulet was stagnant. The temperature of the liquid was held constant82

at 25◦C. Coumarin 152 (max. excitation wavelength 395 nm, max. emission83

wavelength of 510 nm) and Fluorescein (max. excitation wavelength 490 nm,84

max. emission wavelength of 520 nm) were used as the fluorescence dyes85

with a concentration of about 50 mg/l in conjunction with the silicon oils86

respectively the aqueous systems. Note, that while the maximum excitation87

wavelength of Fluorescein is not found near the peak wavelength of the light88

source (as it is the case for Coumarin 152), sensitive measurements using Flu-89

orescein were still possible due to the high sensitivity of the dye and its wide90

excitation spectrum. In this work, the whole setup was flushed and rinsed91

before every single experiment. Then the measurement cell was flooded with92

the fluid using a high inlet mass flow to fully wet the surface. Afterwards, the93

inlet mass flow was slowly adjusted to the target mass flow. After a stable94

rivulet was formed, the remaining droplets and liquid accumulations without95

any connection to the rivulet were carefully removed from the plate by hand.96

In this way, the conditions on the flat surface were very well reproducible97

and the risk of a meandering rivulet was low.98

DC10 Ka 37.7 36.4 36.0

Re 0.3 1.1 1.9 2.8 4.4 7.2 11.9 19.3 30.7

DC5 Ka 96.1 92.6 91.6

Re 1.0 3.3 5.9 9.0 14.7 24.0 38.3 59.6 90.0

Wat./Surf. Ka 1207.4 1164.2 1150.9

Re 29.6 72.8 141.9 193.1 269.8 372.2 500.1 653.6 832.7

Water Ka 2172.2 2094.6 2070.5

Re 27.2 76.5 127.1 176.1 249.8 347.9 470.6 617.9 789.6

Table 2: Values of the examined Kapitza (Eqn. 3) and Reynolds numbers (Eqn. 2).

2.3. Post-processing99

The recorded photos were automatically post-processed using custom100

made scripts for the computing environment Matlab. As proposed in [10]101

a photo without any liquid was subtracted per pixel from the images. In this102

way, the grey-level offset from zero fluorescence intensity as well as back-103

ground color inhomogeneities were reduced. Furthermore, it was recognized104

during the post-processing, that subtracting a background image from the105

pictures could lower the reflections from the plate surface. Afterwards, Mat-106

lab’s spatial filter function was applied with a circular averaging filter of107

radius 7 pixel to reduce the noisy character of the light-induced fluorescence108

measurements.109

The per picture obtained grey levels in the calibration cell were matched110

with their corresponding heights or film thicknesses. For every picture, a111

second order polynomial was fitted through the matched values to obtain112

a smooth calibration function, see Fig. 4. The calibration functions were113

linearly extrapolated to grey values representing film thicknesses below the114

minimal height of the calibration cell, i.e., below 0.33 mm. In this way, the115

spatially distributed thickness of the rivulet was obtained and the rivulet was116

extracted from the background. By using this in-situ calibration measure-117

ment per picture errors due to changes of incoming light intensity and dye118

concentration were reduced. Finally, the interfacial area was calculated from119

the geometric properties of the rivulets by:120

Aif,exp =δx

NC−1

i=1

rb(i)

j=lb(i)qδyi,j2+δhi,j2,(1)

in which δyi,j is the distance between the points yi,j and yi,j+1 at the ith cut121

perpendicular to the flow direction and δhi,j is the difference of the rivulet’s122

thickness at these points. lb and rb of the index jat the ith cut are the left123

respectively right bound of the rivulet, δx is the distance between two cuts124

iand NC is the total number of horizontal lines between upper and lower125

edges of the plate. By using this procedure the interfacial area of the rivulets126

could be reliably determined from light-induced fluorescence measurements.127

0 100 200 300 400 500

0.0

0.5

1.0

1.5

2.0

DC5

DC10

0 1,000 2,000

Wat./Surf.

Water

Gray level (counts)

Thickness (mm)

Figure 4: Exemplarily calibration plots for the different liquids and dyes.

128

3. Results and Discussion129

Fig. 5 shows light-induced fluorescence recordings of the liquids at low130

(top) and high Reynolds numbers or flow rates (from left to right: DC5,131

DC10, Wat./Surf., Water). The inclination angle of the plate was 60◦. It is132

clearly visible, that all rivulets spreaded on the plate and exhibited a certain133

entrance flow length. For the silicon oils DC5 and DC10, the spreading134

was not finished by the end of the plate, whereas water/surfactant reached135

a steady width in the first few centimeters after the inlet. As seen in the136

photos, the water rivulets were the most unstable at both low and high137

Reynolds numbers. At low Reynolds numbers, the water rivulet tended to138

meander on the plate, whereas at high Reynolds numbers a steady rivulet,139

but with increasing and decreasing widths, was formed. Furthermore, waves140

started to occur for the silicon oils at high Reynolds numbers.141

In the following, the spatial properties width and thickness of the rivulets

are presented. The thickness was averaged over the horizontal axis or width of

the rivulet. Based on the experimental values, a correlation for the interfacial

area was identified. The discussion as well as the identified model are based

on the Reynolds Re and Kapitza Ka numbers:

Re =4·˙

V·ρ

U·η(2)

Ka =σ·ρ

η4·sin(α)·g1/3

,(3)

with the volumetric flow rate ˙

V, the perimeter of the inlet U, the inclination142

angle of the plate αand the gravitational acceleration gas well as the fluid143

properties density ρ, dynamic viscosity ηand surface tension σ. The usage144

of the Kapitza number is advantageous, because it combines the important145

fluid properties density, viscosity and surface tension. It is constant for a146

given fluid but depends on the current inclination angle. A detailed analysis147

of the dimensions can be found in [15].148

3.1. Thickness and Width149

The Figs. 6 and 7 display the width respectively the horizontally aver-150

aged thickness. The Kapitza number rises from the top to the bottom of151

the figures and the Reynolds number rises in every single plot from the bot-152

tom to the top. The inclination angle rises from the left to the right. The153

corresponding values of the Kapitza number and the Reynolds number are154

given in Tab. 2. The results show a strong dependency of the wetted width155

as well as the average thickness on both the Reynolds and Kapitza numbers.156

The shown values were averaged using between two (silicon oils) and five157

(aqueous systems) photos of the same rivulet and the error bars are the de-158

viation from the mean. For high values of the Kapitza number the width is159

nearly constant along the flow length. In contrast, the width increases along160

the flow length for low values of Ka. The width of the silicon oils increases161

continuously over the flow length for all values of Re. For water/surfactant162

a slight decrease for the lowest Re and an increasing wetted width for the163

highest Re over the flow length occurs. Using water, the wetted width as164

DC10 DC5 Wat./Surf. Water

Ka = 36.4

Re = 0.3

Ka = 92.6

Re = 1.0

Ka = 1164.2

Re = 29.6

Ka = 2094.6

Re = 27.2

Ka = 36.4

Re = 30.7

Ka = 92.6

Re = 90.0

Ka = 1164.2

Re = 832.7

Ka = 2094.6

Re = 789.6

Figure 5: Light-induced fluorescence recordings of the rivulets at low (top) and high

Reynolds numbers (from left to right: DC5, DC10, Wat./Surf., Water). The inclination

angle of the plate is 60◦. Brightness of the image is enhanced for better visibility.

well as the averaged thickness tend to decrease and reflect the oscillating165

character already reported by [16].166

The geometries of the rivulets show general trends in dependence on the167

flow rate, inclination angle and fluid properties. With higher values of Re168

both width and thickness increase for all fluids and inclination angles. A169

contrary behavior is observed for higher values of Ka. Consistent to [17], the170

width is reduced for rising values of Ka for all fluids and inclination angles.171

The thicknesses on the other hand shows a more complex dependency. The172

thickness is greatly reduced for rising Ka but gets to higher values again for173

very high Ka (water). In the whole range of Reynolds and Kapitza numbers,174

the width is reduced for higher inclination angles. This was already reported175

by [7] and [13].176

3.2. Interfacial Area177

Based on the local measurements of thickness and width, the interfacial178

area Aif,exp of the rivulets were reconstructed as described in section 2. All179

experimental results were combined into a single correlation. The interfacial180

areas obtained from the experiments were fitted using fitnlm, which is part181

of the Statistics and Machine Learning Toolbox of Matlab. This resulted in182

the following proportionality1for the interfacial area Aif,corr:183

Aif,corr ∼Re1/5

Ka1/3.(4)

A similar scaling has been recently observed in numerical simulations by [13].184

Fig. 8 shows the correlation 4 against the experimental values. It is visible

from the correlation, that the interfacial area rises with higher Reynolds

numbers. On the other hand, large Kapitza numbers decrease the interfacial

area. This agrees with the general trends observed in subsection 3.1 for

the width and thickness. The correlation matches most of the experimental

data in the entire parameter range within ±30%, as shown in Fig. 8. For

high values of Ka (pure water) the deviation is slightly larger. The overall

1The measured contact angles (see Tab. 1) are not incorporated into the correlation

yet, see subsection 2.2 and the conclusion 4.

normalized root-mean-square deviation given by:

NRMSD =rPn

i=1(Aif,exp,i −Aif,corr,i)2

/(max(Aif,exp)−min(Aif,exp)) ,(5)

with nequals the total number of experimental values, is NRMSD = 0.12.185

As a result, the correlation can be reliably used to compute the interfacial186

area of rivulets for a wide range of the Reynolds number Re (0.3-800) and187

the Kapitza number Ka (36-2170).188

4. Conclusion189

The interfacial area between rivulets and surrounding gas is an important190

parameter for applications in chemical engineering like distillation and ab-191

sorption processes as well as falling film reactors. In this study, the influence192

of surface tension, viscosity, inclination angle and mass flow on the interfa-193

cial area of liquid rivulets was investigated experimentally. A total of 108194

distinct measurements of the interfacial area for four different fluids were con-195

ducted using light-induced fluorescence measurements. The results showed a196

strong dependency on the mass flow and the fluid properties. Subsequently,197

a new correlation based on the Reynolds number and Kapitza number was198

developed. The correlation was found to accurately predict the experimen-199

tal values. The results of this study provide a convenient possibility for the200

validation of detailed numerical simulations as well as for the development201

of advanced models used for the design of film reactors or column packages.202

In future experiments, the influence of the surface tension and the contact203

angle on the interfacial area will examined separately using different plate204

materials and surfaces.205

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Figure 6: Width of the rivulets over flow length.

Figure 7: Averaged thickness of the rivulets over flow length.

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

−30%

+30%

Aif,corr ∼Re1/5·Ka−1/3

Aif,exp

Figure 8: Correlation of the interfacial area for the entire investigated parameter range

(0.324 <Re <800, 36 <Ka <2170, 60◦to 90◦) against the experimental values. All

values are normalized with the maximal interfacial area.