How do the fluid dynamics change for gravity-destabilized film flow on structured surfaces? An experimental investigation using light-induced fluorescence [original]

Hannes Raddant, Georg Brösigke, Christian Hoffmann, Markus Illner,

Jens-Uwe Repke

How do the fluid dynamics change for

gravity-destabilized film flow on structured

surfaces? An experimental investigation using

light-induced fluorescence

Open Access via institutional repository of Technische Universität Berlin

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Journal article | Accepted version

(i. e. final author-created version that incorporates referee comments and is the version accepted for

publication; also known as: Author’s Accepted Manuscript (AAM), Final Draft, Postprint)

This version is available at

https://doi.org/10.14279/depositonce-18944

Citation details

Raddant, H., Brösigke, G., Hoffmann, C., Illner, M., & Repke, J.-U. (2023). How do the fluid dynamics change

for gravity-destabilized film flow on structured surfaces? An experimental investigation using light-induced

fluorescence. In Chemical Engineering Research and Design (Vol. 196, pp. 390–403). Elsevier BV.

https://doi.org/10.1016/j.cherd.2023.06.052.

cbed This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0

International license: https://creativecommons.org/licenses/by-nc-nd/4.0/

How do the uid dynamics change for

gravity-destabilized lm ow on structured surfaces?

An experimental investigation using light-induced

uorescence

⋆

Hannes Raddant

∗

, Georg Brösigke

, Christian Homann

, Markus Illner

Jens-Uwe Repke

Technische Universität Berlin, Process Dynamics and Operations Group, Sekr. KWT 9,

Straÿe des 17. Juni 135, 10623 Berlin, Germany

Abstract

Liquid lm ow is the dominant ow regime in distillation and absorption

processes within structured packings. Extensive research has been carried

out to improve the understanding of the involved uid dynamics. Up to

now, these investigations mainly focused on gravity-stabilized lm ow, i.e.,

the liquid phase ows over a packing in counter-current ow with a gas phase.

In contrast, gravity-destabilized lm ow is studied much less frequently al-

⋆

This is the Accepted Manuscript of: Raddant, H., Brösigke, G., Homann, C., Illner,

M., Repke, J.-U. (2023). How do the uid dynamics change for gravity-destabilized lm

ow on structured surfaces? An experimental investigation using light-induced uores-

cence. Chemical Engineering Research and Design, Vol. 196, pp. 390-403.

https://doi.org/10.1016/j.cherd.2023.06.052

This work is licensed under a Creative Commons Attribution-NonCommercial-

NoDerivatives 4.0 International License,

http://creativecommons.org/licenses/

by-nc-nd/4.0/

∗

Corresponding authors

Email addresses:

raddant@tu-berlin.de

(Hannes Raddant),

georg.broe[email protected]

(Georg Brösigke),

c.hoffmann@tu-berlin.de

(Christian Homann),

markus.illn[email protected]

(Markus Illner),

j.repke@tu-berlin.de

(Jens-Uwe Repke)

though this type of ow applies to about half of the cases in a column with

structured packings. Here, the liquid runs along the underside of the pack-

ing, also in counter-current ow with the gas phase. To close the gap in the

experimental data, this contribution investigates the fundamental uid dy-

namics of gravity-destabilized liquid lm ows on a smooth plate, a 2D wave

texture, and a 3D pyramidal texture. Results on critical convective inclina-

tion angle, drained liquid mass ow, and liquid lm thickness without gas

counter-current gas ow are reported and compared to numerical results from

literature. In the experiments, the Reynolds number is varied from 28.4 to

113.5 using a surfactant-modied aqueous system. The 2D structure showed

decreased liquid ow stability compared to the smooth surface. However, the

3D structure seems to have a stabilizing eect on the ow.

Keywords:

Fluid dynamics, Film ow, Structured surface, Rayleigh-Taylor

instability, Negative inclination angle

Nomenclature

Abbreviations

CCD Charged-coupled device

CFD Computational uid

dynamics

LIF Light-induced

Fluorescence

RTI Rayleigh-Taylor

instabilities

ROI Region of interest

WRIBL Weighed residual integral

boundary layer

2D Two-dimensional

3D Three-dimensional

Dimensionless numbers

Ka Kapitza number

Re Reynolds number

Greek Symbols

Inclination angle in

α∗

Experimental critical

convective inclination

angle in

Inclination angle in

Liquid lm thickness in

δ∗

Liquid lm thickness

normalized to structure

height in

Wavelength in nm

Density in kgm

Kinematic viscosity in

Surface tension in Nm

Inclination angle in

(same orientation as

)

Latin Symbols

Concentration in mgL

Share of drained liquid

mass ow in %

Width in m

Specic volume ow in

Volume ow in m

Indices

Index for component,

c∈1. . . Nc

Subscripts

abs Absorption

c Capillary

calc Calculated

cmc Critical micellar

concentration

crit Critical

em Emission

ex Excitation

exp Experimental

l Liquid

mean Mean value

rel Relative

sim Simulative

1. Introduction

Over the past decades, structured packings have been intensively studied

(Spiegel and Duss, 2014) and used in distillation (Billingham and Lockett,

1999; Kiss, 2019) and absorption processes (Aroonwilas, 2001; Flagiello et al.,

2019). A major research focus has been the impact of structured surfaces

of packing materials on uid dynamics, e.g., in terms of pressure drop and

separation performance.

Film and rivulet ow are the dominating ow regimes in the liquid phase

on a structured surface material (Green et al., 2007; Aferka et al., 2011; Repke

et al., 2011; Janzen et al., 2013). The liquid ow over the packing structure

is mainly present as a lm ow (Aferka et al., 2011; Repke et al., 2011) and

rivulet ow occurs primarily among contiguous packing materials (Janzen

et al., 2013). For overowed plates, used as approximation for packing ma-

terial, the ow changes with dierent inclinations and both ow regimes can

occur as found by Charogiannis et al. (2018). Besides distillation and ab-

sorption processes, lm ows are also present in various other applications,

e.g., heat exchangers, evaporators and condensators (Abdelall et al., 2006;

Kalliadasis et al., 2011). Due to its appearance in so many applications, a

fundamental understanding of lm ow is highly desired to accurately design

and model structured packings or even whole columns.

In general, the uid dynamic behavior of lm ows is based on a complex

balance of forces which permanently aims at minimizing the liquid surface.

Hence, dierent ow regimes are expected, e.g., wavy lm ows (Alekseenko

et al., 1994). Moreover, thermal and morphologic eects inuence the uid

dynamic behavior of a lm ow (Kapitza and Kapitza, 1949; Kalliadasis

Figure 1: Possible regimes regarding the liquid ow on a solid surface for dierent inclina-

tion angles: a) 0

≤β <

, b)

β=

and

α=

, c) 90

< β ≤

180

and 0

< α ≤

et al., 2011). There are three dierent cases for the range of conceivable

inclination angles (Rohlfs et al., 2017):

(a) the gravity-stabilized case;

(b) the vertical case;

These cases are visualized in Figure 1. In case a), the liquid ows on top of

a solid surface with an overlying gas phase. The gravity stabilizes the ow

and Rayleigh-Taylor instabilities can not occur (Rietz et al., 2017). Case

b) represents a vertical fall lm, which is of minor relevance for the ow

regime in structured packings due to the corrugated morphology and not

further discussed. In case c), a liquid phase ows along the bottom of a

solid surface with an underlying gas phase. For this reason the less dense gas

phase ows under the more dense liquid phase causing a density dierence

in the direction of gravity. As a consequence due to induced interaction of

the liquid phase with the gas phase Rayleigh-Taylor instabilities could be

evolved (Chandrasekhar, 1961; Sharp, 1984).

In the next section, we present a brief literature review of experimental

and numerical investigations on gravity-stabilized and gravity-destabilized

ow. Based on this review, we show that there is insucient experimental

data for gravity-destabilized ow on structured surfaces below and above the

critical inclination angle and at various operating conditions as quantied

by the dimensionless Reynolds and Kapitza numbers. Afterwards, Section 3

introduces the measurement cell that was constructed to conduct the ex-

periments. Finally, Section 4 presents the results and contains a thorough

discussion of the obtained data, including the comparison with numerically

determined values by other authors. The contribution concludes with a brief

summary and an outlook to future research.

2. Literature review and theoretical background

The following section extends the theoretical background regarding gravity-

stabilized lm ow in Section 2.1 and for gravity-destabilized lm ow in

Section 2.2. In favor the decisive uid dynamics and there dierences as well

as the inuence of surface structures are explained.

2.1. Gravity-stabilized lm ow

Gravity-stabilized lm ows have been studied extensively, both experi-

mentally and numerically. For the gravity-stabilized lm ow, several ow

regimes may occur depending on the Reynolds number, i.e., laminar (Re

≤5.7

), stable wave (

25 ≤

≤75

), turbulent ow (Re

≥400

) as well as

two transitions ow regimes between these Reynolds numbers (Ishigai et al.,

1972).

The Reynolds number is dened as written in Equation (1). In this

equation,

is the volume ow related to the plate width

wsurface

and

µl

the kinematic viscosity of the liquid phase.

Re = ql

µl

with

ql=Vl

wsurface

(1)

For a smooth surface, the force balance for a laminar lm ow can be

described by the Nusselt correlation (Nusselt, 1916). Equation (2) is based

on the Reynolds number Re, the inclination angle

, and the kinematic

viscosity of the liquid phase

µl

and is valid up to Reynolds number of 400

(Brauer, 1956; Scheid et al., 2016).

δ=3·µ2

g·cos α1

·Re1

(2)

Its solution describes the liquid lm thickness depending on property data

(e.g., viscosity

µl

), the Reynolds number

, and gravitational acceleration

for a smooth

g·cos α

laminar lm ow. The Nusselt correlation was validated

on a smooth surface using dierent measurement methods (Brauer, 1971;

Lel et al., 2005). A theoretical investigation of dierent lm ow regimes

and characteristic wave patterns was presented by Alekseenko et al. (1994).

Furthermore, semi-empirical equations for the lm thickness and velocity

for occurring stable waves in gravity-stabilized lm ows were reported by

Brauer (1971) and Al-Sibai (2004).

For the more generic case of plates with microstructure, Zhao and Cerro

(1992) demonstrated the inuence of the ratio between the liquid lm thick-

ness and microstructure height (

δ∗

) on the uid dynamics. They diered

between the following regimes:

Regime 1:

δ∗<0.1

; the liquid ow follows the contour of the structured

surface.

Regime 2:

0.1< δ∗<1

; the uid dynamics of the liquid lm are highly

inuenced by the structured surface and specic liquid ow phenomena

are expected.

Regime 3:

δ∗>1

; The lm thickness is noticeably higher than the structure

resulting in a minor structure inuence on the liquid ow.

Over the last 20 years, technological improvements in computer science

and laser technology have allowed a temporally and locally resolved analysis

of gravity-stabilized lm ows (Al-Sibai, 2004; Lel et al., 2005). Further-

more, lm ows over structured surfaces of various types were investigated

intensively, e.g., for dierent micro- and macrostructured surfaces (Zhao and

Cerro, 1992), and for an inclined wall with sinussoidal corrugations (Bon-

tozoglou and Papapolymerou, 1997). Moreover, lm ows over an inclined

periodic wall with rectangular corrugations were investigated by Vlachogian-

nis and Bontozoglou (2002). To study the eect of corrugation steepness,

lm ows for inclination angle from 1

up to 15

and Reynolds numbers from

10 up to 450 over an inclined periodic wall with transverse rectangular corru-

gations were investigated by Argyriadi et al. (2006) and found a stabilization

eect on the critical Reynolds number with increased corrugation steepness.

Moreover, one-phase and two-phase lm ows over tetrahedron and lamellar

structures were studied by Paschke (2011). The liquid lm thickness and

liquid holdup on a real packing for dierent liquid loads was investigated

by Leuner et al. (2018b) using two dierent measurement cell designs and a

successful comparison with literature correlations was shown. Likewise, the

local velocity and lm thickness on corrugated structured packing geometries

were presented by Gerke et al. (2018) and found an increased mixing across

the liquid lm, although similar orders for the velocity distribution over the

lm thickness were found experimentally.

In addition to the experimental work, numerical investigations and mod-

eling of the uid dynamics of lm ows over structured surfaces were carried

out. For a wave structure, modeling and validation by experiments was re-

ported by Shetty and Cerro (1993). Szulczewska et al. (2003) performed

simulations of liquid ows over packing structures using computation uid

dynamics (CFD) and validated their simulations with experiments. In ad-

dition, Trifonov (2011) studied the uid dynamics of lm ows on vertical

corrugated plates numerically and experimentally. The inuence of a single

microstructure, with variations in height and form, on vortex formation and

the interfacial area on the lying lm ow was numerically investigated by

Bonart and Repke (2018). The authors found no eect of microstructures

with small heights compared to the lm thickness on the enhancement of the

interfacial area, however large structures could increase the interfacial area.

As evident from this review, there is a vast body of literature on ex-

perimental and numerical investigations for gravity-stabilized lm ows for

various geometries and operating conditions. Consequentially, we excluded

this ow regime from our investigations within this contribution.

2.2. Gravity-destabilized lm ow

In contrast to gravity-stabilized lm ows, the understanding of uid dy-

namics of gravity-destabilized lm ows is limited. Only in the last decade,

signicant contributions in this area have been reported. The liquid phase

ows along the bottom of the packing material and on top of a gas phase

as shown in Figure 2. For this reason the less dense gas phase ows under

the more dense liquid phase causing a density dierence in the direction of

gravity. Two forms of instabilities are possible as shown in Figure 2: the

convective Rayleigh-Taylor instability (case a)) and the absolute Rayleigh-

Taylor instability (case b)). The former is characterized by a predominant

eect of the surface tension on the overall liquid lm stability or an inertia

dominated region as described by Scheid et al. (2016). For lower Reynolds

numbers the surface tension stabilizes the lm ow because the ow veloc-

ity is relatively low. For higher Reynolds numbers, inertia dominates the

ow stability. The convective Rayleigh-Taylor instability is characterized by

more dominant convective transport of perturbations compared to the local

growth rate of instabilities, amplication, or damping with increasing run

time respectively length (Brun et al., 2015). The disturbance can occur at

small run lengths and is transported away from its origin due to the convec-

tive ow. For the latter, gravity dominates the surface tension and inertia

stabilization mechanism so that continuous and regular droplet detachment

from the liquid ow at a spatial xed position occurs. Therefore, the absolute

Rayleigh-Taylor instabilty is present when perturbations grow exponentially

regardless of an existing convective ow, resulting in a spatially xed position

Figure 2: Possible sub-cases of gravity-destabilized liquid ow ow: a) - convective

Rayleigh-Taylor instability,

α < αcrit

; b) - absolute Rayleigh-Taylor instability,

α≥αcrit

of the perturbation and in a propagation speed of the perturbation of zero

Brun et al. (2015). Overall, the critical inclination angle marks the transition

between convective and absolute Rayleigh-Taylor instability.

According to Brun et al. (2015), the capillary length is dened as

lc=rσl

ρlg,

(3)

based on the liquid surface tension

σl

, the liquid density

ρl

and the grav-

ity constant

. First experiments to determine the critical inclination angle

on smooth surfaces for a laminar lm ow performed by Brun et al. (2015)

demonstrated a signicant inuence of the ratio between the liquid lm thick-

ness and the capillary length. An increase of the aforementioned ratio leads

to a smaller critical angle, as reported by (Brun et al., 2015). At this point,

we present the denition of the Kapitza number, as it is the second important

dimensionless number for the entire article.

Equation (4) shows the denition of the Kapitza number (Scheid et al.,

2016; Kofman et al., 2017).

Ka = σl

3·µ

l·ρl

(4)

The denition of the Kapitza number for the vertical case as in Kofman et al.

(2017) is used for the contribution. The numerical simulations of Scheid et al.

(2016) showed that the occurrence of Rayleigh-Taylor instabilities depends

on the Reynolds number, the Kapitza number, and the inclination angle

least for lm ow over a smooth surface. An inuence of the microstructure

on the liquid ow was not investigated. Scheid et al. (2016) developed their

model based on the lubrication method to determine the linear transition be-

tween absolute and convective Rayleigh-Taylor instability designated as the

critical inclination angle numerically. They extended this model by inertial

forces and made it applicable to higher Reynolds numbers up to 60. The

minimal critical inclination angle on a smooth surface for Kapitza numbers

from 2

-3

to 1

was investigated by Scheid et al. (2016). For smaller angles

than the minimal critical angle of 57.4

, absolute Rayleigh-Taylor instabil-

ity is not present. However, it was concluded that the critical inclination

angle depends on the Reynolds number and the Kapitza number overall.

Therefore, convective instability is dominated by two mechanisms: the sur-

face tension dominates the stabilization of the lm ow until a minimum of

the critical inclination angle for increasing Reynolds number is reached. Af-

ter exceeding this minimal angle, an inertia-dominated convective instability

region is reached. Both stabilization mechanisms lead to larger critical incli-

nation angles for a constant Kapitza number. Regardless of the stabilization

mechanism, gravity induces instabilities in both regions. The ow is stabi-

lized with rising Reynolds number, which leads to higher critical inclination

angles for the region above the minimum critical angle reported by Scheid

et al. (2016). The lm ow is convective unstable, even if Scheid et al. (2016)

found an increasing critical inclination angle for increasing Reynolds number

after exceeding the minimal critical angle. Additional direct numerical simu-

lations and another methodology (WRIBL) was applied on the modelling of

gravity-destabilized lm ows for dierent wave types e.g. solitary-like waves

and sinussoidal-like waves and showed eddy formations in wave crests (Rohlfs

and Scheid, 2015; Rohlfs et al., 2017). These eddy formations are promoted

by structured surfaces and can be suppressed by adding surfactants (Davies,

1972).

Besides the work of Scheid et al. (2016), Kofman et al. (2017) carried

out numerical simulations with the Navier-Stokes equation as well as with

another modelling approach (WRIBL) of gravity-destabilized lm ow over

smooth surfaces regarding the two-dimensional dripping onset. Kofman et al.

(2017) concluded, that the linear absolute to convective instability transition

is insuciently predicting the dripping limit and highlight the importance

of non-linear eects. Experimental investigations on a rotating cylinder as

well as numerical studies are carried out by Rietz et al. (2017) and a major

inuence of the ow condition on droplet detachment induced by wave or

rivulet coalescence was observed. Rietz et al. (2017) concluded the transition

of the lm instability mechanism from Kapitza instability to Rayleigh-Taylor

instability is unclear and could be a sharp or a continuous conversion.

The number of experimental studies on gravity-destabilized lm ows is

very limited. Besides the work of Brun et al. (2015), Alekseenko et al. (2007)

observed the velocity eld of a gravity-destabilized lm ow around a circular

cylinder. Charogiannis et al. (2018) investigated gravity-destabilized lm

ow on at soda lime glass plate for inclination angles between

and

for two Kapitza numbers as well as Reynolds numbers between 0.6 and 193.

They found no droplet detachment in the investigated range of inclination

angles and reported data for liquid lm thickness and wavelength for lm and

rivulets ow. Lerisson et al. (2020) studied steady ow patterns of lm ow

on a glass plate for inclination angles

in the range of 20

to 60

, reported the

thickness of the liquid lm, and carried out a stability analysis. They found

cases where rivulets could suppress or signicantly increase liquid dripping

eects. Finally, Ledda et al. (2020, 2021) also performed a stability analysis

and measurements of the lm thickness on a smooth surface for

in the

range from 20

up to 80

. For a forced steady ow, spanwise parallel rivulets

were found. They also discuss a possible destabilizing eect with increasing

ow length caused by traveling lenses on top of the rivulets. Rietz et al.

(2021) investigated numerically the formation of rivulets and the wavelength

in the case of Rayleigh-Taylor instability for inclination angles

between

to 75

. They observed parallel rivulet ow with increasing inclination

angle and compared two scenarios (Re

= 1

, Ka

α=

= 13.1

and Re

= 40

α=

= 330

) to the work of Charogiannis et al. (2018).

So far, the experimental investigations found in the literature studied

exclusively gravity-destabilized lm ows on smooth surfaces and no exper-

imental investigations on structured surfaces were carried out. In addition,

only Ledda et al. (2020) investigated experimentally lm ows for

α > αcrit

Beside that, no experiments for inclination angles

up to 90

were reported.

To ll this gap, we present new experimental data for the full range of incli-

nation angles between 0

and 90

for a wide range of Reynolds numbers (28.4

to 113.5). Next to the critical inclination angle

αcrit

, the relative drained

liquid mass ow

mrel,exp

(i.e., the ratio of drained liquid to feed) is deter-

mined as a function of the inclination angle

and the Reynolds number. For

inclination angles above the critical one

αcrit

, liquid drains o the surface in

a not yet known amount. A quantication of the drained liquid ow

mrel,exp

and the mean liquid lm thickness

δmean

is determined as a function of the

inclination angle

and the Reynolds number as well as the surface structure.

The experiments are carried out for a smooth surface, a 2D wave structure,

and a 3D pyramidal structure. These structures were designed so that

δ∗

lies

between 0.1 and 1.0 as reported by Zhao and Cerro (1992). Finally, we com-

pare our results with the numerically determined critical inclination angles

on a smooth surface for the linear absolute/convective transition published

by Scheid et al. (2016).

In Figure 2, two possible ow conditions for gravity-destabilized lm ows

are shown. The case a) represents the transition state of gravity-destabilized

lm ow for

α < αcrit

. Here, convective Rayleigh-Taylor instabilities can

occur solely until

αcrit

is reached (Scheid et al., 2016). The gravity induced

acceleration of the liquid decreases with increasing

and a higher inuence

of the surface tension or inertia on the formation of the ow is pronounced.

Hence, changes in the ow regime from lm ow regime for

α=

to par-

allel rivulet ow regime for increasing

appear to be found experimentally

(Charogiannis et al., 2018) and numerically (Rietz et al., 2021). As a re-

sult, dierent liquid ow regimes are present: 1) lm ow, 2) ripped lm

ow and 3) rivulet ow which occur simultaneously. The mean liquid lm

thickness

δmean

rises due to less acceleration of the liquid lm in case a) in

Figure 2. Due to additional inertial forces, the Nusselt's solution for lam-

inar lm ow is no longer fullled (Zhao and Cerro, 1992; Vlachogiannis

and Bontozoglou, 2002). Nevertheless, the liquid lm thickness

for smooth

and wavy laminar lm ow can still be calculated with acceptable accuracy

using Equation (2), but denitly for the vertical lm ow. Likewise, the

Reynolds number and the surface structure should have a signicant inu-

ence on the mean lm thickness

δmean

. An increased ripple of the ow and

a higher tendency of droplet formation is observed for increasing inclination

angle and for higher Reynolds number (Kofman et al., 2017). For the case

of convective Rayleigh-Taylor instabilities wave merging can occur (Kofman

et al., 2017) and near

αcrit

random and irregular liquid detachment out of the

ow may appear. The critical inclination angle

αcrit

marks the point of the

change from convective to absolute Rayleigh-Taylor instability. As soon as

α > αcrit

, absolute Rayleigh-Taylor instability occurs, which is shown in case

b) of Figure 2. Instabilities in the ow are not longer dampened or trans-

ported away from there point of origin due to surface forces or convective

impulse transport and perturbations can grow spatially (Brun et al., 2015).

The absolute Rayleigh-Taylor instability is characterized by enduring liquid

droplet detachment immediately after the liquid inlet (Scheid et al., 2016).

Therefore, the instability is spatial xed along ow path due to local growth

of instabilities with a propagation speed of the perturbation of zero (Brun

et al., 2015). Nonlinear eects like 2D and 3D waves in the ow occur due

to grown perturbations until saturation (Kofman et al., 2017). These nonlin-

ear waves develop along the ow direction (Kalliadasis et al., 2011) caused

by wave formation due to convection known as Kapitza instability (Kapitza,

1948). The eect of droplet formation along a rivulet amplies the occurring

instabilities (Ledda et al., 2020, 2021). Thus, the magnitude of the absolute

Rayleigh-Taylor instability is dierent and is described by the determina-

tion of the relative drained liquid mass ow

mrel

which varies regarding the

Reynolds number, the inclination angle

as well as the structured surface.

On this account, the relative drained liquid mass ow

mrel

is expected to be

a function of the aforementioned values.

3. Methodology and materials

In the following section, the constructed measurement cell, the experi-

mental procedures, and the employed materials are described in detail.

3.1. Measurement cell

The measurement cell is shown in Figure 3a). The setup consists of

a liquid cycle and a tiltable cell with optical measurement devices. More

information on the optical measurement systems is provided in Section 3.2.

The main parts of the measurement cell are displayed in the simplied P&ID

in Figure 3b). The liquid is pumped from the storage vessel (V-01) through a

heat exchanger (HE-02) to the measurement cell. To ensure a constant feed

temperature of (25.0

0.1)

C (TI-01), the heat exchanger is connected to a

thermostat Julabo FP-40 (HE-01). The liquid feed ow rate is monitored via

a coriolis mass ow meter Sitrans FC Massow Mass 2100 from Siemens (FIC-

01) and controlled by a frequency-controlled gear pump Verder VGS096 (P-

01), while the pressure is tracked using a Optibar pressure sensor from Krohne

(PI-01). The liquid is pumped through a liquid distributor, which enables

a uniform distribution over the plate width. In order to gain reproducible

inlet conditions at all inclination angles, the liquid distributor is designed as

a mixing chamber with a internal distributor and an outlet with a 1 mm gap

over a length of 220 mm. This design enables lm ow conditions at the

liquid inlet for all investigated inclination angles.

The measurement cell and the optical sensor equipment is mounted on

the same construction frame, pivoted on ball bearings, and connected to a

planetary geared stepper motor. Thus, any plate inclination can be realized.

The theoretical error of the experimental inclination angle is

0.01

. All ex-

periments were conducted on anodized aluminum surfaces with a dimension

of 220mm in width and 500mm in length.

The drained liquid is collected in a vessel (scale) whose mass is measured

via a scale system Combics Pro equipped with Combics plattform CAPS1-

60FE-NCE from Sartorius (MM-01) and recorded over time until the vessel is

full. At this point, all collected liquid is returned to the storage tank (V-01)

using a gear pump BVP-Z from Ismatec (P-02). The liquid that remains on

the plate is conveyed to V-01 using a gear pump BVP-Z from Ismatec (P-03).

The liquid component system is composed of deionized water, Marlipal

24/70 from Sasol, and rhodamine B from Fluka. The non-ionic surfactant

Marlipal was added to ensure a fully wetted surface and its concentration

was kept above

100

mgL

, to ensure optimal wetting properties. The con-

centration is more than two times higher than the critical micellar concen-

tration reported by Paul (2014), resulting in signicant amount of surfactant

Figure 3: Film ow testing rig with optical measurement technology a) schematic overview

of actual facility b) simplied P&ID owsheet of the rig.

in the bulk phase, so that the inuence of Marangoni eects during droplet

formation is reduced. The surfactant addition resulted in a surface tension

σl≤

0.033Nm

. This was regularly veried using a Krüss tensiometer

K100 equipped with the Wilhelmy plate method for a temperature of 25

The kinematic viscosity for water at 25

C was used for calculations based on

(Korson et al., 1969). Simultaneously, the dynamic viscosity of the liquid,

µl

, was measured at 25

C using the rotational viscometer Haake viscotester

iQ. The liquid density

ρl

was determined using the installed coriolis sensor.

Rhodamine B was added as a uorescence tracer at a concentration around

5mgL

. Next to a pratical smooth surface, a 2D wave and a 3D pyramidal

microstructured surface as shown in Figure 4 were designed to study the in-

uence of a structured surfaces on the ow stability. The 2D wave structure

is shown in Figure 4a) and the 3D pyramidal structure is shown in Figure 4b).

Based on the work of Zhao and Cerro (1992), the structure height was chosen

to achieve a ratio

0.1< δ∗<1

for the ratio of expected lm thickness and

structure height. Thus, both surfaces have an amplitude of 1mm, because

an interaction of the surface and the liquid ow is expected. The surfaces are

made of aluminum treated with a black anodized coating without additional

surface sealing and the texture starting directly after the liquid inlet.

3.2. Light-induced Fluorescence

In general, light-induced uorescence (LIF) is an optical measurement

technique based on the excitation of a dissolved uorescence dye with an

external laser or light source. The tracer-specic emission spectrum is de-

tected using an optical detection device. Droplets, rivulets and thin lm ows

were experimentally investigated using LED-induced uorescence (Hagemeier

Figure 4: Geometries of the examined structured surfaces with their geometric dimensions:

a) 2D wave structured surface; b) 3D pyramidal structured surface.

et al., 2012). LIF was successfully applied for lm thickness measurement on

structured packing materials in our previous work Leuner et al. (2018a,b).

The experimental setup contains two green-light bar LEDs LB250 from

iiM AG (see Figure 3a)) with a wavelength of

λex =

528nm to ensure con-

stant and reproducible illumination during the measurements. Using this

excitation light source, the dissolved uorescence tracer rhodamine B is illu-

minated and then emits light at wavelengths

λem

between 550 and 650nm.

A lens with 50mm focal length and a variable aperture was assembled on

the camera (type: charged-coupled device, CCD). For pixelwise intensity de-

tection, a pco.2000 CCD camera from Excelitas PCO GmbH was used. The

camera was protected by a splash guard with optical access. To avoid spec-

tral interference eects of the tracer, an optical red-light bandpass lter was

mounted onto the lens.

3.3. Measurement procedure

Dened initial conditions are substantial for reproducible experimental re-

sults. For this reason, all experiments were started with a fully wetted surface

and an initial inclination angle of 0

(for visualization, see Figure 1). Then,

the relative drained liquid mass ow

mrel,exp

was determined for varying incli-

nation angles by rotating the plate. Once the liquid begins to drip from the

plate, a measurement time of at least 180s was recorded to ensure a steady

state. The determination of

αcrit

is directly linked to the weighing of drained

liquid, without the optical measurement technique being able to detect liquid

draining positions more than 200 mm from the liquid inlet. Additionally, the

velocity eld of the liquid lm ow was not investigated experimentally. In

the experimental setup, the convective and absolute Rayleigh-Taylor insta-

bility are superimposed. Based on the previous listing of the experimental

boundary conditions and the detailed discussion on the transition of absolute

to convective instability in Section 2, we dene the critical inclination angle

α∗

crit,exp

. This angle marks the beginning of an unstable lm ow with

droplet detachment due to convective Rayleigh-Taylor instability in combi-

nation with a possible superimposed Kapitza instability Rietz et al. (2017).

Due to the investigated region of inclination angle, absolute Rayleigh-Taylor

instability denitely occurs. To experimentally investigate both eects sep-

arately, a very high resolving measurement technique would be necessary.

Moreover, the application of such technique would be related to very high

eort. The focus of this work was to investigate the inuence of structured

surfaces on lm stability (i.e. droplet formation) on a larger scale. Likewise,

an evaluation based on the position of liquid detachment is dicult, because

"occurring already in the vicinity of the inlet" as mentioned by Scheid et al.

(2016), is a relative broad denition for the experimental determination of

the onset of absolute Rayleigh-Taylor instability without investigating the

velocity eld experimentally. The critical inclination angle

α∗

crit,exp

is found

when

of the liquid feed drains from the plate into the tub. Two experi-

mental runs were performed to determine the critical inclination angle

α∗

crit

and the relative drained liquid mass ow

and the mean value of both runs

is reported in the gures of Section 4. In the experiments, a ow path of 500

mm was investigated with LIF measurements covering a maximum recording

area of 200 mm in length.

The measurement of the lm thickness was carried out as a single de-

termination according to a pre-dened methodology: images for background

correction and illumination correction had been taken before a measurement

sequence was started. The uorescence intensity signal was calibrated us-

ing a similar wedge calibration cuvette as described by Bonart et al. (2017).

The LIF measurements were carried out on a xed region of interest with a

size of 10cm

(square in the middle of the surface structure and marked in

Figure 3a)). The post-processing is carried out as follows:

1. Loading of raw image data;

2. Pixel-wise calculation of mean values based on raw image data;

3. Oset calibration (due to minimal mechanical distortion) based on the

actual inclination angle;

4. Setting of ROI;

5. Pixel-wise substraction of background noise from the current image

with average values;

6. Pixel-wise illumination correction (less excitation light intensity leads

to lower uorescence signal);

7. Calculation of calibration curve based on measured intensity along the

wedge cuvette in consideration of temperature changes during the cal-

ibration procedure (no temperature control of the cuvette);

8. Pixel-wise calculation of the lm thickness based on the calibration

curve;

9. Calculation of average value for lm thickness for the whole ROI;

To conclude the Section 3, we investigated ve dierent Reynolds num-

bers between

28.4

up to

113.5

for various inclination angles over three dif-

ferent surfaces. For the investigation of the critical convective inclination

angle

α∗

crit,exp

and the drained liquid mass ow

mrel,exp

an inclination angle

α=

was set and the critical convective angle with large angular changes

was determined. Afterwards, the critical convective inclination angle was

investigated with an increment of the inclination angle of 1

starting at lower

inclination angles than the expected critical angle up to 90

, to be able to pre-

cisely detect the dripping onset as well as the drainage ow experimentally.

Likewise, the lm thickness

was determined for inclination angles between

and 90

and additionally compared to the critical convective inclination

angle and the drained liquid mass ow for

α=

, 20

, 40

and 60

4. Results and discussion

First, the observed critical convective inclination angles for the smooth

surface are shown. Then, the measurements of the relative drained liquid

mass ow and lm thickness are presented. In the last step, the impact of

the microstructure on the lm ow stability is discussed.

4.1. Critical convective inclination angle on smooth surface

The experiments on the smooth surface were carried out for an aque-

ous system with a Kapitza number of 1707 to be able to compare them to

the numerical results obtained by Scheid et al. (2016). In addition to the

measurement procedure outlined in the previous section, this also required a

minimum Reynolds number of 28.4 to always ensure a fully wetted surface

for a vertical liquid falling lm.

Experiments for ve dierent Reynolds numbers were performed to iden-

tify the critical convective inclination angle

α∗

crit,exp,smooth

on the smooth sur-

face and to determine the inuence of dierent Reynolds numbers on the

begin of droplet detachment due to convective Rayleigh-Taylor instabilities.

In Table 1, the experimentally determined critical convective inclination an-

gles of this work, correlation-based results for

αcrit,calc

(Equation (5) given by

Scheid et al. (2016)), and the simulation results of Scheid et al. (2016) for

the critical inclination angle

αcrit,sim

on the smooth surface are reported.

The calculation of the critical inclination angle

αcrit,calc

is carried out

using the correlation Equation (5) derived by Scheid et al. (2016) under the

boundary condition of an imposed ow rate on a smooth surface with no

inertia and no viscous extensional stress.

tan(α(0)

crit)4qsin(αcrit)(0) ≈2.1086Ka3

(5)

The results for

α∗

crit,exp,smooth

and

αcrit,calc

show a decreasing critical in-

clination angle for rising Reynolds numbers. In addition to these values,

Table 1: Comparison of critical convective inclination angles

α∗

crit,exp

with literature data

(Scheid et al., 2016) for various Reynolds numbers Re and a Kapitza number

≈

1707 on

a smooth surface.

Quantity Values

Re 28.4 35.5 42.6 85.1 113.5

α∗

crit,exp,smooth

(this work) 84.4

84.4

82.5

80.5

αcrit,calc

(Scheid et al., 2016) 83.3

82.9

82.6

81.2

80.6

αcrit,sim

(Scheid et al., 2016) 84.4

85.0

85.3

87.3

87.9

simulation results (read from a diagram of Scheid et al. (2016) for Kapitza

number of 1000) show an increase of

αcrit,sim

as the Reynolds number in-

creases. Scheid et al. (2016) predicted a minimum of the critical inclination

angle for a Reynolds number

≈

8. However, we could not ensure a fully

wetted surface for these operating conditions and could therefore not verify

this minimum experimentally.

Overall, the simulation results show an opposite trend to our experimen-

tal results. In the simulation study of Scheid et al. (2016) the velocity eld

and the propagation speed of perturbations were investigated. Exclusively

for Reynolds numbers

28.4

and

35.5

, an agreement with the experiments

could be found, which lies directly in the region of the validity of the model

from Scheid et al. (2016). The stabilization eect of higher Reynolds num-

bers in the region of prevailing inertia (above the minimal critical angle) as

reported by Scheid et al. (2016) could not be reproduced without measur-

ing the velocity eld. In contrast, the transition modeled by Scheid et al.

(2016) is based on a linear transition method while omitting any experimen-

tal non-idealities as well as non-linear instability phenomena. On the other

hand, the results match well with the correlation-based results in terms of

trend and deviation (Scheid et al., 2016). Over the entire range of Reynolds

numbers, the deviation across all measured critical convective inclination an-

gles is below 2

. These minor deviations between the experimental results

and the correlation-based data can be explained with non-idealities in com-

parison with the numerical results, e.g., the practical smooth surface in the

experiment compared to an ideal smooth surface in the simulation. In ad-

dition, minor liquid maldistributions can lead to slightly asymmetrical ow

patterns, which then result in variation of the lm thickness with the width.

Also, signicant changes in the ow regime along the ow direction appear

and rivulet conuence is observed in our experiments, which could result in

earlier liquid dripping (Rietz et al., 2017). The experimental studies by Brun

et al. (2015) showed a decrease of

αcrit

for an increasing ratio of initial lm

thickness to capillary length. The overall trend of our experimental data for

α∗

crit,exp,smooth

conrm the experimental results from Brun et al. (2015).

The decreasing critical convective inclination angle observed in this work

can be explained by a change in the ow pattern as the inclination angle

increases. Therefore, the lm tears open and parallel rivulets are formed (as

will be shown in Figure 6. Convective Rayleigh-Taylor instabilities are grow-

ing along the ow direction and perturbations increase with higher Reynolds

number. In consequence, both eects lead to a lower critical convective in-

clination angle with rising Reynolds number. For higher inclination angles,

the liquid lm laces up and becomes thicker. In addition, the liquid lm

constricts and lead for a non dripping lm to a smaller ow width, which is

superimposed with the increase in lm thickness. Also the ow morphology

changes to parallel rivulet ow and this results in a velocity eld, which is

not covered by the Nusselt solution. Due to these sidewall eects the cross-

sectional area owed through changes, which inuences the ow velocity.

Nevertheless, without measuring the velocity eld a conclusive statement on

the local Reynolds number cannot be made. In summary, liquid detachment

is inuenced by the Reynolds number, the ow pattern (i.e., lm ow or

rivulet ow), the eective growth of convective Rayleigh-Taylor instabilities

along the ow direction within the dierent ow patterns, and the ratio of

local liquid lm thickness to capillary length as found by Brun et al. (2015).

4.2. Investigation of drained liquid ow and average lm thickness on smooth

surface

After the critical convective inclination angle is exceeded, liquid detaches

from the ow. A quantication of the relative drained liquid mass ow

mrel,exp

was carried out for ve dierent Reynolds numbers and for inclinations be-

tween vertical and horizontal orientation of the surface. For inclination angles

from 0

up to 90

, the mean liquid lm thickness

δmean,exp,smooth

was measured

using LIF.

Figure 5 shows the percentage of drained liquid in relation to the liquid

feed on the smooth surface,

mrel,exp,smooth

, as a function of the inclination an-

gle

under variation of the Reynolds number. Inclination angles signicantly

lower than

α∗

crit,exp

are not shown because no liquid drains o at this incli-

nations. In Figure 5 the relative drained liquid mass ow is shown over the

inclination angle

. For a constant Reynolds number and a rising inclination

angle

, the relative drained liquid mass ow

mrel,exp,smooth

increases contin-

Figure 5: Relative drained liquid mass ow

mrel,exp,smooth

over the inclination angle

for

a Kapitza number of 1707 and dierent Reynolds numbers on a smooth surface at T =

C. Lines between data points added to guide the eye.

uously. For a constant inclination angle the relative drained liquid mass ow

mrel,exp,smooth

increases with increasing Reynolds number. For low Reynolds

numbers, the liquid ow is not dominated by inertia forces, and the liquid

drains almost completely at angles slightly above the critical convective incli-

nation angle. For larger Reynolds numbers, drainage occurs earlier but the

percentage of drained liquid rises less strongly (Figure 5). For inclination

angles above 86

, the drainage behavior becomes almost independent of the

Reynolds number.

From Figure 5 can also be seen, that the critical convective inclination

angle

α∗

crit,exp

decreases with rising Reynolds number. Here, no stabilization

Figure 6: Liquid ow regimes on all investigated surfaces for four dierent values of

and

for a constant Reynolds number of 113.5. In a) to d) the ow regime on a smooth surface,

in e) to h) the ow regime on the 2D wave structure and in i) to l) the ow regime on the

3D pyramidal structure is presented. The white dashed square shows the ROI for the LIF

measurements.

eect with higher Reynolds numbers is found, which is contrary to the simu-

lation results reported by Scheid et al. (2016). However, Scheid et al. (2016)

modeled only the linear transition between both forms of the Rayleigh-Taylor

instability and liquid drainage can be inuenced by non-idealities and non-

linearities. In Figure 6, measured ow regimes at four dierent inclination

angles on all investigated surfaces are depicted at Re

= 113.5

. For inclination

angles between 0

and 60

, a lm ow is observed for the smooth surface and

the 3D pyramidal structure. A disturbed lm ow is found for the 2D wave

structure for

α=

, which then develops into a parallel rivulet ow for

α=

. For

α=

, the quantity of parallel rivulets generally increases, but a

signicant fraction of the ow remains as a lm ow for the smooth and 3D

surface. However, parallel rivulets remain the prevailing ow regime on the

2D structure. Once an inclination angle of 85

is set, the critical convective

angle is exceeded on all surface and a region of permanent liquid detachment

is reached. Therefore, a signicant amount of liquid drains from the rivulets

and the lm ow. Over the length of the plate, it transitions into a ripped

lm ow and until it forms a parallel rivulet ow. In accordance with the

ow regimes shown in Figure 6 a) to d), random and non-controllable rivulet

conuence is observed.

In addition to the experimental results for the relative drained liquid mass

ow

mrel,exp,smooth

and the presented ow patterns, we have investigated the

mean liquid lm thickness

δmean,exp,smooth

. The lm thickness was measured

for the range of

from 0

to 90

at dierent Reynolds numbers on a smooth

surface and is shown in Figure 7. The results for the vertical falling lm

are given in Table 2 and are compared with the theoretical values obtained

from the Nusselt correlation Equation (2). A good agreement between the

values calculated with Equation (2) and our experimental data using light-

induced uorescence was determined, including the expected increase of the

lm thickness with increasing Reynolds number (Table 2). A mean lm

thickness of 0.181mm and a mean lm thickness of 0.294mm for the lowest

(28.4) and largest Reynolds number (113.5) is shown in Table 2. Due to the

good agreement of the measured lm thickness with the theoretical Nusselt

solution, we deem the LIF method suitable to measure the lm thickness at

Table 2: Mean liquid lm thickness calculated with Nusselts solution and measured for

the case of a vertical falling lm

α= 0

at T

= 25

C and a Kapitza number = 1707.

Quantity Values

Re 28.4 35.5 42.6 85.1 113.5

δmean,exp,smooth

in mm (this work) 0.181 0.199 0.211 0.280 0.294

δmean

in mm (Equation (2)) 0.190 0.205 0.218 0.274 0.302

other inclination angles.

From Figure 7 it can be seen, that the lm thickness increases moder-

ately while

α≤60

, which is caused by the stronger inclination of the plate

and the resulting reduced acceleration in the imposed ow direction (Leuner

et al., 2018b). When the inclination is increased beyond 79

, the behavior

of the lm thickness depends highly on the Reynolds number: for the larger

Reynolds numbers (85.1 and 113.5), the lm thickness hardly changes but

then increases almost exponentially up to 87

. For the smaller Reynolds

numbers (28.4 to 42.6), the lm thickness remains almost constant for an-

gles up to 85

. The larger inclination angles also cause a signicant amount

of relative drained liquid mass ow

mrel,exp,smooth

. Because dripping occurs

after longer ow length indicating a convective Rayleigh-Taylor instability,

the lm thickness measurement is not aected. Thus, the liquid lm thick-

ness

δmean,exp,smooth

shows a slight increase due to the chosen ROI of the LIF

measurements: for

between 85

and 87

, droplet detachment from both the

closed liquid lm and the rivulets occurs. This drainage behavior out of the

rivulet ow pattern was also found by Ledda et al. (2020, 2021). The ap-

pearance of droplets increases the measurement error of LIF method because

Figure 7: Mean liquid lm thickness

δmean,exp,smooth

over the inclination angle

for

Kapitza number = 1707 on a smooth surface for dierent Reynolds numbers at T =

C (angles rounded to integer values).

the more intensively illuminated droplets changes the results for the measure

lm thickness. For inclination angles above 85

(small Reynolds numbers)

and 87

(high Reynolds numbers), respectively, a signicant decrease of the

lm thickness is found. For Reynolds numbers between

28.4

up to

42.6

the

percentage decrease for the lm thickness per degree is about 10 %, about 26

% for Re =

85.1

and about 37 % for Re =

113.5

. The measured lm thick-

ness for

between 89

and 90

is caused by still adhering liquid. Shortly

after the liquid inlet, all liquid drips o the surface completely. An increased

measurement time before taking the images would lead to

δmean,exp,smooth ≈0

for each Reynolds number due to the slow drying of the surface.

A destabilization eect of higher Reynolds numbers was found for the

critical convective inclination angle

α∗

crit,exp,smooth

and the relative drained

liquid mass ow

mrel,exp,smooth

. This can be attributed to the increase of

the mean lm thickness

δmean,exp,smooth

, which results in a rising ratio of lm

thickness to constant capillary length (Brun et al., 2015), resulting in lower

critical convective inclination angle. These eects cause droplet formation

on top of the lm, e.g., in the form of rivulets (Fermigier et al., 1992), and

shifts the critical convective inclination angle for higher Reynolds numbers to

lower values. The higher local lm thickness in the region of observed parallel

rivulets shown in Figure 6a) to d), seems to exceed the ratio of lm thickness

to capillary length found by Brun et al. (2015) and no counteracting eect

of ow stabilization due to higher liquid loads is found.

4.3. Investigation of critical convective inclination angle, drained liquid ow,

and average lm thickness on structured surfaces

The inuence of the structured surface on the uid dynamics of gravity-

destabilized liquid is of major importance as it may stabilize or further desta-

bilize the ow.

As shown in Table 3,

α∗

crit,exp

generally decreases with increasing Reynolds

number across all investigated structured surfaces. However, a divergence

from this general trend is found for Re = 35.5 and Re = 42.6 on the 2D wave

structure. The deviation from the general trend found could be explained

by the fact that for very small Reynolds numbers (28.4) the ow is strongly

slowed down resulting in liquid detachment. For larger Reynolds numbers

(85.1 and 113.5), perturbations increase more and convective Rayleigh-Taylor

instability is highly promoted. The other two lie between both conditions, so

that the onset occurs later due to more favorable ow conditions. However,

we found higher

α∗

crit,exp

for the Reynolds numbers (35.5 and 42.6) the general

trend of the dripping behavior is similiar as shown in Figure 8.

Table 3: Comparison of the critical convective inclination angle for the smooth place, a 2D

wave structure (Kapitza number = 1761), and a 3D pyramidal structure (Kapitza number

= 1703) at various Reynolds numbers for T = 25

Quantity Values

Re 28.4 35.5 42.6 85.1 113.5

α∗

crit,exp,smooth

84.4

82.5

80.5

α∗

crit,exp,2D

81.0

83.0

84.0

75.0

69.0

α∗

crit,exp.,3D

86.0

85.0

82.0

83.0

Beside the investigation of

α∗

crit,exp

, the relative drained liquid mass ow

mrel,exp

for dierent structured surfaces and dierent Reynolds numbers was

investigated as shown in Figure 8. In Figure 8a), the drained liquid mass ow

mrel,exp,2D

is shown for varying Reynolds number over the inclination angle

for the 2D wave structure. Just as with the smooth surface,

mrel,exp,2D

also

increases. However, an earlier liquid detachment and a broader range of

are observed. Most importantly, an increase of the Reynolds number causes

a noticeably higher relative drained liquid mass ow

mrel,exp,2D

. Hence, a

negative inuence of the 2D wave structure on the stability of the liquid

ow is found compared to the smooth surface in Figure 5. In terms of ow

regimes, we observed both a lled 2D wave structure and a overow of the

liquid (Figure 6 e) to h)). As the inclination angle increases, a ripped lm ow

and parallel rivulets evolve sooner than on the smooth surface. The 2D waves

Figure 8: Relative drained liquid mass ow over the inclination angle

at T = 25

for dierent Reynolds numbers a)

mrel,exp,2D

on a 2D wave structured surface for Kapitza

number = 1761 and b)

mrel,exp,3D

for Kapitza number = 1703 on a 3D pyramidal structure.

Lines between data points added to guide the eye.

orthogonal to the liquid ow direction lead to stronger ow perturbations due

to the structure, which results in increased Rayleigh-Taylor instability. These

perturbations accumulate much faster along the ow direction compared to

the smooth surface and in the most cases dripping occurs at smaller angles.

For the investigated 3D pyramidal structure, the liquid can ow through

the channels between the pyramids along the convective ow direction. This

ow path between the structures prevents overow of the structures. Con-

sequently, the liquid distribution is better and and fewer perturbations due

to the structured surface occur. In Figure 8b), the drained liquid mass ow

mrel,exp,3D

is plotted over the inclination angle

for varied Reynolds num-

bers for the 3D pyramidal structure. Liquid detachment occurs for a narrow

range of the inclination angle

and the resulting relative drained liquid mass

ow

mrel,exp,3D

depends less on the Reynolds number. Due to the lower de-

pendence of the critical convective inclination angle and the relative drained

liquid mass ow on the Reynolds number, perturbations caused by the con-

vective ow seem to be reduced or damped. This indicates a stabilization

eect on the gravity-destabilized lm ow for 3D pyramidal structures. Due

to the more homogeneous liquid distribution caused by the 3D structure, the

local lm thickness distribution is more uniform and local maxima seem to

be reduced. Thus the ratio between lm thickness to capillary length by

Brun et al. (2015) is exceeded only for larger inclination angles, when the

lm thickness increases signicantly due to the reduced convection and thus

lower liquid velocities.

In Figure 9a), the mean liquid lm thickness

δmean,exp,2D

for varying

Reynolds number over increasing

is shown. At constant Reynolds number,

Figure 9: Mean liquid lm thickness

δmean,exp

for dierent Reynolds numbers: a) for

Kapitza number = 1761 on a 2D wave structured surface; b) for a Kapitza number = 1703

on a 3D pyramidal structure.

the mean lm thickness

δmean,exp,2D

remains almost the same for inclination

angles between 0

and 60

. This constant lm thickness is caused by the

valleys in the wave structure that are always lled with liquid. In addition

to the increasing mean lm thickness with higher Reynolds numbers, more

waves are observed, which are caused by the sharp edges of the orthogonal 2D

edges. Overall, an increase of the mean lm thickness

δmean,exp,2D

compared

to the smooth surface Figure 7 is found.

In the range of

from 80

to 84

, only a minor increase of

is detected.

For even larger inclination angles, a certain amount of liquid remains in

the valleys of the structure for all Reynolds numbers, which justies the

assumption of a constant static liquid holdup. In conclusion, the 2D wave

structure seems to have a negative inuence on the lm stability and seems

to enhance convective Rayleigh-Taylor instabilities.

In Figure 9b), the measured mean liquid lm thickness for varying Reynolds

number for increasing inclination angle

over the 3D pyramidal structure

is shown. For inclination angles in the range of 0

to 60

, only a slight in-

crease of the mean lm thickness is observed. This trend is comparable to

the smooth surface shown in Figure 7.

The mean lm thickness

δmean,exp,3D

at dierent Reynolds numbers and

inclination angles turns out to be larger compared to the smooth surface

and lower compared to the 2D wave structure. Nonetheless, a lower liquid

lm thickness for Reynolds numbers from 28.4 to 42.6 is found and could be

caused by liquid acceleration through the cross-section between the pyramidal

structure along the ow pathway.

For

in the range from 80

to 84

, the largest increase of

is observed

for Re = 28.4. This indicates a high inuence of the structure on the liquid

ow. Thus, a strong increase in the liquid holdup within the structure is con-

ceivable and indicates a partially lled structured surface. Additionally, the

lower increase of the lm thickness with rising Reynolds number compared

to the smooth surface indicates a larger liquid holdup with a plateau starting

at Re

85.1.

Due to the improved liquid distribution caused by the 3D pyramidal struc-

ture, a more uniformly lm ow compared to the smooth and 2D wave surface

is observed (Figure 6i) to l)).

In comparison with the smooth surface and the 2D wave structure, a

larger

α∗

crit,exp,3D

on the 3D pyramidal structure could be found for the investi-

gated Reynolds numbers. Additionally, the narrower range of the inclination

angle

over the whole range of Reynolds numbers for the relative drained

liquid mass ow

mrel,exp,3D

indicates a stabilization eect of the 3D pyrami-

dal structure on the lm ow. Given the improved liquid distribution across

the whole surface due to improved cross mixing, the 3D structure seems to

stabilize the lm ow and suppresses Rayleigh-Taylor instabilities.

5. Conclusion and Outlook

In this contribution, a measurement cell for the investigation of gravity-

destabilized lm ows, i.e., lm ows with negative inclination angles was

introduced and applied. It is aimed at closing the gap in experimentally

characterizing and understanding such ow regimes, which is present up till

now.

Experimental results for the critical convective inclination angle

α∗

crit,exp

the relative drained liquid mass ow

mrel,exp

, and the mean liquid lm thick-

ness

δmean,exp

for a smooth, a 2D wave, and a 3D pyramidal structured solid

surface were presented the rst time. We compared our experimental re-

sults at Reynolds numbers between 28.4 up to 113.5, a Kapitza number of

around 1707, and a smooth surface to literature data by Scheid et al. (2016).

This comparison showed good agreement. A decreasing critical convective

inclination angle with rising Reynolds number was observed experimentally.

The inuence of geometry was incorporated using a 2D wave structure

and a 3D pyramidal structure, which were intended to represent real packing

geometries. A negative inuence of the 2D wave structure on the lm ow

stability and a stabilizing eect of the 3D pyramidal structure regarding

the critical convective inclination angle, which represents the beginning of

droplet detachment from the ow, was found. Mainly a decrease of the

critical convective inclination angle with increasing Reynolds number could

also be observed for the 2D wave structure and 3D pyramidal structure.

Furthermore, the relative drained liquid mass ow for angles above the

critical convective inclination angle was quantied. A stabilizing eect of the

3D structure could be found. The relative drained mass ow was shown to

be almost completely independent of the Reynolds number. In contrast, the

relative drained mass ow for the smooth surface and the 2D wave structure

was highly inuenced by the Reynolds number.

Regarding the mean lm thickness, the 2D structure showed larger values

compared to the smooth and the 3D surface. Overall, a stabilizing eect on

the liquid ow was discovered for the 3D pyramidal structure due to the more

homogeneous lm thickness.

The experimental investigations show a signicant impact of the struc-

tured surfaces on the stability of gravity-destabilized lm ows. These fun-

damental experimental results can be used to better understand the uid

dynamics within packing structures used in distillation and absorption pro-

cesses. In future work, more viscous liquids should be investigated to study

the inuence of uid properties on the critical convective angle and draining

behavior. To this end, Kapitza numbers in the range from 0.1 to 200 are

of interest. To carry out further experimental investigations and to make

comparability between the results possible, the height of the structured sur-

faces has to be designed on the ratio

δ∗

in the range of 0.1 to 1 regarding the

expected lm thickness for higher viscous liquids.

Acknowledgements

Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - 426726119

/ funded by the Deutsche Forschungsgemeinschaft (DFG, German Research

Foundation)  426726119.

References

Abdelall, F.F., Abdel-Khalik, S.I., Sadowski, D.L., Shin, S., Yoda, M., 2006.

On the rayleightaylor instability for conned liquid lms with injection

through the bounding surfaces. International journal of heat and mass

transfer 49, 15291546. doi:

10.1016/j.ijheatmasstransfer.2005.07.

055

Aferka, S., Viva, A., Brunazzi, E., Marchot, P., Crine, M., Toye, D., 2011.

Tomographic measurement of liquid hold up and eective interfacial area

distributions in a column packed with high performance structured pack-

ings. Chemical Engineering Science 66, 34133422. doi:

10.1016/j.ces.

2011.01.022

Al-Sibai, F., 2004. Experimentelle Untersuchung der Strömungscharakter-

istik und des Wärmeübergangs bei welligen Riesellmen. Dissertation.

Rheinisch-Westfälische Technische Hochschule Aachen. Aachen, Deutsch-

land. URL:

https://publications.rwth-aachen.de/record/59863/

files/59863.pdf

Alekseenko, S., Antipin, V.A., Bobylev, A.V., Markovich, D.M., 2007. Ap-

plication of piv to velocity measurements in a liquid lm owing down

an inclined cylinder. Experiments in Fluids 43, 197207. doi:

10.1007/

s00348-007-0322-2

Alekseenko, S., Nakoryakov, V.E., Pokusaev, B.G., 1994. Wave ow of liquid

lms. begell house.

Argyriadi, K., Vlachogiannis, M., Bontozoglou, V., 2006. Experimental study

of inclined lm ow along periodic corrugations: The eect of wall steep-

ness. Physics of Fluids 18, 012102. doi:

10.1063/1.2163810

Aroonwilas, A., 2001. Eects of operating and design parameters on CO2 ab-

sorption in columns with structured packings. Separation and Purication

Technology 24, 403411. doi:

10.1016/S1383-5866(01)00140-X

Billingham, J.F., Lockett, M.J., 1999. Development of a New Generation of

Structured Packings for Distillation. Chemical Engineering Research and

Design 77, 583587. doi:

10.1205/026387699526610

Bonart, H., Marek, A., Repke, J.U., 2017. Experimental characterization of

stable liquid rivulets on inclined surfaces: Inuence of surface tension, vis-

cosity and inclination angle on the interfacial area. Chemical Engineering

Research and Design 125, 226232. doi:

10.1016/j.cherd.2017.07.022

Bonart, H., Repke, J.U., 2018. Direct Numerical Simulations of Liquids

on Microstructured Surfaces: Analysing the Fluid Dynamics on Packing.

Chemical Engineering Transactions 69, 6166. doi:

10.3303/CET1869011

Bontozoglou, V., Papapolymerou, G., 1997. Laminar lm ow down a wavy

incline. International journal of multiphase ow 23, 6979. doi:

10.1016/

S0301-9322(96)00053-5

Brauer, H., 1956. VDI-Forschungsheft 457. Forschung auf dem Gebiete des

Ingenieurwesens .

Brauer, H., 1971. Grundlagen der Einphasen-und Mehrphasenströmungen.

volume 2. Sauerländer, Aarau und Frankfurt am Main.

Brun, P.T., Damiano, A., Rieu, P., Balestra, G., Gallaire, F., 2015. Rayleigh-

taylor instability under an inclined plane. Physics of Fluids 27, 084107.

doi:

10.1063/1.4927857

Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability. Dover

Books on Physics Series, Dover Publications, New York.

Charogiannis, A., Denner, F., van Wachem, B.G.M., Kalliadasis, S., Scheid,

B., Markides, C.N., 2018. Experimental investigations of liquid falling

lms owing under an inclined planar substrate. Phys. Rev. Fluids 3,

114002. URL:

https://link.aps.org/doi/10.1103/PhysRevFluids.3.

114002

, doi:

10.1103/PhysRevFluids.3.114002

Davies, J.T., 1972. Turbulence phenomena at free surfaces. AIChE Journal

18, 169173. doi:

10.1002/aic.690180131

Fermigier, M., Limat, L., Wesfreid, J.E., Boudinet, P., Quilliet, C.,

1992. Two-dimensional patterns in Rayleigh-Taylor instability of a

thin layer. Journal of Fluid Mechanics 236, 349383. doi:

10.1017/

S0022112092001447

Flagiello, D., Parisi, A., Lancia, A., Carotenuto, C., Erto, A., Di Natale, F.,

2019. Seawater desulphurization scrubbing in spray and packed columns

for a 4.35 mw marine diesel engine. Chemical Engineering Research and

Design 148, 5667. doi:

10.1016/j.cherd.2019.05.057

Gerke, S.J., Leuner, H., Repke, J.U., 2018. Experimental investigation of

local lm thickness and velocity distribution inside falling liquid lms on

corrugated structured packings, in: Chemical Engineering Transactions,

AIDIC Servizi S.r.l.. pp. 16. doi:

10.3303/CET1869001

Green, C.W., Farone, J., Briley, J.K., Eldridge, R.B., Ketcham, R.A.,

Nightingale, B., 2007. Novel application of x-ray computed tomography:

Determination of gas/liquid contact area and liquid holdup in structured

packing. Industrial & Engineering Chemistry Research 46, 57345753.

doi:

10.1021/ie0701827

Hagemeier, T., Hartmann, M., Kühle, M., Thévenin, D., Zähringer, K.,

2012. Experimental characterization of thin lms, droplets and rivulets

using LED uorescence. Experiments in Fluids 52, 361374. doi:

10.1007/

s00348-011-1232-x

Ishigai, S., Nakanisi, S., Koizumi, T., Oyabu, Z., 1972. Hydrodynamics and

Heat Transfer of Vertical Falling Liquid Films : Part 1, Classication of

Flow Regimes. Bulletin of JSME 15, 594602. doi:

10.1299/jsme1958.

15.594

Janzen, A., Steube, J., Aferka, S., Kenig, E.Y., Crine, M., Marchot, P.,

Toye, D., 2013. Investigation of liquid ow morphology inside a structured

packing using X-ray tomography. Chemical Engineering Science 102, 451

460. doi:

10.1016/j.ces.2013.08.035

Kalliadasis, S., Ruyer-Quil, C., Scheid, B., Velarde, M.G., 2011. Falling

liquid lms. volume 176. Springer Science & Business Media.

Kapitza, P., 1948. Wave ow of thin viscous liquid lm. Zh. Exper. Teor.

Fiz. 18, 328.

Kapitza, P.L., Kapitza, S.P., 1949. Wave ow of thin layers of viscous liquids.

part iii. experimental research of a wave ow regime. Zhurnal Eksperimen-

talnoi i Teoreticheskoi Fiziki 19, 105120.

Kiss, A.A., 2019. Novel Catalytic Reactive Distillation Processes for a Sus-

tainable Chemical Industry. Topics in Catalysis 62, 11321148. doi:

10.

1007/s11244-018-1052-9

Kofman, N., Mergui, S., Ruyer-Quil, C., 2017. Characteristics of solitary

waves on a falling liquid lm sheared by a turbulent counter-current gas

ow. International journal of multiphase ow 95, 2234. doi:

10.1016/j.

ijmultiphaseflow.2017.05.007

Korson, L., Drost-Hansen, W., Millero, F.J., 1969. Viscosity of water at

various temperatures. The Journal of Physical Chemistry 73, 3439.

doi:

10.1021/j100721a006

Ledda, P.G., Lerisson, G., Balestra, G., Gallaire, F., 2020. Instability of a

thin viscous lm owing under an inclined substrate: the emergence and

stability of rivulets. Journal of Fluid Mechanics 904. doi:

10.1017/jfm.

2020.673

Ledda, P.G., Lerisson, G., Balestra, G., Gallaire, F., 2021. Instability of a

thin viscous lm owing under an inclined substrate: the emergence and

stability of rivulets  CORRIGENDUM. Journal of Fluid Mechanics 926.

doi:

10.1017/jfm.2021.699

Lel, V.V., Al-Sibai, F., Leefken, A., Renz, U., 2005. Local thickness and wave

velocity measurement of wavy lms with a chromatic confocal imaging

method and a uorescence intensity technique. Experiments in Fluids 39,

856864. doi:

10.1007/s00348-005-0020-x

Lerisson, G., Ledda, P.G., Balestra, G., Gallaire, F., 2020. Instability of

a thin viscous lm owing under an inclined substrate: steady patterns.

Journal of Fluid Mechanics 898. doi:

10.1017/jfm.2020.396

Leuner, H., Hapke, M., Sacher, J., Grünewald, M., Repke, J.U., 2018a. An

Advanced Approach to Design Structured Packing Columns using Minia-

turized Experimental Set-Ups. Chemical Engineering Transactions 69,

781786. doi:

10.3303/CET1869131

Leuner, H., Hapke, M., Sacher, J., Grünewald, M., Repke, J.U., 2018b.

Miniaturized Experimental Setups as a Fundamental Element of a Promis-

ing Design Approach for StructuredPacking Columns. Chemie Ingenieur

Technik doi:

10.1002/cite.201800057

Nusselt, W., 1916. Die Oberächenkondensation des Wasserdampfes.

Zeitschrift des Vereins deutscher Ingenieure 60, 16451648.

Paschke, S., 2011. Experimentelle Analyse ein-und zweiphasiger Film-

strömungen auf glatten und strukturierten Oberächen. Ph.D. the-

sis. Technische Universität Berlin. URL:

https://doi.org/10.14279/

depositonce-3069

Paul, N., 2014. Theoretische und experimentelle Untersuchungen von

Transport- und Grenzächenphänomenen in mizellaren Flüssig/üssig-

Systemen. Dissertation. Technische Universität Berlin. Berlin.

URL:

https://api-depositonce.tu-berlin.de/server/api/core/

bitstreams/3b441680-005f-4c3c-98a6-87f58870f8f9/content

Repke, J.U., Kohrt, M., Wozny, G., 2011. Untersuchungen mehrphasiger

Strömungen an einfachen geneigten Platten als potenzielles Ausle-

gungswerkzeug für technische Trennapparate. Chemie Ingenieur Technik

83, 11071114. doi:

10.1002/cite.201100048

Rietz, M., Kneer, R., Scheid, B., Rohlfs, W., 2021. Spanwise structuring and

rivulet formation in suspended falling liquid lms. Physical Review Fluids

6. doi:

10.1103/PhysRevFluids.6.084805

Rietz, M., Scheid, B., Gallaire, F., Kofman, N., Kneer, R., Rohlfs, W., 2017.

Dynamics of falling lms on the outside of a vertical rotating cylinder:

waves, rivulets and dripping transitions. Journal of Fluid Mechanics 832,

189211. doi:

10.1017/jfm.2017.657

Rohlfs, W., Pischke, P., Scheid, B., 2017. Hydrodynamic waves in lms

owing under an inclined plane. Physical Review Fluids 2. doi:

10.1103/

PhysRevFluids.2.044003

Rohlfs, W., Scheid, B., 2015. Phase diagram for the onset of circulating waves

and ow reversal in inclined falling lms. Journal of Fluid Mechanics 763,

322351. doi:

10.1017/jfm.2014.684

Scheid, B., Kofman, N., Rohlfs, W., 2016. Critical inclination for abso-

lute/convective instability transition in inverted falling lms. Physics of

Fluids 28, 044107. doi:

10.1063/1.4946827

Sharp, D.H., 1984. An overview of rayleigh-taylor instability. Physica D:

Nonlinear Phenomena 12, 318.

Shetty, S., Cerro, R.L., 1993. Flow of a thin om over a periodic sur-

face. International journal of multiphase ow 19, 10131027. doi:

10.1016/

0301-9322(93)90075-6

Spiegel, L., Duss, M., 2014. Structured Packings, in: Distillation. Elsevier,

pp. 145181. doi:

10.1016/B978-0-12-386878-7.00004-8

Szulczewska, B., Zbicinski, I., Górak, A., 2003. Liquid Flow on Structured

Packing: CFD Simulation and Experimental Study. Chemical Engineering

& Technology 26, 580584. doi:

10.1002/ceat.200390089

Trifonov, Y., 2011. Counter-current gas

−

liquid ow between vertical corru-

gated plates. Chemical Engineering Science 66, 48514866. doi:

10.1016/

j.ces.2011.06.044

Vlachogiannis, M., Bontozoglou, V., 2002. Experiments on laminar lm ow

along a periodic wall. Journal of Fluid Mechanics 457, 133156. doi:

10.

1017/S0022112001007637

Zhao, L., Cerro, R.L., 1992. Experimental characterization of viscous lm

ows over complex surfaces. International journal of multiphase ow 18,

495516. doi:

10.1016/0301-9322(92)90048-L