Open Physics 2020; 18: 104–111
Research Article
Tibor Bešenić*, Milan Vujanović, Jakov Baleta, Klaus Pachler, Niko Samec, and
Matjaž Hriberšek
Numerical analysis of sulfur dioxide absorption
in water droplets
https://doi.org/10.1515/phys-2020-0100
received October 23, 2019; accepted March 17, 2020
Abstract: Mass transfer between the phases is a cornerstone
of many technological processes and presents a topic whose
understanding and modelling is of high importance. For
instance, absorption of gases in liquid droplets is an
underlying phenomenon for the desulfurization of flue
gases in wet scrubbers. Wet scrubbing is an efficient
cleaning method where the liquid is sprayed in a stream
of rising gases, removing pollutants due to the concentration difference between the gas phase and droplets. A
model for absorption in water droplets has been developed
to describe the complex physical and chemical interactions
during the exposure to flue gases. The main factors affecting
the absorption are the mass transfer of pollutants through
the gas–droplet interface and the aqueous phase chemistry
in a droplet. The mass transfer coefficient, which has been
modeled with several approaches, is the most significant
parameter regulating the absorption dynamic into the
droplet, while the in-droplet chemistry controls the maximum quantity of dissolved pollutants. Dissociation of
sulfur dioxide and the chemical reactions in seawater
have been described by the equilibrium reactions.
Afterward, the influence of the mass transfer coefficient
* Corresponding author: Tibor Bešenić, The Department of Energy,
Power and Environmental Engineering, Faculty of Mechanical
Engineering and Naval Architecture, University of Zagreb, Zagreb,
Croatia, e-mail: tibor.besenic@fsb.hr
Milan Vujanović: The Department of Energy, Power and
Environmental Engineering, Faculty of Mechanical Engineering and
Naval Architecture, University of Zagreb, Zagreb, Croatia,
e-mail: milan.vujanovic@fsb.hr
Jakov Baleta: Department of Mechanical Metallurgy, Faculty of
Metallurgy, University of Zagreb, Sisak, Croatia,
e-mail: baleta@simet.hr
Klaus Pachler: AVL List GmbH, Graz, Austria,
e-mail: klaus.pachler@avl.com
Niko Samec, Matjaž Hriberšek: Chair for Power, Process
and Environmental Engineering, Faculty of Mechanical
Engineering University of Maribor, Maribor, Slovenia,
e-mail: matjaz.hribersek@um.si, niko.samec@um.si
Open Access. © 2020 Tibor Bešenić et al., published by De Gruyter.
License.
has been investigated, and the model has been validated
against the literature data on a single droplet scale.
Obtained results are comparable with the experimental
measurements and indicate the applicability of the model
for the design and development of industrial scrubbers.
Keywords: gas absorption, mass transfer coefficient, flue
gas desulfurization, penetration theory, film theory
1 Introduction
A deeper understanding of the underlying physical and
chemical phenomena is essential for designing better, more
efficient, and sustainable technology and equipment.
Improving the processes can help in achieving the crucial
goals of modern society, such as mitigating the harmful
impact to the environment and paving a way to a
sustainable future. For example, the presence of pollutants
in flue gases is an open issue, and sulfur oxides are
among the most harmful gaseous pollutants. They have a
detrimental impact on the environment, leading to respiratory and health effects for humans, causing excessive plant
exfoliation, acidifying the natural waters, enhancing the
corrosion in the exposed metals, and being precursors to the
acid rains. European Union set the goals for SO2 emission
reductions for each country, while the MARPOOL limited
the allowed sulfur levels in the fuel used for ships and
these restrictions will get stricter and include more
territories [1]. sulfur dioxides form during the combustion
from fuel sulfur and exit the combustion chamber with
flue gases. Primary measures for reducing their emissions
include the low sulfur fuel, its removal before combustion,
or influencing the parameters in the combustion chamber to
prevent its formation. However, the more prevalent solution
is exhaust gas after treatment. Removal of sulfur dioxide,
as the dominating among the sulfur species, is usually
done by the wet scrubbing process, where the polluted gas
is placed in contact with the scrubbing liquid. Most common
wet scrubber designs include packed beds, plate scrubbers,
venturi scrubbers, and spray towers. Spray towers are
This work is licensed under the Creative Commons Attribution 4.0 Public
Numerical analysis of sulfur dioxide absorption in water droplets
several meters high vessels, where the liquid is sprayed in
the gas stream from an array of nozzles, to increase the
gas–liquid interfacial surface area. The droplets absorb SO2
from flue gases and are collected at the bottom and treated.
Commonly used liquids include solutions of the alkali in the
water, such as limestone slurry, magnesium hydroxide,
hydrated lime or caustic soda, although even seawater and
pure water have the absorption capabilities [2]. Depending
on the design and the liquid medium used, the SO2 removal
efficiency in spray towers can reach over 90% [3]. Seawater
is convenient as a scrubbing agent in power plants near the
sea [4], as well as in the marine applications on ships
powered by sulfur-containing heavy oil [5].
Previously, researchers had tackled problems pertinent
to the spray scrubber applications, both on the experimental
and modelling sides. The topic of sulfur dioxide absorption
in liquid droplets has been investigated for some years now,
from the atmospherical point of view and the simplified
chemical models in the beginning [6], to the numerical
modelling in the recent years [7]. Maurer [8] investigated
the solubility, and the equilibrium concentrations of sulfur
dioxide in water, Walcek and Pruppacher [6] studied the
atmospherical absorption of SO2 by clouds and raindrops,
while Walcek et al. [9] performed experiments on the freely
falling droplets in rain shaft. Among others, Saboni and
Alexandrova [10] developed a mathematical model for
absorption of SO2 in pure water droplets, while Abdulsattar
et al. [11] modelled chemistry of a sulfur dioxide–seawater
systems. More recently, Andreasen and Mayer [12] compared model predictions of the equilibrium SO2 concentrations in pure and seawater, while Marocco [13] implemented
a model for limestone slurry spray tower in the computational fluid dynamics framework.
A significant scientific focus has been placed on the
mass transfer phenomena between the gas and liquid
phase during the last century, as well as on the aqueousphase chemistry. Numerous scientific articles have been
published concerning the physics and chemistry of
bubbles, drops, liquid sheets, and contactors. However,
reliable experimental data for single droplet absorption of
gases are not abundant, while numerical models are
focused on liquid–liquid systems, and even the pertinent
models include a significant amount of assumptions and
unknown parameters. On the other hand, chemical models
are predominantly focused on the lime slurry sprays, with
only a small number concerning the seawater chemistry.
Taking the above-stated into account, present work
aimed to develop a model for the sulfur dioxide absorption in
pure water droplets and to study the available approaches for
modelling of mass transfer coefficient, to develop a tool for
simulation of the removal of harmful emissions.
105
First, the Mathematical model is presented. It is
divided into the modelling of chemical reactions in the
aqueous phase, and the overview of the approaches
used for the mass transfer modelling. After that, the
simulation results and the comparison with three sets of
experimental data are shown. Finally, the summation of
the work and model is given in the Conclusion.
2 Mathematical model
When considering the sulfur dioxide absorption by liquid
droplets, two main parts must be taken into consideration.
First are the chemical reactions, or the equilibrium concentration of the absorbed pollutant, indicating the maximum
amount of the chemical species that can be absorbed in a
liquid. Second is the mass transfer across the gas–liquid
interface, controlling the absorption dynamics. The idea
behind the approach used in this work was to implement the
single-droplet, lumped-parameters model for pure water,
combining the liquid-phase chemistry and different models
for mass transfer absorption. The model can be used as the
basis of a tool for the design and optimization of the
industrial equipment for the removal of pollutant emissions.
2.1 Equilibrium conditions
When a gas mixture is in contact with the liquid, its
constituents dissolve in the liquid phase, and the amount is
proportional to their partial pressure in the gas phase. This is
Henry’s law, which is valid for dilute solutions and is suitable
for the considered application of sulfur dioxide absorption
from flue gases. Sander [14] gave a detailed introduction to
the Henry’s constant: physical background, variations of
Henry’s law constants and dimensions, dependence on the
temperature and solution composition, and the compilation
of Henry’s constants for various species and water as a
solution. Henry’s solubility constant is defined as:
cp
HSO
= cSO2 P −1.
2
(1)
In the above expression, ca is the concentration of a
species in the aqueous phase (mol/maq3), P is the
pressure (Pa). Temperature dependence of Henry’s
constant can be described with the van’t Hoff equation:
1
−ΔH 1
H (T ) = H ⊖ exp
− ⊖ .
T
R T
(2)
106
Tibor Bešenić et al.
where the exponent ⊖ denotes the reference values of
the temperature and Henry’s constant, while −ΔH/R is
tabulated. Equation (2) and the parameters presented
are valid only for a limited temperature range. At
present, the value of 1.2 × 10−2 is used for H ⊖ and the
value of −ΔH/R is 2,850 K.
2.2 Liquid-phase chemistry
cHSO3− cH3 O+
,
cSO2aq
(3)
with its temperature dependence from Marocco [15].
The dissociation of sulfur dioxide in water occurs in
two stages, and the second one is the reaction of bisulfite
with water, forming sulfite. Below is the equilibrium
constant of this reaction, with its temperature dependence obtained from Rabe and Harris [16].
QII =
c SO32− cH3 O+
cHSO3−
cH3 O+ = cHSO3− + 2c SO32−.
,
(4)
The second dissociation reaction is slower than the
first one and can often be neglected due to the low
concentrations of sulfite ions.
After defining variables and calculation of the
auxiliary values, a loop through the range of partial
pressures calculates the SO2aq using Henry’s constant.
This concentration is the input parameter in the system
of equations calculating the species’ concentration.
Since there are three unknowns (HSO3−, SO3 2 −, and
H3 O+ ), and there are only two dissociation reactions, the
additional equation is needed for “closing” the system.
(5)
After solving the system, the total sulfur concentration is summed up and converted from moles per
kilogram to gram per mole.
Total sulfur = cSO2eq + cHSO3− + c SO32−.
To determine the dissolution of sulfur dioxide in water,
Henry’s law alone is not enough, since it does
not include the influence of chemical reactions in the
liquid phase. SO2 is well soluble in pure water, but
subsequent reactions further increase maximum dissolved concentrations. In aqueous solutions, sulfur
dioxide undergoes dissociation reactions forming bisulfite and sulfite ions.
First reaction is the dissolution of gaseous SO2 into
the unreacted, aqueous form, governed by Henry’s law.
Next is the dissociation of SO2, forming the bisulfite and
the hydronium ion (here, the H3O+ will be used instead
of H+). The reactions below are fast and are in
equilibrium, and the reaction quotient for the first
dissociation is:
QI =
In literature, electroneutrality is commonly used as an
additional condition in aqueous solutions:
(6)
2.3 Mass transfer dynamics
As stated previously, besides the equilibrium concentration, another parameter influencing the mass exchange
across the interface is the mass transfer coefficient. Its
modelling for different technical applications is still an
open topic. Choosing the appropriate model for the mass
transfer is a complex issue, since droplets can display
various intricate physical phenomena on small time and
length scales [17]. The correct model for the mass
transfer in droplets is challenging to develop, since
there are myriad of parameters that influence the
transport, such as droplet velocity and size, turbulence,
droplet pulsations, and internal circulation [9]. Ambient
conditions greatly influence the absorption process and
can modify the mass transfer by orders of magnitude.
Furthermore, interaction with the wall has a major
impact on droplet properties and the gas–fluid interaction [18]. A significant amount of research has been
done on the topic of interface mass transfer during the
better part of the last century [19], but a simple and
comprehensive unified model is still not available. A
major share of the research is focused on general mass
transfer across straight surfaces: in agitated systems, or
for rising bubbles, extension of these models to droplets
falling with terminal velocity is questionable. Usually,
the model can be confidently used on a narrow set of
conditions that it is derived for and changing some of
the dimensionless numbers can lead to major discrepancies with experimental data.
The total mass transfer coefficient KSO2,tot is constituted out of gas (kSO2,g) and liquid (kSO2,l) side mass
transfer coefficients. The resistance of each phase
influences the overall coefficient, and their ratio determines whether the mass transfer is liquid- or gas-side
dominated. This depends on the application and the
parameters, but in the present case, the liquid side
Numerical analysis of sulfur dioxide absorption in water droplets
proved to be the dominating one, which is confirmed by the
literature [20]. In this work, the lumped-parameter model
has been assumed, which is commonly used in the
computational fluid dynamic models for absorption, but
the internal distribution is neglected this way. Thus,
there is a discrepancy between the literature and the
present model for the investigation of mass transfer
controlling side, because detailed models from the
literature assume radial concentration distribution in
the droplet or discretize the interior. In the below
equation, ESO2 is the enhancement factor that accounts
for a higher driving force in the liquid film compared
with the absorption without chemical reactions [21]:
HSO2
1
1
=
+
.
KSO2,tot
kSO2,g
ESO2 kSO2,l
(7)
Here, the gas side mass transfer coefficient is calculated
by the Ranz–Marshall equation [21], which is a commonly used approach:
Sh =
kSO2,g RTd
DSO2,g
0.33 .
= 2 + 0.69 Re0.5
l Sc
(8)
Sh, Re, and Sc are the droplet Sherwood, Reynolds, and
Schmidt dimensionless numbers, respectively; R is the
universal gas constant, T is temperature, d is droplet
diameter, and DSO2,g is the binary diffusivity of SO2 in the air.
One of the first models for liquid side mass transfer
was based on the film theory, described by Whitman [22],
developed for the absorption of gases in horizontal stirred
surfaces. It assumes that the liquid surface is steady, that
no turbulent motion exists inside the droplet, and that the
mass transfer is limited by the diffusion across the film
near the surface. In the equation below, DSO2,l is the
diffusivity coefficient of SO2 in water:
klfilm
DSO2,l
= 10
.
d
(9)
In reality, there is a significant amount of circulation
inside falling droplets caused by shear stress, which
makes the film theory unsuitable for a range of
applications. Internal circulation increases the mass
transfer, and even small droplets can have larger kl
than predicted by film theory [23]. Furthermore, experimental data have shown that the proportionality
between the mass transfer coefficient and diffusivity is
not linear, but it scales with the power of 1/2 [19].
One of the more commonly used approaches is the
penetration theory by Higbie, which assumes that the
107
internal turbulence extends to the surface, with eddies
bringing portions of fresh liquid to the interface [24].
Penetration theory takes into account the unsteadiness
of the mass transfer process with the contact time
parameter, which is difficult to estimate for specific
applications and even harder to generalize.
DSO2,l
tπ
klP.T. = 2
(10)
An additional expansion of the penetration theory is
the surface renewal theory introduced by Danckwerts
[25]. In it, a surface element can be replaced at any
instant of its life time, which accounts for the unpredictability of the turbulent flow inside the droplet:
klS.R. =
DSO2,l s .
(11)
where s is a surface renewal rate, which is not generally
known, just as the contact time t in penetration theory.
These parameters need to be handled carefully since
they significantly influence the values of k, but there is
no general model for all applications and conditions.
Another expansion of the model is possible by including
the influence of droplet oscillations and surface stretch [26].
Angelo et al. developed the model that also scales with D1/2
and includes the stretching of the surface via parameter α,
and models the oscillation time τ by Lamb’s model [27], with
surface tension σ and liquid density ρ:
klAngelo =
π
=
4
τ
DSO2,l
3α 2
1 + α +
8
πt
(12)
ρl d 3
.
σ
Lastly, a model by Amokrane et al. [28] includes the
interfacial velocity u* and oscillation frequency ω, which
relates shear stress with the mass transfer coefficient.
klAmokrane = ω
u⁎
=
DSO2,l u⁎
d
τ
ρl
(13)
A large number of mass transfer models are
available, such as Newman’s model for stationary drops
or Kronig–Brink model for creeping internal flows, but
the models presented here are most common and more
suitable for the application under consideration.
108
Tibor Bešenić et al.
Finally, the molar flux across the interface is
calculated as follows, with P′ being the partial pressure:
NSO2 = KSO2,tot (P′SO2 − HSO2 cSO2 (aq)).
(14)
3 Results
The approach by Andreasen and Mayer [12] has been
followed for the equilibrium approach. They provide
results of their model for the equilibrium concentrations
of SO2 absorbed in pure water. The model is applicable
not only for the water droplet but also for the
equilibrium conditions at the water–gas interface.
However, it does not provide information regarding the
dynamics of absorption into the droplet.
In Figure 1, the sulfur dioxide concentrations are
given for the range of SO2 partial pressures from 0 to 50 Pa
and several temperatures from 273 to 353 K. According to
SO2(g)
SO2(aq)
H2O(l)
HSO3-(l) + H3O+(l)
H2O(l)
+
SO32-(l) + H3O (l)
gas
Figure 1: Chemical reactions in liquid phase.
liquid
Figure 2: Comparison of the MATLAB model with the literature
data [12].
equation (6), sulfur dioxide concentration is the total
sulfur, combining the contribution of the dissolved SO2(aq),
HSO−3 and the SO23−, in gram per kilogram. Comparison of
the MATLAB model with the results by Andreasen and
Mayer is given for the pure water in Figure 2. It can be
seen that the implemented lumped parameter model in
MATLAB reproduces the equilibrium concentrations quite
well. The increase due to the rise of partial pressure is not
linear (as would be with Henry’s law only), but curved due
to the chemical reactions in the liquid phase. The high
concentration at near-zero pressure would be avoided if
the resolution of calculation pressures was increased. It
can be concluded that the agreement is satisfactory and
that the results are correctly reproduced.
Different equations used for the liquid side mass
transfer coefficient are expected to provide varied results
and, as the liquid side has a bigger influence on the total
coefficient in this application, this choice has a critical effect
on the overall mass transfer dynamics. Comparison of the
coefficients by the presented approaches and the literature
values has been made, and they differ significantly. This is
expected since a single equation cannot cover a wide array
of conditions for all the droplets in the spray.
Experiments for determining the absorbed sulfur
dioxide for a single drop are challenging to perform.
There are issues with obtaining droplet sizes, duration
of the contact time with the atmosphere, accurate
measurement on a droplet scale, and at the end, there
is a question of how wide is the range of parameters that
the results are valid. Here, the model results will be
compared with three sets of data.
One is for low sulfur dioxide concentrations in the gas
phase (97 and 1.035 ppm), droplet diameter of 2.88 mm, and
long exposure times by Saboni and Alexandrova [10]. The
second is the experimental investigation by Kaji et al. [29]
Numerical analysis of sulfur dioxide absorption in water droplets
109
for short exposure times, large droplets, and higher SO2
concentrations. The last are results for smaller diameters
and range of SO2 concentrations in the gas phase (300 µm
and 1–100% SO2) by Walcek [9]. The single droplet is
suspended in a stream of gas or dropped from a high shaft,
and after some time in contact with the sulfur-rich
atmosphere, the concentration of transferred sulfur
dioxide is calculated. Reproduced results are presented
in Figures 3 and 4.
It can be seen that for higher concentrations and for
a longer time, the transferred quantity increases and that
the equilibrium concentration is asymptotically approached
after sufficient exposure time. In the second case, with very
small SO2 concentrations, discrepancies between the
models and the literature data are more significant.
A similar experiment by Kaji et al. [29] was
replicated. Here, only short time absorption has been
investigated in a short drop tube, as shown in Figure 5
Figure 3: Comparison with literature data [10]. Single falling
droplet, d = 2.2 mm; 97 ppm SO2.
Figure 5: Comparison with literature data [29]. Single falling
droplet, d = 2.2 mm; T = 20°C; 620, 1,126 and 1,968 ppm SO2,
penetration and surface renewal theory.
the penetration theory and surface renewal theory
display good agreements with the experimental data,
while other approaches showed greater discrepancies
and are therefore not depicted.
For comparison with the literature data by Walcek,
film theory model showed better results than others.
Figure 4: Comparison with literature data [10]. Single falling
droplet, d = 2.2 mm; 1.035 ppm SO2.
For both experimental cases of Saboni and Alexandrova, the
penetration theory with short contact time shows very good
agreement with the literature data. Besides the film theory,
which is theoretically more suitable for long contact times,
all models are meant to be used for the short exposures of
the drop. Experiments above show the contact time around
20 s, which is too high for the application in scrubbers, but
both the literature model and experimental data can be
reached with presented models.
Figure 6: Comparison with literature data [6]. Single falling droplet,
d = 300 µm; 1–100% SO2); film theory.
110
Tibor Bešenić et al.
It produced faster absorption dynamics at the beginning
of the simulation, as well as small overpass at the end, as
shown in Figure 6. The discrepancy between the models
and better results by the film theory can be possibly
explained by the smaller droplet sizes (300 µm). The
results confirm the known fact that the influence of the
droplet size, SO2 concentration, and velocities has a great
and complex impact on the absorption dynamics and that
at present the unified model cannot envelop all the
conditions that can be encountered.
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4 Conclusion
The goal of this work was the implementation of the SO2
absorption model in water droplets, to develop a tool for
the design and optimization of industrial scrubbers. A
single-droplet, lumped-parameter absorption model has
been described with the in-droplet chemistry of sulfur
dioxide in pure water. An analysis and comparison of
the liquid side mass transfer coefficient models were
performed, with models ranging from rudimentary ones
that model diffusive transport only to the more complex
approaches taking into account phenomena such as flow
recirculation and droplet oscillation. In total, four
models for the liquid-side mass transfer coefficient
were investigated and compared with three sets of
experimental data. In the case of low sulfur dioxide
concentrations and big droplets with long exposure
times, penetration theory with short contact time proved
to have the best agreement. When large droplets and
short exposure times in atmosphere of higher SO2
concentrations were observed, penetration theory with
short contact time and surface renewal theory achieved
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in the case of small droplets and high SO2 concentrations film theory proved to be the better approach. In the
presented work, it is shown that the implemented model
reproduces the results comparable with the experimental data, with the temperature, concentration,
droplet size, and velocity conditions representative for
the real industrial cases. It represents a solid basis for
the next phase of the work that will consist of applying
the developed model to the real industrial case within
the CFD framework.
Acknowledgments: This work was supported by the
Croatian Science Foundation.
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