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- *Corresponding Author:
- Hlawitschka MW

Chair of Separation Science and Technology

TU Kaiserslautern

PO Box 3049, Germany

**E-mail:**[email protected]

**Received date: ** June 28, 2016;** Accepted da te:** July 07, 2016; **Published date:** July 17, 2016

**Citation: **Hlawitschka MW, Drefenstedt S, Bart HJ (2016) Local Analysis of CO_{2} Chemisorption in a Rectangular Bubble Column Using a Multiphase Euler-Euler
CFD Code. J Chem Eng Process Technol 7:300. doi:10.4172/2157-7048.1000300

**Copyright:** © 2016 Hlawitschka MW, et al. This is an open-access article
distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided
the original author and source are credited.

**Visit for more related articles at** Journal of Chemical Engineering & Process Technology

Hydrodynamics and reaction process phenomena in bubble column reactors are complex and intrinsic interlinked and not fully understood. To obtain a better understanding, for the first time, a multiphase Euler-Euler solver is developed and validated to experimental data for the absorption of carbon dioxide in water offering the possibility of detailed temporal local and spatial analysis. The open source CFD tool OpenFOAM® with the multiphase solver multiphase Euler Foam is extended: (i) an absorption model was implemented, enabling absorption from different bubble size or phases to the liquid (ii) a chemical reaction model accounts for any number of reactions. The extensions were validated based on literature data and own experimental results at higher pH-value. The simulation results show a satisfactory agreement to experimental and simulative data from literature as well as to own experiments concerning bubble velocity, concentrations and pH profiles

CFD simulation; Bubble column; Chemisorption; Absorption; Reactive flow; Euler-Euler; Multiphase CFD

Reactive bubble columns are widely used in chemical, petrochemical,
biochemical industries [1]. Different intrinsic interlinked phenomena
such as hydrodynamics, absorption, reactions, coalescence and
break-up of bubbles are involved and make predictions and scale-up
difficult. Due to the complexity of the hydrodynamics and its mutual
dependency on chemical species transport (reactions, interfacial
mass transfer), the industrial approach to handle such systems is very
often dependent on a ‘rule of thumb’ basis. The problem is either to
neglect local flow phenomena near stirrers and dead zones (integral
approach) or not to account for the poly-dispersity of the bubble
swarm (pseudo-homogeneity) [2]. State of the art models, such as
axial dispersion models, are still based on an integral description of
the time-dependent hydrodynamics (one axial dispersion coefficient
accounts for all non-idealities), mass transfer (integral mass transfer
coefficient constant over the column height) and an integral bubble size
(such as the constant mean Sauter diameter, d_{32}). Hence, these models
are developed for a specific reactor geometry and chemical system and
thus limited in predictability.

Computational fluid dynamics (CFD) simulations can exactly predict local hydrodynamics considering any changes in apparatus geometry. In addition, population balance equations became a frequently applied tool to describe the changing bubble size distribution due to break-up and coalescence, while the bottleneck is still the predictability of coalescence and break-up kernels. However, recent research indicated a good agreement when coupling simulations with mass transfer models.

Concerning reactive mass transfer, [3] used a one-dimensional two-phase simulation combined with population balance simulation to describe the chemisorption of carbon dioxide into an aqueous solution of sodium hydroxide. The simulations reveal a high sensitivity against small variations in hydrodynamic, mass transfer, and kinetic parameters. By Ref. [4] developed a model for the simulation of hydrodynamics and an irreversible chemical reaction in a gas-lift reactor. A solver capable of calculating compressible two-phase bubbly flows with chemisorption has been introduced by Ref. [5] and the simulations were compared to experimental data of Ref. [3]. While a good fit was found for the pHprofile, the used geometrical design of the column was different to the experimental data [6] developed a transient Eulerian-Eulerian model for hydrodynamics and mass transfer in rectangular bubble columns but chemical reactions were not included. By Ref. [7] introduced a hybrid approach for coupling turbulent mixing and chemical reaction. The mixing of phases is estimated by averaging flow and concentration profiles from preliminary CFD flow field calculations and a numerical tracer experiment. The validation was with data of carbon dioxide absorption into an alkali solution. Darmana et al. [8] used an Eulerian-Lagrangian approach to model hydrodynamics, mass transfer and chemical reactions for twophase flows. The chemisorption process of carbon dioxide in an aqueous solution of sodium hydroxide and the resulting reversible reactions were simulated in a bubble column reactor. A Lagrangian model enables an exact consideration of dispersed bubbles. However, it is not suitable for higher dispersed phase fractions and in general, this type of approach leads to a higher computational effort compared to an Eulerian one.

In this study reaction algorithm are combined with a multiphase
Euler-Euler CFD code based on the open source toolbox OpenFOAM^{®}.
To our knowledge, it is the first coupling of a CFD multi-phase model with
reaction kinetic models to simulate the reactive mass transfer, while the
coupling with population balances is presented elsewhere [9]. The resulting
solver is capable of computing one liquid bulk phase and any amount of
gaseous phase fractions in 3-D simulations. The model will be described in
the next section, followed by a validation against literature test cases and
own experimental data derived in a rectangular bubble column.

The multi-fluid model is based on general conservation equations
such as the conservation of volume and conservation of momentum for
each phase. The continuity equation for the liquid phase *l* is given by:

Here represents the velocity of the continuous (liquid) phase
and is the mass transfer from the liquid phase *l* to the dispersed
(gaseous) phase g and is the mass transfer vice versa. *S _{l}* defines
the source term applied to the liquid phase entry and

with

The conservation of momentum of the liquid phase is represented by:

The viscosity of the liquid phase is given by μl, the gravitation vector is named, and the forcesanddescribe the drag force, lift force and virtual mass force. The term describes the momentum influence due to mass transfer. The lift force is not relevant and not considered but the drag term is given by:

where

The dispersed phase fraction given by *α _{d}* and the corresponding
velocity is described by . The drag coefficient

The second one used is a variant of Tomiyama et al. [12]:

where

The virtual mass force includes the virtual mass effect occurring when a phase accelerates relative to another phase:

The turbulence modelling is based on large eddy simulation [13] with a simulation of large-scale turbulence structures and a finetuned modelling of the smaller scales. In this work the latter is by the Smagorinsky model [14], where the turbulent viscosity is given by:

Where Δ represents the grid size and is the rate-of-strain
tensor. In OpenFOAM 2.3.1 the Smagorinsky constant *C _{s}* is defined as
follows:

The model constant C_{s}=0.1 is taken from literature [8] and in
ordered to ensure comparability the constants C_{k}=0.03742 and
C_{e}=1.048 were used throughout.

**Mass transfer and species transport**

The chemisorption of carbon dioxide in aqueous sodium hydroxide solution starts with the absorption process of carbon dioxide from the dispersed phase into the surrounding continuous phase:

CO_{2,(g)} →CO_{2,(l)}

From a computational point, the presence of gas phase in a mesh cell leads to an increase of the corresponding species in the continuous phase until the solubility limit is reached.

The absorption rates appear in the source terms of the species transport equations, which describe the transport of the species inside the numerical domain:

Only Ns-1 transport equations are solved, where N_{s} corresponds to
the number of species observed in the process. The mass fractions of
the species must fulfill the following constraint:

The mass transfer is based on the two-film theory with a linear
gradient of a species *j* in the liquid (continuous) phase and the dispersed
phase. **Figure 1** shows a schematic concentration profile and the mass
fractions in the bulk phases and at the interface.and represent
the mass fractions in the bulk of the dispersed (gaseous) phase and the
surrounding liquid phase, respectively. The mass fractions and describe the corresponding mass fractions at the interface. Based
on the assumption of a liquid side mass transfer resistance, the mass
transfer is given by:

is the density of the continuous phase, *V _{cell}* is the cell volume and
E is the enhancement factor.

The interface area corresponds to the summation of the interface areas of each single bubble in a numerical cell. Assuming spherical bubbles of constant size, it corresponds to:

The enhancement factor *E* describes the influence of chemical
reaction to the absorption. Fleischer et al. [3] analyzed the dependence
of the enhancement factor *E* on the pH value. A correlation is used to
approximate the results of Fleischer in dependence of the hydroxide
mass fraction as depicted in **Figure 2**.

Up to a pH-value of 10, the enhancement factor *E* is close to unity
(see horizontal dotted line), but in the range of pH 10 to 14 rises rapidly and describes an enhancement of the physical mass transfer. Albeit its
simple structure, the average deviation of the correlation of the findings
of Fleischer is about 4% and never higher than 10%.

The mass transfer coefficient *K ^{j}* is calculated based on the Sherwood

corresponds to the equilibrium mass fraction of species *j* in
the liquid phase. It is defined by the Henry constant *H*.

With the above equations the mass transfer from the dispersed to the continuous phase is then:

Reactions describe the transformation of one set of chemical
substances into another. The developed solver enables a description of
reactions in the liquid phase. The general stoichiometry equation of a
reversible reaction for *J* species taking part in a reaction is:

Xj represents the summation formula of species andare the stoichiometric coefficients of the reaction m for the educt side (left) and product side (right). For a chemical system with M reactions the production rate of a species is:

The production rates are used as source terms in the species
transport equation. The reaction velocity m ω of the m-th reaction
is dependent on the rate coefficients and for the forward
chemical reaction and the backward chemical reaction as well as on the
concentrations *c ^{j}* of the participating species

The concentration *c ^{j}* of the species

The temperature dependent rate coefficients of the forward reaction is described by the Arrhenius equation..

A_{m} is the pre-exponential factor. The activation temperature T_{A} can
be interpreted as thermal energy required to start the reaction. The rate
constant of the backward reaction is commonly derived from the
equilibrium constant GG, m K :

For the rate constants and the equilibrium constant KGG,m different empirical models available (s. chapter 4.3). Besides the existing models in OpenFOAM 2.3.1 an additional model for the rate constant was implemented:

Where A_{m}, B_{m}, C_{m}, D_{m} and E_{m} are reaction specific constants.

**Solver extensions**

The model is embedded in the standard multiphaseEulerFoam
solver in OpenFOAM 2.3.1 and enables the simulation of reactive mass
transfer. Declarations of the different phases *i* as well as hydrodynamic
calculation of these are done within the original multiphaseEulerFoam
environment. **Figure 3** shows the interaction between the phases *i*, the
chemical species *j* and the physical phenomena where one continuous
phase with an infinite number of gaseous phases or different bubble
sizes can be coupled. The composition of the liquid phase is being
described by the OpenFOAM class basic Multi Component Mixture,
where scalar fields Y_{j} represent the mass fractions of the chemical
species. It has no influence on the hydrodynamic behavior of the phases thus neglecting the last term in Eq. (4). Any modification induced by
irreversible, consecutive, parallel or reverse reactions may be defined.
A change in composition of the gaseous phases is not specified as it is
always a pure gas. **Figure 4** shows the computational sequence of the
developed solver for a single time step. Newly introduced extensions to
the original solver are highlighted in grey.

The new model is validated against literature data in three steps:
First, the predicted hydrodynamics are compared with the results of
Deen et al. [15]. The work of Ref. [8] is used to verify the capabilities
of the absorption model and then the chemistry model. The latter
is further validated through own experiments. All simulations
are performed in rectangular columns as shown in **Figure 5**. The
continuous aqueous phase is not being exchanged in the semi-batch
reactors. Our experiments reveal that temperature changes occur in a
range of 1-2 K, which is in accordance to literature [16]. This small
change is neglected and all simulations are performed for 300 K and
1 bar. The utilized discretization schemes and boundary conditions
are identical in all simulations. They are shown in **Tables 1 and 2**. The
virtual mass coefficient C_{VM} is set to be 0.5.

Term | Scheme |
---|---|

ddt Schemes | fixed Value |

grad Schemes | Gauss linear |

div Schemes | Gauss upwind |

Laplacian Schemes | Gauss linear corrected |

**Table 1:** Utilized discretization schemes for the occurring terms in differential equations.

Variable | Inlet | Outlet | Wall |
---|---|---|---|

Phase fraction α | fixed Value | inlet Outlet | zero Gradient |

Mass fraction of chemical species Y |
zero Gradient | inlet Outlet | zero Gradient |

Pressure p |
fixed Flux Pressure | fixed Value | fixed Flux Pressure |

Temperature T |
zero Gradient | inlet Outlet | zero Gradient |

Velocity | fixed Value | pressure Inlet-Outlet Velocity | fixed Value |

**Table 2:** Utilized boundary conditions for different physical variables.

**Validation of hydrodynamics**

Hydrodynamics play a crucial role when it comes to modelling of
reactive mass transfer as it determines the local concentration changes
in the liquid and gas phase. An initial simulation of a rectangular
semi-batch bubble column (0.15 m × 0.15 m × 0.60 m) is based on
the publication of Ref. [15] with a numerical mesh constructed by
30 × 30 × 120 (width × depth × height) hexahedrons. The superficial
gas velocity is set to 4.9 mm/s. Gaseous carbon dioxide enters through
a square pitch of 30 mm side length positioned at the center of the
base area. The bubble diameter is set to the constant value of *d*=4 mm.
At the start of the simulation the filling level is at a height of 0.45 m.
During the simulation, the time step is adjusted to match a Courant
number of 0.3. Turbulence is represented by the Smagorinsky model
with a parameter of C_{S}=0.1. Drag forces are calculated according to the
model of Ishii-Zuber and mass transfer is neglected.

The **Figure 6** shows the time averaged axial velocity of the continuous
water phase in m/s plotted over the normalized side length *x/X* of the
reactor. Velocities are measured at the centerline of the reactor at a
height of 0.25 m. The simulative values are averaged over the entire duration of the calculation and the results are in good agreement with
literature data. On the left hand side velocities are underestimated in
the simulations compared to the experiment, whereas on the right side
values are over estimated. Simulated values of the literature and own
results deviate especially on the left side of the column, in the area of
0.1-0.4 *x/X*. The maximum mean velocity calculated by Ref. [15] is
0.186 m/s whereas our simulation produces a value of 0.176 m/s, which
accounts to a deviation of 5%. Fluid phase velocity is being fixated by
the boundary conditions to 0 m/s at the reactor walls. Concerning the
symmetry of the flow profiles, the performed CFD simulation shows a
high symmetry, while literature values slightly deviate.

**Validation of physical absorption**

The performed simulation is based on the publication of Ref. [8] and the setup is identical to the previous simulations. In addition to hydrodynamics the physical absorption of carbon dioxide is calculated using a Sherwood correlation for moving spheres [17]:

Here initially the terminal rise velocity of the CO_{2}-bubbles and in
preceding simulation a value of =0.231m/s in accordance with
the literature [8] is used. For the later the Sherwood number is being
calculated in each cell for every time step according to the relative
velocity between the gaseous and liquid phases.

In both simulations a rate of 2E-9 m²/s is used for the diffusion of
CO_{2} in water. The solubility of CO_{2} in pure water is approximated with
the formula from Versteeg and van Swaaij [18]:

Darmana et al. [8] estimates the evolution of mass fraction of diluted gas according to:

A_{g,tot} is the total surface area of all gas bubbles inside the reactor.
V_{l,tot} is the volume of the bulk phase. The assumptions of constant
bubble size and quantity are made. The enhancement factor in Eqn. 21
was set to 1. **Figure 7** compares simulations of the physical absorption
process at global constant and a local (variable) *Sh* number with
literature [8] and an absorption estimation according Eq. (32). The
normalized concentration of diluted carbon dioxide in the batch liquid
phase is shown in dependence of simulation time *t* in seconds. Values
are taken at a height of 0.225 m in the center line of the reactor. The
own CFD simulations as well as the simulation by Darmana et al. [8]
show a final deviation of 2% compared to the absorption estimation.
Also, the simulation with a constant *Sh* number (*Sh*=437) based on a
relative gas velocity of =0.231 m/s does not show a huge deviation
to the simulation using a variable *Sh* number (*Sh*=75-850) based on
the local bubble velocity. A detailed study of the *Sh* number revealed,
that the *Sh* ranges from 437 ± 5 in the area, with high gas fraction and
only shows a higher deviation in areas with low gas fraction. For other
geometrical designs and disperser designs, the significance of a local *Sh* number calculation may be more distinct.

**Validation chemisorption**

**Comparison to literature data:** The chemisorption of carbon
dioxide in aqueous sodium hydroxide solution is simulated with respect
to the work of Ref. [16] at reactor measures of 0.2 × 0.03 × 1.5 m^{3}.
Initially the level of the liquid batch phase has a height of 1 m and a pH
of 12.5. All bubbles have a constant diameter of 5.5 mm. The mesh grid
consists of 27 × 4 × 200 (width × depth × height) regular hexahedrons.
Gas is being introduced with a superficial velocity of 7 mm/s and the
central inlet on the bottom has the dimensions 35 × 15 mm2. As above,
the Smagorinsky model with a coefficient of 0.1 is used for modelling
turbulence. For both phases the drag coefficient is being calculated with
the formula

According to Ref. [16] a constant Sherwood number of 562 is chosen to model mass transport. The diffusion rate of carbon dioxide in water is calculated with the help of the formula from Versteeg and Swaalj [18].

In deviation to literature the enhancement factor E is initially constant and set to be 1. Two reversible chemical reactions take place in the reactor:

CO_{2} +OH^{−} HCO_{3}^{−} (35)

The production rates result to:

The derivation of the reaction rate constants *k* follows literature.
The constant of the first forward reaction is estimated with the
relation of Pohorecki and Moniuk [19].

** **

The equilibrium constants *K _{3}* and

Eigen et al. [22] concluded that the reaction rate constants of
processes with proton transfer have a magnitude of 10^{10}-10^{11}. The rate of the second reaction is set to 10^{6} for reasons of CPU time. The
constant for the backward reaction is again calculated with the help
of the equilibrium constant.

For this purpose the approximation of Hikita et al. [23] is used:

** **

Previous to the actual simulation of chemisorption, 120 s of pure hydrodynamic movement had been simulated. This prelude enabled the bubble plume to fully establish itself. After activating the absorption and reaction models another 250 s of simulation time were calculated.

In **Figure 8** shows the evolution of species concentrations during
the conducted simulation of chemisorption and the results of Ref. [16].
The concentrations in kmol/m^{3} are hereby plotted against time t in
seconds. During the first 80 s the production of CO_{3}^{2-} is favored. Then
the equilibrium changes towards the production of HCO_{3}^{-} due to a
drop of the pH-value. After about 100 s all of the hydroxide-molecules
are consumed and CO_{2} starts to accumulate. The simulation is in
very good agreement with the values from literature. A time delay of approximately 10 s can be observed between the two calculations. It is
being assumed that this is a result of the constant diameter model used
in this work as well as the neglect of the enhancement factor *E*.

A shift of equilibrium of the first reaction happens at about 80 s
of simulation time leading to the formation of bicarbonate. After
approximately 200 s the concentration of bicarbonate reaches its
maximum level as the accumulation of carbon dioxide inside the
reactor begins. Both events can be observed as shifts in the evolution
of the pH-value. In **Figure 9 **shows the temporal evolution of the pHvalue
observed in both the experiment and simulation performed by
Ref. [16] as well as own simulation results. All three curves decrease
over time from their initial value of 12.5 to about 6.9 pH after 250 s.
And all of them show the shifts mentioned above. But the simulative
evolutions are delayed compared to the experiment, which is more
serious for the second bump. However, the results of Ref. [16] are
in slightly better agreement with the experiment than our own
simulations. Yet both simulations capture the general trend of the
experimental data. Darmana et al. [16] explain the delay with an
underprediction of the mass transfer due to the chosen model. The time delay between the calculation in literature and the own results
can be explained in the same manner as above when analyzing the
evolutions of the species concentration.

The simulation is improved when considering a variable
enhancement factor *E* using the above introduced Eqn. 18. The result
of the pH-evolution with variable enhancement factor (—) is compared
in **Figure 10** to the data by Darmana et al. [16] (- - -) and our results
with constant enhancement factor (····). The general behavior of the
pH-value is similar to the previous simulations in **Figure 9**. The pH
decreases over time having a value of about 6.9 after 250 s. Also the
two mentioned pH value steps, which refer to changes in chemical
equilibrium, can be observed. In contrast to the previous simulation
the first step is now predicted slightly prematurely and the second step
is again too late. However, it can be concluded that the use of a variable
enhancement factor according Eqn. 18 greatly improves the agreement
with the experimental data.

**Comparison of pH-value to own experimental results:** The
capab ility of the solver to predict evolutions of the pH-value is further
tested in comparison to own experiments. Again the chemisorption
of carbon dioxide in aqueous sodium hydroxide solution is taken
as reference case. The reactor consists of paraglas and measures
0.18 × 0.03 × 1.5 m^{3}. Initially the level of the liquid phase has a height
of 1 m and the solution has a pH of 13. The gas is introduced into the
reactor through 3 × 7 needles in rectangular position at the center of
the bottom plate. The superficial velocity is held constant at 8 mm/s,
resulting in a bubble size of approximated 5.5 mm. Every 30 s the pH
value is determined with the help of the portable measuring device
SP80PI VWR sympHony. Before the measurements, the column
is gassed with nitrogen and to start chemisorption, the gas feed is
instantaneously shifted by a valve to carbon dioxide. Initially both
the liquid and gaseous phase are at room temperature. During the
experiment a rise in temperature from 294.5 K to 296.6 K is being
measured.

A simulation is performed accordingly to the first case (chapter
4.3.1) with adapted boundary conditions as inlet velocity and mesh
geometry. The mesh consists of 27 × 4 × 200 (width × depth ×height)
regular hexahedrons resulting in a total cell number of 21.600 cells.
As in the first case and corresponding to the injection of N_{2} in the
experiments, pure hydrodynamic movement is simulated till 120 s as a
starting point for the CO_{2} injection.

**Figure 11 **shows the temporal evolution of the pH-value observed
in the performed experiment and simulation. The pH values decrease
over time from their initial value of 13 to approximately 10 pH after
250 s. Compared to the first presented case, the higher starting pHvalue
leads to a change of the temporal reaction progress. While OH- and
CO_{3}^{2-} were totally converted after 250 s in the first case, in this
second case, this reaction is in the middle of the process.

While the simulation slightly underestimates the experiment after 120 s, the deviation is never higher than 3%. Hence, for this period of reaction, the simulation results are in very good agreement with the experimental ones. The overall computational time on 10 cores for a simulation of 250 s is 72 h considering a mesh size of 21600 cells.

A new CFD multiphase solver enabling reactive mass transfer has
been developed. The solver enables for the first time a simulation of
different bubble sizes or different gases in a single computational cell
using Euler-Euler framework. The solver has been applied to study reactive mass transfer in bubble column. The results were compared
to experimental and simulative data from literature [8] as well as own
experiments. At no point, model parameters were fitted to match results
with literature. The correct prediction of the hydrodynamic behavior
of the phases was shown in comparison to the results of Ref. [15]. It
could be shown, that the evolution of concentration changes due to
absorption and chemical reaction is in very good agreement with the
simulative data from Ref. [8] for an initial pH value of 12.5, who used
Euler-Lagrangian framework. A small time delay between own results
and literature is being attributed to differences in the models for bubble
size and the enhancement factor *E*. It could be shown, that accounting
a variable enhancement factor in the simulation improves significantly
the agreement with experimental results. The first time comparison to a
second a second case experimental setup, showed a temporal delay in the
overall reaction process, which is related to the higher pH-starting value.
In conclusion, a very good agreement between simulation and experiment
can be observed without any further adjustment, with exception of the
initial starting values that must fit to the experimental setup.

For an independent CFD based column design, the code requires further validation, for example, at higher gas fractions. The use of different gas phases in a gas mixture is already possible but not sufficiently tested. Momentarily, the assumption of spherical bubbles of constant size is being used and requires an adaption to account for changes in the interfacial area due to (elliptical) shape variations of the bubbles. The introduction of an adequate PBM bubble model by combining the developed solver with the work of Ref. [24] enables more sophisticated simulations for heterogeneous flow conditions. With that, bubble coalescence and break-up as well as different and changing bubble shapes can be modelled. This will affect the absorption via the bubble surface area and thus reactive mass transfer. The composition of the gas phases should be specified just as it is being done for the liquid phase to account for a back diffusion of dissolved gas into the bubbles. This would allow simulating the mixing of gases or the absorption of vaporized liquid into a gas phase. Also, the gasliquid interface requires additional investigations. For example, the coalescence of bubbles induces a locally higher mass transfer, which is not accounted in the simulations till now. In order to correctly predict bubble movement, the lift force needs to be considered. Several models have already been developed and tested in literature [25]. Additionally, the code can be extended to allow the simulation of other industrially relevant combinations of systems: solid-liquid, solid-gas and liquidliquid, as the multiphase approach is not restricted to gas fractions. An example to this are liquid-liquid systems, such as (reactive) extraction, when using OpenFOAM with respect to this [24].

This work was supported by the German Research Foundation (DFG), Priority Program SPP1740 “Reactive Bubbly Flows” (DFG HL-67/1-1).

Latin symbols |
|||

c |
concentration | kmol/m³ | |

d |
diameter | M | |

dc |
phase material time derivative for the continuous phase | ||

dd |
phase material time derivative for the dispersed phase | ||

gravitational acceleration | kg/m s² | ||

k |
mass transfer coefficient | m/s | |

k |
reaction rate constant | varies | |

mass transfer | kg/m³ s | ||

p |
pressure | Pa | |

t |
time | S | |

velocity | m/s | ||

A |
interface area | m² | |

Cα |
interface sharpening constant | ||

CD |
drag coefficient | ||

CS |
Smagorinsky coefficient | ||

D |
diffusion coefficient | m²/s | |

E |
enhancement factor | ||

F |
force | N | |

H |
dimensionless Henry constant | ||

K |
drag exchange coefficient | kg/m³ s | |

KGG |
equilibrium constant | ||

S |
source term | kg/m³ s | |

mean rate of strain tensor | |||

T |
temperature | K | |

V |
volume | m³ | |

W |
molar mass | kg/kmol | |

Y |
mass fraction | kg/kg | |

X |
empirical formula | ||

Greek symbols |
|||

α |
volumetric phase fraction | m³/m³ | |

β |
stoichiometry coefficient | ||

µ |
dynamic viscosity | kg/m s | |

µsgs |
turbulent viscosity | kg/m s | |

ρ |
density | kg/m³ | |

ω |
reaction velocity | kmol/m³ s | |

Indices |
|||

comp | compression | ||

g |
gaseous | ||

i |
phase | ||

j |
chemical species | ||

l |
liquid | ||

A |
activation | ||

D |
drag | ||

VM |
virtual mass | ||

Superscripts |
|||

* | interface | ||

‘ | forward reaction | ||

‘’ | reverse reaction |

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