Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 3.6.3 - A VOF-Based Conservative Method for the Simulation of Reactive Mass Transfer from Rising Bubbles
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 Material Information
Title: 3.6.3 - A VOF-Based Conservative Method for the Simulation of Reactive Mass Transfer from Rising Bubbles Multiphase Flows with Heat and Mass Transfer
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Bothe, D.
Kröger, M.
Warnecke, H.-J.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: reactive mass transfer
fluidic interface
volume of fluid
 Notes
Abstract: In this paper numerical results on reactive mass transfer from single gas bubbles to a surrounding liquid are presented. The underlying numerical method is based on the solution of the incompressible two phase Navier-Stokes equations. The Volume-of-Fluid method is applied for the description of the liquid-gas interface. Within the numerical approach the concentration of the transfer component is represented by two separate variables, one for each phase. Numerical results are in good agreement with experimentell data.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


A VOF-Based Conservative Method for the Simulation of Reactive Mass Transfer
from Rising Bubbles


D. Bothe* M. KrOger* and H.-J. Warnecket

Center of Smart Interfaces, Technische Universitit Darmstadt, Darmstadt, Germany
bothe@csi.tu-darmstadt.de and kroeger@csi.tu-darmstadt.de
t Department of Chemical Engineering, University Paderborn, Germany
wamecke@tc.upb.de
SCorresponding author
Keywords: Reactive mass transfer, fluidic interface, Volume of Fluid




Abstract

In this paper numerical results on reactive mass transfer from single gas bubbles to a surrounding liquid are presented.
The underlying numerical method is based on the solution of the incompressible two phase Navier-Stokes equations.
The Volume-of-Fluid method is applied for the description of the liquid-gas interface. Within the numerical approach
the concentration of the transfer component is represented by two separate variables, one for each phase. Numerical
results are in good agreement with experimentell data.


1 Introduction

Many chemical reactions are heterogeneous, in which
case the reaction partners are present in different phases,
e.g. in a liquid and in a gas. At least one of these
species has to cross the interface between the two phases
in order to enable the chemical reaction. This process is
called reactive mass transfer, if the physical transfer of
one chemical component from one phase to the other is
followed by a chemical reaction in this phase. A com-
mon example is the selective oxidation of cyclohexane
where the gaseous oxygen is first dissolved and then re-
acts with the organic liquid to the desired product. This
type of reaction is of high interest, especially for the
chemical process industry. The scale-up from labora-
tory sized model apparatuses to large industrial reactors
often turns out to be quite difficult, because several of
the occurring phenomena in such systems are not fully
understood. Direct numerical simulations have proven
to be a useful addition to small scale experiments and
theoretical analysis. Accurate simulations can give in-
sight to local details which are otherwise not achievable.
Therefore, a lot of work has been devoted to this topic.
VOF-based simulations of purely physical mass transfer
across deforming interfaces without chemical reaction
have been reported in Davidson and Rudman (2002) and
in Bothe et al. (2004). In the latter paper, transfer of


oxygen from air bubbles rising in water or aqueous so-
lutions has been simulated, taking into account the re-
alistic jump discontinuity of the oxygen profiles at the
interface. Onea et al. (2009) used a similar approach as
Bothe et al. (2004) to simulate mass transfer in upward
bubble train flow through square and rectangular mini-
channels. Darmana et al. (2006) performed 3D simu-
lations of mass transfer at rising fluid particles using
the Front Tracking method. There, the transport resis-
tance inside the fluid particle is neglected, i.e. a constant
concentration value inside the bubble is assumed. Radl
et al. (2007) performed 2D simulations of deformable
bubbles and bubble swarms with mass transfer in non-
Newtonian liquids using a semi-Lagrangian advection
scheme. To prevent stability problems, a reduced den-
sity ratio between gas and liquid is used. Recently, first
papers on numerical simulation of reactive mass trans-
fer appeared. In (Khinast et al. 2005), reactive mass
transfer at deformable interfaces is examined using a
2D Front Tracking/Front Capturing hybrid method. In
(Deshpande and Zimmermann 2006a), a Level Set based
method is used to simulate mass transfer across the in-
terface of a moving deformable droplet. This method
is extended to reactive mass transfer in (Deshpande and
Zimmermann 2006b) where an instantaneous chemical
reaction occurs inside a moving droplet, which leads
to a quasi-stationary problem for the mass transfer. In











Radl et al. (2008) 2D simulations are performed using
a Front-Tracking method to investigate the effect of dif-
ferent Hatta and Schmidt numbers on the catalytic hy-
drogenation of nitroarenes for single bubbles and bub-
ble clusters. Based on the numerical approach in (Bothe
et al. 2004), in (Bothe et al. 2009) reactive mass trans-
fer with parallel consecutive reactions at rising bubbles
is analyzed by using a local selectivity. The approach
from (Bothe et al. 2004) employs a single scalar field for
any transfer component and this scalar quantitiy refers to
a normalized concentration for which the jump disconti-
nuity can be removed. This has some disadvantages con-
cerning conservativity of the transfer component which
can lead to artificial mass transfer. The latter is caused
by a relative motion of the interface and the concentra-
tion discontinuity due to the use of a different advection
algorithm. In (Alke et al. 2009) a new VOF-based ap-
proach has been introduced which does not posses these
drawbacks. In the present paper, this approach is ex-
tended with a subgrid scale model and is applied to re-
active mass transfer from single rising bubbles.

2 Governing Equations

In the mathematical model it is assumed that the sys-
tem under consideration consists of two immiscible, in-
compressible fluids, separated through a moving and
deformable interface. The interface between the two
phases is presented as a mathematical (sharp) surface of
zero thickness and is denoted by E(t). The local bal-
ances for mass and momentum can then be written as

V u 0 (1)


for mass and


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


where K = -V ne is the curvature (more precisely,
the sum of the principal curvatures) and expresses the
Laplace pressure jump in stationary cases. It is further
assumed that there is at least one transfer component
with the volumetric molar concentration Ck, being sol-
uble in both phases. In case of ideally diluted systems
with small pressure gradients the components have no
influence on the hydrodynamics and can therefore be
considered as passive scalars, transported by the velocity
field. The local balance for the species thus reads as

C+V.(ce u+jk)= R, in Q (t)UQd(t), (6)
at
governing the species transport inside the bulk phases.
At the interface the transmission condition


[jc] ne = 0


holds and the jump condition


[Ip ] 0


is imposed. The diffusive fluxes are given by Fick's law,


ji = -Dk Vck


with the constant diffusion coefficient Dh > 0, and
the continuity of the chemical potential is expressed by
Henry's law, i.e.
k/c' = Hk (10)
with a Henry coefficient Hk / 1. The source term Rk
on the right-hand side of 6 accounts for chemical reac-
tions. Note that the square bracket stands for the interfa-
cial jump according to


Vp + pg + V S (2)


for the momentum. In this equation S denotes the vis-
cous stress tensor given by


S -Vu+ (VuT).


The interface normal unit vector nE points into the con-
tinuous phase for the remainder of this paper. This set of
equations represents the one-field formulation of the two
phase Navier-Stokes equations where the material prop-
erties p and refer to the phase dependent density and
viscosity. Additionally, the following jump conditions
are satisfied at the interface:

[u] 0 (4)


[pI S] nr = a K cn,


S(x h hns))
(11)


[4] (xi) = limr ( (xe + hns)

where 0 is an arbitrary scalar.


3 Numerical Method


The set of governing equations in Section 2 is solved
with the inhouse code Free Surface 3D (FS3D), Rieber
(2004). A finite volume discretization is used for the
spatial discretization and an explicit Eulerian scheme for
the time discretization. The volumes are arranged on
a Cartesian, staggered grid, where scalar variables like
pressure or concentration are stored as cell centered val-
ues and the velocities are stored on the centers of the cell
faces. The code employs the Volume-of-fluid method.
Here, an additional transport equation is solved to keep
track of the location of the different phases and, thereby,
of the interface. This type of method is called a vol-
ume tracking scheme, since only the different phases


du
p +p(u. V)u







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


are transported and the interface is geometrically recon-
structed from that information. In FS3D, the PLIC al-
gorithm (Rider and Kothe 1998) is employed for inter-
face reconstruction. The code uses a one-field formula-
tion of the two-phase Navier-Stokes equations in which
the volumetric surface tension force is incorporated via
the conservative continuum surface stress (CSS)-model
of Lafaurie (Lafaurie et al. 1994). The volume fraction
transport equation reads as

S+ V (u f) 0, (12)

where f is the phase indicator function of the dis-
persed phase domain d (t). In the FV discretization
scheme employed here, the cell centered value of f cor-
responds to dispersed phase fraction inside a computa-
tional cell. Within the VOF-method, the phase related
material properties are given by

S= fPd + (1 f)pc (13)

and
i = frd + (1 f)rq. (14)

Transport of molar species mass

For the computation of the transport of a transfer species
k, the concentration is represented by two separate scalar
variables according to

.,d { ck(x,t) for xE d(t)
'x, t) 0 for x E Q,(t) (15)

and

(x, t) 0 for xE Qd(t)
'>'= ck(x,t) for x E Qc(t). )

This allows for a representation of the individual one-
sided limits of the concentration at the interface. Within
the FV discretization, these scalars are related to the cell
volume V, i.e. the cell centered values are given as


(d 1
(M = I


SIVI
(1 =- f
|V| Jvno-W)


Ck(t)dV



Ck (t)dV.


The computation is carried out with a directional split-
ting. This means, that for each of the three dimensions
a one dimensional transport step is calculated. The or-
der of the directional steps is changed in every time step
to reduce systematical errors. Inside the bulk phases the
convective transport of the volume fraction f is calcu-
lated with a simple first order upwind scheme. Since


f has only one discrete value in the bulk phase, this is
accurate. The concentration of a chemical species, how-
ever, can take arbitrary, non negative values. A first or-
der upwind scheme applied in that case would lead to
unacceptable numerical diffusion. Therefore the convec-
tive transfer is based on the limiter scheme of Van Leer
(Van Leer 1979). In this scheme the concentration in-
side a computational cell is approximated by a linear
function instead of a constant value. The slope of this
linear function is then used to extrapolate the concentra-
tion onto the cell face, where the convective fluxes are
calculated. This is restricted in a manner that no new
maxmia or minima are created in the solution. The lim-
iter scheme reduces numerical diffusion in areas where
the concentration has steep gradients, especially in the
wake of the bubble. In areas with flat gradients it falls
back to the first order upwind scheme. This allows us to
use the advantage of a higher order scheme without tak-
ing the drawback of over- or undershoots, respectively,
which are often created by higher order schemes. At
the interface the same PLIC algorithm is used as for the
f-field. Since both one-sided concentration limits are
known, a clean distinction between the convective fluxes
for the two phases can be made. This ensures that no ar-
tificial mass transfer due to convection occurs. Diffusive
fluxes inside the bulk phases are obtained by a standard
central differencing scheme. In interfacial cells, differ-
ent cases have to be distinguished. Diffusion between
an interfacial cell and a bulk phase cell is calculated for
the appropriate scalar only, using the according diffusion
coefficient for that phase. For the diffusion between two
interface cells two diffusive fluxes are calculated, one
for each phase with the respective diffusion coefficients.
The effective area in that case is approximated by the
cell face area multiplied by the fraction of the respec-
tive phase. To ensure conservativness inside cells with
very small values of f, the diffusive fluxes are restricted.
Since diffusion can only take place until the concentra-
tion gradient is zero, this can be used as a limit for the
diffusion in one time step.


Mass transfer source term

The mass transfer between the phases takes place in the
cells carrying a part of the interface and is accounted for
by means of source terms for the two scalars represent-
ing the full concentration field. In the following, a brief
description of two possible ways to calculate the inter-
facial source term is given. For more details please we
refer to (Kroeger et al.). The calculation of the source
terms is restricted to interfacial cells. Inside those cells,
the interface is assumed to be planar. Further, it can
be assumed that inside an interfacial cell, the concen-
tration in the dispersed phase is homogeneous due to







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


the high diffusion coefficient. In a one dimensional ap-
proach, one can then calculate the one-sided interfacial
limit value of the concentration for the dispersed phase
by simply extrapolating the cell centered value onto the
interface. The limit value for the continuous phase can
be obtained by using Henry's law (10). Together with
one neighboring value from the continuous phase, a lin-
ear approximation for the gradient of the concentration
at the interface can be obtained from this value. Insert-
ing this gradient into Fick's law ( 9) yields the molar flux
of the transfer species across the interface for the contin-
uous phase. Since (7) states that both fluxes have to be
equal, this also gives directly the flux for the dispersed
phase, leading to a conservative algorithm. In a dimen-
sional splitting scheme, the fluxes are calculated for each
direction separately. The second method is quite similar
to the linear gradient method. Here, instead of a linear
approximation a different function is used to estimate the
gradient. The type of function used in this subgrid scale
model is obtained analytical from the mass transfer for
an overflown plane (Bird et al. 2002) and has the form


o. 25
-J


S0 15
o,.z
0
8
5


Sc = 1000


dial distance om paicle ce in m
radial distance from particle race in m


Sc 10000


radial distance rom particle surface in m


Ck(x,y) = ck(


erf))
69 n


8y 2 D Y
V

It can be shown in simulations that in case of
fine grids, both methods yield the same results. I
ever, with the subgrid scale model the result can air
be obtained with a coarser resolution. Therefore
method is to be preferred. The only drawback he
that in this form it can be only used in situations, w
mass transfer resistance in the dispersed phase is si
All numerical results in the next sections are obta
employing the subgrid scale model.


Figure 1: Comparison of exact (lines) and with the
(19) VOF-method obtained concentration profiles at the
equator of a rising fluid particle with Re = 0.284, Sc =
1000 (top) and Re=0.284, Sc=10000 (bottom).


The Reynolds number is 0.284 for these cases. The pro-
very files show very good agreement in this case.
low- As a measure for the overall mass transfer the integral
eadv Sherwood number


this
re is
There
mall.
lined


4 Validation

In this section, comparisons between theoretical and nu-
merical as well as between experimental and numeri-
cal result are used to validate the described numerical
approach. The first comparison employs the flow field
obtained by Hadamard (1911) and Rybczynski (1911)
which is valid for Reynolds number below 0.3. In Fig-
ure 1 comparisons between the theoretical and simulated
concentration profiles at the equator of a rising bubble
are shown. In the numerical setup the bubble equivalent
diameter is set to dc = 4 mm. The computational do-
main for this 3D simulation has the dimension of 4x2x2
dp and is discretized with 64 cells per diameter. The vis-
cosity of the continuous phase is set to 460 mPa s. Due
to the high viscosity, the Schmidt number is quite large.


SL
Sh = (20)

can be used. There are different theoretical correlations
for the Sherwood number for certain special cases:

Sh= 1 + (1 + 0.546Pe2) (21)

which is assumed to be valid for Re 0 and arbitrary
Peclet numbers Pe = ReSc (Clift et al. 1978),
pel1.72
Sh = 2 + 0.651 1.22 (22)

which is assumed to be valid for Re 0 and Sc --
oc (Oellrich et al. 1973). For physical mass transfer
the numerical results compare reasonably well with the
experimental data as can be seen in Figure 2. Since the
setup for the experiment differs slightly from that of the
simulation, the pictures are only qualitatively compared.

5 Results

The method described above can be applied to simu-
late reactive mass transfer with simple and complex re-


o. OO







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Re 0.284 Re 0.284
Sc 1000 Sc 10000
Numerical Result 18.5 64.1
Eq. (21) 12.0 35.1
Eq. (22) 13.0 36.7

Table 1: Comparison of integral Sherwood numbers

















Figure 2: Distribution of oxygen in the wake of a rising
bubble. LiF experiment (left) dp 1.6 mm, numerical
3D Simulation (right) dp 1.8 mm.


actions inside one of the phases. As an example, the
simulation of the metal catalized oxidation of sulfite is
considered in this section. The overall reaction scheme
for this reaction is given as


HS03 + 0, -i HS04 (23)

with k* given as the gross reaction rate assumed con-
stant for this reaction. Note that the complete reaction
scheme is far more complex, hence there is a radical re-
action mechanism involved. For simplification, the gross
reaction is taken into account. In a more abstract nota-
tion this results in a reaction scheme of the type


A + B k* P. (24)

In this scheme A denotes the transfer component (oxy-
gen), B is the dissolved component in the liquid phase
(the hydrogen sulfite ion) and P is the desired product
(the hydrogen sulfate ion). In the left part of Figure 3 a
snapshot of an LiF-experiment for the sulfite oxidation
is shown. In the center of the wake it can be seen that the
oxygen is depleted because of the reaction, thus creating
a gap in the wake. The numerical result reproduces this
gap very well, where so far only a qualitatively compar-
ison is possible.


Figure 3: Concentration distribution of oxygen


6 Conclusions and Outlook

A numerical approach for the simulation of reactive
mass transfer is introduced in this contribution. It is vali-
dated for the case of physical mass transfer. The method
is applied to the case of a simple reaction and the numer-
ical results are compared with experimental data. Vari-
ation of the gross reaction rate constant can be used to
simulate the effect of rising temperature on this reaction,
which we are currently doing. Since it is quite difficult
to measure the velocity at high resolution in an exper-
iment, it is also planned to combine the simulated ve-
locity fields with measured concentration fields in order
to obtain integral Sherwood numbers from the experi-
ments.


7 Acknowledgements

The authors gratefully acknowledge financial sup-
port from the Deutsche Forschungsgemeinschaft (DFG)
within the DFG-project "Reactive mass transfer from
rising gas bubbles" (PAK-119). We also thank Prof.
Michael Schiiter (TH Hamburg-Harburg, Germany) and
Prof. Norbert Rabiger (University Bremen, Germany)
for providing the experimental results.


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