ORI GI NAL

CFD modeling of hydrodynamics and mass transfer

of Rhodamine B in annular reactor
Jatinder Kumar

Ajay Bansal
Received: 17 December 2011 / Accepted: 2 July 2012 / Published online: 17 July 2012
Springer-Verlag 2012
Abstract The hydrodynamics and mass transfer are the
two crucial issues in annular reactors. An accurate pre-
diction of these issues is required for design, optimization
and scale-up applications. The present study deals with the
modeling and simulation of ow through an annular reactor
at different hydrodynamic conditions using computational
uid dynamics (CFD) to investigate the hydrodynamics
and mass transfer. CFD modeling was utilized to predict
velocity distribution, average velocity and average mass
transfer coefcient in the annular geometry. The results of
CFD simulations were compared with the mathematically
derived equations and already developed correlations for
validation purposes. CFD modeling was found suitable for
predicting hydrodynamics and mass transfer for annular
geometry under laminar ow conditions. It was observed
that CFD also provides local values of the parameters of
interest in addition to the average values for the simulated
geometry.
List of symbols
v ~ Velocity vector (m s
-1
)
P Pressure (N m
-2
)
Y
i
Mass fraction of species i (dimensionless)
J
~
i
Diffusive ux of species i (kg m
-2
s
-1
)
R
i
Rate of production/depletion of species
i (kg m
-2
s
-1
)
D
i,m
Diffusivity of species i in the mixture (m
2
s
-1
)
r Divergence (m
-1
)
v
z
Axial velocity (m s
-1
)
v
z
h i Average velocity (m s
-1
)
r Radius (m)
l Length of the annulus (m)
Sh
av
Sherwood number (dimensionless)
Sc Schmidt number (dimensionless)
Re Reynolds number (dimensionless)
d
e
Hydraulic diameter (m)
Greek symbols
q Density (kg m
-3
)
s Stress tensor (N m
-2
)
j Dimensionless ratio (r
i
/r
o
)
l Viscosity (kg m
-1
s
-1
)
a Dimensionless factor dened as [(1 - j)/j][1/2 - (j
2
/
(1 - j
2
))ln(1/j)]/[((1 ? j
2
)/(1 - j
2
))ln(1/j) - 1]
1 Introduction
Annular reactors have been extensively studied in literature
because of advantageous basic features of this geometry.
Some of the studies on annular geometry include investi-
gation on role of a Pt/Al
2
O
3
catalyst in the oxidative
dehydrogenation of propane [1], kinetics of carbon mon-
oxide oxidation at high temperature [2], partial oxidation of
methane [3], treatment of dye manufacturing plant efuent
using UV/H
2
O
2
and multi-UV lamps [4], photocatalytic
degradation of some of VOCs in the gas phase [5], role of
gas-phase chemistry in the rich combustion of H
2
and CO
over a Rh/Al
2
O
3
catalyst [6], simulation of degradation of
perchloroethylene in air [7], inuence of ns on photo-
catalytic removal of formaldehyde [8], photocatalytic
degradation of gaseous 1-propanol: kinetic modeling and
pathways [9], modeling of annular ow [10] and direct
J. Kumar (&) A. Bansal
Department of Chemical Engineering,
Dr. B. R. Ambedkar National Institute of Technology,
Jalandhar 144011, Punjab, India
e-mail: replytojk@yahoo.com
123
Heat Mass Transfer (2012) 48:20692077
DOI 10.1007/s00231-012-1052-4
conversion of methane to formaldehyde under high tem-
perature and short residence time [11].
The hydrodynamics and mass transfer are the two most
important issues in the annular reactors depending upon the
kinetics, operating conditions, and geometrical properties
of the system. Hence modeling, design and scale-up of
these reactors require an accurate prediction of hydrody-
namics and mass transfer. The traditional methods of pre-
dicting uid ow and mass transfer in annular reactors
depends heavily on theoretical modeling and empirical
correlations. Many correlations in terms of dimensionless
numbers have been developed in past for estimation of
mass transfer in annular geometry under developing and
developed ow conditions [1214]. These correlations
have disadvantages of being applicable to certain range of
hydrodynamic conditions, suitable only to specic reactor
congurations and not taking into account the local
effects [15].
A new method to estimate the hydrodynamics and mass
transfer is required which should also be able to predict the
velocity and concentration elds in the annular reactor for
better design and scale-up applications. A very effective
approach to tackle this challenge is computational uid
dynamics (CFD). CFD is a well established technique for
the analysis of systems involving uid ow, mass transfer,
heat transfer, reaction and associated phenomena. There
are several advantages of using CFD such as ability to
study hazardous system in a safe environment and reduc-
tion in the time and cost of analysis [16]. The recent CFD
studies on annular reactors include simulation of a pilot-
scale annular bubble column photocatalytic reactor [17],
analysis of photocatalytic gas phase vinyl chloride oxida-
tion [18], simulation of trichloroethylene (TCE) oxidation
at various pollutant concentrations, ow rates, and reactor
lengths [19] and study of annular photoreactor hydrody-
namics [20]. More recently, Duran et al. [15] applied CFD
to investigate single-phase ow mass transfer prediction in
annular reactor for laminar, transitional and turbulent ow
condition. It was found that laminar model predicted mass
transfer successfully under laminar conditions. For turbu-
lent conditions, AKN (Abe, Kondoh and Nagano) model,
and RSM (Reynolds Stress Model) performed well. Sant-
oro et al. [21] studied the oxidation of tributyl phosphate
(TBP) and tri(2-chloroethyl) phosphate (TCEP) in parallel
and cross-ow annular photoreactor. CFD simulations
enabled the spatial visualization of hydrogen peroxide and
hydroxyl radical distributions in the reactor. Queffeulou
et al. [22] investigated the removal of acetaldehyde in
annular photocatalytic reactor with a thin lm of TiO
2
coated on stainless steel plate. Modeling of uid dynamics
and reaction was realized with a CFD approach. In terms of
conversion yield, model predictions and experimental
results were found in good agreement. Vincent et al. [23]
studied the hydrodynamics and degradation of acetone in
annular reactor. It was observed that CFD modeling esti-
mate the kinetic parameters of the degradation of acetone
very close to the experimental results. To authors
knowledge, no research has been performed to predict the
hydrodynamics and mass transfer of a dye in annular
reactor using CFD.
The aim of the present work was to carry out CFD
modeling of the annular geometry to predict the uid ow
and mass transfer of a dye at various hydrodynamic con-
ditions. The results of the CFD model were compared with
theoretical modeling and empirical correlations for vali-
dation of the model. Rhodamine B was used as model dye.
It is organic in nature and most important xanthene
dye [24].
2 CFD modeling
2.1 Governing equations
In present study, it is assumed that the uid is Newtonian,
incompressible, isothermal and non-reactive with constant
physical properties. The hydrodynamics and transport of
Rhodamine B was modeled by solving mass, momentum
and species conservation equations using commercial CFD
code Fluent 6.3.26 (Fluent Inc., USA). The general forms
of the governing equations for modeling the system are as
follows:
Mass conservation equation:
r: qv ~ 0 1
Momentum conservation equation:
r: qv ~v ~ rP r s
_ _
qg 2
Species conservation equation:
o
ot
qY
i
r qv ~Y
i
r J
~
i
R
i
3
In Eqs. (1)(3), q is density, v ~ is velocity vector, P is
pressure, s is stress tensor, g is acceleration due to gravity,
Y
i
is mass fraction of species i. The species conservation
equation includes convective, diffusive and reactive terms.
In Eq. (3), R
i
is the rate of production or depletion of the
species i by chemical reaction. The diffusion ux J
i
can be
derived as follows using Ficks law with molecular
diffusion coefcient of the species in the medium:
J
~
i
qD
i;m
rY
i
4
In Eq. (4), D
i,m
is the diffusion coefcient for species i in
the mixture. Detailed description of the conservation
2070 Heat Mass Transfer (2012) 48:20692077
123
equations along with the associated correlations and
parameters are provided in the Fluent Manual [25].
2.2 Hydrodynamic model
The simulation of the present system was performed with a
three dimensional, steady state laminar ow model. The
laminar ow model has been successfully used for CFD
simulations of annular photocatalytic reactors for air
treatment [18], experimental and CFD analysis of photo-
catalytic gas phase vinyl chloride oxidation [19], CFD
modeling of mass transfer in annular reactors [15] and
Three-dimensional CFD modeling of a at plate photo-
catalytic reactor [26]. This model utilizes Eqs. (1) and (2)
combined with Newtons law of viscosity for computation
of velocity eld within the reactor domain. The Eq. (3) has
been used to nd out the concentration prole in the reactor
geometry. In the present case, there is no chemical reaction
taking place and only diffusion of chemical species hap-
pens. The R
i
term in Eq. (3) is zero. Further information
and examples of laminar ow model can be found in Bird
et al. [27]. The same model has been used in the present
work to evaluate CFD modeling of hydrodynamics and
mass transfer of Rhodamine B dye at ow velocities of 1.8,
3.6, 5.5, 7.3 and 9.1 mm/s. These ow velocities corre-
spond to Reynolds numbers (Re) of 20, 40, 60, 80 and 100
respectively through the annular section.
2.3 Geometrical model
The annular reactor geometry studied in the present
research is shown in Fig. 1. The geometry was created in
commercial software Gambit. The reactor consists of
39 mm outer tube diameter, 27 mm inner tube diameter
and 500 mm total length with 12 mm diameter inlet and
outlet tubes. The inlet and outlet tubes were placed 1 mm
from each respective end to form a U-shape annular reac-
tor. The length of both the inlet and outlet tubes were
chosen 50 mm to ensure fully developed ow at the
entrance of the reactor. The 450 mm middle length on the
inner wall of the outer tube was set at constant concen-
tration of Rhodamine B. This length was utilized to study
the mass transfer from inner surface of the outer tube to the
bulk of the uid in the annular region.
2.4 Mesh design
The commercial mesh generator Gambit was used to create
the grid. The hexahedral cells were used to discretize the
reactor domain where the mass transfer was to be studied
(middle annular region of length 450 mm). The unstruc-
tured cells were used in other part of the reactor geometry
as shown in Fig. 2. The utilized grid for the reactor had
approximately 1.2 million elements and they were veried
to give mesh independent results. Figure 3b shows the
meshing in longitudinal center plane of the area of annular
geometry shown in Fig. 3a.
2.5 Boundary conditions
The boundary conditions for the CFD model were dened
as follows. At the inlet, velocity of the uid was specied.
The direction of the ow was dened normal to the
boundary. At the outlet, boundary condition pressure-outlet
was specied with a value of 1 atm. At all the walls, a
no-slip boundary condition was imposed. Also, zero dif-
fusive ux of species was specied at the walls, except for
the walls where a constant concentration of Rhodamine B
was xed with a value of 0.05 (mass fraction). This con-
centration corresponds to the saturation concentration of
Rhodamine B in water at 300 K [28].
2.6 Physical properties
The physiochemical process studied in this investigation is
the isothermal ow and diffusive mass transfer of Rhoda-
mine B in water at 300 K. At this temperature the satura-
tion concentration of Rhodamine B in water is very low so
the physical properties of water can be assumed for the
Fig. 1 Schematic diagram of annular geometry Fig. 2 Regions of structured and unstructured meshing
Heat Mass Transfer (2012) 48:20692077 2071
123
system. The density and viscosity of water considered are
998.2 kg/m
3
and 10.03 9 10
-4
Pa s respectively. The
diffusion coefcient of Rhodamine B was considered to be
3.6 9 10
-10
m
2
/s [28].
2.7 CFD solution
Commercial CFD code Fluent 6.3.26 was used to perform
simulations. The governing equations were solved using
pressure based three dimensional solver. Second order
upwind discretization scheme was employed except for
pressure. PRESTO was selected as discretization scheme
for pressure. The SIMPLE algorithm was chosen for the
pressurevelocity coupling. The already mentioned under-
relaxation factors were considered except for dye where a
value of 0.6 was selected. Convergence of numerical
solution was ensured by monitoring the scaled residuals to
a criterion of 10
-4
for the continuity and momentum
variables, and 10
-6
for the concentration. The solution of
the model was utilized to nd out the velocity vector eld,
concentration eld, velocity prole, average velocity and
mass transfer coefcient.
3 Mathematical modeling
The mathematical modeling of velocity distribution, aver-
age velocity and mass transfer coefcient was also done for
the reactor. The results of CFD modeling were compared
with the outcome of mathematical modeling for evaluation
purpose. The expression for velocity distribution and
average velocity for the reactor geometry was obtained by
applying momentum balance on a cylindrical shell of
innitesimally small thickness within the reactor domain.
The assumptions of laminar ow, constant density and no
end effects were also considered. The Eqs. (5) and (6)
represent the derived expressions for velocity distribution
and average velocity respectively.
v
z

P
0
P
l
r
2
0
4ll
1
r
r
0
_ _
2

1 j
2
ln
1
=
j
_ _
_
_
_
_
ln
r
r
0
_ _
_
_
_
_
5
v
z
h i
P
0
P
l
R
2
8ll
1 j
4
1 j
2
_ _

1 j
2
ln
1
=
j
_ _
_
_
_
_
_
_
_
_
6
where r
o
; r
i
; l; l are the inner radius of outer cylinder, outer
radius of inner cylinder, viscosity of uid and length of
reactor respectively. P
0
and P
l
are the pressures at inlet and
outlet of the annulus respectively. j is a dimensionless
ratio (r
i
/r
o
). Further details of derivations of such expres-
sions may be seen in Bird et al. [27]. The mass transfer
coefcients were determined using the correlations for
laminar ow developed by Ross and Wragg [12], Ould-
Rouis et al. [13] and Mobarak et al. [14]. These correla-
tions for laminar ow are represented in Table 1.
4 Results and discussion
4.1 Hydrodynamics
The hydrodynamics of an annular reactor is characteristic
of its overall performance and it helps to evaluate the
nature of ow patterns in the reactor. In the present
Fig. 3 Meshing a exit region of the annular geometry, b meshing of longitudinal center plane of the exit region of the annular geometry
Table 1 Correlations for mass transfer coefcient
Hydrodynamic
condition
Correlation Source Eqs. no.
Fully developed
laminar ow
Sh
av
= 1.614
(Re Sc a d
e
/l)
1/3
Ross and
Wragg [12]
(7)
Developing
laminar ow
Sh
av
= 1.029 Sc
1/3
Re
0.55
(d
e
/l)
0.472
Mobarak
et al. [14]
(8)
Sh
av
= 0.66 Sc
1/3
(Re a d
e
/l)
0.52
Ould-Rouis
et al. [13]
(9)
2072 Heat Mass Transfer (2012) 48:20692077
123
research, hydrodynamics have been characterized in terms
of axial velocity prole, average velocity, velocity contours
and velocity vector eld to assess the ow structures within
the reactor. CFD simulations of the annular reactor oper-
ating at ow velocities of 1.8, 3.6, 5.5, 7.3 and 9.1 mm/s
were performed. Figure 4 depicts the CFD modeled vari-
ation in axial velocity with respect to radial distance within
the annular region at various Reynolds number. The results
correspond to a line between the inner and outer cylinder at
the exit of the annular section under study. The velocity
distribution depicted by CFD simulations along the annular
space is parabolic which is the characteristic of laminar
ow in agreement with Bird et al. [27]. Figure 4 also
represents a comparison between mathematically devel-
oped Eq. (5) and CFD modeling at various Reynolds
numbers. CFD simulations also show that the velocity is
minimum at the walls and maximum near the middle of
annular region which is in accordance with Eq. (5). CFD
modeling closely maps the velocity proles calculated by
mathematically developed Eq. (5) as obvious from Fig. 4.
The average relative error in prediction of axial velocity by
CFD modeling is around 2.1 % which is acceptable from
industrial point of view. Figure 5 represents the average
velocities predicted by CFD modeling. It also shows a
comparison of CFD modeling with mathematically devel-
oped Eq. (6) in determination of average velocity. It is
obvious that the outcome of CFD modeling is in agreement
with Eq. (6) with a little deviation. The average relative
error in prediction of average velocity is 5.8 %.
Figure 6 shows the contours of velocity magnitude (m/s)
for the annular geometry for a ow velocity of 9.1 mm/s
through the annulus. The results correspond to longitudinal
center plane of the reactor. Similar velocity magnitude
contours (different in magnitude) were obtained in the
simulations at other ow velocities and hence not shown
here. It is very clear from the contours that the velocity
through the inlet and outlet is more because these tubes
have smaller cross-sectional area of ow as compared to
annulus. There is non-uniformity in the ow at the entrance
of annulus region due to sudden expansion and change in
direction of ow. The ow is uniform through the annular
region of interest (i.e. middle annular region of length
450 mm).
The velocity vector elds obtained from CFD simula-
tions performed at ow velocities 1.8 and 9.1 mm/s have
been shown in Fig. 7. The results correspond to the lon-
gitudinal center plane of the exit region of the annular
geometry. Similar velocity vector patterns (different in
magnitude) were obtained in the simulations at other ow
velocities and hence not shown here. As seen in Fig. 7,
Fig. 4 Axial velocity within the
annular region of reactor
Fig. 5 Average velocity through the annular region of the reactor
Heat Mass Transfer (2012) 48:20692077 2073
123
the velocity along the annular space is uniform and it varies
parabolically in radial direction. The velocity decreases
from a maximum near the middle of the annular region to a
minimum at the walls of the reactor. The ow is fully
developed throughout the reactor. The thickness and length
of the vectors indicate the magnitude of velocity. The
similar kind of observations have been recorded by Duran
et al. [15] under laminar ow conditions during his study of
mass transfer of benzoic acid in annular reactors.
4.2 Mass transfer
Computational uid dynamics simulations of molar con-
centration of Rhodamine B were performed over the
Fig. 6 Velocity magnitude (m/s) contours within the reactor for ow
velocity of 9.1 mm/s
Fig. 7 Velocity vectors a ow
velocity 1.8 mm/s, b ow
velocity 9.1 mm/s
2074 Heat Mass Transfer (2012) 48:20692077
123
studied range of ow velocities. Figure 8 shows the con-
tours of molar concentration (kmol/m
3
) of Rhodamine B at
ow velocities of 1.8 and 9.1 mm/s. The contours corre-
spond to the longitudinal center plane of the exit region of
the annular geometry. Similar concentration contours have
been observed for other studied ow velocities and hence
not given here. At all Reynolds number the concentration
gradient in the axial direction occurs only due to diffusion
as there is no reaction taking place. Figure 8 depicts that
the highest concentration is near the internal wall of the
outer pipe where constant concentration of Rhodamine B
was maintained. The lowest concentration (almost zero) is
at the wall of inner pipe indicating no diffusion to this part
of the annular geometry. The diffusion of Rhodamine B to
a very small distance within the reactor geometry is
attributed to its lowdiffusion coefcient (3.6 9 10
-10
m
2
/s)
in water. A comparison of concentration contours at ow
velocities of 1.8 and 9.1 mm/s indicates that at low ow
velocity the stream has enough time to stay in the reactor to
get more dye diffused into the uid as obvious from Fig. 8a,
b. At ow velocity of 1.8 mm/s the maximum concentration
of Rhodamine B in the annular reactor is 6.23 9 10
-2
Fig. 8 Contours of molar
concentration of Rhodamine B
(kmol/m
3
). a Flow velocity
1.8 mm/s, b ow velocity
9.1 mm/s
Heat Mass Transfer (2012) 48:20692077 2075
123
(kmol/m
3
) in comparison to a value of 2.2 9 10
-2
(kmol/
m
3
) at ow velocity of 9.1 mm/s.
The average mass transfer coefcients obtained from
CFD simulation of annular geometry under various
hydrodynamic conditions are presented in Fig. 9. As
expected, mass transfer coefcients increased monotoni-
cally with ow velocities. The mass transfer data obtained
in this work showed good agreement with those from other
reported investigations where similar annular conguration
was used [12, 15].
Figure 10 compares the average mass transfer coef-
cients predicted by CFD simulations with the values cal-
culated by the correlations tabulated in Table 1. The CFD
modeled average mass coefcients coincide well with the
outcome of already proposed correlations. As already dis-
cussed in the Sect. 4.1 that the ow through the annulus is
fully developed, the CFD prediction has close agreement
with the correlation proposed for fully developed ow
(Eq. 7). The average relative error in prediction of average
mass transfer coefcients by CFD simulations with respect
to Eq. (7) is 10 %.
5 Conclusion
The hydrodynamic and mass transfer of Rhodamine B in
annular reactor under various operational conditions has
been investigated using CFD method. The results of CFD
modeling have been compared with the results of mathe-
matical modeling and already developed correlations. The
velocity distribution and average velocity predicted by
CFD were in close agreement with the outcome of math-
ematical modeling of annular reactor. The average mass
transfer coefcients estimated using CFD at ow velocities
of 1.8, 3.6, 5.5, 7.3 and 9.1 mm/s were in close agreement
with the correlations developed by Ross and Wragg [12]
for fully developed ow through annulus. It was found that
CFD also provides local values of the parameters of
interest in addition to the average values for the simulated
geometry. In the present study, detailed local information
(velocity and concentration proles) was obtained using
CFD modeling of the system which provided a qualitative
understanding of the process to better explain the results.
The detailed predicted ow eld gave an accurate insight to
the uid behavior and presented information which cannot
be obtained from correlations and experiments. It has been
observed that CFD modeling is capable of predicting the
hydrodynamics and mass transfer for annular geometry
under laminar ow conditions.
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