Hybrid Numerical Simulation of Fluid Flow and
Light Distribution in a Bubble Column Photobioreactor
Christopher McHardy
1
, Giovanni Luzi
2
, Jose Rodriguez Agudo
2
,
Antonio Delgado
2,3
and Cornelia Rauh
1,2,3
1
Institute of Food Biotechnology and Food Process Engineering, Technische Universität Berlin, Königin-Luise Str. 22,
Berlin, Germany
2
Institute of Fluid Mechanics, FAU Campus Busan, University of Erlangen-Nuremberg, 618-230 Busan, Republic of Korea
3
Institute of Fluid Mechanics, University of Erlangen-Nuremberg, Cauerstr. 4, Erlangen, Germany
Keywords: Photobioreactor, Microalgae, Light-Matter Interaction, Computational Fluid Dynamics, Radiation
Transport, Lattice Boltzmann, Hybrid Method.
Abstract: Cultivation of phototrophic microorganisms occurs often in closed photobioreactors (PBR). Thereby, the
distribution of light inside PBR is a key factor for phototrophic growth and reactor productivity. To predict
local light intensities, it is often assumed that the absorption rate is constant in space, and scattering by
microorganisms is negligible. The present contribution aims to present a hybrid model to simulate fluid flow
characteristics and its impact on light fields in a bubble column PBR. First, numerical simulations of bubble
column flow have been performed. Afterwards, the computed local air volume fractions have been used to
obtain local radiation characteristics of the gassed suspension, and polychromatic light fields were
computed and compared to the optically homogeneous case.
1 INTRODUCTION
Phototrophic microorganisms are characterized by
the ability to use light energy to drive their cellular
metabolism by means of photosynthesis. By doing
so, the energy of light is used for the conversion of
carbon dioxide into biomass. The ability to
accumulate huge amounts of proteins or
triacylglycerides into the biomass makes
phototrophs a promising option for sustainable
production of food, feed and fuels (Williams and
Laurens, 2010).
Technical cultivation of phototrophs commonly
occurs in closed PBR. Since the rate of
photosynthesis is directly linked to the light intensity
a cell is exposed to (Williams and Laurens, 2010),
the distribution of light inside PBR is a key factor
for phototrophic growth and productivity. The
distribution of light is primarily determined by the
presence of microorganisms. Light is absorbed along
the path and consequently intensity is attenuated.
Thereby, the local rate of absorption is directly
proportional to the density of cells in the culture
suspension (Pilon et al. 2011). Moreover, anisotropic
scattering of light causes a major difficulty for an
accurate prediction of local light intensities because
of two reasons. First, the scattering characteristics of
the microorganisms must be measured (directly or
indirectly), which is already a non-trivial task
(Dauchet et al., 2015; Kandilian et al., 2016).
Second, the computation of scattering requires an
adequate discretization of the light angular
distribution (Hunter and Guo, 2015). Due to these
difficulties, a common approach in bioengineering is
to neglect scattering and approximate light
propagation by Lambert Beer’s law instead of
solving the full Radiative Transfer Equation (RTE).
It is often assumed that cells are homogeneously
distributed in a PBR, and consequently also the
radiation characteristics of the suspension are
assumed to show no spatial variance. However, in
most PBR gassing occurs to supply carbon dioxide
to the suspension, remove oxygen and provide
energy for mixing of the liquid phase (Olivieri et al.,
2014). Since the light absorption of gas bubbles is
negligible, and their scattering characteristics
deviate from those of phototrophic cells, the
assumption of an optically homogeneous suspension
does not hold anymore in presence of bubbles.
304
McHardy C., Luzi G., Agudo J., Delgado A. and Rauh C.
Hybrid Numerical Simulation of Fluid Flow and Light Distribution in a Bubble Column Photobioreactor.
DOI: 10.5220/0006226003040311
In Proceedings of the 5th International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS 2017), pages 304-311
ISBN: 978-989-758-223-3
Copyright
c
2017 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
The present contribution aims at presenting a
methodology for computing radiation fields in
multiphase flows. It is based on the assumption that
the radiation characteristics of a mixture can be
obtained by superimposing those of single
components. Therefore, the hydrodynamic
characterization of a PBR is required, for which
Computational Fluid Dynamics (CFD) is a suitable
tool. Two dimensional numerical simulations of
fluid flow inside bioreactors are among the firsts that
have been performed, since they require a lower
numerical effort. However, they were found to be
highly grid dependent (Bech, 2005). Three
dimensional unsteady simulations are more adequate
to reproduce the real complex flow patterns of
bubble column flows. Examples of numerical
simulations of fluid flow in a cylindrical bioreactor
are reported in literature (Pfleger and Becker, 2001;
Lobaton et al., 2011), but the so called “non-drag
forces” are absent in their model. In recent and more
complete models lift, virtual mass, wall lubrication
and turbulent dispersion forces have been
considered, since they strongly affect the flow field
(Masood and Delgado, 2014). More sophisticated
models integrate CFD, light distribution and kinetics
growth of algae cells. Combination of CFD with
Lambert Beer law and Aiba model (Zhang et al.,
2015), or with optical ray tracing simulations based
on an empirical three-parametric model (Krujatz et
al. 2015) are reported. In other contributions
compartmental modelling approach and
photosynthetic factory model (Nauha et al., 2013),
considering also light directionality and day and
night conditions (Nauha et al., 2013) are presented.
In the present contribution, first, the bubble
column flow inside a cylindrical PBR is simulated.
Next, the local radiation characteristics of the gassed
turbid suspension are computed. The RTE is solved
afterwards by means of a Lattice Boltzmann solver
(McHardy et al., 2016).
2 THEORY
Here, the mathematical models of fluid flow and
light distribution are presented. The Eulerian-
Eulerian formulation of two-phase flow and the
applied Lattice Boltzmann method are described in
detail.
2.1 Fluid Flow Model
An Eulerian-Eulerian approach is chosen to simulate
the bubble column flow. Both, the continuous and
the disperse phase are modelled as an interpenetrated
continuum, where the inter-phase forces are taken
into account by an extra term in the momentum
equation. The corresponding mass conservation
equations, both for the liquid and the gas phase read

0
kk kkk
t
 

u
(1)
where
,kLG
. Here,
L
stands for liquid and
G
for
gas. The other symbols
k
u
,
k
and
k
denote the
velocity vector, the volume fraction and the density
of each phase, respectively. The momentum
equations are


,
kkk kkkk
kkkkksk
t
p
 



uuu
gM

(2)
where
,
s
LG
. The terms on the left-hand side
represent the temporal and convective acceleration,
while those on the right-hand side represent the
pressure gradient, the turbulent stress tensor, gravity
and interphase forces (Masood and Delgado, 2014).
The stress tensor is defined as

,
T
kkeff k k

uu
(3)
where incompressibility has been assumed for both
phases. The effective viscosity for each phase is the
sum of the molecular and the turbulent one
,, ,keff kLam kTurb


(4)
Finally, the last term of Eq. (2) reads
,, ,,,
,,, ,
ADL
sk LG LG LG LG
A
VM WL TD
LG LG LG GL


MM FFF
FFF M
(5)
where the terms on right hand side of Eq. (5) are the
drag
, the lift, the virtual mass, the wall lubrication
and the turbulent dispersion forces (Masood and
Delgado, 2014)
. The drag force is due to viscous
shear stress and pressure distribution around the
bubble surface

,
3
4
D
D
L
GGLGLGL
B
C
d


F
uuuu
(6)
Hybrid Numerical Simulation of Fluid Flow and Light Distribution in a Bubble Column Photobioreactor
305
where
D
C
is the drag force coefficient and
B
d
is the
bubble diameter. The Grace correlation is used to
model
D
C
(Grace et al., 1976). The lift force acting
on the gas phase due to a rotational liquid phase can
be written as
,
L
LG G L L G L L
C

Fuuu
(7)
where
L
C
is computed according to the Legendre
Magnaudet model (Legendre et al., 1998). The
virtual mass force represents the added inertia to gas
bubbles since they are moving through the liquid
phase
,
D
D
DD
VM
G
L
LG G L VM
C
tt





u
u
F
(8)
where
0.5
VM
C
is the virtual mass coefficient. The
wall lubrication force is an artificial force that
models the situation where bubbles concentrate in a
region close to a wall
2
,
ˆ
WL
L
GWLGLLGW
C

Fuun
(9)
where
ˆ
W
n
is the normal to a reactor surface and the
coefficient
VM
C
is computed with the Frank model
(Frank et al., 2004; Frank et al., 2008). Finally, the
turbulent dispersion force is responsible for the
dispersion of phases and it can be expressed as
,
,
,
LTurb
TD
G
L
LG TD D
LTurb L G
CC





F
(10)
according to the Favre averaged model (Burns et al.,
2004). Here,
,
L
Turb
is the turbulent Schmidt number
for the continuous phase, and
1
TD
C
is a multiplier.
2.1.1 Turbulence Model
A two-equation model is able to accurately compute
the turbulent viscosity of the continuous phase
,LTurb L
k

(11)
where
k is the turbulent kinetic energy and
is the
turbulent frequency.
These two quantities are here
computed according to the Shear Stress Transport
(SST) model (Menter, 1994) by solving two
independent scalar transport equations for
k and for
. A simple algebraic turbulence model is chosen
for the gas phase
,
,
,
L
Turb
G
GTurb
L
LTurb

(12)
2.2 Radiative Transport Model
The RTE is a Boltzmann-type transport equation,
which balances the spatio-temporal evolution of
intensity of radiation (or radiance)

ˆ
,,Itxs
.



ˆ
,,
ˆˆ
,,
ˆˆˆˆ
,, , , , d
4
4
It
It
ct
It I t





xs
sxs
xs xs' s's
(13)
In Eq. (13),
c
, t ,
,
,
ˆ
s
and
x
denote speed of
light, time, solid angle, extinction coefficient, unit
direction and position vector. The scattering albedo
s
ca
is defined as
sca
(14)
where
and
denote the absorption and
scattering coefficients, which are related to the
extinction coefficient by


.
is the
scattering phase function, which describes the
angular distribution of scattered light. In Eq. (13),
the emission of radiation by microorganisms is
neglected, due to its minor relevance for the light
field in photobioreactors. Properties of the RTE are
described in detail elsewhere (Modest, 2013) and
need not to be repeated here.
2.2.1 Radiation Transport Lattice
Boltzmann Method (RT-LBM)
To solve Eq. (13), a lattice Boltzmann method
(McHardy et al., 2016) is applied in this work.
Following the usual lattice Boltzmann formalism, a
discrete representation of Eq. (13) reads

,,
,,
(,)(,)
(,) (,)
ii
eq
ii i
IttIt
ItIt


 

xx x
xx
(15)
PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology
306
Herein,
,i
I
is discretized in the momentum space,
which consists of directional and frequency
components. Therefore, a monochromatic radiation
propagating along a trajectory, which is given by the
connection of two nodes in a cubic lattice, is
considered. The scattering from the
th
j
into the
th
i
trajectory is captured by the function

,
,
eq
i
I
t
x
,
which is a discrete representation of the in-scattering
integral in Eq. (13).
 
,,,,
1
,,
Q
eq
iijijij
j
I
twItf

xx
(16)
Thereby, the integral is replaced by a summation and
the scattering phase function is represented by a
symmetric
Q
by
Q
matrix which is computed by
means of an algebraic method, proposed in
(McHardy et al., 2016). A correction function
,ij
f
considers that the propagation distance depends on
the direction of propagation on a cubic lattice. The
absorption and scattering coefficients may have
spectral dependency, so
is also spectrally
distributed. For a D3Q26 model (3 dimensions, 26
discrete trajectories), the nodes and weights
i
w
of the
quadrature are shown in table 1. The quantity
,ii


x
(17)
is the collision frequency and relates grid spacing to
the mean free path of radiation. The collision
frequency depends on direction, because the
probability of collision increases with increasing
path length.
The macroscopic quantities of the radiation field,
namely mean intensity
J
and radiation flux
F
, can
be obtained by computing moments of
,i
I
.
,
1
Q
ii
i
J
wI d
(18a)
,
1
Q
ii i
i
wI d
Fc
(18b)
While the integration over solid angles is
approximated by the summation over directions, the
integration across the spectral component of
momentum is approximated by composite
Simpson’s Rule, using a step length of
30
 nm.
Table 1: Lebedev quadrature on a unit sphere, used for
discretization of the RTE. The nodes of the quadrature
result from all possible permutations of the typical nodes.
i typical node (x,y,z) weights
1 - 6 (1,0,0) 1/21
7 - 14 1/3 (1,1,1) 4/105
15 - 26 1/2 (1,1,0) 9/280
2.2.2 Radiation Characteristics of
Multiphase Suspensions
A major assumption of the presented methodology is
that the radiation properties of a mixture can be
obtained by superimposing those of the single
components. This is true, if single particles scatter
independently of each other. For suspensions of
point-like scatterers, several criteria for estimating
scattering-regimes can be found in literature. Baillis
and Sacadura (2000) as well as Modest (2013) refer
to a scattering regime map, which estimates the
scattering regime with respect to the particle volume
fraction and a size parameter. Jonasz and Fournier
(2007) suggest as a criterion the relation
11g
x
, where
g
is the cosine of the mean
scattering angle. For suspensions of microalgae at
concentrations in the single-digit gram-per-litre
range, typical for PBR, the suggested criteria are
generally fulfilled. Therefore the radiation
characteristics of a microalgae suspension can be
obtained from
A A ABS
cA
(19a)
AASCA
cA
(19b)
where
A
c
is the mass concentration of scatterers in
the suspension and
A
is the mass-specific
absorption or scattering cross-section, respectively.
For dispersed bubbles, the absorption and
scattering coefficients can be computed analogously,
by multiplying the number density of gas bubbles
B
n
by the absorption or scattering cross-section. It is
assumed that bubble absorption is negligible
compared to absorption by microalgae, therefore
0
B
. The number density of bubbles inside a
control volume can be computed from the local air
volume fraction and the volume of a single bubble
B
V
, in case of monodisperse bubbles.
The scattering
cross-section can be obtained by the geometrical
Hybrid Numerical Simulation of Fluid Flow and Light Distribution in a Bubble Column Photobioreactor
307
cross-section of a bubble
B
A
and the scattering
efficiency
s
ca
Q
. Therefore, the scattering coefficient
for dispersed bubbles becomes
 
s
ca
B
B
G
B
A
Q
V

xx
(20)
In a gassed suspension of microalgae, the effects
of the single components superimpose (Pilon et al.,
2011; Berberoglu et al., 2007). The effective
absorption and scattering coefficients of the mixture
are given by
1
AG

xx
(21a)
1
A GB

 xx
(21b)
In Eq. (21) it is considered that the liquid volume
and therefore the volumetric concentration of
microalgae are reduced by the presence of air. Since
microalgae scatter light differently compared to gas
bubbles, an effective scattering phase function must
be computed. Pilon et al. (2011) suggests to use the
scattering coefficients of microalgae and gas bubbles
as weighting coefficients, so that the effective
scattering phase function becomes



1
1
GAABB
GAB





x
x
x
(22)
3 PHOTOBIOREACTOR MODEL
Here, details concerning simulation set-up are given.
The physical parameters are described first, followed
by the outline of geometry and grid generation, as
well as numerical solutions of fluid flow and light
distribution.
3.1 Physical Parameters
An isothermal cylindrical bubble column PBR of
diameter
9.4D cm and height 50H cm with an
air headspace of 10 cm is considered. The reactor is
aerated from a small inlet of diameter
4d
mm,
located at the bottom of the column. Spherical
monodisperse bubbles have a constant mean
diameter
7
B
d
mm and mass flow rate
6
9.85 10
G
m

kg/s. The PBR is illuminated by red
LED from four sides. The spectrum of the LED is
approximated by a Gauss distribution with peak
emission at
655 nm and 15 nm standard deviation.
A suspension of microalgae is located inside the
PBR. The absorption and scattering cross-sections of
the microalgae are assumed to be the same as
measured by Kandilian et al. (2016) and cells are
assumed to be present as individuals and no
agglomeration occurs. Mie-scattering of cells is
expressed by the Henyey-Greenstein phase function


2
1.5
2
11
4
12cos
g
gg


(23)
where
0.98g
(Kandilian et al., 2016). For gas
bubbles, the asymmetry factor
B
g
was set to 0.86
and the scattering efficiency
1
sca
Q
was chosen as
an approximation for Mie-scattering of large spheres
(Pilon et al., 2011).
3.2 Fluid Flow Simulation
Both, the geometry and the grid have been created
using the commercial software ANSYS ICEM
®
. The
domain is covered with a structured mesh, obtained
by using two O grids, and a mesh of
59059
volumes was selected. All numerical simulations
were carried out using the commercial software
ANSYS CFX
®
15.0 which is a finite volume based
solver. At the inlet, an air mass flow rate was set,
and an opening boundary condition was set at the
outlet. The remaining parts of the geometry have
been set as walls, and a no slip condition for both air
and water has been imposed.
Simulations with an air mass flow rate
6
4.85 10
G
m

kg/s were run up to 60 s to initialize
the flow field. A time step of 5·10
-4
s was chosen to
advance the solution in time. Afterwards, the time
step has been increased to 10
-3
s, and the mass flow
rate has been raised to the value of
6
9.85 10
G
m

kg/s. Simulations have been run up to 140 s, the cut-
off value of the residuals was set equal to 10
-4
and a
maximum number of thirty iterations for the inner
cycle was imposed. The air volume fraction at every
grid node has been extracted at the time step
110t
s.
3.3 Simulation of Light Distribution
Solutions of the D3Q26 RT-LBM were computed in
MATLAB
®
, using an in-house code. The computed
local air-volume fractions were interpolated on a
PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology
308
lattice with 201 nodes along the column diameter. In
total, the lattice was composed of approx.
7
1.16 10
nodes. The lattice was proven to produce converged
solutions in a grid refinement study and is also used
for computation in the RT-LBM.
At the side walls of the reactor parallel light
enters the turbid suspension from four sides.
Focussing of light by the curved reactor wall is not
taken into account for the boundary modelling. It is
assumed that the walls are non-reflective so that
backscattered light can leave the computational
domain. For each concentration of biomass
A
c
(
0.6
g/L, 1.0 g/L), 7 monochromatic solutions were
computed and integrated by means of Simpson’s
Rule to obtain the polychromatic solution. All
parameters are shown in table 2. Achievement of
steady-state in monochromatic simulations was
estimated by introducing a relative error criterion.


6
,
,
,
max 1 10
,
i
i
Itt
It


x
x
(24)
Table 2: Spectral radiation characteristics of biomass and
light source at the simulated wavelengths.
λ [nm] 610 625 640 655
A
abs
[m
2
/kg] 136 156 176 259
A
sca
[m
2
/kg] 1628 1599 1538 1360
I
0,λ
/I
0
1E-04 [-] 3 36 161 266
λ [nm] 670 685 700
A
abs
[m
2
/kg] 358 311 70
A
sca
[m
2
/kg] 1139 1305 1941
I
0,λ
/I
0
1E-04 [-] 161 36 3
4 RESULTS AND DISCUSSION
In this section, preliminary results are shown to
demonstrate the applicability of the proposed hybrid
methodology. It is shown how different biomass
concentrations and presence of bubbles affect the
light distribution inside a PBR. While the former has
a strong impact on the intensity profile, the latter
exhibits only a weak effect on it, probably due to the
low mass flow rate used in this work.
4.1 Flow Field
Figure 1 shows a contour of air-volume fraction in a
cross-sectional (x-y) plane at 0z at the time step
110t s. It is an oscillating air bubble “plume”,
typical of an aerated bioreactor. The air volume
fraction at each grid node is extracted at the selected
time instant, so that optical properties of the
suspension can be computed.
Figure 1: a) Air volume fraction on an x-y plane at
0z
at
the time instant
110t
s. The mass flow rate is
6
9.85 10
G
m

kg/s.
4.2 Radiation Field
The local air volume fractions were interpolated on a
cubic lattice and grid independence was found for a
grid resolution higher than 50 nodes along the
column diameter (results not shown).
To evaluate the radiation field, profiles on the
plane
0.3y
m of the bubble column were
computed. Figure 2 shows profiles of normalized
monochromatic radiation at different wavelengths.
Near the boundaries, the radiation emitted from
different lamps overlap so that the magnitude of the
normalized radiation field becomes greater than 1.
Figure 3 shows the polychromatic intensity
profiles for two different concentrations of
microalgae. For comparison, simulations of an
optically homogeneous suspension (without gas)
were performed. Although the profiles do not
deviate much, slight differences (up to 6 %, not
shown) were found in presence of gas.
Hybrid Numerical Simulation of Fluid Flow and Light Distribution in a Bubble Column Photobioreactor
309
Figure 4 shows the effect of gas bubbles on the
light field more in detail. It can be seen that increase
of biomass concentration damps the effect of the gas
on the light field and biomass absorption becomes
the dominant characteristic. However, the light field
becomes asymmetric due to differences in local gas
concentrations as it can be seen by comparing
intensity profiles along the x- and z-axes or rather by
the difference of the profiles (Dash-dotted and
dotted lines in Figure 4).
Generally, the effect of gas bubbles on the light
field is weak in the present case. However, the air
volume fraction is higher at higher gas mass flow
rates, and further investigations will address this
effect. Moreover, the present study considers red
LED lamps, which emit in the absorption maximum
of microalgae (Williams and Laurens, 2010). It can
be expected, that the effect of bubbles increases if
light across the full visible spectrum is considered,
due to the lower absorption in other spectral regions.
Further model improvement will also consider the
directional emission of light sources as well as
focussing effects at the boundaries.
It is well known that flashing light has great
potential to improve the productivity of PBR
(Williams and Laurens, 2010). From a Lagrangian
point of view, flashing light is realized if cells travel
between bright and dark regions in the reactor.
Investigations on this effect in PBR require transient
simulations techniques, since optical accessibility
and therefore experimental investigations are
limited. The presented hybrid model offers the
required functionality to investigate transient effects
of the flow on the light field, and to trace virtual
particles through the transient light field.
Figure 2: Profiles of monochromatic light intensity along
the x-axis at
0.3y m and 0z in the PBR for
0.6
A
c
g/L. δ is the radial coordinate.
Figure 3: Profiles of polychromatic light intensity (log-
scale) along the x-axis at
0.3y
m and 0z in the PBR
for different biomass concentrations with (solid lines) and
without gas (dashed lines). δ is the radial coordinate.
Figure 4: Effect of gas bubbles on light intensity. Solid
and dashed lines compare the effect of gas bubbles to an
optically homogeneous suspension at
0.3y m along the
x- and z-axis, respectively. Dash-dotted and dotted lines
show the difference of light intensity along x and z axis in
the presence of gas bubbles. δ is the radial coordinate.
5 CONCLUSIONS
In this work a simulation framework was presented
to couple fluid dynamics and radiation transfer. The
method is innovative and no coupling of CFD and
RT-LBM has been reported in literature before. This
is, in particular, due to the fact that the development
of RT-LBM is still a new field of research.
The methodology was applied to simulate flow
and light fields in a bubble column PBR. Although
the effect of gas bubbles on the light field was weak,
further investigations will proof the generality of this
result. To make use of the full power of the
modelling framework, future work should also
address transient impacts of gas bubbles on the light
field to realize flashing-light regimes in PBR.
PHOTOPTICS 2017 - 5th International Conference on Photonics, Optics and Laser Technology
310
The link between CFD and RT-LBM was
realized by computing local radiation properties
from air-volume fractions and radiation properties of
bubbles and single cells. If the cell density is not too
high, the underlying assumption that radiation
properties of a mixture can be obtained by
superimposing those of single components remains
valid. This enables to take local cell concentrations
into account if mass transport of cells in
incorporated into the CFD model. As shown by
Dauchet et al. (2015), information as particle size
distributions can also be considered for the
calculation of radiation properties to obtain more
realistic representative values, if desired. Similarly,
it seems to be possible to treat mixtures of different
cell species with different particle size.
REFERENCES
Baillis, D., Sacadura, J.-F., 2000. Thermal radiation
properties of dispersed media: theoretical predication
and experimental characterization. In J. Quant.
Spectrosc. Ra., 67(5): 327-363.
Bech, K., 2005. Dynamic simulation of a 2D bubble
column. In Chem. Ing. Sci., 60(19): 5294-5304.
Berberoglu, H., Yin, J., Pilon, L., 2007. Light transfer in
bubble sparged photobioreactors for H
2
production and
CO
2
mitigation. In Int. J. Hydrogen Energ., 32(13):
2273-2285.
Burns, A.D.B., Frank, Th., Hamill, I., and Shi, J-M., 2004.
The Favre Averaged Drag Model for Turbulent
Dispersion in Eulerian Multi-Phase Flows. 5th
International Conference on Multiphase Flow
Dauchet, J., Blanco, S., Cornet, J.-F., Fournier, R., 2015.
Calculation of radiative properties of photosynthetic
microorganisms. In J. Quant. Spectrosc. Ra., 161: 60-
84.
Frank, Th., Shi, J.M., Burns, A.D., 2004. Validation of
Eulerian Multiphase Flow Models for Nuclear Safety
Applications. In 3rd International Symposium on
Two-Phase Flow Modelling and Experimentation,
Pisa, Italy.
Frank, Th., Zwart, P.J., Krepper, E., Prasser, H.-M.,
Lucas, D., 2008. Validation of CFD models for mono-
and polydisperse air-water two-phase flows in pipes.
In J. Nuclear Engineering & Design, 238(3): 647–659.
Grace, J.R., Wairegi, T., Nguyen, T.H., 1976. Shapes and
velocities of single drops and bubbles moving freely
through immiscible liquids. In Trans. Inst. Chem.
Eng., 54 (3): 167-173.
Hunter, B., Guo, Z., 2015. Numerical smearing, ray effect,
and angular false scattering in radiation transfer
computation. Int. J. Heat Mass Transfer, 81: 63–74.
Jonasz, M., Fournier G.R.., 2007. Light Scattering by
Particles in Water, Academic Press. Amsterdam, 1
st
edition.
Krujatz, F., Illing, R.
.
, Krautwer, T., Liao, J., Helbig
K., Goy, K., Opitz, J., Cuniberti, G., Bley, T., Weber,
J., 2015, Light-field-characterization in a continuous
hydrogen-producing photobioreactor by optical
simulation and computational fluid dynamics. In
Biotechnol Bioeng., 112(12): 2439-2449
Kandilian, R., Pruvost, J., Artu, A., Lemasson, C.,
Legrand, J., Pilon, L., 2016. Comparison of
experimentally and theoretically determined radiation
characteristics of photosynthetic microorganisms. In J.
Quant. Spectrosc. Ra., 175: 30-54.
Legendre, D., Magnaudet, J., 1998. The lift force on a
spherical bubble in a viscous linear shear flow. In J.
Fluid Mech., 368: 81-126.
Lobaton, H.F.Q., Suarez, C.A., Molina, A., 2011. CFD-
facilitated flow field analysis of bubble columns with
concentric plates for biological applications. In Chem.
Eng. Technol., 34(9):1490-1496.
Masood, R.M.A., Delgado, A., 2014. Numerical
investigation of the interphase forces and turbulence
closure in 3D square bubble columns. In Chem. Eng.
Sci., 108(28), 154-168.
Menter, F.R., 1994. Two-equation eddy-viscosity
turbulence models for engineering applications. In
AIAA Journal, 32(8): 1598-1605.
McHardy, C., Horneber, T., Rauh, C., 2016. New lattice
Boltzmann method for the simulation of three-
dimensional radiation transfer in turbid media. In Opt.
Express, 24(15): 16999-17017.
Modest M.F., 2013. Radiative Heat Transfer, Academic
Press. Amsterdam, 3
rd
edition.
Nauha, E., K., Alopaeus, V. 2013. Modeling method for
combining fluid dynamic and algal growth in a bubble
column photobioreactor. In Chem. Eng. Journal, 229:
559-568
Nauha, E., K., Alopaeus, V. 2015. Modeling outdoors
algal cultivation with compartmental approach. In
Chem. Eng. Journal, 259: 945-960
Pilon, L., Berberoglu, H., Kandilian, R., 2011. Radiation
transfer in photobiological carbon dioxide fixation and
fuel production by microalgae. In J. Quant. Spectrosc.
Ra., 112(17): 2639-2660.
Pfleger, D., Becker, S., 2001. Modelling and simulation of
the dynamic flow behavior in a bubble column. In
Chem. Eng. Sci., 56(4): 1737-1747.
Olivieri, G., Salatino, P., Marzocchella, A., 2014.
Advances in photobioreactors for intensive microalgal
production: configurations, operating strategies and
applications. In J. Chem. Technol. Biotechnol., 89:
178-195
Williams, P.J.le B., Laurens, L.M.L., 2010. Microalgae as
biodiesel & biomass feedstocks: Review & analysis of
the biochemistry, energetics & economics. In Energy
Environ. Sci., 3(5): 554-590.
Zhang, D., Dechatiwongse, P., Hellgardt, K., 2015.
Modeling light transmission, cyanobacterial growth
kinetics and fluid dynamics in a laboratory scale
multiphase phot-bioreactor for biological hydrogen
production”, Algal Research 8, pp. 99-107.
Hybrid Numerical Simulation of Fluid Flow and Light Distribution in a Bubble Column Photobioreactor
311