ON THE MODELING OF AN AIRLIFT PHOTOBIOREACTOR Christo Boyadjiev,
Jose Merchuk Introduction Mathematical model Average concentration
model Hierarchical approach Conclusions INTRODUCTION
Photobioprocesses include dissolution of an active gas component
(CO 2, O 2 ) in liquid and its reaction with a photoactive material
(cells). These two processes may take place in one environment
(mixed bioreactors ; bubble columns) or in different environments
(air lift photobioreactor). The comparison of these systems shows
apparent advantages in the use of airlift photo bioreactors,
because the possibility of manipulation of the light - darkness
history of the photosynthetic cells. The hydrodynamic behavior of
the gas and the liquid in airlift reactors is very complicated, but
in all cases the process includes convective transport, diffusion
transport and volume reactions. That is why convection - diffusion
equation with volume reaction may be use as a mathematical
structure of the model. MATHEMATHICAL MODEL Lets consider an
airlift reactor with a horizontal cross-sectional area F 0 for the
riser zone and F 1 for the downcomer zone. The length of the
working zones is l. The gas flow rate is Q 0 and the liquid flow
rate (water) - Q 1. The gas and liquid hold - up in the riser are
and (l ). The concentrations of the active gas component (CO 2 ) in
the gas phase is c(x,r,t) and in the liquid phase c 0 (x, r, t) for
the riser and c 1 (x 1,r,t) for the downcomer, where x 1 = l - x.
The concentration of the photoactive substance in the downcomer is
c 2 (x 1,r,t) and in the riser - c 3 (x 1,r,t). The average
velocities in gas and liquid phases are: The interphase mass
transfer rate in the riser is: The photoreaction rates in the
downcomer and the riser are taken, respectively, as:, J = J (x 1,
r, t), J 1 = J 1 (x 1, r, t).. Let consider cylindrical surface
with radius R 0 and length 1 m, which is regularly illuminated with
a photon flux density J 0. The photon flux densities over the
cylindrical surfaces with radiuses r R 0 is:. r RoRo rr J The
increasing of the photon flux density between r and r - r is:. The
volume between the cylindrical surfaces with radiuses r and r - r
is:, and the decreasing of the photon flux density as a result of
the light absorption in this volume is: The different between
photon flux densities for r and r - r is:,. As a result where J(R 0
) = J 0. The solution is:. The equations for the distribution of
the active gas component in the gas and liquid phases in the riser
are:, = const. The equations for the distribution of the active gas
component in the gas and liquid phases in the downcomer are: x 1 =
l x. Photochemical reaction may take place in riser too, and the
equation for the cell concentration is:. The initial conditions
will be formulated for the case, of thermodynamic equilibrium
between gas and liquid phases, i. e. a full liquid saturation with
the active gas component and the process starts with the starting
of the illumination : where c (0) and c 2 (0) are initial
concentrations of the active gas component in the gas phase and the
photoactive substance in the liquid phase. The boundary conditions
are equalities of the concentrations and mass fluxes at the two
ends of the working zones - x = 0 (x 1 = l) and x = l (x 1 = 0).
The boundary conditions for c (x, r, t) and c 0 (x, r, t) are: The
boundary conditions for c 1 (x 1, r, t), c 2 (x 1, r, t) and c 3
(x, r, t) are: The radial non - uniformity of the velocity in the
column is the cause for the scale effect (decreasing of the process
efficiency with increasing of the column diameter) in the column
scale-up. In the specific case of photo-reactions, an additional
factor is the local variations of light availability. Here an
average velocity and concentration in any cross - section is used.
This approach has a sensible advantage in the collection of
experimental data for the parameter identification because
measurement the average concentrations is very simple in comparison
with local concentration measurements. AVERAGE CONCENTRATION MODELS
For the velocity and concentration of the gas phase in the riser:
After introducing of average velocity and concentration in the
equation of the gas phase in riser and integrating over r in the
interval [0, r 0 ] is obtained: Lets use a property of integral
average functions: where As a result is obtained: with boundary
condition: must be obtained from the continuity equation after
integrating over r in the interval [0, r 0 ] : The parameters in
the model are of two types - specific model parameters ( D, k, , )
and scale model parameters ( A, B, G ). The last ones (scale
parameters) are functions of the column radius r 0. They are a
result of the radial non-uniformity of the velocity and the
concentration and show the influence of the scale - up on the
equations of the model. The parameter may be obtained beforehand
from thermodynamic measurements. From the model follows that the
average radial velocity component influences the transfer process
in the cases, i. e. when the gas hold - up in not constant over the
column height. As a result for many cases of practical interest and
the radial velocity component will not taken account in this case.
The hold - up can be obtained using: where l and l 0 are liquid
level in the riser without gas motion. The values of the parameters
D, k, A, B, G must be obtained using experimental data for measured
on a laboratory column. In the cases of scale - up A, B and G must
be specified only (because they are functions of the column radius
and radial non-uniformity of the velocity and concentration), using
a column with real diameter, but with small height ( D and k do not
change at scale up). The model for the liquid phase in the riser is
: where A 0, B 0 and are obtained in the same way as A, B and G.
The concrete expressions of A, B and G are not relevant because
those values must be obtained, using experimental data in any case.
In the equations of the downcomer must put the average velocity,
concentrations and photon flux density: After integration over r in
the interval [ r 0, R 0 ] the problems has the form and are
obtained in a similar way as, but taking into account that the
limits of the integrals are [ r 0, R 0 ]. may be obtained after
integration of the photon flux density model :,. The equation for
and A 3, B 3 and G 3 are obtained in a similar way. may be obtained
after integration: For many cases of practical interest and the
number of parameters of the model decreases, i.e. The
photo-chemical reaction rates equation shown in are acceptable when
J and c 1 are very small. A more general form of the photo-chemical
reaction rate equation is (written here for the downcomer): Another
possible form for these equations could be: where the kinetic
parameters k 0, , 1, 2 must be obtained wsing experimental data.
Applying the last expressions in the model equations, the
photo-chemical reaction rate equations could be: The parameters c
(0), c 2 (0), J 0, , , , , k 0, , 1, 2, k, D, D 0, D 1, D 2, D 3
are related with the process (gas absorption with photobioreaction
in liquid phase), but the parameters R 0, r 0, A, A 0, A 1, A 2, A
3, B, B 0, B 1, B 2, B 3, M, M 3, P 1, P 2 are related with the
apparatus (column radius and radial non-uniformity of the
velocities and concentrations). The equations allow to obtain ( k,
D, D 0, A, B, A 0, B 0 ) without photo-bioreaction if it can be
assumed that. The equations allow to obtain ( k, D, D 0, A, B, A 0,
B 0 ) without photo-bioreaction if it can be assumed that.
HIERARCHICAL APPROACH The obtained equations are the mathematical
model of an airlift photobioreactor. The model parameters are
different types: The obtained equations are the mathematical model
of an airlift photobioreactor. The model parameters are different
types: beforehand known ( c (0), c 2 (0), R 0, J 0, r 0 );
beforehand obtained ( , , , , k 0, , 1, 2 ); obtained without
photo-bioreaction ( k, D, D 0, A, B, A 0, B 0 ); obtained with
photo-bioreaction ( D 1, D 2, D 3 ), because diffusion of the gas
and photoactive substance is result of the photobioreaction;
obtained in the modelling and specified in the scale - up (A, A 0,
A 1, A 2, A 3, B, B 0, B 1, B 2, B 3, M, M 3, P 1, P 2 ), because
they are functions of the column radius and radial non-uniformity
of the velocity concentration. CONCLUSIONS The results obtained
show a possibility of formulation airlift photobioreactor models
using average velocities and concentrations. These models have two
type parameters, related to the process and to the apparatus (scale
- up). This approach permit to solve the scale - up problem
concerning the radial nonuniformity of the velocity and
concentration, using radius dependent parameters. The model
parameter identification on the bases of average concentration
experimental data is much simpler than considering the local
concentration measurements. The results obtained show a possibility
of formulation airlift photobioreactor models using average
velocities and concentrations. These models have two type
parameters, related to the process and to the apparatus (scale -
up). This approach permit to solve the scale - up problem
concerning the radial nonuniformity of the velocity and
concentration, using radius dependent parameters. The model
parameter identification on the bases of average concentration
experimental data is much simpler than considering the local
concentration measurements.
