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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 (CO2 , O2 ) 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
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Let’s consider an airlift reactor with a horizontal cross-sectional area F0 for
the riser zone and F1 for the downcomer zone. The length of the working zones
is l. The gas flow rate is Q0 and the liquid flow rate (water) - Q1. The gas and
liquid hold - up in the riser are ε and (l – ε) .
The concentrations of the active gas component (CO2) in the gas phase is
c(x,r,t) and in the liquid phase – c0 (x, r, t) for the riser and c1 (x1 ,r ,t) – for the
downcomer, where x1 = l - x.
The concentration of the photoactive substance in the downcomer is c2 (x1,r,t)
and in the riser - c3 (x1 ,r ,t).
The average velocities in gas and liquid phases are:
uo 
Q0
Q
Q
, u1  1 , u  1 .
F0
F0
F1
The interphase mass transfer rate in the riser is: I 0  k c  c0  .
The photoreaction rates in the downcomer and the riser are taken,
respectively, as:
I  k0 c1c2 J , I 1  k0 c0 c3 J 1 , J = J (x1, r, t) , J1 = J1 (x1, r, t) .
Let consider cylindrical surface with radius R0 and length 1 m, which is
regularly illuminated with a photon flux density J0. The photon flux densities over the cylindrical
surfaces with radiuses r  R0 is:
i r  
Ro J o .
r
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J
Ro
r
r
The increasing of the photon flux density between r and r -  r is:
J o Ro J o Ro
J o Ro r .
J 1 


r  r
r
r r  r 


The volume between the cylindrical surfaces with radiuses r and r - r is: V  r  1 
r 
,
2r 
and the decreasing of the photon flux density as a result of the light absorption in this volume is:
 r 
J 2  J c2 r1 -  ,
 2r 
The different between photon flux densities for r and r - r is:
J o Ro r
 r  .
 J c 2 r  1 
r r  r 
 2r 
J J Ro J o

 2   c2 J , where J(R0) = J0. The solution is:
As a result lim
r
r 0 r
r
J  J 1  J 2 
Ro
 Ro
 
 Ro

1




J  x1 , r ,t   exp   c 2  x1 ,,t  d  J o  Ro J o  2 exp   c 2  x1 ,,t  d d .
r 

 r


 


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 c0
1   
t
u1 v1

x r
The equations for the distribution of the active gas component in the gas and
liquid phases in the riser are:
  2 c 1 c  2 c 
c
c
c

  u0
  v0
  D 2 
 2   k c   c0 ,
t
x
r
r r  r 
x
uo vo vo

  0,
x
r
r
  2 co 1 co  2 co 
  c0
 c0 
  k c   c0 ,
  1    D0 
 1    u1
 v1


2
2 

x 
r r
r 
 r
 x
v
 1  0 , ε = const .
r
The equations for the distribution of the active gas component in the gas and liquid phases in
the downcomer are:
  2 c1 1 c1  2 c1 
 c1
 c1
 c1
   k 0 c1 c 2 J , x1 = l – x .
u
v
 D1 


2
2 
t
 x1
r
r r
r 
 x
  2 c 2 1 c 2  2 c 2 
 c2
 c2
 c2
  k 0 c1c 2 J , u  v  v  0 ,
u
v
 D2 


2
2 
t
 x1
r
r

r
x1 r r

x

r


Photochemical reaction may take place in riser too, and the equation for the cell concentration is:
  2 c3 1 c3  2 c3
  c3
 c3
 c3 
  1    D3 
1   
 1    u1
 v1
  x 2  r r   r 2
t

r

x



J 1 ro J  x1 , ro ,t 

  c3 J 1 , r  ro , J 1  J x1 , ro ,t , x1  l  x .
2
r
r

  k o co c3 J 1 ,


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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 :
t  0 , c x , r ,0   c
0 
c 0 
c 0 
, c2  x1 , r ,0   c20  ,
, c0  x , r ,0  
, c1  x1 , r ,0  


where c(0) and c2(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 (x1 = l) and x = l (x1 = 0).
The boundary conditions for c (x, r, t) and c0 (x, r, t) are:
 c 
x  0 , uo c 0   u0 c0 , r ,t   D  ,
  x  x 0
x  l , cl , r ,t    c0 l , r ,t  ;
x  0, c0 0 , r ,t   c1 l , r ,t  ,
  c0
c 1 l ,t u  c0 0 , r ,t u1  D0 
 x
c co
r  0,

 0; r  ro ,
r r


;
 x 0
c co

 0.
r r
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The boundary conditions for c1 (x1, r, t) , c2 (x1, r, t) and c3 (x, r, t) are:
x1  0 , c1 0 , r ,t   c0 l , r ,t  ,
  c1 

c0 l ,t u1  c1 0 , r ,t u  D1 
;

x
 1  x1 0
c1
c1
r  ro ,
 0; r  Ro ,
 0;
r
r
c 
x1  0 , c2 0, r , t   c3 l , r , t  , c 3 l , t  u  c2 0, r , t u  D2  2  ;
  x1  x1 0
c 2
c2
r  ro ,
 0; r  Ro ,
 0;
r
r
c 
x  0 , c3 0, r , t   c2 l , r , t  , c2 l , t  u1  c3 0, r , t u1  D3  3  ;
  x  x 0
c3
c3
r  0,
 0; r  ro ,
 0.
r
r
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
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Let’s use a property of integral average functions:
~
1
f x, y   f x . f  y , where f x  
y0
y0

f x, y  dy ,
0
1
y0
y0
~
 f  y  dy  1.
0
For the velocity and concentration of the gas phase in the riser:
uo x, r   u o x u~o r , vo x, r   vo x v~o r , c x, r , t   c x, t c~ r ,
r
r
r
1 o
1 o
1 o
u o x    u o x, r  dr, vo x    vo x, r  dr, c x, t    c x, r , t  dr,
ro 0
ro 0
ro 0
ro
~
 u o r  dr  ro ,
ro
~
 vo r  dr  ro ,
ro
0
0
0
~
 c r  dr  ro .
After introducing of average velocity and concentration in the equation of the gas phase in riser
and integrating over r in the interval [0 , r0] is obtained:
  2c
 k
c
c
 Aro  uo
 G1 ro  vo c  D  2  B ro  c   c  co ,
t
x
 x
 
r
ro
ro
~
1 o~
1
1

c
1
c~
~
~
Aro    uo r  c r  dr, Bro   
dr, G1 ro    vo
dr.
ro 0
ro 0 r r
ro 0 r
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vo must be obtained from the continuity equation after integrating over r
in the interval [0 , r0] :
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uo
 G2 ro  vo  0,
x
v~o ro   v~o 0  1 ro v~o
G2 ro  
  dr.
ro
ro 0 r
As a result is obtained:
  2c
 k
uo
c
c
 A ro  uo
 G ro  c
 D  2  B ro  c   c  co ,
t
x
x
 x
 
G ro  
ro
G1 ro ,
3
with boundary condition:
t  0 , c x ,0   c 0  ;
 c 
x  0 , uo c 0   A ro  uo 0  c 0 ,t   D  ;
 x  x o
x  l,
c l , t    c o l , t .
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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 r0 . 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 uo / x  0 , i. e. when the gas hold - up in not constant over the column
height. As a result vo  0 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:

l  lo Fo  F1 
,
l  lo Fo  F1   Fo lo
where l and l0 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
c  x ,t  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 :
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co
c
u
 Ao ro  u1 o  Go ro  co 1  Do
t
x
x
c 0 
t  0, co x,0 
;
  2 co

k


c  co ,

B
r
c

o
o
o
 2
 x
 1 

x  0, co 0, t   c1 l , t ,
 co 
c1 l , t  u  A ro  u1 0 co 0, t   Do 
 ,

x

 x o
where A0 , B0 and Go 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:
ux,r   u x u~ r , c1 x,r ,t   c1 x,t  c~1 r ,
~
c x, r , t   c x, t  c~ r  , J x, r , t   J x, t  J r ,
2
2
2
1 Ro
1 Ro
u x  
 u  x , r  dr , c1  x,t  
 c1  x , r ,t  dr ,
Ro  ro ro
Ro  ro ro
1 Ro
1 Ro
c2  x,t  
 c2  x , r ,t  dr , J  x,t  
 J  x , r ,t  dr .
Ro  ro ro
Ro  ro ro
After integration over r in the interval [r0 , R0] the problems has the form
  2c1

c1
c1
u
 A1 ro , Ro  u
 G1 ro , Ro  c1
 D1  2  B1 ro , Ro  c1   ko M ro , Ro c1c2 J ;
t
x
x
 x

c 0 
t  0, c1 x1 ,0 
;

 c1 

x1  0,
co l,t  u1  A 1 ro ,Ro u 0  c1 0 ,t   D1 
,

x
 1  x11 o
  2c2

c2
c2
u
 A2 ro , Ro  u
 G2 ro , Ro  c2
 D2  2  B2 ro , Ro  c2   ko M ro , Ro c1c2 J ;
t
x
x
 x

0 
t  0, c 2  c2 ;
 c 
x1  0, c 2 0, t   c 3 l , t , c3 l,t  u1  A 2 ro ,Ro  u 0  c2 0,t   D2  2  ,
x1  x o

1
1 Ro ~ ~ ~
M ro , Ro  
 c1 c2 J dr
Ro  ro ro
c1 0, t   c0 l , t ,
and A1 , A2 , B1 , B2 , G1 , G2 are obtained in a similar way as A, B, G, but taking into account that the
limits of the integrals are [r0 , R0].
J may be obtained after integration of the photon flux density model :
J 
1
,
N1  βN 2 c2
~
Ro
Ro
1

J
1
2
2~ ~
,
N 1 ro , Ro  
r
dr


N
r
,
R

r

 c2 J dr .
2 o
o
Ro J o Ro  ro  ro
r
Ro J o Ro  ro  ro
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The equation for c 3
c3
u
 A3 ro  u1c3  G3 ro , c3 1 
t
x
  2 c3

k
 D3  2  B3 ro  c3   o M 3 ro  co c3 J 1 ;
 x
 1  ε 
0 
t  0, c3 x,0  c2 ;
 c 
x  0, c3 0,t   c2 l,t , c2 l,t  u  A3 ro  u1 0 c3 0,t   D3  3  ,
 x  x o
Ro
ro
1
1 ~~ ~
c3 
 c3  x , r ,t  dr , M 3 ro  
 co c3 J 1 dr
ro ro
ro 0
and A3, B3 and G3 are obtained in a similar way.
J 1 may be obtained after integration:
J x1 , t 
J1 x, t  
P1  P2 c3
1 ro 2 ~
1 ro 2 ~ ~
P1 ro   2 ~
 r J 1 dr , P2 ro   2 ~
 r c3 J 1 dr .
ro J ro  0
ro J ro  0
u u o u1


0
For many cases of practical interest
x
x
x
and the number of parameters of the model decreases, i.e. G  Go  G1  G2  G3  0.
The photo-chemical reaction rates equation shown in are acceptable when J and c1 are very
small.
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A more general form of the photo-chemical reaction rate equation is (written
here for the downcomer):
I
I m axJ
J2
kJ  J 
k inhb
c1c2
.
k c  c1
Another possible form for these equations could be:
I  ko c11 c2 2 J  , I 1  ko co1 c3 2 J 1 ,
where the kinetic parameters k0 , γ, γ1 , γ2 must be obtained wsing experimental data. Applying
the last expressions in the model equations, the photo-chemical reaction rate equations could be:
1
 k o M ro , Ro c1 1 c2 2 J  , M ro , Ro  
Ro  ro
 k o M ro , Ro c1 1 c2 2 J  ;
Ro
~  1 c~  2 J~  dr ;
c
1 2
ro
ko
1 ro ~ 1 ~  2 ~ 
1  2 

M 3 ro co c3 J 1 , M 3 ro    co c3 J 1 dr .
1 
ro 0
HIERARCHICAL APPROACH
The obtained equations are the mathematical model of an airlift
photobioreactor. The model parameters are different types:
– beforehand known (c(0) , c2(0) , R0 , J0 , r0);
– beforehand obtained (ε , χ , α , β , k0 , γ , γ1 , γ2);
– obtained without photo-bioreaction (k , D , D0 , A , B , A0 , B0);
– obtained with photo-bioreaction (D1 , D2 , D3), because diffusion of the gas
and photoactive substance is result of the photobioreaction;
– obtained in the modelling and specified in the scale - up (A, A0 , A1 , A2 , A3 ,
B, B0 , B1 , B2 , B3 , M, M3 , P1 , P2), because they are functions of the column
radius and radial non-uniformity of the velocity concentration.
The parameters c(0), c2(0), J0 , ε, χ, α, β, k0 , γ, γ1 , γ2 , k, D, D0 , D1 , D2 , D3 are related with
the process (gas absorption with photobioreaction in liquid phase), but the parameters R0 , r0 , A,
A0 , A1 , A2 , A3 , B, B0 , B1 , B2 , B3 , M, M3 , P1 , P2 are related with the apparatus (column radius
and radial non-uniformity of the velocities and concentrations).
The equations allow to obtain (k, D, D0 , A, B, A0 , B0) without photo-bioreaction if it can be
assumed that c1 l , t   co l , t  .
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.
BAS - IChE
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