BAS - IChE
LABORATORY
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ON THE MODELING A THREE PHASE
CATALYTIC AIRLIFT REACTORS
Chr. Bojadjiev
Bulgarian Academy of Sciences, Institute of Chemical Engineering,
“Acad. G.Bontchev” str., Bl.103, 1113 Sofia, Bulgaria,
E-mail: [email protected]
Introduction
Mathematical model
Average concentration model
Hierarchical approach
Conclusions
INTRODUCTION
The hydrodynamic behaviour of the gas and liquid flows in airlift reactors is very
complicated. In these conditions the convective and diffusive transfer with volume reactions are
realized simultaneously. The convective transfer is result of a laminar or turbulent (large - scale
pulsation’s) flows. The diffusive transfer is molecular or turbulent (small - scale pulsation’s). The
volume reactions are mass sources as a result of chemical reactions and interphase mass transfer.
The scale - up theory show, that the scale - effect in mathematical modelling is result of the
radial nonuniformity of the velocity distribution in the columns. In many papers are used
diffusion models, where the scale - effect is considered as an axial mixing increasing.
The creation of the models in these conditions and solving of the scale - up problem require
construction of a suitable diffusion models.
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MATHEMATHICAL MODEL
The investigation of the airlift reactor show, that convection - diffusion equation
with volume reaction may be use as a mathematical structure of the model.
Let consider airlift reactor for alcohol oxidation with a cross - section’s 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 - Q1. The gas and liquid holdup in the riser are ε and (1 – ε).
The concentrations of the active gas component (O2) in the gas phase is c (x, r, t) and in the
liquid phase – c0 (x, r, t) for the riser and c1 (x, r, t) - for the downcomer, where x1 = l – x.
The concentration of the alcohol in the downcomer is c2 (x, r, t) and in the riser - c3 (x, r, t) .
If the active sites concentration of the catalyst particles is sufficiently big it may be put equal to
constant.
Q0
Q1
Q1
, u1 
, u
.
The average velocities in gas and liquid phases are: uo 
F0
The interphase mass transfer rate in the riser is: R = k (c – χ c0).
F0
F1
The alcohol oxydation rates in the riser and the downcomer are:
R1  k0 co1 c3 2 , R2  k0 c11 c2 2 .
The mathematical model of the chemical processes in airlift reactor will be built on the basis of
the differential mass balance in the reactor volume. A convection - diffusion equations with
volume reaction will be used, where convective transfer will be results of the laminar flow or
large scale turbulent pulsations, the diffusion transfer is molecular or turbulent (as a result of the
small scale turbulent pulsations) and the volume reactions are interphase mass transfer and
chemical reaction.
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The equations for oxygen concentration distributions 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
 c0 
1  2

1     1    u1  v1   1    D0  2 




k
c


c

k
c
0
o o c3 ,
2 
t
x 
r r  r 
 r
 x
u1 v1 v1

  0.
x
r
r
The equations for the alcohol and oxygen concentration distributions in the liquid phase in the
downcomer are:
  2 c1 1 c1  2 c1 
 c1
 c1
 c1
1  2

u
v
 D1 



k
c
o 1 c2 ,
2
2 
t
 x1
r

x
r

r

r


  2 c2 1 c2  2 c2 
 c2
 c2
 c2
1  2

u
v
 D2 



k
c
o 1 c2 ,
2
2 
t
 x1
r

x
r

r

r


x1  l  x .
The equation for alcohol concentration distribution in the riser is:
  2 c3 1 c3  2 c3 
  c3
 c3
 c3 
1  2

  1    D3 
1     1    u1
 v1



k
c
o o c3 .
2
2 
t
x 
r r  r 
 r
 x
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The initial conditions will be formulated for the case, when at t = 0 the
process starts with the gas motion beginning:
t  0 , cx, r ,0   c 0  , c x,r ,0   0,
0
c1 x1 , r ,0  0 , c2 x1 ,r ,0  c20  , c3 x, r ,0  c20  ,
where c(0) and c2(0) are initial concentrations of the oxygen in the gas phase and of the alcohol 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 , u o 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   c2 l , r ,t  ,
 c
c2 l ,t  u  c0 0 , r ,t u1  D0  0
 x
c co
r  0,

 0;
r r
c co
r  ro ,

 0.
r r


;
 x 0
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The boundary conditions for c (x, r, t) and c0 (x, r, t) and c0 (x, r, t) are:
  c1 
 ;
x1  0 , c1 0, r , t   c3 l , r , t  , c3 l , t  u1  c1 0, r , t  u  D1 
  x1  x 0
c1
c1
r  ro ,
 0; r  Ro ,
 0;
r
r
c 
x1  0 , c2 0,r,t   co l,r,t  , co l,t  u  c2 0 ,r,t  u  D2  2  ;
  x1  x 0
r  ro ,
c2
 0; r  Ro ,
r
c2
 0;
r
c 
x  0 , c3 0,r,t   c1 l,r,t  , c1 l,t  u  c3 0,r,t  u1  D3  3  ;
  x  x 0
c3
c3
r  0,
 0; r  ro ,
 0.
r
r
The radial nonuniformity of the velocity in the column apparatuses is the cause for the scale
effect (decreasing of the process efficiency with increasing of the column diameter) in the column
process scale - up. That is why must be use average velocity and concentration for the of the
cross – section’s area. It lead to big priority beside experimental data obtaining for the parameter
identification because measurement the average concentration is very simple in comparison with
the local concentration measurement.
AVERAGE CONCENTRATION MODELS
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Let’s consider equation for oxygen concentration in the gas phase c (x, r, t) in
cylindrical coordinates.
If use a property of the integral average functions for velocity and
concentration:
1 ro
1 ro
1 ro
uo x    uo x , r  dr ,  o x    vo x , r  dr , c x ,t    c x , r ,t  dr ,
ro 0
ro 0
ro 0
the velocity and concentration may be expressed as a:
uo x , r   uo x  u~o r , vo x , r   vo x  v~o r , c x , r ,t   c x ,t  c~ r ,
ro
~
 u o r  dr  ro ,
ro
~
 vo r  dr  ro ,
ro
0
0
0
~
 c r  dr  ro .
If put these expressions in the convection – diffusion equation and integrate over r in the
interval [0 , r0] is obtained:
 2c
 k
c
c
 Aro  u o
 G1 ro  vo c  D  2  B ro  c   c  co ,
t
x
 x
 
1 ro ~
1 ro 1 c~
1 ro ~ c~
~
Aro    u o r  c r  dr, Bro   
dr, G1 ro    vo
dr.
ro 0
ro 0 r r
ro 0 r
If put average velocity in the continuity equation and integrate over r in the interval [0 , r0] :
v~o ro   v~o 0  1 ro v~o
u o
  dr.
 G2 ro  vo  0 , G2 ro  
r
ro 0 r
x
o
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As a result :
  2c
 k
uo
c
c
 A ro  uo
 G ro  c
 D  2  B ro  c   c  co ,
t
x
x
 x
 
r
G ro   o G1 ro .
3
The boundary conditions of has the form:
 c 
t  0 , c x ,0   c 0  ; x  0 , uo c 0   A ro  uo 0  c 0 ,t   D  ; x  l , c l ,t    co l ,t .
 x  x o
The parameters in this model are 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 result of the radial nonuniformity of the velocity and concentration, and show the
influence of the scale - up on the model equations. The parameter  may be obtained beforehand
as a result of thermodynamic measurements.
From this model follows, that the average radial velocity component influence the transfer
process in the cases uo / x  0 , i.e. when the gas hold - up in not constant over the column
height. For many practical interesting cases ε = const, i.e.  u o / x  0 and the radial velocity
component did not take account vo  0  .
The hold - up  must be obtained using:

l  lo Fo  F1 

,
l  lo Fo  F1   Fo lo
where l and l0 are liquid levels in the riser with and without gas motion.
The parameter values D, k, A, B, G must be obtaining using experimental data for c  x ,t 
measured on the laboratory column. In the cases of scale - up must be specified A, B and G only,
using a column with real diameter but with small height (D and k do not change at (scale – up)).
The same procedure may be used for c0 , c1 , c2 , c3 :
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co
c
u
 Ao ro  u1 o  G0 ro  co 1 
t
x
x
  2 co

k
c  co   M o ro  ko co11 c3 2 ;
 Do  2  Bo ro  co  
1
 x
 1
r
1 o ~1 ~ 2
M o ro    co c3 dr ;
ro 0
t  0, co x,0  0; x  0, co 0, t   c1 l , t ,
 c 
c1 l ,t  u  A ro  u1 0  co 0 ,t   Do  o  ,
 x  x o
  2 c1

c1
c1
u
 A1 ro , Ro  u
 G1 ro , Ro  c1
 D1  2  B1 ro , Ro  c1   ko M ro , Ro c11 c2 2 ;
t
x
x
 x

t  0, c1 x1 ,0  0; x1  0, c1 0, t   c3 l , t ,
 c 
c3 l ,t  u1  A 1ro , Ro  u 0  c1 0 ,t   D1  1  ,
 x1  xo
1
u x  
Ro  ro
Ro
1




u
x
,
r
dr
,
c
x
,
t

1
r
Ro  ro
o
Ro
1




c
x
,
r
,
t
dr
;
M
r
,
R

o
o
r 1
Ro  ro
o
Ro
c
~ 1 c~  2 dr .
1
ro
2
c2
c
u
 A2 ro , Ro  u 2  G2 ro , Ro  c2
 D2
t
x
x
  2 c2

1  2





B
r
,
R
c

M
r
,
R
k
c
2
o
o
2
o
o
o 1 c2 ;
 2
 x

t  0, c2  c20  ; x1  0, c2 0, t   co l , t ,
 c 
c3 l ,t  u1  A 2 ro , Ro  u 0  c2 0 ,t   D2  2 
,
 x1  x1 o
1 Ro
1 ro
c2 
 c2  x , r ,t  dr , c3 
 c3  x , r ,t  dr .
Ro  ro ro
ro 0
c3
c
u
 A3 ro  u1 3  G3 ro , c3 1  D3
t
x
x
  2 c3

ko  1  2





B
r
c

M
r
co c3 ;
3 o
3
o o
 2
1
 x

t  0 , c3 x,0  c20  ; x  0 , c3 0 ,t   c1 l ,t ,
 c 
c1 l , t  u  A3 ro  u1 0 c3 0, t   D3  3  .
 x  x o
For many practical interesting cases the specific volume (m3.m-3) of the catalytic particles or
gas hold - up are constant over the column, i.e.
u uo u1


 0 , v  vo  v1  0
x
x
x
and model parameters number decrease, i.e. G = G0 = G1 = G2 = G3 = 0 .
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The obtained problems are mathematical model of an airlift three phase
reactor. The model parameters are five types:
– beforehand known (c(0), c2(0), R0 , r0);
– beforehand obtained (ε , χ , α1 , α2 , k0);
– obtained without chemical reaction (k , D , D0 , A , B , A0 , B0);
– obtained with chemical reaction (D1 , D2 , D3 , M , M0);
– obtained in the modelling and specified in the scale - up (A , A0 , A1 ,
A2 , A3 , B , B0 , B1 , B2 , B3 , M , M0).
The problems for c and c 0 permit to obtain without chemical reaction if put
c1 l ,t   c0 l ,t  .
CONCLUSIONS
The result obtained show a possibility to build three phase catalytic airlift reactor models,
using average velocities and concentrations. This approach permit to solve the scale - up problem
as a result to the radial nonuniformity of the velocity and concentration, using radius dependent
parameters. The model parameter identification on the bases of average concentration
experimental data lead to big priority in comparison with the local concentration measurements.