Download contents introduction qualitative analysis of the model diffusion

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DIFFUSION MODELS AND SCALE – UP
by
Chr. Bojadjiev
Bulgarian Academy of Sciences, Institute of Chemical Engineering,
“Acad. G.Bontchev” str., Bl.103, 1113 Sofia, Bulgaria,
E-mail: [email protected]
CONTENTS
INTRODUCTION
QUALITATIVE ANALYSIS OF THE MODEL
DIFFUSION MODEL
MODEL WITH RADIAL VELOCITY COMPONENT
AVERAGE FUNCTION VALUES
INTERPHASE MASS TRANSFER MODEL
A PROPERTY OF THE AVERAGE FUNCTIONS
STATIONARY PROCESSES
AVERAGE CONCENTRATION MODEL
CONCLUSION
INTRODUCTION
Many heat and mass transfer processes in column apparatuses may be described by the
convection – diffusion equation with a volume reaction . These are gas absorption in column with
(or without) packet bed , chemical reactors for homogeneous or heterogeneous reactions , air - lift
reactors for biochemical or photochemical reactions .
The convective transfer in column apparatuses is result of a laminar or turbulent (large - scale
pulsations) flows . The diffusive transfer is molecular or turbulent (small - scale pulsations) The
volume reaction is mass sours as a result of chemical reactions or interphase mass transfer .
The scale - up theory show that the scale effect in mathematical modeling is result of the radial
nonunformity of the velocity distribution in the column .
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DIFFUSION MODEL
Let’s consider liquid motion in column apparatus with chemical reaction
between two of the liquid components . If the difference between component
concentrations is very big , the chemical reaction order will be one . In the case
of liquid circulation , the process will be nonstationary . If suppose for velocity
and concentration distribution in the column: u  ur , c  ct,r, z  ,
  2 c 1 c  2 c 
c
c
u
 D 2 
 2   kc;
z
r r r 
 z
the mathematical t
description has
t  0 , c  c0 ;
the form:
c
c
r  0,
 0; r  R,
 0;
r
r
c
z  0 , c t ,0   c t ,l , u c  uc  D
,
z
where u is velocity distribution, c - concentration of the reagent (with small concentration), k chemical reaction rate constant, t - the time, r and z - radial and axial coordinate, D - diffusivity,
co - initial concentration, c and - average values of the concentration and the velocity at the inlet
(out let) of the column R and l - column radius and height.
The radial nonuniformity of the velocity is the cause for the scale effect (decreasing of the
process efficiency with increasing of the column diameter) in the column scale - up. That is why
average velocity and concentration for the cross section’s area must be used. It lead to big priority
beside experimental data obtaining for the parameters identification because the measurement of
the average concentration is very simple in comparison with the local concentration
measurement.
AVERAGE FUNCTION VALUES
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Let consider cylinder with R =R() where  is an angle in cylindrical
coordinates (z, r, ). The average values of a function f (z, r, ) for the cross
section’s area is:
f z  
 f  z , r ,dS
S 
,
S
where
2
S

0
R 2 d ,
2

 
S
2
 R  

1
f  z, r,  dS   R    f  z, r,  dr  d .
2 0
 0

For a circular cylinder R = const:
R
S  R ,  f z, r,  dS  R  f t , r , z  dr ,
2
S 
0
For the average velocity and concentration:
1R
1R
u   u r  dr , ct , z    ct , r , z  dr .
R0
R0
1R
f t , z    f t , r , z  dr .
R0
A PROPERTY OF THE AVERAGE FUNCTIONS
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Let present the function f (x,y) as a:
f o x , y  
f x , y   f x  f o x , y ,
If f  x  is the average function:
f x  
yo
f x , y 
.
f x 
yo
1
 f x , y  dy , 0 f  x , y   y o f  x   f  x  0 f o  x , y  dy , i.e.
yo 0
yo
yo
 f x, y dy  y .
o
o
0
The results obtained permit to presents the velocity and concentration as a:
ur   u uo r , ct,r, z   ct , z  co r  ,
R
R
 u r  dr  R,  c r  dr  R
o
and u and ct , z are average velocity and concentration.
0
o
0
The result obtained permit to obtain equations for the average concentrations.
AVERAGE CONCENTRATION MODEL
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The average concentration model may be obtained if it integrate the
equations over r in the interval [0,R]:
2 c

c
c
 R  u
 D  2  R  c   k c;
t
z
 z

t  0,
z  0,
c  co ;
ct ,0   ct ,l ,
c ,
u c   R  u c  D
z
where
1R
R    u o r  co r  dr ,
R0
1 R 1 co
R   
dr .
R 0 r r
In the model is average velocity of the laminar or turbulent flow in the column, D is diffusivity
or turbulent diffusivity as a result of the small scale pulsations. The model parameters  and  are
related with the radial nonuniformity only and show the influence of the column radius on the
mass transfer kinetics. If radial nonuniformities and are different for different cross sections,
  R, z ,   R, z .
The parameters in the model are , , D and k, where k may be obtained beforehand as a result
of the chemical kinetics modeling. The identification of the parameters ,  and D may be made
by using experimental data for c (t , z) , obtained on the laboratory model. In the cases of scale up must be obtained (R) and (R) only (using real column) because the values of D and k are
the same.
QUALITATIVE ANALYSIS OF THE MODEL
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For the theoretical analysis of the model must be used dimensionless variables:
t  t oT ,
z  l Z,
c  co C , where to is characteristic time of the process.
  2C



 Z 2   C   KC ;


D C






Z

0,
C
T,0

C
T,1
,
C


R
C

,
T  0 , C  1;
u.l Z
ut
where   R  o ,
  l 2 R , K  kto .
l
For long time process to is very big and in the cases   102 the process is stationary. If
chemical reaction rate is very big and K  102 the process is stationary too, but if   1 the effect
C
C D t o

 2
T
Z
l
of the convective transport is negligible. Another situations are possible if compare the values of
the dimensionless model parameters  , Dt o / l 2 and K.
MODEL WITH RADIAL VELOCITY COMPONENT
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In the cases u = u (r , z) the velocity has a radial component and the model
equation has the form:
  2 c 1 c  2 c 
c
c
c
u v
 D 2 
 2   kc,
t
z
r
r r r 
 z
where v = v (r , z) is radial velocity component and satisfy the continuity equation :
u v v

  0.
z r r
The boundary conditions are identical. Let’s present the radial velocity component as a:
R
1R
vr , z   v z  vo r , v   vr , z  dr ,  vo r  dr  R.
R0
0
After integration over r in the interval [0,R] , the equation for c has the form:
1 R co
 2 c

c
c
dr .
  R u   1 R vc  D  2   R c  k c ,  1 R    vo
R0
r
t
z
 z

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The integration of the continuity equation over r in the interval [0,R] lead to:
vo R   vo 0  1 R vo
u
  dr .
  2 R  v  0 ,  2 R  
R
R0 r
z
On the solid interphase vo(R) = 0 and as a result:
R
vo 0   2  2a 2

vo 0 1 vo





v
r


R

r

ar
 2 R  
  dr , o

, i.e. vo(0) = 0.
2


R  R
R
R0 r


The constant a may be obtained using the boundary condition vo(R) = 0 and as a result a = 2 :
2 

vo R    2 R   2r - r 2  .
R 

The constant γ2 (R) may be obtained, using:
R
 vo r  dr  R ,
0
 2 R  
3
3
2 
, v o r    2r  r 2 ,
R
R
R 
v
R u
.
3 z
As a result:
R
c
c
u
2 c
  R u   R c
 D 2  K c ,  R    1 R , K  k  D R .
3
t
z
z
z
The model equation has an additional parameter γ (R) and identical boundary conditions. From
this model follow, that the average radial velocity component influence the transfer process in the
cases u / z  0 , i.e. when the specific volume (m3.m-3) of the solid phase in column (packed by
catalyst particles) or the gas hold-up is not constant over the column height. As a result v  0 for
many practical interesting cases.
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INTERPHASE MASS TRANSFER MODEL
In the cases of interphase mass transfer between gas - liquid or liquid liquid phase in the model equation may be to introduce convection - diffusion
equations for the two phases and the chemical reaction rate must be replaced
with interphase mass transfer rate:
k c1  c2 ,
where k is interphase mass transfer coefficient c1 - concentration of the transferred substance in
the gas (liquid) phase, c2 - the concentration of the transferred substance in the liquid phase,  Henry’s number (liquid - liquid distribution coefficient).
As a result the diffusion model for interphase mass transfer in the column apparatuses has the
form:
  2 ci 1 ci  2 ci
ci
ci
i
  i ui
  i Di  2 
 2
t
z
r r
r
 z

   1i 1 k c1  .c 2 ,


where εi (i = 1 , 2) are hold - up coefficients (ε1+ ε2 = 1). The boundary conditions are similar, but
a difference is possible depending on the conditions for contact between two phases.
Let consider counter-current gas-liquid bubble column, where c1 z1 , r ,t 
conditions
c2 z2 , r ,t of
, have
z1  the
l  form:
z2 
and
t  0 , c1  c o , c 2  0 ;
  c1 
 ;
z 1  0 , u 1 co  u 1 c1 0 , r ,t   D1 
  z 1  z1  0
z 1  l , c1 l , r ,t   c 2 0 , r ,t ;
c
z 2  0 , u 2 c 2 l ,t   u 2 c 2 0 , r ,t   D2  2
  z2
z 2  l , c 2 l ,t   c 2 0 ,t ;

 ;
 z 2 0
c1 c2
c1 c2
r  0,

 0; r  R ,

 0,
r
r
r
r
where l is the column height, Di , u i , ci i  1,2  are diffusivities, average velocities
and concentrations in gas and liquid phases.
The average concentration model may be obtained on the analogy of:
  2 ci

ci
ci
i 1 k
c1  c2 
 Ai R  ui
 Di  2  Bi R  ci    1
t
z i
i
 z i

with boundary conditions:
t  0 , c1  c o , c 2  0 ;
c 
z 1  0 , u 1 co  A1 R u1 c1 0 ,t   D1  1  ;
  z 1  z1  0
z 1  l , c1 l ,t   c 2 0 ,t ;
c 
z 2  0 , u 2 c 2 l ,t   A2 R u 2 c 2 0 ,t   D2  2  ;
  z 2  z 2 0
z 2  l , c2 l ,t   c2 0,t ;where
R
R
1 ~ ~
1 1  c~i
Ai R    ui r  ci r  dr , Bi R   
dr , i  1,2
R0
R 0 r r
~ r , c~ r , i  1,2 are velocity and concentration radial nonuniformities
and u
i
i
in the column.
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STATIONARY PROCESSES
In the cases of stationary processes u  const , c  c z  and
c
2 c
u
 Deff 2  k eff c ,
z
z
where
Deff  Deff R  
D
D k
, k eff  k eff R  
 .

 
The boundary conditions are:
c
z  0 , c  co , co u   u c  D .
z
In the cases of radial velocity component u  u  z , c  c z  and
 c  u
2 c
u  c
 Deff 2  k eff c.
z  z
z
The results obtained show that stationary model equations are equivalent to convection diffusion models with chemical reaction , where diffusivity and chemical reaction rate constant
are functions of the column radius.
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CONCLUSION
The results obtained show that the diffusion models in column
apparatuses permit to change the velocity and concentration radial
distributions with the averages velocities and concentrations for the area
of the horizontal column sections.
These new models permit to identify the model parameter using a
hierarchical approach. As a first step must be obtained chemical reaction
rate constant. The next step is identification of the parameters α , β , γ
and D using model experimental data, obtained with real liquids. The
scale - up is a specification of the parameter values of α , β and γ for the
real apparatus. For these aims are necessary experimental data, obtained
with model liquid in the column with real diameter and small height. As
a result the mathematical model may be used for the simulation of real
column apparatus processes simulate the real column apparatus
processes.