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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 .

DIFFUSION MODELS AND SCALE – UP

by

Chr. BojadjievBulgarian Academy of Sciences, Institute of Chemical Engineering,

“Acad. G.Bontchev” str., Bl.103, 1113 Sofia, Bulgaria, E-mail: chboyadj@bas.bg

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

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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: ,

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, 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.

;0r

c R,r ;0

r

c ,0r

;cc ,0t

;kcr

c

r

c

r

1

z

cD

z

cu

t

c

0

2

2

2

2

z

cDucc u ,l,t c0,t c ,0z

zr,t,cc ,ruu

the mathematical description has the form:

,

c

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AVERAGE FUNCTION VALUES

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:

S

dS,r,zf

zf S

.ddr r,z,f R2

1dS r,z,f ,d

2

RS

R

0

2

0S

2

0

2

.dr z,r,tfR

1z,tf ,dr z,r,tfRdS r,z,f ,RS

R

0

R

0S

2

R

0

R

0.dr z,r,tc

R

1z,tc ,dr ru

R

1u

where

For a circular cylinder R = const:

For the average velocity and concentration:

,

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A PROPERTY OF THE AVERAGE FUNCTIONS

Let present the function f (x,y) as a:

,y,xfxfy,xf o xf

y,xfy,xfo

,dy y,xfy

1xf

oy

0o ,dy y,xfxfxfyy,xf

oy

0oo

oy

0

.ydyy,xf o

y

0

o

o

,rc z,tczr,t,c ,ru uru oo Rdr rc ,Rdr ruR

0

o

R

0

o

If is the average function: xf

.

i.e.

The results obtained permit to presents the velocity and concentration as a:

and and are average velocity and concentration.

The result obtained permit to obtain equations for the average concentrations.

u z,tc

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AVERAGE CONCENTRATION MODEL

The average concentration model may be obtained if it integrate the equations over r in the interval [0,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,

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.

;cc ,0t

;ckc Rz

cD

z

cu R

t

c

o

2

2

z

cDc u Rc u ,l,tc0,tc ,0z

R

0

oR

0oo dr

r

c

r

1

R

1R ,dr rc ru

R

1R

where

,

.

.z,R ,z,R

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QUALITATIVE ANALYSIS OF THE MODEL

For the theoretical analysis of the model must be used dimensionless variables:

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 of the convective transport is negligible. Another situations are possible if compare the values of the dimensionless model parameters and K.

2o l/Dt ,

Ccc Z,lz ,Ttt oo

1;C ,0T

;KCC Z

C

l

t D

Z

C

T

C2

2

2o

,Z

C

l.u

DC RC ,T,1CT,0C 0,Z

.ktK ,Rl ,l

t uR o

2o where

, where to is characteristic time of the process.

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MODEL WITH RADIAL VELOCITY COMPONENT

In the cases u = u (r , z) the velocity has a radial component and the model equation has the form:

,kcr

c

r

c

r

1

z

cD

r

cv

z

cu

t

c2

2

2

2

where v = v (r , z) is radial velocity component and satisfy the continuity equation :

.0r

v

r

v

z

u

The boundary conditions are identical. Let’s present the radial velocity component as a:

.Rdr rv ,dr z,rvR

1v ,rv zvz,rv

R

0o

R

0o

After integration over r in the interval [0,R] , the equation for has the form:c

ckcRz

cDcvR

z

cuR

t

c2

2

1

, .dr

r

cv

R

1R

R

0

oo1

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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 , 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 for many practical interesting cases.

0v 0z/u

The integration of the continuity equation over r in the interval [0,R] lead to:

,0v Rz

u2

.drr

v

R

1

R

0vRvR

R

0

ooo2

On the solid interphase vo(R) = 0 and as a result:

,

10

0

2 R

oo drr

v

RR

vR

,arr R

a22

R

0vRrv 2o

2o

i.e. vo(0) = 0.

The constant a may be obtained using the boundary condition vo(R) = 0 and as a result a = 2 :

2

2o rR

2-2r RRv

The constant γ2 (R) may be obtained, using:

R

0o Rdr rv ,

22

3 v,

3 2o2

r

Rr

Rr

RR .

3 z

uRv

,cK z

cD

z

ucR

z

cuR

t

c2

2

.RDkK ,R

3

RR 1

As a result:

.

,

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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:

,cck 21

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:

,c.ck1r

c

r

c

r

1

z

cD

z

cu

t

c21

1i2i

2i

2i

2

iii

iii

i

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 and are concentrations of the absorbed substance in the gas and liquid phases. In the case of a circulation of the liquid phase, the boundary

conditions of have the form:

t,r,zc 11

2122 zlz ,t,r,zc

;t,r,0ct,r,lc ,lz

;z

c Dt,r,0cucu ,0z

;0c ,cc ,0t

211

0z1

1111o11

2o1

1

;t,0ct,lc ,lz

;z

c Dt,r,0cut,lcu ,0z

222

0z2

2222222

2

,0r

c

r

c ,Rr ;0

r

c

r

c ,0r 2121

where l is the column height, are diffusivities, average velocities

and concentrations in gas and liquid phases.

2,1i c ,u ,D iii

The average concentration model may be obtained on the analogy of:

21i

1iii2

i

i2

ii

iii

i cck

1c RBz

cD

z

cu RA

t

c

with boundary conditions:

;z

c Dt,0cuRAt,lcu ,0z

;t,0ct,lc ,lz

;z

c Dt,0cuRAcu ,0z

;0c ,cc ,0t

0z2

22222222

211

0z1

11111o11

2o1

2

1

;t,0ct,lc ,lz 222 where

2,1i ,dr r

c~

r

1

R

1RB ,dr rc~ ru~

R

1RA

R

0

ii

R

0

iii

and are velocity and concentration radial nonuniformities

in the column.

2,1i ,rc~ ,ru~ ii

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In the cases of stationary processes , and constu zcc

ckz

cD

z

cu eff2

2

eff

.kD

Rkk ,D

RDD effeffeffeff

where

The boundary conditions are:

.z

cDc u uc ,cc ,0z oo

In the cases of radial velocity component and zcc ,zuu

.ckz

cD

z

uc

z

cu eff2

2

eff

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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E N G IN E E R IN G 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.