Catalytic Reactor

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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

pulsations) flows. The diffusive transfer is molecular or turbulent (small - scale pulsations). 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 requireconstruction 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 - sections area F0

for the riser zone and F1 for the downcomer zone. The length of theworking zones is l. The gas flow rate is Q0 and the liquid flow rate - Q1. The gas and liquid hold-

up in the riser are and (1).

The concentrations of the active gas component (O2) in the gas phase is c(x, r, t) and in the

liquid phasec0(x, r, t) for the riser and c1(x, r, t) - for the downcomer, wherex1 = lx.

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.

The average velocities in gas and liquid phases are: .F

Qu,

F

Qu,

F

Qu

1

1

0

11

0

0o

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 andchemical reaction.

The interphase mass transfer rate in the riser is:R = k(c c0).The alcohol oxydation rates in the riser and the downcomer are:

.cckR,cckR 2121 21023o01


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The equations for oxygen concentration distributions in the gas and liquid

phases in the riser are:

,1

02

2

2

2

00cck

r

c

r

c

rx

cD

r

cv

x

cu

t

c

,0 rv

r

v

x

uooo

.0

r

v

r

v

x

u 111

,1

111 21302

2

2

2

0

0

1

0

1

0 cckcck

r

c

r

c

rx

cD

x

cv

r

cu

t

coo

ooo

The equations for the alcohol and oxygen concentration distributions in the liquid phase in the

downcomer are:

,cckr

c

r

c

r

1

x

cD

r

cv

x

cu

t

c21

21o2

1

2

1

2

1

2

1

1

1

11

,cckr

c

r

c

r

1

x

cD

r

cv

x

cu

t

c21

21o2

2

2

2

2

2

2

2

2

1

22

.xlx1

The equation for alcohol concentration distribution in the riser is:

.cckrc

r

c

r

1

x

cD1

x

cv

r

cu1

t

c1 21 3oo23

2

323

2

331313


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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:

,00,r,xc,c0,r,xc,0t 00

,c0,r,xc,c0,r,xc,00,r,xc 0230

21211

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) andx = l(x1 = 0).

The boundary conditions for c (x, r, t) and c0(x, r, t) are:

;t,r,lct,r,lc,lx

,x

cDt,r,0cucu,0x

0

0x

0

0

o

;x

cDut,r,0cut,lc

,t,r,lct,r,0c,0x

0x

00102

20

;0r

c

r

c,0r

o

.0rc

rc,rr oo


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The boundary conditions for c (x, r, t) and c0(x, r, t) and c0(x, r, t) are:

;,,0,,,,,,0,001

1

1113311

x

x

cDutrcutlctrlctrcx

;x

cDu,r,tcul,tc,l,r,tc,r,tc,x

;rc,R; r

rc,rr

x

oo

oo

01

2

2221

11

000

00

;x

cDu,r,tcul,tc,l,r,tc,r,tc,x

;r

c

,R; rr

c

,rr

x

oo

0

3

313113

22

000

00

.0r

c,rr;0

r

c,0r 3o

3

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

crosssections 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.


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AVERAGE CONCENTRATION MODELS

Lets consider equation for oxygen concentration in the gas phase c (x, r, t) incylindrical coordinates.

If use a property of the integral average functions for velocity and

concentration:

,drt,r,xcr

1t,xc,drr,xv

r

1x,drr,xu

r

1xu

or

0o

or

0o

o

o

or

0o

o

o

the velocity and concentration may be expressed as a:

,rc~t,xct,r,xc,rv~xvr,xv,ru~xur,xu oooooo

.rdrrc~,rdrrv~,rdrru~ or0

o

or

0oo

or

0oo

If put these expressions in the convectiondiffusion equation and integrate over rin the

interval [0 , r0] is obtained:

,cck

crBx

cDcvrG

x

curA

t

c

oo2

2

oo1oo

or

0o

o

o1

or

0o

o

or

0o

o

o dr.r

c~v~

r

1rGdr,

r

c~

r

1

r

1rBdr,rc~ru~

r

1rA

If put average velocity in the continuity equation and integrate over rin the interval [0 , r0] :

,0vrGxu

oo2o

dr.r

v~

r

1

r

0v~rv~

rG

or

0

o

oo

oooo2


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As a result :

,cck

crBx

cD

x

ucrG

x

curA

t

coo2

2

o

ooo

.rG3

rrG

o1

o

o

The boundary conditions of has the form:

;x

cDt,0c0urAcu,0x;c0,xc,0t

ox

oo

0

o

0

.t,lct,lc,lxo

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 , i.e. when the gas hold - up in not constant over the column

height. For many practical interesting cases = const, i.e. and the radial velocity

component did not take account .

,

1

1

oooo

oo

lFFFll

FFll

The parameter valuesD, k,A,B, G must be obtaining using experimental data for

measured on the laboratory column. In the cases of scale - up must be specifiedA,Band Gonly,using a column with real diameter but with small height (Dand kdo not change at (scaleup)).

0x/uo 0x/uo

0vo The hold - up must be obtained using:

where land l0 are liquid levels in the riser with and without gas motion. t,xc


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The same procedure may be used for :,,,3210

cccc

;drc~c~r

1rM

;cc

1

krMcc

1

kcrB

x

cD

x

ucrG

x

curA

t

c

o

21

211

r

0

3o

o

oo

3o

o

oooooo2

o

2

o

1

oo0

o

1oo

o

,,,0,0;00,,01oo

tlctcxxct

,x

cDt,0c0urAut,lc

ox

ooo1o1

;ccR,rMkcR,rBx

cD

x

ucR,rG

x

cuR,rA

t

c21

21ooo1oo12

1

2

11oo1

1

oo1

1

,,,0,0;00,,0 31111 tlctcxxct

,x

cDt,0c0uR,rAut,lc

ox1

111oo113

drccrRRrMdrtrxcrRtxcdrrxurRxu

o

o

o

o

o

o

R

roo

oo

R

roo

R

roo 21

2111 ~~

1

,;,,

1

,,,

1

.


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;cckR,rMcR,rBx

cD

x

ucR,rG

x

cuR,rA

t

c21

21ooo2oo22

2

2

22oo2

2

oo2

2

,,,0,0;,021

0

22tlctcxcct

o

,xcDt,0c0uR,rAut,lc

o1x1

222oo213

.drt,r,xcr

1c,drt,r,xc

rR

1c

or

03

o

3

oR

or2

oo

2

;cc1krMcrB

xcD

xuc,rG

xcurA

tc 21 3oooo3o32

3

2

31

3o33

1o33

,t,lct,0c,0x;cx,0c,0t13

0

23

.,00r, 3331o31

oxx

cDtcuAutlc

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.

0vvv,0x

u

x

u

x

u1o

1o

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 ( , , 1 , 2 , 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 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.

The problems for and permit to obtain without chemical reaction if put

t,lct,lc 01

CONCLUSIONS

c c0

HIERARCHICAL APPROACH

.

Catalytic Reactor

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