diffusion models of a fluidized bed reactor chr. bojadjiev bulgarian academy of sciences, institute...

B A S - I C h EL A B O R AT O R Y

P R O C E S SS Y ST E M S

E N G IN E E R IN GDIFFUSION MODELS OF A FLUIDIZED BED REACTOR

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

“Acad. G.Bontchev” str., Bl.103, 1113 Sofia, Bulgaria, E-mail: [email protected]

IntroductionHydrodynamic modelConvection - diffusion modelAverage function valuesAverage concentration modelHierarchical approachConclusions

Many technological processes, as a catalytic chemical reactions, burning, heating, drying, etc, are realized in column apparatuses with fluidized bed. The hydrodynamic behavior of the fluidized bed like to the motion in counter – current flows. A cylindrical surface, where velocity is equal to zero, is the border between the rising and the downcomer flows. In this case the fluidized bed reactors are similar to the airlift reactors, where the solid phase plays the role of the liquid phase. On this base is possible to use the methods for modeling of the airlift apparatuses for fluidized bed modeling.

Introduction



E N G IN E E R IN G

Hydrodynamic model

Let’s consider an fluidized bed column with a horizontal cross - sections area f, for the riser zone and (F - f ) for the downcomer zone, where F is the horizontal cross - sections area of the column.

The average velocities of the solid particle in the riser and downcomer are and , where is the average sedimentation velocity of the solid particles.

Let’s assume that velocities of the solid phase and are known beforehand as a result of experimental or computing methods.

From the mass balance of the solid phase is obtained:

ovfFfv Fvv

vf

o

o

,

Let’s assume that gas phase in the riser away the solid phase while in the downcomer solid flow carry away the gas phase and the average gas velocities in the riser and downcomer are and .

The gas flow rate in the riser zone Q is result of the inlet gas flow Q0 and circulation gas flow Q1 :

fuQQQ 010 )(1 fFvQ o

As a result was obtained:F

Q

v

vvvuo

0

0

0

If accept a cylindrical form of the riser and downcomer zone, the radiuses may be obtained from:

222 , oo rRfFrf

vov ov

v ov

uov

.

,

.

.

Convection - diffusion modelB A S - I C h EL A B O R AT O R Y


E N G IN E E R IN G Let’s consider fluidized bed column with catalytic chemical reaction on the solid interface. The axial and radial gas velocity components in the riser are u and w:

,0 ,0 ,, ,, lzrrzrwwzruu o where l is fluidized bad height.A component of the gas phase react on the catalyst particles interface and its

concentration has radial and axial non - uniformity: .0 ,0 ,, lzrrzrcc o

The convection – diffusions equation for this case is:

,1

2

2

2

2

kcr

c

r

c

rz

cD

r

cw

z

cu

where the velocity components satisfy the continuity equation:

.0

r

w

r

w

z

u

The convective transfer is result of a laminar or turbulent (large – scale pulsations) flow, the diffusive transfer is molecular or turbulent (small – scale pulsations) and the order of the catalytic chemical reaction is n.

In the downcomer the convection – diffusion equation has the form

,1

121

21

21

12

11

1

11

nkcr

c

r

c

rz

cD

r

cw

z

cu

011

1

1

r

w

r

w

z

u.

where z1 = 1 – z , u1 = u1 (r , z) , w1 = w1 (r , z) , c1 = c1 (r , z) are velocity components and concentration in the downcomer.



E N G IN E E R IN G

The boundary conditions are obtained on the bases of the assumption for an ideal mixing in the two ends of the column:

; ),( , ,0 110

z

cDculrc

f

Qc

f

Qccz oo

; )0,( ),()0,( ,01

1111111 z

cDcurcvlcrcz o

0,;0 , ;0 ,0 11

r

cRr

r

c

r

crr

r

cr o

where c0 is initial gas concentration, - are average gas concentration:

cc0 ,

0r

00

dr)z,r(cr

1)z(c

),(11

00

0 lrcfQ

Qc

fQ

Qc

The radial non-uniformity of the velocity is the cause for the scale effect (decreasing of the process efficiency with increasing of the column diameter). That is why an average velocity and concentration for the horizontal cross-sectional area are used.

,

.



E N G IN E E R IN G

Average function values

For the average velocity and concentration in the riser zone may be obtained :

. ,1

, ,1

, ,1

000 ooo r

o

r

o

r

o

drzrcr

zcdrzrwr

zwdrzrur

zu

If use a property of the average functions, the velocity and concentration may to present as a:

, ~ , , ~ , , ~ , rczczrcrwzwzrwruzuzru

ooo r

o

r

o

r

o

drrcr

drrwr

drrur 000

.1 ~1 ,1 ~1

,1 ~1

The same results for the average velocity and concentration is obtained in the downcomer zone:

)(~).(),(),(~).(),( 111111 rwzwzrwruzuzru

1)(~1),(~).(),(

0

10

111

drrurR

rczczrcR

r

1)(~1,1)(~1

00

10

10

drrc

rRdrrw

rR

R

r

R

r

,

,

.



E N G IN E E R IN G

Average concentration model

The average concentration model may be obtained after integration of the equations over r in the intervals [0 , r0] and [r0 , R].

; 2

2

ckcrz

cD

z

ucr

z

cur ooo

; ; ,0 00 z

cDcurcuccz oo

,wrz

uoo

oo r

oo

r

oo dr

r

c

rrrdrrcru

rr

00

,~

11

, ~ ~1

oo r

oo

or

o drr

w

rr

wrwdr

r

crwr

0

o

0

. ~10~~

, ~3

1



E N G IN E E R IN G

In many cases the concentration of the solid phase is constant, i.e. and i.e. γ = γ1 = 0.

A similar models may be obtained for burning or heat and mass transfer processes in fluidized bed (heating or drying of the solid phase), but must be used two - phase models.

0/ zu ,0/ 11 zu

,drr

)r(C~

)r(w~3

1)R,r( 1

R

r

101

0

,dr)r(c~)r(u~rR

1)R,r( 1

R

r

10

01

0

.w)R,r(z

u10

)0(1

1

.z

cDcu)R,r()0(cv),l(c)0(c,0z

1

1111011011

,ckc)R,r(z

cD 11012

1

12

1

1

1101

1

1101 z

uc)R,r(

z

cu)R,r(

drr

)r(c~

r

1

rR

1)R,r( 1

R

r001

0

drr

w~

rR

1

rR

)r(w~)R(w~)R,r(

R

r

1

00

0110

)0(1

0

,

.



E N G IN E E R IN G

The results obtained show a possibility to present the fluidized bed catalytic reactor model as a airlift reactor model. A diffusion model for the fluidized bed catalytic reactor is used. The introducing of average velocity and concentration permit to solve the scale-up problem. The proposed model allow an hierarchical approach to be used for the model parameter identification.

Hierarchical approach

The presented problems are mathematical model of a fluidized bed catalytic reactor. The model parameters are different types:

– beforehand known – F, R, Q0 , c0;

– beforehand obtained - ;

– obtained with chemical reaction - D, D1 , α, β, α1 , β1;

– specified in the scale-up - α, β, α1 , β1 .

Conclusion

kculvvrf ,,,,,,, 0000

As a result an hierarchical approach in mathematical modeling (model parameters identification) is possible to be used. The parameters D, D1 , α, β, α1 , β1 , may be obtained using experimental data for and after solution of the inverse identification problems. In many cases D and D1 are small parameters and inverse problem is incorrect (ill-posed).

c 1c

diffusion models of a fluidized bed reactor chr. bojadjiev bulgarian academy of sciences, institute...

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