Chemical Reactors - American Chemical Society

2 An Initial Value Approach to the Counter-Current Backmixing Model of the Fluid Bed V. K. JAYARAMAN,

B. D. KULKARNI, and L . K. DORAISWAMY

Downloaded by UNIV LAVAL on July 11, 2016 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch002

National Chemical Laboratory, Poona 411 008 India

The counter-current backmixing model of F r y e r and P o t t e r has been modified by assuming mixed f l o w in the emulsion phase. The terminal conversions obtained w i t h the present model a r e compared with those of the original model and found to agree w e l l except a t v e r y low v a l u e s of bubble diameter. The assumption of complete mixing in the emulsion phase converts the original two-point boundary v a l u e problem i n t o a simpler initial v a l u e problem, t h e r e by c o n s i d e r a b l y reducing the mathematical complexity. The i n t e n s i v e gas mixing that occurs in a fluid bed due to the presence of bubbles and the a s s o c i a t e d c i r c u l a t o r y movement of s o l i d s has been recognized f o r q u i t e some time (1, 2) . The r i s i n g bubbles c a r r y wakes of s o l i d s along with them and r e l e a s e them subsequently on b u r s t i n g a t the s u r f a c e (3^, 4_, 5). The released s o l i d s then move downwards f o r reasons of c o n t i n u i t y and a simple c i r c u l a t o r y p a t t e r n of movement of s o l i d s is s e t up. The s t u d i e s on p a r t i c l e movement in deep fluidized beds (6) have i n d i c a t e d that s o l i d s move upwards in the center r e g i o n of the bed and downwards at the p e r i p h e r y . The i n t e n s i t y of c i r c u l a t i o n of s o l i d s i n c r e a s e s with i n c r e a s e in the f l u i d i z i n g gas v e l o c i t y , and a t a critical v e l o c i t y U the v e l o c i t y of down flowing s o l i d s exceeds t h e i n t e r s t i t i a l gas v e l o c i t y , so that the i n t e r s t i t i a l gas is c a r r i e d downwards as described by (7-10). A simple mechanism f o r gas mixing t h e r e f o r e seemed p o s s i b l e and s e v e r a l models - t h e s o - c a l l e d counter-current backmixing models that take i n t o account this f l o w r e v e r s a l have been proposed (8, 11, 12). I t should, however, be noted that the s o l i d s movement p a t t e r n as mentioned above has been observed in beds with s u f f i c i e n t l y l a r g e v a l u e s of l e n g t h to diameter r a t i o ( L / d » l ) . I n d u s t r i a l fluid beds normally operate with L^/d^ v a l u e s l e s s than or c l o s e t o u n i t y and the s o l i d s f l o w p a t t e r n could be e n t i r e l y d i f f e r e n t . More recent experimental s t u d i e s such as f

0097-6156/81 /0168-0019$05.00/0 © 1981 American Chemical Society Fogler; Chemical Reactors ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

t

CHEMICAL REACTORS


20

those of Okhi and Shir a i Ç13) in shallow beds i n d i c a t e a d i f f e r e n t f l o w p a t t e r n . Their expérimental measurements have confirmed the f a c t that s o l i d s move downward in the c e n t r a l r e g i o n of the bed. E a r l i e r , Whitehead e t a l (14) had a l s o made measurements of s o l i d s movement and demonstrated in some cases a strong down f l o w of s o l i d s in a small area a t the center of the bed. Such s o l i d s c i r c u l a t o r y p a t t e r n has a l s o been reported by Werther (15) and Schmalfeld (16). Nguyen and P o t t e r (9, 10) experimenting w i t h a 30 cm diameter column have a l s o observed that gas mixing is at i t s maximum in the center. Bubbles move in the area between the center and the w a l l , f o r c i n g the s o l i d s and the backmixed gas to move downwards in the c e n t r a l and n e a r t o - w a l l r e g i o n . The more recent experiments of Nguyen et^ al^ (17) in a l a r g e s c a l e fluidized bed confirm this f a c t ; however at v e r y h i g h v e l o c i t i e s the stream becomes more u n s t a b l e and f l o w is d i f f i c u l t to d e f i n e . It is c l e a r from the foregoing d i s c u s s i o n that a c o n s i d e r a b l e extent of gas backmixing r e s u l t s due to the presence of bubble t r a c k s and the a s s o c i a t e d s o l i d s movement. Besides this, the i n d u s t r i a l u n i t s are normally operated w i t h b a f f l e s and i n t e r n a l s to remove the heat of r e a c t i o n . The hinderances from these would lead to f u r t h e r enhancement of gas mixing in the emulsion phase. The common assumption of plug f l o w in the emulsion phase t h e r e f o r e seems incompatible with the s i t u a t i o n p r e v a i l i n g in i n d u s t r i a l r e a c t o r s , and in the present work the original F r y e r - P o t t e r model (12) has been modified to take this r e a l i t y i n t o account. This has the a d d i t i o n a l advantage of converting the boundary v a l u e nature of the F r y e r - P o t t e r r e p r e s e n t a t i o n i n t o an initial v a l u e problem, thus c o n s i d e r a b l y s i m p l i f y i n g the mathematical treatment. T h e o r e t i c a l Development Let us consider a simple r e a c t i o n A * R and make the f o l l o w i n g assumptions: the bubbles are uniform in s i z e and f r e e of p a r t i c l e s . The emulsion phase voidage is constant, with the voidage of the bubbling bed equal to t h a t at i n c i p i e n t fluidization. The voidage in the cloud is the same as in the emulsion. Plug flow p r e v a i l s in the bubble and cloud phases, w i t h the emulsion phase completely mixed. With these assumptions the m a t e r i a l balance equations may be w r i t t e n as f o l l o w s : dC b

+

- U Gb

dz

-U

-—— +

(c c

- C ) = 0 b

(1)

dC Gc

dz

£JL

b DC

-kf w

(C

b

- C ) + c

6, Κ (C b ce e

C ) c

e,c = ο b c

Fogler; Chemical Reactors ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

(2)

2.

JAYARAMAN E T A L .

Counter-Current Backmixing Model

21

(3)

-k a - e .Ci+f )) L tCe = 0 D w r

The a p p r o p r i a t e I n i t i a l c o n d i t i o n s can be w r i t t e n as C.(0)

=

C

D

C4)

Ο

Cu-u ) c + C-u ) c C0) = u Downloaded by UNIV LAVAL on July 11, 2016 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch002

Gb

0

Ge

e

G c

c (0)

C5)

c

Equations 1-5 can be w r i t t e n in dimensionless forms as dC -

— dl

+

A (C 1 2

- C ) = 0 1

(6)

dC ~

— dl

2

+ A (C - C ) + A (C - C ) 2 1 2 3 3 2

- AC = 0 4 2

(7)

1

- CC (D - C ) + A j 3

2

CC - C ) d l - Α ^

5

3

2

=0

3

(8)

0 where the constants A - A 1 6

a r e defined as f o l l o w s

S V f 1

"

=

J

A

=

The s e t

U

M

A

=

4

€ Κ L^ JL£*L

a

U Ge

=

= 1 at

1 + B C l

3

is

f

Gc 6

L

w b f υ Gc

k Cl-6, Cl4f ) ) L . b w _ X

6

of Equations 6-8 C

=

L

S ce f υ Gc

5

W

2

«Gb K

A

.

:

accompanied by

initial

conditions

L= 0 =

B C 2

( 9 )

U Ge

(10) 2

at

1= 0

where


(11)

CHEMICAL REACTORS

22

- U Ge

Gc

Β

and

(12)

2 The assumption of complete mixing in the emulsion phase renders t h e c o n c e n t r a t i o n C constant in the bed, and Equations 6 and 7 can be rearranged aâ 2

OC

i

+ (^+A +A + A )

DC +A (A +A ) 0 - A ^ ^ = 0

4

JL

1

3

4


where t h e operator D r e f e r s to d/dl . equation can be r e a d i l y obtained as X1

χ

(13)

The s o l u t i o n to this

X l

X

2

R e 1

+

+

R e 2

(14)

R

where λ and λ a r e t h e r o o t s of Equation 13 w i t h the constant term (A*A C ) removed, and R is th.e p a r t i c u l a r s o l u t i o n of 13 3 3 Equation 13 g i v e n by

(15) R

3

" A

3

+

A

4

Equation 14 can be s u b s t i t u t e d in Equation 6 to o b t a i n X

2

R Οί e 11

1

x i R οί e + 22

L

9

+

(16)

where the CL s are defined as χ 1 1

λ

+ A 1

and

2

Of,

+A 1

(17)

1

I t is i n t e r e s t i n g to note t h a t Equations 14 and 16 r e q u i r e a knowledge of C which can be obtained a f t e r some a l g e b r a i c manipulations by s u b s t i t u t i n g these equations in 8 as C = 3 where

AAR + 7 8 1

A , A and A 7 8 9

(18)

AAR 7 92

a r e constants d e f i n e d as

A

1-A -A 6 5

A

5 3 A

V 4

A

V4


2.

JAYARAMAN ET AL.


A

d e 1

d 5 1

d

)

e

}

d

A

=

Cl - e

23

5

2 Cl

e

2

Cl 9}


Equations 14 and 16 along with 18 give the concentration profiles in the bubble and cloud phases. The constants R^ and R appearing in these equations can be evaluated subject to initial conditions given by Equations 9 and 10 and can be written in matrix form as

(19a)

where parameters Β - Β are defined as 3 6 3

1 +

*3

7

8

1 +

A A A 3 7 9 + A,

+ A,

*3

B A A

= Β d + A 2 1 8

Β

Β

a 2 2

+

-^JLZ A

+

3 L

A A

.

4 0

1

-

(20)

Β A

17

+ A

3

Β A

17

A

4

/

The gas concentration at the bed exit is given by

CCD =

c CD 2

C2D

The concentration profiles in the bubbles, cloud and emulsion phases are plotted in Figure 1 for a set of parameter values. For the sake of comparison, the profiles for the same values of parameters obtained using the Fryer-Potter model are shown in Figure 2. Figures 3-6 show the influence of parameters such as bubble diameter, U/U Η and rate constant on the extent of mf ο


24


CHEMICAL REACTORS

Figure 2. Concentration profiles using Fryer-Potter model (12): H = 50 cm, di — 5cm,U = 10 cm/s, U = 1 cm/s, k = O.5s , e = O.5, D =O.2cm /s 0

1

mf

2

mf

e


JAYARAMAN ET AL.



1-0

RATE

CONSTANT

Figure 3. Comparison of the present and the FP model effect of rate constant: H = 50, d = 5 cm, U = 10 cm/s, U = 1 cm/s, D =O.2cm /s, e = O.5, (X) FP model, ( ) present model 2

0

6

m /

e

mf

to

HEIGHT

Figure 4. Comparison of the present and the FP model effect of the bed height: d = 5 cm, U = 10 cm/s, U = 1 cm/s, e =O.5,D =O.2cm /s, k ==O.5s' , (X) FP model, ( ) present model 2

6

m /

mf

e


1


CHEMICAL REACTORS

U/Umf

Figure 5. Comparison of the present and the FP model effect of U / U on conversion: H = 50 cm, d = 5 cm, U = 1 cm/s, D =O.2cm /s, k =O.5s' , c = O.5, (X)FP model, ( ) present model m /

2

0

b

m /

1

e

m/

1-0

10 BUBBLE

15 DIAMETER

Figure 6. Effect of bubble diameter on conversion: H = 50 cm, kj =O.5s~ D =O.2cm /s, U = 10 cm/s, JJ = 1 cm/s, e =O.5,(X) FP model, ( present model 0

2

e

mf

mf


2.

JAYARAMAN ET

AL.


27

conversion. Again, f o r the sake of comparison, the r e s u l t s obtained using the F r y e r - P o t t e r model are a l s o presented. I t can be seen from these f i g u r e s that the r e s u l t s obtained using the two models are almost i n d i s t i n g u i s h a b l e from each other except a t smaller values of d^. The smaller bubble diameters are however u n l i k e l y in l a r g e i n d u s t r i a l fluid beds, and t h e r e f o r e f o r a l l p r a c t i c a l purposes the p r e d i c t i o n s of the two models are i d e n t i c a l . The l a r g e i n d u s t r i a l fluid beds are normally operated w i t h U/U exceeding 10, so that a l a r g e p o r t i o n of the gas bypasses the bed in the form of bubbles. A l s o the diameter of the bubbles is f a i r l y l a r g e , so that interphase mass t r a n s p o r t is small compared to the r a t e of r e a c t i o n . Under these c o n d i t i o n s the extent of mixing in the emulsion phase is r a t h e r an unimportant parameter as f a r as the p r e d i c t i o n of conversion is concerned. I t would, however, have s i g n i f i c a n t i n f l u e n c e when non f i r s t - o r d e r r e a c t i o n s are i n v o l v e d . The f o r m u l a t i o n of the model as above has the advantage that mathematically it p i c t u r i z e s the bed as an initial value problem in c o n t r a s t to the more complicated boundary v a l u e r e p r e s e n t a t i o n of the F r y e r - P o t t e r model. The i m p l i c a t i o n s of this reduced complexity become more evident (and c o n s i d e r a b l y more important) when the r e a c t i o n s involved are n o n l i n e a r . While the initial v a l u e problem can be r e a d i l y solved f o r such a case, the boundary v a l u e p r e s e n t a t i o n leads to severe s t a b i l i t y and convergence problems.


mf

Conclusions The behavioural f e a t u r e s of the fluidized bed have been modeled based on a modified r e p r e s e n t a t i o n of the F r y e r - P o t t e r model. The r e s t r i c t i v e assumption of plug f l o w of the emulsion gas has been removed, and model equations developed based on complete mixing of the emulsion gas. This s i m p l i f i c a t i o n , in a d d i t i o n to b r i n g i n g the model c l o s e r to r e a l i t y , has l e d to the conversion of a boundary v a l u e problem ( F r y e r - P o t t e r model) to a simpler initial v a l u e problem. Except at v e r y low bubble diameters, the p r e d i c t i o n s of the two models (based on terminal conversion) agree c l o s e l y w i t h each other. On the other hand, agreement between the average concentration p r o f i l e s in the bed p r e d i c t e d by the two models is l e s s s a t i s f a c t o r y . While t h e r e f o r e the modified model proposed in this work has the advantage of s i m p l i c i t y and is perhaps c l o s e r to r e a l i t y , f u r t h e r experimental work on i n d u s t r i a l s i z e equipment is necessary f o r a f i r m e r o p i n i o n on the l a t t e r (nature of gas flow in the emulsion phase).


28

CHEMICAL REACTORS

Legend of Symbols A to A A* to A Β , Β B* £ C , C , C V t

o

B


, C^, C ( 1 ) , C (1) 1 2 C(l) d^ d f t

K

c e

L^ I k R^, R R^ U U ΪΓ

7

Ζ

constants defined by Equation (9) constants defined by Equation (19) constants defined by Equation (12) parameters defined by Equation (20) c o n c e n t r a t i o n in the bubble, cloud-wake and emulsion phase r e s p e c t i v e l y dimensionless concentration in the bubble, cloud and emulsion dimensionless bubble phase and cloud-wake phase c o n c e n t r a t i o n a t the bed e x i s t dimensionless gas c o n c e n t r a t i o n at the e x i t bubble diameter cm diameter of the bed cm r a t i o of wake volume to bubble volume height of the bed a t i n c i p i e n t fluidization v o l u m e t r i c r a t e of gas exchange between bubble and cloud-wake per u n i t bubble volume s~^ v o l u m e t r i c r a t e of gas exchange between _^ cloud-wake and emulsion per u n i t volume s height of bubbling bed cm dimensionless height above d i s t r i b u t o r f i r s t order r e a c t i o n r a t e constant, based on u n i t volume of dense phase, s parameters defined by Equation (19a) parameter defined by Equation (15) s u p e r f i c i a l gas v e l o c i t y cm s'^ critical v e l o c i t y cm s~* _^ s u p e r f i c i a l v e l o c i t y in bubble phase cm s _^ s u p e r f i c i a l v e l o c i t y in cloud-wake phase cm s s u p e r f i c i a l v e l o c i t y in emulsion phase cm s"~* l e n g t h parameter along the bed height

Greek L e t t e r s d^y & g X, y\ l ' 2 € mf v

Λ

constants defined by Equation (17) f r a c t i o n of bed volume occupied by bubbles r o o t s of Equation (13) v o i d f r a c t i o n in bed at minimum fluidization conditions

n

Acknowledg em en t The f i n a n c i a l support r e c e i v e d from Indian Petrochemicals Corporation L i m i t e d , Baroda, is g r a t e f u l l y acknowledged.


2.

J A Y A R A M A N E T AL.


29

L i t e r a t u r e Cited 1. 2. 3. 4. 5. 6. 7.


8. 9.

10. 11. 12. 13.

14. 15. 16. 17.

Gilliland, E . R . ; Mason, E . A . End. Eng. Chem. 1949, 41, 1191. Gilliland, E . R . ; Mason, E . A . Ind. Eng. Chem. 1952, 44, 218. Noble, P.J. "Mineral Dressing Research Symposium Annual Conference"; A u s t r a l i a n I n s t n . M i n . Met. 1962. Rowe, P . N . T r a n s . I n s t . Chem. E n g r s . 1961, 39, 175. W o o l l a r d , I . Ν . M.; P o t t e r , O . E . AIChE J. 1968, 14, 388. Marscheck, R . M . ; Gomezplata, A . AIChE J. 1965, 11, 167. Stephens, G . K . ; Sinclair, R.J.; P o t t e r , O . E . Powder Tech. 1967, 1, 157. Latham, R.L.; Hamilton, C.J.; P o t t e r , O . E . Brit. Chem. Eng. 1968, 13, 666. Nguyen, H . V . ; P o t t e r , O . E . "Advances in Chemistry S e r i e s No.133"; American Chemical Society : Washington, DC. 1974; p 290. Nguyen, H . V . P h . D . t h e s i s , Monash U n i v e r s i t y , A u s t r a l i a , 1975. K u n i i , D.; L e v e n s p i e l , O. I n d . Eng. Chem. Fundam. 1968, 7, 446. F r y e r , C.; P o t t e r , O . E . I n d . Eng. Chem. Fundam. 1972, 11, 338. Ohki, K . ; S h i r a i , T . " F l u i d i z a t i o n Technology I"; Hemisphere P u b l i s h i n g Corporation : Washington DC, 1976; p 95. Whitehead, A.B.; G a r t s i d e , G.; Dent, D . C . Chem. Eng. J. 1970, 175. Werther, J. P r e p r i n t , paper presented at GVC/AIChE M e e t i n g , Munchen, 1974. Schmalfeld, V.J. V.D.I-Z. 1976, 118, 65. Nguyen, H . V . ; Whitehead, A.B.; P o t t e r , O . E . AIChE J. 1977, 23, 913.

1,

R E C E I V E D June 3, 1981.


Chemical Reactors - American Chemical Society

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