Chemical Reactors - American Chemical Society

3 Predictions of Fluidized Bed Operation Under Two Limiting Conditions: Reaction Control and Transport Control Downloaded by UNIV OF MICHIGAN ANN ARBOR on October 31, 2014 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch003

H. S. FOGLER Department of Chemical Engineering, The University of Michigan, Ann Arbor, MI 48109 L. F. BROWN Department of Chemical Engineering, The University of Colorado, Boulder, CO 80309 Some aspects of fluidized-bed r e a c t o r performance are examined using the K u n i i - L e v e n s p i e l model o f fluidized-bed r e a c t o r behavior. An ammonia-oxid a t i o n system is modeled, and the conversion pred i c t e d is shown to approximate that observed exp e r i m e n t a l l y . The model is used to p r e d i c t the changes in conversion with parameter v a r i a t i o n under the l i m i t i n g c o n d i t i o n s of r e a c t i o n c o n t r o l and t r a n s p o r t c o n t r o l , and the ammonia-oxidation system is seen to be an example of r e a c t i o n control. F i n a l l y , it is shown that s i g n i f i c a n t differences in the averaging techniques occur f o r height to diameter r a t i o s in the range of 2 to 20. There has been increased i n t e r e s t in recent years in the science and engineering of fluidized-bed r e a c t o r s . Part of this i n t e r e s t can be a t t r i b u t e d to the p r o j e c t e d extensive use of fluidized-bed c o a l g a s i f i e r s , but the development of m a g n e t i c a l l y stabilized fluidized beds and c e n t r i f u g a l beds a l s o has c o n t r i b uted s i g n i f i c a n t l y to r e j u v e n a t i n g fluidized-bed research and modeling. Some of the many recent reviews and e v a l u a t i o n s o f fluidized-bed modeling are those of Bukur (1974), Chavarie and Grace (1975), Yates (1975), Van Swaaif (1978), Weimer (1978), and P o t t e r (1978). Of these, Yates gives an unusually good comparison of the t h e o r e t i c a l s i m i l a r i t i e s and d i f f e r e n c e s among c u r r e n t l y popular models, while Chavarie and Grace compare the pred i c t i o n s of v a r i o u s models with the experimentally-observed i n t e r n a l behavior of a fluidized-bed r e a c t i n g system. These l a t t e r authors conclude that the K u n i i - L e v e n s p i e l (K-L) model g i v e s the most r e a l i s t i c estimate of behavior w i t h i n a fluidized bed. Yates p o i n t s out that the countercurrent-backmixing model of Fryer and P o t t e r , not considered by Chavarie and Grace, is more r i g o r o u s l y founded than the K-L model. On the other hand, P o t t e r shows that when the average bubble s i z e is smaller than 8-10 cm, there is l i t t l e d i f f e r e n c e between the countercurrent-

0097-6156/81/0168-0031 $06.00/0 © 1981 American Chemical Society In Chemical Reactors; Fogler, H.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

Downloaded by UNIV OF MICHIGAN ANN ARBOR on October 31, 2014 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch003

32

CHEMICAL REACTORS

backmixing and the K-L models, both of which give good p r e d i c t i o n s of fluidized-bed performance. The K-L model, because of i t s greater s i m p l i c i t y , thus seems to be the model of choice f o r systems with smaller bubbles. In this paper we s h a l l show how the K-L model can be used to p r e d i c t the experimental r e s u l t s obtained by M a s s i m i l l a and Johnstone (1961) on the c a t a l y t i c o x i d a t i o n of ammonia. I t w i l l be seen that the performance of t h e i r system was l a r g e l y c o n t r o l l e d by r e a c t i o n l i m i t a t i o n s w i t h i n the bed's phases. The e f f e c t s of v a r i o u s parameters on bed performance a r e examined f o r such a r e a c t i o n - l i m i t e d system, and then the e f f e c t s of these parameters f o r a t r a n s p o r t - l i m i t e d system a r e a l s o d i s c u s s e d . F i n a l l y , we consider the e f f e c t of using average values of the bubble diam eter and transport c o e f f i c i e n t s on model p r e d i c t i o n s . Applying the K u n i i - L e v e n s p i e l Model The K u n i i - L e v e n s p i e l Model w i l l be used in c o n j u n c t i o n w i t h the c o r r e l a t i o n s o f Broadhurst and Becker (1975) and Mori and Wen (1975) to analyze the ammonia o x i d a t i o n of M a s s i m i l l a and John stone (1961). The r e a c t i o n 4NH + 7 0 + 4N0 3

2

2

+ 6H 0 2

was c a r r i e d out in an 11.4 cm diameter fluidized-bed r e a c t o r con t a i n i n g 4kg of c a t a l y s t p a r t i c l e s . The p a r t i c l e s had a diameter, dp, of 105 ym, and a d e n s i t y , p , of 2.06 g/cm . The p a r t i c l e s p h e r i c i t y , φ, was taken to be O.6 as is t y p i c a l of published values ( K u n i i and L e v e n s p i e l , 1969). A mixture of 90% oxygen and 10% ammonia was fed to the r e a c tor a t a r a t e of 818 cm^/s, a temperature of 523 K, and a p r e s s u n of O.11 MPa (840 t o r r ) . The r e a c t i o n is f i r s t order in ammonia. The r e a c t i o n is apparently zero order in oxygen owing to the excess oxygen. Thus 3

From fixed-bed s t u d i e s , k =O.0858 cm gas/[(cm catalyst)(s)]. The c a t a l y s t weight, W, and corresponding expanded bed height, h, necessary to achieve a s p e c i f i e d conversion, X, a r e cat

W = Ah(l-e )(l-6)p m f

p

h =

in which

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

(2)

(3)

3.

FOGLER A N D BROWN

Reaction Control & Transport Control

33

A is the c r o s s - s e c t i o n a l area KR is the o v e r a l l dimensionless r e a c t i o n r a t e constant is the v e l o c i t y of bubble r i s e , cm/s mf -*- the bed p o r o s i t y a t minimum fluidization condi tions 6 is the f r a c t i o n of the column occupied by bubbles e

s


C a l c u l a t i n g the F l u i d i z a t i o n Parameters. The p o r o s i t y a t minimum fluidization is obtained from the Broadhurst and Becker c o r r e l a t i o n (1975):

ε

= O.586ψ

f

-O.72

Γ P

μ

2 Ί O.029 3

nd g Ρ

M

O.021

Ν

(4)

r e s u l t i n g in e f=O.657. At f i r s t s i g h t , this v a l u e appears higher than v o i d f r a c t i o n s of O.35-O.45 normally encountered in packed beds (Drew et a l . , 1950). The c a t a l y s t used by M a s s i m i l l a and Johnstone used an impregnated c r a c k i n g c a t a l y s t , however, and a v a l u e of e f of O.657 is c o n s i s t e n t with the numbers reported f o r m a t e r i a l s o f this type by Leva (1959) and by Zenz and Othmer (1960). The corresponding minimum fluidization v e l o c i t y is m

m

2

0Η ) η ρ

mf

150 μ

-mf 1-ε mf

Re < 20 ( K u n i i and L e v e n s p i e l , 1969)

(5)

which gives u ^ = 1.48 cm/s. ^ The e n t e r i n g v o l u m e t r i c flow r a t e of 818 c m / s corresponds to a s u p e r f i c i a l v e l o c i t y o f 8.01 cm/s. Therefore

5.4 u

mf

In order to c a l c u l a t e the expanded bed height, h, f o r the given c a t a l y s t weight of 4 kg, one needs to c a l c u l a t e the f r a c t i o n o f bed occupied by bubbles, 6. From the K-L model u -u . ο mf U

V mf

(1+a)

(6)

For O.1 mm p a r t i c l e s , K u n i i and L e v e n s p i e l (1969) s t a t e that α = O.4 is a reasonable estimate. At this p o i n t , however, there is a d i f f i c u l t y . To c a l c u l a t e the v e l o c i t y o f bubble r i s e , u^, the bubble diameter a t the midpoint in the column, d^, is r e q u i r e d :


CHEMICAL REACTORS

34 u, = u -u -+U, D o mf br

(7)

and u,

D r

1/2 = (O.71) (gd.)

(u, = r i s e v e l o c i t y of a br . _ , - ν s i n g l e bubble)

D

1

In Mori and Wen s (1975) c o r r e l a t i o n , the bubble diameter is a f u n c t i o n of d i s t a n c e up the column, L: d

b = b - bm- bo t P and conversion (X = X J are tne same for Cases 1 and 2, the r a t i o of equation (28) and (29) y i e l d a t

c a t

0

a

W u ,-u 5u ---u 2 _ o l mf2 _ m f l mf2 W4u 4u 1 mfl mfl r

R e c a l l i n g Eq.

i

(5)


3.

FOGLER AND BROWN

(

mf

ψ

ά


)

2

ρ 150μ

η

£

43

3

mf 1-ε mf

(5)

and n e g l e c t i n g the dependence of ε on d we see that the onlyparameters which vary between Case 1 (d ) a n d Case 2 (d =2d -) Ρ p2 p l are u and W mf f

P

r


u

u

2d

mf2

J*

mfl

Pl

= 4 Pl

J

and t h e r e f o r e W

0

5u

4u

m f l " mf 1 4u mfl

O.25

Thus in the s i t u a t i o n we have p o s t u l a t e d , with a f i r s t - o r d e r r e a c t i o n and r e a c t i o n l i m i t i n g the bed behavior, doubling the p a r t i c l e s i z e w i l l reduce the c a t a l y s t by approximately 75% and s t i l l maintain the same conversion. The s l o w - r e a c t i o n s i t u a t i o n has been t r e a t e d before (Grace, 1974), using a model of bed performance developed w e l l b e f o r e the K-L model (Orcutt e t a l . , 1962). T h i s e a r l i e r work concluded that when the r e a c t i o n was very slow, the hydrodynamics and the way the hydrodynamics were modeled were unimportant. The a n a l y s i s given above, u s i n g the more s o p h i s t i c a t e d K-L model, shows that the hy drodynamics can be v e r y important indeed, even when the r e a c t i o n is slow. In the s i t u a t i o n c i t e d , a r e d u c t i o n of 75% in c a t a l y s t requirement can be a t t a i n e d by e x p o i t a t i o n of the bed hydrodynam ics. The Rapid Reaction. To analyze this l i m i t i n g s i t u a t i o n we s h a l l assume the p a r t i c l e s a r e s u f f i c i e n t l y small so that the e f f e c t i v e n e s s f a c t o r is e s s e n t i a l l y one and that the r a t e o f t r a n s f e r from the b u l k fluid to the i n d i v i d u a l c a t a l y s t p a r t i c l e s is r a p i d in comparison w i t h the r a t e o f t r a n s f e r between the fluidization phases. For the case o f r a p i d r e a c t i o n k

cat-

-t , ca and — » ce k

Ί

1

Using these approximations in the equation f o r K

R

which is


44

CHEMICAL REACTORS

K

R

+

=

k — - ^ cat Κ , cb γ

; 1

+

+

c

'

k

-

cat , +

—

Y

e

ce


one observes the f i r s t term to be neglected is

—1

l

= ο

k 1 cat _1_ (Large No. ) + — Κ γ e ce e Then n e g l e c t i n g the r e c i p r o c a l of w.r.t. Y

K

R

Y

" b

+

ΪΓ~~

ι

cat

"

Y

b

+

c

a

t

/

K R

gives

—

Resistance to transport l a r g e w.r.t. r e s i s t a n c e to r e a c t i o n in s i d e the bubble

cat

Only s i t u a t i o n 1 w i l l be analyzed in the t e s t and the a n a l y s i s of s i t u a t i o n 2 is l e f t as an e x e r c i s e f o r the i n t e r e s t e d reader. Assuming very few p a r t i c l e s a r e present in the bubble phase

K

R

a ^

(32) cat

The c a t a l y s t weight is given by combining Eqs. (2) and (32) Au p (1-6) ( l - e

W=

b P

m f

)p

n

,

2>L_E> * n ( ^ ) DC

N e g l e c t i n g δ w.r.t. 1 in the numerator


(33)

3.

FOGLER AND BROWN


AIL p ( 1 - ε . )

w

= V

-

On observing the equation f o r K terms A and Β

D


C

K

m

= ^ + ^ b

b c

DC

=

A„ Ό

A

+

1 / 2

AB S 5 , U a.

A >>B Q

B

Case B:

>

>

> d i f f u s i o n term in Eq. (12), is a somewhat r e s t r i c t e d one. For AQ >> BQ in s m a l l - p a r t i c l e systems, the b i n a r y d i f f u s i o n c o e f f i c i e n t must be on the order of O.01 cm^/s or less. Systems i n v o l v i n g heavy hydrocarbons f r e q u e n t l y have d i f f u s i o n c o e f f i c i e n t s this low, but systems w i t h l i g h t e r components do


3.

FOGLER AND BROWN


47


not. Systems u s i n g l a r g e r p a r t i c l e s a l s o have AQ >> B^, but then Re > 20, Eq. (5) cannot be used, and the example a n a l y s i s given above is not d i r e c t l y a p p l i c a b l e . Thus the example is l i m i t e d to systems with small p a r t i c l e s and low b i n a r y d i f f u s i o n c o e f f i c i e n t s , E v a l u a t i o n o f the Average Transport C o e f f i c i e n t and Bubble S i z e . A constant bubble s i z e is used when e v a l u a t i n g the proper t i e s o f the fluidized bed, and s i n c e bubbles in r e a l beds vary in s i z e , it is important to ask what bubble s i z e should be used. Fryer and P o t t e r (1972), u s i n g the model o f Davidson and H a r r i s o n , reported that a bubble s i z e found a t about O.4h could be used as the s i n g l e bubble s i z e in that model. E a r l i e r in this paper, the bubble s i z e found a tO.5hwas used a r b i t r a r i l y in c a l c u l a t i n g the conversion in an ammonia o x i d a t i o n system u s i n g the K-L model. The average bubble s i z e d in a bed can be found using Eq. ( 8 ) : b

d, -d, = (d, -d, b bo bm bo

)

h

(l-e-°'

3L/D

)dL

/ \ dL

(43)

Integrating : [(l-e-°( d

b

3h/D

)/(O.3h/D)]

- « W ^ b m - V (44)

P

1 - (1-e )£ At midpoint,

( d

b

d

- bo>

-d^) / ( d

/ ( d

b m

-d^)

b ~ bo> = t " d

1

1-e

, and t h e r e f o r e

β

(1-β" )/β]/(1-β-

β/2

)

(45)

A p l o t o f the r a t i o of the mean bubble s i z e to the bubble s i z e evaluated a t the midpoint in the column is shown in F i g . 3 as a f u n c t i o n of h/D. The mean bubble s i z e is a t l e a s t 90% o f the bubble s i z e evaluated a t h/2 f o r almost a l l the height-to-diameter r a t i o s of p r a c t i c a l interest. E v a l u a t i o n of the Transport C o e f f i c i e n t We now wish to determine the d i f f e r e n c e between the average exchange c o e f f i c i e n t evaluated a t the midpoint in the columns. The dependence of the t r a n s p o r t c o e f f i c i e n t between the bubble and the cloud on the bubble diameter,

American Chemical Society Library 1155 16th St. N. w. In Chemical Reactors; Fogler, H.; Washington, D. C. 20038 ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

CHEMICAL REACTORS

1/4 D

4.5u . + 5.85 mf

"be

1_

AB «

(46)

h

w i l l be approximated as


he

(47)

'

h

owing to the weak dependence o f K and di> in the second term in Eq. (46). From the Mori and Wen c o r r e l a t i o n b c

h

^bm

where χ = L/h.

(^bm

(48)

e

hc?

The l o c a l transport c o e f f i c i e n t takes the form

"bc"

(49) d. -(d, -d. )e-βχ bm bm bo μ

Λ

. bo. -βχ 1-(1- - , — ) e

hra. At

the midpoint in the column, L = h/2, χ = 1/2 (50) 1-(1- - ^ ) e bm

The average t r a n s p o r t rl

B x

bm

coefficient

0 hi

dx (51)

"be dx

he

dx

= A„ 1-(1-

he

1 + l/βΐη

(52)

^ ) e " bm

3

x

hm ho

- 1 "bo


(53)


3. FOGLER AND BROWN

49

Taking the r a t i o n of Eq. (50) to Eq. (50) and l e t t i n g r be the r a t i o n of maximum to minimum bubble diameter, d, , bm bo

he

P

l+l/ein[r-(r-l)e ]

[l-(l-l/r)e

6 / 2

]

(54)

he


as

3 he

(55)

1.0

he A p l o t of the r a t i o of the transport c o e f f i c i e n t s is shown in F i g u r e 4 as a f u n c t i o n of 3 f o r various values of the parameter r . For the ammonia o x i c a t i o n discussed e a r l i e r , bm

(56)

= 55

So For

l a r g e values of r

fbç = he and

P

1 + 1/3 l n [ r ( l - e ) ]

(1-e

)

(57)

"

f o r l a r g e r values of 3

1 +

1/3 In r

(58)

he One notes the greatest d i s p a r i t y between the two transport coeff i c i e n t s f o r l a r g e r a t i o s of the maximum to minimum bubble diameter and f o r columns with h/D r a t i o s in the range of 2 to 20 (.6

Chemical Reactors - American Chemical Society

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