Chemical Reactors - American Chemical Society

5 A Model for a Gas-Solid Fluidized Bed Filter MICHAEL H. PETERS, THOMAS L. SWEENEY, and LIANG-SHIH

FAN

Downloaded by PURDUE UNIVERSITY on March 10, 2013 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch005

Department of Chemical Engineering, The Ohio State University, Columbus, OH 43210

A general mathematical model f o r s i m u l a t i n g particulate removal in g a s - s o l i d fluidized beds is presented. Model p r e d i c t i o n s of the fluidized bed filtration e f f i c i e n c i e s , which i n c l u d e the possibility of electrical e f f e c t s , are shown to compare w e l l to the experimental r e s u l t s of v a r i o u s i n v e s t i g a t o r s . Because of the general formulation of the proposed model it is b e l i e v e d to be a p p l i c a b l e in the design of both s i n g l e and multistage fluidized bed filters. F l u i d i z e d beds have been employed in many i n d u s t r i a l processes such as c o a l combustion, g a s i f i c a t i o n and l i q u e f a c t i o n , s o l i d residue p y r o l y s i s , c a t a l y t i c c r a c k i n g and reforming, and polymer production. In a d d i t i o n , the p o s s i b i l i t y of u s i n g fluidized beds f o r f i n e p a r t i c u l a t e removal has r e c i e v e d growing a t t e n t i o n over recent years (1 - 12). T y p i c a l l y , the fluidized bed is of the gass o l i d type and the p a r t i c u l a t e s may be l i q u i d or s o l i d a e r o s o l s . Note that in this a p p l i c a t i o n the bed medium s o l i d s f u n c t i o n as the c o l l e c t i n g medium and p a r t i c l e removal is accomplished through p a r t i c l e - c o l l e c t o r contacting. Our approach to the problem of p r e d i c t i n g the performance of fluidized bed f i l t e r s i n v o l v e s l o g i c a l l y c o u p l i n g models that des c r i b e the flow behavior of the fluidized s t a t e with models that d e s c r i b e the mechanisms of p a r t i c l e c o l l e c t i o n . The c o l l e c t i o n mechanisms a n a l y s i s leads to expressions f o r determining the c o l l e c t i o n e f f i c i e n c y of a s i n g l e f i l t e r element. An example of a c o l l e c t i o n mechanism is i n e r t i a l impaction by which a p a r t i c l e dev i a t e s from the gas stream l i n e s , due to i t s mass, and s t r i k e s a collector. I t should be noted that because p a r t i c l e c o l l e c t i o n mechanisms are f u n c t i o n s of the fluid flow behavior in the v i c i n i t y of a c o l l e c t o r , there e x i s t s an interdependency between fluidi z a t i o n mechanics and p a r t i c l e c o l l e c t i o n mechanisms. In a previous paper, the importance of fluidization mechanics on the performance of fluidized bed f i l t e r s was demonstrated (13). To accomplish this, c l a s s i c a l methods were employed f o r e v a l u a t i n g 0097-6156/81/0168-0075$05.00/0 © 1981 American Chemical Society In Chemical Reactors; Fogler, H.; ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

76

CHEMICAL REACTORS

the s i n g l e s p h e r i c a l c o l l e c t o r e f f i c i e n c i e s . In the present paper our a n a l y s i s is extended by c o n s i d e r i n g more r e a l i s t i c methods f o r estimating p a r t i c l e removal e f f i c i e n c i e s f o r a s i n g l e c o l l e c t o r element.


Model

Background

The model presented here f o r q u a n t i t a t i v e l y d e s c r i b i n g the mechanics of the fluidization process is a s i m p l i f i e d v e r s i o n of a more complex scheme r e c e n t l y proposed by Peters et a l (14), and is l a r g e l y based on bubble assemblage concepts (15). In b r i e f , the bubble assemblage concept considers an aggregative fluidized bed to be d i v i d e d a x i a l l y i n t o a number of compartments. Each com partment c o n s i s t s of a bubble, cloud, and emulsion phase. The s i z e of each compartment, which v a r i e s throughout the fluidized bed, is based on the cloud diameter computed at a given bed height. The key features of the present a n a l y s i s l i e in the r e d u c t i o n in inde pendencies among the r e l a t i o n s h i p s as w e l l as e l i m i n a t i o n of major two phase theory assumptions (14). Model Figure 1 shows the present model r e p r e s e n t a t i o n of the gassolid fluidized bed. Making a steady-state m a t e r i a l balance on p a r t i c u l a t e s over the n compartment r e s u l t s in the equation t n

U. S(C. is ι n-1

C. ) + F . , . _ . V(C.,, ι i(i+l) 1 l + l η η η J

Λ

1

-C. ) ι n n (1)

3 ( 1 - ε . ) V. +

F

( i -, Λ l ),i

„

V

. l (C.° i -_l n n v

v

Λ

- C., " i )' = η, " iC., " iÏÏ, is n n n n

2D

c

Where, i = 1 f o r the bubble phase, i = 2 f o r the cloud phase, and i = 3 f o r the emulsion phase. Note from the term on the r i g h t hand s i d e of Eqn. (1) that a f i r s t order r a t e equation f o r p a r t i c u l a t e c o l l e c t i o n is assumed (10). The i n l e t gas corresponds to the z e r o compartment, thus, t n

C_ 1 C

= C ο = C

9

2

ο

ο

ο

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

(2)


5.

PETERS ET A L .

Gas-Solid Fluidized Bed Filter

Outlet Gas Velocity : U Concentration : Com

CoutUo

0

Final ICompartmentl

]

*

C

c=D> Compartment

c Inlet Gas Velocity U Concentration: C :

0

up

0

Compartment Number Phase

] *

-X-TJ c

Bubble

Cloud

3o

Emulsion

Figure 1. Schematic of the present model


78

CHEMICAL REACTORS

E s t i m a t i o n of the Parameters of the Model As presented below, the parameters in the model may be e s t i mated in terms of a r e l a t i v e l y small number of fundamental parame t e r s that c h a r a c t e r i z e e i t h e r the bubbling phenomenon, mass con s e r v a t i o n , or p a r t i c u l a t e c o l l e c t i o n mechanisms. For those p a r a meters not based on average p r o p e r t i e s the s u b s c r i p t η has been omitted f o r c l a r i t y in many cases. A. S u p e r f i c i a l gas v e l o c i t y , U . The s u p e r f i c i a l gas v e l o c i t y can be expressed as


U

= ϋ

ο

+ ÏÏ + 2s

Ί

0

Is

ÏÏ

(3)

Q

3s

where U^ , υ , and U are based on average p r o p e r t i e s in the fluidized bed. _ B. S u p e r f i c i a l gas v e l o c i t y in the bubble phase, U^ « super f i c i a l gas v e l o c i t y in the bubble phase is r e l a t e d to the average l i n e a r bubble phase gas v e l o c i t y and the average bubble phase v o l ume f r a c t i o n by s

2 8

3 g

T

n

e

g

D

l e

- D T 1

ε

1

(4)

χ

where and 6^ are computed from the r e l a t i o n s h i p s given in sec t i o n s E. and Μ., r e s p e c t i v e l y . Note that Eqn. (4) represents the s o - c a l l e d v i s i b l e bubble flow r a t e . _ C. S u p e r f i c i a l gas v e l o c i t y in the cloud phase, · Since a bubble and i t s a s s o c i a t e d cloud r i s e together at the same l i n e a r v e l o c i t y , the s u p e r f i c i a l gas v e l o c i t y in the cloud phase is given by δ

-

ε

2 2 -r z. Is δ 1

2s

1

where ό / ό ι is given in s e c t i o n F. _ D. S u p e r f i c i a l gas v e l o c i t y in the emulsion phase, U3 . Substi t u t i n g Eqns. (4) and (5) i n t o Eqn. (3) gives the s u p e r f i c i a l gas v e l o c i t y in the emulsion phase, as 2

S

ÏÏ

3s

U

ÏÏ

= o "

e

1

*2 2>

( 6 )

subject to the s t i p u l a t i o n that U

> ϋ Ο

Ί

I

(Ε ζ-

+ 7 ε )

Λ

±

i

0

0

Δ

Δ

(7)

Ε. L i n e a r gas v e l o c i t y in the bubble phase, U]_. The l i n e a r gas v e l o c i t y in the bubble phase may be computed from the commonly ac cepted r e l a t i o n s h i p proposed by Davidson and H a r r i s o n (16).


5.

PETERS E T AL.

υ


79

= ( U - U ) + O.71 ^GD^

χ

Q

(8)

F

The average l i n e a r gas v e l o c i t y in the bubble phase may be expressed as

Û ,1 = ( Uο - UmfJ +

1

O.7IVGDI

1

(9) T l i e

v

u m e


F. Volume r a t i o of cloud t o bubble phases, 6 2 / ^ 1 · °l r a t i o o f the cloud phase to the bubble phase may be estimated from the model of Murray (17) δ

U -

9

ΊΓ δ

ΓΓ^-Π—

=

(

1

0

)

U

1

Snf V m f

and the average volume r a t i o may be expressed as ~6 -£

U -

0

=

δ1

J5Î

(11)

ε -U- - U mf 1 mf

G. Bubble Diameter, D-. A recent c o r r e l a t i o n by Mori and Wen (18), which c o n s i d e r s trie e f f e c t s o f bed diameters and d i s t r i b u t o r types, is u t i l i z e d . T h i s c o r r e l a t i o n , based on the bubble d i ameter data appearing in the l i t e r a t u r e p r i o r to 1974 is

D

i

-

D

l

m n

= exp (-O.3h/D_)

n

1 ~ m

(12)

R

1 ο

where D.

O.652 [S (U - U J ] o mf

2

/

5

(13)

m and S (U - U

D

= O.347 Ε ο

°

m

-) f

Ν

1

]

(14)

D

(for perforated d i s t r i b u t e r plates) D1

ο

= O.00376 (U - U J ο mr

2

( f o r porous d i s t r i b u t o r p l a t e s )


(15)

CHEMICAL REACTORS

80

This c o r r e l a t i o n is v a l i d over the f o l l o w i n g v a r i a b l e ranges: O.5 < U < 20 mf

, cm/s *

r

O.006 < D c U

ο

exp

-0 L - Lmf mf

ν

(25)

for L _ < h mf M. Volume f r a c t i o n of each phase, 6.. The volume f r a c t i o n of the bubble, cloud and emulsion phases may be computed as


CHEMICAL REACTORS

82

6

V

i

= i η

η

/ S D

(26)

2

η

where i = 1 f o r the bubble phase, i = 2 f o r the cloud phase, i = 3 f o r the emulsion phase, and V-

(27)

= Ν (1/6) πϋ-

mf ε -ϋ -U mf 1 mf

(28)

-V 2

(29)


Ί

V

= SD

3 η

2 η

V

V

η

- V l

η

Note t h a t , 6

χ

= V /SD 1

(30)

2

and = Ν (1/6) τ\Ώ

(31)

1

where Ν is evaluated a t h = l ^ f / 2 and is given by Eqn. (19). N. S i n g l e S p h e r i c a l C o l l e c t o r E f f i c i e n c i e s . Four c o l l e c t i o n mechanisms are considered in the present a n a l y s i s : i n e r t i a l im paction, i n t e r c e p t i o n , Brownian movement and Coulombic f o r c e s . A l though in our previous a n a l y s i s the e l e c t r i c a l forces were c o n s i d ered to be of the induced nature (13), there is evidence that it is the Coulombic forces which dominate the e l e c t r i c a l i n t e r a c t i o n s between the p a r t i c l e and c o l l e c t o r (7, 12^, 22) . Taking the net e f f e c t as the simple summation of each c o l l e c t i o n mechanism r e s u l t s in the s i n g l e s p h e r i c a l c o l l e c t o r e f f i c i e n c y equation, n

lMP

+

η

ΐΝΤ

+

n

BD

+

(32)

\

where n

r v m

J Mr

= -O.19133 + 1.7168 Stk

- 1.2665 Stk

+ O.31860 Stk f o r ε. = O.4 and Stk = O.12 ι c


(33)

5.

ΊΝΤ

n

η for ζ Downloaded by PURDUE UNIVERSITY on March 10, 2013 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch005


PETERS ET AL.

B D

Ε

= 1.5

1 31

= 4 ( i ^ i ) Pe i - 4.4 K ° -

83

(34)

Ν

2

/

3

(35)

8 7

(36)

c

= O.4

The p a r t i c l e d i f f u s i o n c o e f f i c i e n t is c a l c u l a t e d E i n s t e i n equation (24)

from the Stokes-

= -r-^ i l + — [1.257 + O.4 exp (-1.10 r /λ)]} (37) 6π μ r t r p ) Ρ Ρ Equations (33) - (35) are taken from Tardos et a l (23), and a r e based on a low Reynold's number a n a l y s i s . Eqn. (33) is the r e s u l t of a " b e s t - f i t " o f the t h e o r e t i c a l l y computed v a l u e s taken from Figure 7 of that same work. S i m i l a r l y , Eqn. (36) f o r the e l e c t r i c a l d e p o s i t i o n , is obtained from a " b e s t - f i t " o f the t h e o r e t i c a l l y computed values taken from Figure 3 of Tardos and P f e f f e r (21). Note that i f the p a r t i c l e and c o l l e c t o r charges are o f the same s i g n , the e l e c t r i c a l d e p o s i t i o n e f f i c i e n c y becomes the negative o f Eqn. (36). Consistent w i t h the flow f i e l d models used in the de velopment o f Eqns. (33) - (36), the v e l o c i t y employed is an assem b l y averaged v e l o c i t y f o r each phase. For the multi-phase s i t u a t i o n that e x i s t s in the fluidized bed, this is given by the super f i c i a l or empty-tower v e l o c i t y d i v i d e d by the phase volume f r a c tion, _ U. U. = (38) ι ο. η

r

G

1

Note from Eqn. (38) that s i n c e the volume f r a c t i o n o f each phase v a r i e s throughout the bed, so w i l l the assembly average v e l o c i t i e s and hence, the s i n g l e c o l l e c t o r e f f i c i e n c i e s . O. Volumetric average p a r t i c u l a t e c o n c e n t r a t i o n a t the e x i t o f the bed, C , and the o v e r a l l c o l l e c t i o n e f f i c i e n c y , X. The volumetric average p a r t i c u l a t e c o n c e n t r a t i o n a t the e x i t of the bed is expressed by o u t

ÏÏ

C

out »

C

ÏÏ

ÏÏ

ls 2s 3s l IT 2 ΪΓ +

Ο

C

+

Ο

C

3 IT

· o

(

3

9

)

and the o v e r a l l c o l l e c t i o n e f f i c i e n c y , in percent, is defined as


84

CHEMICAL REACTORS

C - C X = 100 ( ° ο


Q

U

t

)

(40)

At r a t i o s of s u p e r f i c i a l to minimum fluidization v e l o c i t i e s great er than three to f i v e , l o c a l flow r e v e r s a l of gas in the emulsion phase can occur (14). In the present a n a l y s i s the d i v i s i o n s of gas flow among the phases are based on average v a l u e s , and thus are taken to be constant throughout the fluidized bed. Equation (7) s t a t e s that only an average upward flow of gas in the emulsion phase is considered here. I t is assumed that the equations de s c r i b i n g the flow of gas in a fluidized bed are a l s o a p p l i c a b l e to the flow of p a r t i c u l a t e s , and that the p a r t i c u l a t e s c o n t a c t i n g a c o l l e c t o r adhere to it and are not r e - e n t r a i n e d by the gas flow. R e l a t i v e changes in p a r t i c l e v e l o c i t i e s due to the motion of the c o l l e c t o r s in the fluidized bed are neglected. Method of S o l u t i o n C a l c u l a t i o n s of the o v e r a l l c o l l e c t i o n e f f i c i e n c y f o r the fluidized bed f i l t e r begin w i t h s p e c i f i c a t i o n of the values of the s u p e r f i c i a l gas v e l o c i t y , U , minimum fluidization velocity, U p bed height at minimum fluidization, L £> v o i d f r a c t i o n at minimum fluidization, c f , column diameter, D , gas v i s c o s i t y , μ, c o l l e c tor diameter, D , d e n s i t y of p a r t i c u l a t e , p , and p a r t i c u l a t e d i ameter, D . There are no a d j u s t a b l e parameters in the present mo del. The charge acquired on both the p a r t i c u l a t e s and c o l l e c t o r s , Qp and Q^> r e s p e c t i v e l y , remain as e x p e r i m e n t a l l y determined in put parameters in the present a n a l y s i s . Because bubble diameter is a f u n c t i o n of the height from the d i s t r i b u t o r , and the height from the d i s t r i b u t o r is taken to the center of the bubble in question, an i t e r a t i v e procedure is used to determine Dj. The initial guess is taken to be the bubble d i a meter computed f o r the previous compartment. For each compartment there are three m a t e r i a l balance equations with three unknowns, the c o n c e n t r a t i o n s in each phase (bubble, cloud and emulsion). The t o t a l number of equations then is three times the t o t a l number of compartments. These may be solved by a matrix r e d u c t i o n scheme or a t r i a l and e r r o r procedure. The average s u p e r f i c i a l gas v e l o c i t i e s in each phase are f i r s t determined from Eqns. (4) - (6). The computational sequence f o r the remaining parameters in Eqn. (1) is given in Table 1. It is assumed here that the s i z e of the l a s t compartment is determined from the d i f f e r e n c e between the cummulative compart ments s i z e and the height of the expanded bed. However, f o r con s i s t e n c y , gas interchange c o e f f i c i e n t s and the l i n e a r bubble phase gas v e l o c i t y are based on a h y p o t h e t i c a l bubble diameter p r e d i c t e d from Eqn. (12). The computational scheme a l s o takes i n t o c o n s i d e r a t i o n the p o s s i b i l i t y of o n l y two phases in any compartment. This can r e s u l t from both c l o u d l e s s and cloud overlap compartments, Q

m

m

m

c

R

p

P


5.


PETERS ET AL.

Table I .

85

Computational sequence f o r parametric e v a l u a t i o n a t the n*-* compartment. Calculated Sequence Eqn. Number Parameter


1

1

12

D

l

2

8

U

l

3

10

4

22a

5

24, 25

2 ε

6

23

Ν

7

27, 28, 29

8

26

9

16, 17

10

38

11

32

δ /δ 2

D

1

Yr V

v

3

δ. 1

F , 1 2

F

Û , u 2

n

2 3

3

i

t y p i c a l l y o c c u r i n g f o r l a r g e r minimum fluidization velocities. Figure 2 shows a t y p i c a l s i t u a t i o n that can occur along with the app r o p r i a t e s i m p l i f i e d equations. Gas interchange in a two phase compartment is taken to be s o l e l y Eqn. (16), based on the s o - c a l l e d i n v i s i b l e bubble flow r a t e . The unsteady-state d i f f u s i o n a l c o n t r i b u t i o n , Eqn. (17), is neglected. Results and D i s c u s s i o n The potency of the present model l i e s in p r e d i c t i n g the p e r f o r mance o f fluidized bed f i l t e r s over a r e l a t i v e l y wide range o f ope r a t i n g c o n d i t i o n s . Our p r e v i o u s l y reported s e n s i t i v i t y s t u d i e s and comparisons with experimental r e s u l t s (13) a r e extended here. Comparisons with the Experimental Results of Tardos^et a l (12).

Figure 3 shows a comparison o f the model p r e d i c t i o n of the overa l l c o l l e c t i o n e f f i c i e n c y as a f u n c t i o n of s u p e r f i c i a l gas v e l o c i t y versus the experimental data o f Tardos et a l (12). Since the charge acquired on the c o l l e c t o r s was not reported, assumed values shown in Figure 3 were employed. I t should be noted that this a s sumed f u n c t i o n a l dependency between Q^. and U was not e n t i r e l y a r b i t r a r y , but q u a l i t a t i v e l y suggested by experimental measurements of the e l e c t r i c p o t e n t i a l in the fluidized bed (12). An important aspect of Figure 3 is both the model p r e d i c t i o n and experimental Q



86

CHEMICAL REACTORS

Umf

1

-ι *mfU, Umf

8, = 1-8, D =Uo-U

1 1

2s

h-

ls

»

83=1-83 U3s=Uo-U,

Umf

s

1

"I 1

Figure 2.

General Case

' ]

Bubble

1 I ι

Cloud

1

I

Emulsion 1 ι

Compartments representation of cloudless and cloud overlap compart ments



PETERS E T AL.


Lmf=9cm D =360/im D = ΙΟΙ μ.π\ Qp=l.3xlO"'°C/cm c

p

2

0

I 4

1 10

1 20

1 30

1 40

U ,cm/s 0

U ,cm/s 0

Figure 3. Comparison of ( ) model prediction and ( O ) experimental data (12) for the overall collection efficiency as a function of superficial gas velocity: Dp = 1.01 μπι


88

CHEMICAL REACTORS


observation of a maximum in the o v e r a l l c o l l e c t i o n e f f i c i e n c y as a f u n c t i o n of s u p e r f i c i a l gas v e l o c i t y . Model a n a l y s i s shows that this is due to the competing e f f e c t s of e l e c t r o s t a t i c c o l l e c t i o n and gas by-passing. In Figure 4, the same charge d i s t r i b u t i o n assumed in Figure 3 is employed f o r comparisons at a s l i g h t l y l a r g e r p a r t i c l e diameter. Model a n a l y s i s i n d i c a t e s that the higher e f f i c i e n c i e s observed in Figure 4 over Figure 3 are due s o l e l y to the higher p r e d i c t e d i n t e r c e p t i o n e f f i c i e n c i e s . Increases in the s i n gle c o l l e c t o r e f f i c i e n c i e s due to increases in the s p e c i f i c charge d e n s i t y outweigh gas by-passing e f f e c t s up to a s u p e r f i c i a l gas v e l o c i t y of about 18 cm/s in Figures 3 and 4. Comparisons w i t h the Experimental Results of Gutfinger Tardos (11).

and

In a d d i t i o n to the e f f e c t s of s u p e r f i c i a l gas v e l o c i t y on the o v e r a l l c o l l e c t i o n e f f i c i e n c y , the d i r e c t e f f e c t s of p a r t i c l e d i a meter are a l s o of importance. Figure 5 shows the present model p r e d i c t i o n s of the o v e r a l l c o l l e c t i o n e f f i c i e n c y as a f u n c t i o n of p a r t i c l e diameter compared to the experimental data of Gutfinger and Tardos (11). Since experimental care was taken to n e u t r a l i z e e l e c t r i c a l e f f e c t s f o r this system, these were not included in the model p r e d i c t i o n s . Thus, only three mechanisms were considered in Figure 5, namely, i n e r t i a l impaction, i n t e r c e p t i o n and Brownian motion. In Figure 5 reasonable agreement is seen at small p a r t i c l e diameters (< O.3 ym) where Brownian motion is prédominent, and at l a r g e p a r t i c l e diameters (> 3 ym) where i n t e r c e p t i o n e f f e c t s are controlling. In the v i c i n i t y of the minimum o v e r a l l c o l l e c t i o n e f f i c i e n c y (~ 1 ym) the agreement is not as good. I t is a l s o in this region that the p r e d i c t e d r e s u l t s are very s e n s i t i v e to the values of the s i n g l e c o l l e c t o r e f f i c i e n c i e s . In Figure 5 the experimenta l data would i n d i c a t e higher s i n g l e c o l l e c t o r e f f i c i e n c i e s in the v i c i n i t y of the minimum than p r e d i c t e d by the equations employed here. For completeness it should be noted that the minimum o v e r a l l c o l l e c t i o n e f f i c i e n c i e s in Figures 3 and 4 occur f o r p a r t i c l e d i a meters l e s s than O.5 ym. Thus, the p a r t i c l e diameters employed in Figures 3 and 4 are s u f f i c i e n t l y d i s p l a c e d from the minimum so that the r e s u l t s are not considered f o r t u i t o u s .

and

Comparisons w i t h the M u l t i s t a g e E f f i c i e n c i e s of Patterson Jackson (8).

For h i g h l y r e a c t i v e systems in which the m a j o r i t y of p a r t i c u l a t e c o l l e c t i o n in the emulsion phase occurs in a r e l a t i v e l y short d i s t a n c e from the d i s t r i b u t o r p l a t e , m u l t i s t a g e fluidized beds have been employed (8, 4·) . Because of the general formulation of the present model, it may be employed f o r determining m u l t i s t a g e fluidized bed f i l t r a t i o n e f f i c i e n c i e s . This i n c l u d e s a v a r i a t i o n in the c h a r a c t e r i s t i c s of each stage such as bed depth and c o l l e c t o r size.


5.

PETERS ET AL.


89


Umr II cm/s L =9cm mf

D =360/im c

Dp = 2/i.m Q =l.3xlO C/cm p

20

_,0

2

30

U ,cm/s 0

Figure 4. Comparison of ( ) model prediction and (O) experimental data (12) for the overall collection efficiency as a function of superficial gas velocity: Op = 2 μτη

Op, μη Figure 5. Comparison of ( ) model prediction and (O) experimental data (11) for the overall collection efficiency as a function of particle diameter



CHEMICAL REACTORS

100 r

Q^ôxIO-^C/cm Qp=l.3xlO C/cm € = O.5 Umf = 1.5cm/s Uo=3.45cm/s D = 15.25cm D = O.0135cm 2

80

HO

2

mf

R

:rSr:.

c

Model Prediction Dp = 1.0 μπ\ Dp =O.67^m Dp =O.50/im 20 h

Experimental (Patterson and Jackson, 1977) Ο Dp = 1.0 μ(Τ\ • D =O.67 tm AD =O.50^m p

A

p

Second

Stage

Figure 6. Comparison of the predicted multistage efficiencies and the experimental results (8j. Model prediction: ( ) D = 1.0 μ*η; ( ) Ό = O.67 fim; (· · -) Dp = O.50 pm. Experimental: (Ο) Ό = 1.0 μ/η; (Π) D = O.67 m; (Α) Ό = O.50 μ-m. p

ρ

ρ

p

M


ρ

5. PETERS E T A L .

91


Under the s i m p l i f i c a t i o n s that each stage has i d e n t i c a l char a c t e r i s t i c s and that the p a r t i c u l a t e s are of a s i n g l e s i z e , model p r e d i c t i o n s o f the s i n g l e stage e f f i c i e n c i e s may be d i r e c t l y used to c a l c u l a t e m u l t i s t a g e e f f i c i e n c i e s by (8)

*M=

(41)

In F i g . 6 the present model p r e d i c t i o n s o f the m u l t i s t a g e e f f i c i e n c i e s c a l c u l a t e d from Eqn. (41) a r e shown to compare c l o s e l y to the experimental data of Patterson and Jackson ( 8 ) . Because of the im portance o f e l e c t r i c a l e f f e c t s noted f o r this system, (12) the Coulombic f o r c e term in Eqn. (32) was i n c l u d e d . Values of Qp and ^AC a r b i t r a r i l y set as shown in F i g u r e 5. I t should be noted that along with pressure drop information the present model may be used f o r o p t i m i z i n g the depths of each stage in a m u l t i s t a g e f l u i d i z e d bed f i l t e r . w


M

100[l-d-3^) ]

e

r

e

Conclusion In the present paper our previous a n a l y s i s o f fluidized bed f i l t r a t i o n e f f i c i e n c i e s has been extended by c o n s i d e r i n g more r e a l i s t i c methods f o r e s t i m a t i n g the s i n g l e c o l l e c t o r e f f i c i e n c i e s as w e l l as more r e c e n t l y reported experimental r e s u l t s . In general the p r e d i c t e d values o f the fluidized bed f i l t r a t i o n e f f i c i e n c i e s compare f a v o r a b l y to the experimental v a l u e s . For e l e c t r i c a l l y ac t i v e fluidized beds, d i r e c t measurements of the p a r t i c l e and c o l l e c t o r charges would be necessary t o s u b s t a n t i a t e the r e s u l t s g i v en here. The present model appears t o be u s e f u l in the design of fluid i z e d bed f i l t e r s . I t does not address questions concerning the q u a l i t y of fluidization, s t i c k i n e s s o f the p a r t i c l e s , s o l i d s regen e r a t i o n r a t e s and agglomeration e f f e c t s . In order t o optimize the fluidized bed f i l t e r these e f f e c t s must be considered in conjunc t i o n w i t h those aspects to the problem e l u c i d a t e d here. Legend o f Symbols th C.

=

c o n c e n t r a t i o n o f p a r t i c l e s in η g/cm^

=

i n l e t p a r t i c l e c o n c e n t r a t i o n , g/cm

=

o u t l e t p a r t i c l e c o n c e n t r a t i o n , g/cm^

=

c o l l e c t o r diameter, cm 2 molecular d i f f u s i o n c o e f f i c i e n t of p a r t i c u l a t e , cm /s p a r t i c u l a t e diameter, cm fluid bed diameter, cm

η

compartment in phase i ,

3 C

q

C

Q

D

U

T

C

Dç Dp D κ

= = =


92

CHEMICAL REACTORS

=

equivalent s p h e r i c a l bubble diameter having the same v o l ume as that of a bubble, cm

=

equivalent s p h e r i c a l cloud diameter, cm

=

average e q u i v a l e n t s p h e r i c a l bubble diameter, cm

=

average equivalent s p h e r i c a l cloud diameter, cm

=

initial


= F

F

bubble diameter, cm

maximum bubble diameter, cm

12

=

g

a

s

i

n

t

e

r

c

n

a

n

e

23

=

g

a

s

i

n

t

e

r

c

n

a

n

e

g c o e f f i c i e n t between phase 1 and phase 2 per u n i t volume of phase 1, 1/s g c o e f f i c i e n t between phase 2 and phase 3 per u n i t volume of phase 1, 1/s 2

G

=

gravitational acceleration,

cm/s

h

=

height from d i s t r i b u t o r p l a t e ,

Κ Kç

= =

Boltzman's Constant, 1.38 χ 1 0 ~ erg/molecule °K dimensionless c h a r a c t e r i s t i c p a r t i c l e m o b i l i t y f o r Coulombic force,

cm 1 6

D

Q

P AC

Q

P

3wÛ.

ε

£

L

=

expanded bed height, cm

L^

=

bed height at U ^ ,

Ν

=

number of bubbles in a compartment

Ν

=

average number of bubbles in a compartment

=

number of o r i f i c e openings on the d i s t r i b u t o r

Pe

=

Q

= =

P e c l e t number, U. D /Ό„ ι c G dimensionless i n t e r c e p t i o n parameter, P D^/D C 2 · assumed of opposite charge on c o l l e c t o r , C/cm^, signs throughout charge on p a r t i c l e , C/cm this work p a r t i c l e r a d i u s , cm 2 c r o s s s e c t i o n a l area of bed, cm'

A

Qp

=

r^

=

S

=

Stk

=

Stk

c

=

1

Stoke s number 1 9

cm

^ U. p p ι Ρ uD 2

D

Ρ

c

1

critical S t o k e s number, below which there can be no c o l l e c t i o n by i n e r t i a l impaction


5.

PETERS E T A L .

u\


93

=

assembly averaged v e l o c i t y of gas in phase i , cm/s

=

average l i n e a r gas v e l o c i t y in bubble phase,

=

average s u p e r f i c i a l v e l o c i t y of gas in phase i , cm/s

=

minimum

= =

s u p e r f i c i a l gas v e l o c i t y , cm/s th volume of phase i in η compartment,

=

average volume of bubble phase,

X

=

overall collection efficiency

X^

=

overall collection efficiency for M

u\

g

U ^ V^

n

fluidization

velocity,

cm/s

cm/s

cm

3

cm^

(%)


th Greek

stage (%)

Symbols =

void

ε mr ε^

=

void

=

permittivity

6^

=

volume f r a c t i o n o f bed occupied by phase i

=

average bubble phase volume

=

single

c

η__. η

η

=

f r a c t i o n of gas in phase i f r a c t i o n in bed at U

ΐΜΡ

s

n

*- êl

e

ΐΝΤ

=

s

n

e

n

spherical

*- 8l

2

2

C /dyne - cm

fraction

c o l l e c t o r e f f i c i e n c y in phase i

single spherical

=

f

mf of free space, 8.85 χ 10

c o l l e c t o r e f f i c i e n c y f o r Brownian motion

spherical

c o l l e c t o r e f f i c i e n c y f o r impaction

spherical

collector efficiency for interception

=

single

spherical

c o l l e c t o r e f f i c i e n c y f o r Coulombic f o r c e s

=

p a r t i c l e density,

=

gas d e n s i t y ,

μ

=

gas v i s c o s i t y , g/cm-s

λ

=

mean free path of gas, ~ 6.5 χ 10 ^ cm f o r a i r at 20°C

3

E

p p

Ρ

g/cm

g/cm 3

Acknowledgement s The authors wish to acknowledge G. Tardos and R. P f e f f e r f o r t h e i r h e l p f u l comments during the course of this work. L. - S.F. was f i n a n c i a l l y supported in part by the B a t t e l l e Memorial I n s t i t u t e under the U n i v e r s i t y D i s t r i b u t i o n Program. Literature 1. 2.

Cited

K n e t t i g , P.; Beeckmans, J.M. Can. J. Chem. Eng. 1974, 52, 703. Tardos, G.; Gutfinger, C.; Abuaf, N. I s r a e l J. Tech. 1974, 12, 184.


CHEMICAL

94

3. 4. 5. 6. 7. 8. 9.


10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

REACTORS

McCarthy, D.; Yankel, A.J.; P a t t e r s o n , R.G.; Jackson, M.L. Ind. Eng. Chem. Proc. Des. Dev. 1976, 15, 266. Svrcek, W.Y.; Beeckmans, J.M. Tappi 1976, 59, 79. Tardos, G.; G u t f i n g e r , C.; Abuaf, N. AIChE J. 1976, 22, 1147. Zahedi, K.; Melcher, J.R. J. A i r Poll. Cont. Ass. 1976, 26, 345. Ciborows, J.; Zakowski, L. I n t . Chem. Eng. I . 1977, 17, 529. P a t t e r s o n , R.G.; Jackson, M.L. AIChE Symp. Ser. No. 161 1977. Zahedi, K.; Melcher, J.R. Ind. Eng. Chem. Fund 1977, 16, 248. Doganoglu, Y.; Jog, V.; Thambimuthu, D.V.; Clift, R. Trans. Int. Chem. Eng. 1978, 56, 239. G u t f i n g e r , C.; Tardos, G.I. Atm. Env. 1979, 13, 853. Tardos, G.; G u t f i n g e r , C.; P f e f f e r , R. Ind. Eng. Chem. Fun. 1979, 18, 433. P e t e r s , M.H.; Fan, L.-S.; Sweeney, T.L. AIChE J. 1981, ( i n press). P e t e r s , M.H.; Fan, L.-S.; Sweeney, T.L. Reactant Dynamics in C a t a l y t i c F l u i d i z e d Bed Reactors with Flow Reversal of Gas in the Emulsion Phase, presented at 1980 AIChE Meeting, Chicago, Ill. Kato, K.; Wen, C.Y. Chem. Eng. Sci. 1969, 24, 1351. Davidson, J.F.; H a r r i s o n , D., " F l u i d i z e d P a r t i c l e s , " 1963, Cambridge U n i v e r s i t y Press. Murray, J.D. J. F l u i d Mech. 1965, 21, 465. Mori, S.; Wen, C.Y. AIChE J. 1975, 21, 109. K u n i i , D.; L e v e n s p i e l , O. Ind. Eng. Chem. Fund. 1968, 7, 446. L o c k e t t , M.J.; Davidson, J.F.; H a r r i s o n , D. Chem. Eng. Sci. 1967, 22, 1059. Chavarie, C.; Grace, J.R. Chem. Eng. Sci. 1976, 31, 741. Tardos, G.I.; P f e f f e r , R. Proc. 2nd World Filt. Cong. Sept. 18-20, 1979, London, U.K. Tardos, G.I.; Yu, E.; P f e f f e r , R.; Squires, M. J. C o l l . I n t . Sci. 1979, 71, 616. F r i e d l a n d e r , S.K. J. C o l l . I n t . Sci. 1967, 23, 157.

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


Chemical Reactors - American Chemical Society

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