Cell Model Studies of Radial Flow, Fixed Bed Reactors

45 Cell Model Studies of Radial Flow, Fixed Bed Reactors J. M .

CALO

Downloaded by UCSF LIB CKM RSCS MGMT on November 18, 2014 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch045

Department of Chemical Engineering, Princeton University, Princeton, NJ 08540

The radial flow, fixed bed reactor (RFBR) was originally de veloped to handle large gas flow rates in the catalytic synthesis of ammonia. Since then, RFBRs have been used, or considered for, catalytic reforming, desulfurization, nitric oxide conversion, catalytic mufflers, and other processes in which fluids must be contacted with solid particles at high space velocities. The prin cipal advantages of the RFBR over the more conventional tubular, axial flow, fixed bed reactor (AFBR) have been outlined elsewhere (1,2). Although usually cylindrical in geometry, spherical RFBRs have also been considered (3). Perhaps the first published analysis of an RFBR was by Raskin, et al. (4), who developed a quasicontinuum distributed parameter model for a radial ammonia synthesis reactor. General conclusions were limited, however, since the model was specifically concerned with ammonia synthesis and later carbon monoxide conversion (5), where both processes are second order and reversible. However, these authors did note that "radial reactors are anisotropic", i.e., they observed higher ammonia yields for centripetal radial flow (CPRF -- periphery to the center) than for centrifugal r a d i a l flow (CFRF -- center to periphery) (4) . In the present work, a packed bed c e l l model i s used to calcu late temperature and concentration profiles i n the adiabatic RFBR for exothermic c a t a l y t i c reactions with interphase resistance to mass and heat transfer. In p a r t i c u l a r , differences between the RFBR and the AFBR, operated at the same space v e l o c i t y , are explor ed with respect to uniqueness, m u l t i p l i c i t y , and s t a b i l i t y of the steady state, p r o f i l e location, s e l e c t i v i t y i n p a r a l l e l and series reactions, and transient behavior. RFBR C e l l Model Following Vanderveen, et a l . (6), the mass and energy conser vation equations for the f l u i d and the p a r t i c l e i n the j t h c e l l for a f i r s t order, i r r e v e r s i b l e , exothermic reaction are M.(v. . - v.) - (v. - v.) =a fdv./dt) D 3-1 Ύ 3 Τ 1 1 J

© 0-8412-0401-2/78/47-065-550$05.00/0

In Chemical Reaction Engineering—Houston; Weekman, V., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

[1]

45. CALO

Radial Flow, Fixed Bed Reactors

551

( V J - v.) - k.v. = a ( d v . / d t )

C3]

3

(y. - y . l + 3k .v. = a.fdy./dt) [4] 3 3' 3 3 4 3 s u b j e c t t o the i n l e t c o n d i t i o n s VQ = VQ(t) = 1.0 and yo = y o ( t ) = 1.0 and 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 (see N o t a t i o n and Vanderveen, e t a l . (β) f o r parameter d e f i n i t i o n s ) . In r a d i a l f l o w , the f l u i d i n t e r s t i t i a l v e l o c i t y , u j , i s a f u n c t i o n o f p o s i t i o n i n the bed, and hence, c e l l number, j . The interphase mass and heat t r a n s f e r c o e f f i c i e n t s vary approximately as R e (£, 1_) , and thus Mj and Hj are f u n c t i o n s o f Re® ·** which increase i n CPRF and decrease i n CFRF w i t h bed depth, k j i s a l s o a f u n c t i o n o f U j due t o i t s dependence on k g . With the assumption o f constant mean f l u i d d e n s i t y throughout the bed (_5) , the c o n t i n u i t y equation f o r c y l i n d r i c a l geometry i n t e grates t o ur = U^Ri = U2R2; w h i l e f o r s p h e r i c a l geometry, u r = 1 1 = U2 2 I n t e g r a t i o n o f the v e l o c i t y , u = ±dr/dt (+ f o r CFRF and - f o r CPRF), from the bed entrance t o the bed e x i t y i e l d s the space time d i s t r i b u t i o n along the bed l e n g t h . R e s u l t s o f t h i s c a l c u l a t i o n r e v e a l t h a t i n r a d i a l flow the l o c a l residence time near the o u t e r p e r i p h e r y o f the bed i s increased a t the expense o f residence time near the i n n e r core o f the bed as compared to a x i a l flow a t the same t o t a l space time. The steady s t a t e temperature p r o f i l e s presented i n F i g u r e 1 were determined by an i n i t i a l value c e l l - b y - c e l l c a l c u l a t i o n f o r the same feed c o n d i t i o n s , average space v e l o c i t y , and bed depth. A l l the parameter v a l u e s are those o f Vanderveen, e t a l . (6) ex cept as otherwise noted. As can be seen, the CPRF p r o f i l e appears e a r l i e r , and the CFRF p r o f i l e appears l a t e r i n the bed than the ax i a l flow p r o f i l e as a r e s u l t o f residence time reapportionment. This " e a r l y - l a t e " phenomenon i s accentuated by d e c r e a s i n g aspect r a t i o and by s p h e r i c a l over c y l i n d r i c a l geometry a t the same aspect ratio. Uniqueness, M u l t i p l i c i t y , and S t a b i l i t y


V

0 , 6

U

R

2

R

2

A l g e b r a i c m a n i p u l a t i o n o f the steady s t a t e forms o f the con s e r v a t i o n equations [104] i n the same manner as E r v i n and Luss (8) yields y. * y- -, = (M.+l)k.(y . - y.)/M. = G. (y.) 3 3-1 3 3 m3 3 3 3 3 Κ

J

,

[5]

Κ

f

where y j i s the maximum p a r t i c l e temperature i n the j t h c e l l (8}. Equation [5] can be r e w r i t t e n as m #

1 = G.CyJ/ly. - y.^)

Ξ F. (y.)

[6]

and the steady s t a t e s o f the system are determined by the i n t e r s e c t i o n o f t h e h o r i z o n t a l l i n e a t u n i t y and the f u n c t i o n F j ( y j ) . For s m a l l y j , d F j ( y j ) / d y j i s n e g a t i v e , and i f A



552

CHEMICAL REACTION ENGINEERING—HOUSTON

0

10

20

30

40

50

60

70

80

90

100

C e l l Number, j

Figure 1. Unique steady state temperature profiles for CPRF and CFRF ( and axialflow(P = 9kP , T — 667 K , T = 667 Κ,β = 5, M/H = 1.67)

P

0

a

0

0


=

0.25)

45.

CALO


dF. (y.)/dy. < 0 3 3

,

j

J

J

y.e(y

3

2

m,3

KJ

553 ., y. _)

[7]

3~1

J

then t h i s i s a s u f f i c i e n t c o n d i t i o n f o r uniqueness of the steady s t a t e . I f c o n d i t i o n [7] i s v i o l a t e d , then F j ( y j ) w i l l e x h i b i t extremum a t


y-j

= ΙΎΙΥ_ ^

1

j [ ^'ί

where

Q

=

Ύ2

y4_J

+

+Y

j-1

±

) 2 +

ôJ/ [y^ + Ύ - y^_J 2

4Yy

j-l m,ji j-l 5

y

"Y

w

" Vj)]*

In order t o i n s u r e c o n d i t i o n [ 7 ] , the argument of Qj must be nega tive , or Inf 4 l

1

>

> F

/ t

t

F


r

554


a x i a l flow because the values o f Aj and k ( i n k j ) a t the t r i f u r c a t i o n p o i n t are not known. Nevertheless, i t can be shown t h a t the lowest i n t e r s t i t i a l v e l o c i t y i n the bed, i . e . , a t the outer p e r i p h e r y , y i e l d s an upper bound i n e v a l u a t i n g j ( y j , t ) - Thus, i f j ^ j # t ) u . b . I* then t h i s i s a s u f f i c i e n t , but not necessary con d i t i o n f o r uniqueness o f the steady s t a t e . The d i f f e r e n c e i n behavior between the F j ( y j ) curves f o r CPRF and CFRF i n F i g u r e 2 i s due t o the s c a l e f a c t o r e f f e c t o f (Mj+1)/ Mj and kg ( i n k j ) . These velocity-dependent terms are l a r g e r f o r CPRF and s m a l l e r f o r CFRF i n the i n i t i a l p o r t i o n o f the bed, and they cause the F j ( y j ) curves t o r i s e f a s t e r from c e l l t o c e l l i n CPRF. Therefore, i f m u l t i p l e steady s t a t e s e x i s t , the region o f m u l t i p l i c i t y w i l l be s h i f t e d toward the bed entrance i n CPRF and toward the bed e x i t i n CFRF. This e f f e c t can be b e t t e r appreciated i n F i g u r e 3, which i s a p l o t o f F j (y-j+) and F j (yj-) as a f u n c t i o n of j f o r the same c o n d i t i o n s as i n Figure 2. As shown, the Fj(yj+) and F j ( y j - ) branches cross u n i t y and coalesce a t the t r i f u r c a t i o n p o i n t f i r s t f o r CPRF, next f o r a x i a l f l o w , and l a s t f o r CFRF. The open i n t e r v a l [ F j ( y j + ) , j ( y j ~ ) 3 i s measure o f the pa rameter space f o r which m u l t i p l i c i t y can occur. CPRF presents a l a r g e r i n t e r v a l and CFRF a s m a l l e r i n t e r v a l than a x i a l flow a t the same average space v e l o c i t y . Thus, i n g e n e r a l , CFRF suppresses m u l t i p l i c i t y and CPRF promotes m u l t i p l i c i t y as compared t o a x i a l flow a t the same average space v e l o c i t y . I n a l l the c a l c u l a t i o n s performed, however, CPRF never induced m u l t i p l i c i t y nor d i d CFRF completely suppress m u l t i p l i c i t y when compared t o the " e q u i v a l e n t " a x i a l flow case. On the other hand, the p o s s i b i l i t y t h a t condi t i o n s may e x i s t under which t h i s could occur has not been e l i m i nated. I t should be noted t h a t Hlavacek and Kubicek (1) concluded t h a t CPRF, and not CFRF, tends t o suppress m u l t i p l e s o l u t i o n s f o r the quasicontinuum a x i a l d i s p e r s i o n model without interphase t r a n s port resistance. A s t a b i l i t y a n a l y s i s o f the t r a n s i e n t equations [1-4] y i e l d s the same necessary and s u f f i c i e n t s e t o f c o n d i t i o n s f o r asymptotic s t a b i l i t y as obtained by Vanderveen, e t a l . {6), v i z . g

F


F

Ul

0

;b

D2

>

0

a

2

2

; b. b H > b _b. . + b _ ; b > D l J2 33 3I J4 33 34

0

,

[11]

except, o f course, t h a t f o r r a d i a l f l o w , M and H are f u n c t i o n s o f j . The f o u r t h c o n d i t i o n , b j 4 > 0 , i s simply c o n d i t i o n [ 7 ] , i . e . , d F j ( y j ) / d y j < 0 . Thus a l l steady s t a t e s on the p o s i t i v e slope branch o f F j ( y j ) are unstable w i t h respect t o s m a l l p e r t u r b a t i o n s . Steady s t a t e s on the h i g h and low temperature branches w i t h nega t i v e slope have a chance o f being s t a b l e i f i n a d d i t i o n they s a t i s fy the f i r s t three c o n d i t i o n s i n [ 1 1 ] . For g a s - s o l i d systems w i t h a i 4 / a 3 » 1, the f i r s t two c o n d i t i o n s i n [11] imply the slope c o n d i t i o n (§_ §_,9) . C o n d i t i o n [7] i s stronger than the slope c o n d i t i o n i f f

(l V

+ H . ) M . / ( H . ( M . + δ.)) 3 3 3 3 3 '

> 1

,


[12]



CALO

Figure 2. Fffîj) curves as a function of cell number for CPRF and CFRF ( = 0.5). (P = 9kP , T = 667°K, T = 667% β = 7.5, M/H = 1.1183). The curves are labeled with cell numbers. P

0

a

0

0




C e 1 1 Numbe r ,

j

Figure 3. F/$+) and F/5,—) as a function of cell number for CPRF, CFRF, and axial flow for the same conditions as in Figure 3. The CPRF and CFRF curves are dashed to F/5i,t)„.. since the Fj(yjj) are not known. &



45.

CALO

557


which i s always s a t i s f i e d f o r M/H > l,and thus c o n d i t i o n [7] im p l i e s the f i r s t , second, and f o u r t h c o n d i t i o n s i n [11]. For M/H < 1 and 6 j - 1, i n e q u a l i t y [12] i s not s a t i s f i e d , and the slope c o n d i t i o n w i l l be s t r o n g e r . In any case, f o r a steady s t a t e on the negative slope branches o f F j ( y j ) t o be stable., the t h i r d con d i t i o n i n [11] must a l s o be s a t i s f i e d , and as has been p o i n t e d out (§_r§) ι t h i s c o n d i t i o n i s a complex expression w i t h no obvious p h y s i c a l meaning. A comparison o f r e l a t i v e magnitudes o f c o n d i t i o n [12] f o r M/H > 1 r e v e a l s t h a t r a d i a l flow tends t o d e s t a b i l i z e the bed a t the outer p e r i p h e r y and s t a b i l i z e the bed a t the i n n e r core t o a g r e a t e r extent than a x i a l flow a t the same t o t a l space v e l o c i t y . Selectivity Effects The reapportionment o f residence time i n r a d i a l flow gous t o v a r i a b l e volume CSTRs i n s e r i e s . As such, r a d i a l would be expected t o a f f e c t product s e l e c t i v i t y i n s e r i e s l l e l r e a c t i o n s . For two p a r a l l e l f i r s t o r d e r , exothermic the p a r t i c l e equations [3, 4] become (v. - v.) - (k . + k .)v. = 0 3 3 ID 23 3 K

J

K

[13]

j

[14]

(Yj " Y j ) + ( 3 ^ 1 . + 3 k ) v . = 0 2

i s analo flow and para reactions,

2 j

The r e s u l t s o f a t y p i c a l c a l c u l a t i o n are presented i n Figure 4. For the parameters chosen (see f i g u r e c a p t i o n ) , CPRF i n c r e a s e d the y i e l d o f R and decreased the y i e l d o f S, w h i l e CFRF acted i n the opposite sense. Thus, i n t h i s case, the y i e l d o f R v a r i e d 6.2% and t h a t o f S 6.7% simply according t o whether the r e a c t o r was operated i n CPRF o r CFRF. For two s e r i e s f i r s t order, exothermic r e a c t i o n s , the p a r t i c l e mass balance i s the same as Equation [ 3 ] , but the p a r t i c l e energy balance (Eq. [4]) becomes y

j "h

+

'ΛΛ

+

2 2?j -

&

k

0

-

[15]

where Wj i s the f r a c t i o n o f i n t e r m e d i a t e . A l s o , a d d i t i o n a l p a r t i c l e and f l u i d mass balances are r e q u i r e d f o r the i n t e r m e d i a t e . The r e s u l t s o f a t y p i c a l c a l c u l a t i o n f o r t h i s case are presented i n F i g u r e 5. As compared t o a x i a l f l o w , and f o r the parameter values chosen (see f i g u r e c a p t i o n ) , CPRF i n c r e a s e d the y i e l d o f product S and decreased the y i e l d o f intermediate R, w h i l e CFRF again acted i n the opposite sense. The y i e l d o f S v a r i e d 9.5% and t h a t o f R 77.3% simply a c c o r d i n g t o flow d i r e c t i o n i n c y l i n d r i c a l r a d i a l flow. A l s o , when the crossover p o i n t f o r R and S i s near the r e a c t o r e x i t i n a x i a l f l o w , CPRF can y i e l d a product w i t h S > R, w h i l e CFRF y i e l d s a product w i t h R > S. This occurs when the crossover p o i n t moves toward the bed entrance i n CPRF, i n c r e a s i n g S over R, and moves out o f the r e a c t o r e x i t i n CFRF, thereby i n c r e a s i n g R over S. Of course, d i f f e r e n c e s i n s e l e c t i v i t y due t o flow d i r e c t i o n i n an RFBR are extremely s e n s i t i v e t o the s p e c i f i c k i n e t i c parameters.




558


45. CALO


559

However, i t i s c o n c e i v a b l e , depending on the r e a c t i o n system, t h a t s e l e c t i v i t y e f f e c t s i n r a d i a l flow could be s i g n i f i c a n t .


T r a n s i e n t Behavior When m u l t i p l e steady s t a t e s e x i s t , steady s t a t e p r o f i l e s can be determined o n l y by a t r a n s i e n t a n a l y s i s . T y p i c a l r e s u l t s o f the numerical s o l u t i o n o f the t r a n s i e n t simple c e l l equations [1-4] are presented i n F i g u r e 6. Here f l u i d temperature p r o f i l e s are presented f o r CPRF and CFRF f o r c y l i n d r i c a l geometry (p = 0.25), and a x i a l flow f o r the same average space v e l o c i t y a t 5, 15, and 30 minutes a f t e r s t a r t up. The feed and i n i t i a l c o n d i t i o n s are noted i n the f i g u r e c a p t i o n . S e v e r a l i n t e r e s t i n g features are apparent. The i n c i p i e n t p r o f i l e s a t 5 minutes are much c l o s e r t o gether than a t 30 minutes. I n each case the r e a c t i o n zone i s f o r med w i t h i n 15 minutes, and the p r o f i l e then creeps s t e a d i l y toward the bed e x i t . I t i s q u i t e e v i d e n t t h a t the CFRF p r o f i l e creeps the most, and the CPRF p r o f i l e the l e a s t . I t i s a l s o i n t e r e s t i n g to note t h a t a l l the CFRF and a x i a l flow p r o f i l e s s i g n i f i c a n t l y overshoot the maximum a d i a b a t i c f l u i d temperature, y , w h i l e the CPRF p r o f i l e a l s o does but t o an almost i m p e r c e p t i b l e e x t e n t . The creep behavior i s d i r e c t l y r e l a t e d t o the degree o f temperature overshoot, s i n c e i t i s caused by c o o l i n g o f the over-temperature c e l l s t o l e s s than o r equal t o y , which, o f course, cannot be ex ceeded a t steady s t a t e . Thus i n F i g u r e 6, the a x i a l flow and CFRF p r o f i l e s continue t o creep u n t i l the c e l l s near the bed e x i t c o o l . The degree o f overshoot, and hence, p r o f i l e creep, i s a func t i o n o f the i n i t i a l bed temperature, y-jo- study was conducted i n which the i n i t i a l bed temçerature was v a r i e d keeping a l l the other parameters constant. As yjQ was increased from 1.0 t o 1.17, t h e p r o f i l e s f o r a l l three flow modes i g n i t e d c l o s e r t o the bed en t r a n c e . A l l the p r o f i l e s were s t a t i o n a r y a t 30 minutes from s t a r t up except f o r the CFRF p r o f i l e f o r yjQ = 1.17, which overshot the maximum a d i a b a t i c f l u i d temperature and was s t i l l c r e e p i n g . A l s o , for yjQ = 1.17 the CPRF and a x i a l f l o w p r o f i l e s were the c l o s e s t together, w h i l e the CFRF was f a r t h e s t from the other two and mov ing away. Thus, f o r i n i t i a l bed temperatures exceeding the feed temperature, steady s t a t e m u l t i p l i c i t y tends t o accentuate the " e a r l y - l a t e " r e a c t i o n zone phenomenon observed f o r the unique steady s t a t e p r o f i l e s . m

m

A

Conclusions In g e n e r a l , CPRF i s the b e t t e r mode o f o p e r a t i o n f o r the RFBR when m u l t i p l e steady s t a t e s are p o s s i b l e . I n a d d i t i o n t o e x p e d i t ing the approach t o steady s t a t e , the establishment o f the CPRF p r o f i l e e a r l y i n the bed i s an advantage i n cases where the r e a c t i o n zone g r a d u a l l y creeps toward the bed e x i t due t o c a t a l y s t de a c t i v a t i o n . When o n l y unique steady s t a t e s are p o s s i b l e and/or f o i p a r a l l e l and s e r i e s r e a c t i o n s , the p r e f e r r e d flow mode may depend on other c o n s i d e r a t i o n s .




Figure 6. Transient behavior of temperature profiles in ÇPRF, CFRF (p = 0.25), and axial flow ( = 0.25, P = HkP , T = 667°K, T = 883°K, β = 5 M/H = 1.67) P

0

a

G

0


45.

CALO


561

Notation [see a l s o Vanderveen, e t a l . (6)]


Aj Fj Gj Hj

= = = =

defined i n t e x t Oj d e f i n e d by Eq. [6] R r d e f i n e d by Eq. [5] U,u dimensionless HTU f o r heat t r a n s f e r v,w k j = dimensionless r a t e constant y Mj = dimensionless HTU f o r mass transfer f

3 = (-AH)k /hfT γ = (-ΔΕ)/RgT g

0

0

= defined i n text = radius = i n t e r s t i t i a l f l u i d ve locity = r e a c t a n t mole f r a c t i o n •» T / T Q (dimensionless temperature)

oj = k j / ( l + k j ) ρ = Ri/R

2

S u b s c r i p t s and S u p e r s c r i p t s j m η 0

= c e l l number = maximum = last cell = feed c o n d i t i o n s

= p a r t i c l e phase 1 = inner radius 2 = outer r a d i u s

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9.

Hlavacek, V . , and Kubicek, M . , Chem. Eng. S c i . (1972), 27, 177. Dudukovic, M. P . , and Lamba, H. S., 80th AIChE National Meet ing, Boston (1975), paper #576. Cimbalnik, Z . , et al., 2nd CHISA Cong., Czechoslovakia (1965). Raskin, A. Y a . , et al., Theor. Found. Chem. Tech. (1968), 2, 220. Raskin, A. Y a . , and S o k o l i n s k i i , Yu. Α., Khim Prom. (1969), 45, 520. Vanderveen, J . W., Luss, D . , and Amundson, N. R., AIChE J . (1968), 14, 636. Wicke, E., Chem. Ing. Tech. (1965), 37, 892. Ervin, Μ. Α., and Luss, D . , AIChE J. (1970), 16, 979. L i u , S.-L., and Amundson, N. R., IEC Fund. (1962), 3, 200.


Cell Model Studies of Radial Flow, Fixed Bed Reactors

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