Chemical Reaction Engineering Reviews - ACS Publications

7 Multiplicity, Stability, and Sensitivity of States in Chemically Reacting Systems— A Review

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R O G E R A. SCHMITZ Department of Chemical Engineering, University of Illinois, Urbana, Champaign, Ill. 61801

Attention

to the topics

of multiplicity,

tivity of states in chemical stemmed

principally

Amundson, studies

from

and Aris in the

through

publications 1950's.

primarily

thermic

reaction.

with

problems

More recently,

borne

out some of the theory,

models

used in mathematical

careful

experimental of

application,

electrochemistry, problems subject

tests.

and area.

This

review

and

in

exo-

research

has

distributed

literature

biology,

considerably the

to

in

other

combustion

and

introducing

surveys

nature,

a single

have not been put

A parallel

emphasis

has

Heerden,

but most of the work

sensi-

subsequent

theoretical

experimental

including

topics with particular

and particles

Van

involving

has developed, broadening

by

These

the 1960's were mainly

dealing

fields

stability

reactors and catalyst

the

a variety scope

literature

on the chemical

of on

of this

these

engineering

literature.

h e essential topics of this r e v i e w are t h e m u l t i p l i c i t y ( o r u n i q u e A

ness)

of steady

states

of o p e n

chemically

reacting

systems, t h e

s t a b i l i t y of states to s m a l l a n d large p e r t u r b a t i o n s , a n d t h e s e n s i t i v i t y of t h e m to p a r a m e t e r or i n p u t changes.

T h e topics h a v e t h e i r p r i n c i p a l

a p p l i c a t i o n i n t h e d e s i g n , s t a r t u p , a n d c o n t r o l of t h e v a r i o u s types o f c o n t i n u o u s - f l o w c h e m i c a l reactors e n c o u n t e r e d i n t h e c h e m i c a l a n d p e t r o l e u m i n d u s t r i e s . O t h e r areas of a p p l i c a t i o n , i n c l u d i n g c o m b u s t i o n , b i o l o g y , a n d e l e c t r o c h e m i s t r y , c e r t a i n l y are n o t n e w ; i n fact, some of t h e m h a v e earlier roots t h a n d o reactor a p p l i c a t i o n s i n t h e p u b l i s h e d l i t e r a t u r e . 156 In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

7.

scHMiTZ

Multiplicity,

Stability,

and

157

Sensitivity

C o m m o n usage of t h e w o r d s m u l t i p l i c i t y , s t a b i l i t y , a n d sensitivity i n the l i t e r a t u r e a n d t h r o u g h o u t this p a p e r is a c c o r d i n g to t h e f o l l o w i n g definitions.

T h e m u l t i p l i c i t y o f steady states is t h e n u m b e r of different

sets of state v a r i a b l e s at w h i c h t h e t i m e rate of c h a n g e

of a l l state

v a r i a b l e s is i d e n t i c a l l y zero f o r a fixed set of c o n d i t i o n s or parameters. G a v a l a s ( 1 ) has s h o w n that o n e s h o u l d expect t h e m u l t i p l i c i t y to b e a n o d d n u m b e r f o r r e a c t i n g systems p r o v i d i n g t h e c h e m i c a l k i n e t i c expressions satisfy some r a t h e r l i b e r a l restrictions. T h e s e states are d e s c r i b e d b y t h e w o r d steady o n l y to s i g n i f y t h a t t h e t i m e rate o f c h a n g e of state v a r i a b l e s at s u c h states is z e r o — n o t to s i g n i f y s t a b i l i t y . A steady state is Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

stable i f p e r t u r b a t i o n s w i t h i n a n a r b i t r a r i l y s m a l l n e i g h b o r h o o d s u r r o u n d i n g t h e state d i e a w a y to zero.

I f e v e n t h e smallest s u c h p e r t u r b a t i o n s

g r o w , t h e steady state is u n s t a b l e . I n k e e p i n g w i t h t h e u s u a l v e r n a c u l a r , s t a b i l i t y so defined s h o u l d p r o p e r l y b e t e r m e d l o c a l a s y m p t o t i c s t a b i l i t y , to d i s t i n g u i s h i t f r o m other types s u c h as g l o b a l s t a b i l i t y , w h i c h i m p l i e s s t a b i l i t y to a n a r b i t r a r i l y l a r g e d i s t u r b a n c e . S e n s i t i v i t y is s o m e w h a t less p r e c i s e l y defined. I t describes a g e n e r a l s i t u a t i o n i n w h i c h s m a l l p e r m a nent changes i n a p a r a m e t e r h a v e a large effect o n the steady state. S e n s i tivity m a y or m a y not be connected

w i t h steady-state m u l t i p l i c i t y a n d

instabilities. T h e i n t e n t i n this r e v i e w is to p o r t r a y as m u c h as possible t h e m a n y p h y s i c o c h e m i c a l situations a n d t h e v a r i e t y of i n t r i g u i n g b e h a v i o r a l c h a r acteristics w h i c h h a v e b e e n d e s c r i b e d i n t h e l i t e r a t u r e . T h e m a j o r t h r u s t is t o w a r d c h e m i c a l reactor a p p l i c a t i o n s , b u t a final section o n other a p p l i cations is i n c l u d e d . M o s t of the papers o n c h e m i c a l reactors h a v e f o c u s e d either o n t h e c o n t i n u o u s - f l o w , w e l l - s t i r r e d reactor ( C S T R ) , o n a single catalyst p a r t i c l e , or o n t u b u l a r or

fixed-bed

reactors.

P a p e r s of a v e r y

g e n e r a l n a t u r e h a v e b e e n rare. A c c o r d i n g l y , separate sections a r e d e v o t e d to e a c h of these three subjects. T h e f o u r t h section contains a m o r e g e n e r a l d i s c u s s i o n o f t h e effects o f m i x i n g a n d m a t h e m a t i c a l m o d e l i n g .

A n at-

t e m p t is m a d e at a p p r o p r i a t e places to p u t t h e papers to b e p r e s e n t e d i n S e s s i o n V I I of t h e s y m p o s i u m v o l u m e

(ADVANCES I N CHEMISTRY

SERIES

N o . 133 ) i n p e r s p e c t i v e . R e v i e w papers f r o m p r e v i o u s s y m p o s i a , p a r t i c u l a r l y t h e one b y R a y (2),

fill

some o f t h e gaps i n t h e present r e v i e w .

I n addition, books

d e v o t e d to this subject b y G a v a l a s ( J ) a n d P e r l m u t t e r ( 3 ) a n d a v o l u m e edited b y Oppelt a n d W i c k e

(4)

are g o o d g e n e r a l references.

Two

a d d i t i o n a l books b y A r i s ( 5 ) a n d D e n n ( 6 ) , b o t h o f w h i c h e m p h a s i z e these topics, w i l l soon b e a v a i l a b l e . A s a n estimate, t h e n u m b e r o f l i t e r a t u r e references

to papers o n

m u l t i p l i c i t y a n d s t a b i l i t y i n c h e m i c a l reactors c o n t a i n e d i n this r e v i e w a m o u n t s to a b o u t 4 0 % o f those p u b l i s h e d i n this area of a p p l i c a t i o n t h r o u g h t h e past t w o decades.

T h e percentage

c i t e d i n o t h e r areas o f

In Chemical Reaction Engineering Reviews; Hulburt, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1975.

158

CHEMICAL

a p p l i c a t i o n is m u c h smaller.

REACTION ENGINEERING REVIEWS

Certainly, some important contributions,

p a r t i c u l a r l y those i n t h e R u s s i a n l i t e r a t u r e , are m i s s e d i n this r e v i e w . The

CSTR I n a d d i t i o n to its p r a c t i c a l r o l e as a n i m p o r t a n t a n d c o m m o n t y p e of

i n d u s t r i a l reactor, the C S T R , or m o r e generally, the i d e a l l y m i x e d o p e n r e a c t i n g system, has b e e n the cornerstone for this area of research.

The

necessary m a t h e m a t i c a l theorems a n d methods of analysis are s t a n d a r d ; l a b o r a t o r y studies are r e l a t i v e l y s i m p l e to c o n d u c t , a n d the results are


easily i n t e r p r e t e d .

Consequently, our knowledge

a n d u n d e r s t a n d i n g of

C S T R b e h a v i o r p r o b a b l y set the u p p e r b o u n d o n o u r c a p a b i l i t i e s for e x p l o r i n g a n d u n d e r s t a n d i n g the b e h a v i o r of other o p e n r e a c t i n g systems a n d suggest the questions one n o r m a l l y poses w h e n i n v e s t i g a t i n g m o r e c o m p l e x d i s t r i b u t e d r e a c t i o n models.

[I a m a d h e r i n g to the u s u a l c o n -

v e n t i o n a n d j a r g o n a c c o r d i n g to w h i c h " d i s t r i b u t e d m o d e l s " is t a k e n to m e a n those w h i c h account for s p a t i a l variations of one or m o r e of the d e p e n d e n t v a r i a b l e s — a s i n most models of p o r o u s catalyst p a r t i c l e s a n d t u b u l a r reactors.

I n " l u m p e d m o d e l s , " s u c h as the C S T R p r o b l e m ,

s p a t i a l d e p e n d e n c e is c o n s i d e r e d . ]

I n fact, i t is t e m p t i n g to

no

conjecture

that the q u a l i t a t i v e features of the b e h a v i o r of d i s t r i b u t e d systems m a y a l l b e easily e x p e c t e d b y a n a l o g y to C S T R b e h a v i o r , the l u r e b a s i c a l l y b e i n g the fact that the m a t h e m a t i c a l d e s c r i p t i o n for most d i s t r i b u t e d m o d e l s reduces to that of a C S T R as d i s p e r s i o n parameters b e c o m e large. findings

Those

that d i s p r o v e this conjecture are of the greatest interest i n studies

of d i s t r i b u t e d systems a n d u s u a l l y c a n be t e r m e d s u r p r i s i n g .

Therefore,

a n y n e w c o m e r to this field is a d v i s e d to a c q u a i n t h i m s e l f f u l l y w i t h the status of k n o w l e d g e r e g a r d i n g C S T R b e h a v i o r before e m b a r k i n g o n n e w problems. The Classic Theoretical Problem of a Single Exothermic Reaction. T h e classic C S T R p r o b l e m , i n t r o d u c e d i n papers b y V a n H e e r d e n B i l o u s a n d A m u n d s o n ( 9 ) , a n d A r i s a n d A m u n d s o n (10)

{7,8),

involves a single

homogeneous exothermic reaction occurring i n a well-stirred, continuo u s l y - f e d reactor.

[ T h e p a p e r b y V a n H e e r d e n i n 1953 was n o t a c t u a l l y

the first to treat m u l t i p l i c i t y a n d instabilities i n c h e m i c a l reactors, b u t no p a p e r before i t h a d m a d e a significant i m p a c t b y 1953. L i l j e n r o t h (11)

a n d W a g n e r (12)

Publications by

are a m o n g those w h i c h p r e c e d e d it.]

T h e facts that ( 1 ) so s i m p l e a system c a n e x h i b i t m u l t i p l e steady states, u n s t a b l e states , a n d sustained o s c i l l a t o r y outputs a n d that ( 2 )

the m e t h -

ods of L i a p u n o v a n d Poincaré are w e l l - s u i t e d for i n v e s t i g a t i n g the s t a b i l i t y a n d transient characteristics of s u c h processes w e r e b r o u g h t out i n characteristics w e r e

studied i n

n u m e r o u s subsequent t h e o r e t i c a l papers c o v e r i n g most of the

these

early publications.

These

same

conceivable


7.

scHMiTZ

Multiplicity,

Stability,

and

159

Sensitivity

variations of t h e classic p r o b l e m a n d g i v i n g rise t o a l e n g t h y l i t e r a t u r e c u l m i n a t i n g i n recent p u b l i c a t i o n s b y P o o r e (13) Poore

and Uppal, Ray, and

(14).

For

first-order

A r r h e n i u s k i n e t i c s , t h e m a t e r i a l a n d e n e r g y balances

for t h e classic p r o b l e m take t h e dimensionless f o r m of E q u a t i o n s 1 a n d 2. dci = 1 — ci — Da et exp ΊΓθ


L ^

= 1 -

U + j8 Da c, exp

(D

-

(l

j^J -

Ο

a(U -

(2)

[ I n t h e equations p r e s e n t e d t h r o u g h o u t this p a p e r , I h a v e i n c o r p o r a t e d first-order

A r r h e n i u s k i n e t i c s w i t h a single c h e m i c a l r e a c t i o n f o r purposes

of d i s c u s s i o n a n d i l l u s t r a t i o n .

T h e vast m a j o r i t y of t h e o r e t i c a l m o d e l s

h a v e u s e d this f o r m ; other k i n e t i c descriptions r e q u i r e o b v i o u s a n d straightforward modifications.] T h e s e equations are expressed i n terms of six p a r a m e t e r groups L, Da, γ, — Stable States — Unstable States • · · Stable Limit Cycles xxx Unstable Limit Cycles

ïïa

fflb

ι (e)

(d) A

C

F

Β

A

A

C J ETb

G

A

B

X

(f)

A "ïïl Va (g)

DIB

y ? A Damkohler

(h)

B

A

Number Chemical Engineering Science

Figure 1. Steady-state and stability results for different regions in parameter space for a CSTR with first-order Arrhenius kinetics; L , γ, and t are fixed (14) a



160

CHEMICAL

Conversion


— • Chemical Engineering Science

Figure 2. Classes of phase plots for the different cases indicated in Figure 1. The symbol u designates unstable steady states; s, stable steady states; sic, stable limit cycles; and ulc, unstable limit cycles (14). β, a, a n d f a

P o o r e (13)

a n d U p p a l et al

(14)

e s t a b l i s h e d regions i n t h e

β, 1 + a p a r a m e t e r p l a n e ( f o r fixed values of L , y, a n d t ) &

as s h o w n i n

t h e center p o r t i o n of F i g u r e 1 b y o b t a i n i n g c o n d i t i o n s for the m u l t i p l i c i t y of steady-state solutions, for t h e i r s t a b i l i t y a n d for the existence, a n d s t a b i l i t y of p e r i o d i c orbits ( s u s t a i n e d o s i c i l l a t o r y s t a t e s — l i m i t c y c l e s ) . A l l of this i n f o r m a t i o n is s u m m a r i z e d i n F i g u r e 1, w h i c h is i n t e n d e d to s h o w o n l y q u a l i t a t i v e features.

[ I n c o p y i n g figures f r o m the l i t e r a t u r e , I

h a v e c h a n g e d n o t a t i o n a n d l a b e l i n g f r o m that i n the o r i g i n a l reference to b e consistent w i t h t h e d i s c u s s i o n a n d s y m b o l s t h r o u g h o u t this r e v i e w . ] T h e sketches of steady-state c o n v e r s i o n vs. D a m k o h l e r n u m b e r , w h i c h s u r r o u n d the c e n t r a l figure, t y p i f y t h e b e h a v i o r w i t h i n the v a r i o u s r e g i o n s — I , I I , I l i a , I l l b , etc.

T h e curves b i f u r c a t i n g f r o m the s o l i d a n d

d a s h e d p o r t i o n s of the steady-state curves i n d i c a t e the a m p l i t u d e s of l i m i t cycles; those m a r k e d b y dots r e p r e s e n t stable orbits a n d those b y t h e s y m b o l X , u n s t a b l e ones. N i n e different sections of the steady state curves, e a c h d i s t i n g u i s h e d f r o m a n y of the others b y the m u l t i p l i c i t y a n d s t a b i l i t y of steady states a n d the existence of one or m o r e l i m i t cycles, c a n b e i d e n t i f i e d . T h e s e sections are d e s i g n a t e d A , B , . . . , H , a n d J o n o n t h e abcissas of the steady-state plots i n F i g u r e 1. F o r e a c h of these sections there is a c h a r a c t e r i s t i c phase p o r t r a i t i n the c o n v e r s i o n - t e m -


7.

scHMiTz

Multiplicity,

Stability,

and

161

Sensitivity

p e r a t u r e p l a n e . Sketches of these are s h o w n i n F i g u r e 2. If, for e x a m p l e , the p a r a m e t e r s β a n d a ( a g a i n for fixed values of other p a r a m e t e r groups ) c o r r e s p o n d e d to a p o i n t i n r e g i o n H l b of the c e n t r a l s k e t c h i n F i g u r e 1, the b e h a v i o r a l features of t h a t reactor are c h a r a c t e r i z e d b y the s k e t c h i n F i g u r e I d . P h a s e p o r t r a i t s of t y p e A , B , C , a n d F are p o s s i b l e , d e p e n d i n g o n the p a r t i c u l a r v a l u e of the D a m k o h l e r n u m b e r .

A s the D a m k o h l e r

n u m b e r is decreased f r o m l a r g e values, the b r a n c h of h i g h c o n v e r s i o n states is stable t h r o u g h section A a n d bifurcates to a l i m i t c y c l e i n section B.

T h e l i m i t c y c l e persists t h r o u g h section F , e v e n t h o u g h t w o other

states—one

a stable l o w - c o n v e r s i o n state, the other a s a d d l e

point—


e m e r g e i n the pase p l a n e . F i n a l l y , the l i m i t c y c l e d i s a p p e a r s as i t i n t e r feres w i t h séparatrices i n t h e phase p l a n e ( a t t h e r i g h t e n d of section C ) , a n d e v e n t u a l l y o n l y l o w c o n v e r s i o n states at l o w values of Da are possible. A sufficient c o n d i t i o n for u n i q u e n e s s of a steady state is that the p o i n t c o r r e s p o n d i n g to g i v e n values of β a n d a l i e b e l o w C u r v e M i n the c e n t r a l s k e t c h of F i g u r e 1. A l l steady states for a n y p o i n t b e l o w C u r v e M satisfy the s o - c a l l e d slope c o n d i t i o n for s t a b i l i t y or static c o n d i t i o n , as i t w a s t e r m e d b y G i l l e s a n d H o f m a n n (15).

( T h i s is the c o n d i t i o n that the d e

t e r m i n a n t of the coefficient m a t r i x i n l i n e a r i z e d transient equations

be

p o s i t i v e or that the slope of the heat r e m o v a l c u r v e e x c e e d t h a t of the heat g e n e r a t i o n c u r v e at the steady state.)

F o r any point above C u r v e M ,

m u l t i p l e states w i l l exist over some r a n g e of Da.

S t a b i l i t y of a l l steady

states c a n be assured i f a n d o n l y i f the p o i n t i n the β, 1 - f « p l a n e lies b e l o w b o t h C u r v e s M a n d S—i.e., i n R e g i o n I. I n fact, i t c a n b e s h o w n that steady states for s u c h cases are g l o b a l l y stable. A l l steady states for p o i n t s b e l o w C u r v e S satisfy the c o n d i t i o n t h a t the trace of the coefficient m a t r i x be n e g a t i v e — c a l l e d the d y n a m i c c o n d i t i o n b y G i l l e s a n d H o f m a n n (15).

I t c a n b e s h o w n that o n l y R e g i o n s I a n d I I are accessible i n the

s p e c i a l case of a n a d i a b a t i c r e a c t o r (i.e., a =

0).

It is not feasible to elaborate here o n the a b u n d a n t a d d i t i o n a l i n f o r m a t i o n c o n t a i n e d i n F i g u r e s 1 a n d 2. T h e p a p e r b y U p p a l et al. contains a m u c h m o r e extensive d i s c u s s i o n of reactor b e h a v i o r for the different cases a n d of the methods of c o n s t r u c t i o n of F i g u r e 1. It also i n c l u d e s n u m e r i c a l examples a n d s i m u l a t i o n s to a d d q u a n t i t a t i v e

fidelity

to the

sketches

s h o w n here. T h e w o r k of P o o r e a n d U p p a l et al. m a i n l y serves to tie together a l l of the p r i o r f r a g m e n t a r y i n f o r m a t i o n r e p o r t e d o n this classic p r o b l e m . M o s t of t h e separate features w h i c h t h e y d e s c r i b e h a v e b e e n s t u d i e d . T h e w o r k of V a n H e e r d e n (7,8)

m a d e i t c l e a r that e i t h e r one or t h r e e

steady states exist, d e p e n d i n g o n p a r a m e t e r v a l u e s , a n d t h a t t h e i n t e r m e d i a t e states are unstable. S u b s e q u e n t studies (10,15,16)

demonstrated

t h a t states other t h a n t h e i n t e r m e d i a t e state c o u l d b e u n s t a b l e i n n o n a d i a b a t i c systems, a n d the phase d i a g r a m s of A r i s a n d A m u n d s o n ( J O )


162

CHEMICAL


s h o w e d that b o t h stable a n d u n s t a b l e l i m i t cycles m i g h t b e

expected.

S c h m i t z a n d A m u n d s o n ( 16) s h o w e d that a l l three steady states c o u l d be u n s t a b l e , b u t the f o l l o w i n g facts e s t a b l i s h e d b y U p p a l et al. a p p a r e n t l y h a v e not b e e n expressed i n p r e v i o u s studies of C S T R b e h a v i o r a n d w e r e not g e n e r a l l y a p p r e c i a t e d b y w o r k e r s i n this area.

(1)

F o r parameter

values c o r r e s p o n d i n g to p o i n t s a b o v e C u r v e S i n F i g u r e 1, except for those i n R e g i o n l i b , l i m i t cycles w i l l a l w a y s exist o v e r some r a n g e of values of the D a m k o h l e r n u m b e r ; ( 2 )

the s t a b i l i t y of the l i m i t cycles

near the p o i n t of b i f u r c a t i o n c a n be d e t e r m i n e d b y a n a l g e b r a i c c r i t e r i o n ; ( 3 ) l i m i t cycles m a y d i s a p p e a r as the D a m k o h l e r n u m b e r is c h a n g e d b y


(a)

s h r i n k i n g to zero a m p l i t u d e (as, for e x a m p l e , i n m o v i n g a l o n g the

steady-state c u r v e f r o m S e c t i o n Β to Section A i n F i g u r e s I d , l e , If, l i , a n d l h as stable l i m i t cycles s h r i n k to a stable state, or i n m o v i n g f r o m S e c t i o n H to S e c t i o n C i n F i g u r e l c as u n s t a b l e l i m i t cycles s h r i n k to a n u n s t a b l e steady s t a t e ) , ( b ) coalescence of stable a n d u n s t a b l e l i m i t cycles ( as i n m o v i n g f r o m R e g i o n D to R e g i o n A i n F i g u r e s I f a n d l g — g e n e r a l l y r e f e r r e d to as " h a r d " b i f u r c a t i o n s ) , a n d ( c )

interference of a l i m i t c y c l e

w i t h a separatrix (as i n m o v i n g f r o m section H to section Ε i n F i g u r e l c ) . A t least t w o other papers (17,18)

h a v e p r e s e n t e d a n extensive a n a l y

sis of this classic C S T R p r o b l e m i n p a r a m e t e r space.

N e i t h e r is as ex

h a u s t i v e ( p a r t i c u l a r l y i n the treatment of the a p p e a r a n c e a n d d i s a p p e a r a n c e of p e r i o d i c orbits ) as the p u b l i c a t i o n b y U p p a l et al. O t h m e r ( 19 ) r e c e n t l y p r e s e n t e d a s i m i l a r analysis of a s i m p l i f i e d k i n e t i c m o d e l of the i s o t h e r m a l B e l o u s o v - Z h a b o t i n s k i i r e a c t i o n — a n a u t o c a t a l y t i c system i n w h i c h m a l o n i c a c i d is o x i d i z e d i s o t h e r m a l l y b y b r o m a t e i n the presence of a m e t a l i o n .

( Other works w h i c h focused on this fascinating reaction

are c i t e d later. ) A l s o , the d i a g r a m s a n d analyses s i m i l a r to those u s e d b y U p p a l et al. c a n easily b e c o n s t r u c t e d for other t w o - v a r i a b l e p r o b l e m s . W i t h q u i t e different k i n e t i c expressions, n e w q u a l i t a t i v e features

could

b e i n t r o d u c e d . F o r e x a m p l e , U p p a l et al. m e n t i o n e d the p o s s i b i l i t y of a greater n u m b e r of l i m i t cycles existing f o r a g i v e n set of parameters, b u t t h e y f o u n d no e v i d e n c e of these i n t h e i r c o m p u t a t i o n s A r r h e n i u s k i n e t i c s . O t h m e r (19)

for

first-order

p r e d i c t e d the a p p e a r a n c e of three l i m i t

cycles i n a p h a s e p l a n e , t w o of w h i c h w e r e u n s t a b l e orbits a r o u n d stable steady states a n d the t h i r d a stable o r b i t s u r r o u n d i n g a l l three

steady

states. N o s i m u l a t i o n s for s u c h a case w e r e g i v e n . T h e exact c o n s t r u c t i o n of the phase d i a g r a m s s h o w n i n F i g u r e 2 a n d the q u a n t i t a t i v e d e t e r m i n a t i o n of the a m p l i t u d e s of l i m i t cycles ( i n d i c a t e d s c h e m a t i c a l l y b y the dots a n d X ' s i n F i g u r e 1)

still r e q u i r e d i g i t a l or

a n a l o g c o m p u t e r solutions of the n o n l i n e a r E q u a t i o n s 1 a n d 2. A p p l i c a tions of the d i r e c t m e t h o d of L i a p u n o v to establish regions of a s y m p t o t i c s t a b i l i t y i n the phase p l a n e h a v e consistently r e s u l t e d i n v e r y conservative estimates, as w e l l i l l u s t r a t e d i n a recent p a p e r b y S h a s t r y a n d F a n


(20).

7.

Multiplicity,

scHMiTZ

Stability,

and

163

Sensitivity

O t h e r references to s u c h studies a n d a d e s c r i p t i o n of m e t h o d s of

con

structing L i a p u n o v functions m a y be found i n the book b y Perlmutter (3).

M e t h o d s of o b t a i n i n g the exact regions of a s y m p t o t i c s t a b i l i t y b y

c o m p u t i n g that separatrix ( b y b a c k w a r d n u m e r i c a l i n t e g r a t i o n ) i n the p h a s e p l a n e w h i c h passes t h r o u g h the s a d d l e p o i n t h a v e b e e n d e s c r i b e d (10,15)

a n d are r e l a t i v e l y easy to a p p l y .

A n u m b e r of papers h a v e b e e n d e v o t e d to m e t h o d s of a p p r o x i m a t i n g the l i m i t cycles i n the phase p l a n e (see, H e b e r l i n g et al. (21)

for e x a m p l e , R e f s . 21, 22,

23).

c o m p a r e d the v a r i o u s a p p r o x i m a t e solutions w i t h

those generated b y n u m e r i c a l s o l u t i o n , i n c l u d i n g u n s t a b l e l i m i t cycles Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

w h i c h w e r e r e n d e r e d stable b y r e v e r s i n g the d i r e c t i o n of t i m e i n the n u m e r i c a l i n t e g r a t i o n . M u c h of the w o r k o n the c o m p u t a t i o n of l i m i t cycles has b e e n m o t i v a t e d b y the f a c t [first p o i n t e d out b y D o u g l a s a n d R i p p i n (24)~\ that the t i m e - a v e r a g e p e r f o r m a n c e g i v e n b y a n o s c i l l a t o r y o u t p u t m a y be better t h a n that of the steady state a b o u t w h i c h t h e o u t p u t oscillates. T a k i n g a l l experiences into account, most researchers i n this field

w o u l d p r o b a b l y r e c o m m e n d s t r a i g h t f o r w a r d n u m e r i c a l or a n a l o g

s i m u l a t i o n s u s i n g the n o n l i n e a r system equations to s t u d y the l a r g e scale transient effects for a g i v e n s i t u a t i o n . [ T h e p e r i o d i c o p e r a t i o n of c h e m i c a l reactors caused b y the d e l i b e r a t e m a n i p u l a t i o n ( c y c l i n g )

of one or

m o r e parameters to i m p r o v e reactor p e r f o r m a n c e is not w i t h i n the scope of this r e v i e w .

A recent r e v i e w b y B a i l e y

(25)

covers this

subject

extensively.] Further Theoretical Work. A m o n g v a r i a t i o n s of or departures f r o m the classic p r o b l e m h a v e b e e n t h e o r e t i c a l studies of systems i n v o l v i n g m o r e t h a n one phase, m o r e t h a n one r e a c t i o n , reactors i n series, a n d others. G e n e r a l l y the features s t u d i e d i n these w o r k s are s i m i l a r to those d e s c r i b e d a b o v e , b u t w h e n the c o m p l e x i t y of the system equations is i n c r e a s e d ( p a r t i c u l a r l y as a result of a n increase i n the o r d e r of t h e sys tem—i.e., i n the n u m b e r of d e p e n d e n t v a r i a b l e s ) , the m u l t i p l i c i t y steady states is often i n c r e a s e d .

S c h m i t z a n d A m u n d s o n (16)

of

showed,

for e x a m p l e , that a single e x o t h e r m i c r e a c t i o n m a y g i v e rise to as m a n y as n i n e steady states w h e n i t occurs i n b o t h phases of a system w i t h t w o fluid phases a n d w i t h transfer resistances b e t w e e n the phases. I n a s i m i l a r situation involving multiphase polymerization, Goldstein and A m u n d s o n (26)

e n c o u n t e r e d as m a n y as 25 different steady states.

( I n connection

w i t h m u l t i p h a s e C S T R systems, there h a v e b e e n a f e w t h e o r e t i c a l studies of steady-state m u l t i p l i c i t y a n d s t a b i l i t y i n t w o - p h a s e p h a s e (28)

m o d e l s of

fluidized-bed

(27)

and three-

reactors w i t h the dense p h a s e p e r

f e c t l y m i x e d a n d of a p e r f e c t l y m i x e d s p r a y reactor ( 2 9 ) , a l l for exo t h e r m i c processes. ) A systematic analysis of t w o s e q u e n t i a l reactions A —» Β - » C , b o t h e x o t h e r m i c , i n a single-phase C S T R has b e e n r e p o r t e d b y H l a v a c e k , K u b i -


164

CHEMICAL


cek, a n d V i s n a k ( 3 0 ) ; they s h o w e d that five steady states are p o s s i b l e , as are s u s t a i n e d o s c i l l a t o r y outputs for a n o n a d i a b a t i c s i t u a t i o n .

The

s t r u c t u r e of the t h r e e - d i m e n s i o n a l phase space for this system w a s c a l c u l a t e d b y Sabo a n d D r a n o f f (31),

w h o s e w o r k also i n c l u d e d the e s t i m a t i o n

of regions of a s y m p t o t i c s t a b i l i t y b y a p p l i c a t i o n of L i a p u n o v ' s

direct

method. I n a theoretical study w h i c h accompanied experimental work, G r a z i a n i et al. (32)

a n a l y z e d the B e l o u s o v - Z h a b o t i n s k i i r e a c t i o n u s i n g a k i n e t i c

s c h e m e v e r y s i m i l a r to the 10-reaction m o d e l of F i e l d , K o r o s , a n d N o y e s (33).

T h e results, s h o w n i n F i g u r e 3, p r e d i c t " h a r d " o s c i l l a t o r y b i f u r c a -


tions (that is, the e l i m i n a t i o n of stable l i m i t cycles b y coalescence w i t h unstable ones). M a n y p u b l i c a t i o n s h a v e f o c u s e d o n t h e p r o b l e m of a u t o m a t i c c o n t r o l , specifically the p o s s i b i l i t y of s t a b i l i z i n g a n i n h e r e n t l y u n s t a b l e state b y a f e e d b a c k c o n t r o l scheme. Some of the v e r y e a r l y papers this p r o b l e m ( J O , 34, 35, 36).

considered

M o r e r e c e n t l y , s t a b i l i z i n g c o n t r o l has b e e n

c o n s i d e r e d b y D e m o et al. (37),

w h o a n a l y z e d the p o s s i b i l i t y of s t a b i l i z -

i n g a n u n s t a b l e state b y m a n i p u l a t i n g the s t i r r i n g s p e e d i n a n o n a d i a b a t i c reactor; b y H y u n a n d A r i s ( 3 8 ) , w h o e x a m i n e d t h e effects of hysteresis i n the f e e d b a c k l o o p ; a n d b y L u y b e n ( 3 9 ) , i n a s t u d y of the effect of the r e a c t i o n v e l o c i t y constant o n reactor s t a b i l i t y . Interest i n the m u l t i p l i c i t y a n d s t a b i l i t y of steady states i n i s o t h e r m a l

0.001

0.01 F/V

0.1 (sec )

1

-1

AlChE Meeting

Figure 3. Predicted behavior for the Belousov-Zhabotinskii reaction in a CSTR at 25°C. Solid curves represent stable states; dashed sections, unstable states. Dots represent the amplitudes of stable limit cycles (amplitudes not shown to scale) and X's, amplitudes of unstable ones (32).


7.

scHMiTz

Multiplicity,

Stability,

and

Sensitivity

165

systems has i n c r e a s e d n o t a b l y i n recent years. T w o studies of the B e l o u s o v - Z h a b o t i n s k i i r e a c t i o n have a l r e a d y b e e n c i t e d . I n a d d i t i o n , H i g g i n s (40)

discussed v a r i o u s i s o t h e r m a l k i n e t i c m o d e l s i n d i c a t i n g m u l t i p l i c i t y

of states a n d o s c i l l a t o r y b e h a v i o r i n m a n y cases. H i s p a p e r w a s a i m e d at b i o l o g i c a l r e a c t i o n m e c h a n i s m s , as w e r e others (see, 41, 42, 43).

for e x a m p l e , Refs.

F u r t h e r d i s c u s s i o n of m u l t i p l i c i t y a n d s t a b i l i t y i n b i o l o g y is

i n c l u d e d later. M a t s u u r a a n d K a t o (44)

presented a t h e o r e t i c a l s t u d y

s h o w i n g m u l t i p l e states i n a n i s o t h e r m a l C S T R , b a s i n g t h e i r k i n e t i c m o d e l o n the a u t o c a t a l y t i c o x i d a t i o n of i s o p r o p y l a l c o h o l . K n o r r a n d O ' D r i s c o l l (45)

p o i n t e d out that v i s c o s i t y effects o n the t e r m i n a t i o n of the p o l y m e r i -


z a t i o n of styrene gives the a p p e a r a n c e of a u t o c a t a l y c i t y a n d causes m u l t i p l e steady states. F u t u r e w o r k i n this area w i l l p r o b a b l y b e addressed m o r e a n d m o r e t o w a r d c o m p l i c a t e d systems of reactions. H o p e f u l l y this w i l l l e a d to some means of c a t e g o r i z i n g or c l a s s i f y i n g r e a c t i o n systems so t h a t o n l y the t y p i c a l b e h a v i o r of e a c h class n e e d b e e l u c i d a t e d . S i n g l e reactions s e e m to b e a p p r o p r i a t e l y d i v i d e d i n t o t w o classes: ( 1 ) those for w h i c h c h e m i c a l rates increase w i t h a n increase i n the extent of r e a c t i o n over some range of the extent ( a c c e l e r a t i n g r e a c t i o n s ) a n d ( 2 ) those for w h i c h rates m o n o t o n i c a l l y decrease w i t h i n c r e a s i n g extent ( d e c e l e r a t i n g r e a c t i o n s ) .

To

the

f o r m e r class b e l o n g e x o t h e r m i c a n d a u t o c a t a l y t i c reactions as examples and

to the latter, e n d o t h e r m i c

and isothermal mass-action

reactions.

B a r r i n g the p o s s i b i l i t y of some v e r y u n u s u a l p h y s i c a l effects, a sufficient c o n d i t i o n for the u n i q u e c o n d i t i o n a n d s t a b i l i t y of steady-state solutions is that the r e a c t i o n b e l o n g to the d e c e l e r a t i n g class. A n o t h e r i m p o r t a n t d i s t i n c t i o n b e t w e e n the t w o classes is t h a t the progress of a c c e l e r a t i n g reactions m a y be f a v o r e d b y b a c k m i x i n g w h i l e d e c e l e r a t i n g reactions are best d o n e u n d e r segregated or p l u g - f l o w c o n d i t i o n s . T h e r e does not seem to b e a u s e f u l a n a l o g y i n terms of a c c e l e r a t i n g or d e c e l e r a t i n g systems of reactions, a n d the task of d e f i n i n g easily i d e n t i f i a b l e classifications appears formidable.

R e c e n t w o r k b y H o r n (46),

H o r n a n d J a c k s o n (47),

F e i n b e r g a n d H o r n ( 48 ) represents a step t o w a r d s u c h a goal.

and

Reaction

m e c h a n i s m s c o n s i d e r e d i n those studies w e r e i s o t h e r m a l systems w h i c h o b e y mass-action kinetics. T h e p r i n c i p a l notions i n v o l v e d w e r e those of w e a k r e v e r s i b i l i t y a n d deficiency of a m e c h a n i s m .

T h e essential r e s u l t

of interest here is a t h e o r e m w h i c h states t h a t a sufficient c o n d i t i o n for a m a s s - a c t i o n m e c h a n i s m to h a v e a u n i q u e a n d g l o b a l l y stable steady state i n a n o p e n , p e r f e c t l y m i x e d system is that i t h a v e a deficiency of z e r o a n d be w e a k l y r e v e r s i b l e . B o t h t h e deficiency a n d w e a k r e v e r s i b i l i t y c a n b e r e a d i l y d e t e r m i n e d f r o m the r e a c t i o n m e c h a n i s m a n d are i n d e p e n d e n t of the r e a c t i o n v e l o c i t y constants. T h e p r o o f i n v o l v e s a n elegant m a t h e m a t i c a l structure essentially a i m e d at e s t a b l i s h i n g c o n d i t i o n s u n d e r w h i c h the n a t u r e of steady a n d transient states i n a n o p e n r e a c t i n g system is the


166

CHEMICAL


same as that d i c t a t e d b y t h e r m o d y n a m i c l a w s for closed systems n e a r equilibrium.

F o r t u n a t e l y for most interested readers the g e n e r a l results

h a v e b e e n a p p l i e d i n a m a n n e r most easily a p p r e c i a t e d i n the last of the a b o v e three references as w e l l as i n three other p a p e r s b y H o r n (49).

The

results seem to b e u s e f u l for m o d e l s i n w h i c h the system is o p e n as a c o n s e q u e n c e of a s s u m i n g t h a t c e r t a i n c o m p o n e n t s

are present i n

ficti-

t i o u s l y constant c o n c e n t r a t i o n — a n a s s u m p t i o n c o m m o n l y i n v o k e d i n b i o l o g i c a l a p p l i c a t i o n s . H o w e v e r , this author's l i m i t e d testing of the t h e o r e m w i t h c o n t i n u o u s flow r e a c t i o n models suggests that i t is q u i t e conservative i n its present f o r m . C e r t a i n l y v a r i a t i o n s a n d modifications of this a p p r o a c h


as w e l l as e n t i r e l y different approaches w i l l be f o r t h c o m i n g . Experimental Studies. T h o u g h l o n g d e l a y e d i n t i m e f r o m the theor e t i c a l w o r k of V a n H e e r d e n a n d others, a n u m b e r of papers p r e s e n t i n g e x p e r i m e n t a l d a t a o n steady-state m u l t i p l i c i t y a n d s t a b i l i t y i n the C S T R h a v e a p p e a r e d , m o s t l y w i t h i n the past three or f o u r years.

[ F o r purposes

of o r g a n i z a t i o n h e r e , papers w h i c h i n c l u d e e x p e r i m e n t a l d a t a are r e f e r r e d to i n separate subsections.

M o s t of t h e m also c o n t a i n some c o n t r i b u t i o n

to or d i s c u s s i o n of a n u n d e r l y i n g theory.]

T a b l e I gives a list of t h e

a b o v e - m e n t i o n e d papers a l o n g w i t h a b r i e f d e s c r i p t i o n of e a c h system s t u d i e d a n d the observations r e p o r t e d .

[ T o m y k n o w l e d g e , the lists of

e x p e r i m e n t a l reactor studies g i v e n i n tables i n this r e v i e w are c o m p l e t e . H o w e v e r , the b o l d c l a i m of no omissions is not m a d e a n d m o s t l i k e l y w o u l d not be correct.]

T h e list contains studies of single-phase systems

( l i q u i d a n d g a s ) , of t w o - p h a s e systems ( e m u l s i o n p o l y m e r i z a t i o n , g a s l i q u i d a n d g a s - s o l i d ) , of e x o t h e r m i c , a n d of i s o t h e r m a l processes. gas-solid cases represent s o l i d - c a t a l y z e d reactions.

These

[The

experiments

u s e d a r e c i r c u l a t i o n reactor ( or r e c y c l e or l o o p reactor ) w h i c h approaches

Table I. Experimental Studies of Steady-State Multiplicity, Instabilities, and Control in Well-Mixed Reactors" Reference

Experimental

System

0

1. H o f m a n n , D e c o m p o s i t i o n of H2O2 i n l i q u i d phase 1965 (50) in nonadiabatic C S T R 2. H a f k e a n d G i l l e s , L i q u i d phase o x i d a t i o n of e t h y l a l c o h o l 1968 (51) by H 0 i n nonadiabatic C S T R D e c o m p o s i t i o n of N 0 o n copper oxide 3. H u g o , 1968 (52) catalyst i n adiabatic circulating reactor V a p o r phase c h l o r i n a t i o n of m e t h y l 4. B u s h , 1969 (53) chloride i n nonadiabatic C S T R 5. F u r u s a w a et al., H y d r a t i o n of p r o p y l e n e oxide i n l i q u i d phase i n a d i a b a t i c C S T R 1969 (54) 6. B a c c a r o et al., H y d r o l y s i s of a c e t y l chloride i n l i q u i d 1970 (55) phase i n n o n a d i a b a t i c C S T R 2

2

2


m

c

7.

scHMiTZ

Multiplicity,

Stability, T a b l e I.

(56)


8. V e j t a s a a n d Schmitz, 1970 (57) 9. G e r r e n s et al., 1971 (58) 10. H o r a k et al., 1971 (59)

11. H a n c o c k a n d Kenny, 1972 (60) 12. H o r a k a n d Jiracek, 1972 (61) 13. H u g o a n d Jakubith, 1972 (62) 14. L o a n d C h o l e t t e , 1972 (63) 15. D u b i l a n d G a u b e , 1973 (64) 16. E c k e r t , H l a v a cek, a n d M a r e k , 1973 (65) 17. E c k e r t et al., 1973 (66) 18. G r a z i a n i et al., 1973 (32) 19. M a r e k , 1973 (67) 20. C h a n g a n d Schmitz, 1974 (68) 21. D i n g , S h a r m a , and Luss, 1974 (69) 22. G u h a a n d A g n e w , 1974 (70) 23. W e i s s a n d J o h n , 1974 (71)

167

Sensitivity

Continued

Experimental

Reference 7. H u g o , 1970

and

System

o

E x o t h e r m i c d e c o m p o s i t i o n of N 0 o n copper oxide c a t a l y s t a n d i s o t h e r m a l o x i d a t i o n of C O o n p l a t i n u m c a t a l y s t i n c i r c u l a t i n g reactor L i q u i d phase r e a c t i o n between N a S 0 and H 0 i n adiabatic C S T R 2

2

2

2

m

c

χ

x

3

2

E m u l s i o n p o l y m e r i z a t i o n of styrene i n three-stage series of isothermal C S T R ' s (three stable states observed) L i q u i d phase a u t o c a t a l y t i c r e a c t i o n of bistrichlormethyl trisulfide w i t h a n i line i n one- a n d two-stage i s o t h e r m a l C S T R system C h l o r i n a t i o n of m e t h y l a l c o h o l i n i s o thermal gas-liquid system

χ

χ

χ

H y d r o g e n - o x y g e n reaction on p l a t i n u m catalyst i n adiabatic recirculating re actor I s o t h e r m a l o x i d a t i o n of c a r b o n m o n oxide o n p l a t i n u m c a t a l y s t i n r e c i r c u l a t i n g reactor L i q u i d phase r e a c t i o n between N a S 0 a n d H 0 i n series of a d i a b a t i c C S T R ' s Oxo reaction i n a nonadiabatic reactor m o d e l e d as a C S T R O x i d a t i o n of C O o n C u O / A l 0 c a t a l y s t i n a d i a b a t i c c i r c u l a t i n g reactor 2

2

2

χ

χ

χ

χ

x

3

2

2

3

χ χ

χ

χ

χ

O x i d a t i o n of C O o n C u O / A l 0 c a t a l y s t i n a d i a b a t i c c i r c u l a t i n g reactor L i q u i d phase B e l o u s o v - Z h a b o t i n s k i i r e action i n isothermal C S T R L i q u i d phase B e l o u s o v - Z h a b o t i n s k i i r e action in isothermal C S T R L i q u i d phase r e a c t i o n between N a S 0 and H 0 i n nonadiabatic C S T R 2

3

2

2

2

3

χ χ χ

χ

χ

2

C h l o r i n a t i o n of n-decane i n g a s - l i q u i d mixture i n nonadiabatic C S T R

χ

L i q u i d phase r e a c t i o n between N a S 0 χ and H 0 i n adiabatic C S T R L i q u i d phase f o r m a l d e h y d e condensaχ t i o n t o c a r b o h y d r a t e s is i s o t h e r m a l CSTR Column headings: ο stands for sustained oscillations, m for steady-state multi plicity, and c for feedback control of unstable states; χ in the columns indicates the behavior observed. 2

2

2

3

2

β


168

CHEMICAL


C S T R b e h a v i o r at h i g h r e c i r c u l a t i o n rates. F o r purposes of m a t h e m a t i c a l analysis, t h e y are f r e q u e n t l y d e s c r i b e d b y t h e u s u a l C S T R

equations.]

B o t h s u s t a i n e d o s c i l l a t o r y b e h a v i o r a n d hysteresis effects r e s u l t i n g f r o m m u l t i p l e steady states h a v e b e e n s t u d i e d , b e a r i n g out some of t h e theo r e t i c a l results. G e n e r a l l y i n t h e studies o f steady-state m u l t i p l i c i t y , t w o stable states w e r e o b t a i n e d .

E x c e p t i o n s w e r e r e p o r t e d b y H o r a k et al.

( 5 9 ) , i n w h i c h t w o C S T R ' s i n series r e s u l t e d i n t h r e e stable states; b y H o r a k a n d J i r a c e k ( 6 1 ) , i n w h i c h t h e h i g h - c o n v e r s i o n state as w e l l as the i n t e r m e d i a t e state w a s u n s t a b l e — a s i t u a t i o n r e s e m b l i n g t y p e C of F i g u r e s 1 a n d 2; a n d i n a s t u d y o f t h e i s o t h e r m a l s o l i d c a t a l y z e d o x i d a t i o n Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

of c a r b o n m o n o x i d e b y H u g o a n d J a k u b i t h ( 6 2 ) , i n w h i c h s u s t a i n e d oscillations w e r e o b s e r v e d

around h i g h conversion

states, r e s e m b l i n g

situations o f t y p e F . I n a l l other studies, o n l y situations of types A , B , a n d Ε have been reported. T h e d a t a s h o w n i n F i g u r e 4 are t y p i c a l of t h e results of experiments b y G r a z i a n i et al. (32)

w i t h the Belousov-Zhabotinskii reaction. T h e

e x p e r i m e n t a l c o n d i t i o n s i n c l u d e t h e r a n g e over w h i c h b o t h stable a n d unstable l i m i t cycles w e r e p r e d i c t e d a c c o r d i n g to t h e t h e o r e t i c a l c u r v e i n

1.10 h

1.05

? >

100

0

2

—

oscillations (peak to peak)

•

experimental stable states

4

6

10

8

Flow Rate (ml/min) AlChE Meeting

Figure 4. Experimental results showing the readings from a platinum electrode immersed in a CSTR (volume 25 ml) for the BehusovZhabotinskii reaction at 25°C (32)


7.

SCHMITZ

Multiplicity,

Stability,

and

169

Sensitivity

F i g u r e 3. T h e d a t a suggest t h a t t h e oscillations a r e h a r d since t h e y d o not a p p e a r to s h r i n k to zero a m p l i t u d e at t h e e n d of t h e o s c i l l a t o r y r e g i o n , a n d hence t h a t situations o f t y p e D exist; h o w e v e r ,

extensive

e x p e r i m e n t a l testing d i d n o t e s t a b l i s h this c o n c l u s i v e l y . 60 h 50 h 40


3

ο 30 α> α. Ε |2 20 α> ο 55

CO

10

Theoretical — Steady States u Region of Unstable States for L=0 Experimental Δ • 30

J

Stable Unstable (Controlled) I

40

L

J

50

60

I

70

L _ J

L

80

Residence Time (sec) Chemical Engineering Science

Figure 5. Experimental results of Chang and Schmitz (68) showing unique unstable (controlled) states for the thiosulfate-peroxide reaction in a nonadiabatic CSTR C h a n g a n d S c h m i t z (68) u s e d a s i m p l e f e e d b a c k c o n t r o l s c h e m e i n a n o n a d i a b a t i c system to s t a b i l i z e u n s t a b l e states—i.e., to e l i m i n a t e t h e l i m i t cycles i n cases of s u s t a i n e d oscillations a n d to s t a b i l i z e t h e i n t e r m e d i a t e states ( s a d d l e p o i n t s ) i n t h e regions o f m u l t i p l e steady s t a t e s — i n a n e x p e r i m e n t a l a n a l o g of t h e e a r l y t h e o r e t i c a l studies b y A r i s a n d A m u n d s o n (10).

F i g u r e s 5, 6, a n d 7 illustrate some e x p e r i m e n t a l obser

vations of C h a n g a n d S c h m i t z a n d s h o w some c o m p a r i s o n s w i t h theo r e t i c a l p r e d i c t i o n s f o r the second-order

reaction between sodium thio-

sulfate a n d h y d r o g e n p e r o x i d e . E x c e p t f o r t h e a u t o m a t i c c o n t r o l aspect, these observations r e s e m b l e those d e s c r i b e d i n m a n y o f t h e other entries i n T a b l e I . S t e a d y states o n t h e range o f residence

times s h o w n i n

F i g u r e 5 are u n i q u e , b u t over a p o r t i o n of t h a t residence t i m e r a n g e t h e y are p r e d i c t e d to b e unstable. T h e u n s t a b l e states w e r e o b t a i n e d e x p e r i -


170

CHEMICAL


mentally b y a conventional proportional-integral feedback controller used as s h o w n s c h e m a t i c a l l y i n F i g u r e 5. C h a n g a n d S c h m i t z u s e d t w o different reactors w h i c h d i f f e r e d o n l y i n t h e i r w a l l t h i c k n e s s — t h a t is, i n the effective v a l u e o f L i n E q u a t i o n 2. F o r the d a t a s h o w n i n F i g u r e 5, L was e s t i m a t e d to b e a b o u t 1.02. F o r the other r e a c t o r L w a s ca. 1 . 1 — l a r g e e n o u g h to cause a l l o f t h e steady states i n F i g u r e 5 to b e stable w i t h o u t f e e d b a c k c o n t r o l . F i g u r e 6 shows

a n experimental limit cycle for the uncontrolled

r e a c t o r o n t h e phase p l a n e a n d t h e course o f t h e t r a n s i e n t f o l l o w i n g t h e initiation of feedback control action.

[ T h e phase p l a n e shows t h e t e m


p e r a t u r e vs. its t i m e d e r i v a t i v e as o p p o s e d to t h e u s u a l t e m p e r a t u r e c o n c e n t r a t i o n plots. T r a n s i e n t concentrations w e r e not m e a s u r e d i n these experiments.]

F i g u r e 7 presents t h e o r e t i c a l a n d e x p e r i m e n t a l results f o r 50 r-

d. 25 feedback control J

I

I

I

I

I

I

I

I

20

L

40

Time (min)

I

0

I

I

ι

ι

ι

I

I

25

I

I

I

50

Temp (°C) Chemical Engineering Science

Figure 6. Sustained oscilla tions and the effect of feedback control with the thiosulfateperoxide reaction in a CSTR. In the lower diagram, Curve (1) represents the limit cycle, and the dashed curve starting at (2) shows the trajectory fol lowing the initiation of feed back control (68).


7.

SCHMITZ

Multiplicity,

Stability,

and

c o n d i t i o n s u n d e r w h i c h m u l t i p l e steady states exist. p o i n t s for the i n t e r m e d i a t e states w e r e

171

Sensitivity

T h e experimental

again obtained b y stabilizing

f e e d b a c k c o n t r o l . B o t h the h i g h a n d l o w t e m p e r a t u r e states i n F i g u r e 5 w e r e i n h e r e n t l y stable. I n those cases for w h i c h c o m p a r i s o n s w e r e m a d e , agreement b e t w e e n t h e o r e t i c a l p r e d i c t i o n s a n d e x p e r i m e n t a l results h a v e b e e n g e n e r a l l y satis f a c t o r y , at least i n a q u a l i t a t i v e sense.

A n a c c u r a t e d e s c r i p t i o n of the

c h e m i c a l k i n e t i c s is u n d o u b t e d l y the l i m i t i n g factor i n this r e g a r d i n almost a n y s i t u a t i o n . A g o o d e x a m p l e is the w o r k of H u g o a n d J a k u b i t h (62)

o n the o s c i l l a t o r y b e h a v i o r i n the s o l i d - c a t a l y z e d o x i d a t i o n of c a r b o n


monoxide.

O t h e r p a p e r s , t h o u g h not r e p o r t i n g e x p e i m e n t a l d a t a , h a v e

closely l i n k e d t h e i r analysis to a c t u a l r e a c t i o n systems or at least u s e d r e a l k i n e t i c d a t a (36, 44, 45, 72, 73, 74, 75, 76, 77, 78).

L i m i t e d space does

n o t p e r m i t a d e s c r i p t i o n or l i s t i n g of the v a r i o u s topics c o v e r e d i n these papers. The Catalyst

Particle

Problem

M a t h e m a t i c a l Models a n d Theoretical Studies. T h e starting point for most t h e o r e t i c a l studies of c h e m i c a l r e a c t i o n w i t h t h e r m a l effects i n porous catalyst p a r t i c l e s has b e e n the f o l l o w i n g system of dimensionless diffusion equations, w r i t t e n here for

$ - ( Ε * Μ δ

first-order

Arrhenius kinetics:

+ ϊδ)-*-ρ|>(ι-£)] + ΐ£)+'*Η:('-έ)]

3

(4)

where 0 for slab geometry

!

1

cylindrical

2

spherical

w i t h boundary conditions:

2

=

0 : ^

=

3^

= 0 (for s y m m e t r i c profiles)


(5)


172

CHEMICAL

^


Experimental Δ Stable • Unstable (controlled)

\γ

10

•Α

Λ

-

Δ

Α

Ά

_L

0

20

40

60

80

Coolant Flow Rate (ml/sec) Chemical Engineering Science

Figure 7. Stable and unstable (controlled) states for the thiosulfate-per oxide reaction in a CSTR. The arrowheads extending from the unstable states indicate the extent of fluctuations in the reactor temperature dur ing controlled operation (68).

T h e q u a n t i t y u l t i m a t e l y of interest i n most a p p l i c a t i o n s is t h e p a r t i c l e effectiveness factor, η, w h i c h is o b t a i n e d f r o m t h e s o l u t i o n profiles a n d d e p e n d s , of course, o n a l l parameters i n t h e d e s c r i b i n g equations. F e w studies h a v e u s e d these equations as w r i t t e n .

Various special

cases are d e d u c i b l e d e p e n d i n g , f o r e x a m p l e , o n w h e t h e r ( 1 ) t h e L e w i s n u m b e r , Le, is t a k e n to b e u n i t y , ( 2 ) s l a b , c y l i n d r i c a l o r s p h e r i c a l g e o m e t r y is a d o p t e d , ( 3 ) t h e N u s s e l t a n d S h e r w o o d n u m b e r s are finite, ( 4 ) t h e Nusselt a n d Sherwood numbers are equal, ( 5 ) internal concentration a n d / o r t e m p e r a t u r e gradients are c o n s i d e r e d a n d ( 6 ) t h e a s s u m p t i o n of s y m m e t r i c profiles, e m b o d i e d i n E q u a t i o n 5, is r e t a i n e d .

M o s t of the

c o m b i n a t i o n s of these s p e c i a l cases either m a k e sense p h y s i c a l l y i n c e r t a i n


7.

SCHMITZ

Multiplicity,

Stability,

and

Sensitivity

173

a p p l i c a t i o n s or l e a d to m a t h e m a t i c a l l y convenient p r o b l e m s .

Obviously

the n u m b e r of different versions of the basic p r o b l e m is l a r g e ; a l i t e r a t u r e search w o u l d s h o w that questions r e g a r d i n g steady-state m u l t i p l i c i t y a n d s t a b i l i t y h a v e b e e n p u r s u e d for m a n y of these. S o m e versions l e n d t h e m selves to s p e c i a l m a t h e m a t i c a l methods

not a p p l i c a b l e to others, a n d

c e r t a i n d i s t i n c t i v e properties of solutions of some are not f o u n d i n others. A l l i n a l l , i t is difficult i n a b r i e f r e v i e w to o r g a n i z e a n d sort out the a p p l i c a b l e m a t h e m a t i c a l m e t h o d s , as w e l l as the effects of the m a n y parameters, a n d to i n c l u d e c o n c l u s i v e statements, m a n y of w h i c h m u s t b e q u a l i f i e d b y a list of restrictions. T h e f o l l o w i n g a c c o u n t is i n t e n d e d o n l y Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

to b r i n g out essential facts a n d , w h e r e possible, to relate b e h a v i o r a l c h a r acteristics for this p r o b l e m to those of the C S T R . A v e r y c o m p r e h e n s i v e t r e a t m e n t of the p o r o u s catalyst p a r t i c l e p r o b l e m m a y b e f o u n d i n the book by Aris (5). F o r t y p i c a l c o m m e r c i a l p o r o u s catalysts u s e d i n i n d u s t r i a l reactors some of the s p e c i a l cases i n d i c a t e d a b o v e w o u l d not b e r e a l i s t i c . F o r e x a m p l e , the r a t i o Sh/Nu

c a n c o v e r s e v e r a l orders of m a g n i t u d e w i t h

the l o w e r l i m i t b e i n g a p p r o x i m a t e l y 10. u n i t y — a n average of p e r h a p s 30.

Le is also m u c h greater t h a n

( P a r a m e t e r s values for some specific

e x o t h e r m i c reactions w e r e t a b u l a t e d b y H l a v a c e k et al. (80).)

Such con-

siderations l e a d to a p r a c t i c a l w o r k i n g m o d e l i n w h i c h the t e m p e r a t u r e is a s s u m e d to be u n i f o r m t h r o u g h o u t the p a r t i c l e , a n y heat exchange resistance b e i n g confined to a n outer transfer resistance is i n t e r n a l .

fluid

film,

a n d i n w h i c h a l l mass

B e c a u s e of the l a r g e Le, it is r e a s o n a b l e

i n the w o r k i n g m o d e l also to assume that the c o n c e n t r a t i o n profile is i n a p s e u d o - s t e a d y state at a l l t i m e s — t h a t is, that it is g o v e r n e d b y the steady f o r m of E q u a t i o n 3. M o r e is s a i d later r e g a r d i n g this m o d e l . M o s t theor e t i c a l studies h a v e not b e e n r e s t r i c t e d to these p r a c t i c a l ranges of c o n ditions a n d j u s t i f i a b l y so, for one s h o u l d a d o p t a b r o a d v i e w of

the

p r o b l e m as essentially b e i n g a m a t h e m a t i c a l d e s c r i p t i o n of a n y t w o state variables i n a general reaction-diffusion problem.

F o r example,

with

a p p r o p r i a t e k i n e t i c expressions a n d i n t e r p r e t a t i o n of the v a r i o u s p a r a m e ter g r o u p s , the equations s h o w n a b o v e m a y d e s c r i b e the c o n c e n t r a t i o n profiles of t w o species i n a n i s o t h e r m a l s y s t e m for w h i c h the m a g n i t u d e s of p a r a m e t e r groups c o r r e s p o n d i n g to Le

a n d Sh/Nu

m a y be

greater

t h a n , less t h a n , or e q u a l to u n i t y . T h e catalyst p a r t i c l e p r o b l e m d e s c r i b e d b y E q u a t i o n s 3 - 6 bears a great d e a l of s i m i l a r i t y to the C S T R p r o b l e m , the flow terms i n the latter r e p l a c e d b y d i f f u s i o n terms i n the f o r m e r .

Aris

similarities i n s t u d y i n g the p r o b l e m for Nu, Sh->

(81) oo.

explored

these

S o m e parameters

s u c h as Nu, Sh a n d a, do n o t h a v e o b v i o u s counterparts i n the C S T R p r o b l e m , b u t one w o u l d expect a priori that, l i k e the C S T R p r o b l e m , t h e catalyst p a r t i c l e w o u l d possess m u l t i p l e s t e a d y states a n d u n s t a b l e states


174

CHEMICAL

u n d e r some conditions.

If, i n fact,


i n t e r n a l gradients

are

neglected

e n t i r e l y or i f r e a c t i o n is a s s u m e d to o c c u r o n l y o n the e x t e r n a l surface a n d the t e m p e r a t u r e of the s o l i d is a s s u m e d u n i f o r m , the m a t h e m a t i c a l p r o b l e m takes o n a f o r m i d e n t i c a l to t h a t of the C S T R . T h e m u l t i p l i c i t y of states a n d s t a b i l i t y considerations for this v e r s i o n of the p r o b l e m w e r e studied b y W a g n e r (12) 86).

a n d others (see,

for e x a m p l e , Refs. 82, 83, a n d

C o m p u t a t i o n s of steady-state profiles b y W e i s z a n d H i c k s

O s t e r g a r d (88),

A m u n d s o n a n d R a y m o n d (89),

(87),

a n d others, f r o m e q u a -

tions s i m i l a r to E q u a t i o n s 3 a n d 4, h a v e d e m o n s t r a t e d the m u l t i p l i c i t y of states a n d h e n c e the m u l t i p l i c i t y of the effectiveness

factor w h e n i n t r a -


p a r t i c l e effects are i n v o l v e d . S o m e studies h a v e b e e n a i m e d at e s t a b l i s h i n g sufficient c o n d i t i o n s for u n i q u e n e s s a n d i n some instances sufficient conditions for s t a b i l i t y as w e l l . S u c h c o n d i t i o n s are p o t e n t i a l l y v a l u a b l e because the c o m p u t a t i o n s i n v o l v e d i n s e a r c h i n g for a l l possible steady profiles r e q u i r e the s o l u t i o n of a n o n l i n e a r b o u n d a r y v a l u e p r o b l e m — a f r e q u e n t l y difficult a n d e x p e n sive task. A s c e r t a i n i n g the s t a b i l i t y of a g i v e n profile is even m o r e p r o b lematic.

I n a p r a c t i c a l s i t u a t i o n , i f one c a n r u l e out the p o s s i b i l i t y of

m u l t i p l e solutions a n d i n s t a b i l i t i e s b y easily a p p l i e d sufficient c o n d i t i o n s , the d e s i g n a n d o p e r a t i o n of a c a t a l y t i c reactor is s o m e w h a t s i m p l i f i e d . M a n y s u c h c o n d i t i o n s are a v a i l a b l e — a l l r e s t r i c t e d to c e r t a i n cases a n d some untested insofar as t h e i r c o n s e r v a t i s m is c o n c e r n e d .

It is o n l y a

slight exaggeration to c l a i m that the l i s t i n g a n d c o m p a r i n g of these a n d m e n t i o n i n g the v a r i o u s m a t h e m a t i c a l m e t h o d s p u t to use i n d e r i v i n g t h e m w o u l d r e q u i r e a r e v i e w p a p e r itself. T o cite some of these: c r i t e r i a g i v e n b y L u s s (90)

for Nu, Sh->

oo a p p e a r to be the least conservative a n d the

most g e n e r a l i n the sense t h a t t h e y a p p l y to the various geometries that a n a r b i t r a r y f o r m of t h e rate expression c a n be J a c k s o n (91)

and

accommodated;

p r e s e n t e d sufficient c o n d i t i o n s for uniqueness for a case of

finite Nu a n d Sh for slab geometry.

K a s t e n b e r g (92)

showed that L u s s '

c r i t r i o n for uniqueness w i t h Nu, Sh —» oo also g u a r a n t e e d g l o b a l s t a b i l i t y for Le = for

1. L i o u et al. (93)

o b t a i n e d sufficient c o n d i t i o n s for s t a b i l i t y

Le^l. A w a r e of the results of the C S T R p r o b l e m , one w o u l d expect the

m u l t i p l i c i t y of solutions of the steady-state forms of E q u a t i o n s 3 - 6 to be three over s o m e ranges of the parameters a n d u n i t y over others.

While

this is the case w i t h Nu, Sh - » oo for the infinite slab a n d i n f i n i t e c y l i n d e r geometries, i t is not necessarily the case for spheres. T a k i n g Nu, Sh —» oo f o r a s p h e r i c a l p a r t i c l e , C o p e l o w i t z a n d A r i s (95)

f o u n d as m a n y as 15

steady s o l u t i o n profiles, a n d M i c h e l s e n a n d V i l l a d s e n ( 9 6 ) f o u n d 21 p r o files.

T h i s great m u l t i p l i c i t y was f o u n d o n l y over a v e r y n a r r o w p a r a m e t e r

r a n g e i n b o t h cases. E a r l i e r , H l a v a c e k a n d M a r e k (97)

had found

five

steady profiles f o r s p h e r i c a l geometry w i t h a z e r o - o r d e r k i n e t i c m o d e l .


7.

SCHMITZ

Multiplicity,

Stability,

M i c h e l s e n a n d V i l l a d s e n (96)

and

175

Sensitivity

s h o w e d that there m a y b e infinitely m a n y

states i n the s p h e r i c a l p a r t i c l e , a c o n c l u s i o n r e a c h e d also b y G e l r a n d a n d F u j i t a (99)

(98)

i n studies of a closely r e l a t e d m a t h e m a t i c a l p r o b

l e m . A l l of the solutions o b t a i n e d b y C o p e l o w i t z a n d A r i s a n d b y M i c h e l sen a n d V i l l a d s e n h a v e b e e n s h o w n s u b s e q u e n t l y to be u n s t a b l e except for the h i g h a n d l o w c o n v e r s i o n profiles (96,

100).

E v e n t h o u g h this

great m u l t i p l i c i t y exists o n l y over a v e r y n a r r o w a n d i m p r a c t i c a l range of p a r a m e t e r s a n d a l l b u t t w o profiles are u n s t a b l e , the fact t h a t t h e y do o c c u r is of great interest because ( 1 ) n e i t h e r g e o m e t r y effects nor m u l t i p l i c i t i e s greater t h a n three for a p r o b l e m of this t y p e are suggested Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

C S T R results a n d ( 2 )

by

they alert researchers to the fact that w h i l e the

well-studied C S T R p r o b l e m provides tremendous theoretical insight, not a l l q u a l i t a t i v e features of the b e h a v i o r of o p e n c h e m i c a l l y r e a c t i n g sys tems i n g e n e r a l are c o n t a i n e d t h e r e i n . O t h e r t h e o r e t i c a l discoveries, l a b e l l e d as s u r p r i s i n g w h e n against C S T R b e h a v i o r , a r e :

(1)

viewed

t h a t five steady-state solutions exist

u n d e r some c o n d i t i o n s w h e n Nu/Sh

< 1 (101)

and (2)

that i f ( a )

the

i m p o s e d c o n d i t i o n i n E q u a t i o n 5 at the p a r t i c l e center is r e m o v e d a n d the c o n d i t i o n g i v e n i n E q u a t i o n 6 is i m p o s e d i n s t e a d at χ = geometry,

(b)

Sh > Nu,

a n d b o t h are finite a n d ( c )

—1 for s l a b

the p r o b l e m as

d e s c r i b e d b y E q u a t i o n s 3 - 6 has three steady-state solutions, t h e n the slab m a y h a v e six a d d i t i o n a l profiles ( a c t u a l l y three sets of m i r r o r - i m a g e p a i r s ) w h i c h are not s y m m e t r i c a b o u t the centerline. T h e p o s s i b l e existence of s u c h profiles was s h o w n b y P i s m e n a n d K h a r a t s (102) (103).

a n d H o r n et al.

T h e a s y m m e t r i c profiles h a v e also b e e n the subject of

recent papers (104,

105, 106, 107, 108, 109).

other

T h i s r e s u l t is a p p a r e n t l y

g e o m e t r y d e p e n d e n t ; n o analogous findings for the infinite c y l i n d e r o r the sphere h a v e b e e n r e p o r t e d . A r i g o r o u s s t u d y of the s t a b i l i t y of steady-state solutions is u n w i e l d l y , c a l l i n g for a n analysis of the eigenvalues of a n o n s e l f - a d j o i n t system i n w h i c h coefficients are functions of steady-state profiles. T h e p r o b l e m is s i m p l i f i e d i n cases f o r w h i c h Le is t a k e n to b e infinity, u n i t y or z e r o w i t h Nu = Sh because a l l eigenvalues c a n be s h o w n to be r e a l , a n d w e l l - k n o w n m a t h e m a t i c a l theorems

can be

applied.

C o n s i d e r a t i o n of

large-scale

transients r e q u i r e s s t u d y i n g the solutions of c o u p l e d n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations i n general. S o m e examples of s t a b i l i t y analyses a n d transient s i m u l a t i o n s i n v o l v i n g v a r i o u s a n a l y t i c a l a n d n u m e r i c a l tools a n d b a s e d o n a n u m b e r of v a r i a t i o n s of the p r o b l e m expressed b y E q u a t i o n s 3 - 6 are c o n t a i n e d i n Refs. 83, 84, 85, 89,100,110, c l u d e d i n some of these (112, 113)

111, 112, 113,114).

In

are n u m e r i c a l solutions for cases w i t h

Le < 1 h a v i n g a u n i q u e u n s t a b l e profile a b o u t w h i c h the state v a r i a b l e s oscillate c o n t i n u o u s l y . T h e oscillations r e s e m b l e l i m i t cycles i n the C S T R problem.

I n g e n e r a l , studies of steady-state s t a b i l i t y a n d of l a r g e scale


176

CHEMICAL


transient b e h a v i o r f o r t h e catalyst p a r t i c l e p r o b l e m h a v e s h o w n n o b e h a v i o r that is not analogous to C S T R results. T h i s posteriori

o b s e r v a t i o n is

n o t meaningless because ( 1 ) there is n o g e n e r a l t h e o r e m f o r d i s t r i b u t e d m o d e l s w h i c h ensures that l i n e a r i z e d equations y i e l d t h e correct answers e v e n f o r i n f i n i t e s i m a l p e r t u r b a t i o n s as d o L i a p u n o v ' s theorems f o r l u m p e d m o d e l s a n d ( 2 ) Poincaré's theorems g u a r a n t e e i n g l i m i t cycles i n a phase p l a n e f o r second-order l u m p e d m o d e l s h a v i n g a u n i q u e u n s t a b l e state d o n o t necessarily c a r r y over to d i s t r i b u t e d p r o b l e m s . C e r t a i n p h y s i c a l effects n o t d e s c r i b e d i n E q u a t i o n s 3 - 6 h a v e b e e n s t u d i e d i n some p u b l i c a t i o n s . T h e s e i n c l u d e n o n u n i f o r m e x t e r n a l c o n d i Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

tions (94, 115) to w h i c h a p a r t i c l e i n a fixed b e d reactor w i l l g e n e r a l l y b e s u b j e c t e d a n d changes i n m o l a r d e n s i t y as a result of c h e m i c a l r e a c t i o n (116,

117).

T h e s e a n d other s u c h effects w o u l d b e e x p e c t e d to g i v e r i s e

o n l y to q u a n t i t a t i v e changes, w h i c h m a y b e i m p o r t a n t i n p r a c t i c a l s i t u a tions b u t w o u l d n o t b e e x p e c t e d to ( a n d h a v e n o t b e e n s h o w n t o ) a d d or t a k e a w a y a n y c h a r a c t e r i s t i c features of t h e p r o p e r t i e s of solutions to E q u a t i o n s 3 - 6 . T h e p a p e r b y J a c k s o n (117) is of f u n d a m e n t a l interest, h o w e v e r , because i t points o u t that a d d i t i o n a l terms m a y b e l o n g i n t h e transient equations i f a change i n m o l a r i t y is associated w i t h t h e c h e m i c a l reaction. I t w a s p o i n t e d out a b o v e that e v e n t h o u g h w i d e p a r a m e t e r ranges h a v e b e e n e x p l o r e d i n t h e o r e t i c a l studies, p r a c t i c a l considerations these ranges c o n s i d e r a b l y f o r n o n i s o t h e r m a l p r o b l e m s .

limit

A very useful,

s i m p l e a n d p l a u s i b l e m o d e l f o r p r a c t i c a l purposes m a y b e d e d u c e d

from

E q u a t i o n s 3 - 6 i f i t is a s s u m e d that a l l of t h e resistance to heat exchange resides i n t h e external fluid film a n d t h a t t h e species profile is a l w a y s i n a pseudo-steady

state r e l a t i v e to t h e instantaneous t e m p e r a t u r e profile.

I n v o k i n g these assumptions, one m a y integrate E q u a t i o n 4 t e r m b y t e r m over t h e p a r t i c l e v o l u m e to o b t a i n

L

e

ΪΡ

= ( Β —> C a n d f o u n d five steady-state profiles i n some cases. C r e s s w e l l a n d P a t e r s o n (120) ( 121,122,

and McGreavy and Thornton

126 ) also w o r k e d w i t h m o d e l s w h i c h a c c o u n t e d f o r m o r e t h a n

one r e a c t i o n i n s t u d y i n g m u l t i p l i c i t y , s t a b i l i t y , a n d s e n s i t i v i t y p r o b l e m s . H a r t m a n et al. ( 128 ) i n c o r p o r a t e d a d u a l - s i t e i s o t h e r m a l L a n g m u i r - H i n s h e l w o o d m o d e l i n t o t h e diffusion equations a n d s h o w e d that three steady states w e r e possible.

Similar problems were worked b y Mitshka a n d

S c h n e i d e r (129, 130).

S u c h reactions, n o t o b v i o u s l y a u t o c a t a l y t i c i n a

strict sense, s h o w a c c e l e r a t i n g b e h a v i o r as t h e r e a c t i o n extent increases o w i n g to a d e p l e t i o n of a reactant w h i c h tends to d o m i n a t e t h e a c t i v e sites.


178

CHEMICAL

M e h t a a n d A r i s (131,

132)


p r e s e n t e d a d e t a i l e d s t u d y of the p t h - o r d e r

i s o t h e r m a l r e a c t i o n a n d s h o w e d m u l t i p l i c i t y for ρ < 0. Lumped Models.

T h e emphasis t h r o u g h the p r e c e d i n g section was

o n t h e o r e t i c a l studies w h i c h u s e d E q u a t i o n s 3 - 6 deduced therefrom)

( o r of s p e c i a l cases

a n d w h i c h u s e d m e t h o d s of a n a l y z i n g t h e m or of

g e n e r a t i n g n u m e r i c a l solutions.

I n recent years m u c h effort has

been

d e v o t e d to the d e v e l o p m e n t a n d a p p l i c a t i o n of m e t h o d s w h i c h c o n v e r t the b a s i c d i s t r i b u t e d m o d e l to a n a p p r o x i m a t e l u m p e d m o d e l — a process to w h i c h the t e r m l u m p i n g has b e e n a p p l i e d .

[Some workers have used

the t e r m l u m p i n g to i n d i c a t e the r e p l a c e m e n t of a p a r t i a l d i f f e r e n t i a l


e q u a t i o n i n a n a p p r o x i m a t e sense b y a single o r d i n a r y d i f f e r e n t i a l e q u a t i o n . I t seems m o r e consistent w i t h u s u a l t e r m i n o l o g y to refer to l u m p i n g g e n e r a l l y as the process of c o n v e r t i n g a d i s t r i b u t e d system to a l u m p e d system, e v e n t h o u g h the n u m b e r of equations i n the latter m a y exceed t h a t of the f o r m e r . ]

E s s e n t i a l l y these m e t h o d s i n v o l v e r e p l a c i n g those terms

c o n t a i n i n g d e r i v a t i v e s w i t h respect to p o s i t i o n b y a l i n e a r c o m b i n a t i o n of t h e d e p e n d e n t v a r i a b l e e v a l u a t e d at specific s p a t i a l positions. T h e n u m b e r Ν of s u c h positions is a r b i t r a r y except insofar as l i m i t a t i o n s m a y b e p l a c e d o n the a c c u r a c y of solutions; p r e s u m a b l y as Ν is i n c r e a s e d , t h e exact s o l u t i o n is a p p r o a c h e d .

T h u s a single p a r t i a l d i f f e r e n t i a l e q u a t i o n is

c o n v e r t e d to Ν o r d i n a r y d i f f e r e n t i a l equations. T h e advantages of w o r k i n g w i t h the l u m p e d v e r s i o n are o b v i o u s : t h e k n o w l e d g e g a i n e d f r o m studies of the C S T R p r o b l e m c a r r y over d i r e c t l y , a n d m a t h e m a t i c a l tools f o r s t u d y i n g n e w b e h a v i o r are r e a d i l y a v a i l a b l e . F o u r m e t h o d s of l u m p i n g h a v e b e e n d e s c r i b e d : (2)

difference m e t h o d s , ( 3 )

(1)

linearization,

orthogonal collocation, a n d (4)

averaging.

A l l f o u r h a v e b e e n a p p l i e d to the catalyst p a r t i c l e p r o b l e m i n c o n n e c t i o n w i t h studies of steady-state m u l t i p l i c i t y a n d s t a b i l i t y .

The

orthogonal

c o l l o c a t i o n m e t h o d has r e c e i v e d the greatest a m o u n t of attention. It was first u s e d for s t u d y i n g t h e catalyst p a r t i c l e p r o b l e m b y V i l l a d s e n a n d S t e w a r t ( 1 3 3 ) , w h o s h o w e d that s p a t i a l d e r i v a t i v e operators c o u l d

be

r e p l a c e d b y m a t r i c e s , the elements of w h i c h d e p e n d o n the c o l l o c a t i o n points.

T h e c o l l o c a t i o n p o i n t s are u s u a l l y t a k e n to b e the roots of a n

o r t h o g o n a l p o l y n o m i a l . O r t h o g o n a l c o l l o c a t i o n is one of t h e m e t h o d s of w e i g h t e d r e s i d u a l s a n d is d e s c r i b e d i n d e t a i l , w i t h examples, b y F i n l a y s o n ( 134 ). T h e first a p p l i c a t i o n of the m e t h o d to steady-state m u l t i p l i c i t y i n a catalyst p a r t i c l e w a s r e p o r t e d b y S t e w a r t a n d V i l l a d s e n (135).

It has

s u b s e q u e n t l y b e e n u s e d i n m a n y studies of the catalyst p a r t i c l e p r o b l e m , i n c l u d i n g those r e p o r t e d i n Refs. 136, 137, 138, 139, 140, 141, 142,

143,

w h i c h a d d r e s s e d p r o b l e m s of steady-state m u l t i p l i c i t y , s t a b i l i t y , a n d t o t a l transient b e h a v i o r .

I n most cases tested, the o r i g i n a l p a r t i a l d i f f e r e n t i a l

equations m a y b e r e p r e s e n t e d b y a r a t h e r l o w o r d e r system of o r d i n a r y d i f f e r e n t i a l equations b y the c o l l o c a t i o n m e t h o d .

R a r e l y are m o r e t h a n


7.

SCHMITZ

Multiplicity,

Stability,

and

179

Sensitivity

three or f o u r c o l l o c a t i o n points r e q u i r e d , a l t h o u g h V a n D e n B o s c h a n d P a d m a n a b h a n (144)

f o u n d it necessary to use a 14-point r e p r e s e n t a t i o n

to attain g o o d a c c u r a c y i n c a l c u l a t i n g v e r y steep steady profiles a n d a seventh-order system for s t a b i l i t y d e t e r m i n a t i o n s . L i n e a r i z a t i o n a n d difference H l a v a c e k a n d others

methods

have been used m a i n l y b y

at the P r a g u e s c h o o l

(79, 145,

146,

147,

148).

[ L i n e a r i z a t i o n here is the t e r m associated w i t h the p a r t i c u l a r l u m p i n g m e t h o d p r o p o s e d b y H l a v a c e k a n d c o - w o r k e r s i n the papers c i t e d .

It

has n o t h i n g to do w i t h l i n e a r i z i n g n o n l i n e a r terms i n the system e q u a tions.] Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

T h e a v e r a g i n g m e t h o d was i n t r o d u c e d b y L u s s a n d L e e (149)

in

e x a m i n i n g the s t a b i l i t y of the steady-state profiles for a p a r t i c l e w h o s e t e m p e r a t u r e was a s s u m e d u n i f o r m .

It b a s i c a l l y involves r e p l a c i n g the

terms i n the o r i g i n a l d i s t r i b u t e d m o d e l b y q u a n t i t i e s a v e r a g e d over the p a r t i c l e v o l u m e or over subsections of i t . T h e s p a t i a l d e r i v a t i v e operators take o n the f o r m of o v e r a l l transfer coefficients. u s e d b y L e e et al.

T h i s m e t h o d w a s also

(150).

I n m a n y of the p u b l i c a t i o n s c i t e d above, comparisons are m a d e of one l u m p i n g m e t h o d vs. others or of results a n d c o m p u t a t i o n a l l a b o r asso c i a t e d w i t h l u m p i n g methods vs. those of other m e t h o d s of h a n d l i n g the o r i g i n a l p a r t i a l d i f f e r e n t i a l equations (see, v a t i o n is that

first-order

for e x a m p l e , Refs. 79, 80,

140,

A v e r y i n t e r e s t i n g a n d p r o m i s i n g obser

141,142,143,144,146,147,149).

l u m p i n g is s u i t a b l e for m a n y purposes.

In

first-

order l u m p i n g w i t h the c o l l o c a t i o n m e t h o d , for e x a m p l e , one w o u l d use o n l y one i n t e r i o r c o l l o c a t i o n p o i n t . I n this a p p r o a c h the p a r t i a l differen t i a l equations i n E q u a t i o n s 3 a n d 4 are e a c h r e p l a c e d b y a single o r d i n a r y d i f f e r e n t i a l e q u a t i o n , the s p a t i a l d e r i v a t i v e t e r m i n E q u a t i o n 3 r e p l a c e d b y A ( l — c ), c

s

and in Equation 4 by A ( l — f ). t

s

T h e coefficients A a n d c

A d e p e n d o n the p a r t i c u l a r l u m p i n g m e t h o d u s e d , a n d c a n d t are c o n H

t

s

s i d e r e d to be average values or values at a specific p o i n t . I n the o r t h o g o n a l c o l l o c a t i o n m e t h o d , c a n d t are a p p r o x i m a t i o n s to t h e exact solutions s

s

at the single i n t e r i o r c o l l o c a t i o n p o i n t , a n d A a n d A are g i v e n b y the c

t

following relationships

λο

where n i and n

2

(10)

=

d e p e n d o n the p a r t i c l e geometry.

Equations 3-6 then

r e d u c e s i m p l y to = X (1 c

c) -

φ c 2

s

exp

and


(11)

180

C H E M I C A L

Le ~p = λ ,

(1

-

R E A C T I O N

Q + β' φ c exp 2

a

[ 7

E N G I N E E R I N G

R E V I E W S

( l -

(12)

T h e s e are i m m e d i a t e l y r e c o g n i z e d as b e i n g e q u i v a l e n t to t h e a d i a b a t i c C S T R p r o b l e m ; i n fact, t h e y c a n r e a d i l y b e p u t i n t h e f o r m of E q u a t i o n s 1 a n d 2 w i t h L r e p l a c e d b y Le Ac/At, Da b y A , a n d β b y / ? ' A / A . T h e n 2

c

C

t

a l l of the i n f o r m a t i o n c o n t a i n e d i n F i g u r e s 1 a n d 2 a n d i n t h e e a r l i e r d i s c u s s i o n r e l a t e d to t h e m c a r r y o v e r to the present s i t u a t i o n . I n fact, several of t h e p u b l i c a t i o n s r e f e r r e d to earlier h a v e u s e d equations of t h e f o r m of E q u a t i o n s 11 a n d 12 a n d h a v e d i s p l a y e d m a n y of t h e transient


features p r e v i o u s l y e x p l o r e d for the C S T R . I f o n e judges the a d e q u a c y of s i m p l i f i e d versions of the p r o b l e m , s u c h as t h a t d e s c r i b e d b y E q u a t i o n s 11 a n d 12 o r of other l o w - o r d e r l u m p i n g forms i n terms of q u a n t i t a t i v e agreement w i t h t h e o r i g i n a l p r o b l e m , h e w o u l d c o n c l u d e that they are i n a d e q u a t e i n m a n y situations (see,

for

e x a m p l e , s u c h c o m p a r i s o n s i n t h e papers b y V a n d e n B o s c h a n d P a d m a n a b h a n (144).)

R e m e m b e r , h o w e v e r , that t h e o r i g i n a l equations are

themselves h i g h l y i d e a l i z e d m a t h e m a t i c a l descriptions of a c o m p l i c a t e d u n d e r l y i n g p h y s i c a l system.

W h i l e they m a y be basically sound for

r e a c t i o n i n a system of p a r a l l e l pores of e q u a l cross-section, l e n g t h , a n d a c t i v i t y , there has b e e n v e r y l i t t l e e x p e r i m e n t a l testing of t h e i r a d e q u a c y i n d e s c r i b i n g t h e q u a n t i t a t i v e b e h a v i o r of a h i g h l y n o n l i n e a r r e a c t i o n process i n structures r e s e m b l i n g t y p i c a l c o m m e r c i a l catalysts. T h e r e f o r e , the o r i g i n a l d e s c r i p t i o n itself s h o u l d b e v i e w e d as u s e f u l m a i n l y f o r q u a l i t a t i v e purposes at present, a n d there is l i t t l e justification f o r t e r m i n g as i n a d e q u a t e t h e s i m p l i f i e d l u m p e d versions as l o n g as t h e y p r e d i c t t h e same q u a l i t a t i v e trends. F r o m E q u a t i o n s 11 a n d 12, one w o u l d c o n c l u d e , f o r e x a m p l e , b y a n a l o g y to t h e C S T R p r o b l e m t h a t s u s t a i n e d o s c i l l a t o r y b e h a v i o r is n o t possible as l o n g as Le > 1 i f Sh = Nu o r i f Nu, Sh - » oo ; i n these cases t h e slope c o n d i t i o n is necessary a n d sufficient f o r s t a b i l i t y d e t e r m i n a t i o n s . T h e y l e a d to the same c o n c l u s i o n for finite a n d u n e q u a l values of t h e N u s s e l t a n d S h e r w o o d n u m b e r s unless Nu > Sh.

These

results a r e consistent w i t h present k n o w l e d g e of t h e b e h a v i o r of solutions of E q u a t i o n s 3 - 6 .

C e r t a i n q u a l i t a t i v e features of t h e solutions of the

o r i g i n a l d i s t r i b u t e d system, h o w e v e r , c a n n o t b e p r e d i c t e d b y t h e s i m p l e s t l u m p e d f o r m , a n d i t is i m p o r t a n t t h a t researchers o r p o t e n t i a l users of t h e s i m p l i f i e d forms b e a w a r e of these.

F o r e x a m p l e , s u c h features d i s

cussed e a r l i e r as a s y m m e t r i c steady profiles a n d a m u l t i p l i c i t y greater t h a n three f o r a

first-order

Arrhenius reaction require a higher-order

a p p r o x i m a t i o n . I t also appears t h a t e x t r e m e l y steep c o n c e n t r a t i o n profiles are n o t easily d e s c r i b e d b y these m e t h o d s .

F o r these, a t w o - z o n e m o d e l

d e s c r i b e d b y P a t e r s o n a n d C r e s s w e l l (139)

appears to b e a g o o d a l t e r n a

tive. I n this m o d e l , a r e a c t i o n z o n e t h r o u g h w h i c h t h e c o n c e n t r a t i o n of


7.

SCHMITZ

Multiplicity,

Stability,

and

181

Sensitivity

1.0 0.8 h

Vto

0.6 0.4 h 0.2 h 0

0

0.2

0.4

0.6

0.8

10


X Chemical Engineering Science

Figure 8. Multiple steady-state concentration profiles in a catalyst slab obtained through numerical integration ( ) and orthogonal collocation: · ( N = 1), A ( N = 2), M(N = 3). L e = J ; γ = 20; β' = 0.7; φ = 0.16; N u = Sh = oo (140).

Figure 9. Sustained oscillations in a spherical particle obtained through or thogonal collocation ( ) with Ν = 1 by Hellinckx et al. (140) and through numerical solution (·) by Lee and Luss (113). L e = 0.1; y = 30; β' = 0.15; φ = 1.1; N u = Sh = oo. a l i m i t i n g reagent decreases to z e r o is a s s u m e d to exist n e a r t h e p a r t i c l e surface a n d is d e s c r i b e d m a t h e m a t i c a l l y b y l o w - o r d e r c o l l o c a t i o n e q u a tions. C h e m i c a l r e a c t i o n does not take p l a c e i n the i n n e r z o n e . V a n d e n B o s c h a n d P a d m a n a b h a n (144)

f o u n d this t e c h n i q u e to b e u s e f u l i n

d e s c r i b i n g steep gradients a n d i n p r e d i c t i n g t h e results of H a t f i e l d a n d A r i s ( m e n t i o n e d p r e v i o u s l y ) w h i c h s h o w a m u l t i p l i c i t y of

five.


182

CHEMICAL


E v i d e n c e that q u a l i t a t i v e l y a n d q u a n t i t a t i v e l y accurate i n f o r m a t i o n m a y b e o b t a i n e d i n s o m e cases w i t h s i m p l e l u m p e d m o d e l s w a s s h o w n w e l l b y H e l l i n c k x et al. (140),

from w h i c h Figures 8 a n d 9 were taken

for illustration. A l l i n a l l , t h e results of studies of s i m p l i f i e d m o d e l s are e n c o u r a g i n g . L i t t l e significant progress is i m m i n e n t a l o n g t h e lines of e x t e n d i n g o u r g e n e r a l k n o w l e d g e to m o r e c o m p l e x c h e m i c a l reactions or of p u t t i n g o u r present k n o w l e d g e to p r a c t i c a l use i n d e s i g n w o r k unless these or other simplifications c a n b e p r o v e d u s e f u l . Related Problems. C l o s e l y r e l a t e d to t h e p r o b l e m s of porous catalyst Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

p a r t i c l e s are those w i t h r e a c t i o n o n e x t e r n a l surfaces i n c l u d i n g c a t a l y t i c w i r e s a n d gauzes. T h e latter at least h a v e l o n g b e e n of c o n c e r n i n i n d u s t r i a l a p p l i c a t i o n s . I n fact, t h e t h r u s t of L i l j e n r o t h ' s p a p e r i n 1918

(11)

was t o w a r d t h e i g n i t i o n a n d e x t i n c t i o n characteristics of a m m o n i a o x i d a t i o n o n a c a t a l y t i c gauze.

I n recent years, L u s s a n d co-workers

studied

with

various problems

m u l t i p l i c i t y a n d s t a b i l i t y (151),

catalytic wires temperature

a n d s t a n d i n g w a v e solutions (155). d a t a ( 151,153,154).

flickering

have

steady-state

(152, 153,

154),

S o m e of these i n c l u d e e x p e r i m e n t a l

F l i c k e r i n g , a cause of inefficient i n d u s t r i a l o p e r a t i o n ,

is a p p a r e n t l y c a u s e d b y c o n c e n t r a t i o n or m i x i n g p i n g i n g stream.

including

R a y et al. (156)

fluctuations

i n the i m -

p o i n t e d o u t that t h e l a r g e t h e r m a l

c a p a c i t y of t h e w i r e s a n d gauzes w o u l d most l i k e l y cause a l l oscillations a r i s i n g f r o m i n t r i n s i c i n s t a b i l i t i e s to d a m p . M a t h e m a t i c a l solutions w h i c h c o r r e s p o n d to s p a t i a l l y s t a n d i n g waves a l o n g t h e w i r e l e n g t h i n cases of a n infinite l e n g t h o r i n s u l a t e d ends are possible b u t h a v e b e e n s h o w n to b e u n s t a b l e (155,

157).

T h e p r o b l e m of steady-state m u l t i p i l i c i t y a n d s t a b i l i t y w i t h diffusion a n d c o n d u c t i o n to a s p h e r i c a l c a t a l y t i c surface has b e e n s t u d i e d i n d e t a i l ( 158,159,160).

A cognate subject is that of n o n c a t a l y t i c

fluid-solid

reac-

tions. I t is n o t possible here to p r o b e the d i s t i n g u i s h i n g characteristics of the m a t h e m a t i c a l models a n d the solutions f o r s u c h systems.

Discussions

of t h e m as w e l l as l i t e r a t u r e references are i n Refs. 161, 162, 163. Experimental Studies.

There have been only a few published re-

ports of experiments w i t h single catalyst particles t h a t h a v e d e m o n s t r a t e d steady-state m u l t i p l i c i t y a n d / o r i n s t a b i l i t i e s . T h o s e of w h i c h t h e a u t h o r is a w a r e a r e l i s t e d a l o n g w i t h b r i e f descriptions i n T a b l e I I . E x c e p t for the s t u d y of c a r b o n m o n o x i d e o x i d a t i o n b y B e u s c h et al. (first e n t r y i n the t a b l e ) , a l l of the studies i n v o l v e d l a r g e t h e r m a l effects, a n d t h e p r i m a r y e x p e r i m e n t a l m e a s u r e m e n t w a s of t h e t e m p e r a t u r e at the center of the catalyst p a r t i c l e . C o m m o n l y the e x p e r i m e n t a l setup w a s one i n w h i c h a gas s t r e a m c o n t a i n i n g t h e reactants

flowed

catalyst p a r t i c l e b y f o r c e d c o n v e c t i o n .

I n some of t h e experiments b y

past a freely

suspended

W i c k e ' s g r o u p , t h e p a r t i c l e w a s i m b e d d e d i n a l a y e r of i n e r t p a r t i c l e s .


7.

SCHMITZ Table II.

Multiplicity,

Stability,

and

183

Sensitivity

Experimental Studies of Steady-State Multiplicity and Instabilities with Single Catalyst Particles"

Reference

Experimental

1. B e u s c h , F i e g u t h , a n d W i c k e , 1970, 1972 (164,165)

System

o

m

(a) O x i d a t i o n of h y d r o g e n o n suspended P t / s i l i c a / a l u m i n a particle (b) O x i d a t i o n of h y d r o g e n o n single p a r ticle i m b e d d e d i n a l a y e r of i n e r t particles (c) O x i d a t i o n of C O (nearly isothermal) o n suspended P t / A l 0 3 pellet O x i d a t i o n of h y d r o g e n o n a suspended P t / A l 0 catalyst particle E t h y l e n e hydrogénation o n a suspended pellet of A d k i n ' s c a t a l y s t O x i d a t i o n of h y d r o g e n o n a P t / a l u m i n a pellet


2

2. H o r a k a n d J i r a c e k , 1970 (166) 3. F u r u s a w a a n d K u n i i , 1971 (167) 4. J i r a c e k , H a v l i c e k , and Horak, 1971 (168) 5. H o r a k a n d J i r a c e k , 1972 (61) 6. J i r a c e k , H o r a k , and H a j k o v a , 1973 (169) a

2

Oxidation particle tubular Oxidation particle

3

of h y d r o g e n o n P t / a l u m i n a i n a b a c k m i x reactor a n d flow reactor of h y d r o g e n o n a P t / a l u m i n a i n a b a t c h recycle-flow reactor

Column headings are defined in Table I. A l t h o u g h these e x p e r i m e n t a l reports d o not c o n t a i n t h o r o u g h q u a n t i -

tative c o m p a r i s o n s w i t h t h e o r e t i c a l p r e d i c t i o n s , the q u a l i t a t i v e features of the l a b o r a t o r y observations are r e a d i l y a p p r e c i a t e d i n v i e w of k n o w n theoretical behavior. T h e c a r e f u l e x p e r i m e n t a l w o r k of W i c k e a n d his c o - w o r k e r s 165)

warrants special comment.

(164,

F i g u r e s 10 a n d 11 demonstrate some of

t h e i r results, i n c l u d i n g t e m p e r a t u r e oscillations w i t h the h y d r o g e n - o x y g e n reaction

( F i g u r e 10)

a n d m u l t i p l e steady states i n experiments

c a r b o n m o n o x i d e o x i d a t i o n ( F i g u r e 1 1 ) . A recent p a p e r b y W i c k e

with (170)

s u m m a r i z e s these a n d other c o n t r i b u t i o n s f r o m his g r o u p at M u n s t e r . T h a t o s c i l l a t o r y states exist is n o t s u r p r i s i n g i n v i e w of

aforementioned

t h e o r e t i c a l studies, b u t the oscillations o b s e r v e d i n these experiments do n o t seem to b e e x p l a i n a b l e i n terms of interactions of t r a n s p o r t processes — i n t e r n a l a n d e x t e r n a l — a n d w i t h a s i m p l e one-step k i n e t i c m e c h a n i s m of the t y p e a s s u m e d i n a l l t h e o r e t i c a l studies of p a r t i c l e oscillations to date. B e u s c h et al. (164,165) suggested that c o m p l i c a t e d surface k i n e t i c s , n o t r e d u c i b l e to a single rate expression, are i n v o l v e d . I t is also possible that the a d s o r p t i v e species c a p a c i t a n c e of the c a t a l y t i c surface, w h i l e n o t e n t e r i n g i n t o the steady-state p i c t u r e , affects the s t a b i l i t y characteristics a p p r e c i a b l y . T h i s c a p a c i t a n c e , w h i c h is u n a c c o u n t e d for i n t h e p r e c e d i n g equations, is c o m m o n l y o v e r l o o k e d .


184

CHEMICAL



Time (min) Figure 10. Temperature oscillations in a single 8-mm spherical catalyst particle (Ft/ silica/alumina) during hydrogen oxidation in air (3.14 vol % H,) (164, 165)

0

1

2

3

4

5

6

vol. % CO in feed Figure 11. Multiple steady states (hysteresis) in the oxidation of CO in air in a single 3X3 mm cylindrical catalyst particle (Pt/Al 0 ) (164, 165) 2

3

A n e x p e r i m e n t a l s t u d y b y B e n h a m a n d D e n n y (171)

deserves

com-

m e n t because i t i n c l u d e s q u a n t i t a t i v e c o m p a r i s o n s of e x p e r i m e n t a l measu r e m e n t s of u n s t e a d y t e m p e r a t u r e profiles w i t h t h e o r e t i c a l p r e d i c t i o n s . T h e experiments i n v o l v e d t h e o x i d a t i o n of c a r b o n m o n o x i d e o n a c y l i n d r i c a l p e l l e t of C u O / A l 0 2

mocouples

3

catalyst i n f o r c e d c o n v e c t i o n

w e r e i m b e d d e d i n the pellet.

flow.

Seven ther-

T h e w o r k w a s n o t a i m e d at

s t u d y i n g steady-state m u l t i p l i c i t y or i n s t a b i l i t i e s ; i t is p e r t i n e n t to this d i s c u s s i o n because i t demonstrates the d i f f i c u l t y a n d p e r h a p s t h e a c c u r a c y


7.

SCHMITZ

Multiplicity,

Stability,

and

Sensitivity

185

w h i c h one m i g h t expect to find i n a t t e m p t i n g to p r e d i c t catalyst p a r t i c l e behavior quantitatively. B e n h a m a n d D e n n y point out that variations i n transfer rates o v e r the p a r t i c l e surface l e a d to m a r k e d l y n o n i s o t h e r m a l a n d a s y m m e t r i c p a r t i c l e profiles. T h e s e measurements w i t h a r e a l catalyst p a r t i c l e therefore a d d emphasis to some of t h e c o m m e n t s a b o v e to the effect that the u s u a l m a t h e m a t i c a l m o d e l g i v e n b y E q u a t i o n s 3 - 6 be considered

u s e f u l p r i m a r i l y for q u a l i t a t i v e purposes

should

a n d that this

s h o u l d be r e m e m b e r e d w h e n s i m p l i f i e d versions are e v a l u a t e d . I n o t h e r e x p e r i m e n t a l studies w i t h a single catalyst p a r t i c l e , K e h o e a n d B u t t (172, 173)

d e m o n s t r a t e d reasonable, t h o u g h r o u g h ,

agreement


w i t h results c a l c u l a t e d f r o m a m o d e l s i m i l a r to that g i v e n b y E q u a t i o n s 3 a n d 4. N o steady-state m u l t i p l i c i t y or i n s t a b i l i t i e s w e r e o b s e r v e d i n those experiments.

C e r t a i n l y there is a n e e d for f u r t h e r c a r e f u l e x p e r i m e n t a l

w o r k o n these p r o b l e m s . Tubular

and Fixed-Bed

Mathematical

Reactors

Models and Theoretical Studies.

N e a r l y a l l of

the

m a t h e m a t i c a l models w h i c h h a v e b e e n u s e d i n studies of m u l t i p l e states, s t a b i l i t y , a n d s e n s i t i v i t y of t u b u l a r or

fixed-bed

reactors

use

balance

equations o n the fluid phase of the f o l l o w i n g f o r m or of f o r m s r e a d i l y deduced

therefrom: dCf m

1

1

dc

d Cf

w. w ~ w ~

=

2

{

D

' n Q

1

(

v

1 • À S' -1 + * - oo, E q u a t i o n s 1 3 - 1 6 , 19, a n d 2 0 h a v e a u n i q u e s o l u c

t

tion for given conditions i f f

a

is constant o r w i t h c o c u r r e n t f l o w of a

coolant. O t h e r reactor configurations, h o w e v e r , s u c h as those w i t h c o u n t e r c u r r e n t c o o l i n g , heat exchange b e t w e e n effluent a n d f e e d , a n d w i t h r e c y c l e streams, w h i c h p r o v i d e a means of t h e r m a l f e e d b a c k , c a n g i v e ris to steady-state m u l t i p l i c i t y a n d s t a b i l i t y p r o b l e m s . V a n H e e r d e n

(7,8)

a n d B i l o u s a n d A m u n d s o n (212) c o n s i d e r e d s o m e of these cases.

There

h a v e b e e n a n u m b e r of other t h e o r e t i c a l studies of these p r o b l e m s ( 146, 191, 196, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225 I n t h e r e c e n t l i t e r a t u r e these m o d e l s

226, 227, 228, 229, 230, 231, 232). Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

h a v e r e c e i v e d r e l a t i v e l y little a t t e n t i o n , b u t m u l t i p l i c i t y a n d s t a b i l i t y p r o b l e m s c a u s e d b y t h e effects c o n t a i n e d i n t h e m m a y b e m u c h m o r e f r e q u e n t l y e n c o u n t e r e d i n present i n d u s t r i a l processes t h a n b y t h e effects of a x i a l d i s p e r s i o n o r r e a c t i o n i n porous catalyst p a r t i c l e s . A n e x a m p l e c o m m o n l y r e f e r r e d to is the synthesis of a m m o n i a i n p a c k e d reactors w i t h heat exchange b e t w e e n the r e a c t i n g fluid a n d the f e e d (7, 214, 215, 229). I n t h e s i m p l e h o m o g e n e o u s p l u g flow reactor m o d e l w i t h o u t a n y m e c h a n i s m of t h e r m a l feedback,

severe d e s i g n p r o b l e m s w i t h r e a c t o r

s e n s i t i v i t y m a y b e e n c o u n t e r e d even t h o u g h steady states a r e stable a n d unique.

S e n s i t i v i t y to s m a l l changes i n a p a r a m e t e r o r o p e r a t i n g c o n d i

t i o n is v e r y l i k e l y t h e e x p l a n a t i o n f o r m o s t i n c i d e n t s of " r u n a w a y " c o n d i tions i n i n d u s t r i a l reactors. B a r k e l e w (233) first p r e s e n t e d some g u i d e lines f o r d e s i g n i n g reactors to a v o i d r u n a w a y s c a u s e d b y sensitivity. O t h e r c r i t e r i a f o r a v o i d i n g e x t r e m e l y sensitive c o n d i t i o n s h a v e g i v e n b y D e n t e a n d C o l l i n a (234), senaere

a n d F r o m e n t (236). 70

and V a n W e l -

T h e agreement between

these v a r i o u s

1

60 -

ο 0^

50

À

i / //

1

1

€D /DS

- ψ

0.2004-

40 30

been

H l a v a c e k et al. (235),

1 a n d Pe , Pe - » oo, s u i t a b l y c

m o d i f i e d to i n c l u d e m o r e t h a n one r e a c t i o n .

t

T h e results p r e d i c t e d b y

t h e i r m o d e l s h o w v e r y g o o d agreement w i t h t h e i r e x p e r i m e n t a l d a t a . I n


7.

SCHMITZ

Multiplicity,

t h e i r experiments Pe

c

Stability,

and

195

Sensitivity

v a r i e d f r o m 300 to 1200, c o n s i d e r a b l y b e y o n d the

v a l u e f o r w h i c h one w o u l d expect to encounter a n y a p p r e c i a b l e effect of a x i a l d i s p e r s i o n a n d , therefore, steady-state m u l t i p l i c i t y . I n the experiments of H a n s e n a n d Jorgenson, the P e c l e t n u m b e r w a s s o m e w h a t l o w e r (Pe

c

~ 270, Pe

t

~ 1 1 5 ) , a n d these w o r k e r s u s e d a h o m o -

geneous a x i a l d i s p e r s i o n m o d e l to describe transients i n t h e h y d r o g e n o x y g e n r e a c t i o n i n a n a d i a b a t i c reactor p a c k e d w i t h P t / a l u m i n a catalyst. A c c o r d i n g to the c a l c u l a t i o n s of H l a v a c e k et al. ( 189 ) for Pe =^= Pe , c

v a l u e of Pe

c

t

the

i n these experiments is a b o u t t w i c e the v a l u e at the u p p e r

l i m i t of m u l t i p l i c i t y .

H a n s e n a n d J o r g e n s o n do not r e p o r t a n y o b s e r v a -


tions of steady-state m u l t i p l i c i t y . T h e authors p o i n t out t h a t the e x p e r i m e n t a l l y o b s e r v e d transients i n the last p a r t of the reactor are n o t w e l l d e s c r i b e d b y t h e i r m o d e l . T h i s o b s e r v a t i o n suggests questions as to the a d e q u a c y of the d i s p e r s i o n m o d e l , p a r t i c u l a r l y for d e s c r i b i n g the u n s t e a d y state. A s i n most r e a l c h e m i c a l systems, s u c h d i s c r e p a n c i e s c a n b e the result of a n i n a c c u r a t e r e p r e s e n t a t i o n of c h e m i c a l r a t e processes, as the authors suggest. E i g e n b e r g e r describes a n i n t e r e s t i n g t h e o r e t i c a l s t u d y of the effects of the t h e r m a l c a p a c i t a n c e a n d c o n d u c t a n c e of t h e reactor w a l l o n the d y n a m i c b e h a v i o r i n h o m o g e n e o u s p l u g - f l o w t u b u l a r reactors.

Both a

l i q u i d r e a c t i o n a n d a gas-phase r e a c t i o n are c o n s i d e r e d , a n d the w a l l is n o t a s s u m e d to be i n t h e r m a l e q u i l i b r i u m w i t h the

fluid.

T w o results are

p a r t i c u l a r l y i n t e r e s t i n g : ( 1 ) for a l i q u i d r e a c t i o n , l o n g i t u d i n a l c o n d u c t i o n i n the w a l l is not significant; yet, o w i n g to the heat c a p a c i t y of the w a l l , some features of the d y n a m i c response are s i m i l a r to the w a n d e r i n g p r o files d e s c r i b e d earlier for fixed b e d s ; ( 2 )

for a gas phase r e a c t i o n , w a l l

c o n d u c t i v i t y effects are i m p o r t a n t a n d l e a d to m u l t i p l e steady states u n d e r some c o n d i t i o n s . A recent p u b l i c a t i o n b y E i g e n b e r g e r (252)

gives a d d i -

t i o n a l a t t e n t i o n to w a l l effects o n t u b u l a r reactor d y n a m i c s . H l a v a c e k a n d V o t r u b a r e p o r t some i n t e r e s t i n g results o n the o x i d a t i o n of C O . T h e i r experiments i n v o l v e d v a r i o u s catalysts i n fixed beds a n d i n m o n o l i t h i c h o n e y c o m b structures.

S u c h reactors h a v e b e e n

discussed

extensively earlier i n this s y m p o s i u m i n the r e v i e w p a p e r o n c a t a l y t i c mufflers b y W e i a n d are also the p r i n c i p a l t o p i c of a p a p e r i n Session I X b y Y o u n g a n d F i n l a y s o n . T h e y differ f r o m the u s u a l c o m m e r c i a l c a t a l y t i c r e a c t o r i n that the P e c l e t a n d R e y n o l d s n u m b e r s are s m a l l ( l a m i n a r Hlavacek

a n d V o t r u b a r e p o r t e x p e r i m e n t a l observations

flow).

of m u l t i p l e

steady states i n b o t h types of reactors. T h e o b s e r v a t i o n of m u l t i p l e states i n the flow of reactants t h r o u g h channels w i t h c a t a l y t i c w a l l s ( as i n the h o n e y c o m b s t r u c t u r e ) does not seem to h a v e b e e n r e p o r t e d p r e v i o u s l y , t h o u g h the experiments b y K i l g e r c i t e d i n T a b l e I I I w e r e s i m i l a r . T h i s p a r t i c u l a r o b s e r v a t i o n b y H l a v a c e k a n d V o t r u b a is i n t e r e s t i n g i n v i e w of t h e t h e o r e t i c a l results of Y o u n g a n d F i n l a y s o n ; the latter authors s h o w


196

CHEMICAL


essentially that w h i l e a s i m p l e o n e - d i m e n s i o n a l

heterogeneous

model

i n c o r p o r a t i n g constant heat a n d mass transfer coefficients c a n h a v e m u l t i p l e steady solutions, a m o r e exact t w o - d i m e n s i o n a l s o l u t i o n u s i n g a f u l l y d e v e l o p e d v e l o c i t y d i s t r i b u t i o n is a l w a y s u n i q u e . Y o u n g a n d F i n l a y s o n d i d not a c c o u n t for l o n g i t u d i n a l heat c o n d u c t i o n i n the reactor w a l l , b u t t h e y p o i n t out t h a t this m a y b e i m p o r t a n t i n some cases.

The

t h e o r e t i c a l results of Y o u n g a n d F i n l a y s o n as w e l l as those of E i g e n b e r g e r ( c i t e d a b o v e , r e g a r d i n g w a l l effects), suggest that w a l l c o n d u c t i o n ( w h i c h i n effect is a n a x i a l d i s p e r s i o n m e c h a n i s m ) is the p h y s i c a l process r e s p o n sible for m u l t i p l i c i t y i n the experiments w i t h the h o n e y c o m b structure.


H l a v a c e k a n d V o t r u b a do n o t discuss t w o - d i m e n s i o n a l m o d e l s b u t c o n s i d e r s e v e r a l v a r i a t i o n s of the g e n e r a l o n e - d i m e n s i o n a l m o d e l g i v e n above. T h e p a p e r i n this session b y M c G r e a v y a n d A d d e r l e y focuses o n the p r o b l e m of d e s i g n i n g

fixed-bed

reactors to a v o i d p a r a m e t r i c s e n s i t i v i t y

a n d m u l t i p l e steady states. E a r l y w o r k o n the s e n s i t i v i t y p r o b l e m dealt e n t i r e l y w i t h h o m o g e n e o u s m o d e l s a n d m o r e recent w o r k b y M c G r e a v y a n d c o - w o r k e r s e m p h a s i z e d possible i m p o r t a n c e of t r a n s p o r t l i m i t a t i o n s i n this r e g a r d . I n t h e i r p a p e r at this s y m p o s i u m , M c G r e a v y a n d A d d e r l e y extend this n o t i o n to o b t a i n c r i t e r i a i n g r a p h i c a l f o r m i n terms of o p e r a t i n g p a r a m e t e r s . T h e results p e r m i t a designer to d e t e r m i n e w h e t h e r o p e r a t i n g p a r a m e t e r s are s u c h t h a t the reactor w i l l b e r e l a t i v e l y insensitive to s m a l l p a r a m e t e r changes a n d w i l l h a v e u n i q u e states.

It is f u r t h e r suggested

t h a t these results m i g h t be r e a d i l y i m p l e m e n t e d i n the c o m p u t e r c o n t r o l or o p t i m i z a t i o n of a g i v e n reactor. Mixing

and Modeling—Effects

on Multiplicity

and

Stability

B y h e u r i s t i c a r g u m e n t s , V a n H e e r d e n ( 8 ) a d v a n c e d the n o t i o n t h a t a necessary c o n d i t i o n for m u l t i p l i c i t y of steady states i n a n e x o t h e r m i c r e a c t i n g system w a s that some p h y s i c a l m e c h a n i s m for heat f e e d b a c k to exist. H e d e m o n s t r a t e d this p o i n t b y w o r k i n g o u t examples of a C S T R , a p l u g - f l o w t u b u l a r reactor w i t h feed-effluent heat exchange, a n d a t u b u l a r reactor w i t h axial dispersion.

O b v i o u s l y the general requirement

of

f e e d b a c k s h o u l d n o t be r e s t r i c t e d to e x o t h e r m i c systems b u t m u s t a p p l y to a c c e l e r a t i n g reactions i n g e n e r a l . purpose

T h e n o t i o n has s e r v e d a u s e f u l

a l t h o u g h n o precise d e f i n i t i o n of f e e d b a c k has b e e n

offered.

T h e n o t i o n is i n t e r w o v e n i n m u c h of the f o l l o w i n g d i s c u s s i o n of m i x i n g effects a n d m o d e l i n g c o n s i d e r a t i o n s — a d i s c u s s i o n a p p r o p r i a t e l y b r i e f a n d l i m i t e d to i n t r i n s i c h y d r o d y n a m i c effects. C l e a r l y i n o r d e r for i n t r i n s i c f e e d b a c k i n a g i v e n r e a c t i n g flow system to b e possible, the residence times m u s t b e d i s t r i b u t e d so that fluid e l e ments of different ages c a n i n t e r m i x . R e s i d e n c e - t i m e d i s t r i b u t i o n i n f o r m a t i o n , h o w e v e r , describes o n l y m a c r o s c o p i c m i x i n g features a n d b y itself


7.

SCHMITZ

Multiplicity,

Stability,

and

197

Sensitivity

tells n o t h i n g a b o u t m i x i n g at t h e m i c r o s c o p i c level—i.e., a b o u t t h e state of segregation as d e s c r i b e d b y D a n c k w e r t s (253).

I f f l u i d elements i n

a n y r e a c t i n g f l o w s i t u a t i o n are c o m p l e t e l y segregated w i t h r e g a r d to b o t h heat a n d mass exchange, there c a n b e n o f e e d b a c k regardless of t h e macroscopic

picture.

( M a t h e m a t i c a l l y , c o m p l e t e l y segregated

systems

g i v e rise to p r o b l e m s of the i n i t i a l - v a l u e type. ) T h u s b o t h m a c r o s c o p i c a n d m i c r o s c o p i c effects w a r r a n t c o n s i d e r a t i o n . N e v e r t h e l e s s , almost a l l m a t h e m a t i c a l m o d e l s w h i c h h a v e b e e n u s e d to s t u d y m u l t i p l i c i t y , s t a b i l i t y , a n d s e n s i t i v i t y of states are b a s e d o n m a c r o s c o p i c considerations. I n t h e a x i a l d i s p e r s i o n m o d e l of a t u b u l a r or fixed-bed reactor, f o r e x a m p l e , t h e Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

effective a x i a l diffusivity c a n b e chosen to d e s c r i b e m a c r o s c o p i c

effects,

a n d once i t is chosen, the l e v e l of m i c r o s c o p i c m i x i n g is also fixed. I n Session V I I Y a n g a n d W e i n s t e i n describe a w a y to s t u d y t h e o r e t i c a l l y b o t h m a c r o s c o p i c a n d m i c r o s c o p i c effects.

T h e y maintain that

t w o adjustable p a r a m e t e r s , η a n d R, i n a h y p o t h e t i c a l reactor m o d e l c o n sisting of η C S T R ' s i n series a n d a r e c y c l e s t r e a m ( w i t h r e c y c l e r a t i o R ) c a n b e separately a n d s i m u l t a n e o u s l y m a n i p u l a t e d so that t h e m o d e l c a n describe given residence-time distribution data a n d still provide flexibility i n c o v e r i n g a r a n g e of m i c r o m i x i n g c o n d i t i o n s b e t w e e n c o m p l e t e segreg a t o n a n d m a x i m u m m i x e d n e s s — t h e extreme m i x i n g situations d e s c r i b e d b y Z w i e t e r i n g (254).

F u r t h e r m o r e , t h e m o d e l is q u i t e a m e n a b l e to

m a t h e m a t i c a l analysis a n d seems w e l l s u i t e d f o r e m p i r i c a l reactor m o d e l i n g i n m a n y p r a c t i c a l situations. Y a n g a n d W e i n s t e i n present results of c a l c u l a t i o n s s h o w i n g t h e effect of m i c r o m i x i n g o n steady-state m u l t i plicity i n an exothermic reaction. A p p a r e n t l y the o n l y p r i o r s t u d y of the effects of m i c r o m i x i n g o n m u l t i p l i c i t y a n d s t a b i l i t y w a s that b y Y a m a z a k i a n d I c h i k a w a (255);

they

e x a m i n e d the extreme cases of c o m p l e t e segregation of b o t h heat a n d mass a n d of m a x i m u m mixedness f o r a n a r b i t r a r y r e s i d e n c e - t i m e d i s t r i b u t i o n . T h e i r c o n c l u s i o n w a s that steady states i n t h e first case are a l w a y s u n i q u e a n d stable a n d that those i n the s e c o n d are stable i f a p e r f e c t l y m i x e d reactor h a v i n g the same m e a n r e s i d e n c e t i m e as t h e g i v e n r e a c t o r is stable. T h e d i s c u s s i o n of m o d e l i n g a n d the n o t i o n of f e e d b a c k i n c o n n e c t i o n w i t h steady-state m u l t i p l i c i t y a n d instabilities c a n b e p u r s u e d f u r t h e r a l o n g s o m e w h a t different lines. A d m i t t e d l y a n y r e a l c h e m i c a l l y r e a c t i n g system has a n u n d e r l y i n g m a t h e m a t i c a l d e s c r i p t i o n w h i c h i n its exact f o r m — i f a n exact f o r m c a n e v e n b e c o n s t r u c t e d — i s f a r too c o m p l e x to a n a l y z e . Simplifications a n d approximations must be made.

Thought-provoking

questions t h e n arise as to h o w one c a n b e r e a s o n a b l y c e r t a i n that t h e s i m p l i f i c a t i o n s a n d a p p r o x i m a t i o n s h e i n t r o d u c e s d o n o t also i n t r o d u c e spurious q u a l i t a t i v e features i n t o t h e solutions of the m a t h e m a t i c a l m o d e l — f e a t u r e s that h a v e n o c o u n t e r p a r t either i n t h e r e a l p h y s i c a l system o r


198

CHEMICAL


i n the solutions of the m o r e exact m a t h e m a t i c a l p r o b l e m . ( S i m p l i f i c a t i o n s , of course, m a y also e l i m i n a t e i m p o r t a n t q u a l i t a t i v e features.)

There

p r o b a b l y is no g e n e r a l a n s w e r to s u c h questions, b u t as a reasonable g u i d e l i n e , i t appears that one s h o u l d b e w a r y w h e n e v e r a n a p p r o x i m a t i o n changes the b a s i c m a t h e m a t i c a l f o r m . S o m e examples are r e a d i l y a v a i l able.

O n e is i n the a f o r e m e n t i o n e d p a p e r b y Y o u n g a n d F i n l a y s o n i n

Session I X . L a m i n a r flow t h r o u g h a c a t a l y t i c d u c t w i t h n e g l i g i b l e a x i a l d i f f u s i o n or c o n d u c t i o n gives rise to a n o n l i n e a r p a r a b o l i c system

of

equations w h e n f o r m u l a t e d p r o p e r l y i n terms of r a d i a l a n d a x i a l p o s i t i o n v a r i a b l e s . F o r this p r o b l e m , as i n most p a r a b o l i c p r o b l e m s , the solutions


are u n i q u e a n d s t a b l e — n o f e e d b a c k m e c h a n i s m exists. If, h o w e v e r , as Y o u n g a n d F i n l a y s o n s h o w , the p r o b l e m is s i m p l i f i e d to a o n e - d i m e n s i o n a l f o r m i n w h i c h r a d i a l v a r i a t i o n s are a c c o u n t e d for o n l y i n terms of c o n stant heat a n d mass transfer coefficients t o the c a t a l y t i c d u c t w a l l , u n i q u e ness c a n no longer be g u a r a n t e e d .

I n fact, this s i m p l i f i e d v e r s i o n m a y

h a v e a n infinite n u m b e r of solutions as discussed i n some p u b l i c a t i o n s c i t e d earlier. T h e r e a l p h y s i c a l s i t u a t i o n m a y i n d e e d h a v e n o n - u n i q u e solutions

(the

experiments d e s c r i b e d i n the p a p e r b y

Hlavacek and

V o t r u b a i n Session V I I s h o w this to b e t r u e ) , b u t these m u s t b e the result of a x i a l w a l l c o n d u c t i o n ( or of m o l e c u l a r c o n d u c t i o n a n d diffusion a x i a l l y i n the t u b e ) a n d not solely of t r a n s p o r t b e t w e e n the c a t a l y t i c w a l l a n d the f l u i d i n l a m i n a r flow. T h e use of constant transfer coefficients is v a l i d o n l y f o r u n i f o r m l y accessible surfaces.

Catalyst particles i n a p a c k e d

b e d m a y p r o v i d e areas t h a t are n e a r l y u n i f o r m l y accessible because of the n a t u r e of the

flow.

I n a p a p e r a d d r e s s e d to these same matters, L i n d b e r g a n d S c h m i t z (256)

c o n s i d e r e d a t h e o r e t i c a l p r o b l e m of

flow

past a n o n u n i f o r m l y

accessible c a t a l y t i c s u r f a c e — n a m e l y b o u n d a r y l a y e r flow past w e d g e s h a p e d solids. T h e c o n c l u s i o n was s i m i l a r to t h a t of Y o u n g a n d F i n l a y s o n ; t h e s o l u t i o n of the b o u n d a r y l a y e r p r o b l e m w a s s h o w n to b e u n i q u e , b u t t h e s o l u t i o n of a s i m p l i f i e d v e r s i o n u s i n g constant heat a n d mass transfer coefficients

was m u l t i v a l u e d . L i n d b e r g and Schmitz i n c l u d e d a dis-

cussion of the g e n e r a l m o d e l i n g p r o b l e m a n d of t h e possible p i t f a l l s t h a t one w o u l d h o p e to a v o i d . C o n s i d e r , as another e x a m p l e , f u l l y

developed

constant-property

l a m i n a r flow i n a t u b u l a r reactor w i t h h o m o g e n e o u s e x o t h e r m i c r e a c t i o n a n d n e g l i g i b l e a x i a l c o n d u c t i o n a n d d i f f u s i o n of h e a t a n d mass b o t h i n the fluid a n d i n the w a l l . T h e t w o - d i m e n s i o n a l steady-state m a t e r i a l a n d e n e r g y balances are of the p a r a b o l i c f o r m

(an initial-boundary value

p r o b l e m ) , a n d solutions are u n i q u e for r e a l i s t i c r e a c t i o n k i n e t i c m o d e l s . N o m e c h a n i s m of t h e r m a l f e e d b a c k exists. I f the n o t i o n of T a y l o r d i f f u s i o n is i n t r o d u c e d (to a c c o u n t for d i s p e r s i o n effects a r i s i n g f r o m r a d i a l v e l o c i t y g r a d i e n t s ) , t h e n the m o d e l is c o n v e r t e d to the o n e - d i m e n s i o n a l


7.

SCHMITZ

Multiplicity,

Stability,

and

Sensitivity

199

a x i a l d i s p e r s i o n f o r m , a n o n l i n e a r b o u n d a r y v a l u e system.

A feedback

m e c h a n i s m has b e e n i n t r o d u c e d t h r o u g h the s i m p l i f i c a t i o n p r o c e d u r e , a n d the uniqueness a n d s t a b i l i t y of steady-state solutions c a n n o l o n g e r b e g u a r a n t e e d . I n l i g h t of these c o m m e n t s , the e x p e r i m e n t a l observations of B u t a k o v a n d M a k s i m o v (see

T a b l e I I I ) , w h i c h i n v o l v e d the l a m i n a r

flow of l i q u i d reactants w i t h n o o b v i o u s i n t r i n s i c m e c h a n i s m of f e e d b a c k other t h a n those of m o l e c u l a r diffusion a n d c o n d u c t i o n , are s u r p r i s i n g . A l l s u c h considerations s h o u l d r e m i n d researchers n o t to lose sight of the u n d e r l y i n g p h y s i c a l p r o b l e m w h e n a t t a c k i n g the m a t h e m a t i c a l one. C l e a r l y they also m u s t b e k e p t i n m i n d i f e x p e r i m e n t a l observations Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

are to be c a r e f u l l y a n d c o r r e c t l y i n t e r p r e t e d . Other Areas of

Application

Steady-state m u l t i p l i c i t y , i n s t a b i l i t i e s , a n d o s c i l l a t o r y b e h a v i o r i n r e a c t i n g systems are also of interest i n b i o l o g y , c o m b u s t i o n , a n d electroc h e m i s t r y . I n e a c h of these areas a large l i t e r a t u r e has d e v e l o p e d — a l l i n p a r a l l e l for the most p a r t w i t h f e w points of contact. F r o m a m a t h e m a t i c a l v i e w p o i n t , at least, p r o b l e m s i n these areas closely r e s e m b l e those c o n n e c t e d w i t h c h e m i c a l reactor d e s i g n . It seems a p p r o p r i a t e , therefore, to d e s c r i b e some of the research a n d the i n t e r e s t i n g p r o b l e m s

encompassed

i n these areas, e v e n t h o u g h the coverage m u s t b e b r i e f a n d i n a d e q u a t e . I n b i o l o g i c a l a p p l i c a t i o n s the major interest has b e e n i n o s c i l l a t i n g systems. T h e p i o n e e r i n g w o r k i n this r e g a r d w a s b y L o t k a (257, 258) i n the e a r l y 1900's o n the oscillations i n p r e d a t o r - p r e y i n t e r a c t i o n s . S i n c e t h e n , o s c i l l a t o r y b e h a v i o r at a l l levels of b i o l o g i c a l a c t i v i t y has

been

o b s e r v e d a n d s t u d i e d . It is f r e q u e n t l y suggested i n these studies t h a t i n t r i n s i c instabilities i n b i o c h e m i c a l reactions are responsible for c i r c a d i a n ( d a i l y ) a n d other r h y t h m s so p r e v a l e n t i n l i v i n g systems. A recent r e v i e w b y N i c o l i s a n d P o r t n o w (259)

covers c h e m i c a l oscillators w i t h p a r t i c u l a r

r e g a r d to b i o l o g i c a l a p p l i c a t i o n s . A n u m b e r of books h a v e b e e n d e v o t e d to b i o l o g i c a l r h y t h m s ; a recent one b y P a v l i d i s (260)

emphasizes

the

m a t h e m a t i c a l analysis of t h e m . Steady-state m u l t i p i l i c i t y has b e e n i n v o k e d i n b i o l o g i c a l p r o b l e m s to e x p l a i n s w i t c h i n g a n d t h r e s h o l d p h e n o m e n a i n b i o c h e m i c a l p a t h w a y s a n d the d e v e l o p m e n t a l processes w h e r e b y a d e v e l o p e d

organism com-

p o s e d of m a n y different types of cells ( different " f i n a l " states ) emanates f r o m a single, n e a r l y u n i f o r m f e r t i l i z e d egg c e l l (see, b o o k b y R o s e n (261)

a n d papers b y E d e l s t e i n (262)

for e x a m p l e , the

a n d L a v e n d a (263).)

E x p l a n a t i o n s for the d e v e l o p m e n t of f o r m a n d s t r u c t u r e i n l i v i n g systems has also b e e n offered i n terms of s p a t i a l or " s y m m e t r y b r e a k i n g " instabilities i n systems i n v o l v i n g diffusion a n d r e a c t i o n . T h i s p h e n o m e n a w a s s t u d i e d first b y T u r i n g (264)

a n d later b y P r i g o g i n e a n d c o - w o r k e r s


200

CHEMICAL

(265, 266, 267, 2 6 8 )


a n d Scriven a n d co-workers

the books b y Glansdorff a n d Prigogine (271)

(269, 270)

(see

and Aris (5)).

also

I n such

t h e o r e t i c a l studies, questions are p o s e d as to w h e t h e r s p a t i a l l y d e p e n d e n t p e r t u r b a t i o n s , i m p o s e d o n a n i n i t i a l l y u n i f o r m steady state, g r o w

(or

d e c a y ) u n i f o r m l y or i n s t e a d g r o w i n t i m e w i t h a c e r t a i n p r e f e r r e d s p a t i a l structure. dicted.

U n d e r c e r t a i n c o n d i t i o n s a s p a t i a l l y p e r i o d i c g r o w t h is p r e -

E v i d e n c e that such

instabilities actually occur

in

biological

systems m a y b e f o u n d i n v a r i o u s d e s c r i p t i o n s of p a t t e r n a n d a g g r e g a t i v e m o v e m e n t of cells i n c u l t u r e s u c h as those d e s c r i b e d b y E l s d a l e


a n d B o n n e r (273)

(see

(272)

also t h e a f o r e m e n t i o n e d b o o k b y P a v l i d i s ) .

Figure 14. Traveling chemical waves in a diffusion tube during the Belousov-Zhabotinskii reaction (with Fe ). The light bands (waves) are blue; the dark regions are red. Waves, initiated by oscillations in the stirred beaker (at left of photo), are about 4 mm apart and are traveling at a speed of about 0.5 cm/min (280). 3+

M o s t m a t h e m a t i c a l m o d e l s u s e d i n studies r e l a t e d to b i o l o g i c a l a p p l i cations are for i s o t h e r m a l k i n e t i c s , a n d t h e y a l l o w for the exchange

of

m a t e r i a l w i t h the s u r r o u n d i n g s , or "openness," b y i n v o k i n g the a s s u m p t i o n t h a t some species concentrations h a v e fixed constant values. T h e r e s u l t i n g m a t h e m a t i c a l descriptions are s i m i l a r to those of t h e u s u a l c o n t i n u o u s f l o w c h e m i c a l reactor m o d e l s , b u t the u n d e r l y i n g p h y s i c a l p i c t u r e differs. T h e r e s u l t i n g steady states are a c t u a l l y pseudo-steady states i f the system is t r u l y closed, or t h e y represent steady states i n extreme cases of

open

systems for w h i c h some components c a n b e e x c h a n g e d w i t h t h e s u r r o u n d ings w i t h o u t resistance.


7.

SCHMITZ Though

Multiplicity, not a

Stability,

biological

and

reaction,

201

Sensitivity the isothermal

liquid-phase

B e l o u s o v - Z h a b o t i n s k i i r e a c t i o n has a t t r a c t e d great interest a m o n g

biolo-

gists a n d b i o p h y s i c i s t s as w e l l as chemists m a i n l y because t h e i n t r i g u i n g b e h a v i o r w h i c h i t d i s p l a y s is r e m i n i s c e n t of b i o l o g i c a l b e h a v i o r .

(Refs.

19 a n d 23, w h i c h c o n t a i n studies of this r e a c t i o n i n a n o p e n , w e l l - m i x e d system, h a v e b e e n c i t e d i n a n e a r l i e r section. ) O s c i l l a t i o n s , i n d i c a t e d b y s h a r p color changes f r o m r e d to b l u e ( w h e n F e

3 +

is u s e d as t h e m e t a l i o n )

w i t h t h e r e a c t i o n p r o c e e d i n g i n a closed s t i r r e d b e a k e r , w e r e b y Z h a b o t i n s k i i (274).

described

S p a t i a l l y p e r i o d i c b e h a v i o r , v i s i b l e as a n a s s e m b l y

of t r a v e l l i n g w a v e s , i n t h e absence of s t i r r i n g , w e r e later r e p o r t e d b y Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

Busse (275)

a n d Z a i k e n a n d Z h a b o t i n s k i i (276).

A detailed physical

d e s c r i p t i o n of t h e f o r m a n d interactions of these t r a v e l l i n g w a v e s has b e e n g i v e n b y W i n f r e e (277, 278, 279),

t h e latter reference c o n t a i n i n g

s o m e b e a u t i f u l color p h o t o g r a p h s of s p i r a l l i n g w a v e s .

T h e propagation

of these waves t h r o u g h a s m a l l diffusion t u b e w a s s t u d i e d e x p e r i m e n t a l l y b y T a t t e r s o n a n d H u d s o n (280)

(see F i g u r e 1 4 ) . A t h o r o u g h s t u d y of

this system b y F i e l d et al. (33)

r e v e a l e d that i t consists of 10 c o u p l e d

reactions i n c l u d i n g a n a u t o c a t a l y t i c sequence.

A t present there is n o

p r o o f that t h e o b s e r v e d s p a t i a l l y p e r i o d i c p h e n o m e n a

are t h e r e s u l t of

s p a t i a l or " s y m m e t r y b r e a k i n g " instabilities of t h e t y p e s t u d i e d t h e o r e t i c a l l y i n references c i t e d above.

Rather the waves

which

propagate

t h r o u g h stagnant m i x t u r e s seem to emanate f r o m b o u n d a r y p e r t u r b a t i o n s , s u c h as dust o r m i n u t e p a r t i c l e s . A g e n e r a l t h e o r e t i c a l analysis of t h e effects of p e r t u r b a t i o n s of this t y p e has b e e n g i v e n b y O r t o l e v a a n d Ross (281) . I n other studies m o t i v a t e d b y b i o l o g i c a l a p p l i c a t i o n s , A r i s a n d K e l l e r (282)

a n d B a i l e y a n d L u s s (283)

h a v e suggested that a s y m m e t r i c c o n -

c e n t r a t i o n profiles t h r o u g h m e m b r a n e s , s i m i l a r to those d e s c r i b e d e a r l i e r for catalyst p a r t i c l e s , r e s u l t i n g f r o m t h e p o s s i b i l i t y of m u l t i p l e states i n e n z y m a t i c reactions m a y have a b e a r i n g o n active transport. A v e r y i n t e r e s t i n g p a p e r i n the c o m b u s t i o n l i t e r a t u r e b y G r a y et al. (284)

crosses the b o u n d a r i e s of c o m b u s t i o n , c h e m i c a l reactor, a n d b i o -

l o g i c a l a p p l i c a t i o n s . A m o n g t h e v a r i o u s topics i n c l u d e d a n d a n a l y z e d are h y p e r t h e m i a i n w a r m - b l o o d e d a n i m a l s — d e s c r i b e d as a t h e r m a l r u n a w a y w h e n the b o d y t e m p e r a t u r e exceeds a c r i t i c a l v a l u e — a n d h i b e r n a t i o n w h i c h is e x p l a i n e d i n terms of the f o r m of t h e heat g e n e r a t i o n curves f o r h i b e r n a t i n g a n i m a l s w h i c h p e r m i t m u l t i p l e states, t h e t w o stable ones b e i n g the h i b e r n a t i n g state a n d t h e active state. M o s t of the t h e o r y of m u l t i p l e states, i n s t a b i l i t i e s , a n d s e n s i t i v i t y f o r c h e m i c a l reactors c a n b e a p p l i e d to c o m b u s t i o n p r o b l e m s .

I n combustible

systems, m u l t i p l e steady states i n o p e n processes are t h e r u l e r a t h e r t h a n the e x c e p t i o n ; the r e g i o n of m u l t i p l i c i t y corersponds

to c o n d i t i o n s o v e r

w h i c h t h e m i x t u r e c a n b e b r o u g h t either to a steady i g n i t e d state or to


202

CHEMICAL


a n e x t i n g u i s h e d o n e b y a p p r o p r i a t e p e r t u r b a t i o n s or start-up c o n d i t i o n s . S t a n d a r d references f o r the t r e a t m e n t a n d d i s c u s s i o n of these a n d r e l a t e d t o p i c s are the books b y F r a n k - K a m e n e t s k i ( 2 8 5 ) a n d V u l i s (286). of the t h e o r y a n d e x p e r i m e n t a l i n f o r m a t i o n p e r t a i n to closed

Much

systems—

i.e., to situations i n w h i c h reactants are c h a r g e d to a vessel, subjected to a p r e d e t e r m i n e d pressure a n d a m b i e n t t e m p e r a t u r e , a n d o b s e r v e d t h r o u g h a n e n s u i n g transient. E v e n f o r s u c h cases, h o w e v e r , t h e t h e o r y u s u a l l y is a p p l i e d to a " p s e u d o " steady-state v e r s i o n of t h e p r o b l e m , a n d w h e n gradients are n e g l e c t e d , t h e results r e s e m b l e those d e s c r i b e d e a r l i e r i n this p a p e r for t h e C S T R (see, Downloaded by COLUMBIA UNIV on June 17, 2013 | http://pubs.acs.org Publication Date: June 1, 1975 | doi: 10.1021/ba-1975-0148.ch007

(287).

f o r e x a m p l e , a recent r e v i e w b y B e r l a d

L o n g w e l l a n d W e i s s (288)

i n t r o d u c e d the c o n t i n u o u s - f l o w , w e l l -

s t i r r e d c o m b u s t o r , s i m i l a r to a C S T R , as a c o n v e n i e n t e x p e r i m e n t a l t o o l a n d d e m o n s t r a t e d its u t i l i t y b y u s i n g e x t i n c t i o n data—i.e., the l i m i t of steady-state m u l t i p l i c i t y c o r r e s p o n d i n g to a t r a n s i t i o n f r o m a h i g h t e m p e r a t u r e state to a l o w o n e — t o d e d u c e k i n e t i c parameters. S i m i l a r theor e t i c a l a n d e x p e r i m e n t a l studies i n d i s t r i b u t e d - p a r a m e t e r flow systems h a v e also b e e n r e p o r t e d (289, 290, 291). Instabilities

a n d m u l t i p l e steady

states

occur

i n electrochemical

systems because of h i g h l y n o n l i n e a r c u r r e n t - p o t e n t i a l r e l a t i o n s h i p s .

A

d e s c r i p t i o n b y W o j t o w i c z (292) is a g o o d reference source. I n i t methods of L i a p u n o v a n d Poincaré are u s e d to a n a l y z e d y n a m i c s , a n d b e h a v i o r a l features are d e s c r i b e d w h i c h h a v e analogs i n c h e m i c a l reactor b e h a v i o r . A c c o r d i n g to this b o o k the first cases of p e r i o d i c b e h a v i o r i n electroc h e m i c a l systems w e r e r e p o r t e d as e a r l y as 1828. A l k i r e a n d N i c o l a i d e s (293, 294)

r e c e n t l y d i s c o v e r e d that t h e equations g o v e r n i n g a c e r t a i n

d i s t r i b u t e d m o d e l of the a e r a t i o n c o r r o s i o n of a m e t a l u n d e r a m o i s t

film

possess at least 13 steady-state profiles for t h e c u r r e n t d i s t r i b u t i o n . F u r t h e r m o r e , some of these l e a d to v e r y h i g h l y l o c a l i z e d r e a c t i o n rates a n d suggest a p o s s i b l e e x p l a n a t i o n or m e c h a n i s m for the l o c a l i z e d c o r r o s i v e attack of metals. P r o b l e m s of i n t r i n s i c i n s t a b i l i t i e s arise i n m a n y other situations w h i c h d o n o t i n v o l v e c h e m i c a l rate processes.

I t is n o t possible to

d e s c r i b e these i n d e t a i l here, b u t t h e y are w o r t h y of n o t e because t h e m a t h e m a t i c a l m e t h o d s u s e d to s t u d y t h e m a n d the p h e n o m e n a of interest are often v e r y s i m i l a r to those d e s c r i b e d here f o r c h e m i c a l systems. A m o n g these a r e p r o b l e m s i n h y d r o d y n a m i c s a n d c r y s t a l l i z a t i o n . D e n n (6)

treats t h e f o r m e r a n d p r o v i d e s s o m e u n i f i c a t i o n of these p r o b l e m s

w i t h those of c h e m i c a l reactors. T h e o c c u r r e n c e of s u s t a i n e d oscillations i n perfectly m i x e d crystallizers was studied from a theoretical v i e w p o i n t b y S h e r w i n et al. i n 1967 ( 2 9 5 ) a n d m o r e r e c e n t l y b y R a n d o l p h et al. (296).

E x p e r i m e n t a l studies of these oscillations h a v e also b e e n r e p o r t e d

(297,298).


7.

SCHMITZ

Concluding

Multiplicity,

Stability,

and

203

Sensitivity

Remarks

O n e n o t i c e a b l e recent t r e n d has b e e n the increase i n e x p e r i m e n t a l i n f o r m a t i o n . U n t i l v e r y r e c e n t l y there was a d e a r t h of e x p e r i m e n t a l d a t a or reports of a c t u a l c h e m i c a l reactor b e h a v i o r to s u p p o r t t h e l a r g e n u m b e r of t h e o r e t i c a l studies. M a n y t h e o r e t i c a l results h a v e n o w b e e n b o r n e out b y e x p e r i m e n t s , a n d l a b o r a t o r y d a t a definitely a d d a t o n e of r e a l i s m to p r o b l e m s i n this area. S t i l l , t h e o r y leads e x p e r i m e n t a l fact b y a c o n s i d e r a b l e m e a s u r e a l t h o u g h s o m e e x p e r i m e n t a l observations h a v e

sug-

gested, a n d p r o b a b l y w i l l c o n t i n u e to suggest, questions f o r f u r t h e r t h e o -


r e t i c a l investigations. T h e n e e d for a d d i t i o n a l l a b o r a t o r y studies is most evident i n distributed problems where some viewpoints need clarification a n d w h e r e p r e d i c t i o n s of the u s u a l m a t h e m a t i c a l m o d e l s c o u l d b e q u a l i tatively incorrect. F r o m a t h e o r e t i c a l v i e w p o i n t , at least, the p r o b l e m of e l u c i d a t i n g the p o s s i b l e b e h a v i o r a l features i n systems of reactions ( w i t h o u t h a v i n g to h a n d l e e a c h case i n d i v i d u a l l y ) is a c h a l l e n g i n g o n e — e v e n for the l u m p e d C S T R case.

U n t i l n o w , the vast m a j o r i t y of w o r k has b e e n w i t h a single

exothermic reaction.

E m p h a s i s has b e e n o n the effects of i n t e r a c t i n g

p h y s i c a l rate processes. S o m e of the e x p e r i m e n t a l observations of o s c i l l a tions i n s i n g l e catalyst p a r t i c l e s a n d of the f a s c i n a t i n g c o n d u c t of

the

B e l o u s o v - Z h a b o t i n s k i i system, d e s c r i b e d i n studies c i t e d a b o v e , as w e l l as a p p l i c a t i o n s i n some k e y areas of b i o l o g y , m a y s t i m u l a t e m o r e effort i n t h e d i r e c t i o n of c o m p l e x r e a c t i n g systems. O n l y a v e r y s m a l l p e r c e n t a g e of t h e papers b e i n g p u b l i s h e d o n the subjects of this r e v i e w are f r o m industry.

A s a result, the present p r a c t i c a l a p p l i c a b i l i t y of

academic

r e s e a r c h to i n d u s t r i a l reactor d e s i g n a n d its i m p a c t t h e r e o n are u n c e r t a i n . Acknowledgments H e l p f u l comments w e r e o b t a i n e d f r o m D a n L u s s , James D o u g l a s , Gerhart Eigenberger, and John Villadsen, who kindly reviewed a prel i m i n a r y v e r s i o n of this p a p e r . Nomenclature

f

g e o m e t r i c constant; 0 for slab, 1 for c y l i n d e r , 2 for sphere area for heat losses p e r u n i t reactor v o l u m e c h a r a c t e r i s t i c l e n g t h ; h a l f - w i d t h for slab, r a d i u s of c y l i n d e r or sphere dimensionless c o n c e n t r a t i o n of reactant i n the b u l k fluid phase,

B

dimensionless c o n c e n t r a t i o n of reactant i n p a r t i c l e pores,

a a b

y

c

CAT/C

c

AO

CAT/CAO

C C

A P

c o n c e n t r a t i o n of reactant heat c a p a c i t y


204

C H E M I C A L

R E A C T I O N

E N G I N E E R I N G

R E V I E W S

b i n a r y diffusion coefficient effective a x i a l d i f f u s i v i t y of r e a c t a n t p o r e d i f f u s i v i t y of reactant D a m k o h l e r n u m b e r g i v e n b y Κτ exp ( —y) or Κ τ a c t i v a t i o n energy v o l u m e t r i c flow r a t e fluid-particle h e a t transfer coefficient b a s e d o n s u p e r f i c i a l exter n a l p a r t i c l e surface area ( — Δ Η ) heat of r e a c t i o n ( p o s i t i v e f o r e x o t h e r m i c r e a c t i o n ) k effective a x i a l fluid t h e r m a l c o n d u c t i v i t y k fluid-particle mass transfer coefficient b a s e d o n s u p e r f i c i a l exter n a l p a r t i c l e surface area k effective t h e r m a l c o n d u c t i v i t y of catalyst p a r t i c l e Κ p r e - e x p o n e n t i a l factor f o r r e a c t i o n rates p e r u n i t of reactor volume V K' p r e - e x p o n e n t i a l factor for r e a c t i o n rates p e r u n i t of p o r e v o l u m e K o , Kq r e a c t i o n v e l o c i t y constants at t e m p e r a t u r e T , g i v e n b y Κ exp ( — γ ) a n d K' exp ( — γ ) , r e s p e c t i v e l y I t u b u l a r reactor l e n g t h

D D D Da Ε F h f

g

0

t

m


B

Q

L

dimensionless c a p a c i t y factor, 1 -\

Le m rii, n Ν

Lewis number, p C D /k mass l u m p i n g constants n u m b e r of s p a t i a l positions ( e.g., c o l l o c a t i o n points ) i n l u m p i n g procedure N u s s e l t n u m b e r ( B i o t n u m b e r ) , hb/k P e c l e t n u m b e r f o r mass d i s p e r s i o n , vl/D P e c l e t n u m b e r f o r heat d i s p e r s i o n , vlc p /k rate f u n c t i o n s u n i v e r s a l gas constant Sherwood number, k b/e D dimensionless t e m p e r a t u r e of b u l k fluid phase, T /T dimensionless t e m p e r a t u r e w i t h i n catalyst p a r t i c l e , T /T temperature o v e r a l l coefficient f o r heat losses f r o m the reactor average i n t e r s t i t i a l fluid v e l o c i t y v o i d reactor volume dimensionless distance, y'/b distance variable measured from reactor inlet distance v a r i a b l e m e a s u r e d f r o m center of catalyst p a r t i c l e time dimensionless coefficient f o r reactor heat losses Ua^r/p±C t dimensionless a d i a b a t i c t e m p e r a t u r e rise i n fluid phase, ( — Δ Η ) C o/ptCptTo dimensionless a d i a b a t i c t e m p e r a t u r e rise i n catalyst p a r t i c l e , (-&H)e D C /T k dimensionless a c t i v a t i o n energy, E/R T r e a c t o r v o i d f r a c t i o n , r a t i o o f b u l k fluid v o l u m e to t o t a l v o l u m e p o r o s i t y of catalyst p a r t i c l e a

2

Nu Pe Pe R Ro R Sh tf t Τ U ν V χ y y' ζ α β c t

u g

s

ps

s

p

*

p a

s

s

t

vi

m

s

{

t

s

t

Q

B

v

A

β'

s

γ c e

s

o

0

e

g

s

Q

0


7.

SCHMITZ

η

Multiplicity,

c

9

t

205

and Sensitivity

effectiveness factor f o r catalyst p a r t i c l e r e l a t i v e to r e a c t i o n rate at C , T dimensionless t i m e f o r C S T R a n d t u b u l a r reactors, ζ/τ dimensionless t i m e f o r catalyst p a r t i c l e , ζ/τ' l u m p i n g constants defined i n E q u a t i o n 10 dimensionless distance, y/l density c h a r a c t e r i s t i c t i m e f o r C S T R a n d t u b u l a r reactors, V/F c h a r a c t e r i s t i c t i m e f o r catalyst p a r t i c l e , b /D T h i e l e m o d u l u s , b y/ K ' / D A f

Θ θ' A, A ξ ρ τ r φ

Stability,

f

2

0

a

s


Subscripts a f ο s

ambient or coolant conditions b u l k fluid phase f e e d conditions state w i t h i n catalyst p a r t i c l e o r s u b m e r g e d s o l i d m a t e r i a l

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7.

SCHMITZ

71. 72. 73. 74. 75. 76. 77.


78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

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Stability,

and Sensitivity

207

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208

121. 122. 123. 124. 125. 126. 127. 128.


129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167.

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7.

SCHMITZ

Multiplicity,

Stability,

and Sensitivity

209

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189. Hlavacek, V., Hofmann, H., Votruba, J., Kubicek, M., Chem. Eng. Sci. (1973) 28, 1897. 190. Vortmeyer, D., Schaefer, R. J., Chem. Eng. Sci. (1974) 29, 485. 191. Cohen, D. Α., Summer Inst. Nonlinear Math. Battelle, Seattle, 1972. 192. Hlavacek, V., Hofmann, H., Chem. Eng. Sci. (1970) 25, 173. 193. Ibid., p. 1517. 194. McGowin, C. R., Perlmutter, D. D., AIChE J. (1971) 17, 831. 195. Ibid., p. 837. 196. Ibid., p. 842. 197. Hlavacek, V., Hofmann, H., Kubicek, M., Chem. Eng. Sci. (1971) 26, 1629. 198. Chen, M. S. K., AIChE J. (1972) 18, 849. 199. Wicke, E., Vortmeyer, D., Z. Elektrochem. (1959) 63, 145. 200. Wicke, Ε., Z. Elektrochem. (1961) 65, 267. 201. Liu, S. L., Amundson, N. R., Ind. Eng. Chem., Fundamentals (1962) 1, 200. 202. Ibid. (1963) 2, 183. 203. Liu, S. L., Aris, R., Amundson, N. R., Ind. Eng. Chem., Fundamentals (1963) 2, 12. 204. Aris, R., Schruben, D. L., Chem. Eng. J. (1971) 2, 179. 205. Farina, I. H., Aris, R., Chem. Eng. J. (1972) 4, 149. 206. Eigenberger, G., Chem. Eng. Sci. (1972) 27, 1909, 1917. 207. Padberg, G., Wicke, E., Chem. Eng. Sci. (1967) 22, 1035. 208. Vanderveen, J. W., Luss, D., Amundson, N. R., AIChE J. (1968) 14, 636. 209. Rhee, Η. K., Foley, D., Amundson, N. R., Chem. Eng. Sci. (1973) 28, 607. 210. Vortmeyer, D., Jahnel, W., Chem. Ing. Tech. (1971) 43, 461. 211. Vortmeyer, D., Jahnel, W., Chem. Eng. Sci. (1972) 27, 1485. 212. Bilous, O., Amundson, N. R., AIChE J. (1956) 2, 117. 213. Douglas, J. M., Orcutt, J. C., Berthiaume, P. W., Ind. Eng. Chem., Fundamentals (1962) 1, 253. 214. Baddour, R. F., Brian, P. L. T., Loglais, Β. Α., Eymery, J. P., Chem. Eng. Sci. (1965) 20, 281.



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