Sheintuch1977.pdf

OsciilfaYms in Cafalytic Raciicns

....................................... 107 .................... 109 a. IS. TZGEL-ZXC2.f. S T T C E S ............................... El Xzacdon a S i q i e Toroua P d c l e ,............ 122 ............. a-i a. Beaction on an a r d CatJlyzic C. T?ie ?roblati of a C a t d p t in a C X R . ................ 136 D. T e t e t l d V e r s u s Expenmertral Besuits ............. 143 N. OSCILUTOXY STA'IZS IX TZlE G&uIIE4yTLEss B u C T O R .....................,......*............... 144 Equa~ons,Norions, and Theorems .................. 145 B. to the Oxtdarion of CO .................... 149 V. C O N C L G D I X G W ............................... 165 I. IXTRODUCTION

SUSVEY OF SX?ERCIEXAL 'XORE A.

scL'i3ce

A.

hpplicarfnn

SmOLS

BEF-C=

............................................ 166 ........................................ 169

L DITBODU~ON T b paper deals wifb rhe of h o w i a d g a amcaxnang open cazalmc r e m a systams whtch, though &posed to consmnc conciiUons, ambit sustvned oscaJla&ry statas. Such behamor arouses 107

GSCILLiTIONS N CAT-lLTTIC RE.4CTIOS

109

(c) Ni foil

asymbrrla uaad fn tMs column are d e a d aa blloara, a, a a u unk reactorr b, conrtrmously fed rtcii(recycle) reactor: c, f k e d bed reactor with single c a u l y s t particle embedded in laper of inart particles, 03 recycle; d. flow past sin&

OUB flow stlrred

Tg: SO-d7O.C. 30-425

3

(a) See Flg. 2(a)

(a) 60 @) 2

a

Conversmon:

20

1

15

i

-

0)C02,g:

see Fig. 3(3). T, : 2' C pedi-b-*&

Conversion: 7 0 4 5 %

T,-Tg:

30-14O'C

Tg: 157-160'C for TF = 157'C. Conversaon: 637 0%

UO'C.

1.3Eo

L50-270'C. in*

0.05 r0 1

AmpUtude: -40% of m e w reaction rate

u'8 to 150

See Figs. 1, 2@), 3@), 5,

co in air

O-lgco

and6 (a) 1 to G U

(a) See T i p . 3jc) and 4

(b)3 to 45

@) See

(c) 6 b 30

(c ) T,: 294406.C

Figs. 3(c) and 4

t

t i*w

-

a min-

Tim

FIG. 1. Ekunples of 'aeariy osclltdnns: (a) wlt!a ondarion of H2 in a spherical catalyst pellet (3.14% Ht in air) [8, 9 1; @) with ofidadon of CO on plathum bil in a C S T B (feed of 1.75% CO i n a s , 254'C; T 8 mln) [la;.

-

h r e a s e or decrease b i b w e d by a U m s span of rel;rripsly slow changes.

In t!m case depicted h Fig. 1, for exampla, tbe rnevnurd C q CoEcentratlon as the s u n o f a perlod changed from Its mtntmrmrvaLte to a maamurn in h u t 15 sec. It then slowly b a y e d back to t b mlnrmnm over a fame spaa of ;ibDuf l20 sac. A mora preciae mazbmdcal deacripaon of m1pndan oscilladolrr Is glpm b a later pare of tb,ta paper. ) Perto& hmng a nmra symmetric s l u e , aach p. Wee shown ln Fig. 2, have a h h e n reported. ObeerPatioru of much mom Lntereadng and compllcared penodic bobmior hvs t#en rrponed in some msumces. For exsmple, la both the oridatton of H2 (rmuotbermal) L3, 9, ~ 9 ; and of CO (bahermal) La, 9, 17, U], there arm c 1 1 ~ c t e r i z e dby a numkmr of peaks on a singis cycle and mtgnt be ~~y a e a c r h d in

t imr

,

+, I I

I I

TIC

I

I

1

1

I I

1 '

FIG. 5. Oddation of CO on pl7rfnum fail (in Gatherma3 CSTX at 329'C, 1.92% CO in air m feed) showmg &e ae.xsunce of oscillatory a d x a l e sfafes (between dashed oercicd Lines) [173. L x s Lidcate pes-to-pesk amplitudes.

a d zuass and the passage of f l u d to downslrean particles, be sui9cient synchronazc the penods of the indivxdual particies? The quesDon cannot be axmvercd with cercilny at tlus time, but those studies uned in Table 1 wmch involved a packed section of particles in the reactor (Enaces 1, 5, and 7) did not seem to yield my o a c i h m r y behanor in the reactor ouzput whch w u cbsacterlstically different from that obsemmd in scaaies with singie pprticles o r cataiytic ails. h three of the studies listed in Tabh 1 for th6 oxidation of CO (6nmer 3, 7 , and 8) a d in one case for the oxlddatlon of H, [19:, tbe 08cillarorp s u f ~was Mt a mume staze. That 1s to say, that omr some of Conrtttbpe there c o e d an oscFtlarory stata and a stable Urndependent st3te. F m 5 depicu ttus situt&7nfmm &sa mported by Pllcbta and Scinnlrr (17:. hperlments conducted by these

7

i1 ETG. 6. of CO o n plaUm.un foil in a C* a2 233*C, 1.34% CO in au m feed [ 17 1. hrrows indicate peak-ro-peats amplimdes. -00

CSCLLATXjX'S I24 CATALPTIC REACTIONS

1 = L2,

... , n

Bormdarp conditions: "i aT z ) = & ---,=o a2 az

123

124

*We refer 0ctasslonaUy tbughorn this secdon t n "y)proOldmlurlt zn.erhobs of analpla or solution witbut fPrthcr description of them. The raierence ia o so-called 'lunpuq" methods wMch in effect convert the orq#aal dstribursd problem to a lumped one. renaerhg it mom a m e d e b analysis. Five -reor lumpmg methods have been described and applied b the i n q a t t i c l e stabillty problem. (See, for example, Fiefs. 23, W-29.) In their stmpkst versions, l'lnm@rq't rnsrbadr would convert each of t b two partial dlflcrcntlatequubru L b e d tnllable ? t o asingle otdfpvp differentialequation In time. To Uusuara. tbr, second entry in Table 2 given rhe "kalumpedt' form of the fLnt enay. The lrrmping uz thrs case PII~LI)~ use o i *he Lzst-order o & g o d collocation tecbplque [ Z j. Nodm t h p t o d y t a n d t . + l a r e & a n g e d . MostImportantly, r l s c b n g e d from a s p a s h l deri~otloeterm b a bear W a c term.

SESECUCH .%XD SCS?IIITZ

126

Prohiem

4.

solp.J;1)us m c l e (uniform pazzlde temperaaae) wltb

w i e r resrstances (see descrqmon LD. Part DI-B) 39 i mt3 negllhible mius m i e r ressupce dfiniteryeofchenusorpdon (see description in P 3 x m-B

x2

1-3

Ts/T

g

cl’cl,g

5. Sane

6 . Gradientless CSTR

1

- XL

l-3

cs’cI,

F

T%

G S C I L U T I O N S IY C.ITALF!C

t

REACTIONS

3

Ca

L27

L

J+l

:'Dl 1

t'3,

s /k

U b

! I

17 X,CC

6 ,L

S P

c

M

&T UA

t'

PCpTF

a k + Y O

C

1+VPC

T

P

- : d x s well ceiow u v t y md hence below ;he c z ~ a c dv d u . (.A reisonable z s t l z u of L, ior a ?omus piaa3m csval::sr is 0.016/d, m e r e d is ihe aver3qee?ore diameter in cecrimeters L l S j. F o r a pre dimeter ox 1030 -1, Lc would be 16OC.) The 2wers '5y 3lavzcek a d coxor:ucs aiso somr aut . a r k critacd V ~ L U ?o i L i2gac c s on 2% T h e i r 30&u 5 . Par suificiendy smail values o i 2 . sscdlations ux sbwn TO be mpossihie ior h e case oi 3 i = 3h o r br &re 3i and Sin tegaxdsss oi the a a p a a e of L. X u m e n c i sinulariona S & O W ~d c a i a s e d o s c t l h m q 3 0 h x n l S for Lnaplte vdues o i 31 and Sh have been presezxeci as U u s t r J t l o n s in 3 a&? of icssuslcbs. Sevenl public3clons by Y?ancek a d comrk;ers ,$, 25, 31. 32: ixiucie such simuiacions ior a s p b n c d 3 ~ q c i eby 3 ,:afierJ, of me&& of soiucton sll mth ;Y > 1 and L < 1. ;C 3ze iixazc& s t o w e d s u s i i e d o s c d a t i o a x o u z a 3 %+conTetsio-, 5 ~ x L~e n *L-,-ee 2 s m d g states ensreti. O t i O i s were reporred ior spcr,-.cd p c . i d e s with L = 0.1 by Lse azc h s s 133 3 (who alSo s t a d k d che line= ~ U F U l i t yproblem by Galarm's merhods) and by is

HeUinc!cx

BK

al. 04:.

T k laaer r e p e a ~ e dc&

srmul3r;oas oi Lee t n d

LLSS,WLO used 3 qhz-cifcreace r n e b d , LD SLOW&at ' l u m p i q " by L-SL-OEXZ s r h ~ q r scoi!oczcc~: i g v e r e s o d i y SCFLWC :SULKS. -. I q u r e 7 ?rerenu .he results oi one oi L& 3m-~Larioas coct3ired m h e piper 'T Lee a d U s :33;. The brge - J ~ ~ ~ U O 1D3 Sthe :enp e r v l u e ?ro&iz b u g n the pamcle are particuiariy mreresung b W s a (1) they smw temperature excmiorrs well beyond the adiaatic 'kzmperarure Ase, and (3) the simulated temperature near t3e particle suriace reacfies a very large value. In dorernentioncd a e r s e n t s by Beusch e t d. 18, 9 1 the temperawe through tbe parcicle (935 nearly ~ 0 n . s ~Nonce ~ . &o tbar b e passage thmqn profiles 3, 4. . , 9 consumes only about 6% of the total cycle tupe, inhctbar hes e particular osclllarions are of tba r e k u t i o n type. The assumptions regarding the values of Bi and Sh were relaxed IDa study by Hlavacek and W i c e k [30], wha also considetsd the &ecr of reactlon order and oi a change rn molar density with reac-an. The suuiy was marrrly concerned wlth wtratber rhesc p-ewrs axxi effects. when properly accouwed for, w c J d lead fD osciyaroxy states for reaListic vdueta of L and JY, Th! principal findings were as follows. (1) The critical v a h of L passed tbmtagh a birr s t i l l mmed l a 8 thn unitp, a the n r t o SWBi acreased Imm uniry. For example. WtthJv = 2, SWBl = 10, and a 0 or 1, the cntical value of L below whwh osdllatoxy s w s could ex12% 0.304. I€, for the same situation, BI and Sh were taken lo be U l i h t C , the d t l c a l v d w of L was oplp 0.075. (2) Increasmg the

..

-

-

I-'-

3

Pmflla

t

I

O.I*IZO

2

amzo

1

0.+5

7 T TC Q : 1.1

J

I

0.2

0

z,

0.4

0.5

0.3

1C .

D u n r m w l i a ~ Affiior h r i t i o n

XC. 7 . i ) cycle ~ of predictad temperature oscrllariona m a spnencal porous i1arncl0 slmubred for 3 flrst-rcer .irr!muus-cype rmctlon by a mte-difference method (1000 g n d p o w ) ; Lee and Lass c33

:.

omer of reamon fmm one ta two decraaaed the critical d u a of L. (3) Volrune con&zction with rcacrion bmaaed t b crftical d u e of L, but &t effacc waa s d . A numetrcal solutlon of the governrng pania). drfferensiatad equa& o m presented rn a RC~SXS p w r by Binder and CJveln [35] for Si r Sh s&wed s u s t v n s d osciUationa about a =qua urrstable r a t e in a spnencal panicle. Here, as in all of the prevynclly cited cases, t b set of parameter valua urvolved In the oscillatory s t a t e would not U d y apply ta realistic situaaons ( P y 8, L = 0.1. SWB1 = 5). especrallyfor k = 0. Slncs fhs restnaace to b a s ~ a n r r p o r within c a catalyst ptu&cle LS usually much s d a r t b n tbe~BlcLezppl resistance, o w u q u) the k g t particia thermal conductivity, tbs system 0quadona may be sunp U e d so~~lewnas K tbo uumpdon u made that tha temperarrve of the p a r d c l e is &rm througntaut. The exparlmontll measuremenrs by

-

SHEPITCCH .L?D SCz?IITZ

130

1.

A

Tht pet effect in the dimezuronless equations is the replacement of x , lp the rau fzqreseloaby x,/(l r K,x$ They 'aok Bi and Sn to he Lrrrlnite, and uskg an, appmrtmatP method of solmon found t b t rhe critical value of L berm wbxh oscillation^ were possrbie tncreaaed 39 either 11, o r Y increased. The upper knit of the crltacal value waa unly. The authors contend that if dl effecrs, induding b %=-Ace m*hbitronerfecrr u Lanpur-iILMhelwood eppe of rate expressions, volume conzracnon witb reacrOn, and mte unequal d u e s of ai and Sb -re accormred for. the3 *hm i o n problem mth a single reacuon scep would lead v) oscillatory wlutions ior r e a s o m l e values oi L. Thay citad Zkie ethyLene hydrogenation r8acQt.m (I < $v c 2.7; Le = 9 C 2 3 ~ ~ a n ~ l C t n w M c h s u c h c o ~ t i o o s m i ~ p ~ brrr thcy pmsenrcd LY) simuladona for it. -US a d cmspp.& u two papem [3a, 391 considered 'mtb tbn surface c;lpacit?ncs and yisorptba enrhalpp effects e%ecrs wphich have besn negllecred rn all pnzvlously cited ~Olmes. Tbe a u b r s maanrarntkat tbe surface capzcitaca is accudlp much larger tban tba capad l l l r s of the mid volume in a typical porous p m f l t . ( merm

.

-

C. The Problem of

3

Camifst In a CSTS

There have been ~ r z s r n g i yfew theoretlcd s d e s of os&tory stares ior the problem of a clralysr in a well-mixed vessel alb u g n it closely resexzbles the classic CSTR pmblem with nomogeneeous reacuon. That classic problem has received exfellsive Reatment through the past cwo decades 133. Some theoretical work, conducted LD conpnction wlth e ~ e r i m e n asoldlee l of baclmwred clulytlc reactors, was described earlier Ln Part D of ths paper. h few of.&= are dted bcrc, a d Parc SV elakmmw furrhar on this pmhlem. 9 o r s d J L x e k ;5l! studied the case 01 3 single porous caralyst particle m a CSTR, assumrag the s u m of the panicle to be umform and paeuav-sready. T h y 3Lso = w e d *&x an exorherrmc reamon, of order U, of rhc hrreolus type occurred EL pmacie and

-

i

.

;

*

1.0

il-oJ/L

I

r

I

I

n l

!FIG. 9. Skeccbes of b e dtfferent classes of p h s e piors ior the cases I9dtc3red on the abscissas in Figs. 8 and 10. The -1 u designates unsrabie steady s u t e s ; 9, stable steady states; slc, s u b l e LLmi t cycles: and ulc, u n s u e limit wcles. Cppal, a y , and Poore (42;.

Figrue 10 is s l m i i a r to Fig. 8 LEI that i t summarazes steady-state, and oscrllafory fe3frrres in t b homogeneous reaction model strr;tied by Uppal, Ray, and Poore. The difference between tlae two fLgaires LS that in Fig. 10 the parameter represenred on the absclssa Ls tb6 residence dme a parameter convenaently -pukred expenr n e d p . (The group Ca on the abbrcassa ui Fig. 8 is not a paramemr wkch can be changed in uqrenmearJ b d e p d e d y of a l l othclr ) bp O d y tb k W mta h 8m8-eRaEtOf. OOB Jh-8 a rhr m g U r n g and d fe-S aMbimd by any ane of the sketchr ~ 1 Fig. 1 10. Nouce thp m some of (a s e p a z z Ul.lt?Ebd closed m e ) e~29ui. Notice also *br= 3 stability,

-

.

FIG. LO. SketEhes of muversion v-s residence t h e sbcharacznstics for e.mtherruc reactann mth ht-order A r r b m u s m d c s 10 a CsfII. t'ppal, Ray, and R o t a [U:.

S t & a f i y - B t Y e , stability, and 0scZll;itorp

t tbs same rections A, B, etc. are W c a e d &ag tne absci9888, reierring as beiore to the different types of phase planes of Fig. 9. It is ForerpstFng to e m the three Bgnres, lneptng in mind the llkely procedures of an experrmeprer and to &cover that same sitmunus (sraarty and o s c i l l a a r y stares) may be missed emrely in an experimeSupposo, for e-xaxzple, p a a m e w weru such that Fig. 1O(C) applhd. h an experiment in which oaly rhs residence times were cbpnsed, the staxes (steady and 0aCill;rtorg) w k c h a;xUt on the twla arould pmnaaly never be replized. Much further discusaton 19 passible mgardbg these resuits, but space for it LS not justiffed here. The papers by Uppal. Ray, idPoor8 contarn mu& more infomatton and includs a mtmber of cornputar simulatbns.

b

Sesides a e s of a uuque s a b l e s w , such as h s e along sec'tons labeled A ui F l g . 8 and 10, sitaations amrch have actually wen o n s e n d xa &.perlmenrswith caralydc reaczions (not dl mth reactor3 oi * h CSI'R we) resemole those along sectloas labeled E (m s w i e staces mttr =a sustaned osd!~tiorrs)[ 8 - C ! , a (a uzllque uns w l e sxam mth sustained oscUlat~ons;see references in Part XI). F ( d u p l e steaay stares with sustaszed oscrLktions about a hgnconversion -table srar.e; sea references m P m It), and C (mula p l e steady 9f3~eswith only the iow canversion s u m W e ; no BUSOSCULUULS) [U:. ObserPatiOm O f ts4k CO= 08cilLatory stazu wera r e p o d by Jiracak a d H o d [:53 3. Additional refererrcea to those experimental obaorwtionn o f steaciy-swc nuldplldty and mstanillri8s in b& caralyUe and noncaralytlc oysans mch did mt repon. any mstaaned oscrllarory berravlor are pven rn

-

rwm~paqer .by S c k n i t z

C33.

IT rates. &Fur-&rzaore, oscalkuions u v e been o o s e m d Wtth b e same reactions m both i s o t f r e d ma lolllsothertnai expennents and UI systems wittr vast difkrecces m rhe r W v e upaatance of the crr;Jytlc suaface for chemasovtbn, suco as pomus paracles and wires o r forls. Also, some of tbe exacnmenral s u e s have pekied very compliwed penodic beruvior wmca suqie-step -tic models (i.e., models ~ a s e don the assumpeion of a srngfe rate-detemuzug nep) are probably mcapmle of desenbang. Tatran together, these kcu strongly suggest thar t&e results of t.&eoretluls t u d l a w b c h k v e employed single-step Ldnedc

ie-ag m.gutuaes oi

-

models have m bear!! on t b experimental obsemations reported '0 dau, ar least for K, and CO oxidation. Tba most recent litarafpre dted above mdica.tes that most researehen i n this m a of sardy are of that same ophon. It is not possible to support such a view re3zrding the oecillatrons reported by Hqp [Sl for mnrsotberm3i X@ aecompositlon thc only ofber reactton for w h c n oscillations have been ooaerPed becanse o e c W o n s with chat reactlon have nor been suadied unner such diverse c n p n d i t i o ~ . It may M e e d prove to b an

--

IV. CSCILWTORY STATES IN TEE GIMDIENTLESS REACTOR

c U OR% ~ iSrpd %, 1 a z ~ ,~ S p e C t i W l y ,t b m~tt3fe of cnemtsorption of species Ai and tbe fate of conversmn of species d i r e d y from the y phase. &L -pie of t b latzer type conversmn would be an Eley-Ridad step ia which A, in the gas pkaae reacLs by collision wlfb other reactants on the suri3ce. A balance on t b same speaes L I ~the &misorkd phase is

T b ~ W OXU~

In o n l e t ta retain the second+rder nature, and .hence to retain rbe paasibiliry of oscillvorp soluzb~ns,the msuicrion is m3de k t F, and F2 rat be linearly rekted. In order that 3 solvdon sausfymg t!ae steady-sua form o f Eqs. (14) ad (15)be sable, it-is nacessarp and suificieot tiaa~tka e i g e n d u s of tbe mamx B have negative red p a n s :

This, in tura, requirrs tb;rr the determinant 01 the nauis be pasittve andtbetracencgativr. Tbuis

In t&a sectton Jle the0re.Uca.l method8 for investl-

oscillvrrrg

states alscusaed in Section A are applied for ilkrscrsim and infa-t h e purposes to a sarcry of the osidatloa of CO in an i s o e h e d pdtendcss reawr a situation m w w e l l known to gim oarcllatarp states

-

over some nngea of conditions. Our procedure m rhts s w was to cornporn four &rent models for tba system and to consider a number o f s p e d c u e s for each one. Each model coamned two dependent or " W c " vaaables, x, and +. T3a models differed basically in the de~iirdt~~m oi the nvo d e s , aa is described Later. The sped cases wese b.rmulated by invoking different sexa of smnptiona and rasrrtctions. The metbds of analfollowed nearly directly from those ouzlined ln the p r s a u s s s c t i o a For each case tba matclr poen by Eq. ( 1 6 ) and t h determinant aad mce in Eqs. (17)and (U) were constrncted h m the far0 descrlblng dFfferePaal equations krr x, and q. As w i l l be seen, rn some canes the ca~acitaacemamx wasnotaragonal. InmnooftbaCYIeudidtkis&ctEq. (L?), butthe trace LII such cam6 was mom ccmpltcvad rb?n the erpttrsion glpm LD Eq. (U).T?M dyau for suswaed oeciUatorg s w e s ynounted to

of 0,. h ttza second model (see Table 4). wMch is based on tbe u*-son tha: oxppn caemrsorptwn and surface reacttons 3pe the rnp ~ wm t steps, s2representr; the surfme coverage of oxygen. Mode! 3 s T d c S a f c o u ~ ~br s Bat.? rases of CO adsorpaon and s e e r e zcnon; the second crpa3mtc m l e ISthe surface coverage of CO. Model 4 (see Table 6) aesczibes a s g s t e m Lo wmch two sutes of chemiwr& Co exist, o d p one of wmch LS reac*Jve. Zxperanental studies of the chemisorption of CO haye revealed tbc exBten~eof m chepusorDed iormrr on pletrmlm t~ the temperaam mnges fndicated for exp m e n r s with CO oadanon UI Table I (601. The two forms are chmmnnly temed "UEear" and "briciged"; the latter is the acfi7e brS. b a d d 4, X i flyb0 b -0 C00e-S O f ths brLdged s t l t e . The sepiua&~cases, bned UI the bomm portton of the tabie for anp OIIB of the modela, were all deollced from thr, gcnenJ morms of that maad by empLoylpg tha ldclltinnll rpoumptiplrs and nsrrktAom Ustad. We shall mt demote space hela to dtsarssion In derail each seprvate

Nott:e tku Cases (e) 3nd {f) of %del 1 admat oscU3torp soludons, brrt fn the fLRt oi U s e we Spud that tba side W o n , Reacdon (5), nrnsr be lars W v e to the maan reactaon. Tbts obviously is rmc zhs case in ewerimem. C u e ( f ) introduces the mtim that the -vation energy of reardon depends on sLzr23ce composition. Dependences of thrs are well !umm to cut, especially for clrermsorprbn and deaorptbn steps. Such dependencies have also been used to dtsmta Surface reacaon. steps aee, br example, Ref. 62 -the usual erplaaadon be= to terms of mnrmrformitp of the activity of SAUS (herrmgeoeity) o r in terns of l o t e ~ c t i o n sof Chemisorbed speclea. Altbu@ oscillatory states are possible in Case ( f ) in Table 3, we found tbat the required rnqrdtude of P wzs large, so Luge in fact rbat the c o n t z l b ~ o nof fi to the yfiwrioa energy seemed unreaLFstlc. The town we reached was thr tbs mecbvnsttc model descnbed Ln Lbs tap parr of Table 3 shows very lflila promise of being able to doscribs tha orcilkmrp oxdatloo oi CO. Consequently tIaa general applicability of some commonly accoeprad r s a forms wUch are based on both0, d CO bslrrg LTJ Ehamlaarption equalibnum far thu nactloo ts cUnpd.

-

A

x

n

n

N

2 7

4

';

L

C*

I

-

X I

4

II

-

Y

L / I

1. = 1

L = I

None

Oscllialory olirlee of typo I.' (see Ylge. 8 nntl 9) arc p t ~ ~ i L l u but ordy c01 uiiraallutlcully lnrgu rulu of Ilcuclloii ( 5 )

n

a II

n I

+

U

.

3

- -

CI ,

i 3

Q

?-

v

P

z

2

M-

=

I

I . I I

II

L

a

m

n

*

-

ma

d

m

<

11 m +

t

6

+

*m

4

e?.

11 E3

n

4

?

In

a 4

Equaktne

Smce Xodel 3 o ~ r some s p o s a i U t g for expplwug susutned oscallanoos, we jmsent a d d f t r o d ~ s u l t sof *hstudy of i t ior Cases (b) a d ( c ) wt!a P,, ui. vl, and v, all zem. Thus only one new paameter M, LS added III tkis case o those conwried ~ 1 Case 1 (a) o i t;?e s-8 quand5es the e x ~ n of t 'te depndence of the modei. The value of w % c e c o v e w e of O2 o n the suxkce reactmn velocitg. W e found the c ~ ~ c value a l of nt. &ve wrzlda osclllvorp states are possbla, to be relarlvelp s d , vvylng beween 1 and 6 dependmg on wherher Case (b)or (cJ ~9 chosen and wnether tbe chermsorbea -0. is assumed to DB LU the dlssoclated or uPdissociared form. (us = 1 causes a chanp in the acrfvation enem of oniy 1 k U m o l e at 500.K for x2 = 1.) The results of s o m e d-ns ar Case @) are presented LII Figs. ll. hm t b 4L2, d 13. F i w U sbo- ~aCi.Uabrp~ t l r e ~ tlple-etue regame, wMle Fig. 12 shows 0 - q Bfafeo on a curve $&t COW only unique s d y m.These similar in fors ta tha shmn in Reaons Vb and VI Ln Fig. 8. Experlmeazd cumus shawn rn Figs. 5 and 6 rearmrhla cbe theoretical a r m s , buz rmponanr qualitative dlftCTences, such the W o n of tbe ampAitode of oadladopb with space vbiocaty, are mdent, An example of

-

x b v axweraloc oi CS ior a reslcence t l n e of LO sea t9 smwn !n Fig. l2(b). T L s oscruvOrg s w e , o b w d by numerical solutloo of the differexid ~ ~ o n sbows s , sharp chnges u1 CO conversion, bat t&e form is ZDE similar @ &at of tbe experLmerrP1 came s k n m LII FIg. 1. ( h c a d e d y , we S o c m e d opt some s ~ m u l a d o ~ far a model wtuch iacluded an e a b l p y balance. S m d z temperamre splkis slrntlaf to b e e shown In Fig. 1 were obramed, but *boecrllarbns xt CO comersmnmre oniy slimy ~I&XUM.) All such qualifeatures, Fncludng the s h e of oecilLauaas a d the eifect of the spa- veincity, were not burid ta be sensitme to tbe set of patuneten ck~sen. Rumexmore, m distincr dlffereaces were found bcrween Cases (b) and (c). Figure 13 presenrs Wt cycles tn the phase plaae for Case @). C w b 4 (k 0) sham a true reLzfatlon oecUuon of tbe type discussed in rhs Q-US secdon. Tbe Wt cycle br % 0 COMISU of horizons "firmen" SUePrarad by relanveiy slow traotl sloop the cprprr W d t 0. CurPer 2 a d 3, computd hr mnzsro ozluce of &, also ertrrbit a relamionin che sense that tha stata changes very rapidly Vong 90Sacdona aad relatively 3lowlp along others. (Nodcs cha time on Curve 3.) Tbs steady-awe solution ior the c u e sbpwn irr Fig. 13 is untque and upecahie. The llmit cycles

& ' o

-

-

-

FIG. E. Steady 3nd osn'!latnry states for the same case described in +&a p a o n of Fig. LL. except apk3,0/C1,F * 0.0088 sec-'; a&/ C,, F = 0.07 sec-1: C, 0.1; K, = 0 . S i m W n a @) is %r t * 10 SeC.

-

I c..

l

2 3 4

412

a io at6 0

T h u g 3 much tbeotetlcal work on % w e d o s n l l r ? stares in hetemgeneous c;stalydc reacdons has been published, them 19 presd y 30 satisi3ctory expiauuinn for gcpertsenul ooservartons of 08cillamxy phemmena. .%E the sumey of lttemmrn in this revfew reveals, t h e o r e c c i a ~untll ~ recedy have eanbmced p r ~ h f - m l v m g a singla h ~ ! a e ~ u - t p preaction, e or other s a l e - a t e p reaction models, e s a r n h n g a p l e b r z of varlatioas of them. Exper?menkl Infamuation now available (mainly far CO and 8, oddatbn) makes It midens that more sapmsdcared k&eUc models am requrred. Such nod& 3pparensly must account for Fnflcacres in t b surface pmcesses aua for m o r e than OM rata-derestep. For r8aCIions wnich exhibit oedllatory states, thcrr?iOra, soldles in whach heareticrl and e r p e r u n d work am Crosdy llrJred may prom fo be vaLuads rn discw n g rival kinetic models. To this po~Jlbllltp,one secction of rbis review was demoted to a sumy of vazious models for ttre oadatlon of CO LD an ~sorhermalm c r r t l a s s re2ctor. 'Emugh the appUcaUan of well-known m&ds of mybematical Wccmatiolq m e f of rbe models arrplysls coupled *th eme~%~~emtal

A

-+

surface area of noonporous ca t al p t parude h e s masfar area of CSTR

Greek Spmbola

169

(11 T. Rrrrrsawa, H. ,

a n d t Endoh, Kq%aaIcDgzka,

s,

949 (1969). (21 C. Nicolis and J. P o m w , C b e n . Rev., 365 (19'73). 131 lt. LL %hmitz, A&. C%erz. S e t . , 148. 1% (1975). CAI E. D. GlUea, Paper Ptesemad at the 4th b r r r u i o d S m s l u m on Chemical Beacaon Engin~enng,Heidebberg, 1976. c i l EL .ms, me Xattiem;rttcal Theory of D l t b i o n d Reacdon in P e m e a b l e Catalysts. Vols. I and P. Clarendon Press, O d d , 1973. la] P. Hugo, 4th European Symwstum on Chemcd Beactton Engineering, Bnasseb. 1968, 1971, p. 458. l2.l (1970). L71 P. Hugo, Ber. aunsenges. Phys. Cbem.. (83 3. 3euacb P. FA -dv. Chm. Ser., 109, 635 (1973). LSJ 3. Beuscq P. Fiegutb, aud E. Wicks, Chem.-Lnq.-Tech.,

n.

2,

-5

(1972).

*,

V. Hlavacek, Id. Sdaicek and X. Jaarek, J. CarA.,

g, 31

(19691.

B. A Fblaysop, Caral. Rev.-Sa. Ebq., g, 69 (1974). Lppal and W. EL Ray, Chem. h e . Sct., 3 l , 305 (1976). B. Van Den Bosch and L. -P h d , 29, 805 (1974). V. Hkvacev and Y. Wica~, J, C m . , 22, 364 (1971). V. Hkoacek, Jd 3dyek. and= Kumcek, Collect. Czech

A.

-

Chem. Chmmnn., 35, 2134 (1970). V. Raxuac d U. Mare& 4th Zraobean Svmwsium 021 chepucd Reacdon EqzxmlSg, Bnrsses, 1968. 1971, p. 107. J. C. lL b e ana D. L-8, N C ‘ Z 3 . . 2,X!l (1970).

-

J. EIorak and F. J h c e k . Collect. Czecb. Chem. Commun., 33, 1790 (1970). J. Boralt a d 7 . Jimcek. Bid., 2 6 , L720 (1971). F. J i r z e k and J. H o d , mld., 40, 3319 (1975). D. L u s , C h e n . Ex. S c i . 7 2 , 'LSSS (1974). h S. Perelson, 31, 170 (1976). K. 0. Frledrlcb, hrfvanced O r c i i z a q Differatid Equations, Gordon and Breach. 9ew YorL 1963. N. Miaar~kp,No&ear OSC&OIU, Van Xasaand, ? P i e t o & Nenv Jersey, 1962.

a.,