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NAME OF AUTHOR DAVED Gt ae NCH LEE DEO AU Wa aR OSCILEATORY (BEHAVIOR IN CATALYTIC REACTIONS: A MATHEMATICAL AND EXPERIMENTAL INVESTIGATION DEGREES PORUWHICH THESIS WAS PRESENTED DOCTOR OF PHILOSOPHY YEAR THIS DEGREE GRANTED SPRING, 1982 Permission is hereby granted to THE UNIVERSITY OF ALBERTA LIBRARY to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author's


written permission.

cuentas eo) wi . " ii

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(C Joaves Peay NCH





SPRING, 1982

Digitized by the Internet Archive In 2022 with funding trom University of Alberta Libraries




The undersigned certify that they have read, and recommend to the Faculty of Graduate Studies and Research, for acceptance, a thesis entitled OSCILLATORY BEHAVIOR IN CATALYTIC REACTIONS: A MATHEMATICAL AND EXPERIMENTAL INVESTIGATION submitted by DAVID T. LYNCH in partial fulfilment of the requirements for the degree of DOCTOR OF



In recent years many theoretical and experimental examinations of the stability of chemically reacting systems have shown that these systems can display oscillatory behav- ior. In this work, a combined modelling and experimental investigation OlmmosScrinatory peneavioreinwtne. oxidation of CO over supported metal catalysts was performed.

The initial modelling studies were of a general nature, and from them it was found that the use of simplifying assumptions in reactor models can result in transient and Steady-State violations of mass conservation. It was also found that complex periodic and chaotic behavior are possi- ble when a reactor model consists of more than two ordinary differential equations.

From the experimental work, the detailed effects of operating conditions on oscillatory behavior in the oxida- Ciongon COMOVerpsupported Pt and iPts=Pd) catalysts were. deters mined. This experimental work enabled some insight to be Gained into the underlying mechanism responsible for oscil- latory behavior.

Using the experimental insights, a specific model for the ene oxidation of CO was developed. For certain parameter values, this model successfully reproduced many of the fine structural details of the experimentally observed



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Table of Contents

Chapter Page eo Or NON AND MO Bertini Sk, 2 6 2 ete ala oe acess @ elaiee ehel aie 8s | ELT OCLC Cr Olgmm cern ots ial ele.s Sd a eee Laas Pett tee 5 1 Lek IRSGES yee in Oleqeiisacy ai phan A eee Ba na ame 3 (POM Ive Shes la VOUtE ms ets Sisue che Mets ane le (oun tet ol oy te Pe enon sce Te sehi ver tes iavoue 4 ee eR SS OTIS S Bret is ie os dvela oie.ekeroder so oie Mary tori: 2 eA A eee renner 0 ese 5 2. GPPECT OF SIMPLIFYING ASSUMPTIONS

ONSAMNONSOSCLUMATORY GMODEIN . sec. Sunny GA ae Bo Doe o Ze nO OUC EECOMMr te. eereisve eee Piaete ee ee AES al cnet D hitiek en hoe fe 8 Pa MEL) REMIT OG Cele ee stats Soot esha crore lah et th ele at acer el te duno woe whee S ase AGU RSG UitOl me ase: Pee D PAUP LUNs atest Pease oe hs daw at ehh a aes 11 Zn SEC AOE Sid COINS SUMP ELON saci. chet eucls eten et es at he oh Hihatid. Mee 3 Foe tee YD dat Drie iee AN SATII EOI) Bees oc eec caretel ot ate al ate eek sl a gi a siccevel obs 14

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ERIS hoses Bee ARRAS Oe a i RA es PR OP Oe ete? ZeGmeCona iti ONS st Oteralrwi Se EQUIVALENCE! oi cm «iets aie etues 18 2 > KO ein GE aaeleWery Jag Arar ates GORE CREO oD Eee oo ey ct oe 20 YE ae’ ASB USS pag laG herbals Tit Coane unk Gara on Ce) LP? OS Gs PC Pato eer a 2h Te Bl) MPASES (Ch DG ( SN ager ire Ga AES ER VET Cay Cote VOL ae Oe eee oe GS 22 EFFECT OF THE EQUILIBRIUM ASSUMPTION ON A MODEL SOR TAN @OSGRRELA TOR WoOREAG LEON stat. tiawie otis Ay ar 23 Seay lh EOCUG MIO Mt fo, oop el eek ce ot de CW iGo HR aN A ee 25 342 “BLrogenbergerwts «Mode Ls nue stn owls trad hl aceee ete sn ate ean cecal ZS o45. Eimear 6 Pel s pe eS Semele tao icc vad ae teehee 27

Siete BUnTG UGH ESSMAN GY S iS aa a ies oc a swliel ere ote oleh ete 28 SS PLS ek NN te AY ATG eV SS ge veh mie vw ah ole «8% Geet hice alate ehobat s 2

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3.4 Elimination of [AX] System #2 ..... PtGtissis a bee OP Becoeme UN iveness@Analys se. lee. Lt. Sy .cteke eins OO Sie ee Ce AGL V'Si Gals Giss o3 a tes os slew 6a 6 ob eu

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Oeieirecmee ELE TMIMant se COUGI Es Olig wis 4 ols a sin ore 40 Seoemmelominationgsof. (BX), —,System #3) .,3.... Le 40 BF DeMUNTOQUCHESSLANDaALYSISy 2.6. «sles oss Mas teasee tors (ovateue se te 40 BECeMCONOat SOM Otm Oye CSM Ail me 2h RONG Boe a sicles » ols eo @ 43 Be eecOlattonmote thes iquilibmiumyAssumption, ...4 ss. aa Seer ime Petiminaemonwor LAN) —,System. £jJA..5.5 2.056 49 Seueen MiaminetuonmeormlAn) System #2A, .... 06. s 51 SPR ceu ll imnarivon: otmEeKkie— Systems sSA0 cee. ot 53

tO) CONC PUSTONSSE msi < « diskere eget Phe cae tg kena te epee ib eiae Gi N eae te 55 Sra POM eCUICM ALU IR: .Afitt aM ales as 2 a a oeatete leurs eetees he aint tel Santee ote) See mere CT OTEG C SRM, ce metveene es Neto tede fecha tris iols isi seis ss ase : sete etek rs 56 EFFECT OF THE REMOVAL OF THE EQUILIBRIUM ASSUMPTION 58 Ure de |) PUG eeierele MN ehe k oboe OA tree ay ri Se Ome BAe reo ch oh Wee enrs hay 58 ePIC ct la MOMC 1. aves secre tomes eer otto wsiuit avant swameraesr ons emai ce Bre te fs eno Va DislTty. Ana LY SUS. 0. seasons eueweretsls lo fodcesqcta lole teauecn)chaneespereee she eo (Numericals Results andeDiscussion 4.). ses beeen Sr eds, Ay RECON CTUSIO TM Sil srs. cis) ocr eects claretal cause fetes cavy age clot stele Orn. es iat eaeey 68 av Gie NOMENG PATULC. tives. cols, cuacemouraepechts siete «kntel si ei GetCae ra snc itis Est 68 ep PeeRE DEL eONGCSUEL . Jrxer ats tei cge ga OReRetanore eLolets fetcbauectckcuct evel el viene ais ie 69 CHAOS. SANDSRE DARED! PHENOMENA RB i, tis cle caisiciecelt cls |s aisis. 6 erate terete ie Seek Pagishe isvolet Sletece le ey peri Sere BY cu Hs card Soe Srtgerenr sacs 70


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ye ee OC Semele VCO MeN Lee es siges cc a sueis Wrevh Paste wie se ss pecs vs. stele 1 SMe eS OG VE An oe es Gua. ot ays brates 3 Sar eee a aieteeeeK 2 ee tect sorters we epeh ee oo eeOCUD Le olLcamronmiLoOmeanUni Guietsteady. State cue 0. eB) SOME COMOCe Xmen Ura MLOnStancd .Chagbic ;BEnd UVLO. wine e ee Oo Sacre CNeCCecni @uracta ly CUnVES BUniins rife. Ren yer ee AA r Ap parete! Se An CCCs eG UNM 2 EO a COu Se he Bie cits pid) «i acto ecw elas 91 Bets) ACCU INUIS MOIS Dee pet par cere Bl, per ee ee eee b haertarsh ct ae 94 rhe ee UCIT TIC Aes CAG BPW i viceh et ote oh hoes ier ct ce 6, ook of ee of shisvastata okeret ce Sra ek On tg COSI ye eksy ole ehemel vv af on ccueta eo onl «bein: «ake ae Sete phe re 296 DESGREPTION OCF THE BAPBRIMENTALIPAPPARATUS® 5 tes. tosses ow» 296 Cem eOemelaleJeGerr aur onloh (phe Equipment ..4. 4. eens ecu 98 ree Me CMEC Cero TCE Ti Cm ele MIE Clie se mares tue nt 6 ex etch al aires ec al vba le Saree are og Crt ee eae UB NeC VC eMC OMA ET MOMEER fits ace ciel sco i ccst meee st sea ee 6 igs 6.4 Catalytic Reactor in Heated Fluidized Sandbath 106 ea OMe re ge Oia CUO Taare nce sh cle he: cunicel sia x’ Hm xb ae ee reds Pe Oar ee Ga Gas Qomousi Gon AnalySiS< «ssees RE: Pa are tig OM eee kG cor GIG OMe (OO Ga Dla ate. ot cokes ae oto. ot eel onne cpien ce 5? shwd oret s 114 GC cei wa CCU BOD EC ODIO LOM CE Wii. ei eteiste¥ela suterens 118 OSCILLATORY BEHAVIOR IN°THE “OXIDATION OF .CARBON MONG ADD Fewer caele uetren ety els iele. aiecs tel ele eh elet ene ccs ole wisheys ei eiese sk) ad eoetwntrate 123 Cn Gh Oar CLOW mets s teal e iohe ater aencceteuere «tehes tice Teletals nes ec ere Li Mind CVs x Del TNE IL tcl peWOIW meets Pept hth: «Pomvketelsus pe tcer 128 7.3 ERrects oreOperating Conga: UuOn sum ser ol ae eee Pee (oth. COseeCO ahnov COoMmOrned «Oscis ba horsmr mer ee 138 Wit e BRE CY Cl Cr Ra CLO sinestopepeis ber ekemed oly hegeie meters tenor ee Ce eee eee 142 pee ce ToiNwa OMI S hel =watsmenbngion ty A yy Canes eA er reds mia cher ee 144 Jeo. CemeGncent ration, Of sCOmun CNne AP SoG. ie. seis os 146 Toe SeavoLlumetnric Feed Ravena0/0, Constant 148

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feo wie Nitrogen ano Helium Di luents ..... RPO AR Aan & 154

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Pies valltati ves effects of Temperature <<...) PEO roma tat a 136

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See MOCO Ue LO VCR GCiO TL Catt ehs cis oleae costars oiece stele. oe) sl ei siete ie ee eles 472 Crema Sis Var iomO NNO Ven eS Gary. 7 pacts Plies Gaels ty fly. ene ols Bele) BP ORG EC a) sinner ha pron She Sih DSi. 4,0. ciate steels es a 0 wie sp Ws Geer GUUS OVS am ye araeauelsiecs’s Gis tepetels. ster. eke 1s ete katet cist setcheuceete 186 eo . LAOS Rela cee is & en Pom Pee OE Pe or ere Pe ee 187 Or Omer Red SY CNCCS2G en eisis tere tne tate ae ty ROS CORPS S Aeros ste) Pee see MM Ne CON Sore teenie tise emails Vales GoGe: sel pia tes eta si akei esis! +4 ¥erw. aie ees 190 ADpengly —AweCOMpULere Programs fOr Chapters 3 and 45.2.3... 2 Appene1 s barCompurcerrPrograms.for. Chapter 25 ois <1cseie see Zu Appendix CC Rotameter and Flowmeter Calibrations «....4.... 224 ADpends ker Gose Chromatogr apheGe Miorat hone: se oncke cranes 230 Appendix Es Computers Programs Lore Chapcer 9G) Fie. cies 239 WAS Ae feces Corer ee ce Gate et ches So emehen st ence ehvkerelcesd sacl cons (ce isan esac nace te 248

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Table Page 2.1 Conditions for Pairwise Steady-State Equivalence 18 2.2 Conditions for Pairwise Dynamic Equivalence 19 Selec oneofe ne Dascraminanc and Ovartic Root Types 23 ae vpesmeOt OO lutions. ton Equations.4.1 4.3 64 Pa eUPSCrDUI Ons OfeOsci| Lat ionss Shown 2anebhigures 7.3 141 8.1 Parameter Values for Figures 8.1 and 8.2 181 8.2 Parameter Values for Figure 8.3 185 Don GGuGald brationsDatae ior CO,-Ne Mixtures 23h Do coecany brabionsbpatar forvOJ-N oO Muxtires 2a2 Dee GeeCais DrationsDatae for CO-N, Mixtores 282


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3.1 System #1 - Regions of Uniqueness and Multiplicity aSear HUNCE VOM Ofeks-. and Ki,

3.2 System #1 - Trace Condition and Stability Regions asmaerunctiton of K.. and i’k,

3.3 System #1 - Combined Stability and Uniqueness Reorons as valeBiinction of K.,, and “Kis

S74 Systeme73 >= "Regions of Differing.-Numbers of Steady-States

3.5 Steady-State Differences between Systems #1, #2, and #3

3.6 Dynamic Differences between Systems #1, #2, and #3

SeMeUSySLEMar ll Art= bt Lecter ths /Keyecr [Ax]

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4,1 Predictions of Eigenberger's Model

4,2 Stability Regions for the General Model as a Bunction) of Ks and K._,

4,3 Predictions of the General Model

4.4 Predictions of the General Model

Dee Stabulsty Regvons "as*a Function of Da, and £2

Seo eHoOpiebhil burcation

Bac Four Maxima per Cycle Oscillation

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5.7 Chaotic Oscillation with Emergence of “a Third Curve on the Next-Amplitude Plot

5.8 Fractal Curves in the Next-Amplitude Plot






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General Description of the Experimental Apparatus

Feed Metering and Mixing Schematic

Heated Recycle Compartment Schematic

Reactor in Sandbath

Reactor Detail

Auxiliary Reactor

Gas Chromatograph Column Arrangement

Typical GC Chromatograms

Infrared Spectrophotometer Sample Cell

PReCGaelvoration Curves for CO) and CO.

Preliminary Observations GM Pt-Pd Catalyst

Preliminary Observations 0.05% Pt Catalysts

Simple Oscillatory Behavior

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Nitrogen and Helium Diluents on

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Qualitative Effects of Temperature

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Reproducibility of the Oscillatory Behavior

Catalyst eActivity Changes at4/3 0h



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Oscillations Similar to Experimental Observations Extreme Forms of the CO Oscillations

Multipeak Oscillations

Rotameter Calibration Curves for Oxygen

Rotameter Calibration Curves for Helium

Rotameter Calibration Curves for Nitrogen

Calibration Curves for Matheson Model ALL-100 Linear Mass Flowmeter (0-100 SCCM)

Calibration Curves for Matheson Model 8116-0153 Linear Mass Flowmeter (0-5000 SCCM)


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Ties LOcuct. On hiversced: Dyejenewcamby work vat Liljenroth [1.1] and van

Heerden [71.2], a considerable amount of research has been

directed at analyzing the stability of chemically reacting

systems. Theoretical and experimental studies, summarized necevera® cvreview arcicies [1.3 - 91.7], have shown that under certain conditions, chemical and catalytic reaction systems can display oscillatory behavior. In a continuous flow reactor, oscillatory behavior is normally manifested through periodic changes in the reaction temperature and/or the concentrations of reacting species with respect to time.

In heterogeneously catalyzed reactions oscillations in

concentrations of adsorbed species and/or oscillations in

the temperature of the catalyst's Surface can occur as well.

These oscillations can occur without accompanying oscilla-

ELONS ite the bulk fluid ‘phase,

The recent interest in oscillatory phenomena in chemical reactors is due to several factors:

1. The fundamental processes occurring in chemical reactors can be more fully understood through a knowledge of the mechanisms by which oscillatory behavior occurs.

2. Large temperature fluctuations on catalyst surfaces can result in rapid catalyst deactivation.

3. Large temperature fluctuations in a chemical reactor can

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present safety problems, hence, a knowledge of how to predict, and suppress, this behavior is necessary for proper design and operation.

4. Increased conversion of reactants to products, and isolation of low yield intermediates, can sometimes be obtained by operating.a chemical reactor in a cyclical


In the field of heterogeneously catalyzed reactions, reports of oscillatory behavior have been mainly restricted COmMOXxTGamronmreact10nS,— 6.07) Ox1dationm ot ammonia: (1.6.1; RyvowCCe rei .o 2 mis) seeCarbolr monox1de fal 2 7.1.91) and Vverocarbonshiie19e-9 1.21 le catalyzed by bulk (wire, gauze CEmLGit ) Oresupported platinum or nickel. Self-sustained ocillations have been observed in recycle, fixed-bed and single-pellet reactors under isothermal and nonisothermal conditions. Various causes for the oscillatory behavior have been postulated; these include competing mechanisms [1.13, 1.14], variations in activation energy with surface coverage [1.22, 1.23], non-reacting adsorbed species [1.24], presence of impurities [1.15], formation of surface oxides eae 1.25], variations in catalyst surface temperature UVecoliemand LOcadeneat) and mass poLanstermenteciuc melee

Many mathematical models which display oscillatory behavior have been constructed using the above postulates. Unfortunately, a large chasm has separated the theoreticians

from the experimentalists, as none of the models which have


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appeared in the literature have been validated through Systematic experimentation. There are two main reasons for this: firstly, an experimental study which systematically examines the effects of the various operating conditions (temperature, concentration, space velocity, etc.) on oscil- latory behavior has yet to appear in the literature, and secondly, most models have been so reduced in complexity through the use of simplifying assumptions that only a

tenuous connection with physical reality remains.

1.2 Research Objectives

As originally conceived, this research project was to encompass a general mathematical and experimental examina- tion of oscillatory behavior in heterogeneous catalytic reactors. Since a total description of this phenomena would indeed be a formidable task, a number of specific research areas, generally consistent with the overall objectives, were chosen.

As a starting point, models which have appeared in the literature were examined with the particular aim of investi- gating the effects of simplifying assumptions, such as the equilibrium assumption. Also, in order to gain modelling experience, simple models displaying oscillatory behavior were developed. In parallel with this work, an experimental apparatus was constructed and oscillatory banavion Of carbon

monoxide oxidation over supported metal catalysts was



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Building on the experience thus gained, a systematic experimental investigation of the effects of operating conditions on oscillatory behavior was performed. In parallel with the experimental study, a new model for carbon monoxide oxidation was developed. The new model was kept as physically realistic as possible, so that all parameters in the model had actual physical counterparts, and thus the model behavior could be directly compared with the actual experimental behavior. However, in a model, increased complexity is the price which is exacted for physical real- ism, and thus a total delimiting of the model behavior has

not yet been achieved.

1.3 Thesis Layout

Since the research project was approached from several directions, this thesis is composed of several quite distinct though related pieces of work. For this reason, references are cited at the end of each chapter.

In Chapter 2, a simple catalytic reaction iS examined SO as to develop the groundwork for determining the effects of simplifying assumptions on reaction models. Building on this development, the effects of the equilibrium assumption on a model of an oscillatory reaction are presented in Chapter 3, and in Chapter 4 the consequences of removing the

equilibrium assumption, and going to a higher order system

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(three ordinary differential equations), are examined. In Chapter 5, an interesting property of high order systems, namely chaos, 1S introduced, and is shown ne exist in a very Simple reaction model.

The next two Chapters, 6 and 7, are devoted to a description of the experimental equipment, and a presenta- tion of experimental observations for carbon monoxide oxida- tion on supported metal catalysts. These experimental observations are then used in Chapter 8 to qualitatively examine the predictions of a new model for carbon monoxide oxidation. Finally in)Chapter 9 .possible' directions for

future work are presented.

1.4 References

Lesa SiehencOt wes G a CHC aMOL ENG 6 me oll On hos AO mn hes Ven HeelOcitnto.) LNG. rida Chema, (953 4550242; ergs Nic Obes eG onOePOrvENOW mu spe COeMomReW. peed og wis we SOON 1.4 SGUVisiez aL aA per ACV.) WCNC erty oie Ol OGL es) Sherrcuch, uM mandy sChinicz t1R Ae meCatalmuRevVeuGO Il | aati, | este Bey nye ss: Pranckse.U obvees Angew... Chelimeln by ehO lm hii Qd ae mel od ocean bee Sinko, OMeG yerana eS lin KoseeMe Mee Gara eRe vee, Erg: we (Camel o.. 1.8 Fliytzani-Stephanopoulos ,. Maw icohmioc = i:..D. and

Gavettar erie munscotel.,» (GOMES 2b.

» Bd s&. SO. URIS Het 125 aoe ie

at, Paella eh ean oa A on Soe ogee ee ear | re 12 zac gem, ate e- 4

ny aisctw ert ie @ ; aA [2 Ani 2g ttosege


et eb oi ean 45? | gale 4 OW: Tepe hm see | . ji eee ay | : de 4 £140 SOR ik o> wore ie | | —P write Wi. “\ > ni ee

"6 e reg a o . .

Puree, « weatombucsi el aneo. Catal. 1976. 53 32. Belyaey, BVaUe more KO. —MIM> “and®*Slin' ko M.G.a.Proc. Bite Conn Gara) wamothmiLonden, 1976), ag. )7158, Chem, SOGapebOnGgon, mis 7.

Kugtan 1c, to, olen nmuchy aM Bandalucs. De, e.7;,sCatal..; POCO ON it.

Beuseh= eH. sFiequth, P.eandsWicke;«E.,;,Chem.oing% Techngs 1972254495445"

Hugomer. mand wakubauhn eM Chema Imge Techn. 4972944 503".

MeCaiony ten mecanradhik mw oe hUCZYASKI .»G.C.. and (Senta Cig te taren ed teicl ey mecle, ane lac pea yey | Gis) ea Oeee He

VaboneSeruir wm CanrDOury, tld. sand WOll) TE. Mw. Corals 97 8).55 76.

PevCnirad whut wand schmutce R.A. Chem, Eng. .-Comnun.., (OSI) Get Cosine

Chew GUC ioneM. seat Ch Bord .t,06 bo Oscars.

Rabhnousky sum Kita &. and: Hiavacek, V., Chem. Eng: Siar WSkedh Shey iss

Cutlip aeM.S-eend Kenney, CN. , Chemicals Reaction

Bugineéring = Houston) “ACS Symp. 9Ser. 65, (p24 i/o pm aGs,

"Washington, 1978.

Hawkins, 0 .R. and Wanke, S.B.7) Can. J. Chem. fng.7 Roos BOA.

Vayenas, C.Guy sLee, By and iMichaelc v7 doa tata. (USTsTeS (IS. Sis

PikiOS paG A amend sbuss, Dj Chem bnuaoCd, lO 2ase,

are ge ais 5 bai erik) Lise anh

4 ~> or * * ya Le ire mit i ne mi 4 ta “@

os pet i‘ d wh h ye r

Soa «(ee


ee a .@ A at . paghnatiok, (e.8 iy 3q' At: hu) (4 40 retin >! vt


+urt ce wt pv pepe

Rais, od. . Side eed! pitye 4 Ree ee : Tas f © Tid ok Ae



EvVeanoOVerE Ate nimakOvVEesonAs es inuko, MG.) Bruns, Dali IOC yy We 7 AGEs (alee Siem ESTs th isp ar soley PUCenvoerGerweG es GNel on. woGl.,) 1970 33. i203. RenotaneG Jae se lnesas « University of P1linois; 1979.

Dacdonnier Re DuMont, Meeand Nuyts, uJ. , 0. Catal. iSS0M6Oe 1 SG.

JVencenen \ tema ngeray . Weis, chem. ENG. aSC1... L980G35


Ml , ePsr4e



eee NcCroavuection

In the modelling of heterogeneously catalyzed reac- tions, whether it be steady-state or dynamic modelling, it is common to begin with fairly complex models which consist of many equations. These large models result from the need to describe the many variables (bulk and surface phase temperatures, bulk and surface phase concentrations of the reactants and products, surface phase concentrations of intermediates, etc.) which are present in heterogeneous catalytic systems.

Since the analytical or numerical solution of large systems of simultaneous non-linear algebraic or differential equations 1S a non-trivial task, it 1S common to reduce the Size of the system of equations by making simplifying assumptions. In the modelling of heterogeneously catalyzed reactions, the two most commonly used simplifying assump- tions are the steady-state assumption and the equilibrium assumption.

These two assumptions have been used, usually without question, throughout virtually the entire history of hetero- geneous catalysis, as shown by examination of the early WOEKS [2 1,022.2 Nelms therfiedd. cist eusceonlyeasctor, Late: that

any questioning of the validity of the assumptions [2.3],

pint rou! 2a SAE. @:

sone SOPs | 4G- err


ay a6


on] >) 10 ie

ous GB eg Vi ah Soe isis ime 4 o* ; -e ieee

seuss @! Stiads cede, vie) eee aneite soc 6s> aaeae as BA Oe F :

, ae ate


- oa ery

7 7

and examination of their effects [2.4 - 2.6] has been performed. However, even in systems in which it can be shown that the surface reaction is the controlling step, the use of the equilibrium assumption in a model can have some unexpected effects. These insidious effects result from the very manner in which the equilibrium assumption is used to eliminate variables from a model.

Rom nem@r Ol lOWInG yt wioll be wshownithat for a very Simple model, extremely different, and in some cases contra- dictory, results can be obtained depending on how the equi- librium assumption is used to eliminate variables from the model. For comparison purposes, the exact analytical solu- tion, and a solution using the steady-state hypothesis, are

also presented.

2.2 A Simple Model The irreversible isomerization of A will be used as an example. The overall reaction is A ——> B rare ie

and the following reaction mechanism will be considered:

yen ee AX C2a24

B+ xX BX G22 8 )

eT ee ee oe a eT

ad is LF ' ' ify = i ¢ : i ¥ ij j te, @ *:

7 , : ae ,@O@Lce ARLE ts poy wet? Se [ 3a

5 Ah Oe sa Or LG" re a , AGL Aoeva : a . 7

r yi Sif As I. hs | -3haa) oh ; sales bh pzoguenme

hae ae iin 1 je) 29h ee i144) 3er Pensa

: ¢ . ; UVOlt Sa jth. at | Sagi: dens 2 at 4 S.4 S¥e..: “Row. sLenre f 23.104 722018

whe & i-<2Ce mi irees

re t 77 ~Aepap


‘= ae. «) ;

7 Bn

pe eee Ce vltee ely hee

If the catalytic sites, X, are conserved,then mass balances

ONAATaDXy anoexooiver

CA xa Axe meet hee) AX aie eke [Ax G2 35) at Cpe eae ih eee kee DBKS 4 ke PAX (2°65) dt OU Ra teae Kee eee eA ak of Bal Ree kes PBX } Oe 287 ) dt

where the normalized site conservation equation is given by: PARE EX ee exe =o 29S)

In dimensionless form, Eqs. 2.5-2.7 are written as:

opAeanks [Rh Sen etait eax) C2098 ould

Cnn) Bake Px aka, [BX aes Ax) (220) aT

ke ee RO XA ce he AS oe Kee Perk SL Bx) 2m lal aT

where, Kae ee eaal PR. (ear

Ke, €2 "hah ke (22125)

K, = Pes ks (PRC)

Key ee heey ke (24 20)

tT. =a (2242e)

In the following it will be assumed that [A] and [B] in

the gas phase are held constant. Conceptually, this could

PRShAlAS Poes, Cea tevreT et eye |e:

i ; id i j B t ‘a ~~ \ x 7 }

; ).



iN u = i { a } tl ra) : wunlh wa . i } } }.4 t, a 7 vu i?

be achieved by having only a very small amount of catalyst Liem Langemneactormmotmarternatively, it could be accom- plished by regulating the flows of A and B to and from the


2.38 exacte solution Since Eq. 2.8 forces the sum of the derivatives to be

G0UawEcOeZero, 1,65:

GQpAx )£e d>EBXdt eal xd = 70 MEER

aT dT aT

then Eqs. 2.9-2.11 are not linearly independent, and any two of these equations are sufficient to completely describe the behavior of the surface concentrations. For example, if [xX] bowe Mimi nareca trom Eqs... 2.9 and 2.10 by using Bq..-2.8, then

the following system of two ordinary differential equations

results: CUA Sema wee ieK tKee) fe OR BX IT KG (Or 1a:)

ar dl Bx] tae 1k oe Axaa—, (Ke FRE.) LBS] bes (2315)


From linear systems theory, the solution of Eqs. 2.14 and

Zoos wOLvVen ibys

aT lou PAS ook (Atk ee iaee™ aller tehe Ub+ hee). eee) a(a-b) bla-b) (2.16) aT Dr eulige aye

+{(atK_,+K,)e [Conk oe thee } [AX], eee = BS, (a-b) (a-b) (a-b)

wiscisi seoget tie Us es . « Wie -0'¥n{ at 99) Saat - | T 7 ag 4 7 -

: 7

et ag ag re Le m4 ‘ica , ton ate fh tee ghas Gare . y ue


fey og ly 4 .WJ¢ shh BO) Serie 6047 Se cs. Vet. Bal aia * pent > ed ro, fated hd ' ace . oot tet) Soderial fe ae wary | > neteye DnNetto sain

: . . 7 See 1): i 7 cars | Prat:

o “Hye a“

Meas WEL I> U96cU.%& Bh), Sate coeza ye \eensy

' Caee Ros

aT lew BBX) Bemaik eke (ar iakoee (tiee ) + {K,4K5 (b+14K.,)}(1-e ) ala-b bla-b at bT +{(at1+K,+K_,)e (btitk,+K2,)e 1 BX],; a-b ak 9pt BiGioKe) hee) -ete)[ Ax}. (9517) a=b5


ans (14K 4K oR, 4K.) [1 /TELR ORR RTT | 2 (1+K,+K_,+K,+K_,)?2 (2. ea)

b = = UU a ol | TI Ea,

2 iad heey gle 20 Rew 25 cen aes 9 ae

| Sees |

C2.upeD) Brom bos. 2. 16,ang 2.17, the values of [AX] and {BZ ] at a

Steady-state are readily found to be:

TAX] = =Ki(atKe,) + K,(btK..) ala-b) b(a-b) [AX] = eke (2.19) Kes at ae ema ( St Ree nh Ret Kone) [Bx] «= Rates Kier Koa) C220)

KECIERS2) RBG. @) ake

ThusyeEqs, «2al6—-2920 completely descruabe thetdynamic jand

steady-state behavior of the reaction system.


2.4 Steady-State Assumption

In the previous reaction system, the steady-state assumption could take the form of assuming that the concen- tration of either of surface species AX or BX is always at a steady-state. If, for example, surface species BX is always at a steady-state, then mathematically this is equivalent to SeutinogshG. 2. 10 equal Lorzerco,

KACk) She UB eA X] (= 20 (So) DEMEG.)2.2i 2S USed 1m Conjunction with the site conserva- MuCnusequati1On) BEC necec, suhen tthe ysyscem of as; 2.9-2, i1.can be reduced to a single equation in one variable. Biiminatingetx’) and LBxX] from Ba. 9 gives:

Gear hee ee PAK Ait Ke 2) +1 enh oe) CR +k. 3s C2: 67:38) aT K,+K_ > (KEK

BoQuation @2)22 ss esolved sto ey ivelid:

cr age eee) a Kee Me Kees) (Kea Ke) (PAS =tek Bie o Girt)

Ree ee) ae Re tke)

eriaert a Rest te ee or ea) te Ke eee +[AX].e ee It should be noted that this result can also be obtained if Eqs. 2.8 and 2.21 are used to eliminate [AX] and [BX] from Eq. 2.11, as the same expression for [AX] always occurs irrespective of the solution path which is followed when the steady-state assumption is used. AMCOMNDar DeoteEOmenG,. 2e23)iwutinethesexact solution egiven

Byeecc. #2 sOmandpaaio Shows “Uhatmtheserequatuons have

to. : : i nn

oi, ae Raat ei" eet © ey ) 8854 "eo" '< AAs 24! “Res tans Vise. HR PS eI ee . al ae = 2 Js i . j . “etety & ve iT F 7 7 ,s iv = srigie@ ‘i . ie . ° i 4 e ¥ . oe Pas | he

be vi . 4 : d =