Review articles on the study of combustion. Combustion Theory and Modelling. Volumes Part 4 1-13

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Heterogeneously catalysed combustion in a continuously stirred tank
reactor—low-temperature reactions
M. Nelson a; G. Wake b; X. Chen c
a Department of Fuel and Energy, The University of Leeds, Leeds, LS2 9JT, UK. b Department of
Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. c
Department of Chemical and Materials Engineering, The University of Auckland, Private Bag 92019,
Auckland, New Zealand.
To cite this Article Nelson, M., Wake, G. and Chen, X.(2000) 'Heterogeneously catalysed combustion in a continuously
stirred tank reactor—low-temperature reactions', Combustion Theory and Modelling, 4: 1, 1 — 27
To link to this Article: DOI: 10.1088/1364-7830/4/1/301
URL: http://dx.doi.org/10.1088/1364-7830/4/1/301

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Combust.TheoryModelling4(2000)1–27.PrintedintheUK PII:S1364-7830(00)02914-4
Heterogeneously catalysed combustion in a continuously
stirred tank reactor—low-temperature reactions
MINelson†,GCWake‡andXDChen§
†DepartmentofFuelandEnergy,TheUniversityofLeeds,LeedsLS29JT,UK
‡DepartmentofMathematicsandStatistics,UniversityofCanterbury,PrivateBag4800,
Christchurch,NewZealand
§DepartmentofChemicalandMaterialsEngineering,TheUniversityofAuckland,
PrivateBag92019,Auckland,NewZealand
Received19March1999,infinalform22November1999
Abstract. Inthispaperamodelfortheheterogeneouslycatalysedreactionofagaseousspecies
undergoingasingle-stepexothermicreactioninawellmixeddiabaticcontinuouslystirredtank
reactorisdevelopedandanalysed. Low-temperaturereactionconditionsareassumed,sothatthe
degradationoftheinflowspeciesinthegasphaseisnegligible. Theadsorptionanddesorptionof
theactivespeciesontoasolidcatalystlayerisexplicitlymodelled.
Undertheconditionsused,theclassiccombustionS-shapesteady-statecurveisnotexhibited.
0
01 Insteadthesteady-statediagramconsistsoftwodisjointsolutioncurves:asolutioncurvecontaining
2
y asteady-statebranchcorrespondingtofullcoverageofthecatalystwithnoconversion;and,an
r
a isolacontainingasteady-statebranchthatcorrespondstolowcoverageofthecatalystwithahigh
u
an conversionoftheinflowspecies. Thesteady-statecurvecontainsoneextinctionpointandno
J
4 ignitionpoints. Consequentlytheinitialconditionsdetermineontowhichsteadystatethesystem
2
evolves. Largevariationsinthecriticalcatalyticsurfaceareaarefoundastheinitialtemperature
0
:5 ofthereactorisvaried.Thisisexpectedtohavesignificantpracticalimplications.
5
0

:
t
A

d
e
d
a
o
nl Nomenclature
w
o
D
The subscripts g and s refer to properties of the gas and solid phases, respectively. The
subscriptsc;d andr refertochemisorption,desorptionandcatalyticreaction. Thesubscript
j cantakethevaluescandex.
A(cid:3)d Non-d(cid:18)imensionalizedp(cid:19)re-exponentialfactorfordesorption (—)
A(cid:3) D LkbTrMAcpg
d h(cid:31)
s:g
A Thegas-phasereactant (—)
A Pre-exponentialfactorofthegas-phasereaction (s−1)
g
A(cid:3) Non-dimensionalizedpre-exponentialfactorforthe
g
gas-ph(cid:18)asereactio(cid:19)n (—)
c V (cid:26)
A(cid:3) D pg g g A
g (cid:31) S g
g:s in
Ar Pre-exponential(cid:20)factor(cid:21)ofthecatalyticreaction (s−1)
E H E
A D r exp r
r RT2 RT
c c
1364-7830/00/010001+27$30.00 ©2000IOPPublishingLtd 1
2 MINelsonetal
A(cid:3) Non-dimensionalizedpre-exponentialfactorofthe
r
catalyt(cid:18)icreaction (cid:19) (—)
A(cid:3) D LMAcpgH
r (cid:31) T
s:g r
[A−M] Theconcentrationofspecies.A−M/ onthe
s s
catalystsurface (molm−2)
B Theproductoftheirreversibleexothermicreactionof
speciesA (—)
CR Conversion(cid:18)ratioof(cid:19)theinflowspecies (—)
a(cid:3)
CRD100 1−
a(cid:3)
0
D(cid:3) Anon-dimensionalizedvariable (—)
R
D(cid:3) D
cpgMA
E Activationenergy (Jmol−1)
i
E(cid:3) Non-dimensionalizedactivationenergy (—)
i
E(cid:3) DE =.RT /
i i r
G(cid:3) Miscellaneousnon-dimensionalizedvariable (—)
(cid:26) V
G(cid:3) D g g
LMASin
H Hypotheticalrampingrateatwhichthecharacteristic
10 temperatureofthecatalyticreactionisdetermined (Ks−1)
0
2
L Densityofactivecatalystsites,perunitsurfaceareaofthe
y
uar catalyst,attimet D0 (molm−2)
n
Ja MA ThemolecularweightofspeciesA (kgmol−1)
0 24 [M]s TheconcentrationofspeciesMs onthecatalystsurface (molm−2)
5:5 M(cid:3)s Thenon-dimensionalizedconcentrationofspeciesMs onthe
0
catalystsurface (—)
:
d At M(cid:3)s D[M]s=LD1−(cid:18)A
de Q Exothermicityofchemisorption (Jmol−1)
a c
wnlo Qc D−1Hc
Do Q Gasphasereactionexothermicity (Jmol−1)
g
Q D−1H
g g
Q(cid:3) Non-dimensionalizedreactionexothermicity (—)
i
Q(cid:3) DQ =.RT /
i i r
Q Exothermicityofthecatalyticreaction (Jmol−1)
r
Q DQ +Q
g c r
R Theidealgasconstant (JK−1mol−1)
S Activesurfaceareaofthecatalyst (m2)
c
S Theexternalsurfaceareaofthereactionvessel (m2)
ex
S Internalsurfaceareaofthereactionvessel (—)
in
S(cid:3) Non-dimensionalizedsurfacearea (—)
j
S(cid:3) DS =S
j j in
T(cid:3) Non-dimensionalizedtemperature (—)
i
T(cid:3) DT =T
i i r
T(cid:3).0/ Thetemperatureattimet(cid:3) D0 (—)
i
T Thetemperatureofthecoolantsurroundingthereactorvessel (K)
a
Heterogeneouslycatalysedcombustion 3
T(cid:3) Thenon-dimensionalizedcoolanttemperature (—)
a
T(cid:3) DT =T
a a r
T Temperature (K)
i
T Areferencetemperature (K)
r
T Temperatureoftheinflow (K)
0
V Volume (m3)
i
a ConcentrationofspeciesAinthereactorvessel (molm−3)
a(cid:3) Non-dimensionalizedconcentrationofspeciesAinthe
reactorvessel (—)
a(cid:3) Da.MA=(cid:26)g/
a TheconcentrationofspeciesAinthefeed (molm−3)
0
a(cid:3) Thenon-dimensionalizedconcentrationofspeciesA
0
inthefe(cid:0)ed (cid:1) (—)
a0(cid:3) Da0 MA=(cid:26)g
a(cid:3).0/ Thenon-dimensionalizedconcentrationofspeciesAin
0
thereactorattimet(cid:3) D0 (—)
c Heatcapacity (JK−1kg−1)
cp(cid:3)i Theratiooftotalheatcapacityofthesolidphasetothe
t
totalheatcapacityofthegasphase (—)
c (cid:26) V
c(cid:3) D ps s s
t c (cid:26) V
pg g g
0 h Planck’sconstant (Js)
1
20 k Boltzmann’sconstant (JK−1)
b
uary ki Rateconstant (mols−1)
Jan nb ThenumberofmolesofspeciesBproducedbythereaction
4 ofonemoleofspeciesAorthereactionofonemoleof
2
:50 species.A−M/s (—)
05 q Feedflowrate (m3s−1)

At: q(cid:3) Non-dim(cid:18)ensiona(cid:19)lizedfeedflowrate (—)

d c (cid:26)
ade q(cid:3) Dq pg g
nlo (cid:31)s:gSin
Dow s Thestickin(cid:20)gprob(cid:21)ability (–)
−E
s Ds exp c
0
RT
s
s Maximumstickingprobability (—)
0
t Time (s)
t(cid:3) Non-dimensionalizedtimescale,baseduponthetimescale
ofNewtoniancooling (—)
(cid:31) S
t(cid:3) D g:s in t
c (cid:26) V
pg g g
t Theresidencetimeoftheflowinthereactor (s)
res
t DV =q
res g
v(cid:3) Non-dimensionalizedmeanmolecularvelocityofspeciesA
A (cid:18) (cid:19)(cid:26) (cid:27)
c (cid:26) RT 1=2
v(cid:3) D pg g r
A (cid:31) 2(cid:25)M
s:g A
1H Enthalpyofchemisorption (Jmol−1)
c
1H Gasphasereactionenthalpy (Jmol−1)
g
4 MINelsonetal
(cid:18) Thenon-dimensionalizedconcentrationofspecies[A−M]
A s
onthecatalystsurface (—)
(cid:18) D[A−M] =L
A s
(cid:18) .0/ Thenon-dimensionalizedconcentrationofspecies[A−M]
A s
onthecatalystsurfaceattimet(cid:3) D0 (—)
(cid:26) Density (kgm−3)
i
(cid:31) Heattransfercoefficientbetweenthegasphaseandthe
g:s
reactorvessel (Js−1m−2 K−1)
(cid:31) Heattransfercoefficientbetweenthereactorvesseland
s:a
thecoolant (Js−1m−2 K−1)
(cid:31)(cid:3) Non-dimensionalizedheattransfercoefficientbetweenthe
s:a
reactorvesselandthecoolant (—)
(cid:31)(cid:3) D(cid:31) =(cid:31)
s:a s:a g:s
Unlessotherwisespecifiedwetakethefollowingtypicalparametervalues: E D 0Jmol−1,
c
L D 1019 molecules=m2 D 0:166 (cid:2) 10−4 mol m−2, MA D 0:1 kg mol−1, Qc D
60(cid:2)103 J mol−1, S D S , S D S , S D 5(cid:2)10−2 m−2, T D 298 K, T D 298 K,
c in ex in in a r
V D 10−3 m3, V D 2:5(cid:2)10−6 m3, b D 0 mol m−3, c D 1040 J kg−1 K−1, c D
g s 0 pg ps
400Jkg−1K−1,q D0:125(cid:2)10−3m3s−1,s D1,(cid:26) D1:1kgm−3,(cid:26) D8:0(cid:2)103kgm−3,
0 g s
(cid:31) D(cid:31) D30.
g:s s:a
The appropriate values for physical constants are: h D 6:63 (cid:2) 10−34 J s, k D
b
010 1:380622(cid:2)10−23JK−1,R D8:31441JKmol−1.
2

y
r
a
u
an 1. Introduction
J

4
2
0 A catalyst is a substance that increases the rate at which a chemical system approaches
5
5: equilibrium, without being consumed in the process. The catalyst does this by providing
0

: an alternative reaction pathway that significantly changes the kinetic parameters (activation
t
A
d energy,orderofreaction,ratecoefficient)ofthesystem. Thishastwoimportantconsequences.
e
ad Firstly,thecatalystincreasestherateofreaction,favourablereactionratesbeingachievedat
o
l
n muchlowerreactiontemperatures. Thiseffectoftenappearstobeabsoluteratherthanrelative:
w
o
D under the conditions concerned the presence of the catalyst is essential for the achievement
of a significant rate of reaction. Secondly, a particular catalyst may promote one reaction
pathwaywhereseveralchannelsmayotherwisebeaccessible,resultinginasubstantialyield
ofspecificproductswhichwouldotherwisebeobtainedininsignificantamounts. Thisfeature
ofcatalysis,wherebythereactionisdirectlyalongaspecificroute,greatlyenhancesitsvalue;
andisutilizedinthemajorityofcatalyticprocessesusedindustrially.
The distinguishing feature of a heterogeneous catalysed reaction is the involvement of
asurface. Thissurfaceisneitherfedintothereactor, norwithdrawnfromit, norconsumed
duringthecourseofthereaction,uponwhichthecatalyticreactionoccurs. Itisdesirablein
heterogeneouscatalysistohavealargesurfaceareaforreaction. Inpracticethesurfacearea
of the catalyst is frequently many orders of magnitude greater than the surface area of the
reactoruponwhichitisdispersed. Thebasicphysicalchemistryofheterogeneouscatalystsis
wellunderstoodintermsoftheadsorptionanddesorptionofactivespeciesuponthecatalyst’s
surface[1–3].
Inthispaperweconsiderthesimplestmodelforthenon-isothermalcatalyticconversion
ofagaseouschemicalspeciesundergoingasinglestepexothermicreactioninacontinuously
stirred tank reactor (CSTR). We explicitly model adsorption and desorption processes on
Heterogeneouslycatalysedcombustion 5
the catalyst. As the steady-state solutions drive the dynamics our approach is to identify
thebifurcationalstructureofourmodelandsubsequentlytoinvestigatehowexperimentally
controllableparametersaffectthebehaviourofthereactor. Althoughourmodelcontainsboth
thehomogeneousandheterogeneousreactions,inthispaperweassumethatthetemperature
ofthevesselremainssufficientlylowthatthehomogeneousreaction,thatis,thereactioninthe
bulkgas, isnotinitiated. Consequently, weinvestigatewhichmultiplicitiesandoscillations
arepurelyduetothecatalyticsurface.
Themodelthatwedevelopisanaturalcounterpointtotheclassicsituationofasingle-step
exothermicreactioninaCSTR.Anexcellentintroductiontothemathematicalmodellingof
theseproblemsisthebookbyGrayandScott[4]whichreviewsthebehaviourofisothermaland
non-isothermalsystems,includinghomogeneouscombustioninaCSTRandheterogeneously
catalysed reactions. We discuss experimental and theoretical results for heterogeneously
catalysedreactionsinsection1.1.
1.1. HeterogeneouscombustioninaCSTR
Heterogeneous catalytic systems have been widely studied, both experimentally and
theoretically. Mathematical modelling of such processes can be classified depending upon
theapproachtakentoaddressthreeissues.
(cid:15) The reactor system: for example, assemblages of catalyst particles in a gradientless
environment,CSTRmodels,differentialreactormodels,plugflowreactormodels,porous
0 catalystparticles/pelletsandtubularreactors.
1
20 (cid:15) Themodellingofphysicalandchemicalprocesses,isothermalornon-isothermaloperating
y
ar conditions,theassumptionofaconstantpartialpressureoftheactivegas-phasespecies,
u
n
a the formulation of the catalytic rate constant, the dependence of the activation energy
J

24 foradsorptionuponsurfacecoverageandnon-uniformdistributionofadsorbatesonthe

50 catalyticsurface.
:
05 (cid:15) Themodellingtechniqueemployed: spatiallystructuredmodels,non-spatiallystructured

:
At models,MonteCarlo-typesimulationsandcellularautomatamodels.

d
e
ad Models for heterogeneous catalysis were comprehensively reviewed by Razon and
o
l
n Schmitz [5]. Recently, very detailed models for heat-transfer processes in heterogeneously
w
o
D catalysed reactions have been developed [6]. Catalytic combustion models combining
multistepsurface-chemistrymechanismswithmassandheattransferhavebeenpresentedatthe
‘catalyticcombustion’sessionsatthe1996[7–9]and1998[10–12]internationalsymposiums
oncombustion(Naples,Boulder).
Homogeneous combustion in a CSTR has been extensively studied by considering
a prototype model in which the combustion process is represented as a single-step
reaction. The archetype for heterogeneous combustion is a single-step reaction on a
catalyst combined with absorption and desorption processes: the fundamental chemical
processes in catalytic combustion. The aim of this paper is to investigate such a model.
IsothermalheterogeneouslycatalysedreactionsinaCSTR,assumingLangmuir–Hinshelwood
kinetics, were comprehensively treated by Chang and Aluko [15,16] who showed that even
simple Langmuir–Hinshelwood mechanisms for bi-molecular reactions can have oscillatory
behaviour. Therearesurprisinglyfewexamplesofthenon-isothermalversionofthisparadigm
in the literature. In fact, most models in the mathematics literature treat heterogeneous
catalysedreactionsasasingle-stepexothermicreaction,ignoringtheadsorptionanddesorption
processes. TheporouscatalystparticlemodeldevelopedbyElnashaieandCresswell[13,14]
appearstobetheonlysimplemodelthatexplicitlymodelsadsorptionanddesorptionprocesses,
6 MINelsonetal
ratherthanmodellingtheseastheproductofamasstransportcoefficientandadifferenceof
concentration levels. They considered the case when the active species is weakly adsorbed
ontothecatalystsurface.
Subramanian and Balakotaiah [17] have classified the steady-state solutions and the
dynamic behaviour of a well mixed two-phase heterogeneous catalytic reactor model. In
additiontoassumingadiabaticconditionstheyrepresentadsorptionanddesorptionprocesses
by a simple mass transfer process and assume that the heat of adsorption is negligible. In
a latter paper Christoforatou et al [18] develop a runaway criterion for adiabatic catalytic
reactors, considering both plug flow and CSTR reactors. Although the latter paper uses the
modeldevelopedinSubramanianandBalakotaiah[17]itiscuriousthatneitherpaperrefers
totheother. Christoforatouetal[18]implythatiftheresidencetimeiskeptbelowacritical
levelcorrespondingtoanignitionlimitpoint,thenthetemperatureofthereactorcannotrise
to an undesirable level. This is only true if perturbations on the reactor parameters cannot
occur,asthehigh-temperatureignitionsteadystatecoexistswiththelow-temperaturesteady
state over a range of residence times between those corresponding to the extinction limit
pointandtheignitionlimitpoint. Furthermore, theresultsofSubramanianandBalakotaiah
[17] show that there are some operating conditions in which perturbations on the reactor
conditionscancausethetemperaturetoreachtheignitionstateforarbitrarysmallresidence
times.
WebelievethatthisisthefirstexemplarmodelforheterogeneouscatalysisinaCSTRto
includenon-isothermaladsorption,desorptionandcatalyticreactionsteps.
0
1
0
2 2. Descriptionofthemodel

y
r
a
anu Wemodelagaseouschemicalspecies(A)flowingthroughadiabaticcontinuouslystirredtank
J
4 reactor. Thefluidandsolidphasesareassumedtobeinaperfectlymixedsystem, i.e.there
2

0 isnospatialvariationofboththetemperatureandthereactantconcentrationinthesolidand
5
:
05 fluidphases.

t: The active species (A) undergoes a single, irreversible, exothermic reaction, producing
A
d n moles of an inert product (B). This reaction can take place either in the gas phase, the
e b
d
a homogeneousreactionoronasolidcatalystthatisdispersedonthewallsofthereactorvessel,
o
l
wn theheterogeneouscatalysedreaction. Inbothcasesweassumethatthedecompositionreaction
o
D canbemodelledusingArrheniuskineticsandtheproductsofeachreactionarethesame.
The dynamics of the chemical and physical processes are modelled by considering the
interactionbetweenthreezones: thewallsofthereactor,uponwhichthecatalystisdispersed;
the gaseous reaction zone within the reactor; and the coolant surrounding the reactor. For
simplicityeachzoneisassumedtobewellmixed. Theresultingmodelcontainsequationsfor
solid- and gas-phase processes which are coupled through inter-phase heat (convective) and
mass(adsorptionanddesorption)transfer.
2.1. Physicsofthereactormodel
TheCSTRisassumedtobecontainedwithinacoolingsysteminwhichtheflowrateofthe
coolantcanbequicklymanipulatedtoobtaintightcontrolofthecoolanttemperature. Wedo
notfollowRussoandBequette[19,20]andmodelthecoolantdynamics.
The reactor walls are assumed to be thermally thin, by which it is meant that there
is no temperature gradient across them. Heat transfer between the reactor walls and the
coolant/reactorinterioraremodelledbyNewtoniancooling. Theassumptionthatthereactor
walls are thermally thin is standard in the literature for simplicity. We use the standard
Heterogeneouslycatalysedcombustion 7
Figure1.Schematicrepresentationofthemodelgeometry.
hypothesesthatthereactorvesselhasconstantvolume,perfectmixingandconstantphysical
0 parameters. Thegeometryofthemodelisshowninfigure1.
1
0
2

y
r
a
u 2.2. Theheterogeneousmodel
n
a
J

24 Thecatalystisassumedtobedispersedalongthewallsofthereactor. Itshouldbenotedthat

50 the surface area of the catalyst is not in general that of the reactor but is many times larger;
:
05 typically the surface area is (cid:24)300 m2 per gram of catalyst. We assume that heat transfer

:
At betweenthedispersedcatalystandthesurfaceofthereactorwallsissuchthattheyareatthe

ed sametemperature.
d
a
o Wemodelthecatalyticchemistrybythereactionscheme
l
n
w
Do A +M (cid:10).A−M/ (1)
g s s
.A−M/ !n B +M (2)
s b g s
where M is the concentration of unoccupied active sites (mol m−2) and .A−M/ is the
s s
concentrationofactivesitesoccupiedfollowingadsorptionofA(molm−2). Weassumethat
theproductspecies(B )isnotadsorbedbythecatalyst.
g
Thisschemecomprisesthreesteps: non-dissociativechemisorptionofspeciesAontothe
surfaceofthecatalyst,reaction(1);removalofspeciesAfromthesurfaceofthecatalystby
desorption, reaction (1); and the degradation of the absorbed species in a surface reaction,
reaction(2). Weassumethattheproductsofreaction(2)arethesameasforthehomogeneous
reaction and that they are not adsorbed by the catalyst. We use the subscripts c;d and r to
refertokineticpropertiesofthechemisorption,desorptionandsurfacereaction,respectively.
Notethatthereisaconservationrelationshipfortheconcentrationofactivesites
M .t/+.A−M/ .t/DM .t D0/DL (3)
s s s
whereListhedensityofactivecatalyticsites.
8 MINelsonetal
2.2.1. Adsorptionkinetics. Therateofnon-dissociativeunimolecularadsorption
A +M !.A−M/ (4)
g s s
isgivenbytheproductoftheimpingementrateatthesurface(theKnudsencollisionfactor),
thestickingprobability(s)andatermrepresentingtheprobabilitythatasiteisempty. Thislast
termismodelledusingLangmuir’ssiteexclusionprinciple. Therateofadsorption(mols−1)
is[3]
(cid:26)(cid:18) (cid:19) (cid:27)
RT 1=2 [M]
k DS g a s s: (5)
a c
2(cid:25)MA L
Thestickingcoefficient(s)iswrittenintheform
(cid:20) (cid:21)
−E
s Ds exp c : (6)
0
RT
s
HereE canbeinterpretedeitherasanactivationenergyforstickingorasanactivationenergy
c
forconvertingtrappedimpingedspeciesintotheirfinalstate. FrequentlyE isnegligible.
c
Adsorption is almost invariably an exothermic reaction. We denote by Q the
c
exothermicityofchemisorption(Q D−1H ).
c c
2.2.2. Desorptionkinetics. Thedesorptionreaction
010 .A−M/s !Ag+Ms (7)
2

y
r ismodelledusingtransitionstatetheory,inwhichitisassumedthatdesorptionproceedsvia
a
u
an anenergeticallyactivatedtransitionstatethatisintermediateinstructurebetweenthereactant
J
24 (.A−M/s)andproducts(Ag andMs). Itisassumedthattheactivatedcomplexexistsinlow

0 concentrationinequilibriumwiththereactants,therebyenablingtheapplicationofstatistical
5
:
05 theory. In addition we assume that the standard entropy of activation of the transition state

t: is zero, which implies that there is no change in molecular structure between the adsorbed
A
d species and the transition state, and that the desorption activation energy is the sum of the
e
ad bindingenergyandactivationenergyforadsorption(−1H +E ).
lo c c
wn Therateofdesorptionisthengivenby[21]
o
D (cid:20) (cid:21)
k T −.Q +E /
k DS b s exp c c [A−M] : (8)
d c s
h RT
s
Theheatofdesorptionis−Q .
c
2.2.3. Thecatalyticreaction. TheArrheniusratecoefficientforthecatalyticreaction
.A−M/ !n B +M (9)
s b g s
isgivenby
(cid:20) (cid:21)
−E
k DA exp r [A−M] : (10)
r r s
RT
s
Wewishtoinvestigatehowthedynamicsofourmodeldependsuponthekineticsofthis
reaction. However,itisnotsensibletocarryoutsuchaninvestigationusingthisformulation;
duetotheexponentialdependenceoftherateconstant,smallchangesintheactivationenergy
(E )havealargeeffectonthereactionrate.
r
Heterogeneouslycatalysedcombustion 9
Wethereforeuseanapproachwhichhasbeenuseful,bothexperimentallyandinmodelling,
inthepyrolysisofpolymersandusetheconceptofa‘characteristicreactiontemperature’and
writetherateconstantintheform
(cid:20) (cid:18) (cid:19)(cid:21)
HE E 1 1
k D r exp r − .A−M/ (11)
r RT2 R T T s
c c s
whereT isthecharacteristictemperatureofthereaction,HisanominalheatingrateandE
c r
remainstheactivationenergy. Thecharacteristictemperatureofapolymerismeasured,ata
givenheatingrate,byusingthemethodologyofthermogravimetricanalysis[22]. Thebasic
conceptofthermogravimetricanalysis,heatingasmallsampleatauniformheatingrate,isthe
sameasfortemperatureprogrammeddesorption,whichisastandardtechniqueinthefieldof
catalysisformeasuringthedesorptionkinetics. Wearenotsuggestingthatthereactionkinetics
forcatalyticreactionscanbeexperimentallydeterminedusingsuchatechnique. Wearewriting
therateconstantintheformofequation(11),ratherthanequation(10),fortheoreticalreasons.
Note that these formulations are equivalent with the pre-exponential factor in equation (10)
definedby
(cid:20) (cid:21)
HE E
A D r exp r : (12)
r RT2 RT
c c
Note that if the pre-exponential factor and activation energy for a catalytic reaction are
knownthenatheoreticalcharacteristictemperaturecanbecalculated,assumingaheatingrate.
10 TheoverallreactionenthalpyfortheconversionofonemoleofAtonb molesofBisthe
20 same,regardlessofwhetherthisoccursinthegasphaseorviatheheterogeneousroute. Hence

y
r wehavetherelationship
a
u
n
4 Ja Qg DQc+Qr: (13)
2

:50 InwhatfollowswefixQg andvaryQc,definingQr bytherelationshipinequation(13).
5
0

:
t
A 3. Modelequations
d
e
d
a
o
l 3.1. Dimensionalizedequations
n
w
o
D
Ourmodelis
(cid:20) (cid:21) (cid:20) (cid:21)
da −E k T −.Q +E /
V Dq.a −a/−V A exp g a+S b s exp c c [A−M]
g 0 g g c s
dt RT h RT
g s
(cid:26)(cid:18) (cid:19) (cid:27)
RT 1=2 [M]
−S s g a s (14)
c
2(cid:25)MA L
(cid:20) (cid:21) (cid:26)(cid:18) (cid:19) (cid:27)
d[M ] k T −.Q +E / RT 1=2 [M]
S s DS b s exp c c [A−M] −S s g a s
c c s c
dt h RTs 2(cid:25)MA L
(cid:20) (cid:18) (cid:19)(cid:21)
HE E 1 1
+S r exp r − [A−M] (15)
cRT2 R T T s
c c s
(cid:20) (cid:21) (cid:26)(cid:18) (cid:19) (cid:27)
d[A−M] k T −.Q +E / RT 1=2 [M]
S s D−S b s exp c c [A−M] +S s g a s
c c s c
dt h RTs 2(cid:25)MA L
(cid:20) (cid:18) (cid:19)(cid:21)
HE E 1 1
−S r exp r − [A−M] (16)
cRT2 R T T s
c c s