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351VOLUME V / INSTRUMENTS
6.3.1 General description and reactor types
Chemical reactions pertaining to the chemical,petrochemical and
oil industries are performed in specialapparatus called reactors.
There are distinct types of reactorsintended to face extremely
varied operating conditions, bothin terms of the nature of the
chemical species involved(reactants and products of the reaction)
and of the physicalconditions under which they operate.
In general, a chemical reactor needs to be able to carryout at
least three functions: provide the necessary residencetime for the
reactants to complete the chemical reaction;allow the heat exchange
necessary; place the phases intointimate contact to facilitate the
reaction.
Thus, reactor types range from large dimensioncontinuous
reactors, like those adopted for catalytic crackingreactions,
specifically for oil refineries, to devices of modestdimensions,
like discontinuous stirred reactors in which fineand secondary
chemistry reactions are performed. Moreover,there are reactors for
sophisticated microelectronicapplications and reactors of
microscopic dimensions(microreactors), designed for biomedical
installations or forin situ production of extremely toxic or
dangerouscompounds. Both converters and burners, catalytic
orotherwise, adopted for energy production can also be listedamong
the reactors.
To classify a reactor, the number of phases in the
reactoritself, whether or not there are agitation systems and
themode of operation (continuous reactor, semi-continuous
ordiscontinuous) need to be taken into consideration. It shouldalso
be noted that most chemical reactors are equipped withheat exchange
apparatus in the form of external jackets orinternal coils with a
fluid flowing through them to act as athermal vector to allow both
heat supply or removal.
Examples of different kinds of reactors are illustrated inFig.
1, while the type of reactions operating inside them aswell as the
most usually adopted operation modes for themost common reactor
types are reported in Table 1; theexamples given show that the
factors mainly influencing thechoice of reactor type are: the
number of phases involved,and thus whether or not it is necessary
to provide particularagitation systems; obtaining and maintaining
the optimal
temperature and pressure for the reaction; and the scale
ofproduction which often determines continuous ordiscontinuous
operation mode.
As far as phases are concerned, the most simple reactorsare
homogeneous reactors, where a single gaseous or liquidphase is
usually stirred to avoid the presence of stagnantzones. The
reaction can be operated in discontinuous mode,by loading the
reactants mixture into the reactor and waitinguntil the process is
completed, or in continuous mode, bymaking a stream containing the
reactants flow into thereactor and extracting another stream
containing the reactionproducts. Typical examples of homogeneous
reactors arethose for thermal cracking and for polymerization
insolution.
Heterogeneous reactors are more complex, in whichreactants,
products and a possible catalyst can be present indifferent phases.
One example is the fluid-solid reactor(gas-solid or liquid-solid),
where heterogeneous catalyticreactions are performed. Another
classic example is thetubular reactor which allows accurate
temperature controlbecause of its very extensive external surface
available forheat exchange; in fact, it is designed as a tube
bundleconfiguration very often, where a large number of reactorsare
connected in parallel, through each of which passes afraction of
the flow rate. Another example is the aerosolreactor, adopted by
the industry of new materials, wheresolid particles are synthesized
from reactants in gaseousphase. Gas-liquid or liquid-liquid
reactors are even morecomplex, in which the main reactants are
distributed betweenthe two phases, immiscible with each other, but
betweenwhich intimate contact has to be realized to make
thereaction progress more easily. Multiphase reactors, like
thegas-liquid-solid reactors, also exist; the trickle-bed
reactor,used to perform hydrogenation reactions or
catalyticoxidation reactions for liquid reactants, is a classic
example.
Reactions involving gaseous reactants are usuallyperformed in
tubular reactors, generally operating inturbulent regime; if a
solid catalyst is involved, it is usuallyarranged as a bed of
particles, generally in a spherical orcylindrical shape. If the
mechanical features of the catalystare appropriate and if efficient
heat exchange is necessary tocontrol the reaction temperature
accurately, mobile bedreactors can be also used; the most important
example of
6.3
Chemical reactors
-
these is the fluidized bed reactor, in which the velocity of
thegaseous phase in contact with the small dimension
catalyticparticles is able to keep these particles in motion,
thusrealizing a high degree of stirring and mixing. Other types
ofbed reactors, depending on the fluid dynamic features of the
bed, are: boiling fluidized beds (where the motion of the
twophases system resembles that of a boiling liquid);
draggedfluidized beds (where the gases drag all the particles in
thebed with their motion) and finally spouted fluidized beds(where
the particles in the bed are dragged at high speed to a
PROCESS ENGINEERING ASPECTS
352 ENCYCLOPAEDIA OF HYDROCARBONS
A
B
C
D
E
F
G I J K
L M N O
H
combustionproducts
g
g
g
g
ll
ll l l
g
g
l
g
H2
H
H2 Cl2
O2
O2NaClNaOH
g
l
g
g
combustionair
electrolyte
cathode anode c(d)d
about
in
crackingreactorregenerationsection
heater
precursor
multiple nozzleburner
flame
preform/target
hydrocarbons feed
products
Fig. 1. Examples of chemical reactors: A, stirred tank reactor;
B, homogeneous tubular reactor; C, heterogeneous tubular reactor;
D, multi-tubular heterogeneous reactor; E, fluidized bed gas-solid
reactor; F, fluidized bed reactors arrangement for the fluid
catalytic cracking process for hydrocarbons; G, bubble gas-liquid
reactor; H, air-lift gas-liquid reactor; I, stirred multi-staged
gas-liquid (or liquid-liquid) reactor; J, gas-liquid-solid
trickle-bed reactor; K, aerosol reactor; L, thin films deposition
reactor; M, fuel cell; N, electrochemical cell; O, membrane
reactor. Letters a and b are reactants; c and d, products; g,
gaseous phase; l, liquid phase.
-
first particular area in the reactor by the gases and allowed
tosettle in a nearby stagnant portion).
Liquid phase reactions are performed principally in tank-shaped
agitated reactors. Agitation is mechanically inducedby propellers
or flat bladed stirrers, depending on the type offluid circulation
to be carried out inside the reactor. Stirringis necessary to
obtain both effective contact between thefluid and the surfaces
installed for heat exchange (thisnecessity is also the case for
homogeneous systems) and,with a resulting close contact, to
guarantee good dispersionbetween the phases in heterogeneous
systems. If a solidcatalyst is present, it can be held in
particular convenientlyplaced baskets to allow contact with the
liquid phase; in
slurry reactors the catalyst can be a suspension within
theliquid phase itself. The agitated tank reactors can be
operatedin discontinuous mode (batch reactors) or continuously.
As has already been seen, reactions involving a gaseousphase and
a liquid one, or two liquid phases, must beperformed in reactors
able to guarantee efficient contactbetween the two phases by means
of appropriate stirrers.Sometimes static mixers, made up of
particular filling beads,can also be used, through which the two
phases to be placedinto contact flow countercurrently (because of
theirdifference in density) inside their channels. Reactors inwhich
static mixers are used are characterized by highvertical
development to enhance the stirring induced by the
CHEMICAL REACTORS
353VOLUME V / INSTRUMENTS
Table 1. Examples of the uses of different reactor types
according to the chemical reaction to be performed,with indications
of the management features more often adopted
Reactor type Features Management Examples of chemical
reaction
Stirred tank Flexibility in operation,good mixing of
reactants
D, S, C Organic reactions of pharmaceuticals or fine
organicchemistry, melamine production, organicnitro-compounds
production, benzene sulphonation,esterification reactions,
saponification reactions, etc.
Stirred multiphase Good mixing betweenphases, good
temperaturecontrol
C, S Emulsion or suspension polymerizations, chlorinationof
organic aromatic compounds, oxidation of organiccompounds (like
p-xylene to terephthalic acid,acetylene to acetaldehyde,
cyclohexane tocyclohexanone and adipic acid)
Multiphase bubble column Possibility of introducingseparation
stages,possibility of operatingboth in co-currentor in
counter-current mode
C, S Ethyl benzene, cumene and isobutane oxidation
tohydro-peroxides, propene oxychlorination tochloropropanol,
nitrous oxides or sulphuric anhydrideabsorption for nitric and
sulphuric acid production,phtalimmide production
Burners Short contact times,high temperatures
C H2S combustion to SO2, carbon black production,acetylene
production, high pressure gasificationfor syngas production
Homogeneous tubular Well-defined residencetime, good
temperaturecontrol
C Thermal cracking of hydrocarbons, visbreaking,delayed cocking
(endothermic reaction), chlorinationreactions of methane, propene
and butadiene, ethylenepolymerization to LLDPE, Linear Low
DensityPolyEthylene (exothermic reactions)
Heterogeneous tubular Well-defined residencetime, good
temperaturecontrol, high fluid-catalystinterfacial surface
C Heterogeneous catalytic reactions (synthesis of NH3,CH3OH,
styrene, etc.), reforming reactions ofhydrocarbons (platforming,
hydrocracking, etc.), ethylbenzene dehydrogenation to styrene
Tubular multiphase High interfacial area,well-defined
residencetime, reduced internalre-circulations
C Chlorination of organic compounds, oxidation oforganic
compounds, adiponitrile production fromadipic acid and ammonia,
nitroalinines production,ethylene oxidation to acetaldehyde
Three phases (trickle-bed) High interfacial area,possibility of
operatingboth in co-currentor in counter-current mode
C, S Liquid organic compounds catalytic oxidation
Fluidized bed reactor High reactants mixingand high
temperaturecontrol
C Roasting reactions of ores, chlorolysis reactions
ofchlorinated hydrocarbons, chlorination of methane,hydrocarbons
catalytic cracking, heavy oils coking,melamine production from
melted urea
C, continuous operation; D, discontinuous operation; S,
semi-continuous operation
-
density difference between the two phases: the lower
densityphase is fed from the bottom and collected at the top,
whilethe other follows the opposite pathway. In gas-liquid
reactorsthis configuration is called bubble column;
differentconfigurations exist, intended to increase stirring and
thuscontact between the phases, without resorting to
mechanicalstirring devices. The most important example is the
air-liftreactor, where the density difference between two
connectedreactor portions is exploited to start a vortical
naturalconvection motion.
In conclusion, it is also important to mention someatypical
reactors, adopted for special applications. First ofall, there is
the catalytic converter, a typical heterogeneouscatalytic reactor
in which the exhaust gases from vehicles areput into contact with a
catalyst whose active element is anoble metal, usually platinum,
supported on a ceramicmatrix, generally a honeycomb monolithic
structure. Insidethis reactor, carbon monoxide oxidizes to carbon
dioxide andnitrogen oxides reduce to elementary nitrogen. The
oxidantused in the first reaction is the oxygen still present in
theexhaust gases, while the fuels for the reduction reaction arethe
uncombusted hydrocarbons traces. The choice to adoptmonolithic
structures allows the reduction of pressure dropsand the
performance of efficient thermal exchanges. Otherimportant
heterogeneous reactors are biological reactors,where the enzymes
catalyzing the fermentation process aresupported on appropriate
solid matrices. In this field, themost innovative applications are
those made in bioreactorswhere selected cell colonies are allowed
to proliferate withina biocompatible polymeric matrix.
Electrochemical reactorsare characterized by having two electrodes,
each at adifferent potential, between which an electric
currenttransported by the ions contained in the electrolytic
solutionflows, where they are both immersed. In this way, it
ispossible to perform important industrial processes based onredox
reactions, like chloro-alkali processes, which useelectric rather
than thermal energy. Electrochemical reactorscan be likened to fuel
cells with polymeric membrane, wherea combustion reaction takes
place involving hydrogen (ormethanol) and oxygen by feeding the two
gases on the twoelectrodes, separated by a ion exchange membrane,
to makeelectric current generation possible. Both the two
reactorsabove can also be classified as membrane reactors, where
asemi-permeable membrane allows the separation of one ofthe
reaction products directly from the reaction environmentthus
contributing to an improvement in the selectivity of theprocesses
under consideration. Currently, the extensiveapplication of these
reactors is limited by the availability ofefficient membranes. Last
but not least, reactors used inmicroelectronics, where chemical
vapour depositionprocesses are performed, are of particular
importance; thesereactors, operating in discontinuous conditions
and at hightemperatures, allow the realization of extremely
controlledprocess conditions so as to obtain highly contained
tolerancelevels both of the thickness and of the
crystallinemorphology of the deposited semiconductor.
6.3.2 Reactor design and simulation
In a chemical plant, the potential and the yield with respectto
the reactants used depend on the size of the reactorsinvolved and
on the mode in which they are operated.
Moreover, in many cases, the reactor cost accounts for
asubstantial amount of the overall plant cost. Thus, correctreactor
design is a fundamental requirement to realizeeconomically
convenient plants.
Reactor design, once largely carried out with asemi-empirical
approach, is today realized by means ofappropriate mathematical
models, based on mass, energyand momentum conservation equations,
intended to simulatereactor behaviour. The analytical formulations
that make thedescription of the main reactor performances and
featurespossible must thus take into consideration not only
thekinetics features of the chemical reactions but also all
thefluid dynamics aspects which influence the transport
anddistribution of the reactants inside the reactor itself.
Reactorworking conditions, when chemical and transportphenomena are
both important, are called macrokinetics.
Such a description is generally very complex; thus, thereactors
are often analysed first in ideal conditions where thefluid dynamic
aspects are described in simple terms andwhere their behaviour is
only dependent on chemicalreactions; these are called microkinetic
conditions. From thispoint of view, the reactors can be represented
by two limitingschemes (ideal reactors): one of a perfectly mixed
systemand the other of a perfectly segregated system. In the
firstcase, composition gradients are absent inside the reactor
andthus the composition is uniform in the whole reactor volume;in
the second case, the concentration of the reactants evolvesin the
reactor volume and consequently the presence of theconcentration
gradient represents its characterizing feature.It is then evident
that the first limiting schematization issuitable for representing
the stirred reactors while the secondone is appropriate for the
tubular ones. Because real reactorbehaviour is at some point
between those of the two limitingschematizations described above,
it is clear why their studyis a good starting point before making
more complexdescriptions later.
Once the ideal reactors have been studied, it is possibleto move
forward to the examination of real ones by removingthe
simplifications introduced, as far as giving a panoramaof the
increasingly rigorous design methods, going throughan analysis of
the residence time distribution function of thefluid within the
reactor to the direct simulation of the flowfield through
Navier-Stokes equations.
Models for isothermal ideal reactors
As has already been mentioned, the study of reactors isusually
initially performed by means of two ideal models,where it is
assumed that the fluid dynamics conditions donot influence the
behaviour of the reacting system. Thatmeans considering a perfectly
mixed volume, for the stirredreactors, and a perfectly segregated
volume, for the tubularreactors. The continuous stirred reactors
(CSTR, ContinuousStirred Tank Reactor) or the discontinuous ones
(batchreactors) belong to the first class, while reactors with
piston-like flux (PFR, Plug Flow Reactor) belong to the second
one.
In the subsequent developments the following compactwriting for
the chemical reactions will be adopted:
[1]
where Ai and nij are the ith chemical species and its
stoichiometric coefficient in the jth reaction,
respectively.
ij ii
NC
A == 0
1
j NR1,,
PROCESS ENGINEERING ASPECTS
354 ENCYCLOPAEDIA OF HYDROCARBONS
-
Usually, the coefficients for the reaction products areassumed
positive and the reactants negative. Accordingly, ifrj indicates
the rate of the j
th reaction, the specific productionterm for the ith chemical
species is expressed as follows:
[2]
where Ri is specific production rate of the ith reacting
species.
Ideal discontinuous reactor (batch)This reactor, illustrated in
Fig. 2 A, is an evolution, on a
larger scale, of the laboratory flask; it is operated by
initiallyloading the reactants and by discharging the products at
theend of the reaction. The discontinuous reactor is thus
moreappropriate apparatus for low, usually very variableproductions
during a year. Being, in its ideal schematization,a perfectly mixed
system, the concentration of the reactantsis the same at each of
its points and thus concentrationgradients do not exist inside
it.
The mass balance for a discontinuous reactor assumesthe
following expression:
[3]
where VR and Ci are the volume occupied by the reactantsand the
concentration of the ith reacting species, expressedon a molar
basis, respectively. In an equation like theprevious one, for each
of the species present in the examinedsystem, ordinary differential
balance equations exist, whereeach one of them needs to be
associated with thecorresponding initial condition indicating the
concentrationvalue at time t0: CiCi
0.If the reaction progress does not significantly alter the
molar density of the reacting mixture, volume VR occupiedby the
reactants in the reactor can be considered constantand thus it can
be omitted from both sides of equation [3].This condition is
verified for elevated dilution systems andobviously for those where
the reaction does not alter thenumber of moles present in the
system. If those conditionsare not verified, it is necessary to
know the molar densitychange as a function of the system
concentration change
r~ f(C) and to write an additional equation whoseintegration
gives the variation in time of the reactionvolume:
[4]
Density changes are more pronounced for gas phasereactions than
those of liquid phase. For systems in gaseousphase, density is
usually calculated by equations of state liker~ p/ZRT, Z(T,p) being
the isothermal compressibilityfactor. The solution of the system of
equations [3-4] can beobtained analytically only if the considered
reactions kineticsare simple, otherwise it is necessary to perform
a numericalintegration.
Ideal continuous stirred tank reactorThis reactor (CSTR) is
suitable to represent a model of
continuous stirred reactors, where there is continuous feed
ofreactants and simultaneous extraction of the products.Usually,
stirred reactors are operated by keeping the reactionvolume
constant, conditions easily realized through a levelcontrol, thus
transferring any possible reaction mixturedensity change effects on
the volumetric flow rate leavingthe reactor. Such a reactor is
shown in Fig. 2 B together withits main instrumentation, including
the temperature,pressure, flow rate and level controls; in this
case, too,perfect stirring is assumed and consequently within
thereactor neither concentration nor temperature gradients
exist.This implies that the composition and the temperature of
thestreams leaving are equal to those existing inside the
reactor.Generally speaking, a continuous reactor can be assumed to
be perfectly mixed when the reactant mixing time is much lower than
the residence time within the reactor(tmixtpermanence).
The mass balance for a CSTR reactor in transientconditions can
be written as follows (where each equation,together with the
corresponding initial condition CiCi
0 fort0, have to be written for any of the species present in
thesystem under examination):
[5]ddt
V C Q C QC RVR iF
iF
i i R( ) = + i NC1,,
ddt
V V RR R ii
NC
( ) ==
1
ddt
V C RVR i i R( ) = i NC1,,
R ri ij jj
NR
==
1
CHEMICAL REACTORS
355VOLUME V / INSTRUMENTS
A B
C
timer
heating/cooling fluid
TC
heating/cooling fluid
heating/cooling fluid
control boundary surfacesectionWR
continuous feed and exit
continuous feed and exit
Fi(V) Fi(VdV)
V VdV
mol/t
TC
TC
in
in
out
out
LC
QRC
QRC
Fig. 2. Diagram of idealreactors together with theirmain
instrumentation: A, perfectly mixeddiscontinuous reactor;B,
perfectly mixedcontinuous reactor; C, plug flow tubular reactor.TC,
temperaturegauge-controller; QRC, flowrate recorder-controller; LC,
level controller; timer, for valvesopening/closure.
-
where Q, VR, Ci and Ri are the volumetric flow rate, thevolume
occupied by the reactants, the concentration and thespecific
production rate for the ith reacting species expressedon a molar
basis, respectively; the F superscript identifiesthe properties
inherent to the feed stream. Usually, the studyof these reactors is
performed in steady state conditions, andthus the balance equations
reduce to NC algebraic equations,generally not of linear type.
In all the continuous reactors, and thus also for theCSTR, the
residence time (or contact time) is defined as theratio between the
reaction volume and the feed volumetricflow rate:
[6]
tR represents the average residence time of the reactantswithin
the reactor.
It is important to note that the volumetric flow rate forthe
leaving stream can be different from that of the feed if, asa
result of the effect of the chemical reactions, a change inthe
number of moles, and consequently of the fluid molardensity,
occurs. If this last property can be consideredconstant (r~ Fr~ ),
the volumetric flow rate is also constant.Whenever it is not
verified, in analogy with what happens inthe discontinuous reactor,
it is necessary to have an equationexpressing the molar density
change as a function of thesystem composition (r~ f(C)) and to
write an additionaloverall mass balance equation expressing the
leaving flowrate change compared to the one entering the
reactor:
[7]
where with FQ/QF the ratio between the leaving and theentering
volumetric flow rates is indicated. It is important tonote that if
the mass balance equation were written on amass rather than a molar
basis, in steady state conditions,equation [7] would be useless
because the entrance massflow rate value is always equal to the one
leaving.
To complete the examination and to justify the CSTRanalysis only
in steady state conditions it is useful toexamine, at least once,
their transient behaviour.Accordingly, it is useful to take a CSTR
which initiallycontains the A reactant at concentration C0A into
examination,in which the irreversible reaction AB, which follows
firstorder kinetics (rkCA) occurs and where only the A reactantis
fed to the reactor. Under these conditions, theconservation of the
volumetric flow rates is respectedbecause no variation in the
number of moles takes place, andthus the mass balance equation for
the A species can bewritten as follows:
[8]
whose analytical integration gives
[9]
where CACFA /(1ktR) is the asymptotic concentration
value, which represents the solution under steady
stateconditions. It can be demonstrated that, under the
consideredhypotheses, the time necessary to reach the steady
statesolution is always the same, independent of the
initialconcentration value. Moreover, this time is equal to
about
three times the mean residence time of the reactants withinthe
reactor.
Ideal piston flow continuous reactorThe ideal continuous reactor
with piston-like flow (PFR)
is a model for the continuous tubular reactor; this reactor
isshown in Fig. 2 C together with its main instrumentation. It isa
perfectly segregated reactor model because compositiongradients
develop inside it, from its entrance to its exit. Thismeans that in
each reactor section a concentration change isobserved compared
both to the previous section and the onefollowing. Thus, in the
typical schematization of idealreactors, the fluid composition is
homogeneous in eachsection of the reactor, while it varies
continuously along itsmain developing axial coordinate.
To write balance equations for a PFR it is then necessaryto
examine an infinitesimal volume of the reactor equal tothe area W
of its section, multiplied by the infinitesimallength dz, which is
dVWdz. The mass balance can bewritten as follows:
[10]
Each NC equation is referred to one component presentin the
system under examination and it needs both aboundary condition,
expressing the inlet conditions (CiCi
F
for V0), and an initial condition, expressing thecomposition
profile along the whole reactor Ci(V)Ci
0(V)for t0. By introducing the ratio between the local flow
ratevalue and the feed one (FQ/QF), and the residence time(tV/QF),
the mass balance equation can be rewritten in thefollowing
form:
[11]
The study of PFRs is also usually performed in steadystate
conditions (Ci/t0); in these conditions the partialdifferential
equations system [11] reduces to the followingordinary differential
equations system:
[12]
As for the CSTR, if the reaction does not alter thenumber of
moles present in the fluid significantly and as aconsequence its
molar density can be assumed constant(r~ Fr~ (t)const), the
volumetric flow rate is also constantand thus F1. Whenever this
condition is not verified, it isnecessary to have an equation
expressing the molar densitychange as a function of the system
composition (r~ f(C ))and to write an additional equation giving
the leaving flowrate change compared to that of the inlet:
[13]
It is important to note that if the mass balance were writtenon
a mass rather than a molar basis, in steady stateconditions, this
equation would also be useless, because themass flow rate values
are always constant along the reactor.
It is important to note that in the case of constant
density,equation [12] has exactly the same form as equation
[3]derived for discontinuous reactors. This means that,
byconsidering the same reaction and boundary conditions(and/or
initial ones), the ideal PFR has an internal
r( )
== Rii
NC
1
d Cdt R
ii
( )= i NC1,,
( )
+ =
CR
Ct
ii
i i NC1,,
+ =
( )
QCV
RCt
ii
i i NC1,,
C C C C eA A A Ak tR R= ( ) +0 1( ) /
C C kC CtAF
A R A RA =
r rF R ii
NC
R + ==
1
0
RRF
VQ
=
PROCESS ENGINEERING ASPECTS
356 ENCYCLOPAEDIA OF HYDROCARBONS
-
concentration profile which has the same behaviour in timeas the
analogous ideal discontinuous reactor. Therefore, itcan be stated
that the composition profile in space(identified by its residence
time) for the PFR is identical tothe corresponding one in time for
the discontinuous reactor.However, it is important to stress that,
even if the residencetime has the dimensions of a time, it is no
other than aconvenient way to identify the reaction volume by using
aspace-time equivalence of kinematic type.
Comparison between ideal reactor performancesThe description of
the different reactor types shows that
stirring devices are usually necessary to obtain correctmixing
of the phases and thus the choice of reactor can beindependent of
factors inherent to volume utilizationefficacy. However, it is
interesting to make a comparisonbetween the two types of continuous
reactor (stirred orsegregated) in these terms; in order to obtain a
simplecomparison, it is possible to refer to a reacting
systemoperating in diluted conditions where two simple reactions
inseries occur, involving three different chemical species
likeA
r1B and Br2C.
If two simple irreversible first order kinetics are assumedfor
the two reactions (r1k1CA, r2k2CB), it is easilypossible to obtain
an analytical solution for the balanceequations and thus to obtain
the evolution of theconcentration leaving the reactor as a function
of theresidence time for both types of reactors. The
comparisonbetween the two types of behaviour is illustrated in Fig.
3,where the trends for the three merit parameters adopted forthe
analysis of reactor performance are reported as afunction of the
residence time, and therefore the mainreactant conversion
xA1CA/C
FA, the yield in the
intermediate product hBCB/CFA and the selectivity between
the two products CB/(CBCC); from an inspection of thefigure, it
is possible to observe that the tubular reactor ischaracterized by
better reactor volume utilization than thestirred reactor
producing, in the same residence time, a lowerconcentration value
leaving the reactor for the A reactant.The difference in behaviour
is negligible in lower residencetimes while it becomes evident in
higher ones; in fact, whilein a stirred reactor all the reactants
are characterized by thesame residence time, in a tubular reactor
this is true only forthe reactants in the feed, while the other
species have lowerresidence times.
Configurations of ideal reactors
One of the advantages of outlining ideal reactors inschemes is
that a combination of them is able to describe anyreal reactor
providing that such an arrangement reproducesthe residence time
distribution of the real reactor (see below).
Reactor arrangements like a CSTR followed by a PFR ora series of
CSTRs are of interest; from the cited examples itis clearly evident
that the reactor combination is alwaysinherent to continuous ones.
Additionally, still in the idealreactor framework, slightly
modified configurationscompared to the basic ones analysed above
(batch, CSTRand PFR) are possible; these are
semi-continuousconfigurations, like the semibatch reactor, or
configurationswith recycling like the PFR with external
recirculation.These arrangements allow the coverage of
intermediatesituations which often occur both in the laboratory and
inindustry.
Ideal semi-continuous reactor (semibatch)The semi-continuous
reactor, illustrated in Fig. 4 A, is a
reactor whose features are somewhere between those of
thediscontinuous and the continuous stirred reactor; in fact, it
isa discontinuous reactor, because it is not provided with
acontinuous leaving stream but it is, however, equipped with
acontinuous reactants feed stream. In the chemical industry,this
configuration is much more widespread than that of thepure
discontinuous reactor, because it is often necessary tofeed in one
or more reagents, even during the progress of thereaction; in the
ideal reactor framework, it is still a perfectlystirred
discontinuous reactor. In principle, the use of thesemi-continuous
reactor is therefore much more similar tothat of the discontinuous
one than to that of the CSTR.
It is easy to derive the mass balance equations for thatsystem
by following what has already been derived fordiscontinuous
reactors, taking into account that in this casean entering fluid
stream is present:
[14]
where VR, QF, Ci
F are the volume occupied by the reactants,the volumetric feed
flow rate, and the concentration of the ith
species in the feed stream; Ci and Ri are, respectively,
theconcentration and the specific production rate of the ith
species expressed on a molar basis. Accordingly, the writing
ddt
V C Q C RVR iF
iF
i R( ) = + i NC1,,
CHEMICAL REACTORS
357VOLUME V / INSTRUMENTS
conv
ersi
on-y
ield
-sel
ectiv
ity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
residence time (h)
CSTR reactorCSTR reactor CSTR reactor PFR reactor PFR reactor
PFR reactor
x
h
S
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Fig. 3. Comparison between CSTR and PFR performancesinherent to
two reactions in series, with irreversible first order kinetics
(k10.5 h
1, k20.7 h1).
-
of the balance equations for these reactors leads to
thestatement of ordinary differential equations, which have tobe
completed with the corresponding initial conditions(CiCi
0 for t0).In principle, it is improbable that the reactor
volume
can be considered constant, unless reactors operating ingas
phase are kept at a constant volume. Thus, theequation expressing
the overall mass balance expressingthe system volume evolution in
time have to be added tothe previous balance equations for the
individual species(together with the equation of state r~ f(C)
giving themolar density variation as a function of the
chemicalcomposition):
[15]
An interesting simplified case regards systemsoperating in
diluted conditions; in these systems thefluid molar density can be
considered constant andequal to that of the solvent; moreover, the
overallnumber of moles in the system is more or less equal tothat
of moles in the solvent because of its great excess.Under these
conditions, the change in the number ofmoles present in the system
as a result of the chemicalreaction effect can be neglected without
committingsignificant errors; thus, the mass balance
equationbecomes:
[16]
Equation [16] can be solved analytically and gives theevolution
of the system volume through the simple relationVRV
0RQ
Ft. It can be inserted into the mass balanceequation for the
individual species, thus obtaining anexpression for the mass
balance of this species which, in anself-compatible way, gives its
time evolution:
[17]
The analytical integration of NC differential equations [17]is
often impossible; in that case numerical integration
isperformed.
Series of CSTRVery often, it is necessary to operate with well
stirred
continuous reactors. This is so, for example, whenever itis
necessary to mix reactants present in different phases.A plant
configuration frequently adopted is a certainnumber of reactors of
this kind connected in series (a battery) as illustrated in Fig. 4
B. Because of thenecessity to minimize the spare parts inventory
stock, thereactors in the battery are usually identical. Only
inparticular cases, when it is necessary to carry out
specificprocess performances, is the battery realized withreactors
of different volumes.
To study the performances of this plantconfiguration, for the
sake of simplicity and withoutover-generalizing, it is possible to
analyse a case whichis carried out under diluted conditions, which
allows theconsideration of the leaving flow rate being equal to
thatentering the reactor. An analysis of this system is easybecause
it is sufficient to write the mass balanceequations for any species
and for all the reactors,balances which can be generalized as
follows:
[18]
where the k index is the reactor being referred to. For
simplekinetics, the algebraic equations systems [18] can be
solvedanalytically and give the evolution of the composition
byproceeding from one reactor to another. To check thepeculiar
overall behaviour of a series of N identical reactors,for example,
simple irreversible first order kinetics (rkCA)can be considered
inherent to the AB reaction performedin diluted conditions. In this
simple case, the concentrationleaving the last reactor in the
series is given by:
[19]
where t indicates the overall residence time in the
batterysystem, calculated with reference to the overall
reactorvolume (tVR,tot/Q
F). If the partition were extended to itsextreme, that is to
consider a series of an infinite number ofCSTRs, whose volume were
of an infinitesimal value,applying equation [19] following result
is obtained:
C Ck NA
N AN
( )( )
/=
+( )0
1
i NC1, , 1, ,k N Q C Q C V Rk i
k kik
k k i( ) ( ) ( ) ( )
,
+ =1 1 0
V Q tdCdt
Q C C RVRF i F
iF
i i R0 +( ) = ( ) + i NC1,,
dVdt Q
R F=
ddt
V Q V RRF F
R ii
NC
r r( ) = +=
1
PROCESS ENGINEERING ASPECTS
358 ENCYCLOPAEDIA OF HYDROCARBONS
A
B
C
timer
heating/coolingfluid
in
(0)(1) (2)
(3)
out
in outCi
(1)
CiFCi
(0) Ci(out)Ci
(2)R
Q Q
Ci(2)
Ci(2)
LCTC
QRC
Fig. 4. Diagram of ideal reactor arrangements together with
their main instrumentation: A, perfectly mixed semi-continuous
reactor; B, series (battery) of perfectly mixed continuous
reactors; C, plug flow tubular reactor with external recycling.
-
[20]
that is, a series of infinite CSTRs is equivalent to a PFR.In
actual fact, in industrial practice, batteries with more
than 4 or 5 reactors are rarely used: indeed, even at
thispartitioning level performances comparable with those oftubular
reactors can be reached.
PFR with external recyclingTo realize a perfectly stirred
reactor (a reactor which
at any point has the same chemical composition) is notan easy
task. One way to realize a perfect CSTR, at leaston a laboratory
scale, is to build a tubular reactor (PFR)with external recycling
as shown in Fig. 4 C. Note thepresence of a mixing node
(feedrecycle) and apartitioning node (where the flow rate leaving
thereactor is split between the real exit stream and therecycling
stream back to the feed). It is important tounderline that the
partitioning node does not alter thecomposition of the fluid
streams involved because it onlysplits the main stream into two
secondary streams; themixing node instead, also alters the
composition besidesthe flow rates. In fact, the stream leaving the
node has aflow rate that is the sum of those of the entrance and
acomposition that is the weighted compositions averageof the two
feed streams whose weights are determined bythe respective flow
rates.
To study the performances of these configurations, forsake of
simplicity and without overgeneralizing, it ispossible to analyse
the case where diluted conditions areexamined. The analysis of this
system is simple because it issufficient to write the mass balance
equations for all thespecies inherent to the tubular reactor and
relative to the feednode:
[21]
[22]
For simple kinetics, like that of the irreversible first
order(rkCA), inherent to the A
B reaction, the system ofdifferential equations [21] can be
solved analytically toprovide the composition evolution along the
reactor. Byanalytically integrating the mass balance equation for
the Areactant and combining it with the node balance [22],
thecomposition value leaving the reactor can be directlyobtained
from that of the entrance as a function of therecycling ration fR/Q
and of the residence time,calculated with respect to the inlet flow
rate to the systemtRVR/Q:
[23]
It is easy to demonstrate that by increasing the recyclingratio
f the reactor performances decrease (that is, theleaving
concentration CA
(2) increases). For values approachinginfinite (f), equation
[23] degenerates in the typicalCSTR expression: CA
(2)CA(0)/(1ktR). In practice, an infinite
recycling ratio is not necessary: in fact, values in between
50and 100 are sufficient to obtain perfectly stirred
reactorperformances. Whenever more complex kinetics areexamined,
the indicated values can submit to significant
changes and thus it is important to verify the soundness ofthe
approximation directly from the case under examination.
Ideal non-isothermal reactor models
Rarely are chemical reactions free from thermal effects.Thus, in
studies of chemical reactors it is also necessary tointroduce, in
addition to the mass balance equations, that ofthe energy balance.
This equation gives the systemtemperature evolution as a function
of the reaction progressand of the amount of heat exchanged with
the surroundingenvironment by means of fluids acting as thermal
vectors;writing the energy balance will be developed here
withreference to the ideal reactors previously examined;
extendingthat equation to the more complex reactors is simple.
The energy balance for a chemical reactor contains
thecontributions due to the energy content of the involvedchemical
species (by taking into account both the breakingand the formation
of chemical bonds during the reaction,both the feeding and the
extraction of the reactants by meansof the entering and leaving
streams), to the heat exchangedwith the surroundings and to viscous
dissipation. Generally,the latter contribution is not such an
important percentageexcept for cases where high viscosity systems
are examined,typical, for example, of some polymerization
reactors.
Discontinuous reactorThe internal energy U of the fluid
contained inside the
reactor can vary over time uniquely due to the energyexchanged
in the unit of time with the external surroundings,under the form
of both heat Q
.and mechanical work W
.(the
latter component is usually negligible):
[24]
The heat power Q.
exchanged with the external surroundings,which is nil only in
the case of adiabatic reactor operation,depends on the difference
between the internal temperatureof the reactor T and that of the
thermal vector fluid Te, on thereactor surface available for heat
exchange SR and on theoverall energy transfer coefficient Ug, which
accounts for allheat transfer resistance:
[25]
In practice, it is useful to directly identify the
systemtemperature and the contributions due to the
chemicalreactions involved; to obtain this result it is sufficient
toexpress the internal energy as a function of the
systemcomposition, UVRiCi(DUi
0TT0cV,idT ), and to substitutemass balance equation [3] in
equation [24]:
[26]
where DUR, j indicates the molar internal energy change
attemperature T associated with the jth chemical reaction.Moreover,
to be compact, the sum of the products betweenthe concentration and
the heat capacity for the ith chemicalspecies at a constant is
usually indicated as rcV, mix.
Perfectly stirred continuous reactorThe approach shown for the
discontinuous reactor can be
immediately extended to the perfectly stirred continuousreactor,
being sure to introduce enthalpic contributions H
F
V C c dTdt
V r U S UR i V ii
NC
R j R jj
NR
R , ,= = = ( )+
1 1
gg eT T W( )+
Q S U T TR g e= ( )
dUdt Q W= +
C C eeA
Ak
k
R
R
( )( ) /( )
/( )2
0 1
11=
+
+
+
QC RC Q R CiF
i i+ = +( ) ( )( )2 1
dCd
R C CiR
i i R i= = = ( ) ( )0 1 i NC1,,
lim lim/
exp(( )( )
( )
N AN
N
AN AC
Ck N
C
=+( )
= 0
0
1 kk )
CHEMICAL REACTORS
359VOLUME V / INSTRUMENTS
-
and H into the balance terms for the entering and the
leavingreactor streams, respectively:
[27]
In practice, in this case too, it is useful to directly
identifythe system temperature and the contributions due to
thechemical reactions involved. To obtain this result, it
issufficient to express the molar enthalpy of the stream as
afunction of the system composition,H iCi(DU i
0T
T0cp,idT ), and to substitute the mass balance
equation [5] in equation [27]; in steady state conditions
thisobtains:
[28]
where DH R, j0 indicates the standard molar energy change
associated with the jth chemical reaction. Moreover, to
becompact, usually rcV, mix(TT
F ) indicates the sum of theproducts of the concentrations of
the ith chemical specieswith the integral of its heat capacity at
constant pressurebetween the temperatures of the entering and
leavingstreams.
Continuous tubular reactorTo write the energy balance equation
for a tubular
reactor it is necessary to refer to an infinitesimal
volumeinstead of the entire reactor volume:
[29]
where aR indicates the heat exchange surface per unit
reactorvolume (which, for a tubular reactor coincides with the
ratiobetween the perimeter and the section of the tube). It is to
benoted that there are no power dissipation terms due tomechanical
devices because there are none in tubularreactors. To obtain the
expression for the temperatureevolution, the previously described
procedure can befollowed, which in steady state conditions leads
to:
[30]
being rcp, mixiCicp,i
Thermal effects: multiplicity of steady statesand runaway
In a chemical reactor, the absorbed, or generated, heatfrom
chemical reactions is given or subtracted through a heatexchanger.
In steady state conditions, the two heat fluxeshave to be
equivalent to keep the reacting systemtemperature constant. This
observation is at the basis of animportant phenomenon typical of
continuous stirredreactors, which can be easily demonstrated by
equations [28]and [5], where, for sake of simplicity, a unique
reaction withirreversible first order kinetics is performed in
dilutedconditions. By taking into account that the reaction
kineticconstant generally assumes the Arrhenius form kk0eE/RT,where
k0 and are the frequency factor and E the activationenergy of the
reaction, respectively; the two contributions forthe heat power
generated and dissipated are the following:
[31]
[32]
As shown in Fig. 5, the curve of the generated thermal poweras a
function of temperature shows a typical S-shaped curve,while the
curve of the dissipated one has almost linearbehaviour. It follows
that the intersections between equations[31] and [32] represent the
possible working conditions forthe reactor. Moreover, it can be
observed that the number ofintersections varies as a function of
the inclination of thestraight line representing the dissipated
heat: either threeintersections or only one intersection is thus
possible.Therefore, in the first case, the continuous stirred
reactorpresents a multiplicity of steady states. The stability
analysisof the working conditions creates an eigenvalues
problem,whose analysis, however, goes beyond the aims of
thisarticle. To sum up, such an analysis definitely states thatonly
the cold point (a in Fig. 5) and the hot point (c in Fig. 5)are
stable working points for the reactor, while theintermediate one
(point b in Fig. 5) is unstable and thus anysmall disturbance
induced in the reactor produces amigration towards one of the other
two working conditions.
Another interesting effect, with dangerousconsequences, is
fugitive (runaway) reactor behaviour. Thisis typical of exothermic
reactions and is independent of thereactor type under
consideration: if a disturbance in itsbehaviour prevents thermal
control, then it causes either atemperature increase or a decrease
in the heat dissipationcapacity (exponential behaviour of the
generated heatcurve); the system is not able to react to the
disturbance andthere is a loss of control in the working
conditions, whichprovokes sharp temperature increases with the
consequentformation of unwanted by-products or dangerous
sharppressure increases. A simple criterion for this analysis
was
Q Q c T T S U T TexF
p mixF
R g e= + r , ( ) ( )
Q r H Vk e C H
k egen R RE RT
iF
RE= =
+
0
0 0
01
/ ( )
//RT RV
Q c dTdV
r H a U T Tp mix j R jj
NR
R g e r
, ,= ( ) + (
= 0
1
))
=
+ ( ) ( ) ( ) r rUt
Q HV
a U T TR g e
V r HR j R jj
NR
,=
= ( )01
+ ( )+S U T T WR g e
Q C c dTF iF
p iT
T
i
NC
F
,
=
=1
dUdt
Q H Q H S U T T WF F F R g e= + + r r ( )
PROCESS ENGINEERING ASPECTS
360 ENCYCLOPAEDIA OF HYDROCARBONS
Q
(c)
(a)
(b)
(d)
Qex
T
.
Qgen.
.
Fig. 5. Behaviour of generated and dissipated heat powers for a
CSTR. Steady states multiplicity analysis: a, cold working point;
b, unstable working point; c, hot working point; d, single
intersection.
-
proposed by Nikolaj Nikolaevic Semnov, by means of thedefinition
of a dimensionless number y, given by the ratiobetween the
generated heat in the reaction volume at thefluid temperature and
the heat dissipation velocity by pureNewtonian cooling, that is by
convective heat transfertowards a constant temperature wall:
[33]
If the Semnov number is lower than the critical valueycre
1, the reactor behaves in a stable manner; otherwise,runaway
conditions take place. The advantage inherent to theSemnov number
is its simplicity. However, today moresophisticated criteria are
available for risk estimation.
Real (non-ideal) reactor models
In practice, it is difficult to obtain perfect stirring
orsegregation conditions and thus the real behaviour of
chemicalreactors is substantially between the two limiting cases.
Themixing effects in reactors are often divided between
thecontributions of two distinct mechanisms: micromixing,
whichdetermines the degree of contact between species at
molecularlevel due to the local turbulent fluctuations of fluid
velocity,and macromixing, which instead is the result of the
differentpaths and by-pass fluxes or of the presence of stagnant
zoneswithin the reactor. The two contributions are independent
ofeach other and therefore a macromixing state does notcorrespond
to an equivalent micromixing state, even thoughthere is an
influence common to both.
An effective way to study such a situation is based on
theresidence time distribution function F(t), which accounts
forflux non uniformity and whose product with dt, F(t)dt,
expresses the fraction of fluid whose residence time isbetween t
and tdt; the F(t) function must be normalizedto obtain:
[34]
As a consequence, the mean residence time of the reactantswithin
the reactor is given by:
[35]
Once the residence time distribution function is known,real
reactor behaviour can be simulated by taking intoconsideration an
ensemble of reactors, each one linked to aspecific residence time,
to finally reproduce the F(t)function. In practice, the F(t) can be
obtained by injecting atracer into the reactor under examination,
whoseconcentration within the reactor can be determined at
anymoment. For a perfectly stirred continuous reactor, after astep
disturbance of the entrance conditions, the residencetime
distribution function has an asymptotic exponentialtrend,
F(t)CSTR
step 1et/tR, while a plug flow reactor has astep trend centred
on the average value of the residence time;the behaviour of a real
reactor is therefore between theextreme limits of two ideal
reactors. A schematicrepresentation of the possible macro- and
micromixingconfigurations as a function of the residence
timedistribution function is reported in Table 2, where
theperfectly segregated and the perfectly stirred situations
areindicated by 0 and , respectively.
The residence time distribution function influences
theconversion of the reactants obtained in the reactor. Theaverage
value for conversion x can be calculated asfollows:
[36]
In simple first order kinetics, by substituting the
expressionfor the conversion and the residence time
distributionfunction, integral [36] gives the solutions found for
the idealreactors exactly. This is indeed a peculiarity of first
orderkinetics in which the conversion depends only on thereaction
time and not on the degree of mixing. For reactionorders different
from unity, different results are obtained andthus the performances
of the real reactor differ from those ofthe ideal one. Some
examples are illustrated in Fig. 6 where,due to a significant
reaction order and degree of mixing, theaverage values of the
conversion at reactor exit as a functionof the mean residence time
are illustrated, placed in a
= =
( ) ( ) ( )( )
0 0
dF dFd
d
R F d=
( )0
F d( ) 0
1
=
=
( )( ) ( )/H k e C
S V U T
E RTiF n
R R gF
F0
CHEMICAL REACTORS
361VOLUME V / INSTRUMENTS
1
1 0.5ideally mixed
reactionordermixing level
2 0.5segregated
3 1ideally mixed or segregated
4 2segregated
5 2ideally mixed
2
x
34
5
tRk(CAF)n1
Fig. 6. Behaviour of the mean conversion value at reactor
exitdepending on dimensionlessresidence time for fiveconfigurations
characterizedby different combinations of reaction orders and
mixinglevels.
Table 2. Schematic representation of limitingmacro- and
micromixing cases relating to residence
time distribution function within the reactor
Mixing degree
macro micro
Shape of the residencetimes distribution
function
0 0 step
0 exponential
exponential
0 step
-
dimensionless form with respect to the pseudo-first orderrate
constant. It can be demonstrated that to know the realperformances
of these systems it is necessary to determinetheir fluid dynamics
behaviour correctly by taking intoaccount both the fluid
macro-circulation and the microscopicmixing phenomena due to system
turbulence.
As will be examined in detail below, today this problemis faced
by means of rigorous reactor simulations based oncomputational
fluid dynamics. However, there are twoapproaches that facilitate a
description of the behaviour of areal reactor, which are more
simple than that mentionedabove. The first originates from the
observation that a seriesof CSTR is able to adjust its performance
to between that ofa single CSTR (perfectly stirred reactor) and
that of a PFR(perfectly segregated reactor); as a consequence, it
ispossible to model real reactor behaviour by considering itsvolume
partitioned into an appropriate number of idealCSTRs, whose number
is intended to reproduce theresidence time distribution function.
The second approach,instead, derives real reactor behaviour from
that of the PFR,by introducing dispersion effects perturbing
perfectsegregation, due to the local diffusion of the reactants;
suchcases are identified as axial dispersion models, whosebalance
equations assume the following form:
[37]
where z, u and DL,i indicate the axial reactor coordinate,
theaverage fluid velocity within the channel (usually called
thesuperficial velocity) and its axial dispersion
coefficient,respectively. The latter, which has the dimensions of
adiffusion coefficient, accounts for the dispersion of
theconcentration front merely due to diffusive
aspects(concentration gradient) and to deviations of the
velocityprofile from the idealized one, constant in the whole
section.
The introduction of dispersion conditions introduces
amodification in the boundary conditions of mass balanceequation
[37] compared to those of equation [10]. In fact, theconditions
inherent to the inlet and the exit sections of thereactor assume
the following form:
[38a]
[38b]
While the latter simply indicates a fully developedconcentration
profile at reactor exit, the first shows theexistence of
discontinuity in concentration value of thereactants in
correspondence with the inlet section, dueprecisely to the
dispersion effects within the reactor. Theseboundary conditions are
usually indicated as Dankwertsconditions, even though they were
formulated for the firsttime by Irving Langmuir.
It is important to point out that both the approachesillustrated
above (a series of CSTRs and an axial dispersionreactor) are able
to reproduce the behaviour of any genericreactor. In fact, it is
possible to shift from one type ofdescription to another by
obtaining, for example, the numberof reactors in a series which
reproduces the value of the axialdispersion coefficient and vice
versa, through appropriatecorrelations. The choice between one or
the otherschematization is usually based on its
computationalcomplexity (in favour of the battery) or on affinity
of themodel to reality (a dispersion model apparently
betterapproximates the behaviour of a tubular reactor than a
stirredreactor). To reproduce the real behaviour of a reactor
morecomplex combinations of reactors are often also examined,like
those illustrated in Fig. 7, for example.
Heterogeneous reactor models
Only single phase reactors have been examined up to thispoint.
As has already been seen, in a real situation, it is veryfrequent
for reactants to be distributed among more than onephase.
Heterogeneous catalytic, gas-liquid and liquid-liquidreactors are
classic examples of this as well as three phasegas-liquid-solid
reactors.
In all the heterogeneous reactors the dispersion of onephase is
carried out inside the other. For the correct reactordefinition it
is therefore necessary to establish which phase iscontinuous and
which is the dispersed phase. Inheterogeneous catalytic reactors,
the catalyst particlesrepresent the dispersed phase while the fluid
phase identifiesthe continuous one. In gas-liquid reactors two
limitingconfigurations are possible: in the first, the most
frequent, the
0=
D CzL ii
L,
u C C D Czi i
FL i
i( ) , = 0
=
+
+Ct
u Cz
D Cz
Ri i L i i i,2
2
PROCESS ENGINEERING ASPECTS
362 ENCYCLOPAEDIA OF HYDROCARBONS
A
D E
B C
Fig. 7. Diagram of the idealreactor arrangements mostfrequently
adopted toreproduce real reactorbehaviour:A, PFRdead volume; B,
CSTRdead volume;C, CSTRby-pass; D, CSTRdead volumeby-pass; E,
PFRCSTR.
-
liquid phase is the continuous one while the gas bubbles arethe
dispersed phase (bubble reactors, whether or not equippedwith
external stirring); in the second case, less frequent, theliquid
phase is dispersed in the gaseous phase (spray reactors,where
liquid drops are sprayed within the gaseous phase).
In all the heterogeneous reactors the heat and masstransfer
between phases plays an important role in definingtheir
performances. In the following, the differenthomogeneous reactor
models will be modified to representthe heterogeneous system.
Heterogeneous catalytic reactorsIn heterogeneous catalytic
reactors the reactions occur
within the catalytic particle; in this framework, it istherefore
more convenient to refer the reaction rate to theunit of catalyst
mass instead of to the unit of volume.Moreover, the catalytic
particle, because of internalresistance to the transport of mass,
usually presents aconcentration profile like that illustrated in
Fig. 8 A whichthus causes a different reaction rate inside the
particle withrespect to its surface. All the above is expressed
aseffectiveness factor h, whose use allows the omission of
theparticle model during the solution of the balance equationsfor
the reactor. Accordingly, before examining how theequations
previously developed for the homogeneousreactors modify, it is
useful to study the behaviour of asingle catalytic particle to
define a functional relationshipexpressing the effectiveness factor
for the particle as afunction of a specific chemical reaction.
As far as catalyst effectiveness is concerned, it isimportant to
remember that the rate of a catalytic reaction isinfluenced,
besides the chemical nature of the reaction, alsoby the physical
phenomena of mass and heat transport. Asillustrated in Fig. 8 B,
the fluid phase components, beforereacting, need to disperse from
the heart of the mixture bulkto the particle surface and then
inside it through its porenetwork; the reverse path has to be
followed by the reactionproducts. Because of the intraparticle
diffusive processes andthe consequent consumption of the reactants
due to thechemical reaction, a concentration profile develops
insidethe particle whose average value can differ greatly from
thatexisting in correspondence with the particle surface. Thus,the
particle effectiveness concept has the aim of reducing itsoverall
behaviour to a unique functional relationship throughthe combined
effect of both the chemical effects occurringinside it and
intraparticle mass transport phenomena.Consequently, catalyst
effectiveness h indicates the ratiobetween the observed reaction
rate and that of the hypothesisof what would occur if the
intraparticle diffusive phenomenawere absent and thus all the
reaction took place at thesurface concentration throughout the
whole particle. It can
be demonstrated that effectiveness depends on adimensionless
parameter, called Thieles module:
[39]
where Vp, Sp, Dieff and k are the volume and the external
surface of the catalytic particle, the effective
diffusioncoefficient of the reactants within the particle and
thepseudo-first order kinetic constant (krrs/Ci
s), respectively.As a result of the findings above, in
heterogeneous
CSTR and PFR, the production rate for the ith species dueto the
effect of the reactions is given by the followingexpression, where
the reaction rate is calculated at aconcentration corresponding to
that of the catalyticparticle external surface and all the
resistance tointraparticle diffusive transport is computed
througheffectiveness factor h:
[40]
To convert such a contribution to that specific to the unit
ofreactor volume it is necessary to multiply it by the
catalystpoured density rs(1e), where rs and e are the density ofthe
catalytic particle and the void fraction of the
bed,respectively:
[41]
In steady state conditions, the mass balance for a CSTR usedto
perform catalytic reactions, where the catalyst isuniformly
dispersed in its volume, is directly obtainable bysubstituting
equation [41] in the equation [5]:
[42]
For the tubular reactor, it is also sufficient to
substituteequation [41] into equation [10] in steady state
conditions toobtain:
[43]
Equation [43] can be generalized analogously to [37] toinclude
axial dispersion contributions; in this case,appropriate
correlations to estimate the axial dispersioncoefficient valid for
packed bed rather than for empty tubeshave to be used.
Gas-liquid reactorIn the gas-liquid reactor simulation, besides
the kinetics
of the chemical reaction, it is important to correctly
describethe amount of matter exchanged through the interface;
forsuch a description it is necessary to know two factors: themass
flux between the phases and the interface extension.
d QCdV
Ri s i s( ) ( ),= r 1
Q C QC R VF iF
i s i s R + =, ( )r 1 0
R Ri s s i= ( ) ,r 1
R rs i ik s k kk
NR
, ,==
1
=VS
kD
p
p ieff
CHEMICAL REACTORS
363VOLUME V / INSTRUMENTS
A B
r
1
2
3
4
56
7Csi
Ci
1- reactants external diffusion2- reactants intraporous
diffusion3- reactants adsorption4- surface reaction5- products
desorption6- products intraporous diffusion7- products external
diffusion
Fig. 8. A, typical behaviour of the concentration Ci of the
i-threactant within a catalytic particle,supposed spherical (r is
the radialcoordinate and Ci
S the concentration on the external surface); B, diagram of
chemical and physical phenomenaoccurring inside a catalytic
particle(steps 1, 2, 6 and 7 of a physical natureand steps 3, 4 and
5 of a chemicalnature).
-
Usually, only some of the reactants and reaction products
aredistributed between the two phases, while in the largemajority
of processes the chemical reaction occurs in theliquid phase. If
the reaction is particularly fast, it is localizedwithin the
interfacial film, but it generally happens in theabsorption
processes (for example, in CO2 absorption inalkaline solutions)
instead of in processes properly devotedto perform a chemical
reaction, like oxidation,hydrogenation or halogenation.
Generally, the flux expression is obtained by applyingthe
two-film theory, which assumes the resistance to masstransfer is
localized both in liquid and gaseous films.Moreover, it is assumed
that the concentrations incorrespondence with the interface are in
thermodynamicequilibrium. Accordingly, under these assumptions, the
massflux expression between the phases takes the following
form:
[44]
where kie and Ki are the overall mass transfer coefficient
and
the equilibrium partition constant, respectively. Theequilibrium
partition constant assumes the form KiHigi/pfor the supercritical
species and Kip
0i gi/p for the subcritical
ones, Hi, gi and p0i being Henrys Law constant, the activity
coefficient and the vapour pressure for the examined
species,respectively. The overall mass transfer coefficient,
inagreement with the two-film theory, assumes the
followingexpression:
[45]
where kG,i and kL,i are indeed the mass transfer
coefficientsinherent to the two phases in contact and E is the
enhancingfactor which accounts for the effect of fast reactions on
themass transfer; this coefficient can be estimated
throughcorrelations based on Hattas module MH
1
Di1
kr1
C1
LB /kL,iwhere Di is the liquid phase diffusion coefficient for
themigrating species, kr is the rate constant for the
reactionbetween the gaseous species and the B reactant in the
liquidphase, and finally kL,i is the mass transfer coefficient in
theliquid phase. In the limiting case in which the reactant
inliquid phase is present in large excess compared to the
onetransferred from the gaseous phase, the reaction can
beconsidered to be one of a pseudo-first order andconsequently the
enhancing factor can be estimated asEMH/tanhMH. To estimate the
mass transfer coefficients,as well as the interfacial area per unit
reactor volume, manycorrelations are available; for those
expressions Chapter 4.2can be consulted.
In the perfectly stirred gas-liquid reactor, if it is possibleto
assume perfect mixing for the two phases (gaseous andliquid) in
contact, the reference scheme for the mass balanceequations is that
reported below; such a formulation matchesboth semi-continuous
configurations (like gas phase fed anddischarged continuously and
discontinuous liquid phase) andcontinuous feed configurations for
both phases; moreover,usually the mass inventory in gas compared to
that of theliquid phase is assumed negligible because of the
largedifference in the density values for the two phases (a ratio
ofabout 1/1,000); accordingly, the overall mass balanceequation for
the liquid phase assumes the following form:
[46]
where rL, VL, as, Ji are the molar density for the liquid
phase,the volume occupied by the liquid phase inside the
reactor,the specific interfacial area (with respect the liquid
volume)and the mass flux between the two phases,
respectively.Instead, the single species mass balance is assumed to
be thefollowing:
[47]
where, CL,i, Ri and QL are the molar concentration in
liquidphase for the examined species, its specific production
ratein the liquid volume as a result of the effect of
chemicalreactions (see equation [2]) and the volumetric flow rate
forthe liquid phase, respectively. Hypothetically, if
chemicalreactions do not occur in the gaseous phase, the
massbalance for the ith chemical species in gaseous phase can
bewritten as follows:
[48]
where CG,i, VG, and QG are the molar concentration ingaseous
phase for the considered species, the volume of thegaseous phase
dispersed within the liquid matrix and thevolumetric flow rate for
the gaseous phase, respectively;consequently, the overall mass
balance for the gaseousphase, necessary to calculate the gas flow
rate leaving thereactor, is given by the following relation, where
the gasinventory is assumed negligible and where rG indicates
thegas molar density:
[49]
To complete the model description, it is necessary todetermine
the volume occupied by the two phases within thereactor, or more
precisely the volume of the gas phasedispersed within the liquid
phase. This amount is usuallyexpressed through the e ratio
(hold-up), between the volumeof the gas phase and the overall
volume:
[50]
The hold-up value is influenced by the fluid dynamicsregime
existing in the reactor and it is substantiallydetermined by the
rising velocity of the gas bubbles withinthe liquid mass; thus, the
larger the superficial velocity forthe gaseous phase (uGQG/WR)
compared to the naturalbubble rising velocity (uG
T), the higher the e value; usuallysemi-empirical correlations
are available for its estimation inthe different reactor
configurations as a function of thephysical properties of the
fluids in contact, of theirvolumetric flow rates and of the reactor
section WR.
Usually, in segregated gas-liquid reactors, to describethe
behaviour of both phases or simply one of them, an axialdispersion
model is used as reference; accordingly, under thesame general
hypotheses adopted to analyse the stirredsystems and in steady
state conditions, the mass balanceequations for the single species
and for the overall systemreduce to:
[51]
[52]d u C
dzD
d Cdz
R a JL L i L iL i
i s i
( )( ) ( ), ,
,= + +1 12
2
d udz
R a JL L ii
NC
s ii
NC( ) ( )r = +
= = 1
1 1
=+V
V VG
G L
Q Q a V JG G GF
G s L ii
NC
r r= =
1
d V Cdt
J VG G,i i L( )
= + a Q C Q Cs GF
G iF
G G i, ,
d C Vdt
RV J VL,i L i L i L( )
= + + a Q C Q Cs LF
L iF
L L i, ,
d Vdt
VL L L
r
r r( )
= + =a J Q Qs ii
NC
LF
LF
L L1
1 1k k
KE ki
eG i
i
L i
= +, ,
J k C K Ci ie
i G i i L= ( ), ,
PROCESS ENGINEERING ASPECTS
364 ENCYCLOPAEDIA OF HYDROCARBONS
-
[53]
[54]
where uL, uG, e, DL,i and DG,i are the superficial velocities
forthe liquid and gaseous phases, the local hold-up and the
twoaxial dispersion coefficients for the two phases,
respectively.The previous equations, being inherent to a
descriptioninvolving axial dispersion, need to be linked to
Danckwerts-likeboundary conditions analogous to equation [38].
Fluidized bed reactors and slurry reactorsThese reactors have
features which are intermediate
between those of the two heterogeneous reactors (gas-solidand
gas-liquid) previously examined; in fact, while thereaction takes
place through the interaction between thegaseous phase reactants
and the solid particles (catalytic ornot), the latter are, however,
free to move within the reactor;thus the solid phase also has
behaviour similar to that of afluid. In fact, if the velocity of a
fluid flowing through a bedof solid particles exceeds a threshold
value (the minimumfluidization velocity), the solid mass starts
moving in asimilar way to that of a liquid. Usually, the stirring
thusobtained is enough to allow its behaviour to be assimilatedinto
that of well stirred system. Moreover, the gaseous phaseflowing
through a fluidized bed for large velocity valuescauses the
formation of by-pass streams similar to thosetypical of the bubbles
rising within a liquid. This is known asboiling fluidized beds.
From the above, it is clear that theformulations obtained for the
gas liquid reactors can also beeasily extended to these
systems.
If the fluid flowing through the solid mass is a liquid, asolid
suspension is originated, often kept stirring bymechanical devices;
these systems are known as slurryreactors.
For both systems, the availability of suitable correlationsto
estimate the fluidization velocity and the by-pass flowrates as a
function of the physicochemical properties of thetwo phases in
contact is particularly important.
New trends in reactor simulation
The great reliability and development of computationalfacilities
and computational fluid dynamics software hasopened new
possibilities in reactor simulation andconsequently in procedures
for their design. In fact, today it ispossible to fully simulate
reactor behaviour by insertingequations appropriate for the
description of fluid motion,written in differential form, within
mass and energy balanceequations. Because in the great majority of
the cases, chemicalreactors operate in turbulent regime, it is
necessary to completefluid motion equations with equations for the
turbulence modelthrough which it is possible to estimate the
transportparameters involved in the balance equations, like
speciesdiffusivities and fluid viscosity. Usually, in chemical
reactorsimulations, the model equations are written from an
Eulerianpoint of view, even though, for some particular systems
wherethe bubbles or the particulate trajectories have to be
accountedfor, mixed Eulerian-Lagrangian approaches are adopted.
To highlight a significant example of the use of
reactivecomputational fluid dynamics in chemical reactor
simulations, a complex case like that of a gas-liquid reactorin
air-lift configuration can be examined. This is a reactorwhere the
fluid motion is induced by density gradientsoriginating from
various gas hold-ups in different reactorzones. Accordingly, by
assuming an Eulerian point of viewfor the two phases, for each of
them it is necessary to writethe continuity equation:
[55]
and the three Navier-Stokes equations, one for eachcomponent of
fluid velocity,
[56]
where with ea, ua, ta, P and Dab the volumetric fraction,
thevelocity vector and the viscous stress tensor for fluid a,
thesystem pressure and the friction force between the
phases,respectively, are indicated. The latter can be estimated
fromthe difference between the two velocities through
dragcoefficient Cab:
[57]
It can be observed that, for each phase, an equation of stateis
also necessary to describe its density as a function of thelocal
physicochemical conditions, rara(Ta,Pa). It isnecessary to impose
the constraint:
[58]
inherent to the volumetric fraction of the phases.As has already
been stated, because these reactors
operate in turbulent regime, equations for the turbulencemodel
have to be added; one of the most commonly usedmodels is the k-e
model, the equations for which have beenomitted for brevity (see
Chapter 4.2). The numericalprocedure for the model solution assumes
that, at the firstattempt, the velocity of the liquid phase, the
pressure and thelocal hold-up values are known; the algorithm then
solvesthe Navier-Stokes equations for both phases and gives
anupdated value for the fluids velocities and for the
hold-ups.These values are then adopted in the next iteration as
newstarting points, until the whole procedure reaches the
desiredconvergence.
Naturally, in chemical reactors, interest is not onlycentred on
fluid dynamics and thus it is also necessary toadd the mass balance
equations for any individual speciesand the energy balance:
[59]
[60]
where Uab and Sab are the heat transfer coefficient
betweenphases and the interface surface between the phases
specificto the unit of volume.
r Hj j+ ( )j
NR
U T T S= + ( )
1
r
CTt
T k Tp T
+
= ( ) +u
+ = ( ) +Ct C D C Ri
i i i i
u
= =
1
1Np
D u u = ( )C
e r ea a a
g= +P ++=Dabb 1
Np
r er ea a aa a a a a
uu u
( )+ ( ) ( ) =t tt
r e
r ea aa a a
( )+ ( ) =t u 0
d udz
a JG G s ii
NC( )r = =
1
d u Cdz
Dd Cdz
a JG G i G iG i
s i
( ),,
,
= 2
2
CHEMICAL REACTORS
365VOLUME V / INSTRUMENTS
-
By means of this kind of simulation it is possible to learnthe
local chemical composition values in great detail as wellas all the
fluid dynamic parameters in the system, even inreactors
characterized by very complex geometries;information about fluid
microsegregation and about deadvolumes are thus obtained. It is
also possible to verify thepresence of zones where important
by-product accumulationsoccur; these can have negative impact both
on system safetyand on conversion, yield and selectivity
performances.
Moreover, computational fluid dynamics allows thecorrect design
of mechanical dispersion systems (like thestirrers) and the study
of the reactant injection modeseffective in obtaining conditions as
close as possible to thoseof ideal reactors; Fig. 9 shows a
simulation of the flow fieldfrom two inclined turbine blades within
a stirred reactor andthe visualization of reactant dispersion
caused by itsinjection into the proximity of the stirrers.
Accordingly, it appears clear that today these
detailedapproaches are playing a greater and greater role in
thedesign of chemical reactors with advanced features
andperformance. The development of computational devices is,to all
intents and purposes, reducing the importance ofintermediate
approaches, and consequently it can bepredicted that the evolution
of chemical reactor designmethods will focus on either the use of
simple models, basedon perfect stirring or piston flow, or the use
of detailedmodels based on reactive computational fluid
dynamics,whose final aim is to design reactors correctly and make
theirbehaviour as similar as possible to that of ideal
reactors.
Dimensionless numbers in reactorengineering
Once the mass and energy balance equations inherent tothe
reactor under examination have been written, it is goodengineering
practice to place all the variables in adimensionless form. By
doing that, groups of variables (thedimensionless numbers) are
highlighted, the aim of which isto compare one phenomenon (of a
reactive, diffusive orconvective nature) to the others, thus
clarifying their relativeimportance and suggesting possible
simplifications: anequation correctly transformed into
dimensionless form, byreducing all the terms to similar values in
order of
magnitude, makes the numerical solution of the modeleasier.
The dimensionless numbers which are most oftenencountered in
chemical reactor engineering practice aresummarized in Table 3,
each one with its definition andphysical meaning.
Bibliography
Butt J.B. (1980) Reaction kinetics and reactor design,
EnglewoodCliffs (NJ)-London, Prentice-Hall.
Carberry J.J., Varma A. (1986) (edited by) Chemical reaction
andreactor engineering, New York, Marcel Dekker.
Carr S. (1994) Reattore chimico, in: Enciclopedia italiana di
scienze,lettere ed arti. Appendice V, 1979-1992, Roma, Istituto
dellaEnciclopedia Italiana, 1991-1995, 5v.; v.IV, 413-418.
Carr S., Morbidelli M. (1983) Chimica fisica applicata.
Principidi termodinamica e cinetica chimica e loro ruolo nella
teoria delreattore chimico, Milano, Hoepli.
Deckwer W.-D. (1992) Bubble columns, Chichester, John Wiley.
Froment G.F., Bischoff K.B. (1990) Chemical reactor analysis
anddesign, Chichester, John Wiley.
Gianetto A., Silveston P.L. (edited by) (1986) Multiphase
chemicalreactors: theory, design, scale-up, Washington (D.C.),
Hemisphere.
Metcalfe I.S. (1997) Chemical reaction engineering: a first
course,Oxford, Oxford University Press.
Varma A. et al. (1999) Parametric sensitivity in chemical
systems,Cambridge, Cambridge University Press.
Westerterp K.R., Wijingaarden R.J. (1992) Principles of
chemicalreaction engineering, in: Ulmanns encyclopedia of
industrialchemistry, Weinheim, VCH, 1985-1993, v.B4: Principles
ofchemical engineering and plant design.
List of symbols
aR heat exchange surface per unit volumeas specific interfacial
surface per unit liquid volumeCab friction coefficientCi molar
concentration for the i
th speciescV,i heat capacity at constant volume for the i
th speciescp,i heat capacity at constant pressure for the i
th speciesDi diffusion coefficient for the i
th speciesDL,i axial dispersion coefficient for the i
th speciesDab friction force between phases E activation energyE
enhancing factorF(t) residence time distribution function H
enthalpy Hi Henrys Law constant for the i
th species in thesolvent under examination
Ji interfacial molar flux K reaction rate constantk0 frequency
factor for the reaction rate constantkci laminar mass transfer
coefficient for the i
th speciesKi equilibrium partition constant between phasesL
reactor lengthMH Hattas moduleN a number of CSTRs in seriesNC
number of chemical speciesNR number of chemical reactions
PROCESS ENGINEERING ASPECTS
366 ENCYCLOPAEDIA OF HYDROCARBONS
A B
Fig. 9. Example of fluid dynamics simulation for a stirred
reactor performed through computational fluid dynamics software: A,
flow field; B, dispersion of reactant after injection close to the
stirring device.
-
p pressurep0i vapour pressure for the i
th speciesQ volumetric flow rateQ.
thermal power exchanged with the environmentR recycled
volumetric flow rateR universal gas constantRi specific molar
production rate for the i
th speciesrj rate for the j
th chemical reactionSab interfacial surface specific to the unit
volumeSR heat exchange surfaceT thermodynamic temperaturet timeU
internal energyUg overall heat transfer coefficient
Uab interfacial heat transfer coefficientu superficial fluid
velocityu fluid velocity vectoruTG bubbles terminal rising
velocityV reactor volumeW.
mechanical power exchanged with the environmentz reactor axial
coordinateZ isothermal compressibility factor
Greek lettersgi activity coefficientDUi
0 standard molar internal energy of formation changefor the ith
species
DHi0 standard molar enthalpy of formation change for the
ith species
CHEMICAL REACTORS
367VOLUME V / INSTRUMENTS
Table 3. Dimensionless numbers typical of reactor
engineering
Number Definition Physical meaning
Damkoler DaR L
C ui
i
= residence time
1111131111133
characteristic reaction time
Grashof Grg L TT=
r2 2
2
buoyancy force3331111133
viscous forces
Knudsen KnL
= mean free path
11331111133
characteristic length
Pclet Pe Sc uLD
= =Reconvective mass transport13333333333331133311133
diffusive mass transport
Prandtl Pr =CpkT
momentum diffusivity1133333333331133311133
thermal diffusivity
Rayleigh Ra Gr= Prbuoyancy forces13333333311133
viscous forces
Reynolds Re =ruL
inertial forces333133311133
viscous forces
Schmidt Sc Di= r
momentum diffusivity33313331111111133
mass diffusivity
Thiele =V
S
k
D
p
p ieff
' characteristic internal diffusion
time3333331111133111111133
characteristic reaction time
Hatta MD k C
kH
i r LB
L i
=,
characteristic film diffusion time33333111133111111133
characteristic reaction time
Semnov = ( ) ( )/H k e C
S UT
E RTiF n
VF
F0 heat generation rate
333111111133
heat dissipation rate
Sherwood Shk d
Dc
i
= convective mass transport33333133111111133diffusive mass
transport
-
DUR, j0 standard molar internal energy change associated
with the jth chemical reactionDHR, j standard molar enthalpy
change associated with the
jth chemical reactione bed void fractionh yield in the producth
catalyst effectivenessnij stoichiometric coefficient for the i
th species in thejth reaction
x reactant conversionr molar density/ Thieles moduleF volumetric
flow rates ratio (exit/feed)f volumetric recycling ratio (recycled
flow rate/feed
flow rate) selectivity between the reaction productst residence
timetta viscous stress tensor for the generic phase y Semnovs
numberW reactor cross section
SuperscriptsF feed conditions 0 initial conditions molar
property e effective property
SubscriptsR reactor L liquid phaseG gaseous phase S solid phase
e external fluid thermal vectormix mixture propertiesp particle
Maurizio Masi
Dipartimento di Chimica, Materiali e Ingegneria chimica Giulio
Natta
Politecnico di MilanoMilano, Italy
PROCESS ENGINEERING ASPECTS
368 ENCYCLOPAEDIA OF HYDROCARBONS