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7/22/2019 08 Introduction to Non-Ideal Reactors....by akash agrawal.itis good for every one: I THINK GIFTING IS GOOD TECH
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Chapter 8
Introduction to
Non-ideal Reactors
Our treatment on reactor design in the
previous chapters is based on the assump-
tion that the reactorsare ideal
What are ideal reactors?
For CSTRs,
they must be well-mixed the concentration of the exit
stream is equal to that of fluid in
the reactor
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For PFRs, theflowin the reactor must
be in order, i.e. there are
no overtaking () no backmixing ()
With these idealbehaviours/character-istics, at steady state, the concentrationof
the fluidflowing outof the reactor is
uniform(or constant)
In reality, however, no reactorsare
ideal, i.e. theflow patternin the reactor
(either CSTRs or PFRs) is non-ideal
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The main reasonsof the non-ideal
flow (or the deviation of the ideal flow) are:
1) the residence time distribution(RTD)
For ideal reactors, it is assumed
that the residence time(or space time, )
ofall substances(all molecules) in the
reactors are identical, i.e. all molecules
that enter the reactor at the same time
must exit the reactor at the same residence
time; no molecules stay in the reactor
shorter or longer than the residence time
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For non-ideal reactors(or real
reactors), there may be some molecules
whose flow rates are slower or faster than
those of others
Channellingof fluid causes some
molecules to havefasterflow ratethan
others, as illustrated below
Channelling of Fluid
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A stagnant(/) zone
in the reactor, on the contrary, delaysthe
flowof some molecules, causing such
molecules to flow out of the reactor slower
than others
Stagnant zone in a CSTR
Stagnant Zone
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2) State of aggregationFor ideal reactors, all molecules
are assumed to be negligibly small(when
compared to the reactor); thus, they can
move freely
In reality, a group ofmolecules
are combined together(i.e.aggregationof
molecules) to form macrofluid molecules
(as apposed to microfluid molecules as
per ideal reactors)
These macrofluid molecules move
in package(or in aggregates), thus causing
non-ideal flowbehaviour to occur
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Microfluid
Macrofluid
3) Earliness or lateness of mixingFor an ideal CSTR, it is assumed
that the reactor is well-mixed, meaningthat the concentrations at any position
(point) of the CSTR are identical throughout
the reactor
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The mixing in the non-idealCSTR,
on the contrary, is NOTideally perfect,
causing the concentrationsin the CSTR
vary from position to position
For an ideal PFR, it is assumed
that there are no overtaking and/or no
backmixing (i.e. the flow is uniform or in
order)
In a real PFR, however, there may
be some overtaking or backmixing, which
results in early mixing or late mixing in the
PFR as illustrated on the next Page
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Uniform Mixing
(Ideal Flow in a PFR)
EarlyMixing
LateMixing
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RTD Function (or The Age Distribution
of Fluid)
As mentioned earlier, the residence
timesof molecules in a real (actual)
reactor are NOTidenticalas per the ideal
reactor
In other words, there is a distribution
of the residence timesoffluidentering
and leaving the reactor
The distribution of these times for the
stream of fluid leavingthe reactor is called
the exit age distribution, E (we shall
discuss about Ein detail later)
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In order to determine the distribution of the
residence times (or the degree of the non-ideal
of the flow), a small amount of substance
(called a tracer) is injected into the reactor,
and the concentrations of the tracer flowing
out of the reactor are recorded
A tracershould be
easy to analyse for itsconcentration(using an ordinary
equipment available in a common
laboratory, e.g., pH meter, gas
chromatograph: GC)
stable(i.e. it is not decomposed easily) inactive(i.e. it does not react with any
other species in the reactor)
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To introduce (or inject) tracerinto the
reactor, it can be done by
apulseexperiment a stepexperiment
The Pulse Experiment
In thepulseexperiment, a known
amount of tracer is injected into the
reactor onlyonceat any instant of time
Assume that a tracer in the amount of
M kg or kmol (kg-mol) is injected into thefluid entering the reactor (or vessel) with
the volumetric flow rate ofv m3/s
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The concentration vstime (which can
be called time-resolved concentration) of
the tracer leaving the reactor is monitored,
and the plot between the concentration of
the tracer ( )pulseC and time ( )t for an ideal
reactor and a non-ideal reactor can beillustrated on the following Pages (Pages
1415)
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For an ideal flow
For the case of an idealflow, after a
certain period of time (which is, in fact, the
residence time), the tracershall leave the
reactor at oncewith the concentration
equalto that enteringthe reactor
Cpulse
tResidence time
=out inC C
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For a non-ideal flow
For a non-idealflow pattern (or a non-
ideal reactor), there is a distributionoftimesthat the tracer leaves the reactor
from = 0t to t=
Cpulse
t
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The area under the pulseC vs t curve for a
non-ideal reactor from = 0t to t= is
( )0
C t dt
The unit of the integral is, for example,
[ ]mol mol s
sL L
=
or
[ ]3 3kg kg ssm m
=
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Multiplying the integral with the
volumetric flow rate gives
( )0
v C t dt
and the unit of the resulting multiplication
is, for example,
L mol s
s L
= mol
or
3
3m kg s kgs m
=
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By considering the unit of the term
( )0
v C t dt
, it leads to the fact that this term
is, in fact, the total amount of the tracer
leaving the reactor, which, by employing
the principle of the mass conservation, is
equal to that entering the reactor
Thus,
( )0
M v C t dt
= (8.1)
which can be re-arranged to
( )
= =
0
Area under
the curve
from 0
M C t dtv
(8.2)
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If the integration is performed from = 0t
to t t= ,
( )0
t
C t dt
we obtain the fact that
( )0
t
m v C t dt = (8.3)
which can be re-arranged to
( )
= ==
0
Area under
the curvefrom 0
tm
C t dtvt
(8.4)
where m is the mass (or moles) of the
tracer leaving the reactor between the time
period of = 0t to t t= as shown graphically
on the next Page
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Cpulse
t t=t
( )0
Area under
the curvefrom 0
t
mC t d t
vt
= =
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Dividing throughout Eq. 8.3:
( )0
t
m v C t dt =
(8.3)
withM
vand re-arranging yields
( )0
t
v C t dtm
M M
v v
=
( )0
t
v C t dt
mvMM
v
=
( )0
t
C t dtm
MM
v
=
(8.5)
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It is defined that
( )
( )0
0
t
t
C t dt
E t dtM
v
= (8.6)
Thus, the meaning ofE (the exit age
distribution) or ( )E t is, in fact,
( )( )C t
E tM
v
= (8.7a)
pulseCE
M
v
= (8.7b)
( )
pulse pulse
0
Area under
the curve
from 0
C C
EC t dt
= =
(8.7c)
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In other words, we can obtain Eat any
instant of time by simply dividing the value
of the concentration of a tracer ( )pulseC by
M
vor by ( )
0
C t dt
or by the area under the
curve from 0t= to t=
From Eq. 8.6, when t= , we obtain the
fact that
( )
( )0
0
C t dt
E t dtM
v
=
(8.8)
but, from Eq. 8.2, ( )0
MC t dt
v
=
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Hence, Eq. 8.8 becomes
( )0
1
M
vE t dt M
v
= = (8.9)
( )E t is also called the RTD functionwhose graph are as shown below
E
t
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It is evident that the E vs tplot is
identical to that ofC vs t, as E is, in fact,
impulseCE
M
v
= (8.7b)
Since the area under the curveof the C
vs tplot from 0t= to t= or ( )0
C t dt
is
M
v(see Page 18), the area under the
curveof the E vs tplot from 0t= to t=
or ( )0
E t dt
or0
Edt
is 1, as illustrated
mathematically on Page 24 (Eq. 8.9)
Note that by dividing Cor pulseC with a
constant (i.e.Mv ), it is called normalisation
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We can employ the RTD function to
calculate the average (or mean) residence
time using the following relationship:
( )0
t tE t dt
= (8.10)
or
( )
( )
0
0
tC t dt
t
C t dt
=
(8.11)
which can be proved mathematically as
follows
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From Eq. 8.8, we obtain the fact that
( )
( )0
0
C t dt
E t dtM
v
= (8.8)
but, from Eq. 8.2,
( )0
M C t dtv
= (8.2)
Combining Eq. 8.8 with Eq. 8.2 results
in
( )
( )
( )
0
0
0
C t dt
E t dt
C t dt
=
(8.12)
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Multiplying throughout Eq. 8.12 with t
gives
( )
( )
( )
0
0
0
tC t dt
tE t dt
C t dt
=
(8.13)
but, from Eq. 8.10, which is defined that
( )0
t tE t dt
= (8.10)
Thus, Eq. 8.13 is, in fact,
( )
( )
0
0
tC t dt
t
C t dt
=
(8.11)
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Example To determine the non-ideal
behaviour of a reactor with a volume of
180 L, a pulse experiment is carried out by
injecting a tracer only once in the amount
ofM = 1.2 kg into the inlet stream flowing
into the reactor with volumetric flow rate of
11 L/min
The concentration vs timedata of the
tracer in the outlet stream of the reactor
within the period of 035 min are as shown
in the Table on the next Page
Determine
a) the mean residence time( )t ofthe flow in this reactor
b) the residence time in this reactorif the flow is ideal
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Time [min] C[g/L]
05
101520253035
03
554210
The plot ofC vs tof this pulse
experiment is as illustrated on the next
Page (Page 31)
Determining the area under the curve
(try doing it yourself) gives
( )0
M
C t dtv
= (8.2)
which can also be written in thepractical
form as
( )iM C t tv
= (8.14)
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0
1
2
3
4
5
6
0 10 20 30 40
Time [min]
Concen
tration
oftracer[g/L]
The meaning of Eq. 8.14 is that the
integral ( )0
C t dt
can practically be obtained
by summing the multiplication ofC (at any
instant of time) and t (or 1n nt t+ )
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The area under the curve of the graph
on the previous Page (Page 31) is found to
be 100 g min
L ; thus,
( )0
g min100
L
MC t dt
v
= =
and
g min100
L L11
min
g min L 100 11L min
M
M
=
=
1,100 g= = 1.1 kgM
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It is given that the tracer is injected
into the reactor in the amount of1.2 kg,
but the data from the outlet streamyield
the fact that the total amountof the
tracer leavingthe reactor is 1.1 kg
Although there is a discrepancy (
), but it seems to be acceptable, as
the error is only1
~ 100 8.3%
12
=
The mean or average residence time ( )t
can be calculated using Eq. 8.11
( )
( )
0
0
tC t dt
t
C t dt
=
(8.11)
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which can also be written in thepractical
form as follows
( )( )
i i
i
t C t t t
C t t
=
(8.15)
Note that the term( )i
C t t
(or the
denominator: ) is, in fact, ( )0
C t dt
,
which is the area under the Cpulse vs tcurve
and is equal tog min
100L
(see Page 32)
The computation of the value of
( )0
tC t dt
or ( )( )i it C t t is illustrated
in the Table on the next Page
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t [min] iC [g/L] t ( ) i it C t 0 0
5 3 5 535 = 75
10 5 5 1055 = 25015 5 5 1555 = 37520 4 5 2045 = 40025 2 5 2525 = 25030 1 5 3015 = 15035 0 5 3505 = 0
= 1,500
Thus, by employing Eq. 8.15, we can
calculate the mean residence time ( )t as
follows
( )
( )2g min
1,500 Lg min
100L
i i
i
t C t t t
C t t
=
=
15 mint =
a)
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The residence timefor the ideal-flow
pattern ( ) can simply be calculated using
the following equation:
180 L
L11min
V
v =
=
16.4 min =
b)
It is evident that the mean residence
time ( )t for the non-idealflow (or the
actualcase) is shorterthan the ideal-flow
residence time (15 min vs16.4 min)
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This implies that, on average, the
molecules stay in the reactor shorter
than the required residence time, which
should lead to the fact that the conversion
may be lower than expected
The reasons that the mean residence
time is lower than the ideal residence time
may include the following:
There is a channellingof the flow inthe reactor
There is a stagnant zonein thereactor, which makes the volume( )V of the reactor smaller than the
actual one; thus,V
v = is lower
than expected
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The Step Experiment
Another way of determining the non-
idealflow pattern of the reactor is to carry
out a step experiment
In the stepexperiment, at any instant of
time ( )or at 0t= , a tracer is introduced
into the inlet stream flowing into the
reactor (volume = V m3) with the volume-
tric flow rate ofv m3/s
The traceris injected continuouslyintothe reactor with the concentration
(relative to the inlet stream) of maxC
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The concentration of the tracer in the
outlet stream leaving the reactor can be
presented graphically as follows
Let
( )
( )
max
C t
F t C=
(8.16)
Cstep
t
Cmax
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The concentration of the tracer at any
instant of time, C or ( )C t , is defined asm
V
(on massbasis) or
( )m
C tV
= (8.17)
where m is the total amount of mass
leaving the reactor from 0t= to t t=
Let concentration of maxC be
max
MC
V= (8.18)
Combining Eqs. 8.17 & 8.18 with Eq.8.16 yields
( )( )
max
mC t VF t
MC
V
= = (8.19)
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For a constant-V system, Eq. 8.19
becomes
( ) ( )
max
C t mF t
C M= = (8.20)
but, as we have learned previously (or from
Eq. 8.3)
( )0
t
m v C t dt = (8.3)
and Eq. 8.1
( )0
M v C t dt
= (8.1)
which can be combined to form
( )
( )
( )0
0
0
t
tv C t dt
mE t dt
Mv C t dt
= =
(8.21)
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Combining Eqs. 8.20 & 8.21 together
yields
( )( )
( )max 0
tC t mF t E t dt
C M= = =
or
( ) ( )0
t
F t E t dt = (8.22)
Eq. 8.22 is the relationshipbetween F
(from a stepexperiment) and E(from a
pulseexperiment)
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Eq. 8.22 can be written in the differential
form as follows
( ) ( )dF t
E tdt
= (8.23)
which implies mathematically that the
value ofEor ( )E t can be obtained by
differentiating the graph ofF vs tat any
given time ( )t
Note that
max
MC
V=
can also be written as
max
MmtC
V v
t
= =
(8.24)
where m
is the mass flow rate of the tracer
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The shaded area on the stepC vs t curve:
is
max
step
0
C
tdC
max
max
0
Shaded
area
C
tdC
=
Cstep
t
Cmax
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Thus, the average residence time( )t for
the stepexperiment can be calculated
using the following equation:
max max
max
step step
0 0
max
step
0
C C
C
tdC tdC
tC
dC
= =
(8.25)
Eq. 8.25 can be written in thepractical
form as
( )
max
i it CtC
= (8.26)
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Conversion in Non-ideal Flow Reactors
In the previous chapters, our treatment
on reactor design is based on the ideal
model, in which the conversionis affected
by
thermodynamic constraint[maximum possible conversion
(how faror how much) or K:
equilibrium constant]
rate (how fast) of the Rxn. (or k:rate constant)
design equation of the reactor (ortype)
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For the non-idealreactors, in addition
to those factors, the following additional
factors must also be taken into
consideration in determining/calculating
the conversion of the system (or reactor)
RTD of fluid in the reactor
Micro- or macro-fluid behaviour Earliness & lateness of mixing
As we have learned previously, these
factors cause the reactor to deviate
() from the ideal behaviours
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We have just learned the effect of the
RTDon the non-ideal behaviours of the
reactor
In the subsequent sections, lets
consider the effects of the earliness & late-
ness of mixingand the state of aggregation
(or the macrofluidbehaviour)
If the early mixingtakes place in the
plug flow reactor (PFR) as shown below
EarlyMixing
the section in which the early mixing is
occurring will behave as a CSTR
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Thus, the concentration in this section
will
reduce substantially be uniform quickly
The resulting lowconcentration of thereactant will then proceed its conversion
along the length of the PFR
On the contrary, if the late mixing
occurs in the PFR:
LateMixing
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the concentration of the reactantwill decrease gradually along the
length of the PFR during the first
stage of the reaction
but it will drop quickly after itenters the mixing zone, whichbehaves as a CSTR
In either case (early or late) of mixing, if
the order of the Rxn. ( )n is lowerthan unity( )1 : e.g., n = 0.5
or
0.5
A A
r kC = (8.27)
the rate of Rxn. ( )Ar would be
decreasing at the rate slowerthan AC (e.g.,
when AC decreases by 2 folds, Ar would be
decreasing at the rate of 0.52 = 1.44 folds)
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higherthan unity: e.g., n = 2 or2
A Ar kC = (8.28)
the rate of Rxn. ( )Ar would be
decreasing at the ratefasterthan AC (e.g.,
when AC decreases by 3 folds, Ar would be
decreasing at the rate of 23 = 9 folds
Hence, to obtain as high conversion as
possible, when
n< 1, we want the reactor to havean earlymixing, in order to enable
AC to drop substantially during the
early stage of the reaction n> 1: we desire to have a late
mixing, to obtain as high rate of
Rxn. during the early stage of the
reaction as possible
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The relationship between the
concentration of A (reactant) at any instant
of time ( )AC and its initial concentration
( )oA
C for a PFR or a BR for the nth order
Rxns. are as follows
1st order Rxn.:
( )expo
A
A
Ckt
C= (8.29)
2ndorder Rxn.:1
1o o
A
A A
C
C kC t =
+(8.30)
nthorder Rxn.:
( )
11 1
1 1 oo
nA n
AA
C
n C kt C
= + (8.31)
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For the non-idealPFR or BR, since the
concentration of A does NOT leave the
reactor at once (i.e. it leaves the PFR or BR
with the different value from 0t= to t= ,
as shown as a graph on Page 15), the
average concentration ofA
C ( )AC that
leaves the reactor during the time interval
of 0 must be used instead of the exact
value of AC
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The average (or mean) concentrationof A
with reference tooA
C o
A
A
C
C
for the non-
idealPFRs or BRs can be calculated using
the following equation:
0o o
A A
A A
C C EdtC C
=
(8.32)
and by employing the same principle, the
average conversionof A ( )Ax can be
computed using the following equation:
0
A Ax x Edt
= (8.33)
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55
Example The reaction with the rate
equation of A Ar kC = , where k = 0.307
min-1, is taken place in a 180-L reactor
with a volumetric flow rate of the inlet
stream of 11 L/min
Determine the conversion of A ( )Ax ifa) the reactor is the idealPFRb) the reactor is the non-idealPFR
(use the data of the Example on
Pages 2930 to determine the non-
ideal behaviour of the reactor)
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For PFRs and 1st order Rxns.,
( )expo
A
A
Ck
C
= (8.29)
Assume that this is a constant-V system
Thus,
( )1oA A A
C C x= (1.24)
Combining Eq. 1.24 with Eq. 8.29 and
re-arranging results in
( )( )
( ) ( )
1exp
1 exp
o
o
A A
A
A
C xk
C
x k
=
=
( )1 expAx k= (8.34)
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For an ideal PFR,
180 L16.4 min
L11min
V
v
= = =
Hence, the conversion of A for the ideal
PFR can be calculated, using Eq. 8.34, as
follows
( ) ( )[ ]1 exp 0.307 16.4
1 0.00651
Ax =
=
0.993Ax =
a)
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If the reactor is non-ideal, the average
conversion ( )Ax can be computed using
Eq. 8.33:
0
A Ax x Edt
= (8.33)
The value ofEcan be calculated using
Eq. 8.7c
( )
pulse pulse
0
Area under
the curve
from 0
C CE
C t dt
= =
(8.7c)
From the previous Example, the area
under the C vs tcurve or ( )0
MC t dt
v
= was
found to beg min
100L
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Thus, in this Example, the value ofEis,
in fact,
pulse
100 100
C CE= =
The conversion at any given residence
time can be computed using Eq. 8.34:
( )1 expAx k= (8.34)
From the given data (on Pages 2930),
we can compute the mean conversion of A
( )Ax using Table as shown on the next
Page
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t CE=
C/100( )= 1 expAx k Ax E t
0
5
101520253035
0
3
554210
0
0.03
0.050.050.040.020.01
0
1 exp[-(0.307)(5)]= 0.7850.9540.9900.9980.999
11
0.785
0.03
5= 0.11780.23850.24750.19960.09990.05
0 = 0.953
Note that the term0
Ax Edt
can be
written in a practical form as follows
( )Ax E t (8.35)
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Thus, the average conversion of A
( )A
x , or0
A
x Edt
or ( )
A
x E t
, for the
non-ideal PFR is found to be 0.953
b)
Note that, in this Example, the
conversion of the non-idealPFR is lower
than that of the idealone (0.953 vs0.993)
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In the case ofCSTRs, whose design
equation is
oA A
A
C C
r
=
(1.40)
If the Rxn. is of 1st order:
A Ar kC =
by combining the rate equation (1st order)
with the design equation (of a CSTR), we
obtain
oA A
A
C C
kC
= (8.36)
Re-arranging Eq. 8.36 gives
oA A
A
C Ck
C
=
1oA
A
C
k C =
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1oA
A
Ck
C= +
1
1o
A
A
C
C k=
+(8.37)
Thus, for a non-idealCSTR and the 1st
order Rxn., we obtain the following
equation:
0o o
A A
A A
C C EdtC C
=
0
1
1o
A
A
CEdtC k
= + (8.38)
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Example Repeat the Example on Page 55
but for ideal and non-idealCSTRs
Since the volume of the reactor (CSTR),
V, and the volumetric flow rate, v, of the
inlet stream are still the same as per the
previous Example, the residence time of
the ideal CSTR is also the same; i.e.
180 L16.4 min
L11min
= =
Substituting 16.4 min = and 0.307k=
-1min into Eq. 8.37 yields
( ) ( )
1
1 0.307 16.4o
A
A
C
C=
+
0.166o
A
A
C
C=
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Re-arranging Eq. 1.24:
( )1oA A A
C C x= (1.24)
results in
( )1o
AA
A
Cx
C=
1o
AA
A
Cx
C= (8.39)
Thus,
1 1 0.166 0.834o
AA
A
Cx
C= = =
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For the non-ideal CSTR,
0o o
A A
A A
C CEdt
C C
=
(8.32)
The above equation (Eq. 8.32) can be
written for the 1st
order Rxn. taken placein a CSTR as
0
1
1o
A
A
CEdt
C k
=
+ (8.38)
Applying Eq. 8.39 for the non-ideal
reactor (can be any type of reactor; BR,
CSTR, or PFR) yields
1o
AA
A
Cx
C= (8.40)
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Combining Eq. 8.40 with Eq. 8.38 gives
0
11
1Ax Edt
k
=
+ (8.41)
which can be written in a practical form as
follows
11 1
Ax E tk
= +
(8.42)
From the given data, we can compute
the value ofo
A
A
C
Cfor a CSTR and 1st order
Rxn. or
1
1 E tk
+ as illustrated
in the Table on the next Page
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t CE=
C/100
1
1+k
1
1 + E tk
0
5
1015202530
35
0
3
55421
0
0
0.03
0.050.050.040.020.01
0
1/[1+(0.307)(5)]= 0.3940.2460.1780.1400.1150.098
0.085
0.394 0.5 5= 0.05910.06150.04450.02800.01150.0049
0 = 0.210
Thus,0
1
1Edt
k
+ or
1
1E t
k
+
oro
A
A
C
Cis
0.210o
A
A
C
C
=
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Hence, the average conversion for the
non-ideal CTSR in this Example can be
calculated, using Eq. 8.40, as follows
1 1 0.210
0.790
o
AA
A
Cx
C= =
=
b)
The conversion of the non-idealCSTR
is found to be lowerthan the idealone(0.834 vs0.790)
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In the case that the fluid in the reactor
behaves as a macrofluid, the residence
time distribution (RTD) of the tracer in the
outlet stream for thepulseexperiment is
as shown in the following Figure
Cpulse
t
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The graph on the previous Page
illustrates that the fluid are combined
together to become a lump or a group of
fluid
This lump of fluid moves together from
one place to another, and, eventually, it
comes out of the reactor as a lump of fluid
The example of the calculations
concerning the non-ideal, macrofluid
behaviour is illustrated in the following
Example
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Example The liquid A in the form of
macrofluidhas an initial concentration
( )oAC of 2 mol/L is decomposed according
to the rate equation of
2 L; 0.5
mol min
A Ar kC k = =
The RTD for the pulse experiment is as
shown below
Determine the conversion of this Rxn. if
it is taken place in a BR
Cpulse
[mol/L]
t[min]
2
1 3
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Since this is a non-ideal, macrofluid
flow pattern, we can employ the following
relationships:
0o o
A A
A A
C CEdt
C C
=
(8.32)
and
1o
AA
A
Cx
C= (8.40)
For the 2nd order Rxn.,
1
1o o
A
A A
C
C kC t =
+(8.30)
Combining Eq. 8.30 with Eq. 8.32 yields
0
1
1o o
A
A A
CEdt
C kC t
= +
(8.43)
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We have learned that
( )
pulse pulse
0
Area underthe curve
from 0
C CE
C t dt
= =
(8.7c)
In the Example, the area under thecurve from 0 - can be computed as
follows
( ) ( )[ ]
pulse
Area under
the curve
2 mol/L 3 1 min
C t
=
=
Area under mol min4the curve L
=
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Hence,
pulse -1
mol2
L 0.5 minmol minArea under4
Lthe curve
CE= = =
Substituting corresponding numericalvalues into Eq. 8.43 and integrating gives
( ) ( )
( )[ ]
( ) ( )[ ]
( )
0
0
3
1
3
1
1
1
10.5
1 0.5 2
10.5
1
0.5 ln 1
0.5 ln 1 3 ln 1 1
0.5 ln4 ln2
40.5ln2
o o
A
A A
CEdt
C kC t
dtt
dtt
t
= +
= +
=+
= +
= + +
=
=
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0.5 ln2 0.347o
A
A
C
C= =
but, from Eq. 8.40, 1o
AA
A
CxC
=
Thus, the average conversion of A for
non-ideal, macrofluid in this Example can
be computed as follows
1
1 0.347
0.653
o
AA
A
A
Cx
C
x
=
=
=
Question: What is the conversion of A forthe ideal BR for the residence time of 3