9.nin ideal flow
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NON-IDEAL FLOW
Residence Time Distribution
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SCOPE:
Design of non-ideal reactors
Identify the possible deviations
Measurement of RTD
Quality of mixing
Models for mixing
Calculating the exit conversion in practicalreactors
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Practical reactor performance deviates from thatof ideal reactors :
Packed bed reactor Channeling
CSTR & Batch Dead Zones, Bypass
PFR deviation from plug flow dispersion
Deviation in residence times of molecules
the longitudinal mixing caused by vortices andturbulence
Failure of impellers /mixing devicesHow to design the Practical reactor ??
What design equation to use ??
Approach: (1) Design ideal reactor
(2) Account/correct for deviations 3
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Deviations
In an ideal CSTR, the reactant concentration is uniform
throughout the vessel, while in a real stirred tank, thereactant concentration is relatively high at the pointwhere the feed enters and low in the stagnant regionsthat develop in corners and behind baffles.
In an ideal plug flow reactor, all reactant and productmolecules at any given axial position move at the same ratein the direction of the bulk fluid flow. However, in a realplug flow reactor, fluid velocity profiles, turbulent mixing,
and molecular diffusion cause molecules to move withchanging speeds and in different directions.
The deviations from ideal reactor conditions pose several
problems in the design and analysis of reactors.
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Possible Deviations from ideality:
Short Circuiting or By-Pass Reactant flows into the tank through theinlet and then directly goes out through the outlet without reacting if theinlet and outlet are close by or if there exists an easy route between thetwo.
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1. Dead Zone 2. Short Circuiting
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Three concepts are generally used to describe the
deviations from ideality:
the distribution of residence times (RTD)
the quality of mixing
the model used to describe the system
These concepts are regarded as characteristics ofMixing.
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Analysis of non-ideal reactors is carried out in
three levels:
First Level:
Model the reactors as ideal and account or
correct for the deviationsSecond Level:
Use of macro-mixing information (RTD)
Third Level: Use of micro-mixing information models for
fluid flow behavior10
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RTD Function:
Use of (RTD) in the analysis of non-ideal reactor
performance Mac Mullin & Weber 1935
Dankwerts (1950) organizational structure
Levenspiel & Bischoff, Himmelblau & Bischoff,
Wen & Fan, Shinner
In any reactor there is a distribution of
residence times
RTD effects the performance of the reactor
RTD is a characteristic of the mixing11
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Measurement of RTD
RTD is measured experimentally by injecting an inert
matrerial called tracer at t=0 and measuring itsconcentration at the exit as a function of time.
Injection & Detection points should be very close to
the reactor 12
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ASSUMPTIONS
1. Constant flowrate u(l/min) and fluid density (g/l).
2. Only one flowing phase.
3. Closed system input and output by bulk flow only (i.e.,no diffusion across the system boundaries).
4. Flat velocity profiles at the inlet and outlet.5. Linearity with respect to the tracer analysis, that is,
the magnitude of the response at the outlet isdirectly proportional to the amount of tracer
injected.
6. The tracer is completely conserved within the systemand is identical to the process fluid in its flow andmixing behavior.
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Desirable characteristics of the tracer:
non reactive species
easily detectable
should have physical properties similar to thatof the reacting mixture
completely soluble in the mixture
should not adsorb on the walls
Its molecular diffusivity should be low andshould be conserved
colored and radio active materials are the
most widely used tracers 14
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Types of tracer inputs:
Pulse input
Step input
Ramp input
Sinusoidal inputPulse & Step inputs are most common
Ramp input
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The fraction of material that has spent an amount of
time between t and t+t in the reactor:
dN = C(t) v dt
0
0 )( dttvCN
For pulse input
0
)(N
NttE
0
)(
)()(
dttC
tCtE
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C curve
1)(0
dttE18
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1
0
1
)( ttimeresidenceahavingFractiondttE
t
1
1
)( ttimeresidenceahavingFractiondttE
t
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The age of an element is defined as the time elapsed
since it entered the system.
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12100
.......(22
)(
nn CCCCCh
dttC
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Disadvantages of pulse input
injection must be done in a very short time
when the c-curve has a long tail, the analysis
can give rise to inaccuracies
amount of tracer used should be known
however, require very small amount of tracer
compared to step input
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Step input of tracer
In step input the conc. of tracer is kept at this
level till the outlet conc. equals the inlet conc.
t
out dttECC0
0 )(
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stepC
tC
dt
d
tE
0
)(
)(
For step input:
Disadvantages of Step input:
difficult to maintain a constant tracer conc.
RTD fn requires differentiation can lead
to errors large amount of tracer is required
need not know the amount of tracer used
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Characteristics of the RTD:
E(t) is called the exit age distribution function
or RTD function
describes the amount of time molecules have
spent in the reactor
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Cumulative age distribution function F(t):
t
dttEtF0
)()(
t
dttEtF )()(1
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Relationship between the E and F curves
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C l ti di t ib ti f ti F(t)
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Cumulative age distribution function F(t):
Washout function W(t) = 1 - F(t): 28
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E and F Curves with bypassing
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E and F Curves with Channeling
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0
32/3
3 )()(1 dttEttS m
What is the significance of these moments ??
Moments of RTD:
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If the distribution curve is only known at a number ofdiscrete time values, ti, then the mean residence time isgiven by:
This is what you use in the laboratory
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Variance:
represents the square of the distribution
spread and has the units of (time)2
the greater the value of this moment, thegreater the spread of the RTD
useful for matching experimental curves toone family of theoretical curves
Skewness:
the magnitude of this moment measures theextent that the distribution is skewed inone direction or other in reference to the
mean 37
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Space time vs. Mean residence time:
0
)( dtttEtm0v
V
The Space time and Mean residence time would be
equal if the following two conditions are satisfied:
No density change
No backmixing
In practical reactors the above two may not be valid
and hence there will be a difference between them.
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Normalized RTD function E():
)()( tEE /t
0
)(1 dttE
0
)(1 dE
What is the significance of E() ??
How does E() vs. looks like for two ideal CSTRsof different sizes ??
How does E(t) vs. t looks like for two ideal
CSTRs of different sizes ??
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Using the normalized RTD function, it is possible to
compare the flow performance inside differentreactors.
If E() is used, all perfectly mixed CSTRs havenumerically the same RTD.
If E(t) is used, its numerical values can change fordifferent CSTRs based on their sizes.
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RTD i id l t
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RTD in ideal reactors:
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RTD f id l PFR
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RTD for ideal PFR:
)()( ttE
00)( twhent0)( twhent
1)( dtt
)()()( gdtttg
0
)()( dtttdtttEtm
0
222 0)()()()( dttttdttEtt mm 42
RTD f id l CSTR:
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RTD for ideal CSTR:
0
/ /)( dttedtttEt tm
0
/2
22 )()()(
dtet
dttEtt t
m
Material balance on tracer st to pulse input:
in out = accumulation0 vC = VdC/dt C(t) = C0 e
-t/
/
0
/
0
/
0
0
)(
)()(
t
t
t
e
dteC
eC
dttC
tCtE
eE )(
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RTD for PFR-CSTR series:
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RTD for PFR CSTR series
For a pulse tracer input into CSTR the output
would be : C(t) = C0e-t/s
Then the outlet would be delayed by a time p at the
outlet of the PFR. RTD for the system would be:
pttE 0)(
p
s
t
te
tEsP
/)(
)(
1/s
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If th l f t i i t d d i t th PFR
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If the pulse of tracer is introduced into the PFR,
then the same pulse will appear at the entrance of
the CSTR p seconds later. So the RTD for PFR-CSTR
also would be similar to CSTR-PFR.
Though RTD is same for both, performance is
different
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Remarks:
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Remarks:
RTD is unique for a particular reactor
The reactor system need not be unique for a
given RTD
RTD alone may not be sufficient to analyze the
performance of non-ideal reactors
Along with RTD, a model for the flow behaviour
is required
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Reactor modeling with RTD:
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Reactor modeling with RTD:
I. Zero parameter models:
(a)Segregation model
(b)Maximum mixedness model
II. One parameter models:(a)Tanks-in-series model
(b)Dispersion model
III. Two parameter models:
Micro-mixingmodels
Macro-mixingmodels
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S ti d l (D k ts & Z i t i 1958)
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Segregation model (Dankwerts & Zwietering, 1958)
Characteristics: Flow is visualized in the form of globules
Each globule consists of molecules belonging
to the same residence time
Different globules have different Res. Times
No interaction/mixing between differentglobules
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M i f l b l di b t t d t dt i th t
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Mean conversion of globules spending between t and t+dt in the reactor =
(Conversion achieved after spending a time t in the reactor) X
(Fraction of globules that spend between t and t+dt in the reactor)
dttEtxxd )()(_
0
_
)()( dttEtxx
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Mean conversion in a PFR using Segregation model:
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Mean conversion in a PFR using Segregation model:
Example: A R, I order, Constant density
OrderIforetx kt1)(
00
_
)(1)()1( dttEedttEex ktkt
kktedttex
1)(1
0
_
Mean conversion predicted by Segregation model
matches with ideal PFR
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Mean conversion in a CSTR using Segregation model:
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Mean conversion in a CSTR using Segregation model:
Example: A R, I order, Constant density
0
/
0
_
/)(1 dteedttEex tktkt
k
kx
1
_
Mean conversion predicted by Segregation model
matches with ideal CSTR
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Mean conversion in a practical reactor using
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Mean conversion in a practical reactor using
Segregation model:
Example: A R, I order, Constant density
00
_
)(1)()( dttEedttEtxx kt
conduct tracer experiment on the practical reactor
measure C(t) and evaluate E(t)
plot and evaluate mean conversion
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Tanks in series (TIS) Model:
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Tanks in series (TIS) Model:
Material balance on the I reactor for tracer:
V1dC
1/dt = -v C
1 C
1= C
0exp(-t/
1)
Material balance on the II reactor for tracer:
V2 dC2/dt = v C1 v C2 dC2/dt + C2/2 = C0exp(-t/2)258
2/0 ttC
C ittC
CSi il l /2
0
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2/
2
02
teC it
i
eCSimilarly
/
2
03
2
it
i
et
dttC
tCtE
/
3
2
0
3
33
2)(
)()(
For n equal sized CSTRs:
it
n
n
en
t
tE
/1
)1()(
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Total = ni = t/ n = t/i
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Total ni = t/ n t/i
nn
t
n
i
n
i e
n
nne
n
tnTEE i
)1(
)(
)1(
)()(1
/1
As the number becomes large,the behavior of the systemapproaches that of PFR
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We can calculate the dimensionless variance 2
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0
2
2
22 )()1(
dE
We can calculate the dimensionless variance
000
2 )()(2)( dEdEdE
1)1(12)1(
)(
0
1
0
12
den
n
den
nn nnn
nn
nnn
nn
n
n
nn
n 11)1(
11
)1(
)1( 22
The number of tanks n = 1/2 = 2/2
If the reaction is I order:n
ik
x)1(
11
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Backmixing or dispersion, is used to represent the combined action of allphenomena namely molecular diffusion turbulent mixing and non
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Ideal Plug flow
phenomena, namely molecular diffusion, turbulent mixing, and non-uniform velocities, which give rise to a distribution of residence times inthe reactor.
If the reactor is an ideal plug flow, the tracer pulse traverses throughthe reactor without distortion and emerges to give the characteristicideal plug flow response. If diffusion occurs, the tracer spreads awayfrom the center of the original pulse in both the upstream anddownstream directions.
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Closed vessel Dispersion Model:
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Closed vessel D spers on Model
Da = Damkohler number = k C0n-1
)1(22
22
2
rPe
rrm
ePePet
2/22/2
2/
)1()1(
41
qPeqPe
Pe
eqeq
qex
PeDq a /41
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The x-axis, labeled macromixing measures the breadth of the residence
time distribution. It is zero for piston flow, fairly broad for the exponential
distribution of a stirred tank, and broader yet for situations involving
bypassing or stagnancy.
The y-axis is micromixing, which varies from none to complete. Micromixing
effects are unimportant for piston flow and have maximum importance forstirred tank reactors.
Well-designed reactors will usually fall in the normal region bounded by the
three apexes, which correspond to piston flow, a perfectly mixed CSTR, and
a completely segregated CSTR.69
Without even measuring the RTD, limits on the performance of most real
t b d t i d b l l ti th f t th th
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reactors can be determined by calculating the performance at the three
apexes of the normal region.
The calculations require knowledge only of the rate constants and the
mean residence time.
When the residence time distribution is known, the uncertainty about
reactor performance is greatly reduced.
A real system must lie somewhere along a vertical line in Normal Region.
The upper point on this line corresponds to maximum mixedness and
usually provides one bound limit on reactor performance.
Whether it is an upper or lower bound depends on the reactionmechanism.
The lower point on the line corresponds to complete segregation and
provides the opposite bound on reactor performance.70
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ANY CLARIFICATIONS ?
Kuhn, Thomas
. . . no theory ever solves all the puzzles with which it is confronted at a
given time; nor are the solutions already achieved often perfect.