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NON-IDEAL FLOW
Residence Time Distribution
R. SATISH BABU
1
SCOPE:
•Design of non-ideal reactors
•Identify the possible deviations
•Measurement of RTD
• Quality of mixing
•Models for mixing
•Calculating the exit conversion in practical reactors
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Practical reactor performance deviates from that of ideal reactor’s :
• 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 and turbulence•Failure of impellers /mixing devices
How to design the Practical reactor ??
What design equation to use ??
Approach: (1) Design ideal reactor
(2) Account/correct for deviations 3
Deviations
In an ideal CSTR, the reactant concentration is uniform throughout the vessel, while in a real stirred tank, the reactant concentration is relatively high at the point where the feed enters and low in the stagnant regions that develop in corners and behind baffles.
In an ideal plug flow reactor, all reactant and product molecules at any given axial position move at the same rate in the direction of the bulk fluid flow. However, in a real plug flow reactor, fluid velocity profiles, turbulent mixing, and molecular diffusion cause molecules to move with changing speeds and in different directions.
The deviations from ideal reactor conditions pose several problems in the design and analysis of reactors.
4
Possible Deviations from ideality:
Short Circuiting or By-Pass – Reactant flows into the tank through the inlet and then directly goes out through the outlet without reacting if the inlet and outlet are close by or if there exists an easy route between the two.
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1.Dead Zone 2. Short Circuiting
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7
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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 of Mixing.
9
Analysis of non-ideal reactors is carried out in
three levels:
First Level:
• Model the reactors as ideal and account or
correct for the deviations Second Level:
• Use of macro-mixing information (RTD)
Third Level:
• Use of micro-mixing information – models for
fluid flow behavior10
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 mixing 11
Measurement of RTD
RTD is measured experimentally by injecting an inert
matrerial called tracer at t=0 and measuring its
concentration at the exit as a function of time.
Injection & Detection points should be very close to
the reactor12
ASSUMPTIONS
1. Constant flowrate u(l/min) and fluid density ρ(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 is directly proportional to the amount of tracer injected.
6. The tracer is completely conserved within the system and is identical to the process fluid in its flow and mixing behavior. 13
Desirable characteristics of the tracer:
•non reactive species
•easily detectable
•should have physical properties similar to that of the reacting mixture
•completely soluble in the mixture
•should not adsorb on the walls
•Its molecular diffusivity should be low and should be conserved
•colored and radio active materials are the most widely used tracers 14
Types of tracer inputs:
• Pulse input
• Step input
• Ramp input
• Sinusoidal input
Pulse & Step inputs are most common
Ramp input
15
Pulse input of tracer
In Pulse input N0 moles of tracer is injected in one
shot and the effluent concentration is measured
The amount of material that has spent an amount of
time between t and t+t in the reactor:
N = C(t) v t 16
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 input0
)(N
NttE
0
)(
)()(
dttC
tCtE
17
C curve
1)(0
dttE18
1
0
1
)( ttimeresidenceahavingFractiondttEt
1
1
)( ttimeresidenceahavingFractiondttEt
19
The age of an element is defined as the time elapsed
since it entered the system.
20
1210
0
.......(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
dtE
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 used24
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
26
Relationship between the E and F curves
27
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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 Dead space
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E and F Curves with Channeling
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33
34
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 of discrete time values, ti, then the mean residence time is given 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, the greater the spread of the RTD
• useful for matching experimental curves to one family of theoretical curvesSkewness:
• the magnitude of this moment measures the extent that the distribution is skewed in one direction or other in reference to the mean 37
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.38
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 CSTRs of different sizes ??
How does E(t) vs. t looks like for two ideal CSTRs of different sizes ??
39
Using the normalized RTD function, it is possible to compare the flow performance inside different reactors.
If E() is used, all perfectly mixed CSTRs have numerically the same RTD.
If E(t) is used, its numerical values can change for different CSTRs based on their sizes.
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RTD in ideal reactors:
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RTD for ideal PFR:
)()( ttE
00)( twhent0)( twhent
1)( dtt
)()()( gdtttg
0
)()( dtttdtttEtm
0
222 0)()()()( dttttdttEtt mm 42
RTD for ideal CSTR:
0
/ /)( dttedtttEt tm
0
/2
22 )()()(
dte
tdttEtt t
m
Material balance on tracer st to pulse input:
in – out = accumulation
0 – 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:
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)(
ps
t
te
tEsP
/)(
)(p
1/s
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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
47
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
48
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-mixing models
Macro-mixing models
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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 different globules
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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
54
Mean conversion in a PFR using Segregation model:
Example: A R, I order, Constant density
OrderIforetx kt1)(
00
_
)(1)()1( dttEedttEex ktkt
kkt edttex
1)(10
_
Mean conversion predicted by Segregation model
matches with ideal PFR
55
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
56
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
57
Tanks in series (TIS) Model:
Material balance on the I reactor for tracer:
V1 dC1/dt = -v C1 C1 = C0 exp(-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/
2
02
te
tCC it
i
etC
CSimilarly
/
2
20
3 2
it
i
et
dttC
tCtE
/
3
2
0
3
33 2
)(
)()(
For n equal sized CSTRs:
itni
n
en
ttE
/
1
)1()(
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Total = ni = t/ n = t/i
nn
tni
n
i en
nne
n
tnTEE i
)1(
)(
)1()()(
1/
1
As the number becomes large, the behavior of the system approaches that of PFR
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0
22
22 )()1(
dE
We can calculate the dimensionless variance 2
000
2 )()(2)( dEdEdE
1)1(
12)1(
)(
0
1
0
12
den
nde
n
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: nik
x)1(
11
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The Dispersion Model:
• The Dispersion Model is used to describe non-ideal PFR
• Axial dispersion is taken into consideration
• Analogous to Fick’s law of diffusion superimposed on the
flow
Da = Diffusivity coefficient; U = superficial velocity;
L = Characteristic length 63
Ideal Plug flow
Backmixing or dispersion, is used to represent the combined action of all phenomena, namely molecular diffusion, turbulent mixing, and non-uniform velocities, which give rise to a distribution of residence times in the reactor.
If the reactor is an ideal plug flow, the tracer pulse traverses through the reactor without distortion and emerges to give the characteristic ideal plug flow response. If diffusion occurs, the tracer spreads away from the center of the original pulse in both the upstream and downstream directions.
64
Closed vessel Dispersion Model:
Da = Damkohler number = k C0n-1
)1(2222
2rPe
rrm
ePePet
2/22/2
2/
)1()1(
41
qPeqPe
Pe
eqeq
qex
PeDq a /41
65
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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 for stirred 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 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 reaction mechanism.
The lower point on the line corresponds to complete segregation and provides the opposite bound on reactor performance.
70
ANY CLARIFICATIONS ?
Kuhn, Thomas. . . no theory ever solves all the puzzles with which it is confronted at
agiven time; nor are the solutions already achieved often perfect.
71
