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Ch E 441 - Chemical Kinetics and Reaction Engineering
Residence Time Distributions
in Chemical Reactors
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Residence Time Distributions
The assumption of a perfectly mixed reactoroften falls short of reality.
Residence time distributions are used to model
the imperfect mixing behavior of real reactors.
Cumulative age, F(t)
External age, E(t)
Internal age, I(t)
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Residence Time Distributions
Gas-liquid CSTR (A(g) + B(l)
C(l)) Reaction occurs at gas-liquid interface
Liquid phase is perfectly mixed
Rate is proportional to bubble surface area
Residence time of gas bubble in reactor is
proportional to bubble volume
Larger bubble escape rapidly
Smaller bubbles may remain in reactor until consumed
Understanding of RTDs is necessary for analysis
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Residence Time Distributions
PBR Sections of the catalyst bed may offer less resistance
to flow, resulting in a preferred pathway through the
bed.
Molecules flowing through the channel do not
spend as much time in the PBR as those taking
another path.
Consequently, there is a distribution of residencetime for the PBR.
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Residence Time Distributions
CSTR Short-circuiting may occur (the direct movement of
material from inlet to outlet.
Dead zones may exist (regions with a minimum of
mixing and thus virtually no reaction takes place).
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Residence Time Distributions
Concepts that must be addressed in approachinga solution to such problems:
distribution of residence times occurs
quality of mixing varies with position in reactor
a model must used to describe the phenomenon
Accounting for nonideality requires
knowledge of macromixing (RTD)
application of the RTD to a reactor (micromixing) to
predict reactor performance.
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RTD Functions
In any reactor, the RTD can affect performance
Ideal Plug Flow and Batch Reactors
Every atom leaving reactor is assumed to have resided inthe reactor for exactly the same duration. No axial mixing.
Ideal CSTR
Some atoms leave almost immediately, others remainalmost forever. Many leave after spending a period of timenear the mean residence time. Perfect mixing.
RTD is characteristic of mixing in a reactor.
RTDs are not unique to reactor type.Different reactor types can have the same RTD.
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Measurement of RTD
RTD is measured experimentally by use of aninert tracer injected into the reactor at t = 0.Tracer concentration is measured at effluent as afunction of time.
Tracer must be non-reactive and non-absorbingon reactor walls/internals.
Tracer is typically colored or radioactive to allow
detection and quantification. Common methods of injection are pulse and step
inputs.
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Pulse Input RTD Measurement
An amount of tracer No is suddenly (all at once)injected into the feed of a reactor vessel with
flow at a steady state.
Outlet concentration is measured as a function of
time.
reactor
injection detection
feed effluent
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Pulse Input RTD Measurement
reactor
injection detection
feed effluent
pulse injection
C
t0 +-
pulse response
C
t0 +-
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Pulse Input RTD Measurement
Injection pulse in system of single-input andsingle-output, where only flow (no dispersion)carries tracer material across system boundaries.
The amount of tracer materialN leaving the
reactor between t and t+t for a volumetricflowrate of is
where t is sufficiently small that the
concentration of tracer C(t) is essentially constantover the time interval.
ttCN
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Pulse Input RTD Measurement
Dividing by total amount of tracer injected, Noyields the fraction of material that has a
residence time between t and t+t:
where E(t) represents the residence-timedistribution function.
ttEtN
tC
N
N
oo
0 dttCtC
tE
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Step Input RTD Measurement
In general, the output concentration from avessel is related to the input function by the
convolution integral (Levenspiel):
where the inlet concentration takes the form of
either a perfect pulse input (Dirac delta function),imperfect pulse injection, or a step input.
dt'tE'ttCtCt
0inout
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Step Input RTD Measurement
t
0o
t
0inout dt'tECdt'tE'ttCtC
Considering a step input in tracer
concentration for a system of constant :
0tC
0t0tC
oo
constant can be broughtoutside the integral
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Divide by Co
F(t) fraction of molecules that have spent a time
t or less in reactor (Cumulative age)
Differentiate to obtain RTD function E(t)
Step Input RTD Measurement
tF'dt'tECC t
0stepo
out
stepo
out
CC
dtdtE
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Step Input RTD Measurement
Advantages Easier to carry out experimentally than pulse test
Total amount tracer in feed need not be known
Disadvantages Often difficult to maintain a constant tracer
concentration in feed.
differentiation of data, often leads to large error.
Requires large amount of tracer, which in some cases
can be expensive.
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RTD Characteristics
E(t) is sometimes called the exit-age distributionfunction.
If the age of an atom is regarded as the amount
of time it spends in the reactor, E(t) is the age
distribution of the effluent.
E(t) is the most often used distribution function
for reactor analysis.
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Fraction of exit stream that has resided in the reactor
for a period of time shorter than a given value of t:
Fraction of exit stream that has resided in the reactorfor a period of time longer than a given value of t:
Integral Relationships
tFdttEt
0
tF1dttEt
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Integral Relationships
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Mean Residence Time
0
0
0m dttEt
dttE
dttEt
t
The nominal holding time, , is equal to the
mean residence time, tm. The mean value of the time is the first
moment of the RTD function, E(t).
can be used to determine reactor volume
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1st moment mean residence time
2nd moment variance (extent of spread of
the RTD)
3rd moment skewness (extent RTD is skewed
relative to the mean)
Other Moments of the RTD
0
2
m
2
dttEt-t
0
3m
13 dttEt-ts 23
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Normalized RTD Function, E()
t
A normalized RTD is often used to allow
comparison of flow profiles inside reactors of
different sizes, where
tEE
etEE
e1
tE tfor an ideal CSTR
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Internal-Age Distribution, I()
Fraction of material inside the reactor that has
been inside for a period of time between and
+
0 dE1
1
I
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RTD in a Batch or PFR
Simplest case
Spike at t = (or = 1) of infinite height and zero
width with an area of one
ttE
0x0x0x
1dxx
gdxxxg
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Effluent concentration is identical to that of
reactor contents.
A material balance for t > 0 on inert tracer
injected as a pulse at t = 0
RTD in a CSTR
dt
dC
VC0
accout-in
t
0eCtC
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RTD in a CSTR
Recall definition of E(t), and substitute:
dt
dC
VC0
t
0eCtC
t
0
t
0
t0
0
e
dteC
eC
dttC
tCtE
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Ideal Reactor Response to Pulse
E
t
Batch/PFR
E
CSTR
1
1
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Laminar Flow RTD
2
2
o
2
maxR
r1
R
2
R
r1UU
Velocity profile in a pipe (cylindrical
coordinates) is parabolic according to:
Time for passage of an element of fluid is
22
o
2
Rr11
2Rr11
2LR
rULrt
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The fraction of total fluid passing between r
and r+dr is d/0:
Laminar Flow RTD
00
rdr2rUd
rdr
R
t4rdr
Rr1
2
R
4dt
2
22
22
2
Rr11
2rt
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Laminar Flow RTD
Combining
dtt2
dtt4R2
tLrdr2
tLd
3
2
2
2
000
00
rdr2rUd
rdr
R
t4rdr
Rr1
2
R
4dt
2
22
22
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Laminar Flow RTD
The minimum time the fluid will spend in the
reactor is
Therefore, the complete RTD function is
22
V
R
R
U2
L
U
Lt
02
2
avgmax
23
22
tt2
t0
tE
5.021
5.00
E 3
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Laminar Flow RTD
The RTD appears graphically as
5.02
15.00
E
3E
1
0.5
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RTD of PFR and CSTR in series
CSTR (s) followed by PFR (p)
CSTR output will be delayed by a time ofp
p
s
t
p
te
t0
tE sp
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RTD of PFR and CSTR in series
PFR (p) followed by CSTR (s)
PFR output will delayed the introduction of the pulse
to the CSTR by a time ofp
Regardless of the order, the RTD is the same. However, theRTD is not a complete description of structure for a particular
reactor or system of reactors (see Example 13-4).
p
s
t
p
te
t0
tE sp