Chemical Reaction Engineering

1
Chemical Reaction Engineering
Lecture 13
Lecturer ???
2
This course focuses on residence time
distribution for chemical reactors.

• Three concepts used to describe nonideal
  reactors
• the distribution of residence times in the system
• the quality of mixing
• the model used to describe the system

3
Residence time distribution (RTD)

• Initially proposed by MacMullin and Weber (1935)
  and extensively studied by Danckwerts (1953).
• The time the atoms have spent in the reactor is
  called the residence time of the atoms in the
  reactor.
• Ideal plug-flow reactor all the atoms of
  materials spend exactly the same amount of time
  inside the reactor.
• The idealized plug-flow and batch reactors are
  the only two classes of reactors that all the
  atoms in the reactor have the same residence
  time.
• The RTD of a reactor
• A characteristic of the mixing within the
  chemical reactor.
• One of the most informative characterizations of
  the reactor.

4
The RTD measurement

• Injection (pulse/step) of a tracer (an inert
  chemical, molecular, or atom) into the reactor at
  time t 0 and measuring the tracer concentration
  in the effluent stream as a function of time.
• Tracer
• non-reactive species
• easily detectable
• similar physical properties to those of the
  reacting mixture
• completely soluble in the mixture
• not adsorb on the walls/other surfaces in the
  reactor
• colored and radioactive materials are most common
  types of tracers

5
Pulse injection An amount of tracer N0 is
suddenly inject in one shot into the feedstream
entering the reactor
injection
detection
C
C
The C curve
t
0
The amount of tracer materials, ?N, leaving the
reactor between t and t ?t is
The effluent volumetric flow rate
The concentration of the tracer in the effluent
?N also represents the amount of material that
has spent an amount of time between t and t ?t
in the reactor.
6
The fraction of material that has a residence
time in the reactor between time t and t ?t .
E(t) the residence time distribution function.
unit time-1 The function that
quantitatively describes how much time different
fluid elements have spent in the reactor.
The fraction of materials leaving the reactor
that has resided in the reactor for times between
t1 and t2
7
Pulse injection example
A sample of tracer at 320K was injected as a
pulse to a reactor. The effluent concentration
was measured as a function of time. (a) Construct
figures showing C(t) and E(t) as a function of
time. (b) Determine the fraction of material
leaving the reactor that spent between 3 and 6
min in the reactor.
C
C(t)
Numerical integration
t
E
E(t)
51 of the material leaving the reactor spends
between 3 and 6 min in the reactor.
t
8
Drawbacks to pulse injection

• The injection must take place over a period which
  is very short compared with residence times in
  various segments of the reactor or reactor
  system.
• There must be negligible amount of dispersion
  between the point of injection and the entrance
  to the reactor system.
• Long tail?

9
Step injection
injection
detection
Cin
Cout
The C curve
t
t
t
t
The concentration of tracer in the feed to the
reactor is kept at this level until the
concentration in the effluent is
indistinguishable from that in the feed.
constant
10
Drawbacks to step injection

• Sometimes difficult to maintain a constant tracer
  concentration in the feed.
• Differentiation of data can sometimes lead to
  large errors.
• The amount of tracer required may be large.

11
More about E(t)

• The residence time distribution function.
• The function that quantitatively describes how
  much time different fluid elements have spent in
  the reactor
• unit time-1
• Also called the exit-age distribution function.
• The age of an atom as the time it has resided
  in the reaction environment.
• The age distribution of the effluent stream.

12
?
?
Nearly ideal PFR
Nearly ideal CSTR
CSTR with dead zones
PBR with channeling and dead zones
The fraction of the exit stream that has resided
in the reactor for a period of time shorter than
a given value t
Danckwerts defined F(t) as a cumulative
distribution function
0.8
80 of the molecules spend 40 min or less in the
reactor
40
13
Mean residence time, tm

• For an ideal reactor, the space time,?, (i.e.,
  average residence time) is defined as V/v.
• The mean residence time, tm, is equal to ? in
  either ideal or nonideal reactors.
• The spread of the distribution (variance)

14
Calculate the mean residence time and the
variance for the reactor by the RTD obtained from
a pulse input at 320K.
t
tE
The mean residence time
Numerical integration
t
(t-tm)2E
The variance
Numerical integration
t
15
Normalized?
The residence time distribution function
Normalized distribution function
Reactors with different sizes can be compared
using normalized RTD distribution functions.
RTD in ideal reactors
Batch and PFR All the atoms leaving such
reactors have spent precisely the same amount of
time within the reactors.
in
out
E(t)
The Dirac delta function
0
?
t
16
Single-CSTR
the concentration of any substance in the
effluent stream is identical to the concentration
throughout the reactor. A material balance on the
inert tracer by pulse injection (at time t 0)
Input - Output Accumulation
B.C. C C0 at t 0
E(?)
1
?
17
Laminar flow reactor
For a laminar flow in a tubular reactor, the
velocity profile is parabolic
The minimum time the fluid may spend in the
reactor
The time of passage of an element of fluid at
radius r is
18
The fraction of total fluid passing between r and
(r dr) is
The fraction of fluid spending between time t and
t dt in the reactor
The RTD function for a laminar flow reactor is
E(?)
0.5
?
19
Modeling the real reactor as a CSTR and a PFR in
series
(1) A highly agitated zone close to the impeller
can be modeled as a perfectly mixed CSTR. (2) The
inlet and outlet pipe can be modeled as a PFR. We
combine them to model a real reactor.
CSTR (?s)
PFR (?p)
E(t)
delay by ?p
pulse injection
t
CSTR (?s)
PFR (?p)
delay by ?p
pulse injection
There is no difference in RTD between (1) CSTR
PFR and (2) PFR CSTR !
20
Modeling a second-order reaction system
A second-order reaction (CA0 1 kmol/m3) is
carried out in a real CSTR that can be modeled as
two different reactor systems (1) CSTR PFR (2)
PFR CSTR Find the conversion in each system
where ?s ?p 1 min
(1) CSTR PFR
CA0
CAi
CA
CSTR
PFR
mass balance
mass balance
21
(1) PFR CSTR
CA0
CAi
CA
PFR
CSTR
mass balance
mass balance
There is a difference in conversion between (1)
CSTR PFR and (2) PFR CSTR ! However, their
RTDs are the same!
The RTD is unique for a particular reactor
however, the RTD alone is not sufficient to
determine the performance of reactors. (Other
characteristics include the quality of mixing,
the degree of segregation ... etc.)
22
Reactor modeling with the RTD

• When the fluid in a reactor is neither well mixed
  nor approximates plug flow (i.e., nonideal), one
  can use RTD data and some model to predict
  conversion in the reactor.
• Frequently used models include
• Zero adjustable parameters
• segregation model
• maximum mixedness model
• One parameter model
• tanks-in-series model
• dispersion model
• Two adjustable parameters
• real reactor modeled as combinations of ideal
  reactors

23
The conversion of the reactants in a
reactor (1) how much time they stay in a
reactor gtgtgt RTD (2) how they move in a
reactor gtgtgt Mixing
For a first-order reaction, the conversion is
independent of concentration
Knowing the RTD is sufficient to predict
conversion.
For reactions other than first-order, the degree
of mixing of molecules must be known in addition
to how long each molecule spends in the reactor
to predict conversion.
Certain model of describing the degree of mixing
is required to predict conversion!
24
Degree of mixing of molecules

• Macromixing a distribution of residence time
  without specifying how molecules of different
  ages encounter one another in the reactor.
• Micromixing describe how molecules of different
  ages encounter one another in the reactor. Two
  extremes
• complete segregation all molecules of the same
  age group remain together during their staying in
  the reactor.
• complete micromixing molecules of different age
  groups are completely mixed at the molecular
  level as soon as they enter the reactor.

25
Segregation model (minimum mixedness)
E(t)
How to model a real PFR?
t

• In a CSTR lumping all the molecules that have
  the same residence time in the reactor into the
  same globule (Fig. 13-14)
• In a PFR batches of molecules are removed from
  the reactor at different locations along the
  reactor (Fig. 13-15)
• Features
• There is no molecular interchange between
  globules, each acts essentially as its own batch
  reactor.
• The reaction time in any globule is equal to the
  time that the particular globule spends in the
  reaction environment.
• The RTD among the globules is given by the RTD of
  the particular reactor.

26
The mean conversion in the effluent stream
average conversion of various globules in the
exit stream
Mean conversion of those globules spending
between time t and t dt in the reactor
Conversion achieved after spending a time t in
the reactor
Fraction of globules that spend between t and t
dt in the reactor
summation
Mean conversion for the segregation model
From design equation of a batch reactor (i.e.,
each globule)
From experimental measurement
First-order reaction
27
Derive the equation of a first-order reaction
using the segregation model when the RTD is
equivalent to (a) an ideal PFR, and (b) and ideal
CSTR.
(1) ideal PFR
The RTD function
Identical!
A first-order reaction in a PFR
(2) ideal CSTR
The RTD function
Identical!
A first-order reaction in a CSTR
28
For a first-order reaction, whether one assumes
complete mixing (ideal CSTR) or complete
segregation (globules) in a CSTR, the same
conversion results. This is because of the
first-order reaction the extend of micromixing
does not affect the reaction. Only RTD is
necessary to calculate the conversion for a
first-order reaction in any type of reactor.
Calculation of the mean conversion using the RTD
Mean conversion for the segregation model
XE
t
29
Maximum mixedness model

• In a PFR as soon as the fluid enters the
  reactor, it is completely mixed radially with the
  other fluid already in the reactor. (Fig. 13-16)
  A PFR with side entrances

? ? 0
? ? ?
v0
V 0
V V0
? the time it takes for the fluid to move from a
particular point to the end of the reactor
The volumetric flow rate of fluid fed into the
side of the reactor in the interval between ?
?? and ? is
The volumetric flow rate of fluid fed inside the
reactor at ? is
The volume of fluid with life expectancy between
? ?? and ? is
30
Mass balance on A between ? ?? and ?
31
A second-order reaction example
The liquid-phase, second-order reaction
The reaction is carried out at 320K and the feed
is pure A with CA08 mol/dm3. The reactor is
nonideal and could be modeled as two CSTRs with
interchange. The reactor is 1000 dm3 and the feed
rate is 25 dm3/min. A RTD test was carried out.
Determine the bounds on the conversion for
different possible degrees of micromixing for the
RTD of this reactor.
C(t)
E(t)
1-F(t)
The bounds on the conversion are under conditions
of complete segregation and maximum mixedness.
(1) complete segregation
E(t)
32
(2) maximum mixedness
Choose ? ? ? (200) , ?? -25, and X0 0 to
start the numerical integration
X1 2
X2 1.46
X
X3 0.192
X 0.56
...
times
The two bounds are segregation 61 and MM 56
33
For two limiting mixing situations, we obtained
the conversion using only the RTD (i.e., we have
no other knowledge about the flow pattern). They
are (1) the earliest possible mixing consistent
with the RTD (Maximum mixedness) (2) mixing
only at the reactor exit (complete
segregation) Calculating conversions for these
two cases fives bounds on the conversions that
might be expected for different flow paths
consistent with the observed RTD.
34
RTD and multiple reactions

• For multiple reaction, we dont want to use X.
• In the segregation model
• the globules are mixed together immediately upon
  exiting
• CA (t) is determined from batch reactor
  calculations. If there are q reactions taking
  place
• In the maximum mixedness model

35
Multiple reactions
E1(t)
E2(t)
t
t
take place in two different reactors with the
same mean residence time tm 1.26 min. When the
RTD is different for each other, determine the
product distribution for the segregation model
and the maximum mixedness model.
Multiple reaction
36
The best fit for E1 (t) is
The best fit for E2 (t) is
(1) the segregation model
All these EQs are solved simultaneously.
(2) Maximum mixedness model
All these EQs are solved simultaneously.
Choose ? ? 6 , ?? -0.2, and CA0 CB0 1 to
start the numerical integration
The results for two different RTDs are given in
Table 13-9.2 on p.858 and Table 13-9.4 on p.859.
Different compositions of products are obtained
for different RTDs.