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# Nonideal Reactor Mixing Patterns Lecture - 10) - Notes, Chemical Engineering Chemical Engineering Notes | EduRev

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## Chemical Engineering : Nonideal Reactor Mixing Patterns Lecture - 10) - Notes, Chemical Engineering Chemical Engineering Notes | EduRev

``` Page 1

10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 10: Non­ ideal Reactor Mixing Patterns
This lecture covers residence time distribution (RTD), the tanks in series model, and
combinations of ideal reactors.
Non­ Ideal Mixing
PFR
CSTR
Figure 1. Ideal PFR with pulse input. A
pulse input will yield an output profile
that is a pulse input.
Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that
is a sharp peak with a tail.
Real mixed tank
stagnant
bypassing
mixing
recirculation
eddies
volumes
Figure 3. A real mixed tank. In a real mixed tank there are portions that are not
well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses.
In a real PFR there is back­ mixing and axial dispersion. In a packed bed reactor
(PBR) channeling can occur. This is where the fluid channels through the solid
medium.
Residence Time Distribution
A useful diagnostic tool is the residence time distribution (RTD). The residence time
is how long a particle stays in the reactor once entering.
E (t) dt = Probability that a fluid element entering the vessel at t=0 exits between
time t and t+dt.
Probability density function for exit time, t, as a random variable.
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Page 2

10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 10: Non­ ideal Reactor Mixing Patterns
This lecture covers residence time distribution (RTD), the tanks in series model, and
combinations of ideal reactors.
Non­ Ideal Mixing
PFR
CSTR
Figure 1. Ideal PFR with pulse input. A
pulse input will yield an output profile
that is a pulse input.
Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that
is a sharp peak with a tail.
Real mixed tank
stagnant
bypassing
mixing
recirculation
eddies
volumes
Figure 3. A real mixed tank. In a real mixed tank there are portions that are not
well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses.
In a real PFR there is back­ mixing and axial dispersion. In a packed bed reactor
(PBR) channeling can occur. This is where the fluid channels through the solid
medium.
Residence Time Distribution
A useful diagnostic tool is the residence time distribution (RTD). The residence time
is how long a particle stays in the reactor once entering.
E (t) dt = Probability that a fluid element entering the vessel at t=0 exits between
time t and t+dt.
Probability density function for exit time, t, as a random variable.
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
t
E t dt Probability that fluid element exits before time t.
?
( )
0
8
E t dt Probability of exiting at time later than t.
?
( )
t
8
mean t =
?
tE ( ) t dt =t
0
8
=
?
E ( ) dt = 1 normalized t
0
8
variance =s
2
= t -t
2
E t dt (measures the broadness of the distribution)
?
( ) ( )
0
E
after t
1
before t
1
t
1
t
Figure 4. E(t) versus t. At a given time point, some material has exited and some
material will still exit at a later time.
Experimental Determination of E(t)
Inflow should be something measurable
­ Absorbance
­ Fluorescence
­ pH
­ salt­ conductivity
Use one of two types of input concentration curves:
Pulse
C
in
Step
C
in
t t
Figure 5. Two types of input. A pulse input is a spike of infinite height but zero
width, ideally. A step input is a constant concentration over a period of time.
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 2 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Page 3

10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 10: Non­ ideal Reactor Mixing Patterns
This lecture covers residence time distribution (RTD), the tanks in series model, and
combinations of ideal reactors.
Non­ Ideal Mixing
PFR
CSTR
Figure 1. Ideal PFR with pulse input. A
pulse input will yield an output profile
that is a pulse input.
Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that
is a sharp peak with a tail.
Real mixed tank
stagnant
bypassing
mixing
recirculation
eddies
volumes
Figure 3. A real mixed tank. In a real mixed tank there are portions that are not
well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses.
In a real PFR there is back­ mixing and axial dispersion. In a packed bed reactor
(PBR) channeling can occur. This is where the fluid channels through the solid
medium.
Residence Time Distribution
A useful diagnostic tool is the residence time distribution (RTD). The residence time
is how long a particle stays in the reactor once entering.
E (t) dt = Probability that a fluid element entering the vessel at t=0 exits between
time t and t+dt.
Probability density function for exit time, t, as a random variable.
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
t
E t dt Probability that fluid element exits before time t.
?
( )
0
8
E t dt Probability of exiting at time later than t.
?
( )
t
8
mean t =
?
tE ( ) t dt =t
0
8
=
?
E ( ) dt = 1 normalized t
0
8
variance =s
2
= t -t
2
E t dt (measures the broadness of the distribution)
?
( ) ( )
0
E
after t
1
before t
1
t
1
t
Figure 4. E(t) versus t. At a given time point, some material has exited and some
material will still exit at a later time.
Experimental Determination of E(t)
Inflow should be something measurable
­ Absorbance
­ Fluorescence
­ pH
­ salt­ conductivity
Use one of two types of input concentration curves:
Pulse
C
in
Step
C
in
t t
Figure 5. Two types of input. A pulse input is a spike of infinite height but zero
width, ideally. A step input is a constant concentration over a period of time.
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 2 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
A pulse input allows for easy interpretation because all materials enter the reactor at
once.
t
C
in
input
detector
t
C
in
curve
Figure 6. Schematic of a residence­ time distribution experiment. The input curve
enters the reactor; a detector detects concentration changes in the output stream.
out
E ( ) t =
t
C (t )
?
C
out
( ) t dt
0
PFR (Ideal)
t
C
in
t
t
C
in
t
0
Figure 7. Pulse input in ideal PFR. A pulse input in an ideal PFR becomes a pulse
output.
E (t)=d (t -t )
?= 0 x ? 0
x d ( ) =
?
?
=8 x = 0
8
?
d ( ) x dx = 1
-8
8
?
f ( ) ( x d x - a) dx = f ( ) a
-8
CSTR (Ideal)
Transient material balance:
In­ Out+Production=Accumulation
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 3 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Page 4

10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 10: Non­ ideal Reactor Mixing Patterns
This lecture covers residence time distribution (RTD), the tanks in series model, and
combinations of ideal reactors.
Non­ Ideal Mixing
PFR
CSTR
Figure 1. Ideal PFR with pulse input. A
pulse input will yield an output profile
that is a pulse input.
Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that
is a sharp peak with a tail.
Real mixed tank
stagnant
bypassing
mixing
recirculation
eddies
volumes
Figure 3. A real mixed tank. In a real mixed tank there are portions that are not
well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses.
In a real PFR there is back­ mixing and axial dispersion. In a packed bed reactor
(PBR) channeling can occur. This is where the fluid channels through the solid
medium.
Residence Time Distribution
A useful diagnostic tool is the residence time distribution (RTD). The residence time
is how long a particle stays in the reactor once entering.
E (t) dt = Probability that a fluid element entering the vessel at t=0 exits between
time t and t+dt.
Probability density function for exit time, t, as a random variable.
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
t
E t dt Probability that fluid element exits before time t.
?
( )
0
8
E t dt Probability of exiting at time later than t.
?
( )
t
8
mean t =
?
tE ( ) t dt =t
0
8
=
?
E ( ) dt = 1 normalized t
0
8
variance =s
2
= t -t
2
E t dt (measures the broadness of the distribution)
?
( ) ( )
0
E
after t
1
before t
1
t
1
t
Figure 4. E(t) versus t. At a given time point, some material has exited and some
material will still exit at a later time.
Experimental Determination of E(t)
Inflow should be something measurable
­ Absorbance
­ Fluorescence
­ pH
­ salt­ conductivity
Use one of two types of input concentration curves:
Pulse
C
in
Step
C
in
t t
Figure 5. Two types of input. A pulse input is a spike of infinite height but zero
width, ideally. A step input is a constant concentration over a period of time.
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 2 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
A pulse input allows for easy interpretation because all materials enter the reactor at
once.
t
C
in
input
detector
t
C
in
curve
Figure 6. Schematic of a residence­ time distribution experiment. The input curve
enters the reactor; a detector detects concentration changes in the output stream.
out
E ( ) t =
t
C (t )
?
C
out
( ) t dt
0
PFR (Ideal)
t
C
in
t
t
C
in
t
0
Figure 7. Pulse input in ideal PFR. A pulse input in an ideal PFR becomes a pulse
output.
E (t)=d (t -t )
?= 0 x ? 0
x d ( ) =
?
?
=8 x = 0
8
?
d ( ) x dx = 1
-8
8
?
f ( ) ( x d x - a) dx = f ( ) a
-8
CSTR (Ideal)
Transient material balance:
In­ Out+Production=Accumulation
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 3 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Since all the material is added at once, In=0. The tracer used is non­ reactive.
Therefore there is no production. This gives:
0 -? C + 0 = V
dC
0
dt

( )
0
-t
t
V

C t = C e , t =
?
0
C t
t
t
t =

?
( )
t

E ( )
8
( )
=
e
-
C t dt
0
CSTR
Figure 8. Pulse input in an ideal CSTR. In an ideal CSTR, a pulse input leads to a
sharp peak with a tail.
8
-t
t
mean residence time =
?
te
dt =t
t
0
CSTR (non­ ideal mixing)
Bypassing: Divide input into 2 streams
0
Figure 9. A bypass is modeled by dividing the input stream into two streams, one of
which does not enter the reactor.
V
0
?
B
?
SB
?
?
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 4 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Page 5

10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. K. Dane Wittrup
Lecture 10: Non­ ideal Reactor Mixing Patterns
This lecture covers residence time distribution (RTD), the tanks in series model, and
combinations of ideal reactors.
Non­ Ideal Mixing
PFR
CSTR
Figure 1. Ideal PFR with pulse input. A
pulse input will yield an output profile
that is a pulse input.
Figure 2. Ideal CSTR with pulse input. A pulse input will yield an output profile that
is a sharp peak with a tail.
Real mixed tank
stagnant
bypassing
mixing
recirculation
eddies
volumes
Figure 3. A real mixed tank. In a real mixed tank there are portions that are not
well mixed due to stagnant volumes, recirculation eddies, and mixing bypasses.
In a real PFR there is back­ mixing and axial dispersion. In a packed bed reactor
(PBR) channeling can occur. This is where the fluid channels through the solid
medium.
Residence Time Distribution
A useful diagnostic tool is the residence time distribution (RTD). The residence time
is how long a particle stays in the reactor once entering.
E (t) dt = Probability that a fluid element entering the vessel at t=0 exits between
time t and t+dt.
Probability density function for exit time, t, as a random variable.
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
t
E t dt Probability that fluid element exits before time t.
?
( )
0
8
E t dt Probability of exiting at time later than t.
?
( )
t
8
mean t =
?
tE ( ) t dt =t
0
8
=
?
E ( ) dt = 1 normalized t
0
8
variance =s
2
= t -t
2
E t dt (measures the broadness of the distribution)
?
( ) ( )
0
E
after t
1
before t
1
t
1
t
Figure 4. E(t) versus t. At a given time point, some material has exited and some
material will still exit at a later time.
Experimental Determination of E(t)
Inflow should be something measurable
­ Absorbance
­ Fluorescence
­ pH
­ salt­ conductivity
Use one of two types of input concentration curves:
Pulse
C
in
Step
C
in
t t
Figure 5. Two types of input. A pulse input is a spike of infinite height but zero
width, ideally. A step input is a constant concentration over a period of time.
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 2 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
A pulse input allows for easy interpretation because all materials enter the reactor at
once.
t
C
in
input
detector
t
C
in
curve
Figure 6. Schematic of a residence­ time distribution experiment. The input curve
enters the reactor; a detector detects concentration changes in the output stream.
out
E ( ) t =
t
C (t )
?
C
out
( ) t dt
0
PFR (Ideal)
t
C
in
t
t
C
in
t
0
Figure 7. Pulse input in ideal PFR. A pulse input in an ideal PFR becomes a pulse
output.
E (t)=d (t -t )
?= 0 x ? 0
x d ( ) =
?
?
=8 x = 0
8
?
d ( ) x dx = 1
-8
8
?
f ( ) ( x d x - a) dx = f ( ) a
-8
CSTR (Ideal)
Transient material balance:
In­ Out+Production=Accumulation
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 3 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Since all the material is added at once, In=0. The tracer used is non­ reactive.
Therefore there is no production. This gives:
0 -? C + 0 = V
dC
0
dt

( )
0
-t
t
V

C t = C e , t =
?
0
C t
t
t
t =

?
( )
t

E ( )
8
( )
=
e
-
C t dt
0
CSTR
Figure 8. Pulse input in an ideal CSTR. In an ideal CSTR, a pulse input leads to a
sharp peak with a tail.
8
-t
t
mean residence time =
?
te
dt =t
t
0
CSTR (non­ ideal mixing)
Bypassing: Divide input into 2 streams
0
Figure 9. A bypass is modeled by dividing the input stream into two streams, one of
which does not enter the reactor.
V
0
?
B
?
SB
?
?
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 4 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
Figure 11. Residence­time distribution for dead volumes. When a dead volume is
present, a decreased amount of material is observed in the output stream.
measureable V=V
SD
+ V
D
V
t =
SD
SD
<t
?
ideal
0
PFR (Non­ideal)
E
bypass portion
E
mixed
t t
combine
E
Perfect mixing t =
V
?
0
V
Bypass
t =
?
t
SB

Figure 10. Residence­time distribution determination for a bypass.
Dead volumes: Stagnant regions not getting mixed
V
D
V
SD
E
ideal
t
present

Channeling
channeling
bed
channel
PFR­like
Figure 12. Channeling. In channeling, the residence­time distribution will show
peaks for each channel as well as the one for the main portion of the reactor.
10.37 Chemical and Biological Reaction Engineering, Spring 2007 Lecture 10
Prof. K. Dane Wittrup Page 5 of 7
Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring
[DD Month YYYY].
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