Review: Nonideal Flow & Reactor
Design
Real CSTRs
• Relatively high reactant conc at
entrance
• Relatively low conc in stagnant
regions, called dead zones
(corners & behind baffles)
Short Circuiting
L23b-1
Real PBRs
• fluid velocity profiles, turbulent
mixing, & molecular diffusion
cause molecules to move at
varying speeds & directions
channeling
Dead Zone
Dead zones
Dead Zone
Goal: mathematically describe non-ideal flow and solve design problems
for reactors with nonideal flow
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-2
Residence Time Distribution (RTD)
RTD ≡ E(t) ≡ “residence time distribution” function
RTD describes the amount of time molecules have spent in the reactor
RTD is experimentally determined by injecting an inert “tracer” at t=0 and
measuring the tracer concentration C(t) at exit as a function of time
Measurement of RTD
↑
Pulse injection
Et 

 E  t  dt  1
0
The C curve
Reactor
Ct

0 C  t  dt

X
↓
Detection
C(t)
tracer conc at exit between t & t+t
sum of tracer conc at exit for infinite time
t
E(t)=0 for t<0 since no fluid can exit before it enters
E(t)≥0 for t>0 since mass fractions are always positive
Fraction of material leaving reactor that has  t2 E  t  dt

been inside reactor for a time between t1 & t2 t1
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-3
E(t)
E(t)

t
Nearly
ideal PFR
E(t)
E(t)

t
t
PBR with
dead zones
Nearly ideal
CSTR
Nice multiple
choice
question
t
CSTR with
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:
F(t) is a cumulative distribution function
0 E  t  dt  F  t 
t

t E  t  dt  1  F  t 
F  t   0 when t<0
0.8
80% of the molecules
spend 40 min or less in
the reactor
F  t   0 when t  0
F   1
40
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-4
Review: Mean Residence Time, tm
• For an ideal reactor, the space time  is defined as V/u0
• The mean residence time tm is equal to  in either ideal or nonideal
reactors

0 tE  t  dt
tm  

0 E  t  dt

0 tE
 t  dt  
V
   tm
u0
By calculating tm, the reactor V can be determined from a tracer experiment
The spread of the distribution (variance):
 2  0  t  tm  E  t  dt
2
Space time  and mean residence time tm 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, hence there will be a
difference between them
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-5
Review: Complete Segregation Model
Mixing of different
‘age groups’ at the
last possible
moment
•
•
•
•
Flow is visualized in the form of globules
Each globule consists of molecules of the same residence time
Different globules have different residence times
No interaction/mixing between different globules
The mean conversion is the average conversion of the various globules in
the exit stream:
X   X t E t t
A
j
A
Conversion achieved after
spending time tj in the reactor
t 0


 X A   X A  t  E  t  dt
0
 j  j
Fraction of globules that spend
between tj and tj + t in the reactor
XA(t) is from the batch
reactor design equation
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-6
Review: 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. Like a PFR with side entrances,
where each entrance port creates a new residence time:
u0E( )
 ∞
0
u0
u 
u
 + 
V=0
V = V0
: time it takes for fluid to move from a particular point to end of the reactor
u: volumetric flow rate at , = flow that entered at + plus what entered
through the sides
u0E: Volumetric flow rate of fluid fed into side ports of reactor in interval
between  +  & 
Volumetric flow rate of fluid fed to reactor at : u     u0  E    d  u0 1  F    
fraction of effluent that in reactor for less than time t
Volume of fluid with life expectancy between  +  & : V  u0 1  F    
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-7
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for (1) an ideal PFR and (2) for the complete segregation model.
CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-8
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for (1) an ideal PFR and (2) for the complete segregation model.
CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
Start with PFR design eq & see how far can we get:
dX A kC A CB2


dV
C A0u0
dX A rA

dV
FA0
dX A kCA0CB0 1  X A 


dV
CA0u0
2
3
CA  CA0 1  X A 
X
CB  CB0 1  X A 
2
V kC
A
dX A
Get like terms
B0 dV
 


3
together & integrate
0 1  X A 
0 u0
X
A

1
1
2
kCB02 
1

1

2kC


X

1

B0


V

A
2
2
1

X
u


2kCB02  1
A
2 1  X A   0
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-9
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for (1) an ideal PFR and (2) for the complete segregation model.
CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02 0.1
0.16 0.2
0.16
0.12
0.08
0.06 0.044 0.03 0.012
0
t*E(t)
0
0.02 0.2
0.48 0.8
0.8
0.72
0.56
0.48 0.396
0
XA  1 
1
2kCB0   1
2
Use numerical method
to determine tm:
How do we
determine ?
0.3
0.144
For an ideal reactor,  = tm
tm  0 tE  t  dt

10
14
0
0
10
tm   tE  t  dt   tE  t  dt   tE  t  dt
1 0  4  0.02   2  0.2   4  0.48   2  0.8   4  0.8  
 tE  t  dt  
  4.57
3  2  0.72   4  0.56   2  0.48   4  0.396   0.3 
0
10
14
 tE  t  dt 
10
2
0.3  4  0.144   0   0.584
3
 tm  4.57  0.584  5.15min
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-10
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for (1) an ideal PFR and (2) for the complete segregation model.
CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02 0.1
0.16 0.2
0.16
0.12
0.08
0.06 0.044 0.03 0.012
0
t*E(t)
0
0.02 0.2
0.48 0.8
0.8
0.72
0.56
0.48 0.396
0
XA  1 
1
2kCB02  1
X A,PFR  1 
0.3
0.144
For an ideal PFR reactor,  = tm
tm  5.15min  
tm  0 tE  t  dt
1
2


L2
mol 
2  176
  0.0313
  5.15min   1
2

L

mol  min  

X A,PFR  0.40
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-11
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2

0.06 0.044 0.03 0.012
0
Segregation model: X A   X A  t  E  t  dt XA(t) is from batch reactor design eq
0
Numerical method
1. Solve batch reactor design equation to determine eq for XA
2. Determine XA for each time
3. Use numerical methods to determine X̄A
Polymath Method
1. Use batch reactor design equation to find eq for XA
2. Use Polymath polynomial curve fitting to find equation for E(t)
3. Use Polymath to determine X̄A
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-12
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2

0.06 0.044 0.03 0.012
0
Segregation model: X A   X A  t  E  t  dt XA(t) is from batch reactor design eq
0
NA0  CA0 V
dX A
3
dX A
2
3
dX A
2


kC
1

X


 kCA0CB0 1  X A  V
NA0
 rA V  NA0
B0
A
dt
dt
dt
Batch design eq:
Stoichiometry:
rA  kC A CB
2
CA  CA0 1  X A 
CB  CB0 1  X A 
XA
 
0

dXA
t
1  XA 3
1
1  X A 2
  kCB02dt
0
X
A

1
2


kC
t

B0
2
2 1  X A   0
 1  2kCB02 t  XA  1 
1
1  2kCB02t
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-13
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min 0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.044
0.03
0.012
0
XA
Segregation model:

X A   X A  t  E  t  dt
XA  1 
0
Numerical method
XA0  1 
X A 1  1 
1

1

1  0.3429min1 t
1  2kCB02t
Plug in each t & solve
1
1
1  0.3429min
1
0 
1
1  0.3429min
1
1min 
0
 0.137
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-14
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min 0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.044
0.03
0.012
0
XA
0 0.137 0.23 0.298 0.35 0.39 0.428 0.458 0.483 0.505 0.525 0.558 0.585

Segregation
X A   X A  t  E  t  dt
model:
0
Numerical method
XA  1 
1

1

1  0.3429min1 t
1  2kCB02t
1

10
14
0
0
10
X A   X A  t  E  t  dt   X A  t  E  t  dt   X A  t  E  t  dt
0  4  0.137  0.02   2  0.23  0.1  4  0.298 0.16 


1
 X A  t  E  t  dt   2  0.35  0.2   4  0.39  0.16   2 0.428 0.12   4 0.458 0.08  
3
0
 2  0.483  0.06   4  0.505  0.044   0.525 0.03 



10
10
 X A  t  E  t  dt  0.35
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-15
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min 0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.044
0.03
0.012
0
XA
0 0.137 0.23 0.298 0.35 0.39 0.428 0.458 0.483 0.505 0.525 0.558 0.585

Segregation
X A   X A  t  E  t  dt
model:
0
Numerical method
XA  1 
1

1

1  0.3429min1 t
1  2kCB02t
1

14
0
10
X A   X A  t  E  t  dt  0.35   X A  t  E  t  dt
14
2
 X A  t  E  t  dt   0.525  0.03   4  0.558  0.012   0.585  0   0.0425
3
10

X A   X A  t  E  t  dt  0.35  0.04  X A  0.39
0
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-16
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min 0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02
0.1
0.16
0.2
0.16
0.12
0.08
0.06
0.044
0.03
0.012
0
XA
0 0.137 0.23 0.298 0.35 0.39 0.428 0.458 0.483 0.505 0.525 0.558 0.585
Alternative approach: segregation model by Polymath:

X A   X A  t  E  t  dt
0
dXA
 XA  t  E  t 
dt
XA  1 
Need an equation for E(t)
1
1  2kCB02t
k=176
CB0=0.0313
Use Polymath to fit the E(t) vs t data in the table to a polynomial
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
time
L23b-17
E(t)
For the irreversible, liquidphase, nonelementary rxn
A+B→C+D, -rA=kCACB2
Calculate the XA using the
complete segregation model
using Polymath
Gave best fit
E(t) = 0 at t=0
Model: C02= a1*C01 + a2*C01^2 + a3*C01^3 + a4*C01^4
a1=0.0889237
a2= -0.0157181
a3= 0.0007926
a4= -8.63E-06
Final Equation: E= 0.0889237*t -0.0157181*t2 + 0.0007926*t3 – 8.63E-6*t4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
Complete segregation model by Polymath
L23b-18
A+B→C+D
-rA=kCACB2
Segregation model by Polymath: XA  0.36
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-19
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, nonelementary rxn A+B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
Maximum mixedness model: dXA  rA  E    X
d
CA0 1  F    A
rA  kCA0CB02 1  XA 
3
=time
0
dF
E
d
F() is a cumulative distribution function
L2
CA0  CB0  0.0313mol L k  176
mol2  min
Polymath cannot solve because →0 (needs to increase)
Substitute  for z, where z=T̅- where T̅=longest time interval (14 min)
E must be in terms of T̅-z.
 rA
 dF
E T  z
dX A
 

XA 
 E  T  z  Since T̅-z= & =t, simply


dz
 CA0 1  F  T  z 
 dz
substitute  for t
E()= 0.0889237*-0.0157181*2 + 0.0007926*3 – 8.63E-6*4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
Maximum Mixedness Model, nonelementary reaction A+B→C+D
-rA=kCACB2
dF
 E  T  z 
dz
L23b-20
 rA

E T  z
dX A
 

XA 


dz
 CA0 1  F  T  z 

Denominator
cannot = 0
z  T   Tz
Eq for E describes RTD function only on
interval t= 0 to 14 minutes, otherwise E=0
XA, maximum mixedness = 0.347
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-21
For a pulse tracer expt, C(t) & E(t) are given in the table below. The irreversible, liquidphase, nonelementary rxn A+B→C+D, -rA=kCACB2 will be carried out in this reactor.
Calculate the conversion for the complete segregation model under adiabatic conditions
with T0= 288K, CA0=CB0=0.0313 mol/L, k=176 L2/mol2·min at 320K, H°RX=-40000
cal/mol, E/R =3600K, CPA=CPB=20cal/mol·K & CPC=CPD=30 cal/mol·K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
dXA
dX A
3
2

X
t
E
t

kC
1

X






Polymath eqs for segregation model:
A
B0
A
dt
dt
E(t)= 0.0889237*t -0.0157181*t2 + 0.0007926*t3 – 8.63E-6*t4
L2
1 

 1
exp
3600K




 320K T  

mol2  min
Need equations from energy balance. For adiabatic operation:
Express k as
function of T:
k  T   176
n
T
 H RX  TR   X A   iCp T0  X A CPTR
i


i1
n


C

X

C

i pi
A
P

i1

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-22
For a pulse tracer expt, C(t) & E(t) are given in the table below. The irreversible, liquidphase, nonelementary rxn A+B→C+D, -rA=kCACB2 will be carried out in this reactor.
Calculate the conversion for the complete segregation model under adiabatic conditions
with T0= 288K, CA0=CB0=0.0313 mol/L, k=176 L2/mol2·min at 320K, H°RX=-40000
cal/mol, E/R =3600K, CPA=CPB=20cal/mol·K & CPC=CPD=30 cal/mol·K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
n
Energy balance for
T
adiabatic operation:
n
 H RX  TR   X A   iCp T0  X A CPTR
i


i1
cal
 iCpi  CpA  CPB  40
mol  K
i1
cal
cal
X A  576
mol
mol
 T
cal
 cal 
2
 XA 

mol  K
 mol  K 
1702
n


C

X

C

i pi
A
P

i1

Cp   30  30  20  20 
dXA
 XA  t  E  t 
dt
k  T   176
Not zero!
cal
cal
 20
mol  K
mol  K
dX A
3
 kCB02 1  X A 
dt
L2
1 

 1
exp
3600K




 320K T  

mol2  min
E(t)= 0.0889237*t -0.0157181*t2 + 0.0007926*t3 – 8.63E-6*t4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
Segregation model, adiabatic operation, nonelementary reaction kinetics
L23b-23
A+B→C+D
-rA=kCACB2
XA  0.93
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-24
The following slides show how the same problem would be solved and
the solutions would differ if the reaction rate was still -rA=kCACB2 but the
reaction was instead elementary: A+2B→C+D
These slides may be provided as an extra example problem that the
students may study on there own if time does not permit doing it in class.
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-25
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, elementary rxn A+2B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
Start with PFR design eq dX A rA
dX A kC A CB2



& see how far can we get: dV F
dV
C A0u0
A0
CA  CA0 1  X A 
0
dX A kC A CB2


d
C A0
b 2
b = =  CB  CB0 1  2XA 
a 1
2
dX A kCA0CB0 1  X A 1  2X A 


d
CA0
2

dX A
2
 kCB02 1  X A 1  2X A 
d
CB0 = 0.0313
k = 0.0313
Could solve with Polymath if we knew the value of 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-26
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, elementary rxn A+2B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02 0.1
0.16 0.2
0.16
0.12
0.08
0.06 0.044 0.03 0.012
0
t*E(t)
0
0.02 0.2
0.48 0.8
0.8
0.72
0.56
0.48 0.396
0
dX A
2
 kCB02 1  X A 1  2X A 
d
Use numerical method
to determine tm:
How do we
determine ?
0.3
0.144
For an ideal reactor,  = tm
tm  0 tE  t  dt

10
14
0
0
10
tm   tE  t  dt   tE  t  dt   tE  t  dt
1 0  4  0.02   2  0.2   4  0.48   2  0.8   4  0.8  
 tE  t  dt  
  4.57
3  2  0.72   4  0.56   2  0.48   4  0.396   0.3 
0
10
14
 tE  t  dt 
10
2
0.3  4  0.144   0   0.584
3
 tm  4.57  0.584  5.15min
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-27
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, elementary rxn A+2B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.02 0.1
0.16 0.2
0.16
0.12
0.08
0.06 0.044 0.03 0.012
0
t*E(t)
0
0.02 0.2
0.48 0.8
0.8
0.72
0.56
0.48 0.396
0
dX A
2
 kCB02 1  X A 1  2X A 
d
0.3
0.144
For an ideal reactor,  = tm
tm  0 tE  t  dt
tm  5.15min  
Final XA
corresponds to
=5.15 min
X A,PFR  0.29
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-28
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, elementary rxn A+2B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
Segregation model X   X t E t dt  dXA  X t E t
 A   
A   
A
with Polymath:
dt
0
0
XA(t) is from
batch reactor
design eq
Batch reactor
dX A
2
2
dX A

N

k
C
C
1

X
1

2X
V



NA0
 rA V
A0
A0 B0
A
A
design eq:
dt
dt
NA0  CA0 V
dX A
2
Stoichiometry:

 kCB02 1  X A 1  2X A 
dt
rA  kCA CB2
Best-fit polynomial line
k=176
CB0=0.0313
CA  CA0 1  X A 
for E(t) vs t calculated
by Polymath (slide 19)
CB  CB0 1  2X A 
2
3
4
E(t)=
0.0889237*t
-0.0157181*t
+ 0.0007926*t
– 8.63E-6*t
Slides courtesy of Prof M L Kraft,
Chemical
& Biomolecular
Engr Dept, University
of Illinois, Urbana-Champaign.
Segregation model, isothermal operation, elementary rxn: A+2B→C+D
L23b-29
XA,seg  0.27
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-30
For a pulse tracer expt, the effluent concentration C(t) & RTD function E(t) are given
in the table below. The irreversible, liquid-phase, elementary rxn A+2B→C+D,
-rA=kCACB2 will be carried out isothermally at 320K in this reactor. Calculate the
conversion for an ideal PFR, the complete segregation model and maximum mixedness
model. CA0=CB0=0.0313 mol/L & k=176 L2/mol2·min at 320K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
dF
Maximum mixedness model: dXA  rA  E    X
E
=time
A
d

d
CA0 1  F   
Polymath cannot solve r  kC C 2 1  X 1  2X 2
L2
A
A0 B0 
A 
A
because →0 (must
k  176
mol2  min
CA0  CB0  0.0313mol L
increase)
Substitute  for z, where z=T̅- where T̅=longest time interval (14 min)
E must be in terms of T̅-z.
 rA
 dF
E T  z
dX A
 

XA 
 E  T  z  Since T̅-z= & =t, simply


dz
substitute  for t
 CA0 1  F  T  z 
 dz
E()= 0.0889237*-0.0157181*2 + 0.0007926*3 – 8.63E-6*4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
Maximum Mixedness Model, elementary reaction A+2B→C+D, -rA=kCACB2L23b-31
dF
 E  T  z 
dz
 rA

E T  z
dX A
 

X
 CA0 1  F  T  z  A 
dz


Denominator
cannot = 0
z  T   Tz
Eq for E describes RTD function only on
interval t= 0 to 14 minutes, otherwise E=0
XA, maximum mixedness = 0.25
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-32
For a pulse tracer expt, C(t) & E(t) are given in the table below. The irreversible, liquidphase, elementary rxn A+2B→C+D, -rA=kCACB2 will be carried out in this reactor.
Calculate the conversion for the complete segregation model under adiabatic conditions
with T0= 288K, CA0=CB0=0.0313 mol/L, k=176 L2/mol2·min at 320K, H°RX=-40000
cal/mol, E/R =3600K, CPA=CPB=20cal/mol·K & CPC=CPD=30 cal/mol·K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
Polymath eqs for
dXA
dX A
2
2

X
t
E
t

kC
1

X
1

2X







A
B0
A
A
segregation model: dt
dt
E(t)= 0.0889237*t -0.0157181*t2 + 0.0007926*t3 – 8.63E-6*t4
Express k as
function of T:
k  T   176
L2
1 

 1
exp
3600K




 320K T  

mol2  min
Need equations from energy balance. For adiabatic operation:
n
T
 H RX  TR   X A   iCp T0  X A CPTR
i


i1
n


C

X

C

i pi
A
P

i1

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
L23b-33
For a pulse tracer expt, C(t) & E(t) are given in the table below. The irreversible, liquidphase, elementary rxn A+2B→C+D, -rA=kCACB2 will be carried out in this reactor.
Calculate the conversion for the complete segregation model under adiabatic conditions
with T0= 288K, CA0=CB0=0.0313 mol/L, k=176 L2/mol2·min at 320K, H°RX=-40000
cal/mol, E/R =3600K, CPA=CPB=20cal/mol·K & CPC=CPD=30 cal/mol·K
t min
0
1
2
3
4
5
6
7
8
9
10
12
14
C
g/m3
0
1
5
8
10
8
6
4
3
2.2
1.5
0.6
0
E(t)
0
0.16
0.12
0.08
0.02 0.1
0.16 0.2
0.06 0.044 0.03 0.012
0
Adiabatic EB:
n
 H RX  TR   X A   iCp T0  X A CPTR Cp   30  30  2  20   20   0
i


i

1
T
n
cal
n


 iCpi  CpA  CPB  40
mol  K
  iCpi  X A CP 
i1
i1

T  288K  1000X A
dXA
 XA  t  E  t 
dt
k  T   176
dX A
2
 kCB02 1  X A 1  2X A 
dt
L2
1 

 1
exp
3600K




 320K T  

mol2  min
E(t)= 0.0889237*t -0.0157181*t2 + 0.0007926*t3 – 8.63E-6*t4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.
Segregation model, adiabatic operation, elementary reaction kinetics
L23b-34
A+2B→C+D
-rA=kCACB2
Because B is completely
consumed by XA=0.5
X̅A = 0.50
Why so much lower
than before?
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois, Urbana-Champaign.