L4-1
Review: Design Eq & Conversion
b
c
d
A  B  C  D
a
a
a
BATCH
SYSTEM:
XA
N j  N j0   jNA0 X A
Ideal Batch Reactor
Design Eq with XA:
FLOW
SYSTEM:
j≡ stoichiometric coefficient;
positive for products, negative
for reactants
moles A reacted

moles A fed
NT   N j  NT0
j
XA
dX A
NA 0
 rA V
dt
Fj  Fj0   jFA0 X A



   j NA 0 X A
 j 
t  NA 0 
0
FT   Fj  FT0
j
Ideal CSTR
Design Eq
with XA:
Ideal SS PFR
Design Eq with XA:
dX A
FA 0
 rA
dV
F X
V  A0 A
 rA
Ideal SS PBR
Design Eq with XA:
dX A
FA 0
 rA '
dW
dX A
 rA V


   j FA 0 X A
 j 
XA
V  FA 0 
0
XA
W  FA 0 
0
dX A
 rA
dX A
 rA '
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-2
Review: Sizing CSTRs
We can determine the volume of the CSTR required to achieve a specific
conversion if we know how the reaction rate rj depends on the conversion Xj
Ideal SS
CSTR
design eq.
Volume is
 FA 0 
FA 0 X A
VCSTR 
 VCSTR  
 X A product of FA0/-rA
 rA
and XA
  rA 
• Plot FA0/-rA vs XA (Levenspiel plot)
• VCSTR is the rectangle with a base of XA,exit and a height of FA0/-rA at XA,exit
Area = Volume of CSTR
FA 0 
V
  X1
rA X
1
FA 0
rA
X
X1

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-3
Review: Sizing PFRs & PBRs
We can determine the volume (catalyst weight) of a PFR (PBR) required to
achieve a specific Xj if we know how the reaction rate rj depends on Xj
X A,exit
X A,exit
Ideal PFR
 FA 0 
dX A
dX A
V

F

V


 
PFR
A0
PFR
design eq.
 rA
  rA 
0
0
Ideal PBR
design eq.
WPBR  FA 0
X A,exit

0
dX A
 WPBR 
 rA
X A,exit

0
 FA 0 

dX A

r
 A
• Plot FA0/-rA vs XA (Experimentally determined numerical values)
• VPFR (WPBR) is the area under the curve FA0/-rA vs XA,exit
Area == Volume
VPFR or W
Area
ofcatalyst,
PFR PBR
FA 0
rA
X1  F X FA 0  X
1
V  A 0 1dX
d
X


FA 0 
V    0 

dX
W



r



r
A
A
0 
0   rA ' 
X1
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-4
Numerical Evaluation of Integrals (A.4)
Trapezoidal rule (2-point):
X1
h
 f x dx  f X0   f X1
2
0
h  X1  X0
Simpson’s one-third rule (3-point):
X2
h
 f x dx  f X0   4f X1  f X2 
3
0
X 2  X0
h
2
Simpson’s three-eights rule (4-point):
X3
X1  X0  h
X1  X0  h X2  X0  2h
3
 f x dx  hf X0   3f X1  3f X2   f X3 
8
0
X3  X0
h
3
Simpson’s five-point quadrature :
X 4  X0
h
 f x dx  f X0   4f X1  2f X2   4f X3   f X 4  h 
4
3
0
X4
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-5
Review: Reactors in Series
2 CSTRs
FA0
If
is monotonically
- rA
increasing then:
VPFR   VPFR   VCSTR
2 PFRs
i
VCSTR1
VCSTR2
VPFR1
CSTR→PFR
VCSTR1
VPFR2
VPFR2
j
 VPFR   VCSTR VCSTR
i
j
PFR→CSTR
VPFR1
VCSTR2
VCSTR1 + VPFR2
≠
VPFR1 + CCSTR2
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-6
L4: Rate Laws & Stoichiometry
t  NA0
XA
dX A

0 rA V
FA0 X A
V
rA
V  FA0
XA
dX A

0 rA
XA
W  FA0 
0
dX A
rA '
• Reaction Rates (–rA )
1. Concentration
2. Temperature
3. Reversible reactions
• How to derive an equation for –rA [–rA = f(XA)]
1. Relate all rj to Cj
2. Relate all Cj to V or u
3. Relate V or u to XA
4. Put together
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-7
Concentration and Temperature
• Molecular collision frequency  concentration
• Rate of reaction  concentration
• At constant temperature : r = f(CA, CB, …….)
CA : Concentration of A
CB : Concentration of B
• As temperature increases, collision frequency increases
• Rate of reaction = f [( CA, CB, ……), (T)]
-rA  k A  T   f  CA ,CB ,... 
Specific rate of reaction, or rate constant,
for species A is a function of temperature
Reaction rate is a function of temperature and concentration
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-8
Elementary Reactions & Rate Laws
• Dependence of reaction rate –rA on concentration of chemical species in the
reaction is experimentally determined
• Elementary reaction: involves 1 step (only)
• Stoichiometric coefficients in an elementary reaction are identical to the
powers in the rate law:
A  B  C
 rA  k A C A CB 
Reaction order:
•  order with respect to A
•  order with respect to B
• Overall reaction order n = 
Zero order: -rA = kA
k is in units mol/(volume∙time)
1st order: -rA = kACA
k is in units time-1
2nd order: -rA = kACA2
k is in units volume/(mol∙time)
3rd order: -rA = kACA3
k is in units volume2/(mol2∙time)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-9
Overall Stoichiometric Equations
• Overall equations describe the overall reaction stoichiometry
• Reaction order cannot be deduced from overall equations
Examples:
2

r

k
C
2NO  O2  2NO2
NO
NO NO CO2
• This reaction is not elementary, but under some conditions it
follows an elementary rate law
• Forward reaction is 2nd order with respect to NO and 1st order
with respect to O2 (3nd order overall)
Compare the above reaction with the nonelementary reaction
between CO and Cl2
rCO  kCCOCCl 3 2
CO  Cl2  COCl2
2
Forward reaction is 1st order with respect to CO and 3/2 order with
respect to Cl2 (5/2 order overall)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-10
Specific Rate Constant, kA
kA is strongly dependent on temperature
k A  T   AeE RT
Arrhenius Equation
Where :
A = Pre-exponential factor or frequency factor (1/time)
E = Activation energy, J/mol or cal/mol
R = Gas constant, 8.314 J/mol K (or 1.987 cal/mol K)
T = Absolute temperature, K
Taking ln of
both sides:
lnk  ln A 
E 1
 
RT
ln k
-E/R
To determine activation energy E, run
the reaction at several temperatures,
and plot ln k vs 1/T. Slope is –E/R
1/T
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-11
Reversible Reactions
aA  b B
kA
kA
cC  dD
a
b
a
b
Rate of disappearance of A (forward rxn): rfA  k A CA CB  rfA  k A CA CB
c
d
Rate of generation of A (reverse reaction): rbA  k  A CC CD
rA,net  rA  rfA  rbA  rA  k A CA a CBb  k  A CCc CDd
At equilibrium, the reaction rate is zero, rA=0
rA  0  k A C A aCBb  k  A CCc CDd  k A CA a CBb  k  A CCc CDd
Thermodynamic equilibrium relationship
CCc CDd
kA


 KC
a
b
k  A CA CB
KC: concentration equilibrium constant (capital K)
KC is temperature dependent K (T)  K (T )exp  HRX  1  1  

C 1
 T T 
(no change in moles or CP): C
R
 1


HRX: heat of reaction
If KC is known for temperature T1, KC for temperature T can be calculated
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-12
L4: Rate Laws & Stoichiometry
t  NA0
XA
dX A

0 rA V
FA0 X A
V
rA
V  FA0
XA
dX A

0 rA
XA
W  FA0 
0
dX A
rA '
• Reaction Rates (–rA )

1. Concentration

2. Temperature

3. Reversible reactions 
• How to derive an equation for –rA [–rA = f(XA)]
1. Relate all rj to Cj
2. Relate all Cj to V or u
3. Relate V or u to XA (Wednesday)
4. Put together (Wednesday)
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-13
1. Relate all rj to Cj
• rA as a function of Cj is given by the rate law
• The rate relative to other species (rj) is determined by stoichiometry
A
b
c
d
B  C  D
a
a
a
“A” is the limiting reagent
 rA 
r
 rB
r
 C  D
b a c a d a
rj is negative for reactants,
positive for products
In general:
rA 
rj
j≡ stoichiometric coefficient
positive for products, negative for reactants
j
B  
b
a
 A  1  c 
c
a
d 
d
a
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-14
For the reaction 2NO  O2  2NO2 the rate of O2
disappearance is 2 mol/dm3•s (-rO2= 2 mol/dm3•s).
What is the rate of formation of NO2?
Hint:  rA 
rj
j
rNO2 = 4 mol/dm3•s
rO 
2
rNO
2
 2 1


 2 rO  rNO
2
2

mol 
mol
 22

r

4
 rNO
 NO2
3
3
2
 dm  s 
dm  s
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-15
2a. Relate all Cj to V (Batch System)
Reaction rate is a function of Cj:  rA  k A C A CB 
N j mol
Batch: C j 

How is Cj related to V and XA?
V
L
b
c
d
A  B  C  D
N j  N j0   jNA0 X A
a
a
a
Put NA in
NA0  NA0  X A 
C

terms of XA: A
V
b
N

NA0  X A 


B0
Do the same for
NB
a
C


B
species B, C, and D:
V
V
N
CA  A
V
c
NC0    NA0  X A 
N
a
CC  C 
V
V
 d
ND0    NA0  X A 
N
a
CD  D 
V
V
Cj is in terms of XA and V. But what if V varies with XA? That’s step 3a!
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
2a. Additional Variables Used in
Textbook
L4-16
b
c
d
B  C  D
a
a
a
b
NB0    NA0  X A 
N
a
CB  B 
V
V
A
Book uses
term Θi:
i 
Ni0
C
 i0
NA0 CA 0
So species Ni0 can be removed from the equation for Ci

NA0  NB0  b  NA 0

X
 
 

1  NA0  a  NA 0 A 
Multiply numerator by NA0/NA0:  CB 
V
b


NA 0  B  X A 
a

  C  C   b X 
 CB 

B
A0  B
V
a A

Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-17
3a. Relate V to XA (Batch System)
Volume is constant (V = V0) for:
• Most liquid phase reactions
• Gas phase reactions if moles reactants = moles products
CO  g  H2O  g
CO2  g  H2  g 
If the volume varies with time, assume the equation of state for the gas phase:
At time t: PV = ZNTRT and at t=0: P0V0 = Z0NT0RT0
P: total pressure, atm Z: compressibility factor
NT: total moles
T: temperature, K
R: ideal gas constant, 0.08206 dm3∙atm/mol∙K
Want V in terms of XA. First find and expression for V at time t:
ZNTRT
PV
 P0   T  Z  NT  What is
 V  V0    


 NT at t?
P0 V0 Z0NT0RT0
 P   T0  Z0  NT0 
d c b 
NT at time t is: NT   N j  NT0      1 NA0 X A  NT  NT0   NA0 X A
a a a 
j
change in total # moles
d c b 
where  =     1   
Moles A reacted
a a a 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-18
3a. Relate V to XA (continued)
NT  NT0   NA0 XA
change in total # moles
d c b 
where  =     1   
Moles A reacted
a a a 
 P0   T  Z  NT  Can we use the eq. for NT above to
V  V0    


 P   T0  Z0  NT0  find an expression for NT/NT0?
NA0
NT0
NA0
NT
S
ubstitut
e:
y
=
=mole fraction of A initially present


XA
A0
NT0
NT0 NT0
NT0

NT
N
 1   y A0 X A Substitute:  y A0    expansion factor  T  1   X A
NT0
NT0
NT
 P0   T  Z  NT 
Plug :
 1   X A into V  V0    


NT0
 P   T0  Z0  NT0 
 P0   T  Z 
 V  V0    
 1   X A 
 P   T0  Z0 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-19
What is the meaning of ε?
 P0   T  Z 
V  V0    
 1   X A 
 P   T0   Z0 
expansion factor:    y A0
If we put the following
equation in terms of ε:

where
 d c b  NA0
     1
 a a a  NT0
NT
 1   XA
NT0
N  NT0
N  NT0
NT
 1   XA  T
  XA  T

NT0
NT0
NT0 X A
N  NT0 Change in total # moles at X A  1
When conversion is
  Tf

complete (XA=1):
NT0
total moles fed
The expansion factor,, is the fraction of change in V per mol A reacted
that is caused by a change in the total number of moles in the system
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-20
4a. Put it all together (batch reactor)
Batch: C j 
N j  N j0   jNA 0 X A
Nj
V
 P   T  Z 
V  V0  0   
 1   X A 
 P   T0   Z0 
Ni0
 Ci0
V0
N j0   jNA 0 X A
Cj 
V
N j0   jNA0 X A
Cj 

V
 P   T  Z 
V0  0   
 1   X A 
 P   T0  Z0 
Nj
C j0   jCA0 X A  P   T0  Z0 
Cj 
   

1   XA
 P0   T  Z 
For a given XA, we can calculate Cj and plug the Cj into –rA=kCjn
What about flow systems?
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-21
2b. Relate all Cj to u (Flow System)
A
b
c
d
B  C  D
a
a
a
Reaction rate is a
function of Cj:
 rA  k A C A CB 
How is Cj related to uand Xj?
Fj
mol s mol

Flow: C j  
u
Ls
L
Fj  Fj0   jFA0 X A
Put FA in
FA0  FA0  XA 
C 
u terms of XA: A
u
b
F

B0   FA0  X A 
Do the same for
FB
a
C


B
species B, C, and D:
u
u
F
CA  A
c
FC0    FA0  X A 
F
a
CC  C 
u
u
 d
FD0    FA0  X A 
F
a
CD  D 
u
u
We have Cj in terms of XA and u, but what if u varies with XA? That’s step 3b!
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
3b. Relate u to XA (Flow System)
Start with the equation of
state for the gas phase:
L4-22
PV  ZNTRT
N
P
Rearrange to put in terms

 T  CT
of CT, where CT = NT/V:
ZRT
V
Can we relate
FT
FT
P
C



T
CT to u?
u
u ZRT
What is CT0 at the
entrance of the reactor?
Put in terms of u0:
Rearrange to put
 1
 FT ZRT    u
in terms of u:
P
CT0 
FT0
u0
 1
 FT0 Z0RT0    u0
 P0 

P0
Z0RT0
Use these 2 equations to
put uin terms of known or
measurable quantities
FT ZRT 1 P 
 FT  Z  T   P0 
u

 u  u0 

   
FT0 Z0RT0 1 P0  u0
F
Z
 T0  0  T0   P 
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
3b. Relate u to XA (continued)
L4-23
 FT  Z  T   P0  substitute in: F  F   F X and simplify
T
T0
A0 A

   
 FT0  Z0  T0   P 
u  u0 
 FT0   FA0 X A  Z  T   P0 
  FA 0 X A  Z  T   P0 

u

u

   

   
0 1 
F
Z
T
P
F
Z

  0  0   

 0  T0   P 
T0
T0
u  u0 

F
N
N V u0 FA0 
Simplify with: y A0  A0 Because y A0  A0  A0


FT0 
NT0 NT0 V u0 FT0 
 Z  T   P0 
    substitute:  y A0  
 Z0  T0   P 
u  u0 1   y A0 X A  
 Z  T   P0 
 u  u0 1   X A  
   
 Z0  T0   P 
When conversion is
complete (XA=1):

NTf  NT0 Change in total # moles at X A =1

NT0
total moles fed
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-24
4b. Put it all together (flow reactor)
Flow:
Cj 
Fj  Fj0   jFA 0 X A
Fj
u
 P   T  Z 
u  u0  0     1   X A 
 P   T0   Z0 
Fi0
u0
 Ci0
Fj0   jFA 0 X A
Cj 
V
Fj0   jFA 0 X A
Cj  
u
 P   T  Z 
u0  0     1   X A 
 P   T0  Z0 
Fj
C j0   jCA0 X A  P   T0  Z0 
Cj 
   

1   XA
 P0   T  Z 
This is the same equation as that for the batch reactor!
For a given XA, we can calculate Cj and plug the Cj into –rA=kCjn
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.
L4-25
4. Summary: Cj in terms of Xj
Batch:
N j0   jNA0 X A
Cj 

V
 P0   T  Z 
V0    
 1   X A 
P
T
Z
   0  0 
Nj
Nj0
V0
 C j0
C j0   jCA0 XA  P   T0  Z0 
 Cj 
  

1   XA
P
T
Z



 0
Flow:
Fj0   jFA0 X A
Cj  
u
 P   T  Z 
u0 1   X A   0    
 P   T0  Z0 
Fj
Fj0
u0
 C j0
C j0   jCA0 XA  P   T0  Z0 
 Cj 
  

1   XA
 P0   T  Z 
This is the same equation as that for the batch reactor!
For a given XA, we can calculate Cj and plug the Cj into –rA=kCjn
Slides courtesy of Prof M L Kraft, Chemical & Biomolecular Engr Dept, University of Illinois at Urbana-Champaign.