Chapter 2
Conversion and Reactor sizing
Overview
• In the first chapter the general mole balance
was derived for different reactors
• In this chapter, these equations are used to
size CSTR and PFR using “Conversion”
• Value and overall conversion of CSTR and PFR
arranged in series
2.1 Definition of Conversion
• Conversion is the number of moles of reactant
A (limiting reactant) that has been reacted per
mole of A fed to the system
mole of A reacted
XA 
mole of A fed
• For irreversible XA=1 complete conversion
• For reversible Xmax=Xe equilibrium conversion
2.2 Batch Reactor Design Equation
• After time t, the number of moles of A remaining is
N A  N A0  N A0 X  N A0 (1  X )
d N A0 (1  X )
dN A
dX

 rAV  
 N A0
dt
dt
dt
dX
N A0
 rAV
dt
differential form
• Batch reactor design equation used for reaction rate data
analysis
t
X
X
N A0 dX
dX
0 dt  0  rAV  t  N A0 0  rAV
Integral form
• This equation gives the time required to achieve a specified
conversion X
• The longer the reactants are left in the reactor, the greater the
conversion
2.3 Design equation for flow reactor
• Mole balance for reactant A around the
reactor
• In liquid phase CA0 is the solution molarity
(moles/volume)
FA0  FA  FA0 X  0



A consumed
FA  FA0 (1  X )
FA0  C A00
• In gas phase
C A0
PA0 y A0 P0


RT0
RT0
ideal gas law
CSTR
• The mole balance for CSTR yields
FA0
FA0  FA
V
 rA
FA0  FA0 (1  X ) FA0 X
V

 rA
(rA )exit
FA
• This equation calculates the CSTR volume necessary
to achieve a specified conversion X
• Because of perfect mixing, the exit conc is identical
to the conc inside the reactor and the reaction rate is
evaluated at the exit conditions
Levenspiel CSTR Plot
Volume =
Area of rectangle
2.3.2 Tubular Flow Reactor (PFR)
• No gradient change in T, CA & -Ra
• The reactants are consumed as they enter and flow
axially down the reactor
dFA

  rA
dV
d ( FA0 (1  X ))

  rA
dV
dX
FA0
  rA
dV
V
X
Differential form
of design for PFR
X
FA0 dX
dX
dV


F
0 0  rA A0 0  rA
Integral form
• used to calculate volume required to achieve
specified conversion X
Levenspiel PFR Plot
Volume=
area under the curve
2.3.3 Packed Bed Reactor
• Packed bed reactors are analogous to PFR
• Differential form of the design equation used
to analyze the reactor pressure drop
FA 0
dX
 r ' A
dW
• Integral form used to determine the catalyst
weight in the absence of pressure drop
When P  0 X  0 W  0
X
dX
 r'A
0
W  FA0 
Applications of the design equations
• We can size the reactor from the reaction rate,
as a function of conversion
• For the first order
 r  kC  kC (1 X )
A
A
A0
1
1  1 



 rA kCA0  1  X 
• For irreversible reactions of greater than zero order
X  1,rA  0, thus,
1
  and V  
 rA
• For reversible reactions
1
X  X e ,rA  0, thus,
  and V  
 rA
2.5 Reactors in series
• For reactors in series where no side stream either fed
or withdrawn, the conversion at point i is defined as
Total moles of A reacted up to po int i
Xi 
Moles of A fed to the first reactor
• The molar flow rate at point i is given by
FAi  FA0  FA0 X i
FA0
FA1
i=1
X1
FA2
i=2
X2
FA3
i=3
X3
2.5.1 CSTR in Series
• For 2 CSTR in series
 1
V1  FA0 
  rA
FA0
FA1
i=1
X1

 X 1

FA1  FA2 ( FA0  FA0 X 1 )( FA0  FA0 X 2 )
V2 

 rA2
 rA2
FA0
V2 
( X 2  X1)
 rA2
-rA1
-rA2
FA2 i=2
X2
Levenspiel CSTR Plot
Volume
of CSTR2
Volume
of CSTR1
For the same overall conversion, the total volume for 2
CSTRs in series is less than that required for one CSTR
CSTR and PFR Comparison
• PFR can be modeled with a large number of CSTRs in series. This
concept can be used in
– Catalyst decay in packed bed reactors
– Transient heat effects in PFRs
V5
1
2
3
4
5
V4
V1
V2
V3
1
2
3
4
5
Reactor Mole Balance Summary