# Mole balance for chemical reaction engineering

```Mole balance for chemical
reaction engineering
(Design Equations for reactors)
Lec 3 week 3
The General Mole Balance Equation
• To perform a mole balance on any system, the system
boundaries must first be specified. The volume enclosed by
these boundaries is referred to as the system volume.
where Nj , represents the number of moles of species j in the system at time t.
• The rate of generation of species j is expressed as The product of the reaction
term and can be written in more familiar terms,
GA = rA V
• V is volume of the system.
• Note that the units for this relation are consistent:
mass
mass

 volume
time volume.time
• If GA (and hence rA) varies with position in the
system volume, we can take this into account by
evaluating this term at several locations. Then
DGA1 = rA1 DV1,
• Summing the reactions over the entire volume yields:
k
k
i 1
i 1
G A   DG Ai   rAi DVi
k 
• As
(that is, as we decrease the size of these cubes and
increase their number) then
DV  0 which gives
V
GA 
r
A
dV
we now replace G in equation (1) to get :
• From this general mole balance equation we can
develop the design equations for the various types of
industrial reactors: batch, semi-batch. and
continuous- flow.
Types of Reactors
• Batch
– No flow of material in or out of reactor
– Changes with time
• Fed- Batch (semi batch)
– Either an inflow or an outflow of material but not both
– Changes with time
• Continuous
– Flow in and out of reactor
– Continuous Stirred Tank Reactor (CSTR)
– Plug Flow Reactor (PFR)
– Steady State Operation ( no change with time)
General Mole Balance in terms of
number of moles
Batch reactor mole balance
• Generalized Design Equation
for Reactors
• No flow into or out of the
reactor, then, FA = FA0 = 0
• Good mixing, constant
volume
dN A
 rAV
dt
Batch Reactor
Batch Reactor
Fed Batch Reactor
• Reactor Design Equation
V
FA0  FA  
dN A
rA dV 
dt
• No outflow FA = 0
• Good Mixing rA dV term
out of the integral
dN A
FA0  rA  V 
dt
Continuous Stirred Tank Reactor
CSTR
• General Reactor Design Equation
V
FA0  FA  
dN A
rA dV 
dt
• Well Mixed
• So
V

dN A
0
dt
rA dV  VrA
FA0  FA  VrA  0
or
FA0  FA
V
 rA
CSTR
Tubular Reactor (Plug Flow Reactor) (PFR)
• Tubular Reactor
• Pipe through which
fluid flows and reacts.
• Poor mixing
• Difficult to control
temperature variations.
simplicity of
construction
PFR Design Equation
• Design Equation
V
FA0  FA  
dN A
rA dV 
dt
• Examine a small volume element (DV) with length
Dy and the same radius as the entire pipe.
Flow of A
into
Element
Flow of A
out of
Element
• If the element is small, then spatial variations in rA
are negligible, and
Assumption of “good mixing”
V

rA dV  rA DV
applies only to the small
volume element
• If volume element is very small, then assume steady
state with no changes in the concentration of A.
dN A
0
dt
• Simplify design equation to:
FA  y   FA  y  Dy   rADV  0
• rA is a function of position y, down the length of the
pipe and reactant concentration
• take the limit where the size of a volume element
becomes infinitesimally small
dFA
 rA
lim
Dv 0 dV
This is the Design Equation for a PFR
• take the limit where the size of a volume
element becomes infinitesimally small
dFA
 rA
lim
Dv 0 dV
• or because Dy A = V,
dFA
 rA
dV
• This is the Design Equation for a PFR
• Bioapplications - Sometimes hollow fiber
reactor analysis is simplified to a PFR
Plug Flow Reactor Mole Balance
PFR:
The integral form is:
dFA
FA 0 r
A
V
FA
This is the volume necessary to reduce the entering molar flow rate (mol/s) from FA0
to the exit molar flow rate of FA.
Packed Bed Reactor
PBR
FA0 FA 
 rdW  dNdt
A
A
The integral form to find the catalyst weight is:
dFA
FA 0 rA
W
FA
Reactor Mole Balance Summary
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