Unit 2
Design Equations for Basic Ideal Reactors
Learning Objectives
General mole balance equations (for species) and
overall mass balance equation
The design (performance) equations for ideal reactors
(batch, PFR and CSTR)
Definitions: conversion, space time and space velocity of
flow reactors
Stoichiometric table analysis
The Mole Balance Equation
System boundary
Fio
Fi
Reactor
Mole (number) balance of species i at any time t
[inflow] - [outflow] + [rate of generation] = [rate of accumulation]
Fi 0 Fi Gi
dN i
dt
3
If the rate of generation varies with location in the reaction
volume V:
Gi dGi ri dV
V
dN i
Fi 0 Fi ri dV
V
dt
The design equations for all types of reactors (batch, semibatch, mixed flow, plug flow) can be derived from this
general relation
There are as many equations as there are species in the
reaction system
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The Mass Balance Equation
A mass balance equation can also be written across the
system boundary. This is an overall (i.e. one) balance
equation and hence should not contain any generation term
(conservation of mass in the absence of relativistic effect)
m 0 m
dM
dt
where m 0 o v 0 and m v
Sometimes the mole balance equation and the mass
balance equation have to be used together for reactor
design
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Ideal Batch Reactor
Fi 0 Fi 0
No inflow and outflow:
Therefore
dN i
Fi 0 Fi ri dV
V
dt
If the reactor contents are well mixed so that composition is
uniform everywhere
r dV ri dV rV
i
V i
or
dN i
rV
i
dt
1 dN i
ri
(differential form of the design equation)
V dt
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The time taken to react species i from Ni0 to Ni:
dN i
t
(integral form of the design equation)
Ni 0 rV
i
Ni
The reaction volume V (the space in which the reaction
occurs) needs not be constant (e.g. the pistons in an internal
combustion engine)
V = constant constant density (volume) system
V constant variable density (volume) system
For a closed system with constant reaction volume V
N i ciV
Since
1 dN i 1 d (ci V ) dci
ri
V dt V
dt
dt
For the more general case where V is not constant
1 dN i 1 d (ci V ) dci ci dV dci
ri
V dt V
dt
dt V dt
dt
Hence a change in concentration is not necessarily indicative of
the occurrence of reactions (a simple but often overlooked truth)
and
dci
the relation ri
is only valid for a constant density system
dt
Since the reaction must be contained by the reactor, the reactor
volume must be equal to or greater than the reaction volume.
dci
Question: For a variable V reacting system, is ri
?
dt
Ideal Mixed Flow Reactor (CSTR)
Assumptions:
Reactor contents are perfectly mixed - no spatial
variations of concentration, temperature or reaction rate
within the reactor
Conditions @ reactor exit = conditions within the reactor
From the general mole balance equation
dN i
Fi 0 Fi ri dV
V
dt
Perfect mixing means V ri dV rV
and the evaluation of this
i
term at the exit conditions
dN i
F
rV
i0
i
i
dt
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CSTR is often run at the steady state (except for start-up,
shut-down or interruptions)
dN i
0 at steady state
Since
dt
or
Fi 0 Fi rV
0
i
Fi Fi 0 Fi
V
ri
ri
This is the design equation for CSTR
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Ideal (Plug Flow) Tubular Reactor
Assumptions
No axial dispersion (back-mixing), concentrations are
uniform over the cross section ideal plug flow
Concentrations vary continuously in the axial direction
from the reactor inlet to the reactor outlet because of
chemical reactions (reaction rates therefore vary axially)
Applying the general mole balance equation for steady
state operations
Fi 0 Fi ri dV 0
V
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Differentiating the equation with respect to V by the Leibnitzs
rule
dFi
ri 0
dV
or
dFi
ri
(differential form of the PFTR design equation)
dV
Fi dF
i
V
(integral form of the PFTR design equation)
Fi 0 r
i
Leibnitzs rule
F ( x)
b( x)
a( x)
f ( z , x)dz
b ( x ) f
dF ( x)
db
da
dz f (b, x) f (a, x)
a ( x ) x
dx
dx
dx
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Constant and Variable Density Systems in
Flow Reactors
For flow systems the reaction volume is fixed and is equal to the
reactor volume. However, the volumetric flow rate through the
reactor is not necessarily constant (consider a gas phase
reaction which changes the total number of moles). The system
is constant density only if the volumetric flow rate is constant
throughout the reactor
M
In batch reactors
hence constant implies constant V
V
(V = V0)
m
In flow reactors
hence constant implies constant v
v
(v = v0)
i.e.
v = vo = constant constant density system
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v constant variable density system
Conversion
The (fractional) conversion of A, xA, for the single reaction
aA bB cC dD
is defined as the number of moles of A reacted per mole of
A delivered to the reactor. i.e.
moles of A reacted
xA
moles of A supplied
For a batch reactor if
N A0 moles of A present initially
N A moles of A present at any time t>0
N A0 N A
xA
N A0
or
N A N A0 (1 x A )
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For a flow reactor (CSTR or PFR),
F FA
x A A0
FA0
or
FA FA0 (1 x A )
Notes
Conversions can be defined with respect to any
reactant and their values can be different
Most meaningful conversion is defined with respect to
the limiting reactant
Conversion has limited utility in the analysis of multiple
reactions
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Stoichiometric Table
Application of stoichiometric table:
To facilitate calculations by:
expressing concentrations of all species in a reactor
(including inert) in terms of a common variable:
conversion (useful for single reactions only)
establishing the relations between concentrations. For
example, for the reaction A B C
c A cB cC constant
Consider
aA bB cC dD
where A is the limiting reactant
Taking A as the basis of calculation and rewriting the
reaction as
b
c
d
A B C D
a
a
a
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If NA0 of A is initially present in the system, NA0xA will be
reacted in time t, leaving NA0 - NA0xA moles of A behind.
We can construct the following table to determine the
numbers of moles of each species left after NA0xA moles of
A have reacted
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where
N N A N B NC N D N I
Define
c d a b
) N A0 xA
N0 (
a
N i0
(i = A,B,C,D)
i
N A0
c d a b
and
N i N A0 (i
i
xA )
A
then
where i = stoichiometric coefficient ( A a, B b, C c, D d )
N N 0 N A0 x A
and
For a constant density batch system where V=V0
N i N i N A0
i
i
(i x A ) c A0 (i x A )
ci
V V0
V0
A
A
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For variable density systems, a relationship between V and
V0 must be sought. For a gaseous system, this can be
obtained from the equation of state (EOS):
PV
At t=0
0 o z0 N 0 RT0
PV zNRT
At t>0
P0 T z N
V V0 ( )( )( )
P T0 z0 N 0
c d a b
Substituting N N 0 N A0 x A ;
a
N
N
1 A0 x A 1 y A0 xA
N0
N0
Define A y A0 (this definition is congruent with that used
by Levensipel prove it yourself). The above equation
simplifies to
P0 T z
N
V
1 A xA ,
( )( )( )(1 A x A )
N0
V0
P T0 z0
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and
N i N i V0 N A0 (i i x A / A ) P T0 z0
1
ci
( )( )( )(
)
V V0 V
V0
P0 T z 1 A x A
P T0 z0 (i i x A / A )
c A0 ( )( )( )
P0 T z
1 A xA
For low pressure operations the effect of z can be neglected
(why?). For negligible pressure drop P=P0. Furthermore if
the system is isothermal T=T0, the concentration expression
is then vastly simplified:
V V0 (1 A x A )
i
c A0 (i x A )
A
ci
1 A xA
This simplified equation only works for constant T,P and z
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The stoichiometric table for a flow system can be similarly
constructed by substituting Fi0 for Ni0, Fi for Ni, and F for N
c d a b
F FA FB FC FD FI F0 (1 A x A ); A y A0 ;
a
i
Fi 0
(i A, B, C , D)
Fi FA0 (i x A ); i
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A
FA0
For constant density systems (v=v0)
Fi Fi FA0
i
i
(i x A ) c A0 (i x A )
ci
v v0
v0
A
A
For variable density systems involving gaseous components
only, the equation of state is changed to Pv zFRT which will
lead to the same concentration expression
P0 T z
v
P T0 z0 (i i x A / A )
( )( )( )(1 A x A ); ci c A0 ( )( )( )
v0
P T0 z0
P0 T z
1 A xA
and its simplification to
v v0 (1 A x A ); ci
c A0 (i
i
xA )
A
1 A xA
For low pressure isothermal systems with negligible pressure
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drop
Short Summary:
For gas phase reactions in batch reactors:
P T0 z0 (i i x A / A )
V V0 (1 A x A ); ci c A0 ( )( )( )
P0 T z
1 A xA
For gas phase reactions in flow reactors:
P T0 z0 (i i x A / A )
v v0 (1 A x A ); ci c A0 ( )( )( )
1 A xA
P0 T z
In both cases A y A0 . The reacting system is therefore
constant density if A = 0, and this is only possible if either or
yA0 is zero.
If 0, A0 if yA0 0 (a forced constant density system which
is useful for the analysis of rate data next Unit)
On the other hand, reactions in condensed phases are natural
constant density reacting systems
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Design Equations in Terms of xA
Batch reactor
or
1 dN A 1 d
rA
[ N A0 (1 x A )]
V dt
V dt
N A0 dx A
(differential form)
V dt
xA
dx A
t N A0
(integral form)
0 ( r )V
A
CSTR
FA FA0 FA0 (1 x A ) FA0 FA0 x A
V
rA
rA
rA
Substituting
FA0 x A c A0 v 0 x A
FA0 c A0 v 0 , V
rA
rA
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PFTR:
dFA
rA
dV
dFA0 (1 x A )
rA
dV
dx A
FA0
rA
dV
or in integrated form
V FA0
xA
x A dx
dx A
A
c A0 v 0
0 r
rA
A
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Space Time
The ratio V / v0 ,where v0 is the volumetric flow rate at the
reactor inlet, is know as the space time (). Space time is
the time necessary to process one reactor volume of fluid
based on the inlet conditions
The design equations for flow reactors are often expressed
in terms of
CSTR:
FA0 x A
cA0 xA
V
v0 v 0 ( rA )
rA
PFTR:
V F
A0
v0
v0
xA
x A dx
dx A
A
cA0
0 r
rA
A
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Space Velocity
Space velocity (SV) is defined as follows
v
SV
V
where v can be the inlet (v0) or exit (v) volumetric flow rate
or their corresponding values at other operating conditions
If v0 is used in the definition of SV
SV
SV is also known as the dilution rate in biochemical
reactors (where v=v0 is almost always valid)
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Commonly Used Space Velocities
Liquid Hourly Space Velocity (LHSV) uses the equivalent
volumetric flow rate of the reactant at a standard
temperature (usually 60F) where the reactant is a liquid
(even though the reactant may be a gas under actual
operating conditions)
Gas Hourly Space Velocity (GHSV) - Here the reactant
gas flow rate to the reactor is converted to an equivalent
volumetric flow rate measured at standard temperature and
pressure (Different industries may have their own
definitions for standard temperature and pressure and it is
always important for an engineer to check the basis of
calculation)
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Residence Time
Residence time (RT) is the actual time spent in the reactor.
For CSTR, since the prevailing flow rate is vexit, the residence
time is
V
RT
v exit
The residence time in a PFTR can be shown to be1
x A dx
A
RT FA0
0 v( r )
A
RT is the same as the space time only for constant density
systems where v=v0=vexit
RT is not a useful parameter for the reactor design
1
This is left as an exercise
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