CHEE 321: Chemical Reaction Engineering
Module 1: Mole Balances, Conversion & Reactor Sizing
(Chapters 1 & 2, Fogler)
Module 1: Mole Balances, Conversion & Reactor Sizing
• Topics to be covered:
– Basic elements of reactor design, terminology/notation
– Development of general mole balance equation with
reaction
– Key characteristics and mole balance equations for common
industrial reactors (batch, CSTR, PFR, PBR)
– Reactor design for single-reaction systems
• Definition of conversion
• Levenspiel Plots
Basic Elements of Reactor Design
• Reactor design usually involves the following:
– Knowledge of nature of reaction
• Catalytic or Non-Catalytic
• Homogeneous or Heterogeneous
• Reversible or Irreversible
– Selection of operating conditions
• Temperature, Pressure, Concentrations
• Type of catalyst (if applicable)
• Flow rates
– Selection of reactor type for a given application
– Estimation of reactor volume required to process given amount
(moles or molar rate) of raw material to desired amount of products
• How fast the reaction occurs (reaction rates) dictates how large the
reactor volume will be
Our approach to reactor design
• Operation of most reactors are relatively complex
– Temperature is not uniform and/or constant
– Multiple reactions can occur
– Flow patterns are complex
• To gain an insight into basic concepts relevant to reactor
design, we will first consider simplified and/or ideal
reactor systems.
• Design of isothermal reactors involves solution of MOLE
BALANCE equation only
– In some cases, pressure drop must also be calculated
• Let us first familiarize ourselves with some common
terminology and notation that we will be using throughout
the course
Ethylene Æ Low Density Polyethylene
Monomer
Feed
Initiator
Feed
- A multi-zone Autoclave is a vertical cylindrical
vessel with large L/D of 10-20. The reaction
mixture is intensely mixed by a stirrer shaft.
- Reactor is divided into separated reaction zones.
- Reactor (and each zone) is considered as wellmixed CSTR in perfect mixing model approach.
- Imperfect mixing of initiator feed can occur due
to very fast initiator decomposition rate, despite
intense agitation
Reversible and Irreversible Reactions
Irreversible Reactions: Reactions that proceed unidirectionally under the conditions of interest
CH4 + 2O2 Æ CO2+2H2O
Reversible Reactions: Reactions that proceed in both
forward and reverse directions under conditions of interest.
SO2 + 0.5 O2 ⇄ SO3
H2S⇄ H2 + 1/xSx
Thermodynamics tells us that all reactions are reversible.
However, in many cases the reactor is operated such that the
rate of the reverse reaction can be considered negligible.
Homogeneous & Heterogeneous Reactions
Homogeneous Reactions: reactions that occur in a
single-phase (gas or liquid)
NOx formation
NO (g) + 0.5 O2 (g) ↔ NO2 (g)
Ethylene Production
C2H6 (g) ↔ C2H4 (g) + H2 (g)
Heterogeneous Reactions: reactions that require the presence
of two distinct phases
Coal combustion
C (s) + O2 (g) ↔ CO2 (g)
SO3(for sulphuric acid production)
SO2 (g) + 1/2 O2 (g) ↔ SO3 (g)
Vanadium catalyst (s)
Material Balances: It all starts from here!
System with Rxn: use mole balances
Output Rate
Input Rate
Rate of INPUT – Rate of OUTPUT + Rate of GENERATION – Rate of CONSUMPTION
= Rate of ACCUMULATION
Note: Rates refer to molar rates (moles per unit time).
Before we get into the details of the mole balance equation, we must introduce
definition for reaction rate as well as associated notation.
Notation: Reaction Rate for Homogeneous Reactions
(– rA) = rate of consumption of species A (A is a reactant)
= moles of A consumed per unit volume per unit time
A+BÆC
(rA)
= rate of formation of species A (A is a product)
DÆA
Units of (rA) or (– rA)
• moles per unit volume per unit time
• mol/L-s or kmol/m3-s
Notation: Reaction Rate for Heterogeneous Reactions
For a heterogeneous reaction, rate of consumption of species
A is denoted as (-rA')
Heterogeneous reactions of interest are primarily catalytic in
nature. Consequently, the rates are defined in term of mass of
catalyst present
Units of (-rA')
•mol per unit time per mass of catalyst
•mol/(g cat)-s or kmol/(kg cat)-h
Reaction Rate and Rate Law
Reaction Rate
• Rate of reaction of a chemical species will depend on the local conditions
(concentration, temperature) in a chemical reactor
Rate Law
• Rate law is an algebraic equation (constitutive relationship) that relates reaction
rate to species concentrations.
• Rate law is independent of reactor type
(-rA) = k ·[concentration terms]
e.g. (-rA) = k CA
or (-rA) = k CA2
where, k is the rate coefficient [k=f(T)]
Note: a more appropriate description
of functionality should be in terms of
“activities” rather than concentration.
We’ll learn more about rate laws in Modules 2 and 4.
General Mole Balance Equation (GMBE)
General mole balance equation is the foundation of reactor design.
Rate of INPUT – Rate of OUTPUT + Rate of GENERATION/CONSUMPTION =
Rate of ACCUMULATION
−
FA0
All terms
with units
of mol/s
GA
FA
+ GA
dN A
dt
=
FA
FA0
System volume V
= (rate of generation of A) · V
V
= ∫ rA dV ′
If A is consumed, add a –ve sign
Need to integrate over
reactor volume, as reaction
conditions (T, CA ) may vary
with position
Common Reactor Types
• Batch Reactor
• Flow Reactors
– Continuous-Stirred Tank Reactor (CSTR)
– Plug Flow Reactor (PFR)
– Packed Bed Reactor (PBR)
• Other Reactor Types
– Semibatch Reactors
– Fluidized Bed Reactor, Trickle Bed Reactor, Membrane
Reactor, …
Batch Reactor
Key Characteristics
• No inflow or outflow of material
• Unsteady-state operation (by definition)
• Mainly used to produce low-volume high-value
products (e.g., pharmaceuticals)
• Often used for product development
• Mainly (not exclusively) used for liquid-phase
reactions
• Charging (filling/heating the reactor) and cleanout (emptying and cleaning) times can be large
For an ideal batch reactor, we assume no
spatial variation of concentration or temperature.
i.e.; lumped parameter system (well-mixed)
General Mole Balance for an Batch Reactor
Input = Output = 0
V
dN A
= ∫ rA dV ′
dt
If well-mixed (no temperature or
concentration gradients in reactor):
differential form
integral form
dN A
= rAV ;
dt
N A = N A0 at t = 0
dN A′
t= ∫
rV
N A0 A
NA
Class exercise:
Derive concentration vs. t profiles for A and B for AÆB with rB=-rA=kCA
for a well-mixed constant-volume isothermal batch reactor.
At t=0, CA=CA0 and CB=0
General Mole Balance for Ideal CSTR at Steady-State
FA0=v0CA0
FA=vCA
NA=VCA
CSTRs are also known as “backmix” reactors, as concentrations
in the outlet stream are the same
as concentrations in the reactor (a
consequence of being well-mixed)
v0, v= volumetric flowrates (L/min, m3/s) of inlet and exit;
if at steady-state and constant density, v0 = v
Average residence or space time of fluid in vessel based on
inlet conditions τ = V/v0
Class exercise:
Derive expressions for concentration of A and B for AÆB with rB=-rA=kCA for
a well-mixed steady-state CSTR with inlet concentrations CA=CA0 and CB=0,
assuming no density change.
General Mole Balance for Ideal CSTR at Steady-State
FA0=v0CA0
FA=vCA
NA=VCA
CSTRs are also known as “backmix” reactors, as concentrations
in the outlet stream are the same
as concentrations in the reactor (a
consequence of being well-mixed)
v0, v= volumetric flowrates (L/min, m3/s) of inlet and exit;
if at steady-state and constant density, v0 = v
Average residence or space time of fluid in vessel based on
inlet conditions τ = V/v0
Class exercise:
Derive expressions for concentration of A and B for AÆB with rB=-rA=kCA for
a well-mixed steady-state CSTR with inlet concentrations CA=CA0 and CB=0,
assuming no density change.
Class Problem
Calculating Reaction Rate in a CSTR
1.0 L/min of liquid containing A and B (CA0=0.10 mol/L, CB0=0.01 mol/L)
flow into a mixed flow reactor of volume 1.0 L. The materials in the reactor
interact (react) in a complex manner for which the stoichiometry is unknown.
The outlet stream from the reactor contains A, B and C at concentrations
of CA= 0.02 mol/L, CB=0.03 mol/L and CC=0.04 mol/L.
Find the rate of reactions of A, B and C at conditions of
the reactor.
Plug Flow Reactor (PFR)
V
dN A
= FA0 − FA + ∫ rA dV ′
dt
Key Characteristics
• Generally a long cylindrical pipe with no moving parts (tubular reactor)
• Suitable for fast reactions (good heat removal), mainly used for gas phase
systems
• Concentrations vary along the length of the tube (axial direction)
For an ideal PFR, we assume:
- constant flowrate
- no variation of fluid velocity or species concentration in radial direction
We also generally assume reactor is operating at steady-state: i.e.; no variation in
properties with time at any position along reactor length
General Mole Balance for Ideal PFR at Steady-State
integral form
dFA′
V= ∫
rA
FA 0
FA
FA V
ΔV
Infinitesimally small
control volume
FA V +ΔV
At steady state: FA V − FA V +ΔV + rA ΔV = 0
differential form
dFA
= rA ;
dV
FA = FA0 at V = 0
Class exercise:
Derive concentration profiles for A and B for AÆB with rB=-rA=kCA for a
isothermal PFR at steady-state, assuming constant volumetric flowrate.
At the reactor inlet, CA=CA0 and CB=0
Packed Bed Reactor (PBR)
FA0
FA
Key Characteristics
• Can be thought of as PFR packed with solid particles, usually
some sort of catalyst material.
• Mainly used for gas phase catalytic reaction although examples for
liquid-phase reaction are also known.
• Pressure drop across the packed bed is an important consideration.
• This is the reactor type used in your integrated design project.
Mole Balance for PBR
FA0
FA
Let W = Weight of the packing
Making the same assumptions as for a PFR:
- no variation of fluid velocity or species concentration in radial direction
- operating at steady-state
differential form
integral form
dFA
= rA′ ;
dW
FA = FA0 at W = 0
FA
W=
∫
FA 0
dFA′
rA′
Same as PFR, but with rate (r ) specified per mass of
catalyst (instead of per unit volume) and using catalyst wt
(W) instead of V as the coordinate
Summary - Design Equations of Ideal Reactors
Differential
Equation
Batch
(well-mixed)
dN j
dt
Algebraic
Equation
t=
V=
(well-mixed
at steady-state)
dF j
(steady-state flow;
well-mixed radially)
dV
∫
N j0
CSTR
= rj
dN ′j
Nj
= (rj )V
PFR
Integral
Equation
(rj )V
Remarks
Conc. changes with time
but is uniform within the
reactor. Reaction rate
varies with time.
Conc. inside reactor is
uniform. (rj) is constant.
Exit conc = conc inside
reactor.
Fj 0 − Fj
− ( rj )
Fj
V=
∫
Fj 0
dFj′
( rj )
Concentration and hence
reaction rates vary
spatially (with length).
Human Body as a System of Reactors
Food
Small Intestine
Mouth
Large
Intestine
Stomach
What reactor type can we represent the various body parts with?
• We can often approximate behaviour of complex reactor
systems by considering combinations of these basic reactor
types (batch, PFR, CSTR)
• Next step (Fogler Ch 2):
¾Formulate design equations in terms of conversion
¾Apply to reactor sizing
Approaches in modeling imperfect mixing in LDPE Autoclave Reactors
Flow of initiator and
side feed
Compartments model
V1
Each reaction zone is considered as a
set of interconnected three CSTR’s.
Recycle flow models the effect of
imperfect mixing in initiator
injection point and backmixing in
the reaction space. The model
parameters are based on geometrical
and flow dynamic of the industrial
system.
Recycle
flow
V2
Flow from
previous zone
V3
Monomer
Feed
Initiator
Marini, L., Georgakis, C., AIChE J. 30, 401 (1984).
Feed
CSTR segment
Feed
Recycle to
previous zone
Plug flow
segment
Segments model
Each reaction zone is
considered as a CSTR
section followed by a plug
flow section. The plug flow is
considered as a series of
CSTR’s in series due to
complex mathematical
difficulties. Recycle streams
show the effects of imperfect
mixing.
Recycle from
the next zone
Chan ,W., Gloor, P. E., & Hamielec, A. E.,
AIChE J. 39, 111 (19xx).
Conversion (X) [Single Reaction System]
•
Quantification of how far a reaction has progressed
Continuous (or Flow) Reactors
Xj =
=
•
mols of species-j reacted
mols of species-j fed
Fj 0 − Fj
Fj 0
Batch Reactors
Xj =
=
mols of species-j reacted
mols of initial species-j
N j0 − N j
N j0
Defined in terms of limiting reactant
Reactants Æ Products
Assume “A” is our limiting reactant
aA + bB → cC + dD
b
c
d
A+ B → C+ D
a
a
a
Design Equation in Terms of Conversion
(limiting reactant A)
IDEAL
REACTOR
BATCH
DIFFERENTIAL
FORM
N A0
dX A
= (− rA )V
dt
dX A
FA0
= (− rA )
dV
INTEGRAL
FORM
t = N A0
XA
∫
0
V=
CSTR
PFR
ALGEBRAIC
FORM
dX A′
− rAV
FA0 ( X A )
(− rA )
V = FA0
XA
∫
0
dX A′
− rA
These equations can be used to size reactors
required to achieve a desired conversion for
a single-reaction system
Æ Levenspiel Plots (Fogler, Ch 2.4-2.5)
Octave Levenspiel
considered to be one of the
founders of Chemical Reaction
Engineering
Basic idea: use plot of
FA0
(− rA )
vs. X to calculate V
Plug Flow Reactor (PFR)
FA0
(− rA )
VPFR =
X PFR
∫
0
X
FA0
dX
−rA
XPFR
Continuous Stirred Tank Reactor (CSTR)
FA0
(− rA )
VCSTR
FA0
=[
] ×[ X CSTR ]
(−rA )
Evaluated at X=XCSTR
X
XCSTR
Class Problem
The following reaction is to be carried out isothermally in a
continuous flow reactor operating at steady-state:
A→B
Compare the volumes of CSTR and PFR that are necessary to
consume 90% of A (i.e. CA=0.1 CA0). The entering molar and
volumetric flow rates are 5 mol/h and 0.5 L/h, respectively. The
reaction rate for the reaction follows a first-order rate law:
(-rA) = kCA where, k=0.0001 s-1
For same conversion, is the CSTR volume
always higher than PFR volume ?
For most cases yes, provided that
rA decreases as X increases.
FA0
-rA
See Fogler Section 2.4 (Ex. 2-2
to 2-4) for using Levenspiel plots
to size reactors (PFR vs. CSTR)
X
The real power of Levenspiel
plots is for reactor networks
(reactor in series)
PFR in Series
FA0
FA1
FA2
X=0
X=X2
X=X1
FA3; X=X3
Let us compare two scenarios
(i) Single reactor achieving X3
(ii) 3 reactors in series achieving X3
FA0
-rA
• How is the total volume of 3
reactors in series related to single
reactor ??
X
CSTR in Series
See Fogler 2.5.1
Compare volume for the following 2 cases
FA0
X=0
(i) A single reactor achieving X3
(ii) 3 reactors in series achieving X3
FA1
X=X1
FA2
X=X2
FA3; X=X3
If you could replace one of the
CSTRs with a PFR, which one would
you choose to minimize the total
volume of the reactor system
We can model a PFR as a series of
“n” equal volume CSTRs
How is the total volume of 3 reactors in
series related to single reactor ??
FA0
-rA
X
Closing Thoughts on Levenspiel Plots
• Levenspiel Plots are useful means to illustrate the difference
between PFR and CSTR behavior
– If the rate law is given in terms of conversion (-rA) = f(X) or can be
generated/derived by intermediate calculations, one can size PFR,
CSTRs, and batch reactors.
– PFR can be modeled as many CSTR in series (strategy used in
UNISIM design software)
• Levenspiel plots are seldom used to design ‘real world’
reactors
– Restrictive conditions: no secondary side streams, single reaction
– Can only be used for scale-up if reaction conditions are kept identical: i.e.;
(-rA) varies with conversion identically in the full-size reactors as in the lab
Class Problem
The following aqueous-phase reaction is carried out isothermally in both a
laboratory-scale reactor and an industrial-scale continuous stirred tank reactor:
A→B
The reaction rate follows a first-order rate law:
(-rA) = kCA where, k=0.1 min-1 at 50 oC
The operating conditions of the reactors are provided below
Industrial CSTR
Lab CSTR
Feed concentration
10%A in solution
2% A in solution
Reactor volume
3600 L
63 mL
Volumetric flow rate
40 L/min
0.7 mL/min
The two reactors, vastly different in scale, and with different feed concentrations
yield similar conversion. Why?
Time is of the Essence
• The extent of conversion of reactants in a chemical reactor
is related to the time the chemical species spend in the
reactor.
• Remember the definition: Average residence or space time
of fluid in vessel is τ = V/v0
– Space time is often used as a scaling parameter in reactor design
• Residence time is chosen to achieve desired conversion
(different for PFR and CSTR!), and can vary from a few
seconds to several hours, depending on the rate of reaction
– See Fogler Table 2.5
Reaction Rates of Some Known Systems
Slow reaction
(requires large residence time)
Fast Reaction
(short residence time)