16222350

Design of Operable Reactive Distillation
Columns
Myrian Andrea Schenk
A thesis submitted for the degree of Doctor of Philosophy
London.
University
the
of
of
Department of Chemical Engineering
University College London,
London WClE 7JE
September, 1999
It.
Abstract
Reactive distillation is an integrated process which considers simultaneous
physical and chemical transformations. It is increasingly receiving attention
both from industry and academia. Significant advances have been made in
the area of modelling and simulation as well as the implementation of such
industrially.
However,
the area of control and optimisation of such
units
has
been
units
not
explored thoroughly.
The thesis presents a general framework for simulation and design, which can
handle reactive and non-reactive systems. Various different aspects of the
distillation
described
have
been
in
modelling and simulation of
order to understand the behaviour of the reactive distillation columns. In the framework
both simulation modes, steady-state and dynamic, are considered and the
described
by
In
is
a
equilibrium and non equilibrium-based models.
process
(or
liquid
between
transfer
and
non equilibrium) model, mass
rates
rate-based
based
Maxwell-Stefan
the
equaon
vapour phase are considered explicitly,
tions. Equilibrium is attained at the phase interface in the non-equilibrium
from
it
A
to
the
to
go
one model
switching policy makes possible
model.
free
by
following
Gibbs
knowledge
based
the
the
gained,
energy as
on
other,
has
helped
determining
in
Tray
function
the
time.
switch
efficiency
also
of
a
between the non-equilibrium and equilibrium models, and has been studied
for various systems.
The existence of multiple steady state has been verified through simulation
diagrams
Bifurcation
hybrid
the
the
also
confirmed
existence of
model.
with
in
the
simulations.
output multiplicity obtained
Analysis of process controllability at the design stage has been shown to proIn
for
thesis
improving
the
operation.
controllability
process
vide guidance
first
tofor
the
studied
are
presented
as
a
step
systems
reactive
measures
wards control structure selection.
design
for
the
A method
of reactive separation columns at miniobtaining
(investment
and
and
which
will
operating
costs)
total
cost
annualised
mum
is
in
the
be able to maintain stable operation
also
presence of variability
presented.
Acknowledgements
I would like to thank my supervisors David Bogle, Rafiqul Gani and Stratos
Pistikopoulos. Their continuous support and guidance has led to the production of this work. It has been a pleasure to work next to them and I also
thank them for the opportunity of this joint research.
The many technical discussions at ICI, especially with Tom Malik, are also
greatly appreciated.
Financial support from the Centre for Process Systems Engineering, CAPECResearch Center and ICI is gratefully acknowledged.
Special thanks go to Carlos, Nicolas, Sonia, Alejandro and Marta for introducing me to the challenge of research.
It has been a pleasure to work amongst different groups, in Lyngby and here
in London. The time would not have been memorable and enjoyable withlike
have
had
I
I
to
the
thank particularly
would
out
nice people
around.
to: Eduardo, with whom I have worked and discussed a lot about reactive
distillation, and with whom I shared the difficult times far away from home;
Cecilia, my office-mate from the old times, it have been a pleasure to work
discussed
Celeste,
her;
I
I
to
piles of work and with whom
with whom
next
keep my Spanish 'up-to-date'; my part-time flat-mates in the 'London time',
the 'adventure guys', Soraya and Geoff; Vik, my desk neighbour at IRC, who
day.
has
to
the
working
start
a smile ready
always
My dearest friends Marianne, Ari, Stefan and Angelika will always be refor
in
friendship
for
their
their
encouragement good
and
sincere
membered
bad
times.
and
I would also like to thank my family for their love and everlasting confidence
through the years.
Last but not least, I would like to thank Jes, my lovely husband, without
infallible
in
been.
His
have
trust
this
my
support and
may never
whom all
these
has
years.
over
all
me
motivated
work
2
Contents
Introduction
1.1
Reactive Distillation
12
1.2
Motivation
14
1.2.1
1.3
2
3
12
.......................
Aim
Work
the
of
and
................
Problem
Overview
Definition
....................
15
..............................
Review
Literature
Background:
18
2.1
Reactive and Non-reactive Distillation
2.2
Modelling- Design- Synthesis Issues
3.1
3.2
Equilibrium
2.2.2
Non-equilibrium
Equilibrium
21
..........
................
Non-equilibrium Models
.............
3.2.1 Maxwell-Stefan Equations
3.4
Model Formulation
3.5
Equations
28
33
Models
Model Equations
3.7
................
....................
(rate-based) Models
18
21
Issues
3.3
3.6
in Process Engineering
Models
2.2.1
Modelling
15
.................
................
.....................
Generation of Models
...............
3.6.1 Tray Efficiency
..............
Condensers and Reboilers
............
3
........
........
........
........
........
........
........
........
........
........
35
35
36
39
41
42
59
66
69
3.8
Computational and Numerical Aspects
.............
3.8.1 Specifications for Equilibrium and Non-equilibrium Sim-
71
ulations
3.8.2
..........................
Solution of Model Equations
3.8.3
Initialisation
3.8.4
3.9
71
...............
74
........................
Switching between Models and Monitoring of Phenomena 76
Conclusions
76
............................
4 Validation
4.1
of models
Systems Studied
....................
4.1.1 MTBE: Methyl- Tert- B utyl- Ether
4.1.2
78
......
......
Ethyl Acetate
.................
4.1.3 Depropanizer Column
............
4.2 Initialisation
......................
4.3 Validation of Models
.................
4.3.1 MTBE
.....................
4.3.2 Ethyl Acetate
.................
4.3.3 Depropanizer Column ............
4.4
Bifurcation Analysis
4.4.1
4.5
.................
Applied Bifurcation Analysis
Conclusions
......
......
......
......
......
......
......
......
......
.........
......
......
......................
79
79
86
91
91
94
94
102
106
107
109
114
116
5.3
Models
Hybrid
with
Monitoring of Phenomena through Gibbs Free Energy .....
Tray Efficiency Analysis
.....................
Illustration of Hybrid Modelling Approach ...........
5.4
Conclusions
132
5 Analysis
5.1
5.2
6
71
Controllability
6.1
............................
Issues and Dynamic
Controllability
6.1.1
Issues
Linear Model
.......................
.......................
4
Optimization
117
119
128
133
134
135
6.1.2
6.2
6.1.3
..................
Controllability Tools
6.1.4
Control Configurations
6.1.5
Examples of Application
Dynamic Optimization
6.2.1
6.3
7
Scaling
Optimization
Conclusions
.........
..........
.........
.........
.........
........
.........
.............
problem
.........
.........
.........
...................
General Conclusions
.........
and Future Work
137
139
140
143
150
153
159
160
Nomenclature
163
References
167
A
Properties
179
A. 1 Enthalpy
179
A. 2 Density
....................
..........
....................
..........
A. 3 Viscosity
....................
A. 4 Surface Tension
................
A. 5 Thermal Conductivity
............
A. 6 Binary Diffusion Coefficients
.........
..........
..........
..........
..........
180
181
181
182
184
B Results
B. 1 Simulation Results for MTBE
..................
B. 1.1 Controllability Analysis for the MTBE production
...
B. 2 Esterification of Acetic Acid with Ethanol
...........
184
189
203
219
C Optimization
CA
180
Objective Function and Disturbances
5
..............
219
L0ist of Figures
1.1
Exemplary reactive distillation
2.1
Residue curve map for MTBE. (P=Iatm)(Ung
1995b)
...............................
2.2
3.1
4.1
column
16
.............
Doherty,
and
25
McCabe-Thiele like diagram for MTBE. (Perez Cisneros et al.,
1997a)
...............................
26
Schematic representation of an stage in the column (KrishnaTaylor,
1985)
murty and
.....................
40
Multiple solutions in the MTBE column
96
............
Temperature profiles for low and high conversionin the MTBE
column ...............................
4.3 Equilibrium constant for MTBE synthesis ...........
Colombo
Hoffmann,
Rehfinger
Simulation
4.4
et
and
results with
for
MTBE
DIPPR
the
column .............
al. and
for
ASPEN
PLUS
those
Simulation
4.5
of
results compared with
the MTBE column ........................
4.2
4.6
Simulation results with the equilibrium and non-equilibrium
for
MTBE
the
column ...................
model
for
different
thermodynamic
Simulation
models
4.7
results with
the esterification column .....................
for
data
the
Simulation
4.8
results compared with experimental
esterification column .......................
4.9 Mass Transfer rate for the esterification column ........
literature
Comparison
transfer
results ....
4.10
rate with
of mass
6
97
98
99
101
102
103
104
105
106
4.11 Simulation results with the non-equilibrium
propanizer column ........................
for
dethe
model
108
4.12 Vapour compositions compared with ChemSep simulations
109
..
4.13 Bifurcation diagram for the esterification column varying reflux flow rate
...........................
4.14 Bifurcation diagram for the MTBE column varying reflux flow
111
112
rate
................................
4.15 Bifurcation diagram for the MTBE
boil-up
...............................
5.1
column varying vapour
113
Excess Gibbs energy for the esterification column. Trays I
7................................
and
118
5.2 Excess Gibbs energy for the MTBE column. Trays 4 and 10. . 119
5.3
Vapour compositions for the esterification column
.......
5.4
Efficiency
for
the esterification
calculated
5.5
Efficiency
depropanizer
for
the
calculated
5.6
Vapour composition
5.7
Efficiency
calculated
for the depropanizer
for the depropanizer
column
column
120
121
.......
122
.......
123
column
.......
from
far
column
steady-state
............................
5.8 Efficiency calculated for the depropanizer, compared with those
Ethane
ChemSep:
of
.......................
5.9 Efficiency calculated for the depropanizer, compared with those
Propane
ChemSep:
of
.......................
Propane
depropanizer
for
the
Efficiency
5.10 Point
column:
....
fixed
5.11 Liquid composition profile with equilibrium model with
model ..............
efficiency and non-equilibrium
fixed
5.12 Vapour composition profile with equilibrium model with
model .............
.
efficiency and non-equilibrium
5.13 Hybrid
the
to
esterification
model applied
column
......
.
124
125
126
127
129
130
131
(10%
linear
from
the
model
Response
non-linear
and
6.1
obtained
138
step change)............................
in
increase
10%
MTBE
reboiler
6.2 High conversion of
as result of
145
heat duty
..............................
7
6.3
Low conversion of MTBE as a result of 15% decreasein reboiler heat duty
146
...........................
6.4 Lower purity of MTBE as a result of adding extra reactive stages147
6.5 Reactive distillation column for the design study
154
........
6.6 Top and bottom compositions of ethyl acetate. Results from
the simulated nominal casewith disturbance
156
...........
6.7
6.8
Ethyl acetate composition. Results from the dynamic optimization ...............................
Controlled variable: Reboiler Heat Duty(* 10'(MJlh))
Re.
from
dynamic
the
sults
optimization ...............
157
158
B. 1 Decreasing 10% methanol feed flow-rate
........
.....
B. 2 Increasing 10% methanol feed flow-rate
.........
.....
B. 3 Increasing 10% isobutene feed composition
.......
.....
B. 4 Decreasing 15% isobutene feed composition
......
.....
B. 5 Decreasing 10% isobutene feed composition
......
.....
B. 6 Poles and Zeroesfor Case I (MTBE)
.........
.....
B. 7 RGA element (1,1) as a function of the frequency for Case I
(MTBE)
..............................
B. 8 Singular values for Case I(MTBE)
...............
B. 9 Step responsefor Case I (MTBE). (10% step size)
.......
B. 10 Poles and Zeroes for Case 2 (MTBE)
..............
B. 11 RGA element (1,1) as a function of the frequency for Case 2
MTBE
of
.............................
185
B. 12 Step response for Case 2 (MTBE). (10% step size) .......
B. 13 Poles and Zeroes for Case 3 (MTBE)
..............
198
B. 14 RGA element (1, I) as a function of the frequency for Case 3
(MTBE)
..............................
B. 15 Singular values for Case 3 (MTBE) ...............
(10%
(MTBE).
for
Case
3
Step
B. 16
step size) .......
response
B. 17 Poles and Zeroes for Case 1 (Esterification) ...........
8
185
187
187
188
190
191
192
193
195
196
199
200
201
202
204
B. 18 RGA element (1, I) as a function of the frequency for Case I
(Esterification)
....
205
......................
B. 19 Step response for Case I (Esterification).
(10% step size)
...
207
B. 20 Poles and zeroes for Case 2 (Esterification)
...........
B. 21 RGA element (1, I) as a function of the frequency for Case 2
(Esterification)
....
......................
B. 22 Singular values for Case 2 (Esterification)
............
B. 23 Step responses for Case 2 (Esterification). (10% step size)
.
B. 24 Poles and zeros for Case 3 (Esterification)
..........
B. 25 RGA element (1, I) as a function of the frequency for Case 3
(Esterification)
.....................
....
B. 26 Singular values for Case 3 (Esterification)
...........
B. 27 Step responses for Case 3 (Esterification). (10% step size)
.
9
209
210
212
.
.
.
.
.
213
215
216
217
218
List
2.1
2.2
3.1
Tables
of
Thermodynamic equilibrium approaches for the
simulation of
distillation
reactive
columns ...................
Design approaches for reactive distillation columns
......
Thermodynamic
3.2
3.3
List of equations per stage in the non-equilibrium
3.4
Various forms of non-equilibrium
3.5
List of variables per stage in the equilibrium
3.6
List of equations per stage in the equilibrium
3.7
Various forms of equilibrium
3.8
Computational
equations
4.1
4.2
models
model ....
model
...
model
model
61
62
63
............
models ...............
times for solution of different
24
47
models
.....................
List of variables per stage in the non-equilibrium
22
......
......
63
64
65
sets of model
75
.............................
Range of Variables
........................
MTBE Column Design. Data taken from Jacobs and Krishna
(1993) except for the tray design which has been calculated in
this work
..............................
82
83
Modelling alternatives for MTBE system
............
4.4 Ethyl Acetate Column Design. Data from Suzuki et al. (1971)
in
been
has
design
this
the
tray
work.
calculated
which
except
86
Modelling alternatives for Esterification system
........
4.6 Depropanizer Column Details. Data taken from Taylor and
Krishna (1993)
...........................
90
4.3
4.5
10
88
92
4.7
Selected set of numerical statistics for the three distillation
systems ..............................
6.1
Basic control configurations
6.2
Ratio control configurations
6.3
Measured variables for the controllability study on MTBE production
...............................
Controllability indices for the MTBE production
........
Measured variables for the controllability study on ethyl acetate production
.........................
6.4
6.5
6.6
Controllability
6.7
Optimization
6.8
Optimization
94
141
...................
142
...................
indices for the ethyl acetate production
148
148
149
149
....
details
.......................
results: nominal and optimal values ......
11
155
.
159
Chapter
1
Introduction
1.1
Reactive
Distillation
Reactive distillation is a hybrid unit which comprises distillation and reaction in a single unit operation.
Recently reactive distillation has become a strong interest in chemical endistillation
has
been
the
gineering research, although
concept of reactive
known for a long time. In the 1920's the technique was applied to esterification processes using homogeneous catalysts (Backhaus, 1921). In 1971,
Sennewald described a development employing solid heterogeneous catalysts.
The most important benefit of reactive distillation lies in the economics: a
in
capital cost, energy saving, raw materials and solvent reducreduction
tion. By carrying out distillation and chemical reaction in the same unit,
is
the
associated pumps, piping and
with
along
one process step eliminated,
(the
This
instrumentation.
area or
gives safer environmental performance
just
hazards
is
one continuous
significantly reduced with
scale of risk and
(the
heat
better
of reaction
energy management
operating unit) as well as
but
increase
better
transfer,
boil-up
no
phase
vapour-liquid
and
more
creates
in temperature; therefore, no cooling is required).
Particularly good candidates for reactive distillation are processesin which
by:
the chemical reactions are characterised
12
unfavourable reaction equffibrium: all chemical reactions have an equilibrium
reaction. There are chemical reactions for which, at operating temperature,
the mixture at chemical reaction equilibrium conditions still contains considerable concentration of reactants. Even if one of the reactants is present
in a high concentration the reaction will not proceed. Such reactions are
For these chemical reactions the connormally so-called equilibrium-limited.
be
increased
by
from
the reacting
version can
continuous removal of products
mixture.
high heat of reaction: some chemical reactions have large heat of reaction
(exothermic or endothermic). If these reactions occur in reactors they will
In
distilthe
temperature
the
change
and modify
reaction progress. reactive
lation the heat of reaction will not modify the temperature, the phases will
boiling
heat
the
the
point, and
of reaction will not affect the reacremain at
tion equilibrium. In the case of exothermic reactions, the heat of reaction is
directly used for the distillation process.
large excess of reactants required: reactive distillation is also potentially attractive whenever a liquid phase reaction must be carried out with a large
large
In
this
situation, conventional processes need
excess of one reactant.
distillation
however,
(for
the
could
reactive
excess of reactant);
rec y7cle cost
be carried out closer to stoichiometric conditions minimising the recycle cost.
Reactive distillation prevents side reactions and overcomes limitations due
to chemical equilibrium by its natural separation.
limitations
imdistillation
the
can overcome
azeotropic conditions: reactive
by
azeotropic mixtures: simultaneous chemical and phase equilibrium
posed
has the most beneficial effect of reacting away some of the azeotropes and
behaviour.
the
thereby, simplifying
phase
is
distillation
to
a
applicable
especially
reactive
catalysts:
systems with solid
The
important
that
points
catalysts.
solid
employ
certain class of reactions
disthe
the
catalyst at
activity of
that characterise the catalyst systems are
the
the
reactants and products.
tillation conditions and
relative volatility of
The balance between these two characteristics makes some chemical systems
for
technique.
this
perfect candidates
highly
is
it
is
distillation
that
disadvantage
system
The main
of reactive
for
be
its
to
each
process.
separately
assessed
needs
suitability
and
specific
13
Reactive distillation has a strong dependency on the properties of the chemical system that is dealt with. The poor knowledge of the chemical reactions
features, (catalyst, kinetics, hold-ups) and distillation (vapour-liquid equilibria, thermodynamics, plate and/or packing behaviour) together with their
in
distillation
difficult
to simcombination
a reactive
unit, makes such a unit
ulate and operate.
Reactive distillation appears to be an interesting process unit. Models capable of dealing with different processes are still not available. Although fully
limits
for
the
equilibrium considerations may give
reaction and separation,
distillation
(real
reactive
columns will not operate at equilibrium
operations
are not at equilibrium).
1.2
Motivation
Aim
Work
the
and
of
The motivation of this study has been the interactions between reaction and
interest
in
for
Increased
the
that
units
a reactive column offers.
separation
the application to fine chemicals and pharmaceuticals, as well as in the comhas
been
incentive
industry
this
of
work.
an
also
modity chemical
The main objective of the present work has been to generate appropriate
tools to derive an optimal design of reactive distillation operations.
The objective has been tackled by:
distillation
for
developing
reactive
rigorous models
e
systems;
best
the
to
dynamic
formulating
choose
optimisation problem able
a
e
for
task;
(from
the
given
a
possible alternatives)
all
structure
the
openusing
studied,
system
on
analysis
controllability
0 performing
loop controllability indicators; and finally,
(dynamic
optimization)
0 applying optimal control
distillation
columns.
active
14
for the design of re-
Problem
Definition
Consider the reactive distillation
column in Fig. 1.1 where a mixture is
to be separated into two product streams. The
mixture of A and B will
react to C and D. Operational constraints, such as product specifications,
be
in
must
met
spite of the presenceof variability in the feed streams over
time (disturbances in composition of A or B). The
is
to design
objective
the reactive column at minimum total annual cost with the
goal of ensuring
feasible operation for all product specifications
over time.
In order to undertake such problem, a number issues
be
to
of
need
addressed:
Rigorous
(equilibrium
based)
models
for
and
non-equilibrium
accounting
highly non-ideal mixtures need to be implemented. Simulation
and sensitivity analysis should be performed to study whether operability problems,
such as multiplicity, can occur.
Optimisation techniques to choose the best design for a
given task should
be considered. An objective function including economics of the
process and
factors,
other
such as operability indicators, should be included.
1.3
Overview
This thesis is structured as follows,
Chapter I gives the introduction, motivation and the aim of this work.
Chapter 2 briefly shows the most relevant literature in the area of reactive
distillation. The Chapter is divided into two Sections. The first one addressesthe literature in terms of equilibrium models presented. The second
the
area of non-equilibrium models.
one addresses
Chapter 3 gives a full description of the hybrid model presented and used in
this work. A brief discussionhighlighting the equilibrium and non-equilibrium
The
is
Maxwell-Stefan
A
detailed
equations are also given.
models presented.
Section with the set of equations describing the hybrid model is presented.
Generation of models from the hybrid framework as well as computational
Chapter.
the
aspects
are given at end of
and numerical
15
aillate
c
B -->C+D
Ittom
D
Figure 1.1: Exemplary reactive distillation
16
column
Chapter 4 presents the validation of models. First the systems studied in this
thesis are presented, then the model validation for each system is studied.
Finally a continuation method integrated into the hybrid approach is also
bifurcation
the
presented showing
analysis.
Chapter 5 gives the application of hybrid models. Whenever computational
time is to be saved, then combination of non-equilibrium and equilibrium
be
from
determine
To
it
is
to
models should
used.
when
possible
switch
one
free
Gibbs
to
tray
tray
energy over the
another,
efficiencies and excess
model
highlighted.
be
Hybrid
time
modelling results are
should
analysed.
simulation
Chapter 6 gives the controllability issues and dynamic optimization.
theory description precedes the results.
A brief
Chapter 7, as last Chapter of this thesis, presents the conclusions and future
directions from this work, as well as highlighting the main contributions of
this thesis.
17
Chapter
2
Background:
Literature
Review
"The idea of evolution is closely associated with an increase of organisation
the
to
giving rise
creation of more and more complex structures"
(Glansdorf and Prigogine, 1971).
2.1
Reactive and Non-reactive
Process Engineering
Distillation
in
Different pieces of equipment configure a chemical plant. Some of these
be
distillation
likely
Distillation
is
to
columns.
one of the
equipment are
The
'stage',
in
concept of
chemical engineering.
which
oldest unit operations
describes distillation columns has been known for over a hundred years. The
description of a distillation column continues by characterising defined flows
flows
heat
These
leaving
they
of
mass
or
and
consist
stage.
any
entering and
feed
flows
draw
flow
be
the
through
the
column,
or any
vapour/liquid
can
flows. Distillation displays different types of multiple phenomena such as
heat
flow,
heat
transfer,
phase equilibrium, mixing, material and
mass and
'equilibrium
distillation
is
in
kind
the
The
column
a
of stage
simplest
etc.
has
the
implies
This
the
that
of
any
component
chemical potential
stage'.
liquid
in
the
phase.
vapour and
same value
big
have
that
It is well-known
advantage: a consisone
equilibrium models
big
is
in
that
the
However
their
solution
tent model.
cases:
most
weakness
is potentially very far from the real process.
18
This weakness has been overcome by chemical engineers with the help of
the efficiency concept. The Murphree efficiency is commonly used for tray
because
its
implementation
in equilibrium models is straight
columns mainly
forward. When using this efficiency we assumethat the vapour composition
obtained with the equilibrium model is not the real composition but that the
fraction
is
tray
real
composition
a
of the tray composition. For binary sephave
the
two
the same value, the concept works
arations, where
efficiencies
but
failing
it
quite well,
starts
when multicomponent mixtures are considered
(Wesselingh, 1997). Efficiencies are often assumed'constant' throughout the
Practical
helps
deciding
'value'
in
be
column.
experience
which
should
used
for a particular example.
What is to be done if we do not want to use the efficiency concept? Recently,
Wesselingh (1997) suggested the 'dream of a chemical engineer', saying: 'subdivide each piece of equipment into zillions of little elements dx. dy. dz, each of
in
Using
the
the
then the
phases
equipment'.
which will contain part of one of
difference form of Navier-Stokes equations for the hydraulics, the MaxwellStefan equation to calculate the diffusion flows, and Fourier equations to
describe the energy transfer, together with the corresponding balances and
boundary conditions we should be able to solve the 'real problem'. However,
be
term
it
the
to
on
a
short
solution
obtain
possible
won't
unfortunately
basis: it is a dream.
Between this 'dream of engineers' and the usual equilibrium approach, there
is the non-equilibrium model, which in these days takes into account the
it
that
the
the
important
vapour
assumes
column:
characteristics of
more
Equilibthermodynamic
liquid
in
the
equilibrium.
column are not at
and
The
interface.
the
is
equations of equilibrium
phase
only assumed at
rium
Well
for
forces
driving
transfer.
the
the
to
mixed
then
mass
obtain
used
are
description
the
liquid and vapour phases are mainly considered and
of the
included.
is
between
heat
the
transfer
phases
mass and
in
1970s
the
been
the
have
area of
Significant advances
early
made since
Simulations
distillation
state
of
steady
columns.
of
modelling and simulation
including
shutdown
and
start-up
operations
state
unsteady
operations and
operations are quite common nowadays.
different
for
the
the
simulation problems
At the end of
models used
century,
including
issues,
the
to
simulation
and
modelling
complex
address
attempt
19
those related to separation of azeotropic mixtures and simultaneous separation and reaction: reactive distillation.
The main technical advantage of reactive distillation is the continuous removal of reaction products. However, as reactive distillation still involves
longer lead time for research, which kind of models should be applied to the
Can
distillation
be
unit?
a
model
extended to consider the simultaneous
physical and chemical phenomena? Would one need a 'special' model to describe the proposed problem?.
Reactive distillation
be
(simultaunits can
modelled as an equilibrium system
neous physical and chemical equilibrium); at physical equilibrium with kineticall y7controlled reaction; or as a non-equilibrium system (separation through
kinetically
transfer
mass/heat
plus
controlled reaction); or even maybe as a
combination of these models which could also be an interesting way of looking at the process.
The selection of the appropriate model type, form and simulation mode
(steady state or dynamic) for a particular simulation problem depends on
the problem being solved. The separation task will provide details for the
being
the
the
separated,
required
simulation
mode,
existence of reacmixture
tions and/or azeotropes and many more. The choice between non-equilibrium
(or rate-based) and equilibrium models, although obvious, is not always followed. That is, since any distillation operation will not attain true equilibis
the
the
most appropriate selection.
always
non-equilibrium model
rium,
The equilibrium-based model, however, predominates in terms of application
A
here
is
implement
the
it
is
to
to
arises
again:
solve.
question
and
easier
as
the
to
all
phenomena occurcapture
non-equilibrium model always needed
in
the
reactive column?
ring
Ideally, what is needed is a computer aided system with a general distillation
be
forms
that
from
the
generated
so
can
necessary model
which all
model
for
If,
be
a particular simulation problem, more
answered.
our question can
becomes
the
type
than one
computer aided system should
necessary,
of model
hybrid
through
the
the
be able to generate
use of a
problem specific model
hybrid
The
generating
consists
of
approach
modelling
approach.
modelling
for
defining
their
hybrid models and
example, modelling of
solution method,
the
trays
distillation
or
use
and
non-reactive
with
reactive
column
single
a
trays
model
equilibrium-based
and
model
on
some
non-equilibrium-based
of
20
on the remaining trays. Also, for any design/ analysis/ control problem, the
hybrid modelling approach will
allow switching between steady-state and
dynamic modes and different forms
of models. This will make the transfer of
data (information) from one simulation
sub-problem to another transparent
'noise-free'.
Smooth
transfer of data also means better initialisation
and
and
consequently, a more robust and efficient solution technique. In this thesis
has
been
an attempt
made to present such a model.
2.2
Modelling-Design-
Synthesis
Issues
This section has been divided into two parts, one is
related to equilibrium models presented in the open literature. The other one treats nonequilibrium-based models.
2.2.1
Equilibrium
Models
The literature review on distillation has been extensively addressed in the
Several
books
past.
offer the possibility of understanding, as well as showing how to model, design and control a distillation unit. Examples are,
King (1980), Kister (1990,1992), Luyben (1974), Skogestad and Postleth(1996),
few.
For
have
focused
to
this
here
waite
name a
reason we
on the
description of the researchover the years of only reactive distillation columns.
In recent years a large number of computational algorithms have been rebalance
to
the
solve
ported
mass and energy
equations together with the
data
describing
distillation
The
the
phase equilibrium
reactive
problem.
unknown variables determined by solving these equations are mole fractions
both
flow-rates
temperatures,
rate of reaction and
of
phases, stage
of each
for
The
is
the
equilibrium model
standard model
simulating reactive
phase.
distillation.
key
The
in
is
these
that the
assumption
models
or non-reactive
leaving
liquid
in
a
non-reactive
stage
reactive or
are
equilibrium
vapour and
(liquid
both
temperatures
the
vapour)
on
stage are assumed equal.
and
and
Table 2.1 summarises some of the simulation approaches using a thermody(many
for
distillation
reactive
more are presented
namic equilibrium model
in the literature, but all of them offer more or less the same characteristics).
Most of the research has focused upon steady-state solution of the model
The
for
earliest algorithms
steady state simulation of staged reequations.
21
Authors
Suzuki et al.
(1971)
Saito et al.
(1971)
Komatsu and Holland
(1980)
Perez Cisneros et al.
(1996)
Table 2.1: Thermodynamic
distillation
columns
active
Equilibrium- Simulation Features
Steady-state model.
Kinetically controlled reaction.
Margules's modified equation for
the thermodynamic model
Experimental results from a pilot
plant.
Steady-state model.
Instantaneous reaction.
Model specific for the esterification
example treated.
Steady-state model.
Kinetically controlled reactions.
Homotopy- Continuation method of
solution.
Dynamic model.
Kinetically controlled reactions.
General model including nondistillation.
reactive
for
the simulation of reequilibrium approaches
(1971)
(general
Suzuki
by
distillation
et al.
columns were reported
alactive
(1971)
Saito
Newton
methods);
et al.
gorithm employing quasi-linearisation
(rather specialised, involves instantaneous reaction equilibrium, etc.); and
Nelson (1971) (employing modified Newton-Raphson).
During the years several other algorithms with different methods of solu(1980),
Holland
HolKomatsu
for
been
have
tion
and
example,
studied,
land (1981), including quasi-Newton methods combined with the 0-method
(1977)
(1976),
Komatsu
Hlavacek
Jelinek
used relaxation
and
of convergence.
(1991)
(1988)
Bondy
Seader
Chang
(i.
false
transient);
and
and
e.
methods
homotopy-continuation
The
homotopy-continuation
methods.
worked on
fail.
Through
Newton
is
the
to
methods
problem when
solve
a way
method
into
is
transformed
the
one
problem
can
solve
a
problem
pararnetrisation
in
that
by
occur
naturally
or
parameters
parameters
additional
using
either
(parametric
the model
continuation).
belong
to tearing methods and take a two tier approach
Inside-out algorithms
22
in which the physical properties are approximated by
in
the
simple models
loop
outer
while the model equations are solved at the inner loop (simultaby
Newton's
neously
methods). These kinds of methods are the subject of
industrial research, as is shown by Venkataraman et al. (1990).
Dynamic equilibrium models for distillation have been presented by Gani
(1986,1989),
Cameron et al.(1986) and Ruiz et al.(1988). Their adet al.
vanced equilibrium stage model including rigorous hydraulics has been applied not only to continuous columns but to shut-down/start-up for columns
Recently,
the model has been adapted to handle reactive distillation
as well.
(Pilavachi et al., 1997).
To the best of our knowledge there are few papers in the literature, apart
from our own work (Perez-Cisneros et al., 1996,1997a; and Pilavachi et al.
1997), that addressed dynamic simulation of reactive distillation columns:
Ruiz et al. (1996) and Schrans et al. (1996), Scenna et al. (1998).
From the 1980's the interest focused mainly on the design aspects of reactive
distillation columns. Several research groups have been dedicated extensively
for many years to this area and to the understanding of the interactions between separation and reactions.
The design of reactive columns has followed different kinds of approaches:
transformed
variable approach;
based
balance
approach;
- element
(including
design
through
techniques
synthesis).
optimization
Table 2.2 highlights some of the main features obtained for the design of
distillation
columns.
reactive
In the two first approaches, full equilibrium is considered. Doherty and coGani
been
the
transformed
have
variable approach, while
working on
workers
based
Both
balance
focused
have
the
approach.
element
on
and co-workers
less
for
in
terms
containing
systems
visualisation
of
excellent
are
methods
than four components.
for
design
been
has
the
Optimization
as well as synthesis of
applied
also
kind
This
distillation
a
superstrucutilises
of
approach
processes.
reactive
23
Authors
Design Features
Barbosa and Doherty (1987,1988a, Developed tools for analysis of reb), Ung and Doherty (1995a,b,c)
active mixtures with a single equilibrium reaction. Later extended to
multiple equilibrium reactions with
inclusion of inerts.
Perez Cisneros et al.(1996,1997a,b) Development of tools based on the
balance
for
element
approach
multiple equilibrium reactions.
Ciric and Gu (1994)
MINLP for the optimal reactive col(minimising
umn
annualised cost)
for multicomponent mixtures and
kinetically controlled reaction.
(MINLP)
Synthesis framework
Ismail (1998)
for multi-component and multi(kinetically
controlled)
reactions
Hold-ups and catalyst loads are
for.
explicitly accounted
Table 2.2: Design approaches for reactive distillation
columns
feed
is
to
ture which
streams, multisystems containing multiple
applicable
kinetic
component mixtures and multiple reactions which are considered as
Pistikopoulos
instead
based
and
of equilibrium reactions.
expressions,
rate
for
developing
been
have
a superstructure approach
working on
co-workers
design and synthesis of reactive and non-reactive distillation columns.
Barbosa and Doherty (1987) introduced a new set of composition variables,
idenis
to
the
which
a
problem
reaction
considering
problem
which reduces
dimensionality
the
the
thus
phase
of
tical to one without reaction
reducing
for
systems.
multicomponent
problem
and chemical equilibria
diadeveloped
Different tools they have
are: computation of reactive phase
for
azeotropes,
derivation
reactive
conditions
general
and
of conditions
grams,
b;
Doherty,
1987,1988a,
(Barbosa
and
maps
curve
of
residue
computation
the
As
the
bc).
1995a,
of
Doherty,
visualisation
Ung and
an example of
for
the
in
transformed
system
reactive
the
variables
curves
residue
method,
is
inert
presented
component
with n-butane as
isobutene-methanol-MTBE
in Fig. 2.1.
24
Image removed due to third party copyright
Figure 2.1: Residue curve map for MTBE.
1995b)
(P=latm)(Ung
Doherty,
and
The element balance based approach was developed originally by Michelsen
(1995) and extended by Perez-Cisneros et al.(1997b). Again, in this apin
dimensionality
is
The
the
the
reduction
problem used.
of
proach,
approach
is based on the fact that the number of 'elements' needed to represent a reless
is
in
than
the
number of components present the reactive
active systems
These
'chemical
but
they
elements
are
not
necessarily
elements',
mixture.
be
the
chemical
single
element,
a
part
of
molecule
or
a
or
even
comcan
The
for
itself.
this
to
the
solution
method
approach
reduces
wellponent
known physical equilibrium problem, which obviously gives a great advantage. Tools developed for this approach for the design of reactive distillation
include:
diagrams,
for
reactive phase
general conditions
element recolumns
McCabe-Thiele-like
(Perez-Cisneros
diagrams
1996,
et al.,
active azeotropes,
25
Image removed due to third party copyright
Figure 2.2: McCabe-Thiele like diagram for MTBE. (Perez Cisneros et al.,
1997a)
1997a, b, Pilavachi et al., 1997). As an example of the methodology and
diagram
like
is
Fig.
McCabe-Thiele
for
in
2.2
the
the
shown
visualisation,
MTBE reactive system.
A design methodology is also described by Pilavachi et al.(1997) which inthree
steps:
cludes
(i) validation of models (including as well selection of thermodynamic models);
(ii) identification of variables sensitive to design, and;
(iii) analysis of these effects on the design and operation of reactive units.
26
Ciric and Gu (1994) developed a synthesis schemefor reactive distillation
fairly
processescombining a
complex equilibrium model with rate-based kiThe
netic expressions.
resulting optimization model is a MINLP. The total
is
function.
The solution of such problems
the
annualised cost
objective
yields the optimal number of trays, reflux ratio, heat exchanger duties and
liquid hold-ups throughout the column.
Recently, Ismail (1998) developed a mass/heat transfer framework within a
superstructure environment which can handle reactive separation systems.
The approach is based on fundamental mass and heat transfer principles.
Units of process alternatives are not prepostulated, but instead these are
generated according to the synthesis objectives.
Summary
Since the 1950s new algorithms are presented frequently for solving the equilibrium stage model equations for distillation columns. However this is not
has
been
Still,
done
implication
in
that
the
everything
area.
many sepaan
Some
hard
them
to
remain.
of
yet
or
are
are not solved
solve
ration problems
(Taylor and Lucia, 1995). The same problems apply to reactive distillation
few
due
fact
in
the
that
to
the
research
area started a
years
more strongly
later (1970s) with equilibrium staged models.
The equilibrium model has been used for reactive distillation without considlack
knowledge
because
the
the
tray
of
about
of
efficiencies, mainly
eration of
data
Experimental
these
are practically not available.
efficiencies.
values of
For the design of reactive distillation columns, the variable transformation
limited
based
balance
to
the
simultaneous
are
approaches
element
as well as
limits
This,
the
the
of
of course gives
chemical and physical equilibrium.
Variable
this
but
condition.
units will rarely work at
possible separation,
From
0-1.
the
'compositions'
the
leads
to
eletransformation
range
outside
is
to
to
there
based
compositions.
component
go
no
way
compositions
ment
both
For
These element compositions are not always useful.
approaches,
four
for
difficult
than
become
techniques
commore
of
mixtures
visualisation
(or
elements).
ponents
in
Using the superstructure approach results
a probably more realistic and
is
The
distillation
this
design
accuracy
of
price
columns.
of
reactive
accurate
27
for
by computationally demanding models. At present, limited
paid
steady
state equilibrium models or limited mass/heat transfer models have been
used. This highlights the need for a complex model, able to capture all the
phenomena present in the reactive distillation process for the superstructure
formulation.
2.2.2
Non-equilibrium
(rate-based)
Models
Rate-based models appear to be the most appropriate for reactive distillation
due
to the fact that the a priori computation of stage efsystems, primarily
ficiency is avoided (Taylor and Krishna, 1993; Sivasubramanian and Boston,
1990).
In general, a non-equilibrium model includes separate mass and energy balfor
each phase, equilibrium relations and mass and energy transfer
ances
The
for
linked
by
models.
conservation equations
each phase are
material
balances around the interface, since there is no accumulation at the interface
(the mass lost by the vapour phase is gained by the liquid phase). Equilibinterface.
to
the
rium relations are used
relate compositions at either side of
Mass and energy are transferred across the interface at rates that depend on
the extent to which the phases are not in equilibrium with each other. These
from
in
transfer
correlations
mass
multicomponent sysrates are calculated
tems (Taylor and Krishna, 1993). Note that equilibrium stage models are a
limiting case of the non-equilibrium model. If non-equilibrium simulations
larger
done
interfacial
100
times
than
they
areas
about
considering
really
are
(Powers
be
the
then
matched
et al.,
results of an equilibrium model can
are,
1988).
Physical reality is more accurately captured through the non-equilibrium
between
Interactions
reaction and separation are considered explicmodel.
itly through mass and heat balances around the vapour and liquid phases,
in
the
case of reactive systems.
respectively,
The non-equilibrium models that have been developed for distillation prointo
interesting
insights
the
have
effect of multicomponent mass
given
cesses
transfer.
The model developed by Krishnamurthy and Taylor (1985) has been the
28
'base' for any other further development of non-equilibrium models. The
model, with or without modifications has been applied for many researchers
to the study of non-reactive columns. Recently reactive columns have also
been studied with the 'base-model'.
Well-known applications of reactive distillation include heterogeneous reactive distillation processes in packed columns such as et herificat ions; processes
involving highly non-ideal mixtures such as esterifications or etherifications;
high
in
least
and systems with a
reaction equilibrium constant
which at
one of
the component concentration is low such as esterification of acetic acid with
(1994)
Taylor
ethanol.
et al.
pointed out that rate-based models are particfor
modelling packed columns; strongly non-ideal systems; sysularly useful
tems with trace components; columns that exhibit minima or change rapidly;
A
or processes where efficiencies are unknown.
non-equilibrium model seems
the way of modelling a reactive distillation column, instead of the well-know
equilibrium
model.
Another key reason for using a non-equilibrium model instead of an equilibfor
is
that
typically
no good predicsystems
reactive
separation
rium model
tive methods exist nor is experimental data available for stage efficiencies.
Kooijman and Taylor (1995) have developed a rate-based model for the sim'second
The
dynamic
tray
generation' of
columns.
operations of
ulation of
behaviour
dynamic
it
is
the
of
called, considers
non-equilibrium model, as
distillation
drop
the
throughout
the column as well as pressure
column.
Gorak and co-workers have done extensive research in the area of packed
for
their
including
with
comparison
obtaining experimental results
columns,
devices
the
Packed
more comand
contact
columns are continuous
models.
differential
is
transfer-based
technique
model using mulmass
used a
mon
Gorak,
(Gorak,
Kenig
1995).
1987,
theory
transfer
and
ticomponent mass
for
is
from
the
developments
this
One of the latest
rate-based model
group
for
the
batch
experiments
scale
as
pilot
well
as
columns,
packed
reactive
(Kreul
1998,1999).
al.,
et
methanol
with
acid
of
acetic
esterification
case of
the
described
(1997)
have
Pelkonen
of
Also,
experimental validation
et al.
distillation;
for
dynamic
multicomponent
of
simulation
approach
rate-based
imple(1999)
in
the
Kenig
simultaneous
which
a
work
presented
al.
et
and,
for
reactive
approach
and
non-equilibrium
model
equilibrium
of
mentation
has
been
distillation simulation
addressed.
29
Although rate-based models for distillation are now
(see
Seader,
quite common
1989; and Taylor and Lucia, 1995, for an interesting review) such
for
models
distillation
in
development
the
The
first
reactive
in
are still
the
stage.
work
field of rate-based model, to the best of our knowledge,
Sawistowski
was
et
(1979)
al.
who modelled a packed reactive distillation column for the esterification of acetic acid with methanol to methyl-acetate. An effective diffusivity
type was used for the mass transfer model along with irreversible kinetics for
the reaction which can lead to a high percentage of error (40%) in the prediction of the mass transfer rates. This may be due to the fact that there is
lack
a
of theoretical and physical understanding with this effective diffusivity
(Taylor
Krishna,
1993).
method
and
Zheng et al.(1992) and Sundmacher (1995) presented rate-based models for
the production of MTBE on a heterogeneouscatalyst, which usesthe MaxwellStefan equations for the description of the mass transfer. The heterogeneous
is
being
(mass
transfer effects to
reaction considered as
pseudo-homogeneous
from
lumped
the
together into a catalyst efficiency term).
and
catalyst are
Just recently more research in the area started to be published (Higler et
b;
Kreul
1997,1998,1999a,
1998,1999)
al.,
et al.,
although of a commercial
(ASPEN
Guide,
PLUS
User
1997) of a generalised steady state rateversion
based model (RATEFRAC) capable of handling reactive systems has been
for
in
industry.
The
has
been
ten
the
almost
years
available
model which
described by Sivasubramanian and Boston (1990) is a steady-state model
based on that of Krishnamurthy and Taylor (1985) for non-reactive columns.
It considers resistance to mass transfer in both phases, but the examples
literature
in
in
to
the
are simplified
resistance
only one phase
open
shown
(liquid or vapour).
Recently, Higler et al.(1997,1998,1999 a,b) presented a non-equilibrium
(applied
distillation
homogeneous
for
the
to
esterification of
reactive
model
by
transfer
simultaneous
accompanied
acetic acid with ethanol), where mass
Maxwell-Stefan
described
is
the
continuity equations.
using
chemical reaction
In their model no expression for calculating the interfacial area is presented
infor
both
These
fixed
is
Lewis
the
phases.
assumptions are
number
and
(liquid
Only
first
in
by
the
the
troduced
phase)
paper.
one phase
authors
is
to
the
the
transfer
case
resistance considered when applying
model
mass
both,
liquid
disregarded,
is
in
Heat
transfer
that
temperatures
so
and
study.
30
vapour phases, are the same.
Summary
Reactive distillation involves multicomponent mixtures, but other distillation
involve
Systems
binary
multicomponent
also
mixtures.
containing
mixtures
(which are practically not used) are more simple to model: efficiencies, for exfor
both
ample, are equal
components. In multicomponent systems, however,
diffusive interaction, reverse diffusion, osmotic diffusion and mass transfer
barriers are of importance. It is clear that efficiencies can be different for
(Taylor
in
Krishna,
Simulation
1993).
tray
each
each component and
and
importance
then
together with a good unand optimization are
of particular
derstanding of the process. It has never been presented when and for which
(such
the
to
the
assumed simplifications
model
as equirange of operation
librium, constant flows, etc. ) are valid. Lack of accurate simulators restricts
for
distillaimprovement
direction,
in
this
reactive
specially
even more any
tion.
Non-equilibrium models will continue to develop. Indeed they will be used
for
the
case of multicomponent non-ideal mixtures, reactive sepamassively
(Taylor
Lucia,
low-efficiency
1995).
and
processes
ration and
The main needs for using a non-equilibrium model at the moment are:
kinetics;
- reaction
for
transport
better
pure
properties
and
physical
of
predicting
methods
dynamic
(specially
for
for
simulation);
mixtures
components as well as
large
for
that
a
range
over
cover
energy
coefficients
and
mass
correlations
the operating conditions;
the
hydraulic
units.
of
performance
be
it
been
though
have
Very sophisticated models
poswill
presented, even
it
is
However,
further.
them
such
whether
clear
not
to
still
extend
sible
in
insights
the
be
developments can
problem.
solving
warranted or give more
Most of the non-equilibrium models can, at this stage, accurately represent
here
that
is
this
it
data.
However,
to
remarking
without
industrial
say
unfair
data
that
find
the
hard
data
that
is
to
of
nature
often
and
very
experimental
is unknown. There comes a point where the value of complex models must
Can
it
be
be compared with what can
obtained using simplified models.
0.05%
large
to
in
tedious
time
get
a
models
to
complex
solving
expend
pay
31
increase in accuracy? Probably the accuracy of the thermodynamic or transdata
lower
is
than that.
port property
Model integration, where equilibrium and non-equilibrium models can be acfor
distillation
be
the
cessed
same
unit; where steady-state simulation can
dynamic
be
to
the
equilibrium model can
applied
switched
simulation; where
for any stage of a column (if thermodynamic equilibrium is attained) while
the rest of the unit will continue in the simulation as a non-equilibrium model.
Model integration also where different types of models, simplified or rigorous
directly.
both
be
In
types
that
of model, equilibrium and
sense,
accessed
can
Still
have
been
integrated.
time
today
computational
never
non-equilibrium,
is an important issue, and the non-equilibrium model requires longer time
to achieve the solution. A model combining both types of model is called
for
To
be
'hybrid
to
a specific column configuration
account
able
model'.
a
dynamic
the
of
model equations
optimization
and control structure as well as
be
these
tasks
be
in
that
implemented
performed.
can
all
a way such
should
In this thesis an attempt to develop a complete hybrid model is presented
In
both
between
together with a way of switching
addition, steadymodels.
be
Dynamic
dynamic
optimization can also
modes are considered.
state and
performed.
32
Chapter
Modelling
3
Issues
Within the chemical industries distillation is the most used unit. Its inherent
(multicomponent
dynamics)
it
difficult
nature
mixtures and complex
make
to model and to simulate.
In this Chapter a detailed hybrid model (dynamic which can be also switched
to steady-state) including both equilibrium and non-equilibrium approaches
for continuous distillation and reactive distillation is presented. In the case
Murphree
the
to
the
model,
efficiency
values
are
added
equilibrium
of
equilibrium stage to account for deviation from ideality while for the rate-based
(or non-equilibrium) model mass transfer is considered explicitly using the
Maxwell-Stefan equations together with binary mass transfer correlations.
The hybrid model approach is presented addressing important characteristics
behaviour
dynamic
the
of multicomprocesses such as steady-state and
of
implemented
framework
The
using
model equations are
ponent mixtures.
the ICAS-integrated simulator (Gani et al., 1997) and gPROMS(PSE Ltd.,
1997).
Validation of the model is highlighted by matching the results with experi(whenever
literature
in
data
the
possible) or against
open
available
mental
for
The
several reactive and nonresults achieved
other simulation results.
is
the
that
the
to
accurately
propredict
show
model
able
reactive systems
Through
time.
comparison of equilibrium and non-equilibrium
cessesover
for
distillation
is
it
to
that
the
show
reactive
possible
simulation results
largest differences occur when the rate of reaction becomes important. The
33
in
bulk
liquid
between
the
transfer
the
the
the phases,
reaction
affects
mass
of
thus moving the equilibrium.
Rate-based models lack appropriate experimental correlations, especially for
heat
(as
transfer;
mass and
equilibrium-based model are not accurate
previKrishna,
hence,
Taylor
both
1993),
ously reported,
and
a model combining
forward
fast
towards
approaches gives a step
and relatively accurate simulation.
In Sections 3.1 and 3.2 some details of equilibrium and non-equilibrium moddescription
Maxwell-Stefan
together
the
with a
of
equations.
els are given,
From Section 3.3 the model developed in this work is described. Assumptions
(with the corresponding justifications) are highlighted.
34
3.1
Equilibrium
Models
Since the 1950's modelling of distillation has been based on the equilibrium
feature
key
is
leaving
The
liquid
that
the stage
vapour and
streams
stage.
are at equilibrium with each other.
Three different types of equilibria are presented in the equilibrium term: material equilibrium (there is no net change of material between the phases),
(there
thermal
are no velocity gradients), and
equimechanical equilibrium
librium (there is no difference between the phase temperatures) (Bird et al.,
1960). Since no distillation unit operates at equilibrium, efficiencies are used
to account for departures from equilibrium. For binary systems, efficiencies
between
This
bounded
0
1.
for
both
they
and
components and
are
are equal
does not hold for multicomponent mixtures, mainly because of the diffusional
interactions, as defined by Toor(1964). The diffusional interactions include:
its
diffusing
diffusion(when
is
own concentraagainst
a component
reverse
has
diffuses
it
if
diffusion(when
tion gradient), osmotic
no
even
a component
barrier(when
transfer
the
a component
mass
concentration gradient), and
does not diffuse despite its concentration gradient).
3.2
Non-equilibrium
Models
When deriving a non-equilibrium model, one of the following theories should
be applied at the interface (if fluid-dynamic equations are not considered),
be
forty
(there
that
theories
than
for example
applied),
can
are more
theory,
two-film
the
model
9
boundary-layer
theory,
the
*
theory,
the
or
penetration
9
theory.
film
penetration
9
because
is
(Whitman,
1923)
experimostly used
The two-film model theory
is
This
theory
is
theory.
to
this
data
reported
also
adapted
available
mental
factors
such as chemical reactions causes
to give accurate results, even when
The
(Taylor
though,
Krishna,
1993).
theory,
transfer
in
the
and
mass
changes
includes various simplifications.
35
3.2.1
Maxwell-Stefan
Equations
The Maxwell-Stefan equation for binary ideal mixtures is written as:
1
df, -= VPJ
p
XIX2(Vl
-
V2)
D12
df
be
be
driving
force
for
diffusion
I
to
the
where 1 can
considered
of species
in an ideal gas mixture is at constant temperature and pressure. The symbol
D12 is the Maxwell-Stefan diffusivity. The Maxwell-Stefan coefficients do not
depend on the compositions, and this gives the Maxwell-Stefan equations an
(Taylor
law
for
Fick's
the
multicomponent systems
advantage over
extended
Krishna,
1993).
and
The theory for binary ideal systems can be extended to multicomponent ideal
systems, so that,
NC
XiX3(Vi
dfi = -1:
j=l
-
Dij
Vj).
(3.2)
If the approach is to be extended to non-ideal systems,then the driving force
for diffusion in non-ideal systems is the gradient of the chemical potential
for
liquid
for
According
to
this,
Krishna,
(Taylor and
1993).
mixtures or
dense gasesexhibiting deviations from ideal gas behaviour, we can apply the
condition,
dfi
Xi
=-
RT
VTPPi
'
(3-3)
from
departure
the
equilibrium
the
give
gradients
chemical potential
where
GibbsDue
the
for
ideal
to
11P
(this driving force reduces to
V pi
gases).
Hence
independent.
the
forces
driving
NC
Duhem restriction, only
are
-1
forces
driving
the
vanishes:
sum of
NC
Edfi
i=l
=0
Eq.
3.3 we can obtain,
Working with
36
(3.4)
NC-1
df i=E
]Pij 7 xj
(3.5)
j=j
for
liquids,
where
non-ideal
dln-yi
6ij + Xi
IT, Pxk,
dxj
k:A3,=l,..,
n- 1
(3-6)
The above formulation can also be used for non-ideal gasesby changing the
for
fugacity
0j,
the
activity coefficients -yj
coefficients,
6ij
dlnoi
+
Xi-IT,
Pxk,
dX3
kOj=l,..,
n-1
(3.7)
The driving force for the diffusive flux j of the component i within a mixture
(Taylor
df
by
Krishna,
is
1993),
NC
and
components, j, given
of
NC
Z
dfý=[F]
jy
- y.Jt
jy
y,
--3;7vDrj
-
(3-8)
Dý3y is the Maxwell-Stefan diffusivit coefficient, which for diluted gases is
identical to the binary diffusion coefficient.
The diffusion fluxes Jý are given by
NC-1
iv
[]P][K]v N7yj + Jtvyi
-
pv
(3-9)
j=l
by,
the
elements given
with
[K]V
Jy;
E"
JtV
= i=1 i and
where,
Ci,
cij
Yi, NC
i -
Di, NC
yi, j
NC
Yi, k
+
k=l, k:oi
=ýý
(IDij
37
Di
Di, NC
(3-10)
k
(3.11)
Because there are NC
-I
flux is calculated by
independent equations, the remaining diffusional
NC-1
iv
(3.12)
NC
(Note that the equations presented here
for
the vapour phase, similar
are
equations are valid for the liquid phase).
Using the two-film theory model requires a value for the film thickness for
both phases.
Different types of phenomena, such as entrainment, backmixing, turbulent
eddies, and maldistribution, have been usually taken into account by means
of mass transfer coefficients, instead of considering only diffusion (given by
the Maxwell-Stefan equations).
These mass transfer coefficients are purely empirical parameters and
are a
function of the diffusion coefficients, among others parameters.
The linearised theory of Toor (1964) and of Stewart and Prober (1964) is a
interesting
for
diffusion
the
very
method
solving
multicomponent
problems,
but it is often limited to situations were the diffusion coefficient does not
during
diffusion
the
the
change significantly as
concentration changes
process
(Taylor and Krishna, 1993). The solution of the linearised model normally
favourably
fluxes
but
the
compares
with
computed with
exact solutions,
com(Krisnhamurthy
differing
Taylor,
For
1982).
position profiles are often
and
the exact solution, the approach of Krishna and Standart (1976) may be used
in conjunction. The method considers that the matrix of mass transfer coefficients can be calculated from Eq. 3.13 in situations where the film thickness
is not known. This method is recommended by Taylor and Krishna (1993)
because it is much easy to solve.
[K]v =
Ciks
[Cks]-l
NC
Yi, NC
ki ,NC
1=1,10i
38
Yi, l
ki,l
(3-13)
(3.14)
Cks
ij
kij
ki,
(3.15)
NC
The binary kij can be calculated as a function of the Maxwell-Stefan diffufrom
from
sion coefficients
a correlation or
a physical model. The approach
Krishna
Standart
is
of
and
used and recommended by Taylor and Krishna
(1993) only because it is 'simple and easy to compute'.
3.3
Model
Equations
In this section a general distillation model is presented. The model allows
the generation of problem specific simulation schemesbased on the use of a
hybrid modelling approach, where equilibrium and non-equilibrium models
be
can
combined.
The main difference between the equilibrium model and the non-equilibrium
is
in
the
way
which the conservation of mass and energy equations
model
For
balance
derived
the
the
are used.
non-equilibrium model,
equations are
for each phase and linked through mass and energy balance equations at the
balance
interface.
For
dethe
the
equilibrium model,
phase
equations are
(distillation
liquid
tray)
that
the
rived around an equilibrium stage
assuming
leaving
Similarly,
the
thermodynamic
stage are at
equilibrium.
and vapour
leaving
for
the
to
stages
relate compositions
equilibrium relations are used
the equilibrium model, K-values are calculated at the composition of two
(usually
both
in
temperature
the
the
same
stage
well-defined streams and at
For
to
the
the
equilibrium
relations
are
used
model
non-equilibrium
phases).
K-values
interface,
the
that
are
phase
so
relate compositions at each side of
In
interface
this
temperature.
general modand
composition
evaluated at
for
framework
the
conservation of mass and energy equations are split
elling
it
to
eachphase, which makes easier consider equilibrium and non-equilibrium
distillation
For
distillation
tray,
the
for
the
neceach
column.
same
models
defining
functions
for
the
corresponding
each phase and
essary equations
(mass transfer flux for the liquid and vapour phase) is the same throughout
A
interface.
is
it
the
the
the tray, or,
schematic representation
same only at
Fig.
in
3.1.
is
in
this
general model given
of an stage
The general distillation model has been incorporated into two computer aided
dynamic
both,
in
mode,
as
well
steady
state
and
simulation
providing
systems
different
models,
of
complexity
and
non-equilibrium-based
and
equilibrium
as
39
Image removed due to third party copyright
Figure 3.1: Schematic representation of an stage in the column (KrishnaTaylor,
1985)
murty and
40
form. Both, equilibrium and non-equilibrium models have reactive and nondifferent
the
forms
reactive options and allow
generation of
model
according
to the specified simulation problem. This means that starting from the gendistillation
eral
model, many of the models reported earlier (for example,
Cani et al., 1986, Krishnamurthy and Taylor, 1985, Rovaglio and Doherty,
1990 and Skogestad, 1997) are available for use with respect to a specified
simulation problem.
The presence of the various model options in the general distillation model
development
the
gradual
allows
of problem specific models such that acceptance of every additional assumption makes the models more simple while
Simulation
the
rejection of an assumption makes
models more complex.
refrom
for
initial
the
a simpler model serve as
estimate
sults
a more complex
hybrid
The
important
modelling approach also allows monitoring of
model.
heat
Gibbs
free
total
transfer
rates, excess
mass and
enphenomena such as
during
tray,
and an efficiency-like parameter
any simulation.
ergy on each
Also, during any simulation, a switch from one model form to another as
be
from
to
made.
well as
one simulation mode another can
3.4
Model
Formulation
In order to present the proposed model, certain simplifying assumptions have
to be made. The simplifications are needed in order to simplify the calculain
for
this
the
tions and are not relevant
work and seem reastudies made
the
Together
the
specific column simulated will
assumptions,
with
sonable.
following:
listed
in
the
have
be assumed to
the conditions and configurations
Liquid and vapour phases are perfectly mixed for the equilibrium model,
(liquid
bulk
for
the
vapour)
and
phases
model
non-equilibrium
while
are well mixed.
leave
liquid
For
each
stage
the
phases
vapour
and
model:
equilibrium
9
For
the
(same
temperature
nonthermal
pressure).
and
equilibrium
at
leave
liquid
the
conditions
stage
at
vapour
and
model:
equilibrium
determined by the mass and heat transfer rates.
Murphree-like
in
for
the
equilibrium
plate
each
efficiencies will apply
model.
41
The feed(s) can enter any tray as liquid, vapour or a combination
of
both. Liquid and vapour products can be withdrawn from
any tray.
No heat is lost from any tray, but heat can be added or taken
out at
any stage in the column.
Pressures
*
varies with time as well as from tray to tray.
The reboiler heat input may be specified or can be used to
control temperature or composition at the bottom of the column. The reflux (or
distillate) flow may be specified or can be used to control temperature
or composition at the top of the column.
The column can handle sieve or bubble-cap trays. Plate hydraulics,
flooding
be
determined.
entrainment, weeping and
can
Physical properties can be determined by using different thermodynamic packages.
Reaction
liquid
in
for
takes
the
the equilibrium model, and
9
place
phase
for the non-equilibrium model the reaction occurs in the liquid bulk.
Reactions are always considered as kinetically controlled.
The list of assumptions hold for the most complex model, however in order
to obtain simplified models from the general model, other simplifications can
be made, and they are stated in Section 3.6. The general hybrid model develin
is
Different
this
work presented as general as possible.
oped
combination
lead
literature.
Different
in
to
the
previous models presented
of equations
for
solving a specific problem are readily available within the
assumptions
framework. The equations and correlations used here have been used before
for equilibrium and non-equilibrium models. The advantage of the general
be
is
different
it
is
the
that
generated
and
can
choice
many
models
model
different
to
the
correlations, mode of simulations, as well as
select
user
of
equilibrium or non-equilibrium model equations.
3.5
Equations
(considering
for
distillation
tray
the
The equations
a
general
column
model
derived
from:
or a section of a packed column) are
42
Material and energy balance equations (ordinary differential equations
for dynamic models and algebraic equations for steady-state models).
Equilibrium
9
relations and physical properties.
Kinetic models.
Mass
*
and energy transfer models.
9
Hydraulics.
*
Defined functions.
The above set of equations gives a system of differential and algebraic equations (DAEs) for a dynamic model or a system of algebraic equations (AEs)
for a steady state model. Typically, the mass and energy balance equations
(ODEs)
differential
AEs.
are ordinary
equations
while all other equations are
The defined functions are the equations that compliment the model, for exhold-ups
the
total
ample
mass and energy
equations and efficiencies.
The full set of equations (including differential and algebraic equations) is
below.
given
1
ODEs
balances:
Mass
Vapour
*
phase:
dmr21P
y
Fvpz,
,p
=
p+
dt
p+i
+ EVp-lyi,
+ Vp + EVp)yi,
-(PVp
i=ly...
Vp+lyi,
p-l
(3.16)
+ nr
p
19P
qp
p=ly
... INP
INCy
Liquid
*
phase:
dm-ý
zip =
dt
FL,
ý
pz,
+L
pp
p-l
+ Lp + ELp)xi,
-(PLp
i=lp
-lxi,
p=ly
... INP
INC,
43
+ ELp+lxi,
p
+ Si,
p+l
p -nL i, p
(3.17)
liquid
hold-up
the
where mýZip'mvlip are
and vapour
of a component
Kmol/h,
Vp
Lp,
in
respectively;
and
i on plate p
are the vapour
liquid
flow-rates
leaving
in
Kmol/h;
F
is
and
stage p respectively
the feed flow (the subscript V or L denotes liquid or vapour feed)
in Kmol/h; x, y, z are the mole fractions of component Z in stage
for
liquid,
liquid
feed;
EV
EL
vapour and
p
and
are
and vapour
liquid
PL,
PV
entrainments;
are
or vapour extracted or added to
the column; Si, is the reaction rate in the liquid bulk in Kmol/h;
p
due
i
loss
interface
to
the
and, ny21p,
net
nýzipare
or gain of species
transport, mass transfer rates (note that transport from the liquid
to the vapour is assumedto be negative) given by
nvi,p
Ný
Pdap
"P
nLi,p
Ný,
Pdap
p
in
twothe
flux
is
Ni,
the molar
point
a
particular
at
1
species
of
p
dispersion.
phase
folit
interface,
the
Since there is no mass accumulation at
phase
lows that,
MT
= nrlip - njL
ýlp
,p
balances:
Energy
Vapour
phase:
*
dHýP
dt
FvpHvfp
=
-(PVp
Hvp-i
EVp-,
Vp+lI-Ivp+l
+
+
Qv
Ov
EVp)Hvp
+
Vp
+
+
+
PP
P=11... pNP
Liquid phase:
44
(3.20)
dHIEP
dt
FL,
+ Lp-,
pHlfp
Hlp-l
+ Lp + ELp)Hlp
-(PLp
+ ELp+,
+
Hlp+l
+ DSp
QL
p
(3.21)
-oLp
p=1,..., NP
HEHE
hold-ups
for
liqthe
the
where V) II are
enthalpic
vapour and
Hvfp,
Hlf,
Hvp,
H1p
uid phase respectively;
and
are the vapourfeed, liquid-feed, vapour and liquid enthalpy respectively; Qp is
heat
DSp
heat
is
to
the
any
extracted or added
stage p; and
of
heat
0'
OL
the
transfer rates.
reaction; PPand
are
Since there is no accumulation at the interface hence:
ET
OV
=
pp
OL = 0.
2 Algebraic Equations (AEs)
Phase
equilibrium:
-
For
the equilibrium model,
*
yi, p -
Ki,
pxi, p
(3.22)
function
is
the equilibrium constant,
of
a
as
expressed
p,
temperature
and pressure.
compositions,
Ki,
NC
(3.23)
yi, p
Interface model for the non-equilibrium model,
(the non-equilibrium model assumesthat at interface physical
is
attained)
equilibrium
II
Ki,
yi, p =
pxi, p
NC
Exii.
NC
yt,
p
i=l
45
(3.24)
(3.25)
,p
Physical
properties:
K-Values: Vapour-liquid
i on tray p is given by,
Ki,
p
or
i=ll
equilibrium constant of component
(T)
pat
sp zip
Oy P
lip
Ki, - lip
P OV
zip
(3.26)
(3.27)
p=ly
INC,
... yNP
Equation 3.26 represents the 7-0 approach, where an activity
liquid
is
the
to
coefficient model
used
calculate
phase activities while an equation of state is used to calculate the vapour
fugacities.
(EOS)
If
is
phase
an equation of state
chosen to
(0
fugacities
for
both
the
the
calculate
approach),
phases
-0
by
is
Eq.
3.27.
vapour-liquid equilibrium constant
given
Different types of thermodynamic models can be used, according to the mixture of the problem studied. For the systems
been
have
in
this
thesis,
used and are
studied
several models
listed below.
For
the
where
-y -0 approach was needed:
systems
The following models for activity coefficients were used,
UNIFAC-vle
(3 parameters)
UNIQUAC-modified
(Suzuki
1970).
Margules
al.,
et
equation
modified
The vapour phasewas considered as ideal vapour, so that,
OV = 1.
0-0
For
the
needed:
was
approach
systems
where
Two different equation of state were used:
Soave-Redlich-Kwong
and,
state;
of
equation
Peng-Robinson
equation of state
The general model of this work is not limited to the thermodynamic models listed above. Table 3.1 give a more complete
46
list of models available within the framework of ICAS (Gani
1997).
et al.,
Activity
Coefficients Models
Equations of State
Ideal Gas
SRK-eos
PR-eos
ALS-eos
SWL-eos
Virial
UNIFAC-vle; Ile
UNIQUAC-original
UNIQUAC-modified
NRTL
Wilson
Margules equation
Van Laar
Table 3.1: Thermodynamic models
Enthalpies:
*
Enthalpies are calculated as a function of temperature, composition and pressure.
Hvp =f
Hlp =f
(yi,
Pp Tp)
pI
7
(xi, Pp, Tp)
pi
(3.28)
(3.29)
See Appendix A for details.
(Antoine
Equation):
Vapour
*
pressure
at =
p is,
p
Bi
exp[Ai
_1
ci + TP
(3.30)
Density:
*
Densities for the liquid phase are calculated through a suitable
(Daubert
Danner,
DIPPR
and
correlation
as
such
correlation
Ideal
by
the
for
the vapour phase are calculated
1986) and
Gas Law or an equation of state (EOS).
TLP =f (Xi,PlTP,PP)
(Yi, TPIPP)
TV
P7
p=f
See Appendix A for details.
47
(3.31)
(3.32)
Molecular
*
weight:
NC
ýWL
p
ML i,
pXi,p
(3-33)
mv i,
pyi,p
(3-34)
NC
vV
p
Viscosity:
*
ViscositY for liquid and vapour phase are calculated through
(Daubert
DIPPR
Danner,
a suitable correlation such as
and
1986)
(Xi,
PL
pf
p iTp)
(yi, TP)
p,
pvpf
(3.35)
(3-36)
See Appendix A for details.
Surface
tension:
*
Surface tension is calculated through a suitable correlation
(Daubert
DIPPR
Danner,
1986)
and
such as
UP
(xi,
=f
p)
(3-37)
See Appendix A for details.
Thermal
*
conductivity:
Thermal conductivity for liquid and vapour phase are calculated through a suitable correlation such as DIPPR (Daubert
Danner,
1986)
and
ALp
f (Xi,
piTp)
-xv
pf
(yi, TP)
pI
(3.38)
(3-39)
See Appendix A for details.
diffusion
Binary
*
coefficients:
'The coefficient of Mterdiffusion of two liquids must be considdepending
on all the physical properties of the mixture
ered as
48
laws
be
to
according
which must
ascertained only by experiC.
Maxwell
in
Encyclopedia
the
Brittanica,
ment'-J.
writing
1952.
Dý3
f (T, M V)
ij
1
Dý3
(xi,
f
T,
M,
V)
13
(3.40)
(3.41)
See Appendix A for details.
Kinetic
models:
Reaction kinetic is assumedto follow, for example, an 'Arrheniustype' equation in the form,
Nr
Np
]I
Il
(T)
k+
(T)
A'
kAO
ri, p =
Ea
k+(k-)
k,,
with
exp(- RT
=
or
i=l)
where
K,,
q
k- -1 K,
(3.42)
(3.43)
(3.44)
q
p=lj
INC,
... )NP
IT
A2
f (T) = K(T,, )exp(A, ((I - -) +
In(-)
T
T,,
TO
2-T,,
3
2) + A5(T _T3) 0
+A3(T - T,,) + A4(T
4
(3.45)
+A6(T _T4)
0
A
be
where
can
compositions, concentrations or activity coeffiNp
Nr
the
reactants,
represents the products;
cients,
represent
NR is the number of reactions occurring in the system.
Total reaction rate (sum over all reactions occurring in the stage)
for component Z in the tray p is given by SP (term added to the
zip
balance
in
tray):
component mass
each
SP
lip
i=l)
NR
E(rk,
=
pVip)
k=l
p=ly
yNC,
... yNP
49
(ç)
(3.46)
In this work, we have assumed the reaction to occur in the bulk
liquid only and not in the mass transfer film, although this aslimits
for
the
the general model. The
sumption
application range
thickness of the liquid film (as well as the vapour film) is very difficult to predict (or estimate) and it is normally considered to be
relatively small, also with respect to the vapour film. So that, the
interactions between diffusion and chemical reaction in the liquid
film have not been taken into account due to the 'predicted' small
thickness of the liquid film. Moreover, in a recent paper, Higler et
(1997)
al.
pointed out that the reaction volume for the mass transfer film is negligible under certain conditions, with respect to the
bulk liquid volume. The main limitation of the model under this
be
for
fast
assumption will
reactions.
Rate
equations
-
Mass
Transfer
*
One of the first assumptions here is that the bulk phases are
has
The
for
been
the
transfer
completely mixed.
model
mass
based on the model of Krishnamurthy and Taylor (1985) for
distillation
steady state simulation of
columns.
The molar fluxes (mass transfer) in each phase are given by,
Ji' + Nýyj,
Ni'
p
21P ttp
NT'
Nil
Jil +
xi,
p
zip
zip
(3.47)
(3.48)
i=l,..., NC-1, p=l,..., NP
The relationship between the molar fluxes and the mass transfer rates is given by Eq. 3.18 for the vapour phase and Eq.
3.19 for the liquid phase.
Since many multicomponent distillation systems and specially
behaviour
by
of
reactive systems are characterised
non-ideal
the mixtures, the influence of the non-idealities on the mass
transfer fluxes should be included when defining the fluxes.
The good representation of the mass transfer fluxes can be
compared with a good representation of the phase equilibria
diffusion
fluxes,
Hence,
the
when equilibrium models are used.
50
J, in our model are given by,
Jiv = ;5v [K'] (yi,
yip)
p
tfp
p
[IN][Kl] (xi,, xi,
i=
;
5L
i"P
p
21P p)
i=ll
(3.49)
(3.50)
NC-1, p=l,..., NP
The mass transfer rates (nY ný, are obtained by combining
P)
'P,
Eqs. 3.47-3.48and 3.49-3.50and multiplying by the interfacial
for
(Krishnamurthy
transfer
Taylor,
area available
mass
and
1985).
nvi, = pva[K'](yi,
p
n',ipi, = TLa[]F'][K'](x,
i=ll
+
y,
n'yi,
i, p)
T p
p
+
xi,
n'
TXi, P
p)
p
(3-51)
(3-52)
NC-1, p=l,..., NP
High flux corrections have not been used because they are
(normally
Nt
for
distillation
the
unity when
case
=0
units).
It can be seen from Eqs. 3.51 and 3.52 that the mass transfer
functions
rates are
of multicomponent mass transfer coeffiQKv],
[K]),
(a),
interfacial
density
cients
area
of the mixture
(p), compositions in the bulk and at the interface as well as the
total mass transfer rate of the components (nvTT n' ), present
,
in the mixture.
A further
is
depending
the
optional
assumption, which
on
be
is
is
that
to
the
to assume
going
problem
solved,
nature of
This
is
transfer,
nvTT =0 and n' =0.
equimolar counter
for
for
distillation
distillation
in
general
and
reactive
valid
in
during
is
total
the
there
number of moles
when
no changes
diffusion with chemical reaction. Following this, Eqs. 3.51
be
3.52
and
can
reduced to:
v
(yi,
ni, p = pvpa[Kv]
yi,
p
p)
1
(xjI,
j5LPa[rli[Kl]
ni, p =
xi,
p)
tip
i=lp
(3-53)
(3-54)
NC-1, p=l,..., NP
from
the genwhich makes another possible simplified model
hybrid
is
dealing
the
eral
model, and
usual approach when
51
distillation.
This
is
with
assumption
equivalent to consider
that the heat of vaporisation of the components are equal to
each other.
When the assumption is not taken into account, an iterative
is
method needed to calculate nTand n'il niv. Repeated subfluxes
(for
from
the
stitution of
starting
an estimate
example,
few
0)
iterations
If
take
to
nt =
will
converge. nTis assumed
fluxes
be
to
then
the
equal
zero,
can
calculated without any
iterations.
The interfacial area is calculated through:
for tray columns the net interfacial area a is (taken from
Taylor and Krishna, 1993):
(3.55)
a'hfAb
froth;
is
interfacial
the
where a'
area per unit volume of
hf is the froth height; and, Ab is the bubbling area.
The interfacial area per unit volume a' is calculated through
the following expression: (Bravo and Fair's correlation)
e.
a
19.78Ap(Rc'Ca
5
1)0.392
HOA
(3.56)
Reynolds
Rev
is
the
the
vapour phase,and
of
number
where,
Cal is the liquid capillary number given by,
v
Re
Cr
PVPU
(3.57)
pvAp
ulill
al =
i5:
(3.58)
The matrix of multicomponent mass transfer coefficients for
[K']),
binary
QK"],
from
transmass
are calculated
mixtures,
fer coefficients, following Krishna and Standart (1976) method.
This method is applicable when the film thickness is not known,
in
is
is
this
the
the
approach.
used
method
reason why
and
The film thickness is (unfortunately) almost never known.
52
for
(Krishna
the
Standart,
1976):
vapour
phase
and
[R]-'
(dimension (NC-l)x(NC-1))
Rý,,
R,
by
following
ý,
the
with
and
given
expressions:
j
j
NC
Yi,NC.
Ri.
t't - ki NC
,
k=l, k:oi
(
-yi, j ikiij
j
R'
Yi,k
ki,
ki
(3-59)
k
(3-60)
NC
i=l,..., NC-1, p=l,..., NP
for
(Krishna
liquid
Standart,
the
1976):
phase
and
[Rll-'
(dimension (NC-l)x(NC-1))
ý
by
following
R,
R(,
the
with
and ,j given
expressions:
,j
NC
Xi, NC
w.
2,2
Rý3-=
i=lj...
ki,Nc
+
Xi, k
Y-d
ki,
k
k=l, k:Ai
I1
-Xi, -'-( ki,
j
ki,
(3.61)
(3.62)
NC
p=lj
jNC-1j
... INP
The binary mass transfer coefficients ki,k are obtained from a
suitable correlation, such as:
(AIChE,
AlChE
Method
1958)
the
for
the
vapour phase,
kv =
(0.776 + 4.57h,, - 0.238F, + 104.89-1)
w
SCO. 5atg
(3.63)
Eq. 3.63 is derived for the case where mass transfer rein
ideal
law
is
the
gas phase, and
sistance exists only
gas
h,,
(m),
is
length
F,
to
the
also assumed apply.
exit weir
is the is the superficial factor (defined by, F, = u,;5vo-5),
Q1 is the volumetric liquid flow-rate (m'/sec), W is the
53
(m),
length
Sc
is
the Schmidt number for the vapour
weir
(defined
by, Se = TvDv
"" ), a is the interfacial area
phase
is
t9
the gas contact time.
and
for
liquid
the
phase,
(1970OD' 0-(0.4F, + 0.17)tl)
kl =
(3-64)
atl
D'
is
Wlsec),
Fick
liquid
diffusivity
the
the
where
of
system
is
liquid-phase
(defined
tj
the
by,
time
tj
and
residence
z
U+I
The AlChE correlations are applicable for bubble cap
trays or sieve trays. It is important to note that these
binary mass transfer coefficients are functions of tray delayout
(or
in
type
sign and
packing
and size,
a packed
This
design
be
that
the
column).
means
column
must
known in advance in order to solve the equations for the
be
However,
these
rate-based model.
parameters can also
determined by carrying out design calculations simulta(Taylor
the
the
model equations
neously with
solution of
1992).
et al.,
The correlation for calculating the binary mass transfer
fluxes is derived under the assumption of vapour-phase
(as
that
this
correlaso
stated earlier),
resistance only
tions should be used with models which considers only rein
the
vapour phase, as a matter of consistency.
sistance
Furthermore, the correlation is derived from experiments
in steady-state processes. The validity of them for dyliterature,
in
been
have
the
reported
not
namic operations
(1998)
Kreul
this
the
where
et al.
work of
with exception of
dynamic
for
has
been
tested
packed columns
correlation
given acceptable results.
In distillation the transfer resistance is almost entirely located on the vapour phase, according to experience and
(Vogelphol,
literature
in
the
experimental work presented
1979; Kayhand et al., 1977, Dribika and Sandall, 1979).
However, there is no evidence that this can hold for redistillation.
Furthermore,
active
some experimental work
54
liquid
in
distillation
the
that
resistance a
shows
column
be
20%
(Chen
the
can
as much as
of
entire resistance
and
Chuang, 1995).
New correlations are neededwhen both phasesresistances
believed
It
is
that new correlations are
are considered.
for
dynamic
also needed
system and specially for reactive
Nevertheless,
in
this work mass transfer correlasystems.
tions as described above are used for dynamic models and
for reactive systems, in an attempt to verify if the correlations lead to acceptable results. Pelkonen et al. (1997)
be
that
transfer
proved
steady-state mass
correlations can
for
dynamic
distilapplied
simulation of multicomponent
lation.
If the stagesof the model are consideredvery closeto each
other then we could used the model to simulate packed
The
columns as well.
sameequations are used with exception of the binary mass transfer coefficients and interfacial
area.
So that, for packed columns the interfacial area is calculated from,
for packed columns the net interfacial area a is (taken
from Taylor and Krishna, 1993):
a= ahA,
(3.65)
h
is
the
interfacial
is
the
volume;
unit
area per
where a'
height of the packing section; and, A, is the cross-sectional
the
column.
of
area
The binary mass transfer coefficients can be calculated
through different expressions,for example,
for
(Onda
1968)
Onda's
random
packal.,
et
correlation
ing in packed columns,
for the vapour phase:
-
55
The binary kij are calculated from the Sherwood number
as:
k.2937S
0.33 (ApDp )-2
(3.66)
Sh
A,,Re'o- Cv
DjjAp
Dp
is
Ap
is
the
the specific
where
nominal packing size and
2/M3)
(m
A,
is
the
that
surface area of
packing
a
constant
.
takes value of 2 if the nominal size of packing is less than
0.012m or 5.23 if Dp is higher or equal to 0.012m. Re'
is the Reynolds number of the vapour phase (Eq. 3.57).
The vapour phase Schmidt number is,
scv
it
-
(3.67)
pv Dv
for the liquid phase:
The binary liquid phase mass transfer coefficient is obtained from the following equation,
/ig)0.333
(;
5L
kl
)0.667SCI
---
0.0051(Re"
-0-5 (apdp
)0.4
(3.68)
liquid
for
Schmidt
Scl
is
the
the
phase
number
where
Scl = pll(; 5LD')
(3.69)
interfacial
based
the
Reynolds
Re"
is
the
on
number
and
area
Re" = pLul/(ttla')
(3.70)
Using this correlation with the expression from Bravo and
k,
different
lead
to
Fair for the interfacial area will
value of
a
is
for
interfacial
Onda's
the
if
area used.
than the
correlation
However, there are no explanations about which one of the
Many
knowledge.
best
the
is
to
the
of our
one
correct
values
'combined'
literature
this
in
taken
the
are
works presented
(Taylor
is
Accuracy
or consistency not reported
equations.
Krishna,
1993).
and
Each stage model represents a section of packing and the
height of this packing section should be specified. Even small
56
heights
therefore,
stages,
more calculameans more
section
tions are as well needed. This is directly reflected in the
increasing of computational time. As more stages are then
included, the model accuracy increases,but how many stages
be
Experience
imshould
used still remains vague.
plays an
determining
'number
the
portant role when
of stages' repreheight
senting a
of packing.
Heat
transfer equations:
*
The energy flux for each phase is defined by,
NC
Ov = a'(Tv - TPI)+
pppi,
(3.71)
nyPH" p
NP
p=1,...,
NC
(TI-Tl)+Enli,
a'
ppppp
Hi,
i=l
(3.72)
P
P:--::
ý'll INP
Wv,
H1
heat
the
transfer
the
are
partial
and
coefficients,
a are
for
in
the
vapour and
p,
stage
1
molar enthalpy of component
liquid phase respectively.
Heat transfer coefficients are calculated following the wellknown Chilton-Colburn analogy. The energy transfer rates for
the vapour and liquid phases are obtained after multiplying
Eqs. 3.71 and 3.72 by the interfacial area and adding the high
flux correction factor expe'vC,-1 as follows
NC
OP
v
= avP a
expev
--v
H
(T' - TI) +
Ppt,
nyp i, P
(3.73)
ni,pHi,p
(3.74)
Pýll ... INP
NC
pP
a' a
expev
NP
p=1,...,
57
(TI - Tpl)+
P
A simplification of these equations (and the model in general)
be
by
can
achieved
considering that there is no differences in
temperature between the phases. So that, Eq. 3.73 and 3.74
be
can
reduced to:
NC
ov = Env
i,P i,p
p
i=l
(3.75)
NC
-1
1:
OPI
nýzip
(3.76)
These are the equations (3.75-3.76) that are used for the simin
this work.
ulations
Hydraulic
equations
The pressure drop between trays is calculated from the pressures
of the adjacent trays:
App= pp-1- pp
(3.77)
Flow-rates of liquid, vapour and entrainments rates (liquid and
functions
hold-up
tray
geometry,
of
and pressure
vapour) are all
drop between trays (Gani et al., 1986)
(AP, holdup, geometry of the tray
Lp =f (holdup, geometry of the tray)
ELp =f (AP, holdup, geometry of the tray)
(3.78)
EVp =f (AP, holdup, geometry of the tray)
(3-81)
V=f
P
(3-79)
(3-80)
functions:
Defined
Hold-ups:
NC
mv
=
p
mvi,p
(3.82)
Mý27P
(3-83)
NC
Mi
=
pE
58
P=11... yNP
Hm'
HE
vp
E%=l
INCmi,vp
HE
lp
p
Brm
J: ýVc Mý
zip
1=1
p
(3.84)
(3.85)
NP
p=1,...,
Efficiency parameter:
pMp
J-Ii,
Yi, p+l
-
Yi, p
(3.86)
yi,p+l - Yý
lip
i=ll
3.6
Generation
NC-1, p=l,..., NP
Models
of
Hybrid models indicate the use of more than one type of model for the simdistillation
For
hybrid
ulation/design of a single
column.
example,
models
having
trays that are at equilibrium and trays that
may represent a column
are not at equilibrium or a column with reactive and non-reactive trays, or
designs.
different
tray
a column with
Also, for a specific simulation of distillation operation, use of different model
forms may require the use of mixed simulation modes (steady state and/or
dynamic simulation modes). Hybrid models are useful for synthesis/ design
how
because
indicate
they
a specified separation can
of separation processes
be achieved.
Note that the same hybrid model is applicable even though different distillation operations may be necessary to achieve the specified separation.
Use of the hybrid modelling approach may result in improved computational
better
because
it
inito
the
can provide
single model
efficiency compared
For
flexibility
in
terms
tialisation and
example, since
of modelling options.
have
the rate-based models usually
more equations than their corresponding equilibrium-based form, if any tray reachesequilibrium, the equilibriumbased form for the remainder of the dynamic simulation run may model it.
Also, most reactive distillation columns usually have non-reactive trays. In
59
be
such cases,simulations may
started with a totally reactive or non-reactive
column and gradually changed to the required hybrid form. In addition, the
have
trays
reactive and non-reactive
may
rate-based or equilibrium-based
models.
The non-equilibrium model is obtained by including massand energy transfer
models and specifying that equilibrium is attained only at the interface of the
liquid and vapour phases. The rate-based model consists of Eqs. 3.16-3.17,
3.20-3.21,3.24-3.25,3.26 or 3.27,3.51-3-52,3.73-3-81, plus defined relations
for
If
be
is
to
and a selected set of equations
properties. reaction
considered,
Eq. 3.46 is added to the system. Models of various degreesof complexity are
by
different
forms
For
the
using
obtained
of
algebraic equations.
example,
for
3.26(or
3.27)
3.51-3.52
expressions
equations
simple
or
or 3.73-3.74 will
lead to simpler versions of the rate based model. The number of variables
(rate-based
for
in
Tables
tray
3.2
a single
and equations
given
model) are
and
3.3 respectively. Note that the procedure equations (for example, representation of property models as library routines where the calculated property
for
specified values of temperature, pressure and/or comvalues are obtained
listed
in
In
these
tables.
they
and
variables
are
not
principle,
can
position)
(temperaknown
be
intensive
the
values of
variables
calculated with
always
ture, pressure and phase compositions). For all the types of non-equilibrium
by
ODE
form
derived
dynamic
the
the
of
mass
models
are
using
models,
derived
by
balance
setequations while steady state models are
and energy
ting the left hand side (LHSs) of the same set of mass and energy balance
belonging
dynamic
Note
that
to
models
all non-equilibrium
zero.
equations
to the general distillation model are represented by a set of differential and
(DAEs)
index
1.
of
algebraic equations
From the general non-equilibrium model, various model forms can be genif
For
there
by
are no
example,
making simplifying assumptions.
erated
Similarly,
drop
((NC
terms
1)NR)
+
out.
variables and reaction
reactions,
if the pressure profile is specified, one variable (tray pressure) and one equaforms
different
Three
drop
(tray
hydraulics)
of ratetion
examples of
out.
based models (reactive and non-reactive) are listed in Table 3.4. Similar
Taylor
by
Kooijman
been
have
and
earlier
reported
models
non-equilibrium
(steady-state
Taylor
Krishnamurty
1995)
(dynamic model,
model,
and
and
1985) for example.
is
The equilibrium-based model a special case of the rate-based model, and
60
[Variable
I Number
Vapour and Liquid flow-rate
Vapour and Liquid compositions
Vapour and Liquid temperature
Vapour and Liquid
interface
compositions
Interface temperature
Mass transfer rates
Energy transfer rates
Stage pressure
Rate of reaction
Heat of Reaction
Total
2
2NC
2
2NC
2NC
2
I
NC*NR
NR
6NC+8+(NC+I)*NR
Table 3.2: List of variables per stage in the non-equilibrium
model
therefore, is a simplification of the rate-based model. Instead of considering
is
it
is
interface,
that
the
equilibrium
attained
assumed
equilibrium only at
Therefore,
it
is
in
liquid
the
come contact.
vapour and
at all points where
in
in
the
the
and
entirely column.
stage
points
assumed equilibrium at all
This means that Eqs. 3.16-3.17 Eqs. 3.20-3.21 are combined as Eqs. 3.88
3.89
respectively.
and
dmT
! I'-P=
dt
Fpzi,
p
-(PVp
+ Vp+lyi,
p+l
+ Lp-lxi,
+ Vp + EVp)yi,
11 NC, p
... I
p-
p-l
(PLp
+ EVp-lyi,
p-l
+ Lp + ELp)xi,
+ ELp+lxi,
p
+ Si,
p
p+l
(3.87)
NP
+,
rny
mil,
mT
where lip =
lipp
from
drops
term
the
hold-up
is
If the vapour
out
vapour
assumed negligible,
(as
for
dynamic
the
thereby,
example,
model
the mT
generating a simpler
P)
z?
for
the
(1986)).
The
Gani
by
at
equilibrium
condition
et
al.
one presented
heat
transfer
(Eq.
3.25)
3.24the
related equations
interface
mass and
and
but
the
3.51-3.52,3.73-3.74)
(Eqs.
equilibrium condition
are not needed,
(Eqs. 3.22 and 3.23) are added.
is
for
balance
oban equilibrium-based model
Again, the energy
equation
61
M
M
E
E
R
R
R
S
H
Q
Equations
Number
Material balances for liquid and
vapour
Material balances at the interface
Heat of reaction
Energy balance equations
Rate of reaction
Transfer rate equations
Energy transfer rate equations
Summation equations
Hydraulic equations
Interface equilibrium calculations
Total
2NC+l
NC
NR
3
NC*NR
2NC-2
2
2
2
NC
6NC+8+(NC+I)*NR
Table 3.3: List of equations per stage in the non-equilibrium
model
tained as a special case of the rate-based model (employing the same asbalance
for
equations).
mass
sumptions as
dHT'
p
+ Vp+, Hvp+l
FpHfp
dt
+ Vp + EVp)Hvp
-(PVp
Hlp-l
+ Lp-,
-
(PLp
i=
where
... I
NC,
Hvp-l
+ ELp+,
Hlp+l
+ Lp + ELp)Hlp
(3-88)
+Qp + DSp
11
+ EVp-,
p
NP
E
HTE
p=
Hip + HýP.
further
is
Eq.
3.89
is
hold-up
If the energy vapour
simassumed negligible,
EE
Hip.
HT
Hence,
drops
term
the
out.
energy vapour
as
plified
P=
If reactive distillation is considered, Eq. 3.46 is added to the equation set.
has
term
the
This form of the equilibrium-based model, without
reaction
(1997),
for
(1994)
Skogestad
Sorensen
by
example.
been earlier reported
and
ODE
form
by
derived
the
is
the
using
The dynamic version of
model again
the
balance
of
version
state
while
steady
equations
the
energy
and
mass
of
by setting the LHSs of the same set of mass and energy
derived
is
model
for
The
the
to
equations
of
variables
and
number
zero.
balance equations
62
Model
Constant
NEQ-1
NEQ-2
NEQ-3
Equal
pressure
Mass
resistance in one
phase
temin
perature
both phases
Number of equations
to solve per stage
yes
no
no
6NC+7
yes
reactive
6NC+7+(NC+1)*NR
for reactive
4NC+5
for
non-
yes
reactive
4NC+5+(NC+I)*NR
for reactive
4NC+4
for
non-
no
yes
yes
yes
for
non-
reactive
4NC+4+(NC+1)*NR
for reactive
Table 3.4: Various forms of non-equilibrium
models
listed
in
Tables
3.5
3.6
general equilibrium-based model are
and
respectively.
Like the non-equilibrium dynamic models, the equilibrium dynamic models
by
DAEs
index
1.
are also represented
a set of
of
Variable
Vapour and Liquid flow-rate
Vapour and Liquid compositions
Temperature
Stage pressure
Rate of reaction
Heat of Reaction
Total
Number
2
2NC
NC*NR
NR
2NC+4+(NC+I)*NR
Table 3.5: List of variables per stage in the equilibrium model
Various forms of the equilibrium-based models can be generated from the
For
Eqs.
3.88
3.89
form
the
of
equilibrium model.
example,
and
general
by
hold-up
is
that
the
be
assuming
molar vapour
negligible
simplified
can
liquid
hold-up.
form
Using
the
the
the
to
molar
general
of
physical
compared
liquid
in
the
two
case of
phases equilibrium with one
equilibrium condition,
(Bossen
(1993),
(1990)).
is
Rovaglio
Doherty
obtained
et al.
and
phase
vapour
be
by
Further simplification can
made
assuming that the rate of change of
63
M
E
E
R
S
H
Q
Number
Equations
Material balances
Heat of reaction
Energy balance equation
Reaction Rate
Summation equations
Hydraulic equations
Equilibrium calculations
Total
NC+1
NR
I
NC*NR
1
I
NC
2NC+4+(NC+1)*NR
Table 3.6: List of equations per stage in the equilibrium model
the energy accumulation term (LHS of Eq. 3.89) is negligible. This assumption will convert Eq. 3.89 to an algebraic equation. Also, this assumption
longer
independent
is
tray
the
that
variable.
pressure
no
an
usually means
Therefore, the tray pressure is removed from the list of variables and the
hydraulic equations are reduced by one (usually, the relationship of pressure
drop, liquid hold-up on tray and vapour flow-rate is not used). Each verfor
further
in
be
terms
these
of models used
simplified
models can
of
sion
lists
Table
3.7
the
hydraulics
models
tray
some of
and physical properties.
Note
that
from
be
the
that can
model.
general equilibrium-based
generated
(for
been
have
these generated models are not new and
reported previously
Doherty,
(1995),
Rovaglio
(1986),
Gokhale
Gani
by
and
et
al.
et
al.
example,
(1990)). Note also that the number of equations in models EQ-2, EQ-3 and
different
AEs
is
ODEs
between
distribution
but
the
EQ-4 do not change
and
has
EQ-2
ODEs
the
largest
has
EQ-4
the
for each model.
while
number of
lowest number of ODEs (not counting EQ-1).
it
distillation
different
for
Hybrid models are useful
operations where may
in
improved
it
be necessaryto achieve a specified separation, and will result
it
because
the
to
procan
model
single
compared
computational efficiency
hybrid
decision
The
flexibility.
model
better
initialisation
a
of
using
and
vide
distillation
for
for
the
with
be
reactive
of
case
example
made a prZorl,
can
for
be
it
trays,
example,
posteriori,
a
made
can
or
reactive and non-reactive
(the
between
simulation
models
equilibrium
and
non-equilibrium
switching
have
determine
trays
to
have
equilibrium).
reached
which
started
should
distillation
the
monitoring of a selected set of variThe general
model allows
For
the
function
tray.
time
(or
example,
and/or
of
as
a
phenomena)
ables
64
Model
EQ-1
EQ-2
EQ-3
EQ-4
Assumption
Number of equations
to solve per stage
Rate of change of en- NC+I
for
nonergy neglected. Con- reactive
stant molar overflow.
NC+I+(NC+1)*NR
for reactive
Rate of change of en- 2NC+3
for
nonergy neglected
reactive
2NC+3+(NC+I)*NR
for reactive
Liquid hold-up and 2NC+3
for nonhold-up
energy
reactive
considered
2NC+3+(NC+1)*NR
for reactive
Liquid and vapour 2NC+3
for
nonhold-up considered
reactive
2NC+3+(NC+I)*NR
for reactive
Table 3-7: Various forms of equilibrium models
V
An,
Ani
general model computes the values of
n'
ntv+l
nt
= nl+,
=
tt
,
Gibbs
G'
free
EM
funcexcess
energy,
and an efficiency-like parameter,
as a
tion of time and tray.
For the rate-based model, the excessGibbs energy for the liquid phase can
be estimated without significant additional computation because the liquid
(see
known
Eq.
3.89).
phase activities are already
GE
p
RTp
i=ll..
i
In
xi, p 7i, p
(3-89)
p=lj
... INP
)NC,
Note that these estimated values of tray excess Gibbs energy change with
time for the dynamic model and that the plots of excessGibbs energy as a
function of time are not necessarily the same as plots of excess Gibbs enfunction
Therefore,
it
is
to
the
of
composition.
possible
monitor
ergy as a
Gibbs
for
Those
trays.
that approach an
excess
energy
transient path of
all
65
asymptotic minimum value may be considered to be at a pseudo-equilibrium
state.
For the non-equilibrium model, the difference of the values of n', nv (repreby
Anj)
Anv
be
function
sented
and
can also monitored as a
of time and tray
Generally,
difference
time.
the
without significant additional computation
of
the values of n', nv in a time step before and the present time should be ex(approaching
decrease
Gibbs
to
the
pected
zero) as
excess
energy approaches
a minimum.
The values of AnI, An, and GE can then be used as a criterion by which the
be
its
to
switched
rate-based model may
corresponding equilibrium-based
form. At equilibrium, the Gibbs energy must be at a minimum. Also, at
Ani
An,
be
and
must
approximately zero.
equilibrium,
Efficiencies are also a way to look for equilibrium.
type of efficiency have been investigated.
3.6.1
Tray
In this work, two different
Efficiency
Two types of efficiencies have been used in this work. 'Standard' Murphree
deBoth
Murphree
types
are
vapour phase point efficiency.
efficiency and
below.
scribed
Murphree
'Standard'
efficiency
o
Taken the Murphree tray efficiency (King, 1980) as a basic, an efficiencylike parameter that is a function of only the simulated bulk and interface compositions from the non-equilibrium model, is proposed,
Em
Yi,p+i - Yi,p
i',p = yi,
Yi,
p+l
lip
(3.90)
i=l,..., NC-1, p=l,..., NP
for
bulk
the
In the above equation, Yki represents
vapour composition
k
in
the
tray
composition
while
equilibrium
y*j
represents
Z
component
k
66
for component Z in tray k (at the interface). With the calculated
comfrom
hybrid
the
positions available
model, the component efficiencies
for all components on each tray can be monitored as a function
of time.
Note that the use of measured bulk compositions in Eq. 3.90 instead
bulk
the
different
(see
of
simulated
compositions may also give
values
Chapter 5).
Murphree vapour phase point efficiency
Point efficiencies for multicomponent systems are given by (Toor, 1964),
(AY)= [Q](Ay')
(Ay)
(y*
(Ayl)
where
yp),
=
pp
[Nov]
by
with
given
(y*
),
[Q]
[Nov]],
=
= exp[- yp+, and,
11+
[Nov]
(3-91)
M(VIL)
[Nv]
(3.92)
[NLI
[Nv], [NL] are the numbers of transfer units for the vapour and liquid
The
be
by,
the
phases respectively.
elements of
matrix can
calculated
Nvi = k'ahf /u,
NLi = klahfZI(QIIW)
(3-93)
(3.94)
hf
froth
height,
is
is
interfacial
Z
the
the
where a
area per unit volume,
is the liquid flow path length, W is the weir length, Q1is the volumetric
liquid flow-rate and u, is the superficial vapour velocity (based on the
bubbling area of the tray). Other models of flow and mass transfer on
[Q],
form
but
different
lead
distillation
to
tray
of calculating
may
a
a
the form of Eq. 3.91 can be retained.
The relationship between the elements of [Q] and the component Murfollows
is
for
a quaternary system:
phree efficiencies as
There are only three independent efficiencies given by
Elov
Qll
Q12 AAY2E
AYlE
67
Q13
AY3E
AYlE
(3.95)
Eov
2
=IEov
=I3
Q22
Q33
-
Q21
AYlE
AY2E
AYlE
-Q
'JJLAY3E
Q23
AY3E
AY2E
AY2E
32AY3E
Q
(3-96)
(3.97)
the fourth dependent efficiency can be calculated with the help of the
(3-95-3.97)
three
other
equations
given,
V
E40
+ AY2 + AY3
141
+,
+
ý42E
1ý41E
4ý43E
(3-98)
If the vapour leaving the stage is assumed to be at equilibrium with
the liquid leaving the stage, then the matrix [Q] is null, and therefore
(as
be
from
Eqs.
the
3.95all
component efficiencies are unity
can
seen
3.98)
[Q] will be different from the 0 matrix whenever the stage is not at
be
There
equilibrium.
could
a case where the approach to equilibrium
is
be
by
That
the
the
the
of all
components
same.
will
represented
[Q] matrix being diagonal with all the elements in the diagonal equal
[Q]
ie.
is
This
is
to
the
a constant.
one
another,
= q[I], where q
have
that
the
the same efficiency:
all
components
same as considering
According to Agarwal and Taylor (1994) this situation
EO' =1-q.
by
happen
the
mixtures are constituted
similar components
when
will
in which the binary pair mass transfer coefficients are equal and the
liquid
is
in
the
transfer
phase entirely negligible.
resistance
mass
[Q] diagonal with unequal diagonal values is the situation most often
data
in
to
some components
match experimental
encountered practice:
[Q]
be
diagonal
different
when
will also
values.
are given efficiencies of
the effective diffusivity model of mass transfer is used. If we assume
that [Q] is a function of [NOV] and, since the elements of the matrix
[NOV] are a function of the mass transfer coefficients of both vapour
diagonal
[Q]
be
longer
in
liquid
then
general.
will not
any
phases,
and
Thus, [Q] will have non-zero off diagonal elements together with unbe
diagonal
These
arbitrarily specified
elements.
elements cannot
equal
(mass
balances
and non negativwithout violating physical constraints
ity of mole fractions) (Agarwal and Taylor, 1994).
68
Efficiencies computed with the corresponding mass transfer rates Eq.
3.91 may be quite different than those calculated with Eq. 3.90.
3.7
Condensers
Reboilers
and
Condensers and reboilers for distillation or reactive distillation are modelled
differ
from
These
the other equilibrium stagesin
stages
as equilibrium stages.
the column becausethey are a heat source or a sink of the column. They can
be total or partial. Efficiencies of reboiler and condensersare usually taken
to be unity (so that the efficiency-like parameter is not applied to these
is
liquid
the
the
the
vapour entering
stages):
condenser at equilibrium with
liquid
is
the
the
entering
reboiler
at equilibrium
condensate and, similarly
leaving
Different
the
the
vapour
with
stage.
specifications are usually made
for these stages (two are needed for the simulation of the full distillation
for
for
the condenser):
the
reboiler and one
column, one
distillate
flow-rate
product stream
of
*
bottoms
flow-rate
product stream
of
9
flow-rate
e reflux ratio or reflux
boil-up
boil-up
ratio
or
e vapour
dutY
heat
of condenser or reboiler
*
distillate
in
the
either
some specific composition of a given component
bottoms
product
or
The independent variables in both casesare the compositions, the stage temfor
both
The
flow-rates.
the
condenser and reboiler
equations
perature and
below.
listed
are
Condenser
from
in
is
The condenser
which vapour
a vapour-liquid equilibrium system
All
heat
flows
the
through
tray
exchange
coil.
a
the top
condenses as coolant
in
is
total
fraction
it
the
or
a
condenser
of
a
case
condensed
of
a
or
vapour
respectively.
condenser,
partial
Equations:
*
69
9
Mass balance:
dT
mj,C VNPYi,
-'":::
dt
NP -
Dxi,
rxi, D
D
(3-99)
Energy
balance:
9
dHT'
C=
VNpHVNP-
DHID - QC
rHID-
dt
(3-100)
Physical
Equilibrium:
9
Yi, D =Ki,
(3.101)
DXi, D
In a total condenser, the vapour compositions (used in the equilibrium relations) are those that would be in equilibrium with the liquid stream that
actually exist.
Reboiler
The reboiler receives the liquid from the bottom tray of the reactive distillation column and partially (or totally) vaporises the liquid using the heat
bottom
is
The
the
to
the
tray
remaining
returned
while
vapour
provided.
liquid is the bottom product of the reactive distillation column.
Equations:
*
9
Mass balance:
T
dm i, B
dt
Ljxj,
VByi,
+
=
B
j -
BXi,
B
(3-102)
balance:
Energy
o
dHTE
B=
dt
LjHIj - BHIB-
VBHVB + QB
(3-103)
Equilibrium:
Physical
o
Yi, B =A,
70
i, BXi, B
(3.104)
3.8
3.8.1
Computational
Specifications
Simulations
Numerical
and
for Equilibrium
Aspects
Non-equilibrium
and
Both models require similar specifications in terms of distillation
decolumn
Feed
flow-rate,
feed composition as well as the state of the
sign variables.
feed need to be specified for a typical simulation problem. The
decolumn
tails include feed location, number of trays, product flow-rates, and product
Additional
specifications.
specifications such as reflux rate and vapour boilfor
both
if
has
the
up are needed
models
column
condenser and reboiler,
A
respectively.
rate-based model also needs the specification of model pafor
the transport properties models needed for calculation of mass
rameters
heat
transfer coefficients. Finally, variables related to column geometry,
and
diameter,
tray geometry, etc., need to be specified for dysuch as column
for
(steady
the
namic equilibrium models and
non-equilibrium model
state
dynamic
In
or
simulation modes).
general, the rate-based model needsmore
properties than the corresponding equilibrium-based model and for some of
these properties, if suitable prediction methods are not available, correlations
be
Obviously,
limit
to
the ability to predict the
need
used.
use of correlations
distillation column behaviour. The correlations used in this work, have been
because
best
but
because
known
they
the
they
chosen not
are
are well
and
be
introduced
Most
to
the
can
easily
model.
of the pure component property
(Daubert
from
databank
DIPPR
Danner,
1986).
taken
and
correlations are
3.8.2
Solution
Model
of
Equations
The model equations listed in Tables 3.4 and 3.7 are solved by a collection of
DAE and AE solvers. Details on the various solution approacheshave been
(see
for
Cameron
1986).
et al.,
example,
given earlier
In general, for the dynamic simulation mode, initial conditions for the ODEs
for
be
Note
AEs
to
the
to
the
need
supplied.
variables related
and estimates
(Gani
be
initial
initial
to
the
that the
condition and
consistent
estimates need
1989).
Cameron,
and
The general model and each of the models which can be generated from it,
defines a set of ODEs and a set of procedures to represent the process:
71
o
ODEs: mass balances
Procedures: thermodynamics, reaction term, mass and energy transfer
rates, etc.
The model is set up in a specific manner so that it can easily be switched
between solving a steady state or dynamic problem. This allows using the
dynamic model as a relaxation method for solving steady state problems
(Gani and Cameron, 1989). A problem is started with a steady-state simulation to initialise all the variables. After that, starting from the initial state
(all the differential variables are known) the solution of the set of ODEs and
for
be
the
equations
representing
algebraic
model
a particular problem will
following:
as
determining
the
algebraic variables using mass and energy transfer
ahydraulic
equations, reaction equations
rates, equilibrium equations,
defined
determine
hand
then,
the
equations and
side of all
and
right
the ODEs.
b- determining values of the differential variables at time tj = tj-l + At,
for
from
end.
aj
and repeating
At each time tj, a set of ODEs and algebraic equations are solved simultain
to
an equation-oriented package.
manner
neously a similar
(Note that here the procedure is described when solving the model in ODEs
form into the WAS simulator. WAS simulator also allows to solve the equations as a DAE system. The model is solved as a DAE system with analytical
Jacobian when gPROMS is used).
The algebraic equation set which includes all the algebraic equations and
ODEs,
the
determine
the
to
algebraic variables of
all
correlations required
been
has
defined
for example, prediction of physical properties,
equations,
divided into subsets for easy numerical solution (Gani et al., 1986).
(most
below
is
the
to
The detailed procedure given
most general model
solve
ICAS
inside
the
simulator.
complex non-equilibrium model)
Proc-edure:
72
Given:
differential
initial
the
the
9
state;
variables at
design
the
o
variables; and,
the 'input-output' variables (for condenser and reboiler): one product
heat
duty,
rate and one
definition
(the
liquid
the
hold-ups)
total
using
equations
molar
and vapour
the bulk-liquid and bulk-vapour compositions can be determined. Then,
1. Assume the plate pressure.
2. Given the plate pressure, the compositions of liquid and vapour phase
and the enthalpy of the plate, the temperature of the phases in the
be
determined.
plate can
3. Given, pressure, temperature, compositions and an estimate of the liqinterinterface,
the
the
the
vapour composition at
uid composition at
face can be calculated through calculation of the physical equilibrium
(using
by
iterative
thermodynamic
package
an
constant
means of any
If
'flash
the
the
to
this
sum
of
vapour comproblem').
method,
solve
from
different
is
interface
the
unity, continue, otherwise go
positions at
to step(5).
4. Assume a new value for the plate pressure (this can be done by using the
methods of successive substitution, secant method or newton-Raphson
method) go to step(3).
5. Given the compositions in the bulk and at the interface, the mass transfer rate can be calculated and new values for the bulk liquid compositions can be obtained. If the new interface liquid compositions minus
the estimated interface liquid compositions are greater than a specified
from
(I
the
04)
then
now
changing
estimated
step(3)
repeat
error -Oeliquid
for
interface
the
the
to
composition.
calculated values
values
6. Use the hydraulic model to determine the liquid and vapour flow-rates,
(if
any).
entrainments rates
liquid
hold-ups
the
the
Calculate
7.
reaction rates, given
and the compobulk
in
the
phase.
sition
73
8. Determine reboiler section variables.
9. Determine condenser section variables.
In the caseof the steady state simulation mode, initial estimates for the set of
be
It
is
important
difference
to
to
the
unknown variables need
supplied.
note
between the initial estimates for the set of AEs and the initial condition for
the set of ODEs. Naturally, if a reference steady state condition is known,
the initial condition (for the ODEs) will be the same as the initial estimate
balance
RHSs
In
the
the
of
equations will approach zero. this thesis, only
and
the use of the hybrid modelling approach with respect to solution efficiency
been
has
investigated.
initialisation
and
Since the dynamic models are represented by sets of DAEs of index 1, except
for initialisation, no attempt has been made to improve the method of solution. The dynamic model employs BDF (backward-differentiation-formulae)
RHS
is
Jacobian
the
the
the
of all
equations
numerically comof
solver and
Jacobian
Jacobian
in
The
the
to
order
save
method retains a copy of
puted.
fast.
The
the
the
model
problem rather
solution of
evaluations, which makes
is not yet optimised for speed or memory requirements.
3.8.3
Initialisation
Initialisation has an important role in the solution of DAE or AE systems.
The hybrid modelling approach provides a gradual, step by step procedure for
(an
first
for
That
is,
initialisation.
a simple model
any simulation problem,
balance
equations and simple propequilibrium-based model with only mass
In
the
is
model
complex
a
more
step,
next
solved.
and
generated
erty models)
is generated and solved using the results from the previous simulation as the
initial estimates and providing additional initial values for only the new variThis
by
introduced
the
(additional
model).
complex
more
are
variables
ables
The
this
is
desired
the
is
of
advantage
reached.
model
repeated until
step
for
be
is
desired
to
by
is
the
time
the
that
simulation,
used
model
procedure
behaviour
the
the
information
to
large
of
with
respect
of
valuable
amount
a
Consequently,
from
is
the
distillation column
a maearlier steps.
generated
distillation
design
for
data
the
column
jor portion of
and analysis of
needed
final
before
the
becomes already available even
model equations are solved.
This
is
becomes
to
Therefore, each new simulation problem
solve.
easier
for DAE systems, where, the simulation problem is solved
true
particularly
(the
AEs
ODE-mode
are solved separately as procedure equations)
in the
74
before switching to the DAE-mode. In this way, the necessaryinitialisation
is obtained for the DAE-mode.
The non-equilibrium model is somewhat more demanding with respect to
computer time than the corresponding equilibrium model. Table 3.8 gives
idea
for
(the
times
total computaan
of computational
required
each model
tional time is divided, for purposes of illustration, into 10 CPU time units).
To compare the simulation times, for the same distillation system, one form
of non-equilibrium model and the corresponding equilibrium model with the
distillation
design
have
been
thermodynamic
same
column
and
models
con(the
depropanizer
is
sidered
column taken as an example). Taylor et al.(1994)
have also reported similar comparison of computational times between equilibrium and non-equilibrium models.
Model
Non-equilibrium
Equilibrium
K-values
0.1
0.1
Mass transfer
Solving equations
(RHS calculation)
2.0
0.0
7.9
5.0
T-5.1
Total (CPU-units)
Table 3.8: Computational
tions
-- TI-O
-0
times for solution of different sets of model equa-
The rate-based models need to solve twice as many mass and energy balMass
transfer
the
rates are only
model.
equilibrium-based
ance equations as
Since
the
total
the
number of equations
rate-based model.
calculated with
for the rate-based models is about three times the number of equations of
the equilibrium based, the solution of the non-equilibrium model equations
takes longer than the corresponding equilibrium model equations. From the
time requirements presented in Table 3.8 it can be noted that the equilibrium
by
the
50%
time
the
non-equilibrium
needed
of
model needs approximately
incentive
Therefore,
to
for
the
reduce
an
same simulation problem.
model
(or
for
the
times
the
the computational
nonmodel
using
non-equilibrium
for
dynamic
is
equilibrium model only when necessary) worth considering
hybrid
By
introduction
the
approach where equilibrium-based
of
simulation.
be
it
the
to
simulapossible reduce
and rate-based models are combined, may
For
in
20
if
trays
time
with
we consider
tion
column
a
example,
significantly.
hybrid
in
the
the
them
10
are equilibrium,
modelling
that
simulation with
of
%
for
75
take
the
the
time
approximately
of
corresponding nonapproach will
75
equilibrium model without any loss of accuracy. This simple analysis serves
indicator
for
the
as an
reduction of computational time. It
of
opportunities
be
however,
that other means of reduction of computational
must
clarified,
time, not considered in this thesis, are also possible.
3.8.4
Switching
between Models and Monitoring
Pheof
nomena
Excess Gibbs energy for a single phase is a function of the chemical potential:
G=
RT
Eni
In tlE
(3-105)
ii
Since during a dynamic simulation (rate-based model), the component comknown
in
positions and chemical potentials are
each phase, evaluation of the
(independent
Gibbs
does
the
tray
time
at any
excess
energy on
variable)
Analysis
time.
not require additional computational
of the transient path of
the Gibbs energy for each tray (with the rate-based model) then indicates
the approach to equilibrium for each tray. If the Gibbs energy is found to
be approximately at a minimum (or asymptotically approaching a minimum
be
While
there
to
the
model
can
made.
equilibrium-based
value), a switch
is no gain in switching from the rate-based model with respect to simulation
behaviour,
dynamic
there
are obvious computational
results or prediction of
for
is
For
the
tray
the
reduced
number of equations
example,
advantages.
by nearly 50 %. Also, from a design and analysis point of view, information
has
distillation
by
the
trays
the
to
column
of
equilibrium
on the approach
importance - the upper limit of conversion will be attained if equilibrium is
Gibbs
is
the
to
To
that
the
a minimum,
energy
close
ensure
approached.
(see
Chapter
Bifurcation
An,
Anj
analysis
are also monitored.
and
values of
4) provides a further verification alternative. If the Gibbs energy is not at a
bifurcation
lower
(that
it
the
is,
if
to
analysis
value),
a
can move
minimum
will point to multiple steady state solutions.
3.9
Conclusions
A general hybrid model has been developed for simulation of distillation
different
between
The
the
of
complexswitch
models
model
allows
columns.
(steady
dynamic).
between
ity as well as
state and
simulation modes
76
This general framework implemented into WAS (Gani et al., 1997) permits
the simulation of reactive as well as non-reactive systems. The framework
Ltd.,
1997) allows the use of optimization techniques
within gPROMS(PSE
for the optimal control objective of this thesis as well as simulation. Both
have
been
implemented
it
models
almost simultaneously and
was not the
idea of this work to compare the results from the two systems. The simulation results obtained from the WAS simulator have been used as the initial
for
However,
has
be
the
to
gPROMS simulation.
estimate
gPROMS
showed
first
has
been
tool
the
once
a very reliable
convergence
achieved.
In the following Chapters the validity of the general model is presented tohybrid
Furthermore,
gether with some applications of
modelling.
controllability analysis, based on a linearised version of the general model is assessed.
Finally the dynamic optimization results, from the gPROMS framework, for
the design of reactive distillation columns is presented.
77
Chapter
Validation
4
of models
In this Chapter the validity of the hybrid model presented previously (Chapter 3) is shown for the systems studied during this work.
First the systems are presented, given some insights into the phase equilibkinetics
distilthe
the
the
mixtures, reaction
rium of
and
characteristics of
lation column where the separation/ reaction is taking place. These systems
have been chosen in order to cover a wide range of examples.
Different characteristics of the systems make them interesting to analyse, for
MTBE
the
production, availability of exexample, reported multiplicity of
data
for
the esterification column.
perimental
Before going to the simulation results, an initialisation procedure is prefaster
This
to
the
more complex modconvergence
allows
procedure
sented.
finding
'slow',
helps
it
look
'tedious'
it
the
soand
may
els, and even when
lution (convergence) particularly for non-equilibrium models. Without this
in
the
it
to
times
more
convergence
attain
possible
was not
procedure, many
hybrid
from
be
the
that
general
model.
generated
can
complex models
Then, simulation results with different models, generated from the general
(whenever
data
hybrid model, are validated against experimental
possible)
literature
results.
or against other open
A detailed study of a vast number of parameters together with simulations
for
is
the
different
which
model
given
shows
more
appropriate
a
models
with
78
system or which set of parameters should be used when simulating the column.
A brief study on bifurcation behaviour of the models is
also presented in
this Chapter. Bifurcation analysis for the reactive systems is investigated.
Bifurcation theory is used to explore all the possible steady states (stable
for
and/or unstable)
reactive systems.
The general hybrid model has shown to be a reliable simulator for distillation
It
has
been
find
to
columns.
always
possible
a model which can match
(or
literature
experimental results
results).
4.1
Systems Studied
Methyl-tert-butyl
*
Ether (MTBE)
production
Esterification
Acetic
9
of
acid with ethanol
Depropanizer
e
column
In the following sections the systems are described.
4.1.1
MTBE:
Met hyl-Tert-B
her
utyl-Et
What is MTBE?
MTBE, or Methyl Tertiary Butyl Ether, is an ether compound in the same
boiling range as gasoline. It is soluble in water, alcohol and other ethers
by
MTBE
is
isobutylene
has
combining
made
a characteristic smell.
and
(isobutene) and methanol in presenceof a suitable catalyst, typically a cation
for
OrigiMTBE
is
gasolines.
mostly used as an additive
exchange resin.
it
function
instead
its
to
the
was raise
octane of gasolinesand was used
nally
It
in
the
the
lead
also
gasolines and
raises
oxygen
content
components.
of
from
by
hydrocarbon
and carbon monoxide emissions
motor vehicles
reduces
MTBE
for
is
the
complete
combustion.
preferred additive
promoting more
in
USA.
has
However,
MTBE
the
gasoline
reformulated
received
producing
health
tests
publicity
as
a
result
of
although some consider
some negative
79
the tests to be inconclusive. The National Academy of Science(N. A. S.) con'Based
does
it
the
cluded:
on
available analyses,
not appear that MTBE
from
the use of oxygenatedfuels are likely to pose a subexposuresresulting
human
health
stantial
risk".
However, MTBE has now been banned in the state of California (25th March,
1999). They acted out of concern for water quality. MTBE has been found
to enter Californian aquifers and show up in drinking water. As this state
goes on environmental regulations so tends the rest of the United States.
Strengths
Weaknesses
High octane
Availability of economical isobutylene
feed stocks is limited
Low volatility
Blending
gasoline
characteristics
similar
Widely accepted in market place by
consumers and refinery
Possible methanol supply constraints
Environmental
totally
issues not
cleared
Reduces carbon monoxide and exhaust hydrocarbon emissions
Economics not dependent on subsidies
MTBE
Production
Methanol and isobutene, under appropriate process conditions, react in the
in
is
isobutene
Typically,
form
MTBE.
the
to
the
a
catalyst
of
presence
isobutene,
50%
from
10%
to
butane
that
about
about
contains
stream
mixed
isobuThe
depending on the source.
other major components are n-butane,
inert
the
These
process
under
tane and n-butenes.
other components are
interest
In
temperature
the
the
of
range
catalyst.
chosen
and
conditions
With
high.
MTBE
is
(31OK-370K), the equilibrium conversion of
an excess
isobutene
is
it
to
theoretically
of
a
conversion
achieve
possible
of methanol
%
two
90-97
in the range of
with a single-stage conventional process, using
by
followed
of
of
excess
recycle
and
external
separation
reactors
series-flow
methanol.
80
Separation of MTBE from the inerts and the excessof methanol is very difficult through distillation. The methanol excessin the distillate is recovered
in another separation step and recycled. MTBE (high purity) is obtained as
bottom
The
isobutene
from
the normal
a
separation of unreacted
product.
butenes by distillation is expensive due to the low relative volatility between
them. Therefore, the unreacted isobutene leaveswith the inerts, thus, a certain amount of isobutene is not converted to MTBE. On the contrary, using
distillation
instead
technology,
is
the
reactive
of
conventional units, possible
to achieve up to 99 % of isobutene conversion.
The MTBE reaction is not severely equilibrium limited. The net reaction
fairly
high
isobutene
is
until
a
rate remains
substantial amount of
converted.
Only when the liquid composition (where the reaction takes place) is close
to the equilibrium compositions the net reaction rate decreaserapidly.
Conventional processes adopted by industries are, in general, made up of:
two liquid-phase catalytic reactors placed in series (the isobutene con%
is
is
90-95
used)
around
and excess of methanol
version
the reactors are followed by one or two distillation
high purity MTBE
fractionation
methanol extraction and
for
recycling.
pure methanol
The MTBE process using reactive distillation
column.
columns to recover
to
recover
columns are required
technology consists of a single
Co.,
(Koch
Engineering
distillation
industrial
A particular
column
reactive
Inc. ) has a diameter of 203cm with a height of 5.2m. It is insulated and
The
for
heat
concolumn
operation.
adiabatic
compensation
with
equipped
Flexipac;
(top
bottom:
beds
four
mid-top and midtains
and
of packing
bed.
distributors
liquid
Katamax),
bottom:
above each
with vapour and
The Katamax packing contains Amberlyst 15 catalyst. The column was opfor
demonstrations
the
for
variables
of
ranges
runs,
and
months
six
erated
).
Co.,
Inc.
Table
Engineering
in
4.1(Koch
studied are presented
81
Variable
Total Feed Rate
Feed Isobutene Concentration
Feed Methanol- Isobutene Molar
Ratio
Average Katamax Packing Temp.
Reflux Ratio
Value
20-30 bbl. /stream day
11-13 wt %
1.0:1 to 2.6:1
333K-353K
1.0:1 to 6.0:1
Table 4.1: Range of Variables
Why Use Reactive Distillation
for MTBE
Production?
Reactive distillation achieveshigher conversionlevels than conventional
for
processes this selected reaction.
(in
the
The
the
temperature
which
catalyst
catalyst
of
range
preferred
9
for
distillation,
be
the
that
a characused more efficiently) match
can
distillation.
for
is
this
that
that
teristic
reactive
process excellent
ensure
less
due
Lower
to
equipment.
operating cost
9
(higher
higher
due
Lower
to
rate of
conversion per pass
operating cost
9
production).
However, one main disadvantage of reactive distillation for MTBE production
is the poor knowledge of the reaction kinetics as well as the poor understanding of the plate or packing behaviour.
Case Study
discussed
industrial
from
differs
the
in
The MTBE column studied this work
the
is
to
The
reactive
simulate
used
program
general simulation
above.
flow
bottom
The
Table
4.2.
in
rate was specified
distillation column specified
industrially
MTBE
the
to
required.
to 197 mol/sec
amount of
obtain
Reaction
Kinetics
isobutene
between
The reaction
on sulfonic catalytic resins
methanol and
(1978)
found
Anccilotti
by
et al.
has been studied
several research groups.
first
in
the
in
the
olefin.
order
alcohol and
that initial rates were zero order
82
[_Variable
Feed Rates:
Methanol
Butenes
Feed Stages:
Methanol
Butenes
Number of Trays
Type of tray
Tray design
Value
215.5 mol/s
535 mol/s
4
11
17
Sieve tray
Diameter = 2.5m
TraySpacing
ActiveArea
Weirlength
WeirheZght
Reactive Section
Catalyst Load
Pressure
Reflux Ratio
Bottom Flow-rate
0.311m
=
2
5.3m
3.66m
0-38m
from tray 4 to tray 11
30OKg/stage
II atm.
7.0
197 mol/s
Table 4.2: MTBE Column Design. Data taken from Jacobs and Krishna
(1993) except for the tray design which has been calculated in this work
The MTBE synthesis reaction catalysed by Amberlyst 15 macroreticular sulfonic resin was also studied by Gicquel and Tork (1983). This catalyst acts
beads
The
bonded
its
to
the
through
porous spherical
resin.
sulfonic groups
4.9
equivalent protons/kg.
contain
The reaction mechanism and the rate expression are dependent on the relative concentrations of isobutene and methanol which affect the activity of the
less
is
isobutene
than
When
to
the
methanol
ratio of
acidic catalyst particles.
hydrogen
(due
highly
its
the
is
to
0.7, methanol adsorbed
polar nature) and
dissociate.
(protons
link
the
bonds that
or catalytic agents)
sulfonic groups
Then methanol becomeshydrogen-bonded to the sulfonic group to react with
it
is
Therefore,
in
the
the
isobutene solution within
asgel phase.
pores and
The
by
Eley-Rideal
MTBE
the
that
mechanism.
synthesis
proceeds
sumed
(Al-Jarallah
by
is
1988):
initial surface reaction rate given
et al.,
83
CIOB5CMe
k., KMeOH
_0I,
KMeOHCMeOH
+
-
15
CMTBEIK
+KMTBECMTBE)l
The rate constant k, is given by:
1013
ks = 1.2x
(-87900/RT)
exp
(4.2)
The liquid adsorption constant for methanol and MTBE are:
KMeOH=
KMTBE:
IXIO-13
5.
---':1.6xl
0-16
(97500/RT)
exp
(4-3)
exp(1190001RT)
(4.4)
The chemical equilibrium constant is given by an expression like:
In[
K(T,, )
1-I)+
A, (
T
T,
+A4(T
2-
T)+
A21n(
A3(T - T,,)
TO
3_
4
T02)+ A5(T
Toý')+ A6(T
_T4)
(4.5)
0
When the concentration of isobutene begins to be significant, the LangmuirHinshelwood mechanism, where both methanol and isobutene absorbed react
to give MTBE, starts to be operative. The sorption mechanism for hetero(Rehfinger
Hoffmann,
is
1990):
geneouscatalysis
and
[aMeOH
aIB
aMTBEj
r. = qkr
2
K aMeOH
(4.6)
is
is
the
the
q
amount of acid
catalyst
mass,
r,
rate of reaction per unit
is
the
the
the
of
component.
activity
groups on
resin per unit mass and a
The rate constant kr is presented as:
012e -11115)
kr = 3.67x1
(4.7)
The chemical equilibrium constant is given as above (Eq. 4.5). When the
is
in
initial
is
isobutene
10
to
the
methanol around
a maximum
rate
ratio of
decreases
Thereafter,
the
the
when
methanol concentration
more,
observed.
dependence
first
dependence
in
isobutene
order
and a
rate shows a zero order
diffusion
Here,
begins
transport
to control methanol transfer
in methanol.
from the bulk liquid phase.
84
The reaction rate is influenced neither by the diffusion inside the micro-sphere
by
the internal surface area.
nor
In order to preserve selectivity the presence of water should be avoided.
Since water is more polar than the alcohol (methanol) it will be preferentially adsorbed on the sulfonic groups. The adsorbed water can react with
the isobutene to give tertiary butyl alcohol.
On the basis of the experimental results from the different authors (as disfollowing
for
the
the reaction system:
cussedabove),
assumptions were made
the amount of methanol feed to the column is close to stoichiometric
isobutene
is
the
to
conditions, so
ratio of
methanol smaller than unity
the column operating conditions are chosen so that the amount of
in
bottom
is
This
the
methanol
very small.
guarantees that the ratio
isobutene
is
in
to
than
the reaction section
of
methanol smaller
unity
in
as well as the stripping section
(dimerization
formation
isobutene,
9 side reactions are negligible
of
of
tert-butyl alcohol from isobutene and water)
With these assumptions the rate of reaction can be accurately described by
the expression proposed by Rehfinger and Hoffmann(1990).
Model Formulation
The following assumptions were made when formulating the model:
(equilibliquid
1. the vapour and
on each stage are at equilibrium or not
rium model or non-equilibrium model)
2. Since this section is actually a packed section when formulating the
is
through
this
a
section
packed
considered as
non-equilibrium model
trays.
number
of
specific
I
(not
liquid
liquid
in
in
the
the
the chemical reaction occurs only
phase
film)
follows
the correspondly kinetic expression: rate4. chemical reaction
based
85
the pressure drop through the column is negligible
6. there are no external heaters or coolers apart from reboiler and condenser.
Phase Equilibrium
The various physical equilibrium modelling alternatives considered in this
in
Table
in
liquid
4.3,
terms
work are given
of
and vapour phase models
together with the sourcesfor their model parameters.
Liq. Phase
Model
Liq. Phase
Model Parameters
Vap. Phase
Model
UNIFAC
UNIFAC
Fredenslund et al. (1977)
Fredenslund et al.
(1977)
Rehfinger and Hoffmann
(1990)
Ideal
SRK
UNIQUAC
SRK
Vap. Phase
Model Parameters
Binary interaction
coefficients set to zero
Binary interaction
coefficients set to zero
Table 4.3: Modelling alternatives for MTBE system
4.1.2
Ethyl
Acetate
Esterification of acetic acid with ethanol to water and ethyl acetate is the
distillation
in
frequently
column.
considered reactive system a reactive
most
In 1921 a patent was published by Backhaus. In the early 70s reactive distillation was described through computer simulations using rigorous equilibrium
different
From
(Suzuki
to
the
1971).
there
algopresent
on
et
al.,
models
this
to
system were presented.
solve
rithms and models
Especially compared with the esterification of acetic acid with methanol, this
be
to
good
very
not
appears
acetate
ethyl
producing
process
esterification
be
it
(Bock
theo1997a)
distillation
for a reactive
and should
et al.,
column
different
high
to
processes.
with
ethyl
acetate
purity
obtain
possible
retically
Bock et al. also show that the equilibrium models, which are normally apdistillation
inadequate
this
to
model
to
are
column
simulations,
reactive
plied
better.
do
task
the
that
model
will
a
non-equilibrium
suggesting
system,
86
Chang and Seader (1988) reported that the main problems in achieving high
in
purity ethyl acetate a reactive column are:
* unfavourable reaction conversion
K-values
o similar
of ethanol, ethyl acetate and water, and
in
temperature
the column.
9
profile
In this work ethyl acetate production is considered despite the disadvantages offered for the system when using reactive distillation. There is one
big advantage of this system: experimental data is openly available in the
literature. This was a major consideration when selecting reactive systems.
Experimental data is hard to find.
Moreover, despite the fact that high purity is not obtained with the reactive
different
interesting
in
this syscolumn, many
as well as
phenomena occur
tem: azeotropes, similar volatilities, low and fast reaction in different trays.
The system can definitely test the general model to its limits, pushing the
its
to
ranges of prediction and accuracy.
model
The general simulation program was used to simulate the reactive distillation
in
Table
4.4.
column specified
Reaction Kinetics
The equilibrium reaction of the esterification is described as follows:
+
+
ý-+
ethyl acetate
water
ethanol
acetic acid
The rate of reaction is given by:
r=
ki exp(-
Ei
RT
) rj
E2
CR
-
k2exp(-
RT
) r, Cp
(4-8)
Both the forward and the backward reaction are second order (the forward
first
in
in
backfirst
the
is
order
acetic
acid
and
order
ethanol,
while
reaction
first
first
is
in
in
order
water
and
order
ethyl acetate, given
ward reaction
87
[.Variable
Value
Feed Rate:
Composition
Acetic acid
Ethanol
Water
Feed Stage:
Number of Trays
Type of tray
Tray design
Section I to 12
4885.0 mol/h
Tray 13
Diameter =6 25m.
.
TraySpacing = 0.911m
2
AchveArea = 54.684m
WeZrIength = 11.684m
Weirheight = 1.193m
Reactive Section
Pressure
Reflux Ratio
from tray 1 to tray 13
1 atm.
10
0.4963
0.4808
0.0229
6
13
Sieve tray
Diameter = 5.60m.
TraySpacing = 0.911m
2
AchveArea = 50.684m
Weirlength = 11.684m
Weirheight = 0.18m
Table 4.4: Ethyl Acetate Column Design. Data from Suzuki et al. (1971)
has
been
in
design
this
the
tray
work.
which
calculated
except
is
for
The
the
the
same
components
all
reaction rate
secondorder reactions).
because the stoichiometric coefficients of all components are equal.
The rate of reaction used in this work is as follows:
E
r=
ki exp(RT
)CAACEt
-
k2eXP(-
where:
483.33,
ki
k2
123-00,
88
E
RT
)CWCEtAc
(4.9)
E= 54970-51
R is the gas constant, and
CAA, CEt, Cw,
CEtA,
(AA),
the
represent
concentration of acetic acid
ethanol
(Et), water (W) and ethyl acetate (EtAC) respectively. The rate
of reaction
is given in Kmol/h.
Model Formulation
The following assumptions were made when formulating the model:
the vapour and liquid on each stage are at equilibrium or not (equilibrium model or non-equilibrium model)
2. the reaction zone (all the column, in this case) is modelled as a series
of trays
3. the chemical reaction occurs only in the liquid phase (not in the liquid
film)
4. the chemical reaction follows the correspond kinetic expression: ratebased
5. the pressure drop through the column is negligible
6. the last tray in the column is bigger than the rest (this facilitates the
distillation
bottom
the
the
column)
of
reactive
reaction at
7. there are no external heaters or coolers apart from reboiler and condenser.
Phase Equilibrium
The system is strongly non-ideal. The following azeotropes are possible:
- ethanol-water
acid
water-acetic
ethanol
acetateethyl
-
89
- ethyl acetate-water
- ethanol-ethyl acetate-water.
Suzuki et al. (1971) determined the phase equilibrium for the system taking
the reaction into account (they fitted coefficients to use in a modified MarSawistowski
(1982)
Pilavachi
gules equation).
and
mention that the phase
do
data.
the
equilibrium
not correspond with
experimental
Bock et al. (1997a) have determined experimentally NRTL parameters withhave
the
they
reaction and
out considering
reproduced the experimental mole
fraction profile quite well.
In our work (Perez-Cisneros et al., 1996,1997; Pilavachi et al., 1997) different
thermodynamic models were employed in order to match the experimental
data. Table 4.5 surnmarises the different models used.
Liquid Phase
Model
Liquid Phase
Model Parameters
Vapour Phase
Model
UNIFAC
Fredenslund et al.
(1977)
Fredenslund et al.
(1977)
Ideal
Fredenslund et al.
(1977)
Kang et al.
(1992)
Kang et al.
(1992)
Virial Eq.
Suzuki et al.
(1970)
Suzuki et al.
(1970)
Ideal
UNIFAC
UNIFAC
UNIQUAC
UNIQUAC
Margules Modif.
Wilson
SRK
Vapour Phase
Model Parameters
Binary interaction
coefficients set
to zero
Hayden and
O'Connell(1975)
Ideal
SRK
Binary interaction
coefficients
set to zero
SRK
Binary interaction
coefficients
set to zero
Table 4.5: Modelling alternatives for Esterification system
90
4.1.3
Depropanizer
Column
Taylor and Krishna (1993) have also studied this depropanizer column. The
distillation column details are given in Table 4.6. This column has been
simulated with the equilibrium model as well as the non-equilibrium model.
Models of types RB-3 and EQ-3 have been used in all simulations. For
initialisation purposes, simpler model forms have been used.
The depropanizer column is selected as a system to study in this work, mainly
due to the availability for data for the column design as well as the property
for
the mixture.
model
Model Formulation
The following assumptions were made when formulating the models:
the vapour and liquid on each stage are at equilibrium or not (equilibrium or non-equilibrium model)
2. the pressure drop through the column is negligible
3. there are no external heaters or coolers apart from reboiler and condenser.
Phase Equilibrium
Since the system is a hydrocarbon mixture an equation of state for both
behaviour
is
Ideal
the
not considmixture quite well.
phases will represent
(15atm.
).
Peng-Robinson
The
high
in
due
the
to
the
column
pressure
ered
both
for
in
is
the
thermodynamic
taken
phases,
model
as
equation of state
(1993).
SRKby
Krishna
Taylor
to
and
compare results presented
order
EOS is also used as a thermodynamic model in order to investigate sensitive
to
simulation.
parameters
4.2
Initialisation
have
been
In
two
the
tried.
For the simulation of
starting points
systems,
known
is
the
first
not
condition
although
a
reference
simulation
the
case,
91
Variable
Feed Rate
Feed composition:
Ethane
Propane
Butane
Pentane
Feed Stage
Number of Trays
Type of tray
Tray design
Section I to 15
-Value
1000.0 mol/S
0.10
0.30
0.50
0.10
16
35
Sieve tray
Diameter =4
TraySpacing
Act2veArea
Weirlength
WeirheZght
82m.
.
0.5m
=
2
14.96m
17.6m
0.05m
Section 16 to 35
Diameter
6 17m.
.
TraySpacing = 0.5m
2
ActMeArea
24.516m
Weirlength
0.037m
0.038m
Weirheight
Reactive section
Pressure
Reflux Ratio
Bottom Flow-rate
none
15 atm.
2.5
600 mol/s
Krfrom
Taylor
Data
Column
Details.
taken
and
Table 4.6: Depropanizer
ishna (1993).
92
problem needsto predict the transient behaviour due to disturbances around
the reference condition -this situation is often encountered in process design
where the reference condition or designed condition of operation needs to
be determined. This means that first an appropriate reference
steady state
be
determined
before
the effect of disturbances can be studied. Since
must
the starting point for dynamic simulation mode is far from the referencecondition, simulation is started with the simplest model and after short
periods
of simulation, a switch to a more complex model is made. This is continued
desirable
has
been
the
In
the second case, a reference
until
model
reached.
is
known
from
this condition to study
condition
and simulations are made
disturbances
in design variables. For these simulations,
effect of changesor
the reference condition provides a very good initial estimate and thus the
initialisation procedure is not needed unless a switch from one model type
be
is
to
or mode
made.
For each distillation system, simulations were started with the equilibrium
EQ-1
type
model of
when the corresponding reference conditions were not
known. After an arbitrary number of integrations, for example 5 integration steps, a switch to equilibrium model of type EQ-2 was made and after
further
integration
5
EQ-3
After
to
a
steps, a switch
model
was made.
a
further integration (t=1 hr), a switch to the steady state mode was made
(this step was only taken if a steady state with the equilibrium model was
for
At
design
the
purposes
or
of
needed
validation).
computed equilibrium
(type
RB-3)
to
the
non-equilibrium model
model steady state, a switch
was
for
formuin
If
is
included
the
simulation problem
a check
stability
made.
lation, dynamic simulation for a further period of time (about 100 hrs) was
(MTBE
distillation
involving
For
reaction,
production and
systems
made.
in
increintroduced
terms
of small
ethyl acetate production) reactions were
final
(Ar
form
0.001)
the
to
the
switch
model
after
ments of reaction rate
=
(equilibrium or non-equilibrium). The integration statistics (NFE: number
NSTEP:
Jacobian
NJE:
function
evaluation;
numnumber of
evaluation;
of
ber of integration steps; Time: time of simulation; and, Residuals: residual
for
differential
three
time)
the
the
the
simulation
equations after
values of
first
Although
in
Table
4.7.
to
this
may
sight,
seem
at
problems are shown
be a lot of extra computation, use of the initialisation procedure did result in
faster convergenceto the desired steady state solution when compared with
(see
Table
3.8).
initialisation
the
also
procedure
simulations without use of
In most cases,without using the initialisation procedure, it was not possible
to obtain the steady state solution with the non-equilibrium model.
93
Model
MTBE col.:
-equilibrium
-non-equilibrium
Esterification col.:
-equilibrium
-non-equilibrium
ODEs I NFE I NJE _ FNSTEP Time
Residuals
68
68
136
136
518
731
2251
2807
7
7
18
19
26
34
198
101
ih.
100h.
ih.
100h.
0.3809e-14
0.5658e-19
0.1824e-08
0.1257e-12
52
52
104
104
393
873
1069
1457
7
8
12
13
26
20
108
103
ih.
100h.
1h.
100h.
0.3753e-13
0.8962e-21
0.5283e-08
0.4016e-13
140
140
280
280
494
678
2938
3558
7
8
22
23
56
48
158
147
ih.
100h.
Ih
,
100h.
0.8872e-14
0.8978e-21
I 0.4027e-08
0.3075e-14
Depropanizer col.:
-equilibrium
-non-equilibrium
Table 4-7: Selectedset of numerical statistics for the three distillation systems
4.3
4.3.1
Validation
Models
of
MTBE
In the open literature many authors have presented the MTBE profile in
Sundmacher
1995,
Hauan
(Jacobs
Krishna,
1993;
the column
and
et al.,
and
Hoffmann, 1996, Ung and Doherty, 1995a,b). In this thesis different models
from the general hybrid model have been used to simulate the process. A set
debeen
in
(Table
has
to
4.3)
different
thermodynamic
order
used
models
of
be
the
to
that
the
simulation results.
termine
sensitive
can
set of properties
be
found
been
have
to
The activity coefficient model parameters
sensitive
have
been
There
(see
1996).
Perez-Cisneros
in
et al.,
some special cases
only
hardly any differences between the simulated results when using UNIFAC,
UNIQUAC or Wilson equation as thermodynamic models.
liquid
for
UNIFAC
the
the
With the combination of
phase activity
model
for
SRK
the
ideal
the
phase,
vapour
gas
of
state
equation
or
coefficients,
dependent
(1990)
for
Hoffmann
the
temperature
Rehfinger and
expression
dynamic
it
term,
through
and
was
steady
state
possible
constant
equilibrium
(including
in
the
the
to
reported
results
multiple
solutions)
match
simulation
94
literature. Fig. 4.1 shows the variation of fractional conversion of isobutene
feed
function
location,
for
dynamic
the
as a
methanol
of
steady state and
Temperature
for
low
high
simulation, respectively.
profiles
and
conversions
(for the multiple solutions) are shown in Fig. 4.2.
In order to corroborate the multiplicity presented by this system, differdependence
the
temperature
ent ways of calculating
of the equilibrium conhave
been
(1990),
Kq,
Rehfinger
Hoffmann
Colombo
stant,
studied:
and
et
(1983);
from
based
Gibbs
free
data
the
the
and
al.
expression
on
energy using
from the DIPPR Data Bank (Daubert and Danner, 1986).
K,
q
for the three expressionsis shown in Fig. 4.3.
Simulation results using the three expressions for Kq, are given in terms of
fractional conversion of isobutene with the methanol feed location in Fig.
4.4. From this figure it can be noted that while the Rehfinger and Hoffmann
the
two
other
expressions multiple
yields
multiple
solutions,
with
expression
be
found.
solutions cannot
From Fig. 4.3, it can be noted that the differences between the three exfrom
Fig.
increasing
for
Kq
increases
4.4
temperature
while
with
pressions
it can be noted that the differences between the different solutions (considFrom
Fig.
in
high
10-14.
the
stages
conversion solution)lies
ering only one
4.3, however, it can be noted that these stages correspond to the 'higher'
for
between
differences
the temperthe
the
temperatures where
expressions
increases
the
temperature
dependence
the
as
equilibrium constant
of
ature
increases.
Somepilot plant experiments (Sundmacher, 1995) tend to confirm these mulHowever,
for
that
MTBE
the
tiple solutions
showed
our
analysis
column.
The
for
the
there are specific cases
multiple solutions can occur.
which
indein
the
these
systems,
remain
specific
conditions,
under
multiplicities,
been
fully
have
Three
types
the
used:
models
of
model
used.
of
pendently
fikinetically
reaction
and
controlled
equilibrium
with
physical
equilibrium,
nally non-equilibrium.
been
has
the
Multiple solutions with
equilibrium model
validated against
Fig.
The
if
is
4.5.
as
shown
equilibrium
results
model
comsimulated
other
95
1.0
0.9
0.8
0.7
I
Rehfinger and
Hoffmann (1990)
0.6
0.5
0.4
0.3
0.2
0.1
13
From 2 to 14
&
From 14 to 2
0.0
234567891011121314
Methanol feed stage
Figure 4.1: Multiple solutions in the MTBE column
96
440
High conversion
Low conversion
420
ii'.
..
1
a..
..
M.
'--'
400
U
380
U
F-4
360
/
340
10
3456789
11
12
13
14
15
16
17
Stage
Figure 4.2: Temperature profiles for low and high conversion in the MTBE
column
97
-6
-5
,;::ý -4
0
-2
-1
0
2.4e-3
2.7e-3
3.0e-3
3.3e-3
l/T (K -1)
Figure 4.3: Equilibrium constant for MTBE synthesis
98
3.6e-3
1.0
0.9
pp
Q 0.8
0.7
Z
CD 0.6
0--ý
Gn
0.5
Z
0.4
0.3
Z
c) 0.2
o. 1
0.0
10 11 12 13 14
METHANOL FEED STAGE
Figure 4.4: Simulation results with Rehfinger and Hoffmann, Colombo et al.
MTBE
for
the
DIPPR
column
and
99
full
(meaning
ASPEN
PLUS
pared against
results where
equilibrium
physical
is
and chemical equilibrium)
assumed.
Multiple solutions still remain when using the non-equilibrium model. This
(1999)
in
Higler
results are agreement with a recent paper of
et al.
where they
also present their non-equilibrium model reproducing the same multiplicities
presented earlier with the equilibrium model.
Composition
in
the column with equilibrium and non-equilibrium
profile
It
in
Fig.
is
interesting
4.6.
deviations
beto
that
models are shown
note
tween the two models are largest on the reactive stages, but the differences
bottom
indistinguishable.
the
top
the
column are almost
of
at
and
The non-equilibrium model used to simulate the MTBE column assumesconFurthermore,
the
the packed section of
throughout
column.
stant pressure
the column has been simulated considering packed column expressionsfor
the binary mass transfer coefficients (Onda et al., 1968) and interfacial area.
Mass transfer resistance has been consider to be predominant in the vapour
has
been
The
liquid
the
that
assumpneglected.
phase resistance
phase, so
tions, however are not limiting the ability to provide a good representation
the
of
system.
The aim of this work has been to determine which model represents acculack
The
little
time
the
as possible.
computational
rately
system using as
knowledge
the
number
regarding
of accurate correlations as well as a poor
height
the
the
to
of
packing section compares with
represent
of stages able
the assumptions made. The complex non-equilibrium model, which considers
both resistances and variable pressure takes around 40 % extra time during
better
it
believed
is
it
that
percannot achieve significantly
calculations and
formance.
due
data
be
to
the
Unfortunately results cannot
compared with experimental
(very
from
the
information
obtained
small) set of results presented
restricted
in the open literature. For this reason, high production of MTBE and the
(in
investigate
to
what actually will
order
existence of a multiplicity region
happen if for any reason the systems gets into that operating conditions)
Considering
differences
between
the
the
this
the
work.
models
of
aim
were
data
in
to
this
lack
the
principle
point any
and
up
experimental
of
and
be
MTBE
the
(equilibrium
to
could
or
non-equilibrium)
used
simulate
model
100
1.0
0.9
0.8
iz
0
14-4
0
0.7
0.6
0.5
0.4
0.3
w
ý4
0.2
0.1
0.0
68
10
Methanol feed stage
12
14
Figure 4-5: Simulation results compared with those of ASPEN PLUS for the
MTBE column
101
1.0
0
0.9
0.8
(D
o
E
0.7
C
0
0.6
V)
0
CL
0.5
E
0
0.4
0.3
cr
w
0.2
0.1
0.0
10
12
14
16
Stage Number
Figure 4.6: Simulation results with the equilibrium
for
MTBE
the
column
model
and non-equilibrium
(Chapter
how
be
it
later
in
5)
However,
to
this
will
show
work
column.
determine which one is the best and which one should be used.
4.3.2
Ethyl
Acetate
Several thermodynamic models (for activity coefficients) were used in order
been
has
by
Suzuki
It
1971.
to reproduce experimental results given
et al.,
for
Margules
the
that
activity coefficients
modified
equation
unless
proved
(1970)
dimerization
is
Suzuki
the
by
of
used or
association and
et al.
given
(acetic
is
into
it
is
taken
to
acid)
account,
not possible
the vapour phase
Fig.
4.7
the
the
equilibrium model.
shows
experimental results with
match
different ethyl acetate composition profiles using the thermodynamic models
Activity
from
4.5.
Table
in
listed
coefficient model parameters estimated
102
0.6
0
0.5
4.0
0.4
0.3
0.2
0.1
0.0
10
12
14
Stage number
Figure 4.7: Simulation results with different thermodynamic models for the
esterification column
data,
in
when used simulations of reacting systems
vapour-liquid equilibrium
and physical equilibrium may not give acceptable results.
In Fig. 4.8, the experimental data given by Suzuki et al. (1971) are compared
(using
UNIFAC
from
thermodythe
as
equilibrium
simulated values
with
final
liquid
Only
the
steady state
namic model) and non-equilibrium models.
Fig.
in
4.8.
the
tray
plots of
are shown
compositions of ethyl acetate on each
Note that no tuning of model parameters has been performed at this stage.
The non-equilibrium model performs better than the equilibrium model, a
has
been
for
the
made with
non-equilibrium model
surprisingly good match
is
the
the
bottom
the
column, and neither of
models able to match
section of
for
the top section of the column when
the reported experimental values
UNIFAC is used as the activity coefficient model. Using the Margules model
103
0.6
0
--0+
Suzuki et al. (calculated values)
Experimental values
-40--0-9-
Non-equilibrium model wiffi UNIFAC
Equilibrium model with UNIFAC
Non-equilibrium model wiffi Margules
0.5
0.4
0.3
Ow
0.2
4..1
w
0.1
0.0
10
12
14
Stage number
Figure 4.8: Simulation results compared with experimental data for the esterification column
fit
better
(1971)
however,
Suzuki
gives a significantly
parameters,
et al.
with
for the entire column with the non-equilibrium model (shown in the plot) as
(not
figure).
in
for
the
the
shown
equilibrium model
well as
Figure 4.9 shows the simulated total mass transfer rates for the four compois
liquid
from
Mass
to
the
the
transfer
in
this
vapour phase
system.
nents
is
be
As
be
the
to
more
can
expected, since ethyl acetate
negative.
considered
behave
differently
(but
multicomponent mixtures often
volatile component
from binary systems), it transfers from the liquid to the vapour phase in all
the trays. The non-equilibrium model, therefore, qualitatively and quantitabehaviour.
transfer
the
tively, gives
correct mass
The mass transfer values obtained with the non-equilibrium model, are com104
12(
10(
8(
0
E
4(
cc
21
LCD
cn
c
-21
co
-41
-61
-81
-100+
0
440
IV
Stages
Figure 4.9: Mass Transfer rate for the esterification column
105
100
Ethýl Acetate
90
Rate-based Model
80
70
i
2 60
,v
CC
5C
cc
2
4C
3C
2C
Higler et al.
10,
1
456789
10
11
12
Stage number
Figure 4.10: Comparison of mass transfer rate with literature results
(1998).
The
is
Higler
trend
those
comparison
of
et al.
rather good
pared with
(ethyl
for
be
from
Fig.
4.10
the
components
one of
acetate).
as can
seen
4.3.3
Depropanizer
Column
Different thermodynamic models have been also used for this system, in order
(as
the
to
identify
to
simulation results
possible set of sensitive parameters
There
are practically
well as good agreement with previous reported results).
EOS
Peng-Robinson
between
the
the
and
results obtained with
no variations
the SRK-EOS. Since both equations of state lead to similar results, the rest
has
Peng-Robinson
for
done
the
this
the
equation of
used
system
work
of
literature
keep
in
to
results.
consistency with
order
state
In Fig. 4.11, the simulated steady state liquid composition profiles with the
for
depropanizer
The
the
comcolumn are shown.
non-equilibrium model
fairly
by
Taylor
those
well with
reported
puted composition profiles compare
(1993).
The
Taylor
Krishna
in
Krishna
results of
and
are not shown
and
Fig. 4.11. These results validate, at least qualitatively, the non-equilibrium
Taylor
Krishna
the
this
and
and
work
model since approximately
of
model
106
behaviour
is
similar simulated
predicted.
Simulation with the equilibrium model has also been performed. The results
differ somehow from those predicted from the non-equilibrium model, but
the results from the equilibrium model are not unacceptable.
The ChemSep package (Kooijmann and Taylor, 1996) has been used to comSimulations
the
the
results obtained with
general model.
pare
with the
ChemSep package are compared with our non-equilibrium model. Fig. 4.12
four
for
The
the
the
the
profiles of
vapour composition
components.
shows
differences are acceptable.
4.4
Bifurcation
Analysis
In order to explore the dynamic behaviour of the reactive systems, a continubeen
has
implemented
the
the
to
general
equations of
use with
ation method
model.
Due to the non-linearity of the general form of the model equations, it is
behaviour
have
distillation
that
or
unstable
columns may
reactive
possible
for
been
has
the
have
case
shown
already
multiple steady states, which
may
MTBE
the
production.
of
Bifurcation diagrams are believed to give insight, especially in the caseswhere
have
for
but
caseswhere multiple solutions
also
multiple solutions are present
determine
furthermore,
the
found.
Bifurcation
been
exisanalysis will,
not
tence or not of multiple solutions.
by
final
the
The
simulation are
reactive systems obtained
steady states of
it
is
To
bifurcation
through
poscontinuation,
perform
analysis.
also verified
(Petersen,
PATH
for
to
example,
use some software already available,
sible
(1995);
Marek,
Schreiber
1991
(Kubicek
V
Marek,
1983;
1989); CONT
and
and
bifurcation,
into
As
introduction
theory
the
1976).
Kubi'c"ek,
several
of
an
Thompson
(Mess,
Solari
1996;
Hilborn,
1994;
1981;
books are available
et al.,
1986).
Stewart,
and
In this work, all continuations are performed with the software program
107
1.0
0.8
Ethane
E
Propane
0.6
Butane
Pentane
c
0
a
0.4
E
0
C)
'0
.B
0.2
AT
-i
0.0
5
10
20
15
25
30
35
Stage number
Figure 4.11: Simulation results with the non-equilibrium
propanizer column
108
defor
the
model
1
0.9
+
0.8
0.7
-0- PropaneC-hewn-Si)
Propane (RB)
Butane ChemSep
+ Butane (RBI
E 0.5
0.4
Co
+'
0.3
+.
Ok
0.2
0.1
0
02468
10
12
14
16 18 20
Stage number
22
24
26
28
30
32
34
Figure 4.12: Vapour compositions compared with ChemSep simulations
CONT (Kub'Cek and Marek). The software has been added to the set of
ODEs,
for
AEs
DAEs.
the
solvers
and
The continuation method-based solver makes it possible to calculate all the
design
to
two
steady-state solutions, stable and/or unstable, with respect
(bifurcation
for
bifurTherefore,
two
any
preselected
parameters).
variables
bifurcation
diagrams
is
it
to
and analyse
possible generate
cation parameters,
the steady state behaviour of the distillation operation.
After defining the feed stream (compositions, temperature, pressure) in the
degrees
freedom.
These
two
there
are remaining
are chosen as
of
column
bifurcation parameters.
4.4.1
Applied
Bifurcation
Analysis
Bifurcation diagrams have been produced for the two reactive systems, ethyl
MTBE
production.
acetate and
109
For the bifurcation analysis, two parameters (design variables) were chosen.
The results show the analysis for a fixed value of one parameter.
For the esterification column, the bifurcation analysis results are shown in
Fig. 4.13. Ethyl acetate composition at the top of the column (product) as
function
for
fixed
heat
duty is shown
the
of reflux rate
a
a
value of
reboiler
in the Figure. As can be seen, only one steady state (stable) is found for all
the range of reflux rate studied. The operating point of the nominal design
is also marked in the Figure, showing that the design can be optimized to
higher
achieve slightly
product purity.
The analysis has also been made varying reboiler heat duty for a fixed value
flow
(stable)
has
been
found
in
the
reflux
rate
of
and again only one solution
the
the
of
all
range
variable studied.
For the MTBE column, the bifurcation analysis results are shown in terms of
MTBE composition in the bottom product as a function of reflux rate for a
fixed value of vapour boil-up in Fig. 4.14 and as a function of vapour boil-up
for a fixed value of reflux rate in Fig. 4.15.
It should be noted that for this column, the results of Figs. 4.14 and 4.15 cordistillation
fixed
the
the
to
column where
reactive
configuration of
respond
the multiplicities have been already reported (the methanol feed is located
in stage 10 and the butenes are fed at stage 11).
It can be noted that while for the esterification case no multiple solutions
heat
for
for
be
found
the
the
range of reboiler
range of reflux ratio or
could
duty investigated, for the MTBE column, multiple steady states were found
for a range of reflux ratios. The region of operation for this system has been
flowrates
from
flowrates
in
the
the
±20%
column, starting
of
calculated as a
Fig.
According
4.14
design
the
to
from
the
the
column.
of
nominal
calculated
have
this
some serious problems, probably changing
column will
operation of
from a high purity of the product to almost no product recovery.
110
0.6
0
*p
0 0.5
co
75
Operating point
,E0.4
CL
0
4-0
cc
(D
4-0
co
cz
0.3
0.2
w
0.1
500
1000
1500
2000
Reflux flow-rate (Kmol/h)
Figure 4.13: Bifurcation diagram for the esterification column varying reflux
flow rate
ill
1.0
c
Co
2
0.0
0
2000
4000
6000
8000
10000
12000
Reflux Flow-rate (kmol/h)
Figure 4.14: Bifurcation
diagram for the MTBE column varying reflux flow
rate
112
1.005
1.000
0
t5
U.Vvo
0.990
E
U) 0.985
E
0
0 0.980
0
w 0.975
CO
Operating point
*
*
0.970
Stable steady state
Unstable steady state
0.965 1111111111-1
10800 11000 11200 11400 11600 11800 12000 12200 12400 12600 12800
Vapour boilup
boilfor
MTBE
diagram
Bifurcation
the
4.15:
Figure
column varying vapour
up
113
4.5
Conclusions
The systems selectedin this work showeddifferent types of phenomena which
helped to demonstrate the strength and the versatility of the general model
its
as well as
applicability.
The non-idealities of the systems have been captured through the general
The
be
the
to
model.
analysis of
mixture
separated or reacted away together
first
with
simulation results gives an identification of sensitive parameters.
The set of sensitive parameters that has been identified for each system helps
in the design of a distillation column.
Systems presenting multiple solutions, highly non-ideal mixtures, as well as
4simple'
hydrocarbons
for
distillation
'normal'
have
a
mixture of
a
column
been selected in order to validate the general (hybrid) model as well as its
handling
capability of
various problems.
Two other systems have been tested with the general model although not
in
The
this
thesis.
presented
systems are, one reactive system: methyl ac(Papaeconomou,
1999)
etate production,
and one non-reactive: methanol(Pilavachi
1999),
the
capability of the genet al.,
enhancing
ethanol-acetone
eral model.
The aim of this work was to cover a wide range of systems. Being able to
definitely
help
different
different
when a new
systems will
aspects of
cover
be
has
to
simulated or analysed.
system
The bifurcation analysis also contributes to the development of a computer
different
facilitates
the
simulation probsolution of
aided environment which
lems.
The hybrid modelling approach helps to avoid problems of mismatch between
(for
between
forms
two
of models
any
model and simulation results as well as
be
forms
two
approximately
of models, a sub-set of simulation results should
the same).
The results presented in this Chapter have mainly focused on highlighting
features
In
following
the
the
the
of
of
general
some
model.
validating
and
114
Chapter, the application of hybrid models, i. e, the solution of a combined
is
Through
detailed
equilibrium and non-equilibrium model,
presented.
a
in
terms of tray efficiency, mass transfer and reaction rate, and
analysis
Gibbs free energy as a function of time a switch between non-equilibrium
be
and equilibrium model can
made.
115
Chapter
Analysis
5
with
Hybrid
Models
Hybrid models are useful in many distillation operations when a product
be
Hybrid
in
improved
should
met.
specification
models result
computational efficiency compared to the single model because they provide better
initialisation and flexibility.
As stated before (Chapter 3) the decision to use a hybrid model can be made
focus
hybrid
Chapter
In
this
on using
we will
models
a priori, or a posteriori.
between
has
the
the
non-equilibrium
started, i. e, switching
simulation
after
to the equilibrium model.
The general distillation model presented and validated previously, allows the
function
tray.
time
of
and/or
monitoring of a selected set of phenomena as a
It computes the values of:
o
GE,
Gibbs
free
and,
energy,
excess
EM.
9 efficiency parameter,
(most
time
be
Both can estimated without significant additional computation
durknown
or calculated
of the needed variables and parameters are already
ing the simulation).
The transient path of excessGibbs energy for all trays is monitored over the
be
Those
that
time.
approach
an
asymptotic
minimum
value
may
simulation
The
difference
be
the
to
a
pseudo-equilibrium
state.
of
values
at
considered
)
by
An,
An,
(represented
the
and
are
also
monitored
at
present
nv
n',
of
116
time and are expected to approach zero when the excessGibbs energy ap(or
Tray
in
the column is calculated
proaches a minimum.
point) efficiency
during the simulation time, or after the simulation has finished. Efficiency
(or
to
values close unity represent an equilibrium
close to equilibrium) stage.
These results can then be used as a criterion by switching between the nonto
model
equilibrium
an equilibrium model.
The main idea of this approach is to obtain useful insights to the reactive
distillation operation as well as to save computational time while capturing
believed
The
is
the
the
non-idealities of
all
system.
non-equilibrium model
to reproduce accurately the behaviour of the mixture in the column but it
is time consuming. Whenever the non-equilibrium model is approaching to
(the
limit
is
of any non-equilibrium model
simulate an equilibrium model
the equilibrium), all the extra equations are no longer needed because an
be
instead.
used
equilibrium model can
5.1
Phenomena
Monitoring
of
Free Energy
through
Gibbs
Since, during a dynamic simulation with the non-equilibrium model, the
known
in
each phase,
component compositions and chemical potentials are
This
is
Gibbs
time
the
tray
the
performed.
at any
energy on
evaluation of
indicate
for
Gibbs
tray
the
transient
will
each
energy
path of
analysis of the
the approach (or not) to equilibrium for each tray.
Studies of tray Gibbs energy as a function of time have been made for the
funcGibbs
in
Fig.
Plots
5.1
the
tray
two reactive systems.
energy as a
show
be
It
(for
7).
I
for
trays
the
noted
can
tion of time
and
esterification system
that while the Gibbs energy value on tray 1 is asymptotically approaching
Gibbs
for
Note
is
7.
that
tray
true
the
of
plots
while
not
same
a minimum,
(possibility
be
function
smooth
of composition may not
energy surfaces as a
be,
function
Gibbs
to
time
local
appeared
of
energy
plots
as
a
solutions),
of
first
in all casesstudied,
order.
The
been
found
behave
I.
have
tray
3
to
the
2
Trays and
also
same way as
The
is
to
free
Gibbs
rest of
a minimum.
energy asymptotically approaching
117
298
297
0
E
-)
LLJ
(D
278
270
0.0
0.1
0.2
0.3
0.4
0.5
Time (h)
Figure 5.1: Excess Gibbs energy for the esterification column. Trays 1 and
7.
keeps
free
Gibbs
The
behaviour
follows
7.
tray
the
the column
energy
on
of
decreasing as a function of time.
The final steady state is stable and the mass transfer rate differences (Anj
likely,
highly
It
is
)
these
therefore
that
An,
are approximately zero.
and
for
1-3.
to
trays
indicate
a state close equilibrium
results
For the MTBE column, the tray Gibbs energies as a function of time (see
Fig. 5.2) show a decreasingtrend, thereby indicating a state still further away
from equilibrium. In this case,mixed model simulation is not necessary.Note
Gibbs
for
does
the
energy
a non-equilibrium process
that at steady state,
not
have to be at a stationary value. This also means that the non-equilibrium
be
At
that
this
the
is
the
should
used
when
simulating
column.
model
model
118
292
285
0
E
-)
w
0
250
245
0.0
0.1
0.2
0.3
0.4
Time (h)
Figure 5.2: Excess Gibbs energy for the MTBE column. Trays 4 and 10.
be
By
Chapter
it
4
analysis
used.
was not clear which model should
end of
dynamics
All
becomes
issue
the
this
the
as well
complex
clear.
system,
of
better
through
the
different
non-equilibrium
captured
non-idealities are
as
model.
5.2
Tray
Efficiency
Analysis
for
for
been
has
the
Efficiency analysis
esterification case as well as
studied
the depropanizer column.
Figs.
5.3,
the
the
For the esterification case, using
composition profiles of
(shown
have
been
in
Fig.
5.4)
parameters
estiefficiency-like
component
119
0.6
-0A
0.5
--0-0-
Water
Acetic acid
Ethanol
Ethyl acetate
14.
0.4
0.3
0.2
0.1
0.0
10
12
14
Stage number
Figure 5.3: Vapour compositions for the esterification column
Figure
5.3
the
mated.
shows
corresponding simulated vapour compositions
Without
introduction
the
with
non-equilibrium model.
any
of noise and with
bulk
in
phases, the component efficiency-like
assumption of perfect mixing
lie
between
0
1.
and
parameters
As previously predicted through the analysis of the excess Gibbs free energy,
for
3
1
to
trays
to
are
very
close
unity, giving an extra
component efficiency
believe
that
these
to
stages are actually at equilibrium.
reason
For the rest of the column (trays 4-13) efficiency values vary between 1 and
0.6. A tray efficiency of 0.7 should predict accurate results for the system.
Tray efficiency analysis is also highlighted for the depropanizer column in Fig.
120
0.8
0.6
0.0
268
10
12
Stage number
Figure 5.4: Efficiency calculated for the esterification column
121
1.00
0.95
.c
0.90
Ethane
Propane
Butane
Pentane
0.85
0.80
3579
11
13
15
17
19
21
23
25
27
29
31
33
35
Stage number
Figure 5.5: Efficiency calculated for the depropanizer column
5.5 where the calculated component efficiency-like parameters are shown. An
important point to note here is that all the component efficiency-like paramlie
between
between
largest
0
0.8
1,
1,
the
with
eters again
and
varying
and
deviations from unity observed at the top of the column. The simulations
for
this system again showed the same phethe
with
non-equilibrium model
for
The
the
calculated vapour compositions
esterification case.
nomena as
in
Fig.
in
the
the
efficiency-like parameters are shown
calculations of
used
5.6. This result is not surprising since perfect mixing in the bulk phases
have been assumed. If measured composition values are used, perfect mixing probably will not be valid, resulting in erratic efficiency values. This
(for
be
by
simulated
adding noise
example, randomly change
condition can
the simulated compositions by a factor of (le - 03) to the simulated steady
(the
).
between
Figure
5.7
efficiencies vary
state compositions
-oo and oo
'noise'
to
the
the
erratic
component
efficiencies
with
added
calculated
shows
for
the
non-equilibrium model.
calculated compositions
If we compare our results with those of Taylor and Krishna (1993) we can
disagreement.
From
large
it
this
practical
experience of
column
observe a
122
1.0
0.9
0.8
0.7
0.6
0.5
E
8
0.4
0.3
0
CL 0.2
cd
0.1
0.0
10
20
15
25
30
35
Stage number
Figure 5.6: Vapour composition for the depropanizer column
123
1.0
0.9
--qr--v
0.8
0.7
Q
0.6
0.
;Oil
c
(D
2
4-w
Ethane
Propane
Butane
Pentane
0.5
v -V"V.
0
0.4
0.3
0
0.2
0.1
0
0.0
02468
10 12 14 16 18 20 22 24 26 28 30 32 34
Stage number
Figure 5.7: Efficiency calculated for the depropanizer column far from steadystate
124
0.8
Ethane
0.6
0.4
0.2
0
-0.2
o Non-equilibdum model
ChemSep model
-0.4
-0.6
-0.8
-1 05
10
15
20
Stage number
25
30
35
Figure 5.8: Efficiency calculated for the depropanizer, compared with those
Ethane
ChemSep:
of
is known that the global efficiency of this column is around 0.7, which are
by
in
less
this
those
the
than
predicted
work.
non-equilibrium
model
actually
Looking back at the assumptions made when formulating the problem, we
decide to include resistance in the liquid phase, which has been considered
both
Re-running
the
resistances,we
simulations, now considering
negligible.
Taylor
KrResults
0.7.
to
and
are close
obtain a global efficiency of around
ishna (1993) but still we get the efficiencies in the bandwidth 0-1. This can
be seen from Figs. 5.8 and 5.9 where the efficiency for ethane and propane
the
the
against
effinon-equilibrium model and plotted
are calculated with
(Kooimann
Taylor,
ChemSep
1996).
the
and
program
ciencYcalculated using
The main point of the difference, is the way yi* is calculated. In this work,
ie.
interface,
the
the
is
yi* = xiK(Z)
composition at
yi* calculated using
bulk
for
in
Krishna,
the
the
Taylor
example, utilise
composition
and
while
This
lead
if
ie.
to
the
liquid,
will not
same value mass
yi* = xjK(i).
of the
liquid
has
been
in
liquid
film
bulk
is
the
the
considered
or
transfer resistance
fact
The
in
that
tray
the
real
efficiencies
should remain
not perfectly mixed.
125
Propane
0.9-
0.8-
0.7-
0.613
50.5-
0.4Non-equilibrium model
ChemSep
0.3-
0.2-
0.11
0
5
10
15
20
Stage number
25
30
35
Figure 5.9: Efficiency calculated for the depropanizer, compared with those
ChemSep:
Propane
of
limit 0-1 is not claimed here. However, efficiencies are very much composition
dependent, so that, depending which compositions have been considered, different results are going to be obtained. With the definition presented in this
limit.
Consider0-1
the
the
efficiency-like parameters will stay within
work,
ing other definitions of efficiency, it is clear that they can go beyond the 0-1
limit, however it is still uncertain if the definition is consistent as well as if the
for
is
bulk
the
correct.
calculating
equilibrium composition
composition
use of
Taylor and Krishna (1993) in the concluding remarks of Chapter 13 (MultiEfficiency
Models)
Distillation:
that
the
compopointed
out
also
component
to
the
computed equilibrium composition,
nent efficiencies are very sensitive
different
it
is
that
to
They
efficienpossible obtain completely
also claim
y*.
is
for
different
by
thermodynamic
computing y*, which
model
using a
cies
different
be
However,
it
to
could even possible get
efficiencies
not surprising.
but
interaction
thermodynamic
the
the
model
only changing
same
utilising
here
both
models or parameter sets give a good
parameters, and surprisingly
data.
the
vapour-liquid equilibrium
representation of
126
0.1
0.1
i o.«,
w
0.0.ý
0.:
0.11
05
111111
10
15
20
Stage number
25
30
35
Figure 5.10: Point Efficiency for the depropanizer column: Propane
This is exactly what it is possible to observe in this work. Even when vapour
liquid
composition are quite well matched, efficiencies are not.
and
In order to compare the well-know Murphree efficiency and efficiencies calhave
knowledge
transfer,
the
through
also
point efficiencies
mass
of
culated
been calculated for the depropanizer column. Figure 5.10 shows the results.
Completely different values are obtain. More experience with this kind of
'right'
is
help
in
the
one.
understanding which one
efficiency could
Efficiency remarks
definition
be
in
few
that
In general, there are a
our
marked
should
points
following
The
conditions, which
of efficiency as well as model assumptions.
follows from the assumption of perfect mixing in the bulk phase, must apply:
0<
therefore
then
if
>
yi*,p
yi, p+,,
yi, p > yi, p+,,
a).
Em < 1.
0<
therefore
b). if yi, < yi, p+,, then yi*
yi,
p+,,
p
,p<
Em < 1.
127
The conditions, a.) and b. ), are fixed by the thermodynamics the
of
mixture.
The equilibrium composition for a light component is always higher than its
corresponding non-equilibrium value. Similarly, the equilibrium composition
heavy
less
is
than the corresponding non-equilibrium value.
of a
component
These conditions are always satisfied in the simulation results when the
conis
vergence tight enough. When composition changes between plates are
small, quite accurate simulation results are neededbecauseseveral places after the decimal point are significant. With all this we can conclude that while
the non-equilibrium models give more accurate composition and temperature
profiles than the equilibrium models, and provide more useful information,
they are not able to match experimental tray efficiency values becausecom)
)
b.
This does not mean that experimental
positions satisfy conditions a. or
it
values of efficiencies are wrong, just means that the non-equilibrium moddeveloped,
including
the one presented in this work, still are theoretically
els
limited by the assumption of perfect mixing of the bulk phase, given by
)
).
b.
conditions a. and
5.3
Illustration
Hybrid
of
Modelling
Approach
With the results obtained in the previous Sections, the solution of hybrid
be
highlighted.
depropanizer
The
models can now
column and the esterifihybrid
The
the
cation column are used as examples of
modelling approach.
for
from
the
the
the
model
results
combined
esterification, which reproduces
less
data
in
20%
compuexperimental
quite accurately using approximately
tational time, are shown.
Using the tray efficiency parameters from the non-equilibrium model in the
been
for
depropanizer
have
the
performed
equilibrium model, simulations
for
is
0.85
An
the
equilibrium model of
used.
specified efficiency
column.
A hybrid model switching between non-equilibrium and equilibrium model
has also been performed. The top section has been simulated with the nonbottom
the
the
section with
equilibrium model, as
equilibrium model and
by
the
efficiency analysis.
suggested
The results in Figs. 5.11 and 5.12 show the simulated liquid and vapour comfor
fixed
HM
the
caseof equilibrium model with
efficiency.
position profiles
in these Figures refers to simulations with fixed column efficiency, while the
128
1.0
0
0.8
as
75
E
c
0
0
aE
0
cr
--0-
0.6
--00.4
--0-
Ethane
Propane
Butane
Pentane
Ethane(HM)
Propane(HM)
Butane(HM)
Pentane(HM)
0.2
0.0
5
10
15
20
25
30
35
Stage number
Figure 5.11: Liquid composition profile with equilibrium
efficiency and non-equilibrium model
fixed
model with
from
A
the
the
other results are
ones obtained
non-equilibrium model.
simfrom
the non-equilibrium model to the equilibrium model at the
ple switch
converged steady state with the non-equilibrium model performs this simulation (time used to convergethe non-equilibrium model: 2.53 min., time for
the hybrid model: 1.25 min. in a SUN-Ultra I Workstation).
With a fixed column efficiency value (for equilibrium model), small deviations can be noted. However, for the hybrid model, not surprisingly, hardly
be
noted.
any variation can
The same procedure has been repeated for the esterification case. In this
129
1.0
0.9
0.8
co
0.7
0
E 0.6
.20.5
0 0.4
0E
0
0.3
0
CL 0.2
co
0.1
0.0
5
10
20
15
25
30
35
Stage number
Figure 5.12: Vapour composition profile with equilibrium
efficiency and non-equilibrium model
130
fixed
model with
0.6
Full non-equilibrium model
Hybrid model
0.5
0.4
0
0.3
0.2
0.1
0.0
10
2468
12
14
Stagenumber
Figure 5.13: Hybrid model applied to the esterification column
Gibbs
free
the
to
the
energy analysis, a
excess
efficiency and
case, according
hybeen
The
has
0.6
for
the
used.
equilibrium model of
specified efficiency
brid model, comprising stages 1 to 3 simulated with equilibrium model and
been
has
4-13
performed.
also
simulated with non-equilibrium model,
stages
The results obtained from the hybrid model are presented in Fig. 5.13. Small
deviations in the composition values (bottom of the column) in the hybrid
model of combined non-equilibrium and equilibrium models, compared with
(time
the
to
full
the
non-equilibrium
used
converge
non-equilibrium model
SUN-Ultra
for
hybrid
1
in
1.46
time
the
2.13
a
model:
min.
min.,
model:
Workstation).
131
5.4
Conclusions
The general(hybrid) model contributes to the development of a computer
distilfacilitates
the
the
aided environment which
understanding of
reactive
lation operation and the solution of different simulation problems. The new
features
have
been
model
validated through reactive and non-reactive systems. The hybrid modelling approach not only allows different model types
in the modelling of a single distillation column, but also the use of different
during
the solution of a specific simulation problem.
simulation modes
The hybrid modelling approach provides a powerful analysis tool through
be
different
importance
in
the
the
terms
models can
established.
of
which
In principle, the non-equilibrium model should provide a more correct and
distillation
behaviour
the
the
reactive
system.
of
consistent prediction of
The non-equilibrium model, however, is more time consuming and requires
information on a greater number of model parameters. Use of a suitable
hybrid model, whenever applicable, makes the solution more efficient without
loss
of model accuracy.
appreciable
In addition to the analysis presented in this Chapter, the hybrid models,
WAS or gPROMS can also be used for other types of analysis, such as bifurcation (which has already been shown in Chapter 4), controllability and
dynamic optimization.
Analysis with the hybrid model in terms of controllability
timization is presented in the following Chapter.
132
dynamic
opand
Chapter
6
Controllability
Issues and
Dynamic
Optimization
Reactive distillation systems require tight process control to ensure that defrom
'optimum'
be
handled
The
development
the
viations
can
adequately.
identification
the
of a good process control scheme requires
of the process
objectives, translation of these objectives to possible or achievable control
integration
the
objectives and
with the control strategy.
Reactive distillation differs from conventional distillation due to the presence
has
has
It
been
influthat
this
of reactions.
proven
a significant
phenomena
(Sneesby
1997).
the
the
al.,
performance of
et
ence on
control system
Obviously, the main objective of any process is to maximise profitability.
The objective for the reaction part of the column should be to maximise the
be
for
to
the
the
should
separation
part
objective
reactant conversion, while
fulfil
full
Hence,
the
the
the
column should
purity of
products.
maximise
the overall objective (maximise profitability) by achieving in each process
(reaction and separation) the particular objective.
Other processobjectives such as contamination problems when working with
dangerous products or to maximise the catalyst life can also be included.
Before starting to implement any controller design some ideas about how
distillation
What
to
is
to
a
measure?
reactive
unit
are
needed.
control
easy
What to manipulate? What are the best pairings?
133
Newells and Lee et al.(1987) presented somequalitative rules answering these
questions:
* control outputs that are not self-regulating
have
favourable
dynamics characteristics
that
e control outputs
inputs
have
large
that
* select
effects on the outputs
inputs
that rapidly affect the controlled variables
9 select
These questions are characteristic of the control of the process itself: controllability. Thus, controllability is independent of the controllers and can only
be changed when the design of the unit is changed (Skogestad et al., 1990).
This is then the main reason for controllability analysis made at the design
stage.
For control system design, many mathematical methods are used, but for
broad
indication.
A
'tethe
only
controllability
results
provide a
number of
dious' simulations normally replaces this analysis.
Important features on controllability of reactive distillation are described in
this Chapter. A linear model is shown. Configuration is defined as well as
disturbances.
measurements,manipulated variables and
In the second section of this Chapter dynamic optimization is addressed.
The optimal control results show that integrated design-control techniques
distillation
be
to
columns.
reactive
applied
should
6.1
Controllability
Issues
There has been very little research in the area of controllability of reactive
has
been
for
batch
disHowever
some research
presented
reactive
columns.
tillation (Sorensen, 1994).
In engineering practice a plant is called ccontrollable' if the plant is able to
(Rosenbrock,
1970).
the
control
objectives
specified
achieve
134
All controllability assessmentsstudies have been proposed with the
main
idea of perfect control. This idea is limited by time delays, right-half plane
zeroes, model uncertainty and manipulated variables constraints. Controllability analysis is an extensive area of research but only some characteristics
here.
are presented
Most of the controllability analysis found in the literature is presented in
terms of linear measures. Although it may seem restrictive usually it is not,
important
the
most
since
non-linearity, given by the constraints in the inbe
handled
linear
(Skogestad
However,
1990).
puts, can
with
analysis
et al.,
non-linear simulations are always required to confirm the results obtained
from the linear analysis.
The procedure for controllability analysis in this work is described by the
following main steps (Wolff et al., 1992) starting with the linearised model,
o scale the plant;
* compute controllability
measures; and,
9 analyse controllability.
To analyse the controllability of reactive columns a control configuration
be
There
is
be
the
that
probably no single configuration
must
assumed.
can
best for all the columns (Skogestad et al., 1990). Because of the large number of possible control configurations there is a clear need for tools that can
best
in
the
the
selection of
possible configuration.
assist
In the following Sections a linear model, some issues about scaling, controllability tools and possible control configurations are presented.
Linear
Model
Applying linear control theory to nonlinear systems is an approach adopted
by several authors who have demonstrated its success on a number of realistic
by
Morari
Perkins
The
Barton
works reported
and
et al(1985),
examples.
(1986), Perkins and Wong (1985) among others, confirmed that the closed
loop performance of the control system was consistent with the predictions
theory.
the
of
135
Almost all the tools developed for controllability analysis are in terms of linRGA,
).
ear models:
poles, zeroes, condition number, etc. So that, in order
to analyse the controllability of reactive columns a linear model is needed.
Based on the non-linear model of Chapter 3, a linear model has been deby
linearising
This
the
equations
veloped
at an operating point.
operating
is
linearisation
done
the
is
the
condition
steady-state
and since
point
around
this point, the linearised model will only be valid for small deviations from
the steady-state and the validity of the model should be probed.
A linear model of any process can be described by the following equations,
i=
Ax + Bu + Ed
Cx + Du + Fd
(6.1)
(6.2)
where,
y=
x=
[measurements
d=
[allstatesIT
(6-3)
variableS]T
(6.4)
T
[control
u=
variables]
[disturbance variableS]T
Applying Laplace transformation
(6.5)
(6-6)
to the linear model, we get,
Gd(s)d(s)
G(s)u(s)
+
y(s) =
(6.7)
describes
functions
the
Gd(s)
G(s)
transfer
efmodels which
are
and
where
fect on the output of the input and disturbances.
The objective is to keep the outputs close to their set-points (r), i. e, e=y-r,
by
inverting
The
ideal
disturbances.
this
control will accomplish
and to reject
become,
inputs
that
the
the process such
manipulated
G-'r - GW'd
d
(6-8)
(control)
linear
the
deriving
When
varimodel, measurements, manipulated
have
been
defined
disturbances
as,
and
ables
136
The measured variables are temperatures and/or, composition of the
(it
is
assumed that these variables can be practically meaproducts
).
Interest
in
from
these
controlling
sured!
variables comes
previous
in
distillation
batch
discontrolling
results
columns as well as reactive
tillation (Skogestad et al., 1990; Sorensen,1994).
The
distillate
flow
the
boil-up
the
manipulated variables are
9
and
vapour
heat
the
and/or
of the reboiler.
Disturbances are in feed flow-rate and in feed composition. The selection of disturbances in the feed composition will offer also an indirect
looking
disturbances
Disturthe
way of
at
on
reaction parameters.
bances in the feed composition affects the reaction stoichiometry.
The linear and non-linear models are compared by looking at Fig. 6.1. A
10% step in distillate flow rate are introduced to both models. The linear
(shown
in
describes
in
the
top
the
graph)
model
responses the same way as
the non-linear model (bottom graph) (the axes for the top graph are the same
bottom
Ethyl
for
the
top
the
the
graph:
acetate composition at
of
column
as
linear
The
for
be
time).
then
model can
controllability and control
used
vs.
far
from
for
linearised
disturbances
i.
too
the
small
point, e. only
studies not
(Skogestad et al., 1990).
6.1.2
Scaling
Why is it important
to scale the plant?
Most of the controllability measures are scale dependent. It is crucial that
following
described
in
is
One
the
way of scaling
variables are scaled properly.
(Skogestad and Postlethwaite, 1996),
its
(u):
to
Inputs
allowed range.
normalise uj with respect
9
its
(y):
to
Outputs
allowed range.
normalise ej with respect
*
its
(d):
dkwith
to
Disturbances
allowed range.
respect
normalise
*
Another way, which also achievesthe same objectives, is to scale the transfer
d,
Gd,
G
the
that
the
signals
e
of
u,
assuming
magnitude
allowed
and
matrices
(Havre
Skogestad,
The
later
frequency
1992).
the
do
with
and
vary
not
r
and
in
this
work.
is the scaling adopted
137
Step Response
x 10,
3.
2.5
2
1.5
1
0.5
0
0.525
IE
0.52
0
0
a
E
0
0.515
Co
w
0.51
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time (h)
(10%
linear
from
the
Response
Figure 6.1:
and non-linear model
obtained
step change)-
138
6.1.3
Controllability
Tools
Which tools for controllability analysis are available?
Functional
e
controllability
The first thing that should be checkedis that the plant is functionally
(Skogestad
Postlethwaite,
1996).
controllable
and
Essentially, a plant is not functionally controllable if the rank of G(S) is
for all s less than the number of outputs that we want to control. This
is extended for square plants to the requirement that det(G(s)) should
be different from 0. For unstable plants it should be checkedthat the
unstable states are state controllable and state observable. Nevertheless, this is not important becausethe states we really care about are
in the output vector y.
half
Right
delays
time
e
plane zeroes and
Right-half plane zeroes (RHP-zero): a zero is defined as the value for
G(s)
(for
for
loses
be
this
s
which
rank
square matrices
may
computed
det(G(s)).
A
G(s)
limits
RHP-zero
the
the achievsolution of
as
of
bandwidth
holds
the
this
able
of
plant and
regardless of the type of
(Holt
RHP-poles
limitaMorari,
1985).
also put
controller used
and
tions on the control system through stability considerations. If there
direction,
is
important
RHP-zeros
in
it
that the
the
and poles
are
same
RHP-pole at p is located at a higher frequency than the RHP-zero at
i.
z, e, p> z-
Time delays have essentially the same effect as RHP-zeros.
decomposition
Singular
value
e
Singular value decomposition (SVD) of any matrix G is:
UH.
(6.9)
There will be rank(G) singular values. The SVD is useful to examine which manipulated input combinations have the largest effect and
largest
disturbances
the
give
output variations.
which
139
Condition
*
number
The ratio between the maximum singular value and the minimum
is
the condition number. Plants with large condition
singular value
'ill-conditioned',
different
input magnumbers are
and require widely
depending
direction
desired
the
(Skogestad
the
nitudes
on
of
output
Postlethwaite,
The
1996).
is
and
process sensitive to plant/model mismatch when the condition number is high.
Relative
o
gain array
The relative gain array (RGA) defined by Bristol (1966) for a square
defined
is
'open-loop'
'closed-loop'
the
bethe
plant
as
ratio of
and
gains
tween input 1 and output Z. Plants with large RGA are 'ill-conditioned'.
The RGA is scaling independent and ideally should be a diagonally
dominant matrix. In this case the control loops are largely decoupled.
The general rule about pairing is to pair the controlled outputs yj with
the manipulated variables uj in such way that the RGA values are positive and as close as possible to unity. RGA can also be calculated as a
function of frequency. If the large values of RGA are at high frequencies
One
'never'
difficult
to
should
control.
use a controller with
plants are
large RGA values (Skogestad, 1992). One big disadvantage of RGA is
that it contains no information about disturbances and dynamic be-
haviour of the plant.
6.1.4
Control
Configurations
The control configurations of a distillation column can be divided into two
categories:
Configurations
Basic
e
Schemes
Control
Ratio
e
basic
the
6.1
Table
configurations:
summarises
feasible
but
lead
bad
level
VB
to
LD
Both
configurations are
control
and
(via
balance)
the
control
overall
mass
and reduces
composition
affects
which
control response.
140
II Configuration
LV
LB
LD
DV
DB
VB
Primary
Variable
Manipulated
duty
reflux rate or reboiler
bottoms rate
distillate rate
distillate rate
distillate rate
bottoms rate
Secondary
Variable
Manipulate I
duty
reboiler
or reflux rate
reflux rate
reflux rate
duty
reboiler
bottoms rate
duty
reboiler
Table 6.1: Basic control configurations
The LV configuration is the most common used in the industry. The main
diis
this
that
the
manipulated variables affect
configuration
advantage of
to
compositions.
redly
The IV configuration is sensitive to disturbances when no control is used but
RGA
is
In
the
insensitive
this
configuration,
with one-point control.
rather
large
large
for
high
be
to
reflux.
purity columns with
expected
The DB configuration it is believed violates the mass balance and is normally
demonstrated
has
been
its
experimeneffectiveness
not considered although
tally (Skogestad et al., 1990). This configuration was considered as a non
does
basis
it
however
basis,
long
term
term
on a short
a
on
viable alternative
long
for
time
then
the
B
If
D
behave
a
and are adjusted
unreasonably.
not
is
This
then
filled
being
by
configuration
up or emptied.
column will end
feed-back
control.
actually rather good when using composition
For the DV configuration the RGA elements are always small contrary to the
(thus,
infinite
RGA
DB configuration where the
at steady state
elements are
for
this
is
RGA
configuration).
the
measure
not a reliable
but
the
more
For the ratio control schememany combinations are possible,
by
been
has
(L/D)(V/B)
Table
in
6.2.
recommended
presented
are
common
The
double
known
'the
Its
Morari,
1987.
Skogestad and
ratio' scheme.
as
but
for
double
is
not
control
composition
advantageous
scheme
ratio control
(as
decoupling
implicit
This
for single composition control.
scheme creates
the
schemes).
ratio
all
well as
141
Configuration I Primary
Manipulated
Variable
(L/D)V
duty
reboiler
(L/D)(V/B)
boil-up
ratio
reflux ratio or
(V/B)L
reflux ratio
Secondary
Variable
Manipulated
reflux ratio
boil-up ratio or reflux ratio
boil-up ratio
Table 6.2: Ratio control configurations
Important differences between configurations are related to interaction when
(Shinskey,
input
1984)
to
and sensitive
using single-loop controllers
gain unfor
both
large
decoupler
In
two-point
control.
casesa
certainty when using
frequency
in
RGA
the
the
range of
closed-loop results
at
control
value of
problems.
Recent work has been presented in the literature for an ETBE (ethyl-tertbutyl-ether) production (Sneesby et al., 1997). They recommended two configurations: LV and LB, both for a single composition control. They found
that the location of the temperature control is critical and can dramatically
due
is
interactions
This
to
the
the
the
control scheme.
stability of
affect
between the chemical and phase equilibria. Dynamic simulation for studying
the responsesof the system was recommended.
A model for reactive distillation columns including control system and the
(1997b).
by
derived
Bock
et al.
possibility of coupling additional columns was
in
disturbances
to
They analysed sensitivities
a
manipulated variables and
for
the
distillation
esterificolumn
column and a coupled recovery
reactive
(to
it
SISO
Based
the
this
make simple) control
on
cation of mystiric acid.
structure was carried out.
Bock et al.(1997b) also found that the influence of the reaction temperature
is
be
known,
has
temperature
the
to
the
when
reactive column
profile within
They
for
the
the
using
system
controlled
purity.
variable
reference
used as a
to
loops,
SISO-control
ensure conversion.
not
sufficient
two
as only one was
by
the
top
the
manipuThey kept the temperature at
column constant
of
fast
leads
to
the
the
top
heat
a very
lating the
column, which
of
supply at
its proximity; and the temperature in the middle of the
due
to
loop
control
In
heat
by
the
the
kept
is
most
of
reboiler.
manipulating
constant
column
A
long
time
disturbances
the
was
quite
well.
were
eliminated
the
cases
of
influence
the
the
the
could
not
to
of
reaction
when
set-point
achieve
required
142
be compensated by a manipulated variable.
For this work the DV configuration has been chosen.
6.1.5
Examples
Application
of
It is known that the interactions between the top and bottom of distillation
large.
Changing
lead
the
top
to changes
columns are very
conditions at
will
in the bottom, and vice versa. Becauseof this distillation columns are known
difficult
impossible
to control.
as very
or
Using two single control loops it may be possible to control both the top and
the bottom of the column.
In the caseof reactive distillation it is desirable to control not only the combut
loss
be
the
the
that
position of
product,
also
of reactants
may
at the
be
the
that
column so
other end of
only one point control will not
able to
achieve this.
The temperature of the reactive section may be another variable desirable to
however,
it
is
in
For
this
the
work
not
examples studied
considered.
control,
it seemsto be sufficient to control compositions or top and bottom temperatures. Controlling these variables the temperature in the reactive section (or
in the column itself, in the case of full reactive column) suffices to achieve
desirable temperature profiles.
The DV-configuration was adopted for this controllability analysis:
DV-configuration: controlling the distillate flow and the reboiler heat duty
(or vapour boil-up) to control the distillate composition (or temperature)
(or
bottom
temperature):
the
composition
and
Controlled variable:
Manipulated variable
XD
XB
TD)
or
(or, TBor
D
XD or XB (or, TBor TD)
V
Procedure
143
Assuming that we have made a decision on the plants inputs and outputs,
G
investigate
to
to
the
we want
analyse
model
what control performance can
be expected. The procedure is described from Skogestad and Postlethwaite
(1996) and is only an extension from the procedure presented above. The
described
below.
indices
controllability
are
Linearise
the model.
*
Scale all the variables (inputs, outputs, disturbances, references) to
obtain a scaled model.
Check
functional
*
controllability.
Compute
the poles.
9
Compute
the zeroes.
e
Obtain
G(jw)
frequency
RGA
the
the
9
response
and compute
matrix.
Compute the singular values of G(jw) and plot them as function of
frequency.
Compute condition number: the ratio between the minimum and maximum singular singular value.
Analyse.
de
For the results shown in this thesis, the hybrid model included into the
WAS simulator has been linearised, however, gPROMS also allows the linhave
been
indices
All
the
obtained using
controllability
earisation of models.
MATLAB.
MTBE
Production
The MTBE production column has been first analysed by a series of simfor
for
The
taken
which multiple solutions exist was
steady state
ulations.
butenes
fed
fed
is
in
10
11.
Methanol
tray
the
while
remain
at stage
study.
Simulations have been performed with the non-equilibrium model and it was
hour
disturbance
for
feed
flows
in
have
I
to
changes
and composiassumed
distillate
flow-rate.
duty
found
heat
Full
be
in
and
tion, reboiler
results can
144
-El
0
Le
E
E
0
0
1
CL
9
w
Co
5
10
15
Time (h)
Figure 6.2: High conversion of MTBE as result of 10% increase in reboiler
heat duty.
Appendix B. The reboiler heat duty was found to be a very sensitive variBelow
high
is
There
the
conversion solution.
a range of values giving
able.
low
drastically
to
the
the
that
conversion
changes
system
range
or above
by
(see
This
Figs.
the
6.2,6.3).
the
conobtained
results
confirms
solution
tinuation method (bifurcation analysis in Chapter 5).
Through simulation it has been shown that adding or reducing the number of
it
in
However
improve
does
the
many cases
column performance.
not
stages
from
Fig
5
be
6.4
where reactive stages
seen
may worsen performance, as can
have been added to the system. The purity of MTBE at the bottom of the
design.
by
half
the
the
is
to
nominal
achieved
of
purity
almost
reduced
column
between
desirable
is
the
in
then
the
If any changes
ratio
number of stages
between
the
the
the size of
reactive section, as well as
stripping section and
(see
be
the
changed.
reactive column should
the rectification section and
detail
for
B
Appendix
explanations).
more
described
following
the
above
Secondly the controllability analysis
procedure
145
0.9 0.8 0.7
E
E 0.6(
0
1
CD
fm 0.5
0
1
0.4
CL
0.3 -
Co
5 0.2 0.1 0.
0
0.5
1
1.5
2
2.5
Time (h)
Figure 6.3: Low conversion of MTBE as a result of 15% decreasein reboiler
heat duty.
Once
in
this analysis, the steady state with multiple sowas studied.
again
lutions is analysed (methanol is fed in tray 10 while the butenes remain fed
for
DV-configuration
be
Three
11).
the
as can
cases were analysed
at stage
Some
indices
from
Table
in
6.3.
obtained are surnmarised
controllability
seen
Table 6.4. Accounting for the process dynamics some controllability indices,
frequencies.
RGA
and singular values were calculated over a range of
such as
The results from the frequency-dependent analysis are the same as obtained
for the steady-state analysis. Detailed calculations are in Appendix B.
From the analysis of all the indicators (steady-state and frequency dependent) it can be seen that:
have
indicating
the
the
systems
negative real parts,
stability
poles of
9 all
the
open-loop system,
of
limiting
RHP-zeroes,
the
the
present
achievablecontrol
studied
cases
" all
performance,
to
to
the
plant/model
very
sensitive
are
mismatch,
according
processes
"
the high condition number,
146
1.0
--oA
0.9
Nominal design
Simulated design with 5 extra stages
0.8
0.7
It:
0.6
0.5
0.4
0.3
0.2
0.1
0.0
05
15
10
20
25
Stage
MTBE
Lower
6.4:
Figure
as a result of adding extra reactive stages
purity of
147
-- FMeasured Variables
Cases
Case I
XMTBE,
D
XMTBE,
B
Case 2
XMTBE,
TB
D
Case 3
TD
I Disturbances
Xisobutene, F
TF
Xisobutene, F
TF
Xisobutene, F
TF
TB
Table 6-3: Measured variables for the controllability study on MTBE production
Case
RGA
1
1.063
2
-0.06
1.047
Singular values
-0.06
1.063
-0.047
1.047
-0.047
0.71 0.29
0.29 0.71
3
mZn
max
mM
max
mZn
max
Table 6.4: Controllability
0.0223
2.8356
0.00387
0.12200
0.000316
0.320000
ondition
Number
RHP-zeroes
127.16
yes
31-52
yes
1012.7
yes
indices for the MTBE production
following
for
from
RGA
the
the
three
the
cases,
control pairings
values
9
be
suggested,
can
Manipulated variable: Controlled variable:
XMTBE, D
D
V
TB or XMTBE,
B
Temperatures are easy to measure in a distillation column, and they
have a linear dependence with the control measure. The pairing of the
bottom
in
done
be
the
temperature
the
of
with
vapour-boil-up should
the column probably in spite of the very small minimum singular value
(compared with that of Case 1).
in
is
the
the
all
cases, suggesting
very small
minimum singular value
9
desired
be
inputs
the
to
large
that
outobtain
may
needed
magnitude
puts.
Ethyl
Acetate Production
described
following
the
procedure
above was also
The controllability analysis
The
Chapter
in
4
for
the
column
column.
studied
using
esterification
studied
148
the equilibrium model with tray efficiency is taken for the linearisation and
Three
for
DV-configuration
the
controllability analysis.
caseswere analysed
Some
in
Table
6.5.
as presented
results showing the controllability indices
(full
in
Table
6.6
Appendix
in
B).
are given
results are shown
Cases
Case 1
Measured Variables
Disturbances
TD
Xethanol, F
TF
TB
Case 2
XEtAc, D
Xethanol, F
TF
XEtAc, B
Case 3
XEtAc, D
Xisobutene, F
FF
XEtAc, B
Table 6-5: Measured variables for the controllability
production
Case
RGA
1
1.17 -0.17
0.17 1.17
1.05 -0.05
1.05
-0.05
1.02 -0.02
1.02
-0.02
2
3
Singular values
min
max
min
max
mZn
max
Table 6-6: Controllability
0.00106
8.08000
0.00018
0.24400
0.0008
0.1317
study on ethyl acetate
Condition
Number
7622.64
yes
1355.56
yes
164.62
no
RHP-zeroes
indices for the ethyl acetate production
From the analysis of all the indicators (steady-state and frequency dependent) it can be seen that:
indicating
have
the
the
stability
negative real parts,
systems
poles of
* all
of the open-loop system,
limiting
but
RHP-zeroes,
the
the
one present
casesstudied
almost all
Case
for
it
3,
is
RHP-zeroes
there
not
achievable control performance,
disturbances
be
that
the
temperature
are
measurements or
noted
may
RHP-zeroes
in
the
the
system,
generating
the processes are very sensitive to plant/model
(very
high),
the condition number
149
mismatch, according to
from the RGA values for the three cases,the following control pairings
be
can
suggested,
Manipulated variable: Controlled variable:
D
XEtAc, D
v
XEtAc, B
the minimum singular value is very small in all the cases, suggesting
that large magnitude inputs may be needed to obtain the desired outputs.
6.2
Dynamic
Optimization
If we wish to determine, for example, the optimal diameter of the reactive
in
column
addition to the optimal way of operating the column over the time
then we need to use dynamic optimization.
Since operation and design problems are of transient nature then both are
dynamic
known
is
applications of
also
optimization, which
as optimal control.
The dynamic optimization in this work is carried out by using gPROMS and
deis
Ltd.,
In
1997).
to
general, gPROMS
able
solve problems
gOPT(PSE
by
following
items,
the
scribed
Mathematical
Problem
We consider a process described by:
i(t),
y(t), U(t), V) =
g(x(t),
i(t)
differential
the
the
are
variables,
and
algebraic
where, x(t) and y(t) are
time derivatives, u(t) the control variables and v the time invariant parameters to be determined by the optimization.
Initialisation
by:
described
is
initial
the
the
that
system
condition of
gPROMS assumes
i(o),
Y(O), U(O), V) =0
g(x(o),
150
By performing dynamic simulation we will completely determine the transient response of the system, once the time variation of the control is fixed.
Objective
Function
determine:
to
gPROMSIgOPT seeks
horizon,
tf,
the
time
*
invariant
the
the
time
* v,
value of
parameters,
horizon
the
the
time
the
time
variation of
9
control variables over
entire
final
to
the
as well as minimise or maximise
value of a single variable, z,
mintf,
u(t), V,tE[t, tf]Z(tf
)
(6.12)
differential
is
the
variables x or the algebraic variables y:
z either
Z(tf) =
10tf
Bounds of the optimal
D(x (t)
Iý
(t)
(t), u (t), v) dt
Iy
decision variables
tmin
f-f
min
<
U
v
min
<
tf
<
<
U(t)
<V<
tmax
(6-14)
Umax
(6.15)
Vmax
(6.16)
Vt E [07tf]
End point constraints
These are conditions that must be satisfied at the end of the operation.
W*
equality =#,. w(tf) =
inequality
=*
W,
in
<
Path constraints
151
W(tf)
<
Wmax
(6.17)
(6.18)
These are conditions that must be satisfied at all the time during the
operation.
mtn
w
<
W(tf)
Vt E [01tf
:5 Wmax
(6-19)
Classes of control variable profile
but
this is the one utilised within this
gPROMS offers several possibilities,
work,
Piecewise
9
constant control: the controls remain constant at a certain
value over a certain part of the time horizon before jumping discretely
to a different value over the next time interval.
Solution
The solution comprises the following:
horizon,
the
the
time
tf,
9
value of
9 the values of the time invariant parameters,
9 the variation of the control variables over the time horizon.
The dynamic optimization
is
problem
solved as:
e parametrisation of the control variables
fix
the parameters values
o
informaDA-SOLV
integrate
the
the
to
system and get
sensitivity
use
tion
NLP
SRQPD
to
the
to
get
new set of values.
solve
9 use
The control vector parametrisation approach (Vassiliades et al., 1994a,b) is
implemented within the gPROMS process modelling (PSE Ltd., 1997). The
dynamic
the
optimization problem
problem of solving a
approach converts
(DAE, initial estimates, junction conditions and point equality and inequality
finite
(NLP).
dimensional
The
to
nonlinear programming
a
soluconstraints)
DAE
NLP
the
the
tion of
solution of a multistage
are calculated via
system
152
in the variable sensitivities. The control parametrisation approach requires
that the DAE system of the problem being solved must have a solution for
decision
the
variables may take.
any set of values of
Other method of solving dynamic optimization has been developedby Biegler
(Cuthrell
Biegler,
In
1988).
differentheir
the
and
approach
and co-workers
tial equations are discretised by using Lagrange form polynomials and orthogThe
onal collocation.
resulting algebraic approximations are then written as
in
NLP.
The
large
NLP's
is
for
an
solution
constraints
of
still under study,
that, the method works quite well for small systems but starts to fail when
large number of equations are neededto represent the physical problem, since
discretised.
the
equations are
all
6.2.1
Optimization
problem
This study addresses some aspects of optimal control of a reactive distillation
is
The
this
the
example
since
esterification column was used as an
column.
system that, using equilibrium models with tray efficiency, can accurately
data.
represent experimental
Problem statement:
The system under consideration is shown in Fig. 6.5. Table 6.7 gives the
structure of the optimization problem.
The overall objective is to design the column (diameter, areasof reboiler and
(Jo
horizon
interest
finite
time
feasible
to
of
operation over a
condenser) give
hours) while maximising profitability, i. e. minimising the total annualised
cost.
minimise
Cost = minimise
(Cshell
+
Ctray
+
Chx
+
Cop)
(6.20)
Chx
shell)Ctray
is
the
tray,
the
is
the
the
cost
of
column
cost
of
whereCshell
C,,
is
(for
the
the
heat
and
is the
reboiler and condenser),
exchangerscost
p
operating cost.
bottom
loss
in
the
the
as well
of product
maximum
There are constraints on
153
]late
> 0.50
Ac
A EtAc
0.30
Figure 6.5: Reactive distillation column for the design study
154
Objective function:
Fixed:
Optimized-variables
Specifications:
Disturbance:
Control variables:
total annualised cost
(13)
trays
number of
feed location (tray 6)
diameter of the column
(3m < D, < 6m)
(4m !ý D2< 8m)
areas of condenser and reboiler
ethyl acetate top composition
(xtop > 0.525)
bottom
ethyl acetate
composition
(Xbot
0.3)
ý,
:!
feed composition
(sinusoidal, 1h period)
distillate flow-rate
(1000 < Dzst < 3800(mol/h))
heat
duty
reboiler
1 (0.25 x 107 <Q
Table 6.7: Optimization
:50.9X106(M
details
is
The
distillate.
in
the
the
subject
system
of
product
purity
minimum
as on
Manipulated
feed
in
disturbance
the
variables
to a sinusoidal
composition.
flow-rate
distillate
the
heat
the
for control are the
according
reboiler and
of
to the suggestions from the previous controllability studies.
function
together
The
2609
with
The problem comprises
objective
variables.
C.
Appendix
in
disturbances and constraints are given
Results
(through
tested
simulation)
The nominal design for the reactive column was
feed
disturbance
The
comwas a sinusoidal
in the presenceof a disturbance.
large
violations.
there
As
constraint
of
number
a
were
expected
position.
be
top
the
seen
composition, as can
The most important was the violation of
limits
beyond
the
from Fig. 6.6, where the composition of ethyl acetate goes
for
highlights
This
the
be
0.525).
need
(minimum purity at the top should
the
system.
controlling
155
0.6
0.5
r. 0.4
0
-,4J4
-A
th
0
a
0
u
0
,
to
4J
4J
pq
0.3
0.2
0.1
0.0
10
Time
Figure 6.6: Top and bottom compositions of ethyl acetate. Results from the
disturbance.
simulated nominal case with
156
0. E
0.5
0.4
0.3
co
-i
0.2
0.1
0.0
56789
10
Time
Figure 6.7: Ethyl acetate composition. Results from the dynamic optimization.
Dynamic optimization was carried out, controlling the reboiler heat duty and
the distillate flow-rate. The results are shown in Fig. 6.7. Table 6.8 comFor
the
the
the
and
optimal
values
obtained.
optimal solution
nominal
pares
there are no constrains violations. The system is rejecting the disturbance
by
is
increasing
heat
the
top
the
product specification maintained,
of
while
(explaining
implies
is
increased
that
the
this
the reboiler,
reaction rate
why
the bottom purity increasesalthough remaining below the upper bound). It
because
is
increased
that
the
the
be
while
operating
well,
cost
of
as
noted
can
duty
increase,
in
is
heat
the
a
notable
reduction
capital cost made
reboiler
by a reduction in the column diameter.
157
0.9
0.8
0.7
CY
0.6
0
U
0.5
0.4
0.3
0.2*
0
56789
10
Time
Figure 6.8: Controlled variable: Reboiler Heat Duty(* 106(MJlh)).
from the dynamic optimization.
158
Results
Variable
Nominal
Optimal
Diameter
Reboiler heat duty
Distillate flow-rate
5.6/6.25m
106
MJ1h
0.58 x
1150Kmol/h
4.5/5m
106
X
0.85
MJ1h
737.87Kmol/h
Table 6.8: Optimization
6.3
results: nominal and optimal values
Conclusions
Controllability studies have been performed for two reactive systems. The
for
from
have
been
the
this
the
applied
solution
of
analysis
optimal
results
control problem.
Following the controllability studies an optimal control problem has been
distillation
design
From
these
the
columns
of reactive
optimal
results
solved.
has been determined.
The optimal solution found is limited to the objective function, proposed disdisturbances
Certainly,
defined.
types
the
to
turbance, and
of
other
model as
however
in
be
included
the
optimization problem,
and uncertainties should
The
data
(or
the
to
results are not available.
corroborate
plant)
experimental
for
is
for
be
this
other
generally applicable
analysis and
used
approach can
similar systems.
large
Nowadays,
for
broad
The results presented open up a
research.
area
trouble,
be
simultaneous
and
much
without
solved
can
problems
optimization
design of reactive distillation columns and their associated control structure,
for
be
the
disturbances
optimal
solved
to
can
uncertainties
as
well
as
subject
solution.
159
Chapter
7
General Conclusions
Future Work
and
The main contributions of this thesis have been:
hybrid
for
distil9A
model
continuous staged reactive and non-reactive
lation columns.
The hybrid modelling approach allows different model types in the moddistillation
It
different
the
elling of a single
column.
also allows
use of
during
the solution of a specific simulation problem.
simulation modes
The framework developed helps to avoid problems of mismatch between model and simulation results as well as between the two forms
of models.
New tools for the analysis of distillation columns.
The importance of terms in the different models can be established.
The analysis of the Gibbs free energy as a function of time has been
distance
from
Bifurcation
the
analysis
equilibrium.
used as a measure of
for reactive columns gives the existence or not of multiple solutions.
Controllability
analysis.
9
For the first time controllability analysis has been addressed for rehelped
The
have
indices
distillation
to
columns.
controllability
active
determine the best control structure for the problem studied.
Dynamic optimization
As well as controllability issues, for the first time, to the best of our
knowledge, dynamic optimization has been performed for continuous
The
here,
distillation
limited
work presented
systems.
although
reactive
160
in scope, opens a new area of study for the interaction between design
and control of reactive distillation columns.
The simulation and design of hybrid (reactive and non-reactive) distillation
different
types of models and combination of them is made
columns with
framework
developed
in this thesis. Different analythrough
the
possible
sis tools, such as tray efficiency values, monitoring of the Gibbs free energy,
bifurcation analysis, are available within the framework. Moreover, controllability indices can be determined as a first step towards developing the control
finally
design
distillation
the
structure and
optimal
of reactive
columns can
be found through dynamic optimization.
The motivation for this study has been the interactions between reaction and
for
Increased
interest
in
that
the
a reactive column offers.
separation
units
fine
in
chemicals and pharmaceuticals manufacturing, as well as
applications
in the commodity industry chemical industry, has also been an incentive of
this work.
The work has been mainly focused in developing models (dynamic and steady
behaviour
illustrate
the
to
of
complex
state) which are sufficiently rigorous
different
The
distillation
thesis
addresses
columns.
continuous staged reactive
distillation
design
columns.
of reactive
aspects of operation, simulation and
A novel integrated model has been presented, given the hybrid modelling
different
The
integration
types
of models as well as modes of
of
approach.
design
for
the
tool
of reactive columns.
simulation gives a new
Controllability issueshave also been addressedfor reactive distillation columns.
it
for
been
investigated
has
However, this analysis
specific casesand
only
in
Assessment
depth.
in
be
process
controllability
of
greater
studied
should
design should include many other aspects such as the ability of the system
to handle process uncertainties.
in
has
been
dynamic
this
Optimal control or
work.
addressed
optimization
future
for
benefit
that
work
This constitutes a major area of work
would
in
this
from
the
leading
work.
obtained
results
161
The following future work is suggested:
First of all the need for experimental data must be highlighted, since
lacking
literature.
in
data
is
the
open
seriously
real
be
The
both
throughly
tested
9
non-equilibrium model should more
when
in
liquid
film
included.
the
transfer
and vapour
mass
resistances
are
Models that consider reaction in the liquid film may be needed in cases
(albe
in
framework
fast
included
the
general
of
reactions, and should
though at the present such models have not given much advantage over
the models which does not consider reaction in the liquid film).
be
investigated.
in
The
the
models should
effect of changing correlations
*
Controllability
e
be
in
terms of non-linear models.
may
carried out
Dynamic optimization should be carried out in order to determine the
design
the
together
of
with
optimal control structure and parameters
the reactive column at minimum annualised cost. It has been shown
that the simultaneous design of the process and the control structure
(Mohideen
design
1996;
et al.,
offers great advantages over sequential
Bansal et al., 1999). Important savings can be made by utilising the
detecting
the
control
possibility of
simultaneous approach as well as
design
is
be
detected
the
that
already made.
when
might not
problems
162
Nomenclature
a
a
Activity coefficient [-]
[M2/M3]
Interfacial area
a'
A
Interfacial area per unit volume of froth [M2/rn3j
Area [m 2]
Ab
[M21
Bubbling area
[M2/Ml]
Ap
Specific surface area of the packing
A, B, C,.. Coefficients in correlations
C
Concentration [kmol/hj
Cal
Liquid capillary number
df
Driving force
21S]
Df
Fick's coefficient [m
[M2/S]
Dij
Maxwell-Stefan diffusivity (Inverse friction coefficient)
Diffusion
low
D?.
i
in
infinite
in
j
coefficient of species
concentration
species
Z3
Dp
Packing size [m]
DSp
Heat of reaction [J/Kmol]
[eq]
Amount
the
of acid groups on
resin per unit mass
eq
Activation energy [J/Kmol]
Ea
EM
Efficiency parameter [-]
EM
Murphree tray efficiency
P
Liquid entrainment [Kmol/hl
EL
EV
Vapour entrainment [Kmol/h]
fij
Friction factor [-]
Feed flow [Kmol/h]
F
Superficial factor [-]
F,
Excess Gibbs free energy [J/Kmol]
GE
Froth height [m]
hf
Exit weir height [rn]
h,,
[m]
Height
the
H
packing section
of
[J/Kmol]
Enthalpy
H
[J/Kmol]
hold
Enthalpic
HE
up
163
[CM 2/sl
HI
HIf
Hvf
Hv
i
K
K
K,
q
L
M
M, M
ni
N
Ni,
p
NC
Np
Nr
NR
P
sat
P
PL
PV
Q
R
Re
Si,
P
Sc, S,
A
t
t9
ti
T
us
V
V
V
V
X
XjXj
Y
Liquid enthalpy [J/Kmol]
Enthalpy of a liquid feed [J/Kmol]
Enthalpy of a vapour feed [J/Kmol]
Vapour enthalpy [J/Kmol]
[MOI/M2,
S]
Molar diffusion flux
Matrix of volumetric mass transfer coefficients [s-1]
Equilibrium constant [-]
Chemical equilibrium constant
Liquid flow rate [Kmol/h]
Hold-up [Kmol/h]
Molecular weight [Kg/Kmol]
Mass transfer rate [Kmol/h]
Number of transfer unit
Molar flux of speciesi at a particular point in the two phase dispersion [Mol/,rn2 S]
Number of components
Products
Reactants
Number of reactions
Pressure [atm]
Vapour pressure [atm]
Liquid extracted from (or added to) any stage [Kmol/h]
Vapour extracted from (or added to) any stage [Kmol/h]
Volumetric flow rate [m'/s]
Gas constant [J/Kmol K]
Reynolds number [-]
Reaction rate in the liquid bulk [Kmol/hj
Schmidt number [-]
Sherwood number
Time [h]
Gas contact time [h]
Liquid-phase residence time [h]
Temperature [K]
21S]
Superficial vapour velocity [m
21S]
Velocity [m
Molecular diffusion volumes
3/M01]
Molar volume at the normal boiling point of the solute [cm
Vapour flow rate [Kmol/h]
[-]
Liquid composition
Factor of concentration weight
[-]
Vapour composition
164
Liquid flow path length [m]
z
Greek letters:
2
[W/m
K]
Heat
transfer
a
coefficient
[-]
LY
Point
efficiency
,
r
Factor accountingfor thermodynamic non idealities
[-]
Activity
7
coefficient
AH
vap
V)
0
OP
P
A
A
Y
Pi
V
a
Enthalpy of vaporisation [J/Kmol]
[W/M2]
Energy flux
Fugacity coefficient
Heat transfer rate [W]
Density of mixture [Kmol/m']
Density of pure components [Kmol/m']
Thermal conductivity [W/(mK)l
Thermodynamic factor
Viscosity [CP]
Chemical potential [J/Kmol]
Stoichiometric coefficient
Surface tension [N/m]
Superscript
Liquid
L
Vapour
v
Subscript
CP
i1i
P
r
AH
Heat capacity
Components
Tray
Reduced
Enthalpy
Mathematical Symbols:
Gradient
'7
Matrix
Inverse of a matrix
Abbreviations:
Acetic acid
AA
Algebraic equation
AE
formulae
differentiation
Backwards
BDF
Differential algebraic equation
DAE
165
EOS
Et
EtAC
MTBE
ODE
RHS
SRK
w
Equation of state
Ethyl
Ethyl acetate
Methyl-tert-butyl ether
Ordinary differential equation
Right hand side
Soave-Redlich-Kwong
Water
166
References
AIChE, 1958, Bubble Tray Design Manual, AIChE, New York.
Agarwal, S., and Taylor, R., 1994, Distillation Column Design Calculations
Using a Non-equilibrium Model, Ind. Eng. Chern. Res., 33,2631.
Al-Jarallah, A. M., Siddiqui, M. A. B., and Lee, A. K. K., 1988, Kinetics of
Methyl Tertiary Butyl Ether Synthesis Catalized by Sulfuric Acid, Chem.
Eng. J., 39,169.
Ancillotti, F., Massi Mauri, M., Pescarollo, E., and Romagnoni, L., 1978,
Mechanisms in the Reaction between Olefins and Alcohols Catalysed by Ion
Exchange Resin, J. Molec. Catal., 46,49.
ASPEN PLUS User Guide, 1997, Aspen Technology.
Backhaus, A. A., 1921, U. S. Patent 1400849.
InterThe
N.,
E.
1999,
Pistikopoulos,
J.
D.,
Perkins,
R.,
Bansal, V., Ross,
and
Control,
J.
Proc.
Distillation,
Effect
Double
Control:
Design
and
actions of
accepted.
Azeotropic
Diagrams
Phase
Theory
F.,
M.
1987,
Doherty,
and
of
Barbosa, D., and
A14,
Soc.
London,
R.
Systems,
Proc.
Reactive
Two-Phase
for
Conditions
443.
ChemThe
Equilibrium
Influence
F.,
M.
1988a,
Doherty,
of
Barbosal D., and
Sci.,
Chem.
Eng.
Diagrams,
43,529.
Phase
Liquid
Vapor
ical Reactions on
167
Barbosa, D., and Doherty, M. F., 1988b, The Simple Distillation of HomogeReactive
Chem.
Mixtures,
Eng.
Sci.,
43,541.
neous
Barton, G.W., and Perkins, J. D., 1986, A Case Study of Dynamic Simulation
in Control System Design, presented at Technology & Industry, Sydney.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena7
John Wiley & Sons Ltd., New York.
Bock, H., Jimoh, M., and Wozni, G., 1997a, Analysis of Reaction Distillation
Using the Esterification of Acetic Acid as an Example, Chem. Eng. Technol.,
20,182.
Bock, H., Wozni, G., and Gutsche, B., 1997b, Design and Control of a ReacColumn Including the Recovery System, Chem. Eng. Froc.,
tion Distillation
367 101.
Bondy, R. W., 1991, Physical Continuation Approaches to Solving Reactive
Distillation Problems, AIChE Meeting, Los Angeles.
Bossen, B. S., Jorgensen, S.T., and Gani, R., 1993, Simulation, Design and
Analysis of Azeotropic Distillation Operations, Ind. Eng. Chem. Res., 32,
620.
Bravo, J. L., and Fair, J. R., 1982, Generalised Correlation for Mass Transfer
in Packed Distillation Columns, Ind. Eng. Chem. Proc. Des. Dev., 21,170.
Bristol, E. H., 1966, On a New measure of Interactions for Multivariable ProAC-11,133.
Automatic
Control,
Trans.
IEEE
Control,
cess
Model
for
C.
Gani,
A
Generalized
DisA.,
R.,
1986,
Ruiz,
T.,
I.
Cameron,
and
Comp.
Chem.
Computational
Aspects,
Numerical
Columns-11,
and
tillation
Eng., 10,199.
Synthesis
Nonequilibrium
Reactive
Gu,
D.,
Distil1994,
R.,
A.
Ciric,
of
and
168
lation Processesby MINLP Optimization, AIChE J., 40,1479.
Colombo, F., Cori, L., Dalloro, L., Delogu, P., 1983, Equilibrium Constant
for the Methyl Tertiary Butyl Ether Liquid Phase Synthesis by Using UNIFAC, Ind. Eng. Chem. Fundam., 22,219.
Chang, Y. A. and Seader, J. D., 1988, Simulation of Continuous Reactive Distillation by a Hornotopy- Continuation Method, Comp. Chem. Eng., 12,
1243.
Chen, GX, and Chuang, K. T., 1995, Liquid-Phase Resistance to Mass
Transfer on Distillation Trays, Ind. Eng. Chem. Res., 34,3078.
Cuthrell, J. E., and Biegler, L. T., 1988, On the Optimization of DifferentialAlgebraic Process Systems, AIChE J., 337 1257.
Daubert, T. E., and Danner, R. P., 1986, DIPPR Data Compilation, AIChE:
New York.
Dribika, M. M., and Sandall, O. C., 1979, Simultaneous Heat and Mass Transfer for Multicomponent Distillation in a Wetted-wall Column, Chem. Eng.
Sci. 34,733.
Fuller, E. N., Ensley, K., and Giddings, J. C., 1969, Diffusion of Halogenated
Hydrocarbons in Helium. The Effect of Structure on Collision Cross Section,
J. Phys. Chem., 73,3679.
Fuller, E. N., Schettler, P.D., and Giddings, J. C., 1966, A New Method for
Prediction of Binary Gas-PhaseDiffusion Coefficients, Ind. Eng. Chem., 58,
19.
Fredenslund, Aa., Gmehling, J., and Rasmussen, P., 1977, Vapour-Liquid
Equilibria Using UNIFAC, Elsevier.
Gani, R. and Cameron, I. T., 1989, Extension of Dynamic Models of DistilComp.
Simulation,
Chem.
Eng.,
State
Steady
13,271.
to
Columns
lation
169
Gani, R., Hytoff, G., Jacksland, C., and Jensen, A. K., 1997, An Integrated
Computer Aided System for Integrated Design of Chemical Processes,Comp.
Chem. Eng., 21,1135.
Gani, R., Ruiz, C. A., and Cameron, I. T., 1986, A Generalized Model for
Distillation Columns-I, Comp. Chem. Eng., 10,181.
Gicquel, A., and Tork, B., 1983, Synthesis of Methyl Tertiary Butyl Ether
Catalized by Ion-Exchange Resin- Influence of Methanol Concentration
and
Temperature, J. of Catalysis, 83,9.
Glansdorf, P., and Prigogine, 1,1971, Thermodynamic Theory of Structure,
Stability and Fluctuations, John Wiley & Sons Ltd., NewYork.
Gokhale, V., Hurowitz, S., and Riggs, J. B., 1995, A Comparison of Advanced
Distillation Control Techniques for a Propylene/ Propane Splitter, Ind. Eng.
Chem. Res., 34,4413.
Gorak, A., 1987, Simulation Methods for Steady State Multicomponent
tillation in Packed Columns, IChemE Symposium Series, A413.
Dis-
Guide,
Systems
User
Process
Enterprise
Ltd.,
London,
U.
K.
1997,
gPROMS
Jacobs, R., and Krishna, R., 1993, Multiple Solutions in Reactive Distillation
for methyl Tertiary-Butyl Ether Synthesis, Ind. Eng. Chem. Res., 32,1706.
Jelinek, J. and Hlavacek, V., 1976, Steady State Countercurrent equilibrium
Stage Separation with Chemical Reaction by Relaxation Method, Chem.
Eng. Commun., 2,79.
Hauan, S., Hertzberg, T., and Lien, K. M., 1995, Why Tertiary Butyl Ether
Production by Reactive Distillation May Yield Multiple Solutions, Ind. Eng.
Chem. Res, 34,987.
Havre, K., and Skogestad,S., 1996, Input/Output Selection and Partial ConCongress,
San
Francisco,
USA,
VOI.
M,
World
181.
IFAC
trol,
170
Hayden, J. C., and O'Connell, J. P., 1975, A Generalised Method for PredictSecond
Coefficients,
Virial
Ind.
Engng.
Chem.
Proc. Des. Dev., 14,209.
ing
Higler, A., Taylor, R. and Krishna, R., 1997, Modeling of a Reactive Separation Process Using a Non-Equilibrium Stage Model, AIChE Annual Meeting.
Higler, A., Taylor, R. and Krishna, R., 1998, Modeling of a Reactive SepaProcess
Using
Non-Equilibrium
Stage
Model,
Comp.
Chem. Eng.,
ration
a
22, S111.
Higler, A., Taylor, R. and Krishna, R., 1999a, Nonequilibrium Modelling of
Multiple Steady States in MTBE Synthesis, Chem.
Reactive Distillation:
Eng. Sci., 54,1389.
Higler, A., Taylor, R. and Krishna, R., 1999b, The Influence of Mass Transfer
Column
for
Mixing
Performance
Tray
Reactive
Distillation,
the
on
of a
and
Chem. Eng. Sci., 54,2873.
Hilborn, R. C., 1994, Chaos and Nonlinear Dynamics (An Introduction to
Scientists adn Engineers), Oxford University Press Inc., USA.
Holland, C.D., 1981, Fundamentals Multicomponent Distillation, Mc Graw
Hill, New York.
Holt, B. R. and Morari M., 1985, Design of Resilient ProcessingPlants. 4.The
Effects of Right Half-Plane Zeroes on Dynamic Resilience, and 5.The Effect
Sci.
Chem.
Eng.
1229.
40,59
Resilience.,
Dynamic
Dead
time
and
on
of
Synthesis
for
Framework
Modular
Generalised
the
A
S.,
1998,
Rahim,
Ismail
UniverThesis,
Separation
Processes,
Reactive
Separation
Nonideal
and
of
1998.
London,
sity of
Equilibria
Vapour-Liquid
K.,
Lee,
W.
Y.,
1992,
Y.
Lee,
W.,
Y.
with
Kang,
and
EthylAcetic-Acid,
Containing
Systems
EquilibriumReaction.
Chemical
Chem.
Japan,
Eng.
25,649.
J.
Ethyl-acetate,
Water
and
alcohol,
Simultaneous
Heat
A.,
D.
O.
C.,
Mellichamp,
1977,
Sandall,
F.,
and
Kayhand,
171
Mass
Transfer
in
Binary
Distillation-II
and
32,747.
Experimental, Chem. Eng. Sci.
Kenig, E., and Gorak, A., 1995, A Film Model-Based Approach for Simulation of Multicomponent Reactive Separation, Chem. Eng. Prog., 34,97.
Kenig, E., Schneider, R., and Gorak, A., 1999, Rigorous dynamic modelling
of complex reactive absorption processes,Chem. Eng. Sci., 54,1347.
King, J. C., 1980, Separation Processes,McGraw-Hill, New York.
Kister, H. Z., 1992, Distillation Design, McGraw-Hill, New York.
Kister, H. Z., 1990, Distillation Operations, McGraw-Hill, New York.
Kooijman, H. A., and Taylor, R., 1995, Nonequilibrium Model for Dynamic
Simulation of Tray Distillation Columns, AIChE J., 41,1852.
Kooijman, H. A., and Taylor, R., 1996, ChemSep Technical Manual, v3.01.
Komatsu, H., 1977, Application of the Relaxation Method for Solving Reactive Distillation Problems, J. Chem. Eng. Japan, 10,200.
Komatsu, H., and Holland, C.D., 1977, A New Method of Convergencefor
Solving Reacting Distillation Problems, J. Chem. Eng. Japan, 10,292.
Kreul, L. U., Gorak, A., Dittrich, C., and Barton, P.I., 1998, Dynamic CatComp.
Validation,
Simulation
Experimental
Advanced
Distillation:
and
alytic
Chem. Eng., 22, S371.
Kreul, L. U., Gorak, A., and Barton, P.I., 1999, Modeling of Homogeneous
Sci.,
Eng.
Chem.
54,19.
Columns,
in
Packed
Processes
Separation
Reactive
Film
Model
InA
Standart,
G.
L.,
Multicomponent
1976,
R.,
Krishna,
and
Maxwell-Stefan
Solution
General
Matrix
Method
the
to
of
a
corporating
J.,
227383.
AIChE
Equations,
172
Krishnarnurthy, R., and Taylor, R., 1982, Calculation of Multicomponent
Mass Transfer at High Transfer Rates, Chem. Eng. J., 25,47.
Krishnamurthy, R., and Taylor, R., 1985, A Non-Equilibrium Stage Model
Multicomponent
Separation
Processes,
Part
I
11,
AIChE
J,
31(3), 449.
of
and
Kubi'c'ek, M., 1976, Algorithm 502. Dependence of Solutions of Non-linear
Systems on Parameter, ACM Trans. Math. Software, 2,98.
V
i
Kub'Cek, M., and Marek, M., 1983, Computational Methods in Bifurcation
Theory and Dissipative Structures, Springer-Verlag, New York.
Luyben, W. L., 1974, Process Modelling, Simulation, and Control for Chemical Engineers, McGraw-Hill, New York.
Mess, A. I., 1981, Dynamics of Feedback Systems, Wiley, New York
EquiChemical
Calculation
in
Concepts
Basic
the
L.,
M.
1995,
Michelsen,
of
Systems,
Separation
Reactive
Conference
Nordic
librium, presented at
on
Technical University of Denmark, Denmark.
DeOptimal
N.,
E.
1996,
Pistikopoulos,
D.,
J.
Mohideen, M. J., Perkins,
and
J.,
AIChE
42,2251.
Uncertainty,
Systems
Dynamics
under
sign of
IEEE
Control,
Integral
Systems
Stability
Morari, M., 1985, Robust
with
of
Transactions on Automatic Control, 30,574.
ReSeparation
Stage
with
Nelson, P.A., 1971, Countercurrent Equilibrium
J.,
17,1043.
AIChE
action,
Coefficients
Transfer
Mass
Y.,
1968,
Okomoto,
Onda, K., Takeuchi, H., and
Japan,
Eng.
Chem.
J.
Columns,
Packed
in
Phases
Liquid
Gas
Between
and
1,56.
Denmark.
DTU,
Report,
CAPEC
Internal
1.,
1999,
Papaeconomou,
173
Pelkonen, S., Gorak A., Kooijman, H., and Taylor, R., 1997, Operation
of a
Packed Distillation Column: Modelling and Experiments, IChemE. Distillation and Absorption 97,269.
Perkins, J. D., and Wong, M. P.F., 1985, AssessingControllability
Chemicalof
Plants, Chem. Eng. Res. Des., 63,358.
Perez-Cisneros, E. S., Schenk, M., and Gani, R., 1997a, A New Approach to
Reactive Distillation Modelling and Design,IChemE Symposium Series, 142,
715.
Perez-Cisneros, E. S., Gani, R., and Michelsen, M. L., 1997b, Reactive SepaSystems.
Part
Computation
1:
Physical
Chemical
Equilibrium,
ration
of
and
Chern. Eng. Sci., 52,527.
Perez-Cisneros, E. S., Schenk, M., Gani, R., and Pilavachi, P.A., 1996, AsSimulation,
Design
Analysis
Reactive
Distillation
Operations,
pects of
and
of
Comp. Chem. Eng., 20, S267.
Petersen, C. K., 1989, PATH User's Guide. Department of Applied MatheStudies
Centre
for
Studies,
Nonlinear
University
Leeds,
U.
K.
matical
and
of
Pilavachi, P.A., Schenk, M., and Gani, R., 1999, Aspects of Chemical and
Physical Equilibrium and Energy Requirements in Distillation, IChemE, submitted.
Pilavachi, P.A., Schenk, M., P e'rez-Cisneros, E. S., and Gani, R., 1997, ModResearch,
Operations,
I&CE
Distillation
Reactive
Simulation
of
elling and
36,3188.
Powers, M. F., Vickery, D. J., Arehole, A., and Taylor, R., 1988, A Nonequilibrium Stage Model of Multicomponent Separation Processes-V.ComputaChem.
Eng.,
Comp.
for
Solving
Equations,
12,
Model
Methods
the
tional
1229.
Gases
B.,
The
Properties
M.,
Poling,
J.
1987,
Prausnitz,
C.,
R.
Reid,
of
and
174
Liquids,
McGraw-Hill, New York-London.
and
Rehfinger, A., and Hoffmann, U., 1990, Kinetics
Methyl
Tertiary Butyl
of
Ether Liquid Phase Synthesis Catalyzed by Ion Exchange Resin-1. Intrinsic
Rate Expression in Liquid Phase Activities, Chem. Eng. Sci., 45,1605.
Rosenbrock,H. H., 1970, State-space and Multivariable Theory, Nelson, London.
Rovaglio, M., and Doherty, M. F., 1990, Dynamics of Heterogeneous Azeotropic
Distillation Columns, AIChE J., 36,39.
Ruiz, C. A., Basualdo, M. S., and Scenna,N. J., Reactive Distillation Dynamic
Simulator, Trans IChemE, 70,363.
Ruiz, C.A., Cameron, I. T., and Gani, R., 1988, A Generalized Dynamic
Model for Distillation Columns-111.Study of Startup Operations, Comp.
Chem. Eng., 12,1.
Saito, S., Michishita, T., and Maeda, S., 1971, Separation of Meta- and
Para- Xylene Mixture by Distillation Accompanied by Chemical Reactions,
J. Chem. Eng. Japan, 4,37.
Sawistowski, H., and Pilavakis, P.A., 1979, Distillation with Chemical Reaction in a Packed Column, IChemE Syrnposium Series, 56.
Sawistowski, H., and Pilavakis, P.A., 1982, Vapor-Li qui d-Equilibrium with
Association in Both Phases - Multicomponent Systems Containing AceticAcid, J. Chem. Eng. Data, 27,64.
Scenna, N. J., Ruiz, C.A., and Benz, S., 1998, Dynamic simulation of startChern.
S719.
Cornp.
Eng.,
distillation
22,
columns,
up procedures of reactive
Schrans, S., de Wolf, S., and Baur, R., 1996, Dynamic Simulation of Reactive
Distillation: An MTBE Case Study, Com. Chem. Eng., 20, S1619.
175
Scheiber, I., and Marek, M., 1995, Chaotic Behaviour
Deterministic
Disof
Systems,
Cambridge University Press, Cambridge.
sipative
Seader, J. D., 1989, The Rate-Based Approach for Modelling Staged Separations, Chem. Eng. Prog., 85,41.
Sennewald, K., Gehrmann, K., and Schafer, S., 1971, Column for Carrying
Out Organic Chemical Reactions in Contact with Fine Particulate Catalysts,
US Patent 3,579,309.
Sivasubrarnanian, M. S., and Boston, J. F., 1990,The Heat and Mass Transfer
Rate-Based Approach for Modelling Multicomponent Separation Processes,
Com. Appl. Chem. Eng., 331.
Skogestad, S., 1997, Dynamics and Control of Distillation Columns- A Critical Survey, Modelling, Identification and Control, 18,177.
Skogestad, S., Lundstrom, P., and Jacobsen, E. W., 1990, Selecting the Best
Distillation Control Configuration, AIChE J., 36,753.
Skogestad, S., and Morari, M., 1987, Control Configuration Selection for
Disti Ration-Columns, AIChE J., 10,1620.
Skogestad, S., and Postlethwaite, 1., 1996, Multivariable Feed-Back Control,
John Wiley & Sons Ltd., England.
Sneesby,M. G., Tade, M. O., Datta, R., and Smith, T. N., 1997, ETBE Synthesis via Reactive Distillation. 2. Dynamic Simulation and Control Aspects,
Ind. Eng. Chem. Res., 36,1870.
Solari, H. G., Natiello, M. A., and Mundlin, G.B., 1996, Nonlinear Dynamics:
from
Physics
Math,
Trip
to
Two-Way
a
Sorensen,E., 1994, Studies on Optimal Operation and Control of Batch Distillation Columns, Phl). Thesis, University of Throndheim, Norway.
Sundmacher, K., 1995,Reaktivdestillation mit katalytischen Fuellkoerperkack176
ungen ein neuer Process zur Herstellung der Kraft stffkomponente MTBE,
PhD Thesis, CUTEC Institute, Clausthal-Zellerfeld, Germany.
Sundmacher, K., and Hoffmann, U., 1996, Development
New
Catalytic
of a
Distillation Process for Fuel Ethers via a Detailed Nonequilibrium Model,
Chem. Eng. Sci., 51,2359.
Suzuki, I., Komatsu, H., and Hirata, M., 1970, Formulation and Prediction
of
Quaternary Vapor-Liquid Equilibria Accompanied by a Chemical Reaction,
J. Chem. Eng. Japan, 3,152.
Suzuki, I., Yagi, H., Komatsu, H., and Hirata, M., 1971, Calculation
Mulof
ticomponent Distillation Accompanied by a Chemical Reaction, J. Chem.
Eng. Japan, 4,26.
Stewart, W. E., and Prober, R., 1964, Matrix Calculation of Multicomponent
Mass Transfer in Isothermal Systems, Ind. Eng. Chem. Fundam., 3,224.
Taylor, R., Kooijman, H. A., and Hung, J. S., 1994, A Second Generation
Nonequilibrium Model for Computer- Simulation of Multicomponent Separation Processes, Comp. Chem. Eng., 18,205.
Taylor, R., Kooijman, H. A., and Woodman, M. R., 1992, Industrial Applications of a Nonequilibrium Model of Distillation and Absorption Operations,
IChemE Symposium Series, A415.
Taylor, R., and Krishna, R., 1993, Multicomponent Mass Transfer, Wiley,
New York.
Taylor, R., and Lucia, A., 1995, Modelling and Analysis of Multicomponent
Separation Processes, AIChE Symposium Series, 91,19.
Thompson, J. M. T., and Stewart, H. B., 1986,Nonlinear Dynamics and Chaos:
Geometrical Methods for Engineers and Scientists, Wiley, New York.
Toor, H. L., 1964, Solution of the Linearized Equations of Multicomponent
Mass Transfer, AIChE J., 10,448.4
177
Ung, S., and Doherty, M. F., 1995a,Vapour-Liquid PhaseEquilibrium in Systems with Multiple Chemical Reactions, Chem. Eng. Sci., 50,23.
Ung, S., and Doherty, M. F., 1995b, Vapour-Liquid Phase Equilibrium in Systems with Multiple Chemical Reactions, Ind. Eng. Chem. Res., 34,2555.
Ung, S., and Doherty, M. F., 1995c, Necessaryand Sufficient Conditions for
Reactive Azeotropes in Multi-Reactions Systems, AIChE J., 41,2383.
Venkatararnan, S., Chan, W. K., and Boston, J. F., 1990,Reactive Distillation
Using ASPEN PLUS, Chem. Eng. Prog., 86(8), 45.
Vogelphol, A., 1979, Murphree Efficiencies in Multicomponent
Symposium Series 2(56), 25.
Wesseling, J. A., 1997, Non-equilibrium
Res. Des., 75,519.
Systems, IChemE
Modelling of Distillation,
Chem. Eng.
Wesseling, J. A., and Krishna, R., 1990, Mass Transfer, Ellis Horwood, ChichEngland.
ester,
Whitman, W. G., 1923, A Preliminary Experimental Confirmation of the
Two-Film Theory of Gas Absorption, Chem. Metall. Eng., 29,146.
in
Coefficients
Diffusion
Correlation
Chang,
P.,
1955,
C.
R.,
Wilke,
of
and
Dilute Solutions, AIChE J., 1 264.
A
ProceW.,
K.
1992,
Mathisen,
M.,
S.,
Hovd,
Skogestad,
Wolff, E. A.,
and
Ulf.
London,
Workshop,
IFAC
Analysis,
Controllability
for
dure
Processes.
Distillation
Catalytic
Study
X.,
Xu,
1992,
Zheng, Y., and
on
QuasiHomogeneous
Processes
Distillation
Catalytic
and
2. Simulation of
A,
Part
465.
E
70,
IChem.
Trans.
Model,
Rate-based
178
Appendix
A
Properties
A. 1
Ent halpy
The component liquid enthalpy is calculated from a polynomial expression
for the heat capacity taken from the DIPPR data bank (Daubert and Dan1986).
ner,
hý =
T
JT.IL
(A,
p
+ B,
pT
2+
+ CpT
DpT
3+E,
4
)dT
pT
Mixture liquid enthalpy is calculated by means of the following equation,
NC
(A. 2)
hL)
xi(hLir -
HL
hL
is
a reference enthalpy.
where r
The vapour enthalpy is calculated through the liquid enthalpy and the heat
(Daubert
heat
bank
data
For
DIPPR
the
the
of vaporisation
of vaporisation.
follows,
The
is
is
1986)
Danner,
expression as
again used.
and
vap
H,
Aal.I'(1 -T
r)
(B, &H+C, &HTr+Da.
'2+EAHTr)
HTr
3
Tr =T
TC
179
(A-3)
(A. 4)
NC
AH
vap
XiHvap
(A. 5)
i
Mixture vapour enthalpy is calculated by,
NC
L-
Hv
A. 2
AH
yi(h i
vap
(A. 6)
Density
Component density of the liquid phase is calculated with the DIPPR data
bank (Daubert and Danner, 1986).
Ap
PLi
(A. 7)
P)
('+('-
CTP),
Bp
Mixture density is calculated through,
PLp
-
ENC
i=l
(A-8)
PLi, pXi, p
Density of the vapour phase is calculated from the equation of state (or ideal
law).
gas
A. 3
Viscosity
Viscosity for liquid and vapour phase are calculated through a suitable cor(1986),
by
Daubert
Danner
relation given
and
Bj,,,
AjjvT
v
P' ':ý (I
+ CA,IT + Div/T2)
Ejj)
C,,,
B,,
In(T)
DAIT
+
+
+
exp(A. 1
IIT
(A. 9)
(A. 10)
Mixture viscosity is calculated through,
NC
YL = exp[E
i=l
In pixi, + 1/2
p
Gij]
xixj
(A. 11)
p
O,
ENC -tyi,
Y, j
(A. 12)
i:Aj
NC
YV =E-
i=l
180
j=l
where
oij
[8(1+
A. 4
(A. 13)
MlMj)]112
Surface Tension
Surface tension for pure components is calculated with DIPPR data bank
(Daubert and Danner, 1986):
(1
A,
*
eri =
Tr)Bý,
+ CGT
(A. 14)
For mixtures an approximate estimate is used:
NC
X, Orr
with r=
to
r=1
-3
A. 5
Thermal
(A. 15)
is usualy used.
Conductivity
data
DIPPR
is
for
Thermal conductivity
calculated with
pure components
bank (Daubert and Danner, 1986):
Bx
Ai =
Mixture thermal conductivity
A,xT
Q\IT + D,\IT2
1
(A. 16)
is calculated through the following expression,
NC NC
E pipjAij
(A. 17)
Aij = 2(Ai + Ai
xivi
Pi .ENC
v
j=l
3
(A. 18)
XL
X: NC
Ay
Y, 2
i=l
NC
j: j=1
yjoi3
for
interaction
is
the
Oij
gas
viscosity.
mixure
parameter
where
181
(A. 19)
(A. 20)
A. 6
Binary
Diffusion
Coefficients
The relationship between the Fick's diffusion coefficient
and the MaxwellStefan diffusion coefficients is given by:
Df = DIP
Df
is
the Fick's coefficient, D is the Maxwell-Stefan coefficient
where
r
and
is a factor accounting for thermodynamic non-idealities. For ideal
systems
both coefficients are identical (IF = 1).
Fick's diffusion coefficient incorporates two aspects: a) significance
inverse
of
drag and b) the thermodynamic non-idealities.
Fick's coefficients are less easy to interpretate physically than the MaxwellStefan coefficients. (Taylor and Krishna, 1993).
o
Vapour phase
Reid ct al. (1987) and Daubert and Danner (1986) recommend the
due
(1966,1969)
Fuller
to
to calculate the
use of a correlation
et al.
diffusion coefficients and this is the one used in this work.
M2)/Mlm2l
+
Df = CT
1.75ý[(Ml
P(, ýýV
+,
12
ýýV)2
(A. 21)
kelvin(K),
T
in
P
M,
C=1.013e-02,
M2in
in
Pa,
with
and
g1mol
and
2/,
V2
diffusion
D will be in (rn S) The terms V, and
the
are
molecular
.
by
in
the
atomic
are
calculated
summing
contribution
volumes and
components.
Liquid
phase
4o
For the liquid phase, Wesselingh and Krishna (1990) suggested the
following model for the limiting diffusivities:
Dij = (D?.Dj'i
.7
182
)1/2
(A. 22)
Do
due
the
the
Wilke
to
are
calculated
coefficients
with
where
method
(1955):
Chang
and
(02M2)
D'12 = 7.4e - 08
1/2
/12VJO.
6
T
(A. 23)
(the
diffusion
Do
is
1
the
coefficient of species
where
solute) present in
21S; M2
infinitely low concentration in species2(the solvent), cm
is the
P2 is the
T
is
K;
temperature,
the
the
solvent, g/mol;
molar mass of
V,
is
the
the
of
molar volume at the normal
viscosity
solvent, cP; and
3/Mol.
boiling point of the solute, cm
183
Appendix
B
Results
B. 1
o
Simulation
Results for MTBE
Changes in Feed Rate
The transient response to a 10% decrease in methanol feed flow rate
boil-up
distillate
held
is
in
Fig.
B.
I.
with vapour
and
constant
shown
Without a decrease in the vapour boil-up input, there was a decrease
in the bottom flow rate which means low MTBE production.
The composition of the bottom shifted to higher MTBE concentration (actually the global isobutene conversion did not considerably
increased
The
in
temperature
the
accordingly as
column
changed).
the reboiler remained in phase equilibrium. After the disturbance the
its
original steady state.
column recovers
Increasing the methanol feed flow rate in 10% makes the system unstable, going to a low conversion solution without recovering from that.
The transient is shown in Fig. B. 2.
The control of the stoichiometric ratio is rather complicated, specially
because the feed composition will not be accurately (it will probably
depends on a process upstream). The unreacted methanol is recovbottoms
MTBE
in
the
the
the
the
and
amount
column,
of
ered with
isobutene
in
increases
the
the
conversion.
reactive zone
of methanol
184
_E,
.2
E
E
0
CD
fi
Z
A:!
E
8
w
10
15
Time (h)
Figure B. I: Decreasing 10% methanol feed flow-rate.
0.9 0.8
s
0
t
0.7 E
E 0.6(
0
1
0.5 0.4 0
CL
I
0.3 LU
In
0.2 0.1 -
Of
%nq
25
Time (h)
Figure B. 2: Increasing 10% methanol feed flow-rate.
185
A compromise must be determined between isobutene conversion (increasesas methanol increases)and ether purity (which falls as methanol
increases). High MTBE purity can be produced with a lower methanol
but
be
excess,
some excessshould
used to suppressside reactions.
Changes
Feed
Composition
in
e
The isobutenes feed composition is fixed by upstream plant operations
between
15 and 55% of isobutene, depending in the
and usually varies
(which
Increasing
process upstream
units and catalyst are employed).
the concentration of isobutene could have some effects on the reactive
distillation columns operations, such as:
in
increase,
the
reaction zone
with a'favourable'
- reactant concentration
in
the reaction equilibrium,
effect
temperature
temsection
gradient
reactive
section
and
reactive
of
increase
the
reaction equilibrium, and
perature
with effect on
duty
be
decreased
to
the
maintain product
specific reboiler
must
specifications or conditions.
Fig. B. 3 shows the transient responseto a 10% of isobutene feed comby
increased
isobutene
The
increased.
the
global
conversion
position
in
MTBE
for
driving
force
the
the
purity
well
as
as
reaction
additional
bottoms. The bottom temperature follows the composition changes.
When the isobutene composition decreasesis interesting to notice that
(15%
A
low
is
the system going to
step change
conversion solution.
decreasing isobutene feed composition) makes the system go to very
low MTBE production and it will not go back to the high conversion
back
is
to
the
the
amount.
normal
composition
when
again,
solution
This effect is shown in Fig. BA.
is
the
is
10%
then
If the step change only
a small change produced and
(Fig.
B.
5).
time
reactive column will recover after some
the
increased
in
the
The temperature
and exceeded
reactive section
limit for the catalyst temperature degradation.
186
1
0.995
-2,0.99
.2
0.985
E
E 0.98
0
t:
W
.8
SO.
.
975
0.97
80.965
w
9
0.96
0.95E
0.95 L
0
5
Time (h)
10
15
Figure B. 3: Increasing 10% isobutene feed composition.
0.9
0.E
T
0.-0
0.E
0.,
-
n
10
5
li
Time (h)
feed
isobutene
Decreasing
15%
BA:
Figure
composition.
187
0.965
0.96
2
ri
20.955
0
i5
E 0.95
E
0.94E
0.94
0.9319
E
8
0.9.
w
c12
0.921
0.91
0.91!
10
15
Time (h)
Figure B. 5: Decreasing 10% isobutene feed composition.
Decreasing the methanol feed flow rate will help to maintain the desired conditions.
Stages
Non-Reactive
Reactive
Effects
and
of
9
Increasing or decreasing the number of reactive stages from the 'opti
between
bad
interaction
the
phase equilibrium and
muml produced a
the chemical reaction which leads to high decomposition of product on
the lower reactive stages.
The rectification zone of a reactive distillation column should remove
from
the
in
(light
this
inert
reactive
the
case)
component, n-butene,
(isobutene)
the
to
secreactive
reactant
unreacted
recycle
section and
tion.
from
MTBE
the
the
the
reactive
The stripping section should remove
the
as
product
to
purify
and
conditions
good
reaction
section maintain
loss
the
the
product.
with
of
reactants
prevent
as
well
188
Even though the separation objectives are clear, increasing in rectification or stripping stages is not necessarily good for the reaction section
conditions.
If the rectification zone is increased too much will result in loss of
isobutene in the distillate, and if the stripping zone is increased too
from
be
the reaction section.
moved away
much methanol will
B. 1.1
Controllability
tion
for the MTBE
Analysis
produc-
Three cases are presented:
Casel
The input variables (u) are given by,
ID1
u=
(B. 1)
vj
The disturbances (d) are considered as follows,
F
bute
ne,
Xiso
TF
(B. 2)
(y)
are,
while the measurementsvariables
[XMTBE,
XMTBE,
D'
(B. 3)
B.
by,
is
(after
G
The transfer function
scaling) given
2.8325 0.1254
0.0388 0.0288
(B. 4)
from
different
is
detG(s)
The
functional
is
controllable.
The reactive column
zero.
189
Pole zero map
Xle
0
3
0
x
oxx
2
x
0
x
cý
x
1
XX
(0
-9 .....................
1137
0( lQ,
........
X
...........
0.. --
AX :
0:
-1
x
c;
x
0
x
-2
x
ox
x
0
-3
0
L
-4
-3
-2
01234
Real Axis
x1o
Figure B. 6: Poles and Zeroes for Case I (MTBE)
Computation
e
of poles and zeroes
All poles are in the left half plane (system stable). Existence of RHPfrom
be
Fig.
B.
As
6.
can
seen
zeroes.
o
RGA
The steady-state RGA matrix is given by,
[ 1.06340
-0-0634
RGA =
1.06340
-0.0634
190
(B. 5)
5
Frequency response of RGA element
1
10
c
................:......................
10-11
10-8
I**II*I..................................................
10 -4
10
10-2
10
0
2
10
10
4
Frequency (rad/h)
25
20
15
10
...............
:
................
................................
................
................
................
I .................................
i...........
....................
5
10-8
10-6
10 -4
10-2
10
0
10
2
10
Frequency (rad/h)
Figure B. 7: RGA element (1, I) as a function of the frequency for Case 1
(MTBE)
The Ij element of the matrix can be plotted as a function of the
frequency, as can be seenin Fig. B. 7. The values of RGA are closeto I
high
high
frequencies
frequency,
takes
very
some smaller
only
at
at any
distillate
is
The
and composition a good choice.
values.
pairing of
Singular
value
o
The singular values are :
0.0223
(B. 6)
2.8356
(B. 7)
The singular values at every frequency are given in Fig. B. 8
191
4
Singular Values
-160
-180
-200
(0
-220
Co
:3
-240
CO
-260
-280
"
-300 1
10
le
10,
le
10
5
Frequency(rad/hr)
Figure B. 8: Singular values for Case I(MTBE)
192
le
10
7
Step Response
From: U(l)
From: U(2)
3
2.5
2
1.5
1
0.5
'a
3
0
X 10-13
0
5
4
0
0.85
1.7
2.55
3.4 0
0.85
1.7
2.55
X 10-3
3.4
x 10,
Time (h.)
Figure B. 9: Step response for Case 1 (MTBE).
(10% step size)
Analysis
The condition number is large (,- 127) but not too large. The RGA values
high
is
it
the
not really
should not
condition number
are small, and since
be a problem to control this column. The pairing of ditillate with the prodbottom
bottom
the
the
the
the
of
with
compostion
at
and
uct composition
logical.
The
best,
is
is
the
only negative effect are
which
physically
column
the RHP-zeroes which implies inverse response,as can be seenfrom Fig. B. 9.
193
Case2
The input variables (u) are given by,
DI
v
(B-8)
The disturbances (d) are considered as follows,
d
Xisobutene,F
(B. 9)
TF
(y)
the
while
measurements variables
are,
[
XMTBE, D
TB
]
The transfer function G is given by,
]
0.1223
0.00410
3.88e - 05 2.88e - 05
(B. 11)
The reactive column is functional controllable. The detG(s) is different from
zero.
Computation
of poles and zeroes
9
All poles are in the left half plane (system stable). Existence of RHPfrom
Fig.
B.
be
10.
As
seen
can
zeroes.
e
RGA
The steady-state RGA matrix is given by,
RGA =
1.0475
-0.0475
-0.0475
1.0475
(B. 12)
function
be
the
the
The 1,1 element of
of
a
as
plotted
matrix can
RGA
tends
to
The
Fig.
B.
be
in
11.
frequency, as can
values of
seen
is
frequencies,
high
that
the
a good
1 at
pairing ul-yl
which confirms
choice.
194
Pole zero map
x 10,
4,1
11111
0
3
0
x
oxx
2
x
X
xP
1
X6X
(0
Co
....................
......
0
x
-1
x
x
x
oxx
-2
x
0
-3
0
L
-4
-3
-2
4
0
Real Axis
(MTBE)
Case
for
2
Zeroes
Poles
B.
Figure 10:
and
195
X 105
Frequency response of RGA element
1
10
10
1
-1
lo
10
10-4
10-6
10-2
10
0
10
2
10
4
Frequency (rad/hr)
30
25
20
cn
...............
L..................
15
0
Co
CL
*................
..................
........................................
......
..................................
..................................
I-
.............................................................................
10
5
n
%1
10 -8
10 -6
10 -4
10 -2
10
z
10
10
Frequency (rad/hr)
Case
for
2
frequency
function
(1,
the
I)
of
Figure B. 11: RGA element
of
as a
MTBE
196
o
Singular value
The singular values are :
0.00387
or =
'57= 0.122
(B. 13)
(B. 14)
Analysis
The condition number is large
31.5), but not too large. The RGA values
A
indicating
interactions
that
the
are small,
negative effect
are not strong.
be
from
RHP-zeroes.
The
Fig.
B.
It
inverse
12.
the
are
response can
seen
be
in
should not
a problem
controlling this column.
Case3
The input variables (u) are given by,
[D]
v
(B. 15)
The disturbances (d) are considered as follows,
x',,
buten, F
TF
(B. 16)
(y)
are,
while the measurements variables
[TD]
TB
(B. 17)
The transfer function G is given by,
0.12230 0.00410
0.00060
-0.0077
(B. 18)
from
different
is
detG(s)
The
functional
is
The reactive column
controllable.
zero.
197
Step Response
1.4
x
10-9
From: U(l)
From: U(2)
1.2
1
0.8
0.6
0.4
a)
CL
0.2
n
X 10-13
6
54
2
0
02
60
26
X 10-3
Xle
Time (hr. )
Figure B-12: Step response for Case 2 (MTBE).
198
(10% step size)
x10
1.5 r-
5
Pole zero map
0
1
0.5
0
OXXx
xx
cb
CO
--0-
0 ..........
..................................
- -cx- x
0x
xgiL:10
- .............................
-
oxx
0x
-0.5
-1
0
-1.5'
-3
11
-2.5
-2
-1.5
-1
-0.5
Real Axis
0
0.5
1
1.5
x lor)
Figure B. 13: Poles and Zeroes for Case 3 (MTBE)
Computation of poles and zeroes
All poles are in the left half plane (system stable). Existence of RHPbe
from
B.
As
Fig.
13.
can
seen
zeroes.
RGA
The steady-state RGA matrix is given by,
RGA
0.71 0.29
0.29 0.71
199
(B. 19)
Frequency response of RGA element
1
10
10
10-11
10-8
I O-b
10 -4
10-2
10
0
10
2
10
4
Frequency (rad/hr)
0
.....................................................
-50
C»
0-100
...............
................
................................
I .................................
cu
m -150
CL
................................
I ..........................................................
I
I .......................
I
-200
-250'lo-
10-6
10-4
10-2
0
10
2
10
4
10
Frequency (rad/hr)
Figure B. 14: RGA element (1, I) as a function of the frequency for Case 3
(MTBE)
The Ij element of the matrix can be plotted as a function of the
frequency, as can be seen in Fig. B. 14. The values of RGA tends to
I at high frequencies, which confirms that the pairing ul-yl is a good
choice.
o
Singular value
The singular values are :
or
0.000316
(B. 20)
'57= 0.32
(B. 21)
Singular values for every frequency are given in Fig. B. 15.
200
Singular Values
-150
-200
-250
cts
co
-300
-350
L-400 1
10
2
10
10
3
10
4
10
5
10
6
Frequency (rad/hr)
Figure B. 15: Singular values for Case 3 (MTBE)
Analysis
The condition number is large (?-.o 1012-93). The RGA values are different
form unity, indicating that the interactions are strong. Another negative effect are the RHP-zeroes. The inverse responsecan be seenfrom Fig. B. 16.
201
Step Response
1.4
From: U(2)
From: U(l)
x 10-9
1.2
0.8
0.6
0.40.2
0
E
10'
0
-1
-2
CSJ3
>.
o -4
I-5
-6
-7
-8
0
2
60
4
26
X 10-3
x 10,
Time (hr.)
Figure B. 16: Step response for Case 3 (MTBE).
202
(10% step size)
B-2
Acetic
Acid
Ethanol
of
with
Esterification
For the case of the esterification we have assumed the 0.00387D-V configuration.
Casel
The input variables (u) are given by,
[D]
(B. 22)
v
The disturbances (d) are considered as follows,
d
Xet han ol, F
(B. 23)
TF
(y)
the
are,
measurements variables
while
[TD]
(B. 24)
TB
The transfer function G is given by,
3.282
-7.387
-0.0015
4.53e - 04
1
(B. 25)
from
different
is
detG(s)
The
functional
The reactive column is
controllable.
zero.
Computation
*
of poles and zeroes
RHPExistence
(system
half
left
in
All poles are the
of
stable).
plane
B.
from
Fig.
17.
be
As
seen
can
zeroes.
o
RGA
by,
is
RGA
The steady-state
given
matrix
[ 1.1726
-0.1726
RGA =
1.1726
-0.17260
203
(B. 26)
Pole zero map
80
0
60
x
x
40
xx0
0
x
20
00
x
x
0
U)
CD
E
cu
.. 0
- -OX.
OX ---...
0xXxx<)-Ox.
x- ä--
0
0)0
- 0.
x.
0x0
... X-OK-><)...
0 >o
XXx
0
.-
8-x
09--)ICD...
X- (lK- 00( X X- - 19 4330
00x0
x
0
-20
0x
0
xx
0
x
-40
x
-60
on
_vv
-180
0
-160
-140
-120
-100
-80
-60
-40
-20
Real Axis
Figure B. 17: Poles and Zeroesfor Case 1 (Esterification)
204
0
20
Frequency response of RGA element
1
10
......................................................................................................
i
ol
10
10 -6
10
10 -4
10-2
10
0
10
2
10
4
Frequency (rad/h)
0
-2
'a
-8
-101-8
10
10 -6
10 -4
10-2
10
0
10
2
10
Frequency (rad/h)
Figure B. 18: RGA element (1,1) as a function of the frequency for Case 1
(Esterification)
The 1,1 element of the matrix can be plotted as a function of the
frequency, as can be seen in Fig. B. 18. The values of RGA tends to
1 at high frequencies, which confirms that the pairing ul-yl is a good
choice.
decomposition
Singular
value
*
The matrix G is decomposedinto it singular value giving,
G=UE
VH
(B. 27)
for our case,
U
0.4064
-0.9137
205
0.9137
0.4064
(B. 28)
4
1
8.08
0
0
1.06e 03
vH
0.9900
05
-0.417e 05
-0-417e -0.9900
(B. 29)
(B. 30)
Where the matrix E gives the singular values in the
main diagonal.
Analysis
Since there are elements in G(s) smaller than I in
magnitude this suggests
that may be control problems with input constraints.
The 1,1 element of the gain matrix G is 3.282. Thus,
increase
in ul (distilan
late) yields a large steady-state change in
(temperature
distillate),
that
yj
of
is the outputs are very sensitive to changes in ul. In
increase
in
contrary an
(vapour
boil-up)
U2
gives Y2 = -0-0015, a small change and in the opposite
direction of that for the increase in ul (the opposite analysis to this is for
U2)-
If both ul andU2 (distillate and vapour-boil-up) are increasedsimultaneously,
the overall steady-state change in yj is (3.282 - 0.0015) = 3.2805, which indicates that the temperature is strongly dependent on internal changes of
flows. This can also be seenfrom the smallest singular value: 0.00106.
From the input singular vector,
0.9137
0.4064
(B-31)
is
in
direction,
that
the
the
to
the
we see
effect
move
outputs
same
which
lead
keep
both
to
to
temperature at the
will
a relative short control action
desired values simultaneously.
due
7620),
The condition number is rather large (e-%.
to
the
very
small
miniO
Although
high
RGA
the
the
values are relatively small,
mum singular value.
RHPimplies
the
the
condition number
control problems as well as
value of
(RHP-zeros
is
to
the
to
the
also
very
close
close
origin are
origin
which
zero,
the most difficult to overcome). The RHP-zero also implies inverse response,
from
Fig.
B.
be
19.
seen
as can
206
Step Response
From: U(2)
From: U(l)
x
>-
0
10,1-11
0
0
x
10-8
-2
-4
A
In
0
0.5
1
20
1.5
0.5
1
1.5
Time (h)
Figure
B. 19: Step
response
for Case 1 (Esterification).
207
(10%
step size)
2
Case2
The input variables (u) are given by,
[D]
(B-32)
v
The disturbances (d) are considered as follows,
dxth
(B. 33)
ano IIF
TF
while the measurements variables (y) are,
[
X4D
(B. 34)
XIBI
For this case the transfer function G is given by,
0.2440
-0.042
Computation
e
03
-1.152e 1.033e - 05
(B. 35)
of poles and zeros
All poles are in the left half plane (system stable). Existence of RHPPoles
zeros.
and zeroes map is shown in Fig. B. 20
e
RGA
The RGA matrix is given by,
RGA =
[ 1.055
-0.055
-0.055
1.055
(B. 36)
The 1,1 element of the matrix can be plotted as a function of the
frequency, as can be seen in Fig. B. 18. The values of RGA tends to
1 at high frequencies, which confirms that the pairing ul-yl is a good
choice.
208
Pole zero map
60
0
x
40
0
x
XX
00
20
x
00
Co
xx
0
0. .............................................
QX x
mx.
0 000xx
0
(aw.
0
xc)
x
-20
00
XX
x
0
-40
x
-P.
A
-450
-400
-350
-300
-250
-200
Real Axis
-150
-100
-50
Figure B. 20: Poles and zeroesfor Case 2 (Esterification)
209
0
50
Frequency response of RGA element
1
10
...............................
........
I. .................
...............................
..................................
.................
.................
...
..........
.................
.....................
....................................................................
......................
................
..................................
..................................
.................
...................................
..................................................
................
.................................
................
.................................................
.................
10
...............................................
...................
..................................
....................................................
...................................................
...................
.................................
.....................
.........................................
..............................................
..................................
...............
..................................................................................................
...................................................................................
10 -11
10
10-6
...............
10 -4
0
10
10-2
2
10
4
10
Frequency (rad/hr)
.................................
cm
0
10 -1
............................
.....................
--8
10 -6
10 -4
..............
-
...........................
....................
.............
10
fN............
*..........
10-2
...................
10
024
10
10
Frequency (rad/hr)
Figure B. 21: RGA element (1,1) as a function of the frequency for Case 2
(Esterification)
210
Singular
decomposition
*
value
The matrix G is decomposedinto it
singular value giving,
VH
(B-37)
for our case,
0.9855
-0-1693
VH =
-0.1693
-0.9855
(B. 38)
0.244
0
0
1.875e- 04
(B 39)
.
0.47le
03
-0.999
03
0.9900
-0.47le -
(B. 40)
Where the matrix E gives the singular values in the main diagonal.
Singular values aa function of the frequency are given in Fig. B. 22.
Analysis
Since there are elements in G(s) smaller than 1 in magnitude this suggests
that may be control problems with input constraints.
The 1,1 element of the gain matrix G is 0.2440. Thus, an increase in ul
(distillate) yields a small steady-state change in yj (composition of distillate),
that is the outputs are not sensitive to changes in ul. Similarly an increase in
(vapour
in
boil-up)
the
a
small
change
and
gives
opposite
Y2
U2
= -0.0012,
direction of that for the increase in ul (the same analysis to this keeps for
U2)-
If both ul andU2 (distillate and vapour-boil-up) are increasedsimultaneously,
(0.244
0.2428,
is
0.0012)
in
the overall steady-state change
which
yj
=
indicates that compositions are weakly dependent on internal changes of
flows. From the input singular vector,
0.9855
(B. 41)
-0.1693
direction.
This
in
is
the
the
to
the
that
opposite
effect
outputs
move
we see
desired
both
large
keep
the
to
lead
to
at
composition
control
a
action
will
211
SingularValues
-160
-180
-200
c12
-220
Co
Z
C»
r_
CO
-240
-260
-280
-300'
10
10-1
lo,
10
1
Frequency(rad/hr)
Figure B. 22: Singular values for Case 2 (Esterification)
212
10
3
10
4
Step Response
;3
10-9
x
From: U(l)
Fran: U(2)
............
2.5
2
1.5
1
0.5
V
0
4-.
CL
10-10
x
-5
-10
-15
-qn 0
....................
2.5
5
7.5
100
2.5
5
7.5
10
Time (hr)
Figure B-23: Step responses for Case 2 (Esterification).
(10% step size)
values simultaneously.
The condition number is rather large (ý,- 1300), due to the very small minihigh
RGA
Although
the
the
values are relatively small,
mum singular value.
RHPimplies
the
the
control problems as well as
condition number
value of
inverse
be
The
is
to
the
response can
seen
origin.
also very close
zero, which
from Fig. B. 23.
Case3
The input variables (u) are given by,
[D]
U=
V
213
(B. 42)
The disturbances (d)
follows,
are considered as
XlF
(B. 43)
X4F.
while the measurements variables (y) are,
X4D]
(B. 44)
XlB
For this case the transfer function G is given by,
1.4e - 02 ]
05
-2.58c -
0.1091
-0-0738
Computation
9
(B. 45)
of poles and zeros
All poles and zeros are in the left half plane (system
Fig.
B.
24.
stable).
o
RGA
The RGA matrix is given by,
RGA =
[
1.0274
-0.0274
-0.0274
1.0274
(B. 46)
RGa as a function of the frequency is given in fig. B. 25.
Singular
4o
value
The singular values as a function of the frequency are given in Fig. B. 26.
Analysis
The condition number is not too large (r--t 165), and the RGA values are
disappear
if
in
diagonal
The
RHP-zero
to
the
the
unity
main
very close
disturbance on the feed temperature is not present. Step changesare shown
in Fig. B. 27.
214
Pole zero map
80
0
60
0
x
40
x
xx
00
20
x
x00
0x
ci)
00
09
0
0
@DOK-X- CK 4e
0
(D ............................................
a)
0e9
E
0,0
0 ..........
0
x
00
x0
x
-20
CýK
00
xx
x
-40
x
0
-60
an
-%. TV
-500
-400
-300
-200
Real Axis
-100
0
Figure B. 24: Poles and zeros for Case 3 (Esterification)
215
100
Frequency response of RGA element
1
10
10
10-11
10-8
10-6
10 -4
10-2
10
0
10
2
10
4
Frequency (rad/hr)
1
0
...
.......................
...............
................................................ .........
\
.............
..................................
............
.................................
«a -2 ................
*..........
0-3
m
CL
................
-4
-5
-%j
-Q
10 -8
...............
.............................
10 -6
*, ****,,
......
..............................
10 -4
-
*, -xf............................
****,
10 -2
10
.............................
10
id
le
Frequency (rad/hr)
for
Case
3
frequency
function
(1,
the
I)
Figure B. 25: RGA element
of
as a
(Esterification)
216
Singular Values
-160
-180
-200
(D
-220
0
CO
-240
U)
-260
-280
'
-300
10-1
id,
101
lcý
10
Frequency(rad/hr)
Figure B. 26: Singular values for Case 3 (Esterification)
217
le
Step Response
6
From: U(l)
x
From: U(2)
a)
4. -
CL
0
x 10-10
-2-3-4
-5
--
................
---
1111
-6 0
0.5
20
1.5
0.5
1.5
Time (hr)
(Esterification).
Case
for
3
Step
Figure B. 27:
responses
218
(10% step size)
2
Appendix
C
Optimization
CA
Objective
Function
Disturbances
and
Objective Function
The objective function is composed by capital and operating costs. The andetermined
by
its
internal,
investment
is
the
the
cost of
column,
cost
nualised
and reboiler and condenser.
minimiseCost
+
= mznzmzse-(Cshell
Ctray
+
Chx
+
(c.
Cop)
1)
Ctray7 Ch., and C,, are given by,
p
whereCshell7
Cshell
M&S
ý1
3(
EA+1:
)
F,
(101.9Dt)(2.18 +
T80 )p
2Apf)
0.802(Zp
-
Zp+, OC. 2)
P/<p
where,
A is a constant equal to 101.9;
800;
Swift
to
Marshall
index
is
the
M&S
equal
and
B is a constant equal to 2.18;
bottom
to
the
top
the
the
equal
column
of
H,, is the sum of
and
spacing at
irn.;
219
Fc is the material of construction factor equal to 1.05;
A is the column diameter (variable); and,
Apf is a dummy variable which is a function of tray spacing and is
set to 1
for a tray spacing of 2 feet.
M&S
Ctray
4.7D'
280 )tp
.
55
Flc E 2Zp
(C. 3)
where,
M&S is the Marshall and Swift index equal to 800;
F, I is the material of construction factor equal to 1.
The column diameter, Dt, its easily calculated from the internal vapour flowbe
for
factors
the
a given
materials of construction
will
constant
rate, and
system.
Chx
M&SI
=1 3[
280
101 (2.29 + F,,
.3
65
) (Ao .
cr
65)
.
+ Ao
(C. 4)
where,
M&S is the Marshall and Swift index equal to 800;
F,, is the material of construction factor for heat-exchangerequal to 1.30;
A, and A, are the condenserand reboiler areas(variables),respectively.
Cop QcCwater
+ QrCsteam
hence,
directly,
is
The cost of steam and cooling water
computed
andCsteam
kgmol);
dollars
(2.8573E-4
per
the cost of cooling water
kg).
dollars
(3.63763E-3
per
cost of steam
(C-5)
Cwater
is
is the
Disturbances
Sinusoidal in feed composition:
(wt)
0.15
sin
zAA ý zAA +
ZEt
ZEt+
0.15sin(wt)
220
(tOti1fl
IJNN
(C.6)
(C. 7)