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Diss. ETHNo. 10666 20. JUll KWH
Ma,
Robust Control of an
Industrial High-Purity
Distillation Column
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
for the degree of
Doctor of Technical Sciences
presented by
HANS-EUGEN MUSCH
Dipl. Chem.-Ing. ETH
born June 19,1965
citizen of Germany
accepted on the recommendation of
Prof. M. Steiner, examiner
Prof. Dr. D. W. T. Rippin, co-examiner
1994
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3
To my grandparents
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Acknowledgments
This Ph. D. thesis was written during my years as a research and educa
tional assistant of the Measurement and Control Laboratory at the
Swiss Federal Institute of Technology (ETH) at Zurich. I would like to
take this opportunity to thank the numerous persons who have
supported this project.
First of all I express my gratitude to Prof. M. Steiner. He arranged this
project and helped to overcome many difficulties with the industrial
environment. Many thanks are also due to him and to Prof. D. W. T.
Rippin for the critical examination of this thesis, which essentially
improved its clarity.
The numerous discussions with my colleagues and their uncountable
suggestions gaverise to
importantcontributions to this work. In this
context, E. Baumann, U. Christen, and S. Menzi must be specially
mentioned.
Last but not least I should emphasize the support of B. Rohrbach. She
never lost her patience with my never ending questions concerning the
English language. Without her willingness to correct the manuscript,the choice of the English language for this thesis would have been
impossible.
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Content
Symbols 13
Abstract 15
Kurzfassung 17
1 Introduction 19
1.1 "Modern Control: Why Don't We Use It?" 19
1.2 Scope and significance of this thesis 21
1.2.1 Distillation as a unit operation example 21
1.2.2 Earlier research 21
1.2.3 Robust control and nonlinear plants 22
1.2.4 Contributions of this thesis 22
1.3 Structure of the dissertation 23
1.4 References 26
2 The Distillation Process
An Industrial Example 29
2.1 Introduction 29
2.2 Column design and operation 29
2.3 Steady-state behavior 32
2.4 Composition dynamics 35
2.5 Control objectives and configurations 37
2.5.1 The 5x5 control problem 39
2.5.2 Control design steps 40
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2.6 Tray temperatures as controlled outputs 41
2.6.1 Pressure-compensated temperatures 42
2.6.2 Temperature measurement placement 44
2.7 References 45
3 ARigorous Dynamic Model of
Distillation Columns 47
3.1 Introduction 47
3.2 Conventions 48
3.3 The objective of modelling 48
3.4 Simplifying assumptions 48
3.5 Balance equations 51
3.5.1 Material balances 51
3.5.2 Energy balance equations 52
3.6 Fluid dynamics 55
3.6.1 Liquid flow rates 55
3.6.2 Pressure drop 57
3.7 Phase equilibrium 59
3.7.1 Vapor phase composition 59
3.7.2 Boiling points 60
3.8 Volumetric properties60
3.8.1 PVT relations 61
3.8.2 Density 61
3.9 Enthalpies 62
3.10 Numerical solution 63
3.10.1 The dependent variables and the equation system... 63
3.10.2 Formal representation of the DAE 66
3.10.3 The index 66
3.10.4 Solution methods and software 67
3.11 Notation 71
3.12 References 74
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4 Linear Models 77
4.1 Introduction 77
4.2 How to linearize the rigorous model? 78
4.2.1 The state, input, and output vectors 78
4.2.2 Handling of the algebraic equation system 80
4.3 Linearization of a simplified nonlinear model 80
4.3.1 The simplified model 80
4.3.2 Analytical linearization 84
4.4 Linearization of the rigorous model 86
4.4.1 Model modifications 86
4.4.2 Numerical linearization 88
4.5 Comparison of the linear models 89
4.5.1 Open loop simulations 89
4.5.2
Singularvalues 92
4.6 Order reduction 94
4.7 Summary 96
4.8 Appendix: Model coefficients 97
4.9 Notation 101
4.9.1 Matrices and Vectors 101
4.9.2 Scalar values 102
4.10 References 103
5 AStructured Uncertainty Model 105
5.1 Introduction 105
5.2 Limits of uncertainty models 106
5.3 Input uncertainty 107
5.4 Model uncertainty 110
5.4.1 Column nonlinearity 110
5.4.2 Unmodelled dynamics 117
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5.5 Measurement uncertainty 118
5.6 Specification of the controller performance 119
5.7 Summary 120
5.8 References 122
6 |0,-Optimal Controller Design 123
6.1 Introduction 123
6.2 The structured singular value 1246.2.1 Representation of structured uncertainties 124
6.2.2 Definition of the structured singular value 126
6.2.3 Robustness of stability and performance 128
6.3 The design model 130
6.4 Controller design with u-synthesis 133
6.4.1
Synthesis algorithms 1346.4.2 Applying the DK-Iteration 137
6.4.3 Applying the uK-Iteration 137
6.5 Design of controllers with fixed structure 148
6.5.1 Diagonal PI(D) control structures 149
6.5.2 PI(D) control structures with two-way decoupling ... 156
6.5.3 PID control structures withone-way decoupling
161
6.6 Summary 164
6.7 References 166
7 Controller Design for
Unstructured Uncertainty
AComparison 169
7.1 Introduction 169
7.2 Diagonal Pl-control 170
7.2.1 The BLT method 170
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7.2.2 Sequential loop closing 172
7.2.3 Optimized robust diagonal Pi-control 174
7.3 Pi-control with decoupling 177
7.4 H optimal design 182
7.5 Summary 187
7.6 References 187
8 Feedforward Controller Design 189
8.1 Introduction 189
8.2 The design problem 190
8.2.1 The design objective 190
8.2.2 One-step or two-step design? 190
8.3 Hro-minimization 192
8.4 Optimization approach 1968.5 Summary 199
8.6 References 200
9 Practical Experiences 203
9.1 Introduction 203
9.2 Controller implementation 204
9.3 Composition estimators 207
9.4 Controller performance 208
9.5 Economic aspects 214
9.6 Summary 214
10 Conclusions and
Recommendations 217
10.1 Introduction 217
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10.2 Controller synthesis 218
10.3 State-space or PID control? 219
10.4 How many temperature measurements? 220
10.5 Column models 221
10.6 Recommendations 221
10.6.1 Academic research 221
10.6.2 Decentralized control systems 222
10.6.3 Cooperation industryuniversity 223
Curriculum vitae 225
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Symbols
8 Uncertainty scalar value
A Uncertainty matrix or deviation from nominal operating point
8 Parameter vector
k Condition number, k = ov /o_.ind.x nun
X Eigenvalue
(j, Structured singular value
p Spectral radius
a Singular value
B Bottom product stream (mol/s)
D Distillate stream (mol/s) or diagonal scaling matrix
d Disturbance signals
e Control error
F Feed flow rate (mol/s)
7t Lower fractional transformation
G(s) Transfer function
Gu Transfer function from control signals u to output signals y
I Identity matrix
K(s) Controller
L0 Reflux (mol/s)
M Joint weighted plant and controller, M (P, K) = ^(P, K)
P Weighted plant
p Pressure (N/m2)
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r Reference signals
Se(s) Sensitivity function at e, Se (s) = [I + G (s) K(s) ]-1
Su(s) Sensitivity function at u, Su(s) = [I + K(s)G(s)]_1
T Temperature (C)
Tr Transfer function from reference signals to output signals
u Control signals
V51 Boilup (mol/s)
W(s) Diagonal matrix of weighting transfer functions
w(s) Weighting transfer function
xrj Top product composition (mol/mol)
xg Bottom product composition (mol/mol)
xF Feed composition (mol/mol)
y Output signals
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15
Abstract
It is well known that high-purity distillation columns are difficult to
control due to their ill-conditioned and strongly nonlinear behavior.
Usually distillation columns are operated within a wide range of feed
compositions and flow rates, which makes a control design even more
difficult. Nevertheless, a tight control of both product compositions is
necessary to guarantee the smallest possible energy consumption, as
well as high and uniform product qualities.
This thesis discusses a new approach for the dual composition control
design, which takes the entire operating range of a distillation column
into account. With the example of an industrial binary distillation
column, a structured uncertainty model is developed which describes
quitewell the nonlinear column
dynamicswith several simultaneous
model uncertainties. This uncertainty model forms the basis for feed
back controller designs by |x-synthesis or u-optimization. The resultingcontrollers are distinguished by a high controller performance and high
robustness guaranteed for the entire operating range. This method
enables the synthesis of state-space controllers as well as the u-optimal
tuning of advanced PID control structures.
The already satisfactory compensation of feed flow disturbances can be
improved even further by use of feedforward control. Even for the design
of the feedforward controllers the basic ideas of the feedback controller
design can be employed. A simultaneous feedforward controller design
for two column models representing the extreme column loads yields
outstanding results. Similar to the feedback controller design, a design
of state-space controllers by Hm -minimization or an optimal tuning of
simple feedforward control structures by parameter optimization is
possible.
Control engineers working in an industrial environment are conscious
of the high effort needed for the implementation of state-space control-
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16
lers in a distributed control system. Therefore a controller design based
on PID or advanced PID control structures is of high relevance for the
industrial
practice. Usually,the
performanceof these PID control struc
tures is expected to lag significantly behind the performance of high-
order state-space controllers. However, comparing the performances of
the state-space controllers with those of the advanced PID controllers,
merely slight advantages of the state-space controllers are detected.
This surprising result, achieved with an unconventional tuning of the
PID control structures, allows the simple implementation of advanced
PID control structures in a decentralized control system without a
significant loss of controller performance.
The good robustness properties and the high performance of the control
schemes are confirmed by the implementation of an advanced PID
control scheme on a real industrial distillation column. An estimation of
the economic benefits made by this project much more than justifies the
effort expended.
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17
Kurzfassung
Bekanntermafien sind Rektifikationskolonnen mit hohen Produktrein-
heiten wegen ihres schlecht konditionierten und stark nichtlinearen
Verhaltens schwierig zu regeln. Haufig werden sie in einem weiten
Bereich unterschiedlicher Zulaufkonzentrationen und -mengen
betrieben, was den Entwurf von Regelungen zusatzlich erschwert.
Dennoch ist eine gute Regelung beider Produktkonzentrationen
notwendig, um einerseits einen moglichst kleinen Energieverbrauch
und andererseits hohe und einheitliche Produktqualitaten sicher-
zustellen.
Diese Arbeit beschreibt einen neuen Ansatz fur den Entwurf von
Konzentrationsregelungen, der den gesamten Arbeitsbereich einer
Rektifikationskolonne berucksichtigt. Am Beispiel einer industriellen
binaren Rektifikationskolonne wird ein strukturiertes Unsicherheits-
modell entwickelt, welches das nichtlineare dynamische Verhalten der
Rektifikationskolonne durch mehrere Modell-Unsicherheiten gut
beschreibt. Dieses Unsicherheitsmodell bildet die Basis fur den
Entwurfvon Reglern mittels u-Synthese oder u-Optimierung. Die resul-
tierenden Regler zeichnen sich durch eine - iiber den gesamten
Betriebsbereich garantierte - hohe Regelqualitat bei sehr grosserRobustheit aus. Dieses Vorgehen erlaubt sowohl den Entwurf von
Zustandsregelungen als auch die Berechnung u-optimaler Einstel-
lungen fur erweiterte PID-Regelstrukturen.
Die bereits zufriedenstellende Unterdriickung von Storungen der
Zulaufmenge wird durch den Einsatz einer Storgrofienaufschaltungnoch verbessert. Auch fur ihren Entwurf kdnnen ahnliche Konzepte
verwendet werden. Ein Entwurf von Storgrossenaufschaltungen, beidem gleichzeitig zwei Modelle der Rektifikationskolonne berucksichtigt
werden, welche die extremen Kolonnenbelastungen wiedergeben, fuhrt
zu hervorragenden Ergebnissen. Vergleichbar mit dem Regelungs-
entwurf konnen sowohl Storgrossenaufschaltungen mit der Struktur
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18
von Zustandsregelungen (durch Minimierung der H^-Norm) als auch
Storgroflenaufschaltungen mit einfacher Struktur (durch Parameter-
optimierungim Zeitbereich) berechnet werden.
In der industriellen Praxis tatige Regelungstechniker sind sich der
Schwierigkeiten, die mit der Realisierung von Zustandsregelungen auf
dezentralen ProzelJleitsystemen verbunden sind, sicherlich bewufit.
Daher ist der Regelungsentwurf auf der Grundlage von PID- oder erwei-
terten PID-Regelstrukturen von hoher praktischer Relevanz. Meist
bleibt die mit solchen Strukturen erzielbare Regelgiite hinter der von
Zustandsregelungen deutlich zuriick. In dieser Arbeit werden dieentworfenen Zustandsregelungen und die optimal eingestellten fortge-
schrittenen PID-Regelstrukturen verglichen. Dabei zeigt sich, dafi auch
mit einfachen Regelstrukturen, die entsprechenden unkonventionellen
Regler-Einstellungen vorausgesetzt, eine Regelqualitat erzielt wird, die
der von Zustandsregelungen nahekommt. Dieses iiberraschende
Resultat erlaubt die einfache Implementierung von erweiterten PID-
Regelstrukturen in dezentralen ProzelJleitsystemen ohne wesentlichenVerlust an Regelgiite.
Die Erprobung eines Regelungsentwurfs auf der Grundlage fort-
geschrittener PID-Strukturen an der industriellen Rektifikations
kolonne bestatigt die grofie Robustheit und die hohe Regelgiite in der
Praxis. Dabei zeigt eine Abschatzung der Wirtschaftlichkeit, dafi der bei
einem solchen Projekt notwendige Aufwand mehr als gerechtfertigt ist.
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1.1 "Modern Control: Why Don't We Use It?" 19
Chapter 1
Introduction
1.1 "Modern Control: Why Don't We Use It?"
"Modern Control: Why Don't We Use It?" is the title of a paper written
by R. K. Pearson in 1984 [1.4]. In the first section of that paper Pearson
states: "Advanced control systems utilizing multivariable strategies
based on process models can outperform traditional designs in broad
classes of application. Yet, in spite of market forces demanding better
process performance and ample evidence showing that the improve
ments can be achieved, the gap between theory and practice in the
industrial sector is not narrowing appreciably."
Ten years later the situation has not changed. The modern control theo
ries provide the process control engineer with increasingly sophisticated
tools for a robust, model-based controller design. The advantages of
these controllers over the PID control structures which are usually
tuned on-line, have been shown in numerous publications. Neverthe
less,more
than90% of all
control loopsin
the process industryuse PID
control, while only a few applications of the modern control theories can
be reported [1.10]. Therefore the mismatch between theory and practice
is still evident. Some of the reasons for this situation are discussed
below.
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20 1 Introduction
Distributed Control Systems
For a control engineer in the process industry, process control in the first
place is a hardware problem. His perspective is the installation and
configuration of a Distributed Control System (DCS) [1.1]. Even the
modern DCS are often limited to PID and advanced PID control. For the
DCS, an implementation of modern state space controllers requires
either the coupling with an external computer or the programming of
software modules. Both ways are troublesome and expensive. The
university research pays little attention to this situation. The design of
robust controllers with fixed structures (e.g., PID control structures) is
a largely unexplored field.
Dynamic Models
Linear dynamic models are the foundation of a modern, robust
controller design. However, no general dynamic models are available for
unit operations. For each plant linear dynamic models must be developed, based on either linearization of nonlinear models or on system
identification methods. Both ways are often expensive and very time-
consuming ([1.5], [1.6]). Furthermore, most plants in the process
industry show a strongly nonlinear dynamic behavior, which is unsatis
factorily described by a single linear model.
Economic benefits
The economic benefits of improved control tend to be significantly
underestimated. Abenchmark study by ICI "indicated that the effective
use of improved process control technology could add more than one
third to the worldwide ICI Group's profits" [1.1]. Another study shows
smaller, but still massive benefits [1.2].
Of course it is not necessary to replace all PID-controllers by modernadvanced control structures. Most control problems in the process
industry are handled well with simple PID control. However, strongly
nonlinear or/and ill-conditioned plants require advanced control tech
niques for a high controller performance.
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1.2 Scope and significance of this thesis 21
1.2 Scope and significance of this thesis
1.2.1 Distillation as a unit operation example
Distillation is one of the most widely used unit operations in the process
industry. In the simplest case, a distillation column separates a feed of
two components into a top product stream (with a high fraction of the
low-boiling component) and a bottom product stream (with a high frac
tion of the high-boiling component). In an industrial setting, the feed
flow rate and the feed composition may vary within a wide range of oper
ating conditions.
This separation consumes a huge amount of energy. A minimization of
the energy consumption and an economic optimal operation usually
require (1) a tight control of both product compositions (dual composi
tion control) and (2) often small fractions of impurities in the product
streams (high purity distillation). However, the strongly nonlinear and
ill-conditioned behavior makes high-purity distillation columns difficultto control. Therefore high-purity distillation columns have become an
interesting test case for robust control design methods.
1.2.2 Earlier research
Without any doubt the distillation process is the most studied unit oper
ation in terms of control. Skogestad estimates that new papers in this
field appear at a rate of at least 50 each year [1.7]. It is practically
impossible to give a review of all these publications. The interested
reader is advised to consult the reviews of Tolliver and Waggoner [1.8],
Waller [1.9], MacAvoy and Wang [1.3], and the recent review of
Skogestad [1.7].
If we focus our interest on the design of linear, time-invariant control
lers, we must state that all the well-known model-based and robustcontrol design methods (LQG/LTR, H^, Normalized Coprime Factoriza
tion, u-synthesis, etc.) have been applied to distillation columns.
However, all these publications discuss the controller design for just one
operating point. The problem designing a robust controller which maxi-
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22 1 Introduction
mizes the controller performance for the entire operating range has not
been addressed as yet.
1.2.3 Robust control and nonlinear plants
The well-known robust control design methods like HM -minimization or
LQG/LTR are based on the assumption of an unstructured, frequency
dependent uncertainty at one location in the plant. Such an unstruc
tured uncertainty may be a multiplicative uncertainty at plant input or
output,or an additive
uncertainty.
A controller design for the entire operating range of a distillation
column using one of these well-known methods has two inherent prob
lems:
Due to the high nonlinearities an estimation of unstructured
uncertainty bounds will lead to very large bounds, prohibiting
any acceptable controller design.
A controller design using any arbitrary, smaller uncertaintybound guarantees robust performance (RP) and robust stability
(RS) for the actual operating point, but not for the entire oper
ating range.
1.2.4 Contributions of this thesis
This thesis presents a new approach for the composition control design
of a binary distillation column (Figure 1.1). The design concept is based
on a structured uncertainty model which describes the column dynamics
for the entire operating range quite well. The resulting controller
designs using u-synthesis (for state-space controller) or u-optimization
(for controllers with fixed structure), respectively, lead to results which
guarantee robust performance and robust stability for the entire operating range of the distillation column. Special emphasis is placed on the
optimal tuning of easy-to-realize PID-control structures. It will be
shown that extraordinary controller performance can be achieved even
with these relatively simple controller structures.
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1.3 Structure of the dissertation 23
Standard approaches
Linear model for a
single operating point
Robust control design
IL LQG/LTR,
Weak point:
Improved approach
Uncertainty model
describing column dynamics
for entire operating range
(i-synthesis
(X-optimization
Advantage:
RS & RP guaranteed
for whole operating range
Figure 1.1: Robust control design approaches
1.3 Structure of the dissertation
A robust, model-based controller design for a distillation column
consists of several steps. A typical course is illustrated in Figure 1.2.
The results and methods of each step influence all the following steps.
The consideration of just one of these design steps, disengaged from all
others, neglects the conceptional coherence. Therefore all of the design
steps are discussed within this thesis. The sequence orients itself to the
natural course of the controller design.
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24 1 Introduction
Nonlinear Model
Uncertainty structure
Controller synthesis
Nonlinear simulations
Tests on plant
Implementation in DCS
Figure 1.2: Steps ofa model based controller design
The following chapter consists of three parts: The first part describes
the design and operating data of the distillation column, followed by an
overview of the steady-state and dynamic column behavior. The second
part discusses the control objectives and control configurationfor this
column, while the third part describes the use of pressure-compensated
temperatures as controlled outputs.
Rigorous nonlinear dynamic models are the basis for simulation studies
and for linearization. They are discussed in Chapter 3.
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1.3 Structure of the dissertation 25
The main subject of Chapter 4 is the derivation of linear models. Two
different methods are presented which lead to linear models which
neglectand include flow
dynamics, respectively.A structured uncertainty model which describes the nonhnear behavior
of the distillation column for the entire operating range is developed in
Chapter 5.
Based on that structured uncertainty model, controllers can be designed
within the framework of the structured singular values. In the first part
of Chapter 6 the theoretical background of the structured singular value
\i is summarized. While the second part of that chapter presents the u-
optimal design of state-space controllers, the third part is dedicated to
the u-optimal design of PID control structures. Simulation studies
confirm the theoretical results.
In Chapter 7 the results of the (i-optimal controller design are compared
with results obtained by more common design methods, based on an
unstructured uncertainty description.
Usually the feed flow rate is a measured disturbance input to a distilla
tion column. Therefore, feedforward control can significantly improve
the compensation of feed flow disturbances, which is discussed in
Chapter 8.
Acontroller design should yield a satisfactory control quality not only in
dynamic simulations but also in the real plant. The results of the prac
tical implementation are presented in Chapter 9.
The conclusions and the recommendation for further research in
Chapter 10 complete this thesis.
The literature references and, if necessary, the special notations are
given at the end of each chapter.
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26 1 Introduction
1.4 References
[1.1] Brisk, MX.: "Process Control: Theories and Profits," Preprints of
the 12th World Congress of the International Federation ofAuto
matic Control, Sydney, July 18-23, 7, 241-250 (1993)
[1.2] Marlin, T. E., J. D. Perkins, G. W. Barton, and M. L. Brisk: "Ben
efits from process control: results of a joint industry-university
study," J. Proc. Cont, 1, 68-83 (1991)
[1.3] McAvoy, T. J. and Y. H. Wang, "Survey of Recent Distillation
Control Results," ISA Transactions, 25,1, 5-21 (1986)
[1.4] Pearson, R. K: "Modern Control: Why Don't We Use It?," InTech,
34, 47-49 (1984)
[1.5] Schuler, H., F. Algower, and E. D. Gilles: "Chemical Process
Control: Present Status and Future NeedsThe View from Eu
ropean Industry," Proceedings of the Fourth International Con
ference on Chemical Process Control, South Padre Island, Texas,
February 17-22, 29-52 (1991)
[1.6] Schuler, H.: "Was behindert den praktischen Einsatz moderner
regelungstechnischer Methoden in der Prozess-Industrie," atp,
34, 3, 116-123 (1992)
[1.7] Skogestad, S.: "Dynamics and Control of Distillation Columns -
a Critical Survey," Preprints of the 3rd IFAC Symposium on Dy
namics and Control of Chemical Reactors, Distillation Columns
and Batch Processes, April 26-29, College Park, Maryland, 1-25
(1992)
[1.8] Tolliver, T. L. and R. C. Waggoner: "Distillation Column Control;
a Review and Perspective from the CPI,"Advances in Instrumen
tation, 35, 1, 83-106 (1980)
[1.9] Waller, K. V.: "University Research on Dual Composition Con
trol of Distillation: A Review", Chemical Process Control 2, Sea
Island, Georgia, January 18-23, 395-412 (1981)
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1.4 References 27
[1.10] Yamamoto, S. and I. Hashimoto: "Present Status And Future
Needs: The View from Japanese Industry," Proceedings of the
FourthInternational
Conferenceon Chemical Process Control,
South Padre Island, Texas, February 17-22, 1-28 (1991)
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28 1 Introduction
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2.1 Introduction 29
Chapter 2
The Distillation Process
An Industrial Example
2.1 Introduction
A distillation column is not just any mass-produced article such as a
toaster or a washing-machine. Each distillation column is a unique
process unit, specially designed for the separation of a particular
substance mixture. Nevertheless, the thermodynamic principles and
basic dynamics are always the same. Therefore it is possible to demon
strate ideas for the controller design by the example of one column
without extensive loss of generality.
First in this chapter, the design and operating data of the industrial
distillation column are outlined, followed by a brief description of the
composition dynamics. The further two sections outline the control
objectives, the control structures, and the use of tray temperatures as
controlled outputs. The literature references terminate the chapter.
2.2 Column design and operation
The distillation column described in this thesis is an industrial binary
distillation column. A synopsis of the most important data for this distil-
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30 2 The Distillation Process An Industrial Example
lation column is given in Table 2.1. The distillation column (Fig. 2.1) is
equipped with 50 sieve trays, a total condenser, and a steam-heated
reboiler. The subcooled feed F enters the column ontray
20(counted
from the top) and for the greater part consists of a mixture of two
substances. Because of the small fraction of impurities, these are
neglected and the distillation column is considered to be a binary distil
lation column. The desired product compositions are 0.99 mol/mol (low
boiling component) for the top product D and 0.015 mol/mol for the
bottom product B. As these product purities are relatively high, this
distillation column can be classified as a "high purity distillation
column."
Table 2.1: Steady-state data
Column data
No. of trays 50
Column diameter (m) 0.8
Feed tray 20
Murphree tray efficiency =0.4
Relative volatility a 1.61
Operating data
Top composition x-q (mol/mol) 0.99
Bottom composition xg (mol/mol) 0.015
Feed composition xp (mol/mol) 0.7-0.9
Feed flow rate F (mol/min) 20-46
Top pressure (mbar) 60
Nominal operating point
Feed composition (mol/mol) 0.8
Feed flow rate (mol/min) 33
Reflux L0 (mol/min) 65
Boilup V51 (mol/min) 104
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2.2 Column design and operation31
Feed
F,xp
20
47
48
49
50
Reflux
Boilup
Vacuum
Condenser
Top product (Distillate)
D,xD
Reflux accumulator
Bottom
product
Figure 2.1: The industrial distillation column
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32 2 The Distillation Process An Industrial Example
Feed disturbances
The distillation column is connected in series following two other distil
lation columns, which operate in parallel. The bottom product streams
of these two columns are buffered by a tank and fed into the column
considered here. The level of the buffer tank is measured periodically
(typical period: 2 hours) and the feed of the column is set to keep the
tank level within specified bounds. Therefore, the feed flow is varied not
continuously but stepwise. In contrast to that, the variations of the feed
composition are always smooth. Even a shutdown of one of the other two
columns cannot cause a sudden increase of the buffer tank's composi
tion.
Top pressure control
The boiling points of the entering substances are high at standard atmo
spheric pressure. Because of a thermal decomposition of the light
component at higher temperatures, the column is operated under
vacuum. Correspondingly, the cooling water flow rate for the condenser
is kept constant and the top pressure is controlled by a vacuum pump.
Top level control
The reflux accumulator level is controlled by overflow. Hence the top
product flow rate D is not available as a manipulated variable for a
composition control system.
2.3 Steady-state behavior
Let us assume a composition control scheme with integrating behavior,
e.g., one PI controller which controls the top composition by manipu
lating the reflux and one which controls the bottom composition by
manipulating the boilup. Then, in steady-state, the product compositions are kept perfectly at their set-points, and an S-shaped composition
profile is developed within the distillation column. Figure 2.2 shows the
simulated composition profiles for different feed flow rates and compo
sitions. While these steady-state profiles are nearly independent of the
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2.3 Steady-state behavior 33
i 1 1 1 1 1i 1ir
xp = 0.7 mol/mol
xp = 0.8 mol/mol
xp = 0.9 mol/mol
F = 20 mol/min
F = 33 mol/min
F = 46 mol/min
i i i i i i i i i i i i i i
0.0 0.2 0.4 0.6
Composition (mol/mol)
0.8 1.0
Figure 2.2: Simulated composition profiles for the industrial distillation column
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34 2 The Distillation Process An Industrial Example
feed flow rate, they depend essentially on the feed composition. This has
a high significance for a controller design: Ifwe want to keep the product
compositionsclose to their
setpoints,we must allow
profilevariations in
the middle of the column. Consequently, we cannot control any composi
tion in the middle of the column.
The internal flow rates can be illustrated in a similar manner. Figure
2.3 shows the simulated liquid and vapor flow rates for the nominal
operating point. As previously mentioned, the reflux as well as the feed
are subcooled, i.e. they enter the column at a temperature below the
boiling point. A fraction of the vapor flow is condensed at the trays
where these two streams are fed into the distillation column. The two
discontinuities of the vapor flow profile at trays 1/2 and 20/21 result
Liquid flow
Vapor flow
Figure 2.3: Simulated vapor and liquidflow rates at nominal operating point
60 80 100 120
Flow rate (mol/min)
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2.4 Composition dynamics 35
therefrom. The reason for the slopes of the two profiles within the strip
ping and rectifying section of the column is the different heat of evapo
rationof
thetwo substances.
2.4 Composition dynamics
The composition dynamics within a distillation column is effectively
described by movements and shape alterations of the composition
profile. In order to illustrate this, let us control the reboiler level of the
distillation column by the bottom product flow rate B, and let us keepthe reboiler heat duty constant. The simulated step responses of the
composition profile to a 5% increase and a 5% decrease of the reflux flow
rate are shown by Figure 2.4. An increase of the reflux (Fig. 2.4 a) raises
the fraction of the light component in the column bottom. Consequently,
the composition profile of the light component moves towards the
column bottom, degrading the bottom product composition from 1.5% to
more than 30% impurity. The opposite effect is observed for a decreaseof the reflux flow rate (Fig. 2.4 b): The composition profile moves
towards the column top, which improves the bottom product composi
tion and debases the top product composition.
These plots illustrate two important properties of the composition
dynamics:
Column nonlinearity: The product compositions are a nonlinearfunction of the reflux, boilup, and the feed condition: A 5%-
increase of the reflux flow rate improves the top product compo
sition by 0.007 mol/mol, but a 5% decrease degrades it by more
than 0.2 mol/mol.
Strong interactions: A change of reflux or boilup alters both
product compositions.
The interaction between both product compositions and reflux and
boilup has a severe consequence for the composition dynamics, usually
called
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36 2 The Distillation Process An Industrial Example
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Composition (mol/mol) Composition (mol/mol)
a) b)
Figure2.4: Simulated
composition profiles (light component)for a
step changeoi
the reflux. Reboiler heat duty, feed flow rate and composition are kept at their
nominal values (see Table 2.1)
a) L0=1.05*L0>nom b) L0=0.95*L0inom
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2.5 Control objectives and configurations 37
Ill-conditioned behavior.
This is best explained by another two examples. If we like to increase
both product purities simultaneously, we have to increase reflux and
boilup by an exact quantity, for example the reflux by +26.5% and the
boilup by +19% (Figure 2.5 a). This keeps the composition profile's posi
tion constant, but it slowly intensifies the S-shape of profile. However a
slightly smaller step size for the reflux completely alters the dynamic
behavior (Fig. 2.5 b): The purity of the top product decreases, the purity
of the bottom product increases, and the dynamic response is much
faster. Therefore an exact direction of the input vector [L, V]T is
required in order to achieve a simultaneous increase of both product
purities. Consequently, even a small uncertainty of the input vector
[L, V]T may lead to undesired results. High condition numbers
K.
-.tq>(2.
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38 2 The Distillation Process An Industrial Example
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Composition (mol/mol) Composition (mol/mol)
a) b)
Figure 2.5: Simulated composition profiles (light component) for a step change of
the reflux and the reboiler heat duty. The feed is kept at nominal condition (see
Table 2.1).
a) Lo=1.265*L0>nora b) L0=1.260*L0,nom
V51=1.19*V51inom V51=1.19*V51>noln
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2.5 Control objectives and configurations 39
setpoints, especially in the presence of disturbances such as variations
of feed flow rate and feed composition. Tight composition control
requires sophisticated control schemes. Their design is the main topic ofthis thesis.
Reflux, boilup, and pressure drop are allowed to vary within a
predefined range. Any operation of a distillation column outside of this
range may cause insufficient separation or even damage of the column.
Each control system must handle such constraints to enable safe opera
tion. This topic is well discussed by Buckley et al. [2.2] and Shinskey[2.4].
2.5.1 The 5x5 control problem
A simple distillation column, such as the industrial example discussed
here, presents a control problem with the five control objectives
Top composition
Bottom composition
Reflux accumulator level
Reboiler level
Top pressure
and the five manipulated variables
Reflux
Boilup (indirectly controlled by reboiler duty)
Top product flow rate
Bottom product flow rate
Cooling water flow rate (or vapor flow rate to vacuum)
This problem is often called the 5x5 control problem. As mentioned
above, the top pressure is controlled by a vacuum pump and the reflux
accumulator level by overflow. Thus the 5x5 control problem is reduced
to a 3x3 control problem. These relations are illustrated in Figure 2.6.
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40 2 The Distillation Process An Industrial Example
Controlled outputs Manipulated inputs
3x3 control problem
Top product xp Reflux L
Bottom product xB Boilup V (Reboiler duty Q)
Reboiler level Mb Bottom product flow rate B
Condenser level MD + Top product flow rate D
Top pressure p * Overhead vapor Vp
(Cooling water flow rate,
vacuum pump)
5x5 control problem
Figure 2.6: The distillation control problems
2.5.2 Control design steps
In principle, the design of a MIMO controller for the 5x5 or in this case
the 3x3 control problem does not cause any particular difficulties.
However, the failure of just one actuator or sensor disables all control
loops. Due to the high sensitivity of MIMO controllers to sensor or actu
ator failure, the inventory control and the composition control usually
are independently designed, thus improving the robustness of the
control system and simplifying the controller design. The corresponding
design approach consists of three steps [2.5]:
1. Choosing the control configuration
In a first step the two manipulated variables for the composition control
are to be chosen. This choice names the control configuration. For
example, if the top composition xrj is controlled by reflux L and the
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2.6 Tray temperatures as controlled outputs 41
bottom composition xjg is controlled by boilup V, the control configura
tion is called L,V control configuration. After the choice of the manipu
lated variables for composition control, the remaining three
manipulated variables are available for level and pressure control.
The choice of the control configuration is often based on configuration
selection methods such as Relative Gain Array (RGA), Niederiinski
Index, or Singular Value Decomposition (SVD). The application of these
indices may lead to very different results (see [2.1], [2.6]), and the reli
ability seems to be low. One reason for the limited reliability may be the
neglect of inventory control: Yang et. al. [2.9] point to the substantialinfluence of inventory control on the composition control dynamics.
Most indices for control configuration selection are based on steady-
state gains. Consequently, perfect inventory control is assumed and
dynamic effects due to the interaction of inventory and composition
control are neglected.
The most common control configuration in the chemical industry is the
L,V configuration [2.7]. This control structure is rather independent of
inventory control dynamics [2.9] and has shown good results within an
experimental comparison of different control structures [2.8].
2. Inventory control design
In general, tight inventory control can be achieved with three simple PI
controllers. Some distillation columns show an inverse response of the
reboiler level to an increase of boilup. In this case, tight level control
with boilup as manipulated variable may be difficult.
3. Composition control design
A 2x2 controller for composition control is to be designed as a third step
of the design. This step is discussed in chapters 5-8.
2.6 Tray temperatures as controlled outputs
On-line composition analyzers are frequently used to determine product
compositions. However, their investment and maintenance costs are
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42 2 The Distillation Process An Industrial Example
prohibitive for distillation columns below a certain size. Provided that
substances with a boiling point difference of at least 10 C are separated
and that the
product purity specificationsare not
extremely stringent,pressure-compensated temperatures may substitute composition
measurements ([2.2], [2.4]).
2.6.1 Pressure-compensated temperatures
For binary mixtures a definite correlation exists between boiling
temperature, pressure, and composition
T = f(p,x) (2.1)
This correlation is illustrated in Figure 2.7 for the two components
entering the industrial distillation column. A substitution of the compo
sition measurements by temperature measurements requires a
compensation for the effect of pressure variations.
If the pressure variations are small, the temperature measurement can
be compensated by a linear function. The nominal pressure and compo
sition are denoted by the index N.
(P-PN) (2.2)N
In case of larger pressure variations, a second-order term has to be
supplied:
(p-pN)2 (2.3)N
Estimation of tray composition
Itis possible to infer the tray composition directly. By regression of
{x, T, p} data, the coefficients of a simple polynomial expression can be
calculated. An example is given by
T = T +Compensated gp
T = T + -Compensated Qp N+5aprT
x = e] + Q2(T: + TCon)+e3p + Q4p2 (2.4)
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2.6 Tray temperatures as controlled outputs 43
Figure 2.7: Boiling points of the two-component-mixture
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44 2 The Distillation Process An Industrial Example
Such an equation in terms of the absolute temperature and pressure is
simpler to implement in a distributed control system than an equationin terms of deviations from reference values
x = e1 + e2(T-TN)+e3(p-pN)+e4(p-pN)2 (2.5)
One problem of the tray composition estimate is a potential bias of the
temperature measurements. Practical experience has shown that a bias
of up to 2 C is to be expected due to heat transport phenomena. In (2.4)
the bias is corrected by theparameter TCoTT
In
practice,however, this
correction is difficult to estimate. In principle, it would be possible to
include cross terms such as 0Tp in the regression model. However,
errors in the absolute temperature may lead to incorrect numerical
values of these cross terms. Therefore, in the regression model, cross
terms should be avoided.
Pressure compensation as well as the estimation of tray composition are
easily implemented in a process control system. Without a pressure
compensation, it is impossible to use tray temperatures in a vacuum
column as controlled variables and expensive composition analyzers are
necessary. For temperature measurements close to the column top, a
linear eompensation is usually sufficient. For trays close to the column
bottom, we have to expect higher pressure variations, and a compensa
tion with a second-order polynomial is recommended.
2.6.2 Temperature measurement placement
The sensitivity of the tray temperatures near the ends of the column to
changes of the product compositions is very small. To make the temper
ature measurement sensitive enough, it has to be located at some
distance from the column ends. Figure 2.2 shows simulated steady-state
composition profilesfor the industrial distillation column. These
profilesillustrate the fact that the effect of a change of operating conditions
increases with growing distance from the column ends. On the other
hand, a deterioration of the correlation between product composition
and tray temperature results from an increasing distance from the
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46 2 The Distillation Process An Industrial Example
trol of Chemical Reactors, Distillation Columns and Batch Pro
cesses, April 26-29, 1992, College Park, MD, 1-25 (1992)
[2.8] Waller, K. V., D. H. Finnerman, P. M. Sandelin, K. E. Haggblom,
and S. E. Gustafsson, "An Experimental Comparison of Four
Control Structures for Two-Point Control of Distillation," Ind.
Eng. Chem. Res., 27, 624-630 (1988)
[2.9] Yang, D. R., D. E. Seborg, and D. A. MeUichamp: "The Influence
of Inventory Control Dynamics on Distillation Composition Con
trol," Preprints of the 12th World Congress of the International
Federation ofAutomatic Control, Sydney, 18-23 July 1993,1, 71-
76(1993)
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3.1 Introduction 47
Chapter 3
ARigorous Dynamic Model of
Distillation Columns
3.1 Introduction
The rigorous dynamic process simulation has become an accepted and
widespread tool in process and even more so in controller design [3.11].
Increasing competition and environmental protection provisions
require an optimization of process and control structures, which can be
obtained only by a substantial knowledge of process dynamics. At the
same time, dynamic experiments on a running plant are less and less
desired. Rigorous dynamic modelling and simulation can replace such
expensive and time-consuming measurements. This has special signifi
cance for high-purity distillation columns. Due to their long time
constants and varying feed flow rates and feed compositions, reproduc
ible operating conditions are difficult to guarantee. Therefore, new
controllers are usually tested thoroughly by dynamic simulation for the
full operating range of the distillation column. The rigorous models of
distillation columns used for that purpose match the reality to a largeextent [3.17].
In this chapter, a rigorous dynamic model for distillation columns is
discussed. This model is used in all nonlinear dynamic simulations
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48 3 A Rigorous Dynamic Model of Distillation Columns
within this thesis. In a special section, the numerical treatment of the
resulting system of algebraic-differential equations is outlined. The
modelling and control fields use very different notations. Therefore the
notation used within this chapter is explained in section 3.11.
3.2 Conventions
Figure 3.1 shows a schematic representation of a distillation column
equipped with nt trays. The column top (condenser and reflux accumu
lator) is denoted by the index 0, the trays with the indices 1, 2,... nt, and
the column bottom (including the reboiler) with the index nt+1. To
simplify the formal mathematical description the reflux stream R is
designated as liquid flow (L0).
The feed of the industrial distillation column, as described in Chapter 2,
is in liquid phase and subcooled. The top pressure is controlled by a
vacuum pump and the condenser is operated with a constant cooling
water flowrate.
Flashcalculations
forthe feed stream
as
wellas
dynamic models for the top pressure of the column are therefore not
considered here. For other applications, the model presented is easily
extended with appropriate model equations.
3.3 The objective of modelling
The control or process engineer is interested in thedynamic
behavior of
various important process variables (e.g., tray temperatures, product
compositions) as a function of the time-varying column inputs. The
objective of a dynamic model is an approximation of the real process
input/output behavior by a system of differential and algebraic equa
tions. These model equations are based on material and energy balances
as well as on thermodynamic and fluid dynamic correlations.
3.4 Simplifying assumptions
Within a distillation column many different physical phenomena occur.
Although it would be possible to include models for the fluid streams on
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3.4 Simplifying assumptions49
nt-2
.1.
.2.
.3.
4
V;
nt-2
nt.:!
nt
R
(=L0)
Si,
5v,nt-l
Vnt+1
QoCondenser
1 Reflux accumulator
D
Qnt+1Reboiler
nt+1
&B
Figure 3.1: Distillation column
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50 3 A Rigorous Dynamic Model of Distillation Columns
the trays, for the dead time caused by the transport time of vapor flow
from one tray to the next one above, or for the heat exchange with the
environment,the
resultingmodel would be of
very highorder. As
mentioned earlier, the aim of modelling the distillation column
dynamics is a sufficient description of the real macroscopic behavior.
This means that we are interested primarily in the dynamics of tray
compositions, temperatures, and pressures etc. rather than in the fluid
streams on the trays. Experience shows that no substantial improve
ment can be achieved with models including effects with more micro
scopic characteristics. Hence thefollowing
assumptions areusuallyintroduced in order to achieve a compromise between model accuracy
and order ([3.3], [3.13], [3.17]):
The holdup of the vapor phase is negligible compared to the
holdup of the liquid phase.
Liquid phase and vapor phase are each well mixed on all trays,
i.e., the composition of the liquid and of the vapor phase are inde
pendent of the position on the tray.
The residence time of the liquid in the downcomer is neglected.
The variation of the liquid enthalpy on a tray can be neglected on
all trays. (This assumption is not applicable to the evaporator.)
In the literature so far, uniform liquid flows and constant holdups for all
trays have often been assumed (equimolar overflow). This assumption
is problematic because it implies a neglect of flow dynamics. Essential
dynamic effects may remain unmodelled, e.g., a non-minimum phase
behavior (inverse response) of the reboiler level and the tray composi
tions in the lower section of the column to an increase in reboiler heat
supply.
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3.5 Balance equations 51
3.5 Balance equations
3.5.1 Material balances
The differential equations describing the dynamics of the holdup for
each component on a tray are derived from a material balance for each
component. The balance border is the single tray with its ingoing and
outgoing streams (Figure 3.2).
Figure 3.2: Balance border for the material balances
Material balance for component k on trayj (k=l, ..., nc;j=l, ..., nt)
dnVi d(n-xt-)
"dT
=
dT1^=
pixF,kj +Vi*kj-i- (VSy)^ (3.+ (Vj + 1-SVij + 1)yk)j + 1-Vjyk>j
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52 3 A Rigorous Dynamic Model of Distillation Columns
In the same way, the balance equations for the column top and the
column bottom are formulated:
Material balance for component k in condenser (k=l, ..., nc)
dnk0 d(n0xk0)
dt dt (Vi-Sv,.)yk,i-(Lo+ D)xk,o (3.2)
Material balance for component k in the evaporator (k=l, ..., nc)
Usually the liquid phases in the column bottom and the reboiler are
mixed either by natural convection or by a pump. Assuming perfect
mixing we obtain
dnk,nt+1=
d(nnt+lxk,nt+l>dt dt (3.3)
= *-'ntXk, nt ~ "Xk, nt + 1~~
%t + 1 ^k, nt + 1
The total holdup on tray j equals the sum of the holdups of the indi
vidual substances:
nc
nj= X nk, (3.4)
k= 1
3.5.2 Energy balance equations
The vapor flows within a distillation column are calculated by an energy
balance. The balance border is the same to the border in Figure 3.2,
which was used for the material balance equations.
Energy balance for tray j:
SW=F^ + V.hH+(VJ + -Sv,] + ,)h"] + , (35)
-(S^ + L^-V^
For the left-hand side of this equation the following holds
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3.5 Balance equations 53
d dni dh'irt(nih'i)=h'jdF+nniF (3-6)
If in (3.6) we substitute the expression for the differential term dn-/dt
according to
^ = VLj-i+vj+i-svj+i-si,rLrvi w
thefollowing energy
balanceequation
holds
A "h'
W= tFi
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54 3 A Rigorous Dynamic Model of Distillation Columns
AQ= AH
,
,AV,
.^v, nt + 1 nt+.
dAT
+ nnt + lVnt+lPnt+lCp,nt+l Jt
nt+1(3.10)
To achieve a first-order differential equation in AVnt+1, the differential
term dATnt+1/dt has to be substituted by a differential term in AVnt+1.The increase of the pressure drop due to a changing vapor flow rate
(assuming a constant total holdup on the tray) can be estimated with
A(APj) =K+ JAV.
j + l(3.11)
Hence the pressure change in the evaporator can be approximated for a
distillation column with nt trays by
A
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3.6 Fluid dynamics 55
lag
with the time constant
nnt+lVnt+lPnt+lcp,nt+l
^-g = T3(Q-Qlag) (3.15)
9Pnt+lnt
UVj + Jlag AH^
(3'16)
Ifwe substitute the total bottom holdup balance equation in the energy
balance equation
dn+, ,
"nt+l-ir1 = Lnth'nt + Qlag-Bh'nt+I-Vnt+1h"nt+1 (3.17)
the following equation holds:
Energy balance for the evaporator
V _Lnt(nnt-h'nt+l> + ^lag , 1Sx
Vnt+1 V5 Ivl ;
"nt+1 n nt + 1
The parameters (e>Tnt+))/(9pnt+1) and (3Ap)/(3V- + 1) canbeeval-
uated numerically or analytically from the appropriate equations (see
sections 3.6.2 and 3.7.2)
3.6 Fluid dynamics
In the previous sections, the equations describing composition and total
holdup dynamics, as well as the vapor flow rates have been derived.
Here the calculation of the liquid flow rates and of the pressure drop is
discussed.
3.6.1 Liquid flow rates
The volumetric liquid flow rate over the weir on tray j can be calculated
according to the Francis weir formula ([3.16], [3.10]):
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56 3 A Rigorous Dynamic Model of Distillation Columns
LV;j = u^2i|bwh^;j (3.19)
For sharp-edged weirs jo. = 0.64 holds. Perfect mixing on the trays,
including the liquid in the downcomers, is assumed. Nevertheless, ifwe
calculate the effective liquid head hLW , above the weir edge, we have
to take the liquid phase fraction ej and the liquid volume in the down-
comer into account (Figure 3.3). The liquid level in the downpipe is the
sum of the liquid head on the tray and of the hydrostatic level due to the
pressure drop according to
p- -p-
Hydrostatic liquid level in downcomer =
Pjg(3.20)
The liquid head hL of the pure liquid on a tray (without a vapor phase
fraction) is equal to the total liquid volume on the tray n-v'- minus the
o o
o o o
Pj
"LWJ
Pj-Pj -l
Pj*1
thLJ
Figure 3.3: Liquid levels on a tray
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3.6 Fluid dynamics 57
liquid volume in the downcomer due to pressure drop
AB (P: - Pj _ j) / (Pjg) > both divided by the total area AA +AB:
Vj
Ki =
Pj-Pj-1,
AA +AB(3.21)
For the application of the Francis weir formula, we have to evaluate the
liquid level of the pure liquid (liquid without vapor phase fraction). For
that purpose, first the height of the two-phase layer is to be evaluated
and second the liquid phase fraction j must be taken into account. The
effective liquid level becomes
Ti.W.j-hw
Vj-Pi-Pi_L-i
j=
Pjg
AA +AB-jhw (3.22)
Substituting (3.22) into the Francis weir formula (3.19), we obtain the
volumetric liquid flow rate of the two-phase mixture. The flow rate from
tray j in molal units is calculated by:
u-v^tv
Lj =
VrPj-p'izi.
pjg
AA + ABjhw
3/2
(3.23)
In many industrial distillation columns, calming zones exist in front of
the weir. For this case, e- = 1 holds at the weir edge. Otherwise, we
have to estimate the liquid phase fraction on the trays. The Stichlmair
correlation is well suited for that purpose [3.18].
3.6.2 Pressure drop
A vapor flow through a tray in a distillation column suffers a pressure
drop. Its amount depends on the vapor flow rate, the tray holdup, and
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58 3 A Rigorous Dynamic Model of Distillation Columns
the geometry of the tray. Usually, the pressure drop is assumed to
consist of three different parts ([3.7], [3.12]):
Dry pressure drop occurring at the flow through the tray without
liquid (Aptr j)
Hydrostatic pressure drop due to liquid head and liquid density
(ApLJ)
Pressure drop by bubble-forming due to surface tension of liquid
(APa;i>The pressure drop by bubble-forming usually is insignificant and can be
neglected.
Dry pressure drop
With sufficient accuracy, the dry pressure drop can be approximated by
the following well-known expression:
AptrJ = ^(Re)^V Ao J
(3.24)
The orifice coefficient (Re) either can be evaluated by measurement
on comparable trays, or it can be estimated with experimentally verified
correlations. During the simulations, the following correlation for sieve
trays is used [3.19]:
Aptr,j
1-
aaJ+ 0.211f
v Ao ;(3.25)
Hydrostatic pressure drop
The hydrostatic pressure drop results from the liquid head and the
liquid density. We have to take the liquid volume in the downcomer into
account (see 3.6.1).
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3.7 Phase equilibrium 59
ApL,i' A.+L p>(3'26)
The total pressure drop consists of the sum of the two parts dry pressure
drop and hydrostatic pressure drop:
APj = Pj + i-Pj = Aptr>j +ApLj (3.27)
3.7 Phase equilibrium
All equations we have discussed in the previous sections are explicitly
or implicitly interrelated with the vapor phase composition. In this
section, the most important correlations concerning the vapor phase
compositions and boiling points are presented.
3.7.1Vapor phase composition
The liquid on each tray and in the evaporator is at boiling-point. Phase
equilibrium thus can be assumed. At moderate pressures up to some few
bar, the concentration of a substance in the vapor flow leaving tray j can
be obtained according to
yEquilibrium = M*Jx = Kk .xk j (3.28)
If the substance mixture exhibits ideal behavior, the activity coefficient
y becomes one, and the vapor phase compositions are equal to the ratios
of the partial pressures of the substances and the absolute pressure on
the tray.
The vapor pressures of the pure substances pk can be calculated with a
high level of accuracy by the Antoine equation (3.29). The parameters A,
B, and C are listed in many tables of substance properties (e.g., [3.5]).
^M^tTC (3"29)
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60 3 A Rigorous Dynamic Model of Distillation Columns
The calculation of the liquid phase activity coefficients yk . can be
effected by one of the well known correlations (Wilson, NRTL,
UNIQUAC etc.).
Murphree tray efficiency
In a distillation column only little contact time exists on each tray for
the mass transfer between liquid and vapor phase. Therefore no perfect
phase equilibrium can be achieved, and the tray efficiency will deviate
from the unit value. This effect can be modelled by the Murphree tray
efficiency for the vapor phase.
-^Equilibrium ,.yk,j ~yk,j + l
3.7.2 Boiling points
The vapor phase composition according to (3.28) is a function of the tray
temperature Tj. At boiling point, the sum of the vapor phase mole fractions calculated becomes one. Hence for a tray j, the following boiling
point equation holds:
X yEquilibrium = ^ .^ p.,^^ . = , (3.31)k=l k=l
The Murphree tray efficiency is not considered for the boiling point
calculation, because it relates to the mass transfer between vapor and
liquid phase rather than to the equilibrium composition.
3.8 Volumetric
propertiesThe fluid dynamic models discussed are interrelated with the molar
volumes of the vapor phase and of the liquid phase, and with the corre
sponding densities. Their calculation is the subject of this section.
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3.8 Volumetric properties 61
3.8.1 PVT relations
The molar volumes of the liquid phase v'- or the vapor phase v". depends
on the pressure pj, the temperature Tj, and the actual compositions x^jor ykj. A great number of different equations of state has been developed
to describe this behavior. They are extremely well documented ([3.5],
[3.6]), and a discussion of their properties is not repeated here.
The PVT behavior is described here by the Soave-Redlich-Kwong equation (SRK equation, [3.15], [3.6]) with the Peneloux correction. This
correction improves the estimate of the molar volumes of the liquid
phase, which is overrated by 10-15% using the SRK equation.
If measurement data of the PVT behavior of the pure substances exist
and their mixing behavior is nearly ideal, a different possibility has
shown good results for the liquid phase:
We can correlate the molar volumes measured with the temperature by
a polynomial regression. The molar volume v'- of the substance mixture
can be approximated as a weighted sum of the individual molarvolumes:
nc
v'j = I xk,/k,j (3-32)k=l
3.8.2 Density
The densities of liquid and vapor phase can be computed from the molar
volume, the molar mass, and the mole fractions.
nc
I xk,jMkLiquid phase density: o'- =
k= ],
(3.33)
nc
I yk>JMkVapor phase density: p" = (3.34)
Vj
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62 3 A Rigorous Dynamic Model of Distillation Columns
3.9 Enthalpies
The quantity not discussed so far is the enthalpy of a substance mixture
in liquid or vapor phase. The enthalpy of a real fluid is estimated by the
sum of an ideal part and the value of a departure function Ah^apdescribing the deviation of the enthalpy from the enthalpy of the ideal
gas state:
T
h = h +
j cjfdT+Ahp (3.35)
T
The ideal part can be calculated by summing the ideal parts for each
component:
( T
KddT= I xkHdkdTT k=l0 "_iV *0Tn(3.36)
The ideal heat capacities c are often approximated by a third-order
polynomial for each component:
cj,dk = Ak + BkT + CkT2 + DkT3 (3.37)
The parameters for equation (3.37) are listed in many tables of
substance properties, or they can be estimated with very high accuracy
by Joback's method ([3.15], p. 154-156).
The real part Ah^ p describes the departure of a mixture from the idealbehavior. It can be evaluated using one of the well-known equations of
state, e.g.,the SRK
equation ([3.15], [3.6]).
If measurement data for the heat capacities and for the heat of vapor
ization are available, a simple solution is possible in a manner similar
to that mentioned in section 3.8.1:
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3.10 Numerical solution 63
f T,
Liquid phase enthalpy: ti = V
k=ll Tft
k,JcP,J,kdT + h (3.38)
Vapor phase enthalpy: h". = V Yk jk = l VTn
+ h (3.39)
3.10 Numerical solution
The complete rigorous dynamic model for distillation columns, as intro
duced above, consists of a system of differential and algebraic equations
(DAE). The complexity of the model is illustrated by Figure 3.4. It illus
trates the interconnection of the model equations for three adjoining
trays. The solution of the differential equations obviously depends on
the solution of the algebraic equation system. Therefore an efficient
numerical integration using standard integration methods is not
possible. This requires special adapted integration algorithms, as
outlined in section 3.10.4.
3.10.1 The dependent variables and the equation system
As a first step for the numerical treatment, we have to decide which
variables should form the vector of the dependent variables. This vector
of dependent variables must at any time completely describe the state
of a distillation column and should be of minimum size to avoid exces
sive computation times.
The vapor phase composition is an illustrative example for the complete
description of the distillation's state: If we know the tray composition,the tray temperature, and the tray pressure, then the vapor phase
composition in equilibrium is easily calculated by an explicit algebraic
equation. Consequently, it is not necessary to insert the vapor phase
composition into the vector of the dependent variables.
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crc?
3oCO
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3.10 Numerical solution 65
As one vector which satisfies the requirements of a complete description
and of minimum order, the following vector is proposed (as a modifica
tion of the vector proposed by Holland & Liapis [3.10]):
y - [QlCond> D> nl,0 > nnc,0> T0> P0> L0>
(Vj, nxj,..., nncJ, (Sy), (SvJ), Tj, pj( Lj}j=1> 2>..., nt
Qlag Q> Vnt+1, nlnt+1, ..., nncnt+1,
B, Tnt+1, pnt+i,
States of the control system]
(): Value is inserted only if it physically exists
The Jacobian matrix of the equation system (as described below) corre
sponding to these dependent variables has a numerically advantageous
block diagonal dominant structure.
For the calculation of these dependent variables y, the following equa
tions are to be solved
Differential equations
nc material balance equations (3.1)
Algebraic equations
1 equation for vapor flow rate (3.9)
1 equation for liquid flow rate (3.23)
1 equation for boiling point (3.31)
1 equation for pressure drop (3.27)
Total: nc +4 equations per tray
and in addition the equations for the evaporator, the condenser, and the
control system. Considering industrial distillation columns which are
often equipped with more than 50 trays, the resulting algebraic differ
ential equations amount to several hundred equations. The model for
the industrial binary distillation equipped with 50 trays gives an
impression of these numbers: It consists of a system of 107 differential
and 210 algebraic equations.
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66 3 A Rigorous Dynamic Model of Distillation Columns
3.10.2 Formal representation of the DAE
We can formally represent the entire dynamic model by the semi-
explicit equation system
^ = f (t, n (t), z (t)) n (t0) = n0 (3.40a)
0 = g(t,n(t),z(t)) z(t0)=z0 (3.40b)
The vector n consists of all tray holdups (for all components), while the
vector z contains all other dependent variables. A different but equiva
lent formal representation is the implicit form:
F(t,y(t),y'(t)) =0 y(t0) = y0 (3.41)
Here the vector y contains all the dependent variables. A simulation of
the dynamic behavior requires a simultaneous solution of the whole
equation system.
3.10.3 The index
The index of a set of differential-algebraic equations (DAE) character
izes the integration problem. The higher the index, the more difficult is
a solution of the DAE. The differential index is the most common defini
tion:
The differential index m of the system F (t, y (t), y' (t)) = 0 is the min
imal number m such that the system ofF (t, y (t), y' (t)) =0 and of the
analytical differentiations
d(F(t,y(t),y'(t)))_
A dm(F(t,y (t), y'(t)))_
dt-U'""
dt
can be transformed by algebraic manipulations into an explicit ordinary
differential system [3.8].
Consequently, a system of ordinary differential equation has an index of
m=0.
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3.10 Numerical solution 67
3.10.4 Solution methods and software
The first general method for the numerical solution of semi-explicit
DAE with index 1 was proposed by C. W Gear in 1971 [3.4] and was soon
extended to the solution of implicit index 1 problems. The method is
based on a special class of the linear multistep methods entitled the
backward differentiation formulas, which are standard algorithms for
the integration of stiff systems. The most important convergence results
may be found in [3.1]. In theory, it is possible to solve problems of higher
indices with the backward differentiation formulas. However, the neces
sary software is not available as yet. The apparently very frequentlyused integrators DASSL and LSODI are based on Gear's method. These
methods are distinguished for their effectiveness in solving continuous
problems. However, the computational effort grows significantly for
systems with discontinuities arising, for example, during the simulation
of the response to several feed flow or feed composition step changes. For
such cases, the one-step methods find more and more interest [3.11].
The one-step methods are extensions of the well-known Runge-Kutta,
Rosenbrock, or extrapolation methods. An extensive discussion of the
properties of these methods is found in [3.8]. However, the development
of the integrators (RADATJ5, LIMEX) is in an early stage, and no imple
mentations are found in any of the widespread Fortran libraries.
For the simulation studies the DASSL integrator, as implemented in the
NAG Fortran Library is used with good success. The differential-algebraic equations (DAE) are solved in an implicit manner according to
(3.41).
The calculation sequence
During the integration, the right-hand sides of the differential and alge
braic equations repeatedly have to be evaluated for a given vector y of
the dependent variables and for a given time t. The algebraic equations,
and often the differential equations as well are solved in an implicit
manner. The equation errors, which have to be supplied to the integra
tion, are the difference between the right-hand sides of the equations
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3.10 Numerical solution 69
a)
b)
c)
d)
e)
( Vector of dependent variables y J
Vapor phase composition for evaporator and trays
(Equation (3.28))
Error for boiling point at evaporator and trays
k
Calculation of the thermodynamic states
h', v', v", p', p"
for the feed
Murphree tray efficiency for trays nt, nt-1,..., 1
yk,J=
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70 3 A Rigorous Dynamic Model of Distillation Columns
u
h)
Error for vapor flow leaving evaporator
Lnt(hnt-hnt+l>+Qlag h" -h' nt+1"nt+1 "nt+1
V
i)
Error for liquid streams
13 p,g B,
3/2
3A + A
J W
AA+ABJ
I )
If
j)
Error for pressure drop
P]+1-P3-AP]
(Equation (3.27))
' '
k)Differential equation for vapor flow lag
(Equation (3.15))
< '
1)Differential equations for holdup of substances
(Equations (3.1), (3.2) and (3.3))
' '
m) C Vect or of differeiitials and errors J
Figure 3.5 continued
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3.11 Notation 71
between supplied and calculated flow rates and pressures, and the
errors of the boiling point equations are combined in one vector and
suppliedback to the
integrationroutine.
3.11 Notation
A0 [m2] Hole area in tray
AA [m2] Tray area without downcomer area
Ab [m2] Downcomer area
bw [m] Length of weir
pidLP
[J/mol-K] Ideal gas heat capacity
CP,1 [J/kg-K] Liquid heat capacity
do [m] Diameter of holes of sieve tray
Fj [mol/s] Feed flow rate to tray j
h [J/mol] Molar enthalpy
h'j [J/mol] Molar enthalpy of liquid phase
h"j [J/mol] Molar enthalpy of vapor phase
hL [m] Liquid level above upper edge of weir
hw [m] Weir height
AHv,k,j [J/mol] Heat of evaporation of component k on tray j
AHvj [J/mol] Heat of evaporation of liquid on tray j
Kkj [mol/mol] Distribution coefficient for comp. k on tray j
LJ [mol/s] Liquid flow leaving tray j
Wj [m3/s] Volumetric flow from tray j
Mk [g/mol] Molar mass of component k
nt H Number of trays in column
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72 3 A Rigorous Dynamic Model of Distillation Columns
nj [mol]
nkj [mol]
nc [-]
Pj [N/m2]
APj [N/m2]
K [N/m2]
P [N/m2]
Q [J/s]
Qlag [J/s]
s [m]
SU [mol/s]
Jvj [mol/s]
t [s]
T [K]
TJ [K]
Vj [mol/s]
VVj [m3/s]
xkj [mol/mol]
XF,ko [mol/mol]
ykj [mol/mol]
Yk [-]
Total holdup on tray j
Holdup of substance k on tray j
Number of components
Pressure on tray j
Pressure drop over tray j
Steam pressure of pure component k
Pressure
Heat supply to evaporator
"active" heat supply
Thickness of sieve tray
Side product flow rate from tray j,
liquid phase
Side product flow rate from tray j,
vapor phase
Time
Temperature
Temperature on tray j
Vaporstream from
tray j
Volumetric vapor stream from tray j
Liquid phase mole fraction of
component k on tray j
Mole fraction of component k
in feed to tray j
Vapor phase mole fraction of
component k above tray j
Liquid phase activity coefficient
of component k
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3.11 Notation
j [m3/m3] Liquid phase fraction on tray j
Tl [mol/mol] Murphree tray efficiency for vapor phase
V [m3/mol] Molar volume
V'j [m3/mol] Molar volume of liquid phase on tray j
V"j [m3/moI] Molar volume of vapor phase on tray j
% H Orifice coefficient
P'j [kg/m3] Liquid density on tray j
P"j [kg/m3] Vapor density on tray j
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74 3 A Rigorous Dynamic Model of Distillation Columns
3.12 References
[3.1] Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical so
lution of initial-value problems in differential-algebraic equa
tions, North-Holland, New York (1989)
[3.2] Byrne, G. D., P. R. Ponzi, Differential-Algebraic Systems, Their
Application and Solution, Comp. Chem. Eng., 12, 5, 377-382
(1988)
[3.3] Gani, R., C. A. Ruiz, and I. T. Cameron: "A Generalized Model for
Distillation Columns," Comp. Chem. Eng., 10, 3, 181-198 (1986)
[3.4] Gear, C. W.: "Simultaneous Numerical Solution of Differential-
Algebraic Equations," IEEE Trans, on Circuit Theory, CT-18, 1,
89-95 (1971)
[3.5] Gmehling, J. and U. Onken: "Vapor-Liquid Equilibrium Data
Collection;' 1, Part 1, XI-XXII, DECHEMA, Frankfurt (1977)
[3.6] Gmehling, J. and B. Kolbe: Thermodynamik, Georg Thieme Ver-
lag, Stuttgart (1988)
[3.7] Grassmann, P. and F. Widmer, Einfiihrung in die thermische
Verfahrenstechnik, 2nd ed., de Gruyter,Berlin
(1974)
[3.8] Hairer, E. and G. Wanner: Solving Ordinary Differential Equa
tions II Stiff and Differential-Algebraic Problems, Springer
Verlag, Berlin (1991)
[3.9] Hajdu, H., A. Borus, and P. Foldes: "Vapor Flow Lag in Distilla
tion Columns," Chem. Eng. Sc, 33, 1-8 (1978)
[3.10] Holland, C. D. and A. I. Liapis, Computer Methods for Solving
Dynamic Separation Problems, Chapter 8, McGraw-Hill, New
York(1983)
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3.12 References 75
[3.11] Marquardt, W.: "Dynamic Process Simulation Recent
Progress and Future Challenges," Fourth International Confer
ence on Chemical Process Control, South Padre Island, Texas
(1991)
[3.12] McCabe, W. L., J. C. Smith, and P. Harriott: Unit Operations of
Chemical Engineering, 4th ed., McGraw-Hill, New York (1985)
[3.13] Najim, K. (Editor): Process Modeling and Control in Chemical
Engineering, Marcel Dekker, New York (1989), Chapter III, 145-
211, S. Domenech, L. Pibouleau, "Distillation"
[3.14] Petzold, L.: "Differential/Algebraic Equations are not ODE,"
SIAMJ. Sci. Stat. Comput, 3, 3, 367-384 (1982)
[3.15] Reid, R. C, J. M. Prausnitz, and B. E. Poling: The Properties of
Gases and Liquids, 4th ed., McGraw-Hill, New York (1988)
[3.16] Retzbach, B.: "Mathematische Modelle von Destillationskolon-
nen zur Synthese von Regelungskonzepten," Fortschritt-Berichte
VDI, Reihe 8: Mess-, Steuerungs- und Regelungstechnik, Nr. 126,
VDI Verlag (1986)
[3.17] Rovaglio, M., E. Ranzi, G. Biardi, and T. Faravelli: "Rigorous Dy
namics and Control of Continuous Distillation Systems Simu
lation and Experimental Results," Comp. Chem. Eng., 14, 8, 871-
887 (1990)
[3.18] Stichlmair, J.: Grundlagen der Dimensionierung des GaslFliis-
sigkeit-Kontaktapparates Bodenkolonne, Verlag Chemie, Wein-
heim (1978)
[3.19] Weiss, S. et. al.: Verfahrenstechnische Berechnungsmethoden,
Teil 2: "Thermisches Trennen", VCH Verlagsgesellschaft, Wein-
heim (1986)
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4.1 Introduction 77
Chapter 4
Linear Models
4.1 Introduction
Robust controllers are designed on the basis of linear process models.
Therefore the elaboration of linear dynamic models for the distillation
column is a central part of control system synthesis. These models
should describe the dynamic behavior of the process within a wide
frequency range. They can be obtained in two ways:
System identification
Linearization of a nonlinear model
It is a big advantage of the system identification that it avoids a compli
cated and expensive nonlinear model. Nevertheless, this approach has
some severe drawbacks, for example:
The time-constants of the composition dynamics are large. A
recording of input/output data for the real plant is very time-
consuming.
Due to the high sensitivity of distillation columns to changes of
the internal flow rates, even for small magnitudes of the input
variation (e.g., 5% of the steady-state value) the response mayfar exceed the linear region.
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78 4 Linear Models
Each experiment causes undesired disturbances of the product
qualities.
It is practically impossible to obtain models for the entire operating range of the distillation column
These disadvantages and some other fundamental problems of the iden
tification itself (see Jacobsen et al. [4.5]) lea