a rigorous dynamic model of distillation columns

8/6/2019 A Rigorous Dynamic Model of Distillation Columns

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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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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


6.2 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


7.2 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


8.2 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


9.2 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



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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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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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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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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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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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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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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


33/225

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


34/225


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)


35/225

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


36/225


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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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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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


39/225


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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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


41/225


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


42/225


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)


43/225


Figure 2.7: Boiling points of the two-component-mixture


44/225


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


45/225


46/225


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


48/225

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


49/225

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


50/225


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.


51/225


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


52/225


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


53/225


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


54/225


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


55/225


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]):


56/225


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


57/225


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


58/225


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).


59/225

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)


60/225


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.


61/225

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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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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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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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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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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3.10.4 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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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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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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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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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

a rigorous dynamic model of distillation columns

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