Modelling and Control of Reactive Distillation Processes

M odelling and C ontrol o f R eactive D istilla tion P rocesses

Nicholas C. T. Biller

A thesis subm itted for the degree of Doctor of Philosophy ofthe University of London

Departm ent of Chemical Engineering

University College London

London W CIE 7JE

September 2003

ProQuest Number: 10014376

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A bstractReactive distillation has been applied successfully in industry where large capital and

energy savings have been made through the integration of reaction and distillation into one

system. O perating in batch mode, in either tray or packed columns, offers the flexibihty

required by pharm aceutical and hne chemical industries for producing low volume/high

value products with varying specifications. However, regular packed or tray columns may

not be suitable for high vacuum operations due to the pressure drop across the column

section and short path distillation may be more applicable.

The objective of this thesis is to investigate the control of reactive distillation in batch

columns, tray and packed, and in short path columns. In order to study control fully,

it is necessary to develop rigorous dynamic models th a t accurately capture the process

behaviour. The higher the degree of rigour, the more accurately the process conditions

and dynamics are captured. However, more rigorous models are more com putationally

expensive to implement and can be prone to numerical errors, introduced for instance

during linearisation. Therefore, in this thesis, the degree of modelling rigour required

for both simulation and control purposes is explored in detail for tray and packed batch

columns and short-path columns.

For batch tray columns, it is dem onstrated th a t to accurately capture the change in

process conditions, it is necessary to model pressure dynamics and employ a dynamic

energy balance. For packed columns, distributed ra te based modelling is compared to

lumped equilibrium modelling and it is found th a t due to the varying conditions within

the packing, the efficiency changes, resulting in m ism atch between the two m ethods. The

short-path distillation column which has h itherto only been modelled at steady-state, is

modelled using a dynamic rate based model, essential for investigating control.

Having developed the dynamic models, the control and controllability of these reactive

distillation processes are examined. General control properties of reactive batch distillation

are discussed and m ethods are presented for applying linear controllability tools to these

non-linear process models. The linear models are then employed to dem onstrate the

implications for control when adopting one of the three processes.

A cknow ledgem ents

I would like to thank my supervisor, Dr. Eva Sprensen, for her guidance and encouragement

througout the course of this work. I would also like to thank the students and staff, past

and present, of the Com puter Aided Process Engineering group for the lively and useful

discussions and the occasional, but necessary, excursions to Huntley St. The financial

support from the Engineering and Physical Sciences Research Council and the Centre for

Process Systems Engineering is gratefully acknowledged. Finally, I would like to thank

Anna for her constant love and support.

C ontents

A b strac t 2

A ck n o w led g em en ts 3

List o f figures 10

List o f tables 14

1 In trod u ction 15

1.1 M otivation ....................................................................................................................... 15

1.2 Reactive distillation ...................................................................................................... 16

1.3 Batch column o p e r a t io n ................................................................................................ 17

1.3.1 Control of reactive batch d is t i l la t io n ............................................................ 18

1.4 Short-path column o p e r a t io n ..................................................................................... 19

1.4.1 Falling film e v a p o ra to r s .................................................................................... 19

1.4.2 W iped him e v a p o r a to r s ........................................................................................20

1.4.3 Short-path d is t i l la t io n ........................................................................................... 20

1.5 Objectives of this w o rk ....................................................................................................... 21

1.6 Outline of this t h e s i s .......................................................................................................... 22

1.7 Main c o n tr ib u tio n s ..............................................................................................................23

4

C O N T E N T S 5

2 L iterature r e v iew 25

2.1 Modelling of reactive batch d is t i l la t io n ........................................................................ 25

2.1.1 Introduction .......................................................................................................... 25

2.1.2 Reactive batch distillation l i t e r a tu r e .................................................................26

2.1.3 Reactive batch distillation c o n c lu s io n s ............................................................. 40

2.2 Modelling of short-path d is t i l la t io n ...............................................................................41

2.2.1 In troduction .......................................................................................................... 41

2.2.2 Short-path l i t e r a tu r e .............................................................................................41

2.2.3 Short-path conclusions......................................................................................... 47

2.3 C o n c lu s io n s ...........................................................................................................................48

3 M odell in g o f R B D in tray co lum ns 49

3.1 Tray column m o d ellin g ...................................................................................................... 49

3.1.1 Rigorous m o d e l .......................................................................................................50

3.1.2 A ssum ptions .......................................................................................................... 51

3.1.3 Initial c o n d it io n s ................................................................................................... 51

3.1.4 In tegration ............................................................................................................. 52

3.1.5 Simplified m o d e l ................................................................................................... 53

3.2 Comparison between simphfied and rigorous models ............................................ 53

3.2.1 E thyl acetate case s t u d y ......................................................................................53

3.2.2 Case study r e s u l t s ................................................................................................56

3.3 C o n c lu s io n s .......................................................................................................................... 60

C O N T E N T S 6

4 M od ell in g o f R B D in packed co lum ns 64

4.1 Rate-based modelling of packed c o lu m n s .....................................................................65

4.1.1 Modelling of mass and energy t r a n s f e r ............................................................. 65

4.1.2 Modelling of hydrodynam ics................................................................................66

4.1.3 Modelling of chemical r e a c tio n s .........................................................................67

4.1.4 Packed column m o d e l l in g ....................................................................................68

4.2 Reactive batch distillation case s tu d y ........................................................................... 69

4.2.1 Column d e s i g n .......................................................................................................69

4.2.2 Column operation ................................................................................................70

4.2.3 Effect of d isc re tisa tio n ......................................................................................... 71

4.2.4 D eterm ination of H E X ? ................................................................................. 72

4.3 Comparison between rate based and equilibrium m o d e l s ......................................80

4.4 Conclusion .......................................................................................................................... 86

5 C ontro l o f react ive batch colum ns 88

5.1 In tro d u c tio n .......................................................................................................................... 89

5.1.1 Control of batch distillation c o lu m n s ..............................................................91

5.2 Controllability m e t h o d s .................................................................................................. 94

5.2.1 Simulation controllability analysis ................................................................. 94

5.2.2 Frequency response controllability analysis ................................................ 97

5.3 M ethods for controllability a n a ly s is ............................................................................ 102

5.3.1 M ethod for simulation controllability a n a ly s is .............................................102

5.3.2 M ethod for frequency response controllability a n a ly s i s ......................... 102

5.3.3 R obust method for linearisa tion ......................................................................103

C O N T E N T S 7

5.4 Controllability of batch distillation c o lu m n s .............................................................. 106

5.4.1 Case s tu d ie s ............................................................................................................106

5.4.2 Scaling .................................................................................................................. 107

5.4.3 Linear m o d e ls ........................................................................................................ 107

5.4.4 Frequency response based con tro llab ility ......................................................110

5.4.5 Controller t u n in g ..................................................................................................I l l

5.4.6 Non-linear model s im u la tio n s ..........................................................................112

5.5 Effect of r e a c t io n ............................................................................................................... 113

5.5.1 Simulation c o n tro l la b il i ty ................................................................................ 114

5.5.2 Frequency response controllability ................................................................114

5.6 C o n c lu s io n s ...........................................................................................................................115

6 M od ell in g and control o f short-path co lum ns 124

6.1 In tro d u c tio n .......................................................................................................................... 124

6.2 Modelling of short-path evaporators ........................................................................... 125

6.2.1 Modelling of him p h e n o m e n a ......................................................................... 126

6.2.2 Modelling of evaporation p h en o m en a ............................................................128

6.3 Dynamic short-path distillation m o d e l ........................................................................130

6.3.1 Modelling a ssu m p tio n s .......................................................................................131

6.3.2 Boundary and initial co n d itio n s......................................................................134

6.3.3 Numerical s o lu t io n ............................................................................................. 134

6.4 Case study .......................................................................................................................... 134

6.4.1 Effect of d isc re tis a tio n .......................................................................................136

6.4.2 Effect of feed h o w r a t e .......................................................................................137

C O N T E N T S 8

6.4.3 Effect of v iscosity ................................................................................................... 139

6.4.4 Effect of feed te m p e ra tu re .................................................................................. 140

6.4.5 Effect of heat of re a c tio n ..................................................................................... 141

6.4.6 Effect of e f f ic ie n c y ............................................................................................... 141

6.5 Control of short-path d is t i l l a t io n .................................................................................... 142

6.5.1 L in e a r is a t io n .........................................................................................................143

6.5.2 Frequency a n a ly s is .............................................................................................. 143

6.5.3 Controlled response ........................................................................................... 144

6.6 Conclusion ...........................................................................................................................145

7 C onclusions and d irections for future work 149

7.1 C o n c lu s io n s ...........................................................................................................................149

7.1.1 Modelling of reactive batch distillation in tray columns ........................150

7.1.2 Modelling of reactive batch distillation in packed c o lu m n s .................... 151

7.1.3 Control of reactive batch d is t i l la t io n ..............................................................152

7.1.4 Modelling and control of reactive short-path e v a p o ra to rs ....................... 153

7.2 Directions for future w o r k ................................................................................................154

7.2.1 Model v a l id a t io n ................................................................................................. 154

7.2.2 Modelling detail ................................................................................................. 154

7.2.3 Further control s tu d ie s ....................................................................................... 155

N o m e n c la tu r e 157

B ib liography 160

C O N T E N T S 9

A P r o c ess M od els 166

A .l Equilibrium tray model .................................................................................................166

A .2 Rate-based model of packing s e c t i o n ......................................................................... 169

A .3 Reboiler m o d e l ..................................................................................................................173

A.4 Condenser m o d e l .............................................................................................................. 174

A.5 Reflux drum m o d e l ...........................................................................................................175

A.6 Accum ulator m o d e l.......................................................................................................... 177

B Linearisation and scaling m eth od s 178

B .l Linearisation of model e q u a t io n s ............................................................................... 178

B.2 Scaling of the linear m o d e l s ..........................................................................................179

List o f F igures

1.1 Reactive batch distillation column ........................................................................... 18

1.2 W iped-Film evaporator ................................................................................................... 20

1.3 Short-path e v a p o r a to r .......................................................................................................21

3.1 Distillate composition (top) and distillate how rate (bottom ) for the constant

rehux ratio study .............................................................................................................. 58

3.2 Accum ulator composition (top) and holdup (bottom ) for the constant rehux

ratio study ............................................................................................................................59

3.3 Distillate composition (top) and distillate how rate (bottom ) for controlled

composition s t u d y .............................................................................................................. 60

3.4 Accum ulator composition (top) and holdup (bottom ) for controlled compo

sition s t u d y ........................................................................................................................... 61

3.5 Distillate how for non-reactive system ................................................................... 62

3.6 Reboiler forward reaction ra te and ethyl acetate composition (Rigorous

Model) .............................................................................................................................. 62

3.7 Distillate composition (top) and distillate how rate (bottom ) for controlled

composition study with reboiler heat input disturbance ( b o t t o m ) ..................... 63

4.1 Mass and energy transfer between phases ............................................................ 66

4.2 Vapour composition of Ethyl A cetate at 3 hrs (constant rehux ratio) . . . 74

10

LIST OF FIGURES 11

4.3 Composition and Distillate Flowrate - (TO P Constant reflux ratio - B O T

TOM Controlled C o m p o s it io n ) .................................................................................. 77

4.4 H ETP packing profile against tim e for Reactive Case Study (C onstant reflux

ratio) ................................................................................................................................. 78

4.5 Mean HETP profile for Reactive Case Study (C onstant Reflux Ratio) . . . 78

4.6 H ETP packing profile against time for Reactive Case Study (Controlled) . . 79

4.7 Mean H ETP profile for Reactive Case Study (C o n tro l le d ) .............................. 79

4.8 Distillate composition (top) and distillate flowrate (bo ttom ) for constant

reflux ratio p o lic y ............................................................................................................. 84

4.9 Distillate composition (top) and distillate flowrate (bo ttom ) for controlled

composition p o l i c y ......................................................................................................... 84

4.10 Distillate composition (top), distillate flowrate (middle) and reboiler dis

turbance profile (bottom ) for controlled composition policy ................................85

5.1 Process for c o n t r o l ......................................................................................................... 89

5.2 Batch Column ............................................................................................................... 91

5.3 Frequency response of a first order s y s te m ............................................................. 98

5.4 Block diagram of feedback control .......................................................................... 99

5.5 Controllability req u irem e n ts .......................................................................................... 101

5.6 Comparison of robust m ethod(Simple) to standard m ethod(Rigorous) . . . 104

5.7 Internal reflux ratio p r o f i le s ..........................................................................................106

5.8 Distillate composition response to step change in i n p u t s ................................... 109

5.9 Reboiler tem perature response to step change in inputs ................................... 110

5.10 Distillate composition response to unit step change in reflux f l o w .................. 116

5.11 Tray column frequency response of distillate composition to reflux flow at

7.5 h r s ................................................................................................................................. 117

L IS T OF FIGURES 12

5.12 Packed column frequency response of distillate composition to reflux flow

at 7.5 h r s .............................................................................................................................117

5.13 Frequency response of distillate composition to reflux flow for 10 tray column 118

5.14 Frequency response of distillate composition to reflux flow for 8m packed

co lu m n ....................................................................................................................................118

5.15 Tray column closed loop response to setpoint change (Linear model) . . . . 119

5.16 Packed column closed loop response to setpoint change (Linear model) . . 119

5.17 M agnitude composition response (10 tray column) ...............................................120

5.18 M agnitude composition response (8m packed column) ........................................ 120

5.19 Composition controller error in response to 10% step increase in reboiler

heat duty (Rigorous Tray C o lu m n ) ..............................................................................121

5.20 Composition controller error in response to 10% step increase in reboiler

heat duty (8m Packed Column) at 3 hours .............................................................121

5.21 Distillate composition response of reactive and non-reactive tray columns . 122

5.22 Reboiler tem perature response of reactive and non-reactive tray columns . . 122

5.23 Composition frequency response of reactive and non-reactive tray columns . 123

6.1 Short-path e v a p o ra to r ..................................................................................................... 126

6.2 Top view of mixing bow wave (adapted from M cKenna 1 9 9 5 ) ........................ 127

6.3 Tem perature profile in a wiped film evaporator (Lutisan et al 2 0 0 2 ).............. 128

6.4 Cross section of E v a p o r a to r ........................................................................................... 131

6.5 Reaction S c h e m e ............................................................................................................... 135

6.6 Base case profiles: compositions (top), tem peratu re ( b o t t o m ) .......................... 136

6.7 O utlet Tem perature profile resulting from disturbance (T O P ) Steady-state

tem peratu re profile ( B o t to m ) ........................................................................................137

L I S T OF FIGURES 13

6.8 Effect of feed flowrate on residence t i m e ............................................................. 138

6.9 Effect of feed flowrate on reactor y i e l d ................................................................ 139

6.10 Effect of film viscosity on reactor y i e l d ................................................................ 139

6.11 Effect of viscosity on residence time .................................................................... 140

6.12 Effect of feed tem perature on reactor y ie ld ..........................................................140

6.13 Evaporator composition profile with no separation of V .....................................142

6.14 Product, B, composition step responses. Non-linear model (T O P ), linear

model (B O T T O M )............................................................................................................ 146

6.15 Short-path column frequency response of product B composition to feed flow 147

6.16 Short-path column frequency response of controller loop and tem perature

d is tu r b a n c e ......................................................................................................................... 147

6.17 Column composition response to a set-point change (controlled scaled linear

m o d e l) ....................................................................................................................................148

6.18 Column composition response to a feed tem perature disturbance (controlled

scaled linear model) .........................................................................................................148

A .l Sieve t r a y ............................................................................................................................ 166

A.2 Packing s e c t io n ..................................................................................................................169

A.3 R e b o i le r ................................................................................................................................173

A.4 C o n d e n se r ............................................................................................................................ 174

A.5 Reflux Drum ..................................................................................................................... 176

List o f Tables

2.1 Summary of papers on reactive batch distillation 1979 to present day . . . . 38

2.2 Summary of papers on reactive batch distillation 1979 to present day (con

tinued) 39

2.3 Summary of papers on short-path d i s t i l la t io n ............................................................46

3.1 Column p a ra m e te rs ............................................................................................................... 56

3.2 Controller p a r a m e te r s ........................................................................................................ 57

3.3 Accum ulator holdup and batch t im e s ............................................................................. 63

4.1 Column dimensions and packing c h a ra c te r is tic s ......................................................... 70

4.2 Comparison between level of discretisation for 8m packed c o lu m n ....................... 72

4.3 Comparison between modelling approaches. (EQ: equilibrium model, NEQ:

rate based model) ..............................................................................................................83

5.1 Control S c h e m e s .................................................................................................................. 93

5.2 Inputs and outputs for linear model (6 Tray column at 3 hrs Un. point) . . 107

5.3 Controller tuning p a r a m e te r s .........................................................................................I l l

5.4 Integrated controller errors ............................................................................................ 113

6.1 Short pa th column configuration ..................................................................................135

14

C hapter 1

Introduction

This thesis is concerned with the modelling and control o f reactive distillation

in tray and packed batch columns as well as in short-path columns. Reactive

distillation offers advantages over separate reaction and separation steps, for

instance through improved yield, as volatile products are removed from the re

action zone. However, by combining these processes, control is made more

difficult. In this chapter, reactive distillation is introduced with its advantages

and disadvantages. Then, more specifically, batch operation in packed and tray

columns and operation in short-path evaporators, is introduced. General com

m ents are made about the control o f these processes. The thesis motivations,

objectives and contributions are outlined. The outline o f the rest o f the thesis

is also presented.

1.1 M otivation

This thesis is concerned with the modelling and control of reactive distillation. Due to

the complexity of combined reaction and separation and the operational constraints on

these processes such as reaction tem perature, they may be difficult to control in practice

(Sprensen and Skogestad, 1994). In this work, reactive distillation in tray and packed

columns as well as short-path columns are considered. For batch columns, the changing

15

C H A P T E R 1. IN T R O D U C T IO N 16

process conditions with time adds a extra dimension to the complexity of the control. For

the short-path column, typically used for the treatm ent of tem pera tu re sensitive products,

tem perature control is extremely im portant. In order to study the control of these pro

cesses, it is necessary to develop rigorous dynamic models which accurately capture the

behaviour of the processes.

1.2 R eactive d istillation

Reactive distillation is the combination of both reaction and separation into a single unit.

This can offer particular advantages as reported in the litera tu re by Taylor and Krishna

(2000), Doherty and Buzad (1992) and others:

• Combining two units into one can lead to significant capital savings

• Improved conversion of reactants, approaching 100% as volatile products are removed

from the reaction zone

• Low product concentrations in the column section, reducing unwanted side reactions

and leading to higher selectivity

• Azeotropes which would otherwise be formed by the reac tan ts/p roducts can be elim

inated through reaction

• Lower reboiler duty with exothermic reactions, as the heat of reaction assists in the

vaporisation

However, caution should also be taken when considering reactive distillation for the fol

lowing reasons:

• The equilibrium enhancements offered by reactive distillation relies on one of the

products being the most volatile component in the system , so th a t it is preferentially

removed from the reaction zone.

C H A P T E R 1. IN T R O D U C T I O N 17

• If the reaction is slow, it may be more economical to carry out the operation in

separate reaction and separation steps as a large column with large reflux wiU be

required w ith large capital and operating costs to achieve the required residence

time.

• The coupling of reaction and separation in one unit can result in a mismatch between

ideal process conditions. The optim al tem perature and pressure for the reaction may

be very different to th a t for the separation. This is also an acute problem in packed

columns where the selection of the packing is often a compromise between separation

and reaction performance.

1.3 B a tch co lu m n operation

In the m anufacture of low volume, high value chemical products, and in situations where

varying specifications of different products are required, the m otivation for employing

batch operations is well known. Reactive batch distillation can be carried out in a column

such as th a t shown in Figure 1.1. Depending on the volatilities of the reactants compared

to the products, the reaction may occur throughout the column or be confined to the

reboiler. The separation section can be either a tray stack or a packing section. The

choice depends on the operating conditions, although packed columns are particularly

suitable if the reaction is to be heterogeneously catalysed in the packing. The reactants

may be initially charged to the reboiler or one or more may be fed during semi-batch

operation. A homogeneous catalyst, such as a concentrated mineral acid, may also be

fed or charged initially. As with non-reactive distillation, reactive columns are normally

operated at to ta l reflux until a steady-state profile is developed before distillate withdrawal

is started . Several production cuts may be made during the operation which may or may

not be recycled.

C H A P T E R 1. I N T R O D U C T IO N 18

Condenser

Reflux Drum

Reflux Distillate

DistillationColumn

Accumulator

iX M " Heat Supply

Reboiler

Figure 1.1: Reactive batch distillation column

1 .3 .1 C o n tr o l o f r e a c t iv e b a tc h d i s t i l la t io n

The coupling of reaction and distillation processes creates a more complex process which

is more difficult to control than either one on its own. In general, the use of autom atic

control systems in batch systems is fairly limited, there are a num ber of reasons for this:

• It is quite difficult, due to the transient nature of the process, to control the unit

based only on regulatory or tracking control of certain variables. The controllers

need to be adjusted according to the current s ta te of the process with the aim of

achieving the desired sta te at the end of the batch.

• The desired sta te at the end of the batch and the performance of the process will

change with different charges and process requirem ents which further complicates

the procedures


$ There can be difficulties in observing some sta te variables, e.g. composition, and if

they are to be controlled they may have to be inferred from quantities th a t are more

easily measured, e.g. tem perature.

1.4 Short-path co lum n operation

It is often not possible to heat many organic compounds to a tem peratu re even close

to their normal boiling points without therm al decomposition occurring. The degree of

decomposition will also depend on the length of time the compounds are exposed to the

heat source. Vacuum distillation enables the separation of these kinds of compounds by

keeping the pressure and hence tem perature low. Traditional batch distillation is some

times unsuitable for vacuum distillation since the pressure drop across the column limits

the am ount of vacuum achievable in the still. A minimum pressure of around 50 mbars

(Erdweg, 1983) can be achieved under ideal conditions. Additionally, there are large res

idence times within the still, which increases therm al decomposition. This has been the

motivation for developing other distillation/ evaporation processes. These include falling

film evaporators, wiped-film evaporators and short-path evaporators.

1 .4 .1 F a l l in g f i lm e v a p o r a to r s

Falling film evaporators are used successfully in many industries. The feed flows down

heated walls forming a film. W hen operated under vacuum and high evaporation rates,

a num ber of problems can occur. Hot spots form where m aterial overheats resulting in

decomposition. The lam inar film restricts the distillation rates and large tem perature

difference can occur in the film.

CH A F T E R 1. I N T R O D U C T I O N 20

FEED VAPOUR T O EX TERN A L C O N D EN SER A N D

V A CET'M PUM P

W IPIN G SYSTEM

RESID U E

I ' i gu re 1.2: W ip e d - F i l i n e v a j i o r a t o r

1 .4 .2 W ip e d film e v a p o r a to r s

In a wi])e(l-lilni e v a p o r a t o r , sho w n in F ig ur e 1.2, t h e p r o d u c t is fed t o t h e inside o f a

s ingle t u b e an d a inecha ii ical , r o t a t i n g , w ip e r sp r e a d s a n d mo ve s t h e p r o d u c t s , avo i d ing

hot sp o t s . T h e m o r e volat i le c o m p o n e n t s gen er a l l y run a g a i n s t t h e p r o d u c t flow, l eav ing

t h e e v a p o r a t o r t o an e x t e r n a l c o nd e n se r . T h e p re ss u re d r o p b e t w e e n t h e e v a p o r a t o r an d

t h e e x t e r n a l c o n d e n s e r d e t e r m i n e s t h e degre e o f v a c u u m ac h i e va b le in th i s t y p e o f un i t .

1 .4 .3 S h o r t -p a th d is t i l la t io n

S h o r t - p a t h e v a p o r a t o r , sho wn in F ig u r e 1.3, is in p r incip le a w iped- f i lm e v a p o r a t o r wi th

an i n te r n a l c o n d e n se r . T h i s e l i m in a te s t h e p r es su r e d r o p a s s o c ia te d w i th t h e p i p e w o r k

c o n n e c t i n g t h e evaporator a n d t h e co n de n se r . In t l ieory, t h e r e is no p r e s s u r e d r o p b e t w e en

th e e v a p o r a t i n g su r fa c e a n d t h e c o n d e n s i n g su r fac e be c a u s e t h e d i s t i l l a t ion g a p is of th e

s a m e o r d e r of m a g n i t u d e as t h e m e a n free p a t h of th e e v a j i o r a t i n g molecu les a t t h e low

o p e r a t i n g p ress u r e .

21

FEED

CONDENSER

HEATING JACKET

WTPING SA'STCM

— COOLING

DISTILLATERESIDUE

Figu re 1 .X: S i ior i- j ia t l i e v a p o r a t o r

1.5 O bjectives of this work

R e ac t i v e b a t c h (listi l latioii is an in l i e rent ly d y n a m i c p roce ss s ince a m o u n t a n d c o m po s i t i o n

o f m a t e r i a l w i t h in t h e c o l u m n ch an g es w i th t i m e as t h e r e a c t i o n p r o c e e d s a n d p r o d u c t

is w i t h d r a w n . As a r e su l t , a d y n a m i c m a t h e m a t i c a l m o d e l is r e qu i r ed to desc r ibe its

o p e r a t i o n . C o n t i n u o u s d i s t i l l a t ion , sucli as s h o r t - p a t h d i s t i l l a t io n , sh ou ld be d y n a m i c a l l y

m o de l l e d if t h e m o d e l is to be used for con t ro l .

T h e o b je c t i v e o f th i s thesi s is to in v e s t ig a te t h e co n t ro l o f r ea c t iv e d i s t i l l a t ion in ba tc h

c o l u m n s , t r a y a n d packed , a n d in sh o r t p a t h co lu m ns . C o n t r o l h a s h i t h e r t o on ly been

s t u d i e d for r ea c t iv e b a t c h d is t i l la t ion in t r a y c o lu m ns , a n d t h e n o n l y e m p l o y i n g simpl i f ied

d y n a m i c mo d e l s w i t h l inear t r a y d y n a m ic s . In o r d e r to s t u d y c o n t r o l fully, it is nec es sa r y

to deve lop r ig oro us d y n a m i c m od e l s t h a t a c c u r a t e l y c a p t u r e t h e p roce ss b e h a v io u r . For

b a t ch p rocesses in p a r t i c u la r , p rocess c o nd i t io n s c h a n g e w i t h t i m e which will affect th e

co n t ro l l ab i l i tv a n d it is th e re fo r e es sent i al t h a t t h e c o n d i t i o n s a r e a c c u r a t e l v mode l l ed .


T he higher the degree of rigour, the more accurately the process conditions and dynam

ics are captured. However, more rigorous models are more com putationally expensive to

implement and they require more param etric da ta which may be unavailable or uncer

tain . Additionally, more numerically complex models can be prone to numerical errors,

introduced for instance during hnearisation.

Therefore, in this thesis, the degree of modeUing rigour required for both simulation and

control purposes is explored in detail for tray and packed batch columns and short-path

columns.

For batch tray columns, it is dem onstrated th a t to accurately capture the change in

process conditions, it is necessary to model pressure dynamics and employ a dynamic

energy balance. For packed columns, distributed rate based modelhng is compared to

lumped equilibrium modelling and it is found th a t due to the varying conditions within

the packing, the efficiency changes, resulting in m ismatch between the two m ethods. The

short-path distillation column which has h itherto only been modelled at steady-state, is

modelled using a dynamic ra te based model, essential for investigating control.

Having developed the dynamic models, the control and controllability of these reactive

distillation processes are examined. General control properties of reactive batch distillation

are discussed and methods are presented for applying linear controUabihty tools to these

non-linear process models. The linear models are then employed to dem onstrate the

implications for control when adopting one of the three processes.

1.6 O utline o f th is thesis

C hapter 2 presents a review of the literature on modelhng and control of reactive batch

distillation in both tray and packed columns. The modelhng of short-path distillation

columns is also considered. It is concluded th a t little work has been undertaken on the

control of reactive batch distiUation in tray columns and no work has been undertaken

on the control of reactive distillation in batch packed columns and short-path distillation

columns.

C H A P T E R 1. I N T R O D U C T I O N 23

As already noted, rigorous dynamic models are required to enable the study of control.

C hapters 3 and 4 deal with the development of rigorous models to describe reactive dis

tillation in batch tray and packed columns. The tray column is compared to a shghtly

simplified, more numerically robust column for a num ber of different operating policies.

As packed columns are commonly modelled using equilibrium models, such as th a t for

the tray column, the two modelling approaches are compared. In order to determine the

equivalent tray column, the Height Equivalent to a Theoretical Plate (H E T P) needs to be

established and a m ethod is presented for extracting this inform ation from the column

packing.

Chapter 5 is concerned with the control of tray and packed batch columns. In order to

use linear control tools it is necessary to generate a linear approxim ation to the non-linear

models. The rigorous tray column model proves to be numerically unstable for this purpose

and an alternative scheme is presented for generating the necessary linear information from

the simplified column model. The effect of the choice of column, packed or tray, the effect

of reaction and the size of the column are investigated during the controllability study.

C hapter 6 considers the modelling and control of reactive distillation in short-path distilla

tion columns. A dynam ic short-path column is presented and used to investigate the effect

of changes in operation on a complex, industrially m otivated, reaction. The composition

control of this process is also investigated.

Finally, in chapter 7, overall conclusions are drawn, and some possible directions for future

work are outhned.

1.7 M ain con tributions

The m ain contributions of this thesis are th a t a rigorous, equilibrium, tray column model

and a rigorous, rate-based, packed column model for reactive batch distillation are devel

oped. A m ethod is presented to determine the Height Equivalent to a Theoretical Plate

(H E T P) from the packed column model and used to analyse how efficiency changes during

operation. The rigorous tray column model proves to generate numerical difficulties when

C H A P T E R 1. I N T R O D U C T I O N 24

linearised. Therefore, a m ethod is presented for coping with these difficulties by means

of a simplified version of the model. A comparison is made between the controllability

of packed and tray batch columns. Finally, a dynamic model for reactive distillation in a

short-path column is presented and used to investigate the controllabhty of the process.

C hapter 2

Literature review

In this chapter, the work that has previously been undertaken within the area

of modelling and control o f reactive hatch distillation and the modelling and

control of short-path distillation is reviewed. It is concluded that, while some

work on the control o f batch tray columns has been done, the models used have

been simple. No work has been undertaken on the control o f packed columns or

short-path columns. It is also noted that no work has been done on modelling

reaction in short-path columns.

2.1 M od elling o f reactive batch d istillation

2.1.1 Introduction

Reactive distillation can be operated in both continuous and batch modes of operation.

As is generally the case, continuous operation is best suited for large production volumes

but, since the continuous design is tailored to a particular reaction /separation system, it

lacks the flexibility afforded by batch operation. Control of both modes is complex but

the batch mode offers additional challenges. In the continuous mode, control is generally

required about a desired steady-state when not considering s ta rt-up or shut-down. Batch

control, on the other hand, is inherently dynamic which adds an ex tra dimension to its

25

C H A P T E R 2. L I T E R A T U R E R E V I E W 26

complexity. The process is not controlled to a desired steady s ta te bu t is controlled to a

desired operating policy which may, and often does, change with tim e. The process itself

changes w ith tim e, the volume in the reboiler is decreasing and the composition profile

in the column changes. Consequently, the response of the process and its control system

to disturbances wiU change with time. Efficient control is im portan t, as minimising the

impact of disturbances reduces batch inconsistencies and hence wastage. Good set-point

tracking of an optim al operating policy ensures more economic operation. In this thesis,

the focus is therefore on investigating the dynamic behaviour and the controllability of

these processes as a more detailed understanding will lead to be tte r control. In this

section, the literatu re th a t has been published on reactive batch distillation (summarised

in Tables 2.1 and 2.2) is examined to identify what work has been undertaken on the

modelling and control of these processes.

2 .1 .2 R e a c t i v e b a t c h d i s t i l la t io n l i t e r a t u r e

Egly et al. (1979) developed a m ethod for optimising the operating policies of batch distilla

tion operations. The simple model used consisted of component mass and energy balances

and a vapour-liquid equilibrium equation. Reaction could be considered throughout the

liquid phase in the column and the still. The holdup of liquid in the trays and condenser

were assumed to be constant and no pressure dynamics were considered. Optimisations

were performed using a modified conjugated gradient m ethod, minimising the batch time

while m aintaining product specifications in term s of yield and composition. Optimisations

considered were constant reflux ratio, tim e variable reflux ratio and tim e variable reflux

ratio w ith feed of a reactan t. This was applied to a theoretical reactive case study with

reaction confined to the still. The optim al, shortest batch tim e, was found for the time

varying reflux ra tio policy with feed of reactant which was 40% shorter than the constant

reflux ratio policy. The authors also developed a non-linear, m ulti-variable control algo

rithm th a t determ ined the required reflux ratio at any instan t from tem peratures in the

column.


Cuille and Reklaitis (1986) considered the simulation of batch distillation with and w ith

out liquid phase chemical reaction in the reboiler. The model was a system of differential

and algebraic equations and the authors discussed the numerical problems associated with

these systems and strategies for tackling these. The system was initialised at steady-state

with to ta l reflux and no reaction and integrated using G ear’s m ethod. C onstant volumet

ric holdup on the trays was assumed and tray efficiencies were included. The reaction

considered was an equilibrium estérification of 1-propanol and acetic acid. The rate of

reaction was simplified by assuming no tem perature dependence. However, the reaction

is not practical for reactive batch distillation since 1-propanol, a reactan t, is the most

volatile component in the system and is hence removed preferentially. The authors com

pared the non-reactive separation of cyclohexane and toluene with experim ental data. The

agreement was good for the distillate composition profiles for a num ber of constant reflux

ratio simulations. However, there were no comparisons m ade for reactive batch distillation.

Reuter et al. (1989) considered the modelling of reactive distillation with control systems.

The model considered non-ideal stages and variable pressure. Assum ptions included: con

stan t liquid hold-up on the trays and in the condenser and reaction only in the liquid phase.

They considered an equilibrium transestérification reaction, although no details were given

on either the com ponents or the reaction. The model was s ta rted from steady sta te , to ta l

reflux with no reaction and the steady sta te profile was calculated using Newton-Raphson

procedures. The dynamic simulation was performed using a modified relaxation m ethod.

The control scheme employed consisted of three controllers: Condenser cooling was used

to control the condenser outlet tem perature, reboiler steam flow for column pressure drop

and distillate ra te for the tem perature at the top of the column. The authors compared

both controlled and uncontrolled operation of the column but did not evaluate the two

cases for disturbances. In the controlled case, the flow ra te of the equilibrium limiting

by-product was greater than th a t for the uncontrolled case, hence the reaction was faster.

Comparisons were m ade with experimental results and it was concluded th a t in practice.


the reflux ratio set by the controller in the simulation was always less th an in the exper

im ent when m aintaining a constant distillate composition. This suggest th a t the model

agreem ent is not very good, although this is diflficult to verify from the results.

Sprensen and Skogestad (1994) investigated control strategies for reactive batch distil

lation. They indicated th a t most authors had considered optim al control in order to

maximise profit or minimise batch time. However, they argued th a t, in some cases, it may

be a more im portant control objective to m aintain product consistency between batches.

They considered the modelling of an industrial esteriflcation process where the reaction

was limited to the reboiler. The reaction produced a separate polymer phase and water,

the m ost volatile component in the reaction m ixture. Modelling assum ptions included:

perfect mixing and equilibrium between the liquid and vapour phases, constant pressure,

negligible vapour holdup, constant liquid enthalpies, linear tray hydraulics, to ta l condensa

tion w ithout sub-cooling in the condenser, R aoult’s law for the liquid vapour equilibrium,

perfectly controlled vapour holdup and imm ediate heat input. The authors highlighted the

differences between non-reactive and reactive distillation by applying typical non-reactive

operating policies to the reactive example. The open loop policies: constant reflux ra

tio, R, and tim e varying reflux ratio, R (t), were found to be inappropriate, although

there was acceptable separation, as the reactor tem peratu re varied too greatly under dis

turbances, resulting in varying polymer composition between batches. Only a constant

product composition pohcy, implemented using feedback control, m aintained the reactor

within permissible tem perature limits. The authors considered the controUabihty of the

process by generating hnear models at different times during the batch. The linerised

models were used to investigate the controllability by means of step responses and RGA

analysis. They identified the reactor tem perature and the distillate composition as being

the m ost im portan t variables to control. They also concluded th a t the same amount of side

product being formed should be removed in the distillate a t any tim e, th a t the system ’s

response varied with time as the conditions in the colum n/reactor change and th a t the

trays had different and varying sensitivity to change and th a t the responses were generally


non-linear. The authors evaluated a series of control schemes: one point bottom control

(reactor tem peratu re directly), two point control (controlling both distillate composition

and reactor tem perature) and one point column control (controlling the tem perature on a

tray in the column). It was concluded th a t the la tte r offered good control of the process

with disturbances and did not have the disadvantages of the interactions encountered with

two point control.

S0rensen et al. (1996) addressed the issues of optim al control of the same case study as

S0rensen and Skogestad (1994). They developed optim al profiles for the operating vari

ables, assessed the controllability properties at optim al conditions, designed controllers

to implement the optim al profiles and verified the stability and control performance. A

series of optim isations were performed for maximum profit w ith and w ithout raw m aterial

costs and for minimum batch time. The im plem entation was performed using the tem

perature on a column tray to control reflux while m aintaining the heat to the reboiler

at the optim um value. The tray selected was the one th a t gave the largest response,

identified in the controllability analysis. The stability of the controlled and uncontrolled

model were assessed by introducing disturbances in the reboiler heat supply and in the

reaction param eters which were used to indicate uncertainty in those param eters. The

uncontrolled case gave significant variations in the reboiler tem pera tu re and breakthrough

of the volatile reac tan t into the distillate. In the controlled case, the controllers were

tuned using a linerised model about a series of operating points and a polynomial was

used to describe th e tem peratu re set point profile. This yielded significant deviations in

reboiler tem pera tu re only towards the end of the batch. The model was also interfaced

to an industrial real-tim e control system and the controllers were implemented using the

systems facilities. R ather than a polynomial description of the set points, a series of set

points were used. This yielded good performance, similar to the continuous controller.

M ujtaba and M acchietto (1997) considered a com putationally efficient m ethod for optim is

ing the operation of reactive batch distillation for m aximum profit. They indicated th a t


this operational condition is strongly dependent on the cost param eters which in tu rn are

dependent on m arket forces. They pointed out th a t to perform a rigorous optim isation

procedure, involving integration of the model equations such as th a t used by Sqrensen

et al. (1996), would be too com putationally expensive to perform every tim e m arket con

ditions change. The authors proposed a less com putationally expensive m ethod th a t used

polynomial approxim ations to estim ate the optim al operating policy from previously per

formed optim isations. Their example was the esteriflcation of ethanol and ethanoic acid

to produce ethyl acetate and water and they used simple models for the VLE and did not

consider azeotropes such as th a t formed between ethyl e thanoate and ethanol. However,

they indicated th a t the m ethodology is general and could be applied to more complex sys

tem s such as azeotropic m ixtures by employing more rigorous Vapour-Liquid equilibrium

models. The model includes reaction throughout the whole column in the liquid phase

and assumes constant molar holdup on the plates and condenser. The energy balance is

algebraic and hence assumes no change in liquid enthalpies. This is simpler than th a t

used by Sprensen and Skogestad (1994) and Sprensen et al. (1996) as the tray holdup is

assumed to be constant which is reasonable as they are not considering dynamics and the

column would rem ain in pseudo steady-state during a large portion of the batch. The

authors asserted th a t in reactive distillation where one of the products, desired or unde

sired, is the most volatile component, finding the maximum yield is equivalent to finding

the maximum production of distillate. This, they indicated from previous work, is equiv

alent to the m axim um profit for fixed batch time. A series of optim um constant reflux

ratios were calculated for maximum product yield for a series of fixed batch times and for

two product specifications using rigorous non-linear program m ing techniques (NLP). The

results to these optim isations were used to develop polynomial descriptions of maximum

conversion, optim um distillate, optimum reflux ratio and to ta l reboiler heat load. These

were all functions of batch time. These were then used in the form ulation of a maximum

profit optim isation which was an algebraic optim isation in only one variable, time. O pti

misations performed using this m ethod were approxim ately 200 tim es faster than using the

rigorous m ethod. They indicated th a t accuracy was very good and th a t this was mostly


due to the accuracy of the regression of the polynomials to the initial optim isation data

which behaved well, justifying their use of low order polynomials.

W ajge et al. (1997) examined the accuracy and speed of numerical m ethods for simulating

both reactive batch distillation and non-reactive distillation in packed columns. They in

dicated th a t packed columns models differ from tray model columns in th a t mass transfer

effects need to be considered. They considered the finite difference technique which in

volves converting the differential equations to algebraic equations of small intervals. This

m ethod was used for the simulation of batch distillation and it was concluded th a t the

finite difference m ethod is very com putationally expensive. They considered an orthogonal

collocation m ethod where the equations are approxim ated to polynomials. They concluded

th a t this was more efficient but the com putational advantages over finite element methods

were lost as the need for greater accuracy required the use of higher order polynomials.

They proposed a hybrid m ethod called collocation on finite elements which perm its high

accuracy while retaining the use of low order polynomials and their shorter solution times.

They also identified th a t com putation also took longer if the composition profiles became

widely separated in the column. They indicated th a t the use of sparse m atrix techniques

in the solution also yielded improved solution times.

Wilson and M artinez (1997a) investigate m ethods for the estim ation of s ta te variables such

as composition from tem perature measurements in reactive batch distillation. The m oti

vation for this was th a t good composition control is essential in reactive batch distillation

but the cost of on-line composition measurem ent is usually prohibitively expensive. The

authors considered two types of s ta te estim ator. Firstly an Extended Luenberger Observer

(ELO) th a t derives its estim ates from a linearised model of the process and secondly an

Extended Kalm an Filter (EKF) which, although also based on a linear model, produces

its estim ates based on the statistical characteristics of the prevailing random process dis

turbances and m easurem ent noise. The two techniques were com pared for an industrial

m ulticom ponent reaction. Two models were used, a very simple binary distillation model


and a slightly more complex multicom ponent distillation model. The simple non-hnear

model was used to generate the Hnear models for the sta te observers as the more complex,

multi-component model was too computationaUy expensive for this purpose. Tem pera

tu re measurements were taken from the more detailed model and m easurem ent noise was

added. The authors dem onstrated th a t the EKF estim ator produced be tte r estimates

th a t the ELO, which had stability problems. They concluded th a t the process mismatch

between the models did result in reduced estim ator accuracy and further refinement was

required. However, they concluded th a t this accuracy was sufficient basis for composition

control.

Wilson and M artinez (1997b) considered the autom ation of batch processes using fuzzy

modelling and reinforcement learning and applied their techniques to reactive batch dis

tillation. They stated th a t, in general, the operation of batch processes rehed heavily on

the skills of operators to achieve the desired product due to the difficulties of developing

effective autom atic control systems. They suggested th a t the abilities of the operator can

be im itated , and even improved on, by the use of embedded autonom ous agents. The

agent “perceives” the sta te of the process at each time step, executes an action and re

ceives a reward or payoff in return . The agent’s task is to react continuously to the process

sta te , influenced by disturbances and events, by determining a sequence of actions which

maximises some cumulative measure of rewards, driving the process towards its “goal

s ta te” . Wilson and M artinez (1997b) concluded th a t the most common algorithm for this

reinforcement learning, Q-Learning, had significant stabiHty problems and com putational

expense when appHed to batch processes where states vary with tim e. They presented a

hybrid approach where Q-Learning was combined with fuzzy rules relating process states

to control actions. This hybrid approach was term ed fuzzy Q-Learning. This technique

was successfully dem onstrated for the control of the industrial reactive distiUation case

study presented in Wilson and M artinez (1997a).

W ajge and Reklaitis (1998) presented a methodology for campaign optim isation of reactive

batch distillations. The authors considered the compositions and am ounts of off-cuts as

C H A P T E R 2. L IT E R A T U R E R E V I E W 33

optim isation variables in an upper level optim isation with reflux ratio profile optim isation

as a lower level decision problem. To simplify the procedure, they introduced the concept of

Distillation Characteristics which, is the composition profile developed at to ta l reflux. They

deduced th a t for reversible reactions where holdup in the column is neghgible, two mixtures

have the same distillation characteristic if the products and reactan ts are in the same

stoichiometric proportions. Thus the optim al operational policy for the column would be

the same in both cases. They considered the recycling of off-cuts, mixed with fresh feed,

which have the same distillation characteristic as the feed charge. The authors illustrated

their m ethodology with the esteriflcation of ethanol, using a packed column based on the

model reported by Wajge et al. (1997). They concluded th a t the distillation characteristic

is relevant in the design of campaign structures which minimise off-cut fragm entation and

reduce operational complexity by minimising the num ber of distinct distillation tasks.

In addition to a better reprocessing strategy, it offers insight into the trade-off between

production rate , reactant utiUsation and waste generation. The authors noted th a t the

feature of distillation characteristics breaks down if the reaction is irreversible and if the

holdup within the column is not negligible. This would Hmit its appHcability to real

processes.

Bollyn and W right (1998) considered the use of experim ental d a ta in developing and

refining a dynamic model describing a fed-batch reactive distillation column. The reaction

considered was the synthesis of ethyl esters of pentenoic acid by substitu tion of aUyl

alcohol for ethanol on triethyl orthoacetate. The reaction was assumed to occur in the

reboiler only. They simphfled the reaction scheme by ignoring reaction steps th a t occurred

sufficiently fast to be considered instantaneous, but were stiU left w ith 3 equilibrium

reactions and 4 ehminations, yielding a to ta l of 10 reactions. Their objective was to

establish an optim al operating policy, through simulation, such th a t a high conversion

of triethyl acetate and a high selectivity for the desired ethyl ester was achieved while

minimising excess alcohol. They also noted th a t in previous work, substantial tim e was

spent ensuring th a t the vapour-liquid equilibrium models were accurate and th a t less effort


was spent on the details of the reaction kinetics. They indicated th a t more a ttention

should be paid to the kinetics as they are more tem peratu re dependent than most other

physical properties. Few details of the modeUing assum ptions were given except th a t the

process was simulated in BatchCAD which uses a rigorous dynamic mass transfer based

distillation model. Experim ental d a ta was collected and analysed in order to refine the

models. This was carried out at several levels, starting w ith a lab batch investigation into

the kinetics moving through to pilot plant. The refined models were used at each stage

to enable effective targeting of further experiments to enhance the model. This led to the

development of an optim al operating policy which was successfully implemented on the

pilot plant resulting in an improvement in selectivity from about 50% to over 98%.

Li et al. (1998) considered the optim isation of a semi-batch distillation process and used

an industrial process to validate their model. The reaction considered was an industrial

transestérification reaction where an alcohol product is the most volatile reactant. A

semi-detailed model was developed including constant holdup, constant tray efficiencies

and constant pressure profile. The pressure on each tray was in terpolated from the top

and bottom pressures in the industrial column and the M urphree tray efficiency was de

duced by trial and error through comparison to the experim ental results. The simulation

results were sufficiently close to the experimental to justify the use of the model for op

tim isation. The optim isation was performed for minimum batch tim e employing control

vector param eterisation (CV P) and sequential quadratic program m ing. They optimised

the feed howrate of the alcohol, the reflux ratio and the switching tim e between the main

and off-cuts. Two scenarios were considered: firstly the optim isation of the process under

present requirem ents, where a 30% tim e saving was achieved, and the optim isation of the

process under slightly lower product purity requirem ents. The authors acknowledged th a t

although their solutions were feasible, it is hkely th a t sub-optim al solutions were found

due to the nonconvexity and complexity of the problem.

Xu and Dudukovic (1999) considered the modelhng of a photo reaction in a semi-batch


reactive distillation column. In their model, they considered reaction in both liquid and

vapour phases and indicated th a t this was im portant for photo reactions because differ

ent kinetic behaviours occur in the two phases. Their model was a staged equilibrium

model where both liquid and vapour phases are considered to be at equilibrium in each

com partm ent. They included constant volumetric holdup of each phase, constant pres

sure and assumed perfect mixing. The reaction considered was the chlorination of toluene

where selectivity for the desired benzyl chloride is enhanced by reactive distillation as the

product is distilled away from the reaction zone. The presence of UV light also enhances

the selectivity as it prom otes the desired chlorination of the m ethyl group and not the

chlorination of the benzene ring. They performed simulations for a series of three different

column configurations and compared it with experim ental data . The comparison with

experim ental d a ta was poor and it was concluded th a t the sim ulation only agreed with

the experim ental da ta in term s of trends. It was only as the real system approached ideal

operating conditions described by the model th a t the best perform ance could be achieved.

Venimadhavan et al. (1999) considered the synthesis of a reactive batch distillation pro

cess for the m anufacture of butyl acetate, an im portant industrial solvent. The reaction

is an esteriflcation of butanol and acetic acid to form butyl acetate and water. They de

veloped a very simple model to capture the essence of the process and to provide insight

for exploring process alternatives. Assumptions included: liquid-phase reaction confined

to the reboiler, constant molar overflow, operating reflux sufficiently large to be approx

im ated as to ta l reflux for the purposes of calculating column profiles, and tray holdups

negligible com pared w ith th a t of the still. They considered a reflux policy such th a t the

instantaneous Dam kohler num ber remained approxim ately constant which is equivalent

to the rate of product removal being kept equal to the ra te of production. The reaction

kinetic param eters were regressed from earher published work. The phase equilibrium

is quite complex w ith organic and aqueous phases being formed. The authors indicated

th a t there was a debate within literature as to which of two azeotropes was the lightest

boiling azeotrope: ternary, butanol-w ater-butyl acetate or binary, w ater-butyl acetate.


Using topological argum ents for the nature of singular points on the residue curve map for

the non-reacting ternary system with acetic acid absent, they concluded th a t the lightest

was the ternary azeotrope. They indicated th a t despite the two azeotropes having close

boiling tem peratures, the detection of the correct azeotrope had large implications for the

process design. In this case, the aqueous phase, containing a small am ount of butanol,

was removed as distillate and the organic phase was returned as reflux. This results in the

exclusion of water and the accumulation of butyl acetate in the still which can be purifled

through non-reactive distillation at the end of the reaction. A second model was proposed

which included constant holdup on the trays. This model was compared favourably with

the simplifled model although the inclusion of traydynam ics resulted in a slower response.

M onroy-Lopereba and Alvarez-Ramirez (2000) commented th a t optim isation approaches,

such as th a t presented by M ujtaba and M acchietto (1997) have an im portan t drawback in

th a t the optim al solution depends strongly on the model and model param eters and there

fore feedback control is essential in order to m aintain optim um profitability in the wake

of uncertainty. The au tho r’s objective was to obtain an output-feedback controller with

guaranteed tracking properties, despite uncertainties in the dynamics of the RED process.

They also wanted to dem onstrate th a t the resulting reflux ratio policy approaches th a t

obtained via optim isation techniques. The controller design is based on an approxim ate

model of the composition dynamics and makes use of a reduced order observer to estim ate

the modelhng error. The resulting controUer is shown to have the same structure as a PID

controUer with anti-reset windup. The controller performance was tested on the column

model as presented by M ujtaba and M acchietto (1997) and dem onstrated th a t the result

ing reflux ratio pohcy approached the optimal.

Schneider et al. (2001) developed a ra te based model for reactive distiUation in a packed

distiUation column. The heterogeniously catalysed reaction was assumed to be pseudo-

homogenious. The column model contained dynamic mass and energy balances. The

MaxweU-Stefan equations were used to describe the interfacial mass transfer within the


structured packing. The liquid holdup in the packing was determ ined using an experi

mentally derived correlation, specific to the packing. The authors neglected vapour phase

holdup due to the low pressure (< 1.2 Bar). No details on the discretisation m ethod em

ployed were given. The model was validated with experim ental results from the synthesis

of methyl acetate in a semi-batch column. Following a sensitivity analysis it was concluded

th a t the reaction kinetics and models of the column periphery have a significant influence

on the simulation results. It was indicated th a t, for the column to be used for control

purposes, some form of model reduction would be necessary.

A u th ors R ea ctio nloca tion

M od el LiquidD yn am ics

P ressu reD yn am ics

W ork

Egly et al. (1979) liquidphase

equilibrium constant molar holdup

constant optim isation of reflux ratio policies

Cuille and Reklaitis (1986) liquidphase

equilibrium + efficiency

constant volume holdup

constant simulation only

Reuter et al. (1989) liquidphase

equilibrium + efficiency

constant volume holdup

variable pressure and tem peratu re control

Sprensen and Skogestad (1994) reboileronly

equilibrium linear constant controllability and control strategies

Sprensen et al. (1996) reboileronly

equilibrium linear constant optim isation and im plem entation of optim al policies

M ujtaba and M acchietto (1997) liquidphase

equilibrium constant m olar holdup

constant m ethods for online optim isation

W ajge et al. (1997) bquidphase

rate-basedpacked

variable from correlations


accuracy and efficiency of spatial discretisation techniques

Wilson and M artinez (1997a) reboileronly


constant sta te estim ation of composition from column tem peratures

Wilson and M artinez (1997b) reboileronly

equilibrium unknown unknown application of neuro-networks to control

gIto

IH

Ig

Table 2.1; Sum m ary of papers on reactive batch distillation 1979 to present day

COoo

A u th ors R eactionlocation

M od el LiquidD yn am ics

P ressu reD yn am ics

W ork

W ajge and Reklaitis (1998) hquidphase

rate-basedpacked



campaign optim isation

Bollyn and Wright (1998) liquidphase

rate-basedpacked

unknown unknown optim isation and model validation

Li et al. (1998) reboileronly

equihbrium + efficiency

constant molar holdup

variable optim isation and model validation

Xu and Dudukovic (1999) liquid and vapour


constant sim ulation only

Venimadhavan et al. (1999) reboileronly


constant development of novel operating policy

M onroy-Lopereba and Alvarez-Ramirez (2000) liquidphase


constant Model based control

Schneider et al. (2001) hquidphase

rate-basedpacked


constant Simulation and model validation

i

(\2

IIg

Table 2.2: Sum m ary of papers on reactive batch distillation 1979 to present day (continued)

COCO


2 .1 .3 R e a c t iv e b a tc h d is t i l la t io n c o n c lu s io n s

Most of the work on reactive batch distillation has focussed on modelling of reactive batch

distillation with some work on optim isation of operating policies and some limited work on

control. The level of modelhng detail has varied but most authors have used equihbrium

modelling with either constant hquid holdup on the trays or hnearised tray hydrauhcs.

Little consideration has been given to pressure dynamics w ith vapour holdup neglected

and pressure assumed constant.

Few papers have considered control, (Reuter et al. (1989), Wilson and M artinez (1997a),

Wilson and M artinez (1997b) and M onroy-Lopereba and Alvarez-Ramirez (2000)) but

mostly for simple models. It is also noted, th a t with the exception of Sqrensen and Skoges

tad (1994) and Sqrensen et al. (1996), no work has been undertaken on the controUabihty

of reactive batch distillation although analysis of controUabihty forms the basis for under

standing the features th a t make control of these processes difficult. The fiexibihty which

is offered by batch operations would be enhanced by be tte r knowledge of how to modify

the process, its operation or control structure, to yield be tte r controUer performance.

Note th a t reactive distiUation in packed columns has rarely been considered in packed

columns, those th a t have (W ajge et al. (1997), W ajge and Reklaitis (1998), Bollyn and

W right (1998) and Schneider et al. (2001)), have im portantly not considered the vapour

phase which is essential for considering pressure dynamics.

It is therefore the objective of this thesis to investigate in more detail the controUabihty

and control of batch reactive distiUation in general. In particu lar, it is felt th a t a more

detailed equilibrium model of the batch distiUation column, in particu lar the inclusion of

pressure dynamics for the study, would give greater insight into the process behaviour.

An investigation into controUabihty of reactive distiUation in batch packed columns is also

proposed, not previously undertaken in any form. Conclusion wiU be drawn which may

influence the selection of separation medium, trays or packing, during design.


2.2 M od ellin g o f sh ort-p ath d istilla tion

2 .2 .1 I n tr o d u c t io n

Short-path distillation is considered in this section and emphasis has been placed on papers

th a t consider the modelling of these processes. As noted earlier, short-path , or molecular,

distiUation occurs at very low pressures. Short-path evaporators are ideal for processing

tem perature sensitive m aterials as much higher levels of vacuum can be achieved than in

a norm al batch column due to the absence of trays. Also the residence tim e of m aterial

in the column is much lower. A sum m ary of the literature is given in Table 2.3.

2 .2 .2 S h o r t -p a th l it e r a tu r e

Cvengros et al. (1995) experimentally evaluated the residence tim e distribution (RTD)

curves obtained in the hquid film formed on the surface of the wiped film evaporator.

The experim ental apparatus was a short-path evaporator w ith a segmented wiper in the

absence of distiUation. The experimental Uquid was Triethylene glycol (TEG ) and an

aqueous solution of NaCl was used as a trace. Residence tim e was m easured via a conduc

tivity probe. The wiping process was modeUed by a cascade of apparatuses with lam inar

falling film ideaUy mixed at the exit from each stage in a mixer with zero residence time.

The authors compared the effect of the Uquid load and Uquid viscosity in both the experi

m ental apparatus and the model. It was concluded th a t the model agreed closely with the

experimental behaviour and provided a m ethod for estim ating the efficiency of the wiping

in a real system.

Lutisan and Cvengros (1995a) indicated th a t in order to achieve molecular distiUation,

the size of the gap separating the evaporating and condensing surfaces should be less

than , or at least comparable to, the mean free path of the molecules at the prevailing

pressure. This would ensure efficient evaporation. They indicated th a t the mean free

path predicted by the kinetic theory of ideal gases gave a significantly smaller mean free


path than experience suggested. They gave an example th a t predicted a mean free path

of 1 mm but in practice, the distillation ra te did not drop for separations up to 50 mm.

They concluded th a t the prediction of the mean free pa th m ust also depend on other

factors such as geometry. In order to get a better prediction, they developed a one

dimensional model for molecular distillation based on Direct Simulation Monte Carlo

m ethod (DSMC) which is used to determine particle velocities w ithin the distillation space.

The da ta was used to calculate mean free path and other macroscopic variables such as

particle density, collision frequency and kinetic tem perature throughout the distillation

space. These phenomena were compared for different separation spaces, and condenser

and evaporator tem peratures. The model was also used to compare the efficiency of

molecular distillation from concave and convex surfaces. The efficiency was defined as the

ratio of the actual diffusion rate to the ideal, Langmuir-Knudsen diffusion. They indicated

th a t the efficiency was lowest for concave surfaces, particularly at small diameters where

the curvature is highest. They also indicated th a t the efficiency was higher for concave

surfaces because the space above expands rapidly so density falls.

Lutisan and Cvengros (1995b) used the one dimensional model of a short path molecular

distiller, (Lutisan and Cvengros, 1995a), to investigate the effect of inert gas pressure on

the molecular distillation process. They concluded th a t the presence of the inert gas at

lower partial pressures than the distilling components has negligible effect but at higher

partial pressure, the effect was quite marked, significantly reducing the distiUation rate.

M cKenna (1995) developed a model for the design of a wiped film evaporator for the

separation of volatile components from polymer solutions. It was noted th a t for highly

viscous systems, the action of gravity might not be sufficient to induce a reasonable flow

along the evaporator. The movement of Uquid can be enhanced by inclining the normaUy

vertical wiper blades providing a pumping action. The mixing of the Uquid film was

considered in detail as well as mass transfer through the Uquid film. The resistance to

mass transfer at the surface between vapour and film was considered to be relatively


negligible and therefore at equilibrium. The model was used to examine the separation

performance and this was found to increase with the speed at which the wiper blades

remix the film. However, the author noted th a t there appears to be a limiting rotational

speed, above which significant gains in mass transfer are obtained only at the expense of

very large increases in power consumption. The results from the model were compared

with da ta pubHshed on commercially available wiped film evaporators and were found to

agree well, both quabtatively and quantitatively.

Lutisan et al. (1998) considered the inclusion of an entrainm ent separator, a sieve inserted

into the distiUation gap to tackle the problem of entrainm ent. W hen the feed enters the

column, there is a rapid evaporation and escape of dissolved gases and low volatiUty sol

vents which can cause splashing. M aterial on the evaporator surface can be transported

into the distillate stream , causing a drop in efficiency. This problem is particularly acute

where the distillation gap is very small. A sieve, inserted between the evaporating and

condensing surfaces, traps both entrained Uquid and evaporated solvents and gases. Re

evaporation of the solvents occurs on the other side of sieve. The Uquid film was modelled

along the same lines as the au th o r’s earlier paper (Micov et al., 1997) and the vapour

phase was modelled using a direct simulation Monte Carlo m ethod. The results obtained

indicated th a t the sieve improves the composition of the distiUate but decreases the dis

tillation rate.

BatisteUa et al. (2000) considered the im plem entation of the non-ideal vapour phase model

developed by Lutisan and Cvengros (1995b) in their DISMOL software which was com

bined with work on the Uquid phase reported in BatisteUa and Maciel (1996). They

indicated th a t this software would enable non-ideal systems to be studied. The authors

indicated th a t the model would support m ulticom ponent systems although only a single

component was considered in their case study. The model was employed to determine

how efUciency is affected by system pressure, condenser tem peratures, separation distance

between evaporator and condenser for different layouts. It is noted th a t for the conditions


investigated the efficiency did not drop below 60 %.

Cvengros et al. (2000a) indicated th a t the film surface tem pera tu re is an im portan t factor

in determining the efficiency of a falling film evaporator. In their paper, they developed

a model to determ ine how the surface tem perature profile varies w ith feed tem perature

and liquid load. The equipment modelled was a falling film evaporator (unwiped) and the

single component film was assumed to be lam inar, modelled by the Nusselt equation. Heat

transfer was considered axially and radially within the film. Resistance in the vapour phase

was neglected and the evaporation rate was assumed to be governed by the Langmuir-

Knudsen equation. The film surface tem perature rises along the length of the evaporator

until a steady-state tem peratu re is reached. The authors concluded th a t the feed should

ideally be preheated to this tem perature to avoid using a portion of the evaporator to

heat the feed. During this post-heat portion of the evaporator, evaporation ra te is lower

than at the steady-sta te tem perature and hence separation efficiency is lower. They also

compared the film tem peratures calculated by the “non-approxim ate” model presented in

this paper with the linear development of tem perature presented in their earlier paper

(Micov et ah, 1997) and concluded th a t the linear assum ption gave a good approxim ation

to the surface tem perature .

Cvengros et al. (2000b) investigated the modelling of fractionation in a molecular evap

orator. The equipm ent modelled was a falling film evaporator (unwiped) with a divided

condenser which allows product to be w ithdrawn at various heights thereby perm itting

fractionation. The model was based on the one developed in their previous paper (Micov

et ah, 1997) which was extended to model the behaviour of the liquid film created on the

condenser surface. They dem onstrated th a t with properly adjusted process param eters,

fractions with different compositions can be obtained. The condenser at the top of the

column will be richest in the more volatile component, the lowest richest in the less volatile

component. They also commented th a t the divided condenser is generally more efficient

due to the lower liquid load as a result of the side drainage. The model was compared to


experim ental results showing a reasonably good agreement.

Kawala and Dakiniewicz (2002) developed a model to describe high vacuum distillation

in an evaporator with ro tating discs. The ro tating discs are arranged perpendicularly to

the flow of liquid providing a greater area for evaporation than conventional wiped film

evaporators. The model agreed well with experimental d a ta and they concluded th a t the

rate of evaporation is highly dependent on the size of the vapour outlet and th a t this

should be as large as possible.

Lutisan et al. (2002) indicated th a t a real wiped film evaporator operates between two

limits. At the lower, lam inar, limit there are large concentration gradients between the

outer, heated, surface and the inner surface where the volatile components evaporate. At

the upper lim it, due to wiping, the concentration and tem pera tu re gradients are elimi

nated. The authors, using models developed earlier (Cvengros et ah, 2000a), explored the

differences between these two regimes. They assumed th a t the pressure was sufficiently

low to neglect vapour phase modelling. They dem onstrated th a t the distillation rate in

the turbulent regime is much higher than in the lam inar regime at the same surface tem

perature of the evaporating cylinder and th a t consequently, due to the lower residence

tim e for a given distillate rate, therm al decomposition would be smaller. They noted th a t

the regime had m inimal effect on the evaporator separation efficiency. They compared the

experim ental values of relative volatility with the model showing qualitative agreement

between experim ents and model.

A u th ors E vap oratorT y p e

Q uantityV ariation

LiquidB eh aviou r

V apour P h a se M od el

W ork

Cvengros et al. (1995) wiped-hlm radial -1- axial

lam inar+ mixing section

N /A Residence tim e distribution

Lutisan and Cvengros (1995a) N /A radialonly

N /A DSMC Effect of mean free path on efhciency

Lutisan and Cvengros (1995b) N /A radial N /A DSMC Effect of inert gases on efficiency

McKenna (1995) wiped-hlm radial complex Equilibrium Effect of mixing

Lutisan et al. (1998) N /A radial N /A DSMC Separation efhciency

BatisteUa et al. (2000) falUng-hlm radial lam inar DSMC Effect of pressure and condenser tem p on efhciency.

Cvengros et al. (2000a) falling-hlm radial + axial

lam inar Langmuir-Knudsen

Effect of feed tem pera tu re

Cvengros et al. (2000b) faUing-hlm radial -|- axial

lam inar Langmuir-Knudsen

Fractionation with divided condenser

Kawala and Dakiniewicz (2002) ro tatingdiscs

N /A N /A Langmuir-Knudsen

Simulation and model vahdation

Lutisan et al. (2002) wiped-hlm axial(radial)

lam inar 4- turbulent

LangmuirKnudsen

Laminar and turbulen t regimes effect on efhciency

IIto

iH

iI

Table 2.3: Sum mary of papers on short-path distillation

4


2 .2 .3 S h o r t -p a th c o n c lu s io n s

As can be seen from the literature review on short-path distillation, the modeUing of

short-path distiUation can be broken into two: the trea tm en t of the liquid phase, both the

evaporator film and the condenser film, and the trea tm en t of the vapour phase and the

resistance it offers to mass transfer.

The liquid film m ay be treated as lam inar, for example in falling film evaporators, where

large concentration and tem perature profiles are to be expected radially through the depth

of the film (e.g. Lutisan et al. (2002)). During wiping the film can be considered to be weU

mixed without a radial profile. McKenna (1995) for example, considered the modeUing

of the mixing process in detail. Lutisan et al. (2002) indicated th a t the mixing produces

higher distiUation rates bu t not significantly higher efficiency. Regardless of the Uquid flow

regime, variations are expected in tem perature and composition along the axial length of

the evaporator.

The vapour phase resistance can be ignored at low pressures when the distiUation gap,

or the distance between the evaporator film surface and the condenser film surface, is

of the same order of m agnitude as the mean free path of the gas molecules. Here, the

ra te of evaporation is governed solely by the Langmuir-Knudsen equation for molecular

evaporation. Otherwise, some resistance is given by the vapour, reducing the distiUation

rate . This can be modeUed by solution of the MaxweU-Stefan equations and authors have

tended to adopt a direct simulation M onte-Carlo m ethod (DSM C) to achieve this (Lutisan

and Cvengros (1995a)(1995b) and BatisteUa et al. (2000)). The efficiency of the system

is also effected by re-evaporation of m aterial from the condenser. TypicaUy, lowering the

tem perature of the condenser has the effect of reducing this effect. It is noted, however,

th a t none of the authors have considered carrying out a reaction w ithin the evaporator

film although, this is done industriaUy. Also, all of the authors have considered steady-

s ta te models, which is reasonable as these units tend to be operated on a continuous basis.


However, these models cannot be used to consider the control of these processes where

a dynamic model is required. It is therefore necessary to develop a dynamic model th a t

considered a possible reaction in order to investigate the control and controllability of

these processes.

2.3 C onclu sion s

In this chapter, the s ta te of the art of reactive batch distiUation modelhng and control

has been presented together with th a t for the modelling of short-path evaporators. In

the following chapters, rigorous dynamic models of reactive distiUation in tray and packed

batch columns and in short-path columns will be presented th a t wiU subsequently be used

for control and controllability studies in the final chapters of this thesis.

C hapter 3

M odelling o f reactive batch

distilla tion in tray colum ns

In this chapter, the modelling of batch tray columns with chemical reaction in

the liquid phase is considered. The features and assumptions o f both a rigorous

model and a simplified model are presented. A case study is presented and the

two models are compared fo r tioo different operating policies, a constant reflux

operating policy and a controlled distillate composition study. The models are

also compared under reboiler disturbances for the controlled case study. It is

concluded that the simplified model is significantly different to the rigorous, es

pecially under varying process conditions. Therefore the rigorous model should

be adopted when simulating reactive batch distillation. However, the rigorous

model has a number o f disadvantages: The initialisation o f the model is more

complicated and the computational expense is high.

3,1 Tray colum n m odelling

In this chapter, two process models are presented for the sim ulation of reactive batch

distillation in tray columns. Firstly, a rigorous model which considers the modelling

49

C H A P T E R 3. MODELLING OF RBD IN T R A Y COLUM NS 50

of both liquid and vapour phases with dynamic mass and energy balances is presented.

Secondly, a simplified model, although more rigorous than models previously used for

control studies, neglecting pressure dynamics and w ith an algebraic energy balance is

considered.

3 .1 .1 R ig o r o u s m o d e l

The h tera tu re review in chapter 2 indicated th a t simple dynamic models have been used

for studying the control of reactive batch distillation in tray columns. In this thesis, a

more rigorous equilibrium tray model is developed and used with the following features:

• Incorporates a dynamic energy balance equation instead of an algebraic energy bal

ance where liquid enthalpy on the tray is assumed constant or constant molar over

flow where the energy balance is neglected

• Both liquid and vapour phase tray holdups are considered, their combined value

being a function of the prevailing pressure and the in ter-tray spacing

• Liquid tray holdup is allowed to vary, the liquid flowrate from the tray being deter

mined by the Francis weir formula

• A detailed pressure drop equation th a t takes account of both dry and wet head loss

on each tray, thereby determining the vapour flowrate

• Rigorous vapour-hquid equilibria, considering non-ideality in both liquid and vapour

phases can be implemented if necessary

• Condenser can operate as partial, to ta l or subcooled

• Reaction is considered throughout the hquid phase and is incorporated into the

component mole balances in the reboiler, trays, reflux drum and accum ulator. Heat

of reaction, where applicable, is incorporated into the energy balance. Reaction is

ra te based

C H A P T E R 3. M ODELLING OF R B D IN T R A Y COLUMNS 51

Dynamic m aterial and energy balances are also used to model the accum ulator, reflux

drum and reboiler drum . In each of these, both Hquid and vapour holdups are taken into

account.

3 .1 .2 A s s u m p t io n s

The assum ptions for the rigorous model are:

1. No entrainm ent effects

2. No downcomer dynamics

3. A diabatic operation

4. Phase equihbrium

5. Perfect mixing

6. Im m ediate heat input

7. Negligible holdup in the condenser

M urphree plate efficiencies can be introduced to take account of imperfect equihbrium

although it is im portan t to note th a t no m ethods have been reported in the Hterature

on how to account for chemical reactions in efficiency calculations (Ruiz et ah, 1995). It

is also worth noting th a t the perfect mixing assum ption is one of the more restrictive in

reactive distiUation since it in effect suggests th a t each tray is a completely mixed stirred

tank reactor. The model equations are given in Appendix A.

3 .1 .3 I n it ia l c o n d it io n s

The m athem atical model of a batch reactive tray distiUation column as used in this work

(Appendix A) forms a set of differential and algebraic equations (DAEs). For integration

of the DAE system , a consistent set of initial conditions is required. The system of DAEs

C H A P T E R 3. MODELLING OF R BD IN T R A Y COLUMNS 52

has an index of one and therefore the num ber of initial conditions required is equal to the

num ber of differential variables or states. Each sub-model contains ( N c P l ) states, arising

from the N c dynamic component balances plus a dynamic energy balance. A practical

initial set of specifications would be the mole fractions of N c — 1 components in the liquid

phase, the to ta l liquid holdup and the tem perature in each unit.

An alternative, adopted in this work, is to make the N c — 1 component specifications

and liquid holdup specifications but instead of specifying the tem pera tu re on all trays and

reboiler, a single tem peratu re specification is made on the top tray and the pressure drop

between each tray and the one below is instead initialised. The practical advantage of

doing this is th a t the initial conditions are more flexible. The initial tray tem peratures

are strong functions of pressure and composition. However, the pressure drops between

trays are much weaker functions of pressure and composition and are instead strongly

dependent on the height of liquid on the tray and the reboiler heat input. Therefore, the

second m ethod is more readily transferable to different m ixtures and different operating

pressures with only one tem perature specification required in the column.

3 .1 .4 I n te g r a t io n

Having determ ined a set of consistent initial conditions as outlined above, it is then nec

essary to solve the DAE system. This has to be done numerically due to the non hnearity

of the equation system . The integration of the DAE system describing batch reactive

distillation, as w ith most process engineering apphcations can be stiff due to phenomena

operating on widely different tim e scales.

Implicit numerical integration techniques are better suited than explicit ones for stiff sys

tems as they have superior stability properties. The process modelling software pPROMS

(Process Systems Enterprises Ltd., 1999), employed in this work, uses the backward dif

ference form ula (BD F - Gear (1971)) family of methods. In the BDF m ethod, the order

C H A P T E R 3. MODELLING DE R BD IN T R A Y COLUMNS 53

of the integration and the size of time step are varied autom atically to ensure th a t the

longest possible tim e steps are taken while satisfying the error tolerances of the user.

3 .1 .5 S im p lif ie d m o d e l

The simphhed model employed in this thesis is closely based on the rigorous model. It is

more numerically robust for the purposes of linearisation, as discussed in Chapter 5. The

following assum ptions are made:

• The variable pressure feature is replaced by a constant pressure drop across the

column stages and constant pressure in the condenser

• The dynamic energy balances on trays, reboiler and reflux drum are replaced by

algebraic energy balances. The assum ption is made th a t the enthalpy of the liquid

phase remains constant.

• Vapour phase holdup is assumed to be neghgible

For the simplified model, the initial condition specifications are the to ta l m aterial holdups

and the N c — 1 composition specifications.

3.2 C om parison b etw een sim plified and rigorous m odels

Having presented the rigorous and simplified models it is necessary to compare the two

approaches. This is in order to justify the adoption of the rigorous model over the sim-

phfied model for simulations. Two operating policies: constant reflux ratio and controlled

distillate composition are used for the production of a 0.6 mole fraction m ixture of ethyl

acetate.

3 .2 .1 E th y l a c e ta t e c a se s tu d y

In order to compare the simplified model with the rigorous model, the production of ethyl

acetate is considered. The reaction is an equilibrium estérification of ethanol (Tj, = 352 K )

C H A P T E R 3. MODELLING OE R BD IN T R A Y COLUMNS 54

and acetic acid (Ti = 391 K ) which forms the ester, ethyl acetate (Tb = 350 K ) and water

(Tb = 373 A").

Acetic A c ld (l) + Ethanol(2) ^ E thyl A ceta te(S) + W ater(4)

As can be seen, the boiling point of ethyl acetate is lower th an any other component

in the m ixture, m aking it suitable for reactive batch distillation. The product ester will

be separated from the reactants preferentially through the distillation as it is produced,

driving the equilibrium towards the products.

The tem peratu re dependent, reaction kinetics, shown in Equation 3.1 (Sm ith, 1956).

- # § ) (3.1)

- 2 . 7 3 1 x 1 0 ^k l = 0.083531 X 10 T-----

Reaction occurs in the hquid phase, in the reboiler, on the trays and in the condenser.

However, it is noted tha t due to the high boihng point of acetic acid it will be largely

confined to the reboiler during the reaction phase and hence the bulk of reaction will

occur in the reboiler. It is also im portant to note th a t the reverse reaction may weU occur

on trays, particularly towards the bottom of the tray stack where large concentrations of

w ater and ethyl acetate will occur. The heat of reaction is neghgible.

The calculation of the physical properties, notably the Vapour-Liquid equihbrium, is per

formed using the Multiflash software (Infochem Ltd., 1998). Both the hquid and vapour

phase are assumed to behave ideaUy; R aoult’s law apphes to the hquid phase and Dal

to n ’s law apphes to the vapour phase. The column configuration param eters and feed

composition are shown in Table 3.1. The feed is charged to the reboiler, and some as

holdup on trays and in the condenser. R ather than assume “perfect” pressure control in

the condenser and level control in the reflux drum; PI controllers wiU be employed to con

tro l these quantities. In-order to avoid overshoot, particularly during composition control

where the controller spends the initial period saturated as composition builds, anti-reset

C H A P T E R 3. M O DE LLING OF R BD IN T R A Y COLUM NS 55

windup features were added to the controller where integral error is not increased while

ou tpu t is sa tu rated .

As is norm al industrial practice, the condenser pressure will be controlled by m anipulation

of the condenser cooling rate. The level in the reflux drum , is controlled by m anipulation

of the reflux flow, Lq. (Further discussion of appropriate pairings of controlled and m anip

ulated variables is m ade in Chapter 5). The controller bias, controller gains and controller

reset times are given in Table 3.2. The condenser pressure is controlled at P = 0.5 bar.

T hroughout the case studies, the reboiler heat duty is m aintained at a constant value

Q = 0.885 X 10^ J/Sy except where it is considered as a source of disturbance in the

system. Two different operating policies are considered for producing a 0.6 mole fraction

ethyl aceta te product, constant reflux ratio and controlled composition.

C on stan t reflux ratio op eratin g po licy

The operation using an open-loop, constant reflux ratio operating policy is considered.

For this operating policy, the column is charged and then operated at to ta l reflux for 3

hours, then the internal reflux ratio ( L / V ) is fixed a t 0.95 with product w ithdrawn into

the accum ulator. This continues until the composition of ethyl acetate in the batch drops

to 60 mol% .

C on tro lled c o m p o sitio n op eratin g p o licy

The operation based on a feedback control strategy for both an undisturbed case and a

case w ith reboiler disturbances is also considered. The column is set up in exactly the same

way as for the constant reflux ratio study except th a t a P I controller with anti-reset windup

properties is introduced to control the distillate composition to 60 mol% ethyl acetate by

m anipulating the distillate flowrate, D. Controller param eters are shown in Table 3.2.

The batch is term inated when the distillate flowrate drops below, 1 x 10“ m o l/s .

The controlled case study is also run with disturbances to the reboiler heat supply. The

C H A P T E R 3. MODELLING OF RBD IN T R A Y COLUMNS 56

In itia l M ateria l H oldu psColumn holdup 21.7 km ol

condenser 100 moltrays 125 molreboiler 20.6 km ol

Feed C om p ositionEthanoic Acid 0.49Ethanol 0.49Ethyl Acetate 0W ater 0.02

Tray P aram etersNumber of trays 8Ytray 0.125 m^hweir 10 m m1wezr 0.50 mA 0.1875 mÂh 0.025 m^

R eb oiler P aram etersYyessel 0.5 m^Heat Duty R885 x l 0 5 J /s

Table 3.1: Column param eters

column is operated identically to the undisturbed case study but the reboiler heat duty is

stepped up and down, as shown in Figure 3.7.

3 .2 .2 C a se s tu d y r e su lt s

Figure 3.1 shows the values of the ethyl acetate composition in the distillate under constant

reflux and the flowrate of the distillate under these conditions. The simplified model

overestim ates the distillate composition, particularly during the first 10 hours of the batch

which may explain why, when the composition tails off, it lags behind the rigorous. The

consequence of the overestimation is shown in in Figure 3.2 which shows the composition

profile for the accum ulator where the composition of ethyl acetate in the accum ulator

remains higher for the entire batch. This is therefore why the batch term inates later

for the simphfied model and the holdup of product is larger th an for the rigorous model

(Table 3.3).

C H A P T E R 3. MODELLING OF RBD IN T R A Y CO LUMNS 57

C on d en ser P ressu re C on trolControlled Var: PcM anipulated Var: Q cGain: K c -1 0 0Reset: T£) 100 5

Set-Point: 0.5 atmR eflu x L evel C on tro lControlled Var: M t

M anipulated Var: LGain: K c -0 .1Reset: T£> 100 5

Set-Point: 100 molD istilla te C om p osition C on tro lControlled Var: X dM anipulated Var: DGain: K c - 1Reset: tc 50 5

Set-Point: 0.6

Table 3.2: Controller param eters

The distillate flow (Figure 3.1) is, as expected for constant reflux profile, approxim ately

constant after the initial to tal reflux period with a slight dip, towards the end of the batch

as the reaction stops and the heavy product, w ater, begins to rise up the column.

Figure 3.3 shows the values of the composition of ethyl acetate with tim e with the values

for the m anipulated variable, distillate flowrate, for the two models for the controlled

composition policy. As can be seen, the composition builds up as reaction proceeds and

ethyl acetate is formed. During this period, the distillate flow is held a t its minimum

value, 0 i.e. no product is removed. As the composition passes through, 0.6, the distillate

flowrate rises, starting at approximately, 1.5 hours., for the rigorous model. The distillate

flowrate peaks after 5 hours then the flowrate dechnes. This distillate profile is quite

different to th a t encountered for a non-reactive batch distillation separation (Figure 3.5)

where the distillate flowrate rapidly increases from zero to its peak within a few minutes

and then slowly decreases over the length of the batch. In the reactive case, there are

two phenomena taking place simultaneously: the reaction which is the generation of ethyl

C H A P T E R 3. M O DE LLING OF RBD IN T R A Y COLUMNS 58

R igorous

0.6

0.4

Q 0.2

5 10 15 20 25 30 350

Time [Mrs]

0.15

IE 0.05

iI 0

-0 .05250 5 10 15 30 3520

Figure 3.1: Distillate composition (top) and distillate flowrate (bottom ) for the constant reflux ratio study

acetate, principally in the reboiler, and the separation phenom ena, extracting ethyl acetate

from the system . Figure 3.6 shows the reaction ra te in the reboiler together with the

concentration of ethyl acetate. During the initial period, the reaction is dom inant and the

quantity of ethyl acetate in the column is low compared to the am ount of ethanol (volatile

reactan t). Hence the distillate rate will be low but will steadily increase as the quantity

of ethyl acetate increases in the column. The distillate ra te reaches its maximum when

the reaction begins to slow as the concentration of reactan ts in the still decreases and the

distillate flow ra te decreases to m aintain the composition of the distillate.

For the controlled policy there is a dram atic discrepancy between the simplified model

and the rigorous model, again resulting from the simplified model over-estimating the

composition of the column. This is reflected in the accum ulator profile of composition

and holdup, shown in Figure 3.4. The composition is held at 0.6 because of the controller

but the am ount of m aterial in the accum ulator builds up faster for the simphfied model.

C H A P T E R 3. M ODELLING OF R BD IN T R A Y COLUMNS 59

1

0.8

; l 0.6

Io 0.4 Ü

— Rigorous Simplified

0.2

00 5 10 15 20 25 30 35

14000

12000

10000

Ê 8000

2 6000

4000

2000

0 5 3510 15 20 25 30Time [hrs]

Figure 3.2: A ccum ulator composition (top) and holdup (bottom ) for the constant reflux ratio study

This results in the, 20.7%, discrepancy in the batch times between the two modelling

approaches (Table 3.3) with an insigniflcant 1.8% discrepancy between the predicted final

accum ulator holdups (Table 3.3).

In the th ird case study, the controlled reactive batch column is subjected to disturbances in

the reboiler heat supply. The distillate composition profile, distillate flowrate and reboiler

heat duty, as a percentage of the initial value, are shown in Figure 3.7. The profile for the

distillate flow and composition are similar to the undisturbed case study. It is evident th a t

there is a m arginally different response in the distillate flows when the system responds

to the reboiler disturbance. This may be due to the fact th a t the disturbances are being

applied at different stages in the profile. For example, the first disturbance occurs while the

distillate flow is still rising for the rigorous case study, but the simplified model has reach

its peak distillate flow a t this stage. Examining the final accum ulator holdup. Table 3.3,

there is an insiguificant discrepancy of 1.4% between the two modelling m ethods. However,

C H A P T E R 3. MODELLING OE RBD IN T R A Y COLUMNS 60

0.7

0.6

0 .5

E 0.4

— R igorous0) 0 .3

0.2

0 2 4 6 8 10 12

a 0.4

0.2

0 2 4 8 10 126Tim e [hrs]

Figure 3.3: D istillate composition (top) and distillate flowrate (bottom ) for controlled composition study

examining the predicted batch times (Table 3.3) the simplifled model simulation is 26.5%

shorter than the rigorous simulation. This is compared to the 20.7% discrepancy between

the batch times for the undisturbed case. It is apparent th a t the presence of disturbances

has resulted in a larger discrepancy between the two modelling approaches.

3.3 C onclusions

The objective of this chapter was to present a rigorous dynamic model for the simulation

of reactive batch distillation. A simphfied and more numerically robust model was also

presented for use in the control studies to be presented later (C hapter 5). In order to

justify the adoption of the rigorous model over the simplifled model for simulations, the

two modelling approaches were compared for the production of ethyl acetate. Two oper

ating policies: constant reflux ratio and controUed distillate composition were used for the

C H A P T E R 3. MODELLING OF R B D IN T R A Y COLUMNS 61

0.7

0,6

E 0,3 Ü

0,2

0,1

0

1 ' 1 1 1 1

— Rigorous-

< " " Simplified

:

1 1 ] 1 _ i_

-

10 12

12000

10000

8000

§- 6000

I 4000

2000

Time [hrs]

Figure 3.4; Accum ulator composition (top) and holdup (bottom ) for controlled composition study

production of a 0.6 mole fraction m ixture of ethyl acetate.

It appears th a t, although the predicted batch holdups are reasonably close, less than 5%

discrepancy between the m ethods, the discrepancies in the predicted batch times are sig

nificant and they increase as the column is operated under varying conditions. For the

constant refiux ratio case study, the reflux fiow remains approxim ately constant th rough

out. For the controlled case study, the refiux flow is changed continuously and for the

disturbed case study, the reboiler disturbances introduce greater variances in the reflux

fiow. This large difference in the performance of the simplified model compared to the

rigorous under control would justify the adoption of the rigorous model for simulating

reactive batch distillation. As reported by Sprensen and Skogestad (1994), reactive batch

distillation cannot be effectively operated without some form of feedback control.

The simulation times for the two modelling approaches were markedly different. The

simplified model took ju st 36 s to solve but the rigorous took 574 s, approxim ately 16

C H A P T E R 3. MODELLING OF R BD IN T R A Y COLUM NS 62

ro.6

0.4

0.2

155 100Time [hrs]

Figure 3.5: Distillate flow for non-reactive system

“ “ R eaction R ate [m ol.m “ s ” ] Ethyl A ceta te C om postion [molfrac]

0.8

0.6

0 .4

0.2

4 10 120 2 6 8Tim e [hrs]

Figure 3.6: Reboiler forward reaction rate and ethyl acetate composition (Rigorous Model)

times longer. The sixteen fold increase in com putational tim e is particularly significant

if the rigorous model were to be employed for optim isation. In dynamic optim isation,

a complete sim ulation is required to evaluate the objective function at each step in the

optim isation m ethod. However, M onroy-Lopereba and Alvarez-Ramirez (2000) indicated

th a t the controlled composition study can be thought of as being equivalent to the optim al

reflux ratio policy. It is im portant to note th a t the “optim al” reflux ratio profile for the

rigorous model is very different to th a t predicted for the simple model. It can therefore

be assumed th a t optim isation results from the two models would also be significantly

different.

C H A P T E R 3. M O DE LLING OF RBD IN T R A Y COLUMNS 63

0.8

o 0 .6

0 .4— R igorous Simplified0.2

2 4 6 8 10 120

E 0.5

110

= 105

® 100

cn 90Tim e [hrs]

Figure 3.7: D istillate composition (top) and distillate flowrate (bottom ) for controlled composition study with reboiler heat input disturbance (bottom )

Case Study Accumulator Holdup [kmol] Batch Time [hrs]Rigorous Simplifled % discr. Rigorous Simplified % discr.

Const. Reflux Const. Comp.Const. Comp. (D isturb)

12.7510.4610.47

13.3210.6510.61

+4.4% + T 8% + L4%

31.8910.5710.64

3R328.397.84

+ + 4%-20.7%-26.3%

Table 3.3: Accumulator holdup and batch times

C hapter 4

M od elling o f reactive batch

d istilla tion in packed colum ns

In this chapter, a dynamic rate based model is presented fo r the simulation

o f reactive distillation in packed batch distillation columns. Packed columns

are often modelled using equilibrium stage models such as that presented in

Chapter 3. Here, a method is presented for determining the Height Equivalent

to a Theoretical Plate (H ETP) from the packed column model. The H E T P is

demonstrated not to be constant, an assumption critical to equilibrium mod

elling. H E T P is shown to be a strong function o f both flowrate and compo

sition which vary with both time and position during batch operation. This

makes it difficult to determine a representative value fo r H ETP. In this chap

ter, a packed column is compared to the rigorous and simplified tray columns

presented in Chapter 3, using an average H ETP. The sim ulations are found to

be close where conditions vary slowly, e.g. under constant reflux ratio. How

ever, when conditions vary widely, fo r instance when the column is suffering

reboiler disturbances, the varying H E T P and different dynamics combine to

produce large differences in performance. This demonstrates the need to adopt

a rate-based method instead o f either the simplified or rigorous equilibrium tray

column models fo r modelling packed columns for control purposes.

64

C H A P T E R 4. MODELLING OF R BD IN PA CKED COLUMNS 65

4.1 R ate-b ased m odelling o f packed co lum ns

The objective of this chapter is to consider the dynamic modelling of packed columns,

so th a t the control of this process can be examined. Therefore it is necessary to inves

tigate the modelling alternatives: equihbrium or ra te based, to estabhsh which is mores

suitable. The equihbrium approach, presented in C hapter 3, has the advantage of signif

icantly shorter simulation times but relies on the assum ption th a t the packing behaves

equivalently to an equihbrium tray column of a fixed num ber of trays. This approach

assumes th a t the efficiency does not change significantly during the operation. The rate

based m ethod aims to model the packed column in a physically more reahstic m anner

than the equihbrium approach by considering the mass and energy transfer across the

boundary between the hquid and vapour phases. The m ost im portan t difference with this

approach is the relaxation of the phase equihbrium assum ption. In general, equihbrium is

rarely attained in either heat or mass transfer as these are ra te controUed processes driven

by gradients in chemical potential and in tem perature. In packed column modelhng, it is

more appropriate to model the phases separately and to consider diffusional interaction

phenom ena between the phases described by the Maxwell-Stefan equations (Bird et al.

(1960) or Taylor and Krishna (1993)), as shown in Figure 4.1. The dynamic model of a

catalytic packed column used in this thesis, is based on th a t presented in Furlonge (2000)

which described a non-reactive packed distillation column. The model equations are given

in Appendix A. In the following sections, the modelhng of the mass and energy transfer

and the modelhng of the hquid and pressure dynamics in the packing, are discussed in

more detail.

4 .1 .1 M o d e l l in g o f m a s s a n d e n e r g y tr a n s fe r

The determ ination of mass and energy transfer is carried out in accordance with the

film theory, which assumes th a t hquid/vapour equihbrium exists only at the interface

between the hquid and vapour phases (Welty et ah, 1984). Johnstone and Pigford (1942)

concluded th a t , in distillation, the resistance to mass transfer is neghgible on the hquid

C H A P T E R 4 M O D E L L I N G O E R B D I N P A C K E D C O L U M N S 66

LIQUID VAPOUR

REACTION

F ig u re 4.1: M a ss a n d en e r g y t r a n s f e r b e t w e e n p h a s e s

side a n d th e re fo re in t h e mo d e l used in th is wo rk , on ly t h e v a p o u r s ide is cons ide red

in t h e ca lc u la t io n o f t h e overal l m a ss t r a n s f e r coefficient . H ea t t r a n s f e r by co n d u c t i o n

is de sc r i be d by h e a t t r a n s f e r coefficients in b o t h t h e licpiid a n d v a p o u r films us ing t h e

C h i l t o n - C o l b u r n a n a l o g y ( W e l t y e t al . , 1984) . H ea t is a l so t r a n s f e r r e d by con vec t ion as a

r esu l t o f t h e m o v e m e n t o f m a s s f ro m o n e p h a s e to t h e o t h e r . D i sp e r s i on is n o t cons ide red

in th i s m o d e l b u t a p p r o p r i a t e t e r m s cou ld be a d d e d t o t h e m a s s a n d e n e r g y b a l anc es to

a c c o u n t for th i s (A ly et a h , 199 0a ,b ) .

4 .1 .2 M o d e l l i n g o f h y d r o d y n a m i c s

In t h e a b s e n c e o f r ig o ro us c o m p u t a t i o n a l fluid d y n a m i c s , it is n e c e s s a r y t o desc r ibe b o t h t h e

liquid p h a s e h o l d u p in t l ie p a c k in g a n d t l ie p r es su r e d r o p r e l a t i o n s h i p s us ing co r re la t io ns .

Ideal ly, one sh o u ld deve lop t h e s e f r o m t h e specific p a c k in g e m p l o y e d , us ing a su i t ab le

r a n g e o f l iquid a n d v a p o u r flows ( K r e u l e t a h , 1998) . In th i s m o d e l , t h e l iquid ho ldup-

l iquid f l owra te r e l a t i o n sh i p for Fall r ings a n d Ras ch i g r ings is t a k e n f ro m P e r r y a n d G re en

(1984) . T h e p r e s s u r e d r o p - v a p o u r f l owra te co r r e l a t io n is t a k e n f ro m B e m e r a n d Kalis

(1978) .

C H A P T E R 4. MODELLING OF RB D IN PACKED COLUM NS 67

4.1 .3 M odelling of chem ical reactions

Typically, when considering reactive distillation columns, reaction is only considered in

the hquid phase and can be homogeneously or heterogeneously catalysed. The modelhng

of homogeneous catalysis is generahy simpler, requiring an appropriate kinetic expression.

There are a number of approaches to the modelhng of heterogeneously catalysed reactions

in reactive columns, some of which are described below. In this chapter, the reaction is

modelled as homogenous, with the reaction confined to the hquid phase. This permits

heterogenous reactions to be modehed as quasi-homogenous if necessary.

H etero g en eo u s reaction ( th ree phase m odel)

The three phase modelhng approach considers the sohd catalyst phase in addition to the

hquid and vapour phases. It is useful as it ahows the characterisation of the various steps

before and after reaction occurs. These are namely:

• Convective and diffusive transport of reactants through the hquid boundary layer to

the outer surface of the catalyst

• Internal diffusion through the catalyst structure

• Absorption of reactants on to the active catalytic sites

• Chemical reaction

• Desorption of the reaction products

• Diffusion through the catalyst structure to the surface

• Convective and diffusive transport through the hquid boundary layer

Experim ental param eters are needed and their determ ination often difhcult. Hydrody

namic conditions within the catalyst have to be determ ined using simphfied models such

C H A P T E R 4. MODELLING OF RBD IN PACKED COLUMNS 68

as the Film Model and associated param eters such as the film thickness need to be de

term ined. It is also necessary to derive the appropriate sorption mechanisms for the

determ ination of the ra te constant.

Sim plified heterogeneous reaction m odel

Some authors, for instance Thiel et al. (1996, 1997), have simplified the mechanism by

assuming th a t the rate of reaction is dom inated by one step in the sequence above. In

their case, they assumed th a t the mass transport effects inside the catalyst particle play no

role in relation to the kinetics of the chemical reaction. In other words, the mass transfer

resistance of the chemical reaction wifi be very much bigger than th a t caused by mass

transport. Another common assum ption is to assume th a t it is the mass transfer step th a t

is ra te determining and the chemical reaction is assumed instantaneous.

Q uasi-hom ogen eou s reaction (tw o phase m odel)

The two-phase model is a simplification of the above models where the internal diffusion

phenom ena are neglected by incorporation into the ra te constant. Kreul et al. (1998) pro

vided justification for the use of this approach for certain reaction systems, particularly

those where the sorption mechanism assum ption breaks down. In some cases, particu

larly for ion exchange resins where catalyst swelling occurs and the protons are evenly

distributed throughout the resin, the system becomes very close to a homogeneous system

and it is simpler to assume this rather than model the system as heterogeneous.

4 .1 .4 P a c k e d c o lu m n m o d e l l in g

The column model is formed from the rate-based packing model for the column section,

reboiler, condenser, refiux drum and accum ulator models as given in appendix A. The

packed column model consists of partial differential equations which are converted into

differential and algebraic equations using an orthogonal collocation on finite elements

C H A P T E R 4. MODELLING OF R B D IN PACKED COLUM NS 69

m ethod (Wajge et ah, 1997) during integration in pPROM S (Process Systems Enterprise

Ltd., 1999).

4.2 R eactive batch d istilla tion case stu d y

In this section, the rate-based model is employed to sim ulate the production of ethyl

acetate in a packed batch column. The case study wiU dem onstrate how the separation

efficiency of the packing is affected during the operation by calculating the H ETP (Height

Equivalent to a Theoretical P late) at various positions in the packing. An equihbrium tray

model rehes on the value of the H ETP to be constant and it is dem onstrated in this case

study th a t this is not the case. It is therefore concluded th a t the equihbrium approach is

not suitable for modelhng packed batch columns for control purposes.

4 .2 .1 C o lu m n d e s ig n

In order to dem onstrate the behaviour of reactive batch distiUation in the packed column

the ethyl acetate case study presented in C hapter 3 wiU be used. As in C hapter 3, the

more rigorous, tem perature dependent kinetics presented by Smith (1956) are employed

and ideal physical properties are adopted, modelled by M ultiflash (Infochem Ltd., 1998).

The synthesis is performed in a regular packed column flUed with 25mm steel Pah rings.

Table 4.1 details the column dimensions and the packing features. The feed is charged

to the reboiler and some as holdup on the trays and condenser. In this model, perfect

pressure control is assumed. However, level control in the reflux drum , necessary to prevent

the drum from overflowing or draining, is performed by a P I controUer m anipulating the

reflux flow, Lq. The controUer bias, controUer gain and controUer reset tim e are as given

in Table 3.2 in C hapter 3. The condenser pressure is fixed at P = 0.5 bar. Throughout the

case study, the reboiler heat duty is m aintained at a constant value, Q = 0.885 X 10^ J /s ,

except where it is considered as a source of disturbance in the system.

C H A P T E R 4. M ODELLING OF RBD IN PACKED COLUM NS 70

Accum ulator Volume 0.4 rmPReflux drum Volume 0.01 rrNPacked bed Total length 8 m

Diameter 0.4 mPacking PaU Rings

Reboiler drum Cross-sectional area 4 m?Volume 0.65 wP

Initial Charge 21.7 km ol E thanoic Acid 49%

Ethanol 49% W ater 2%

Ethyl A cetate 0%PaU Ring Size 25 m m

Characteristics M aterial Steel(Coulson and Critical Surface Tension 75 N /m

Richardson, 1991) Packing Area {Sb ) 210 m ^ /m ^Packing Factor {F) 160 m~^

Void Fraction e 0.94

Table 4.1: Column dimensions and packing characteristics

4 .2 .2 C o lu m n o p e r a t io n

The column is operated under two different operating scenarios, constant reflux ratio and

controlled composition operating poUcy.

C on stan t reflux ratio operating policy

In part of this study, the column is operated using an open-loop, constant reflux ratio

operating policy. For this operating pohcy, the column is charged and then operated at

to ta l reflux for 3 hours, then the internal reflux ratio { L /V ) is fixed at 0.95 with product

w ithdraw n in to the accum ulator. This continues until the composition of ethyl acetate in

the batch drops to 60 mol% (after approxim ately 30 hrs).

C H A P T E R 4. M ODELLING OF RBD IN PACKED COLUM NS 71

C on tro lled com p osit ion operating policy

The operation using a feedback control strategy for both an undisturbed case and a case

with reboiler disturbances is also considered. The column is set up in exactly the same way

as for the constant reflux ratio study except th a t a PI controUer w ith anti-reset windup

properties is introduced to control the distiUate composition to 60 mol% ethyl acetate

by m anipulating the distiUate flowrate, D. The batch is term inated when the distiUate

flowrate becomes negligible, below 1 x 10“ m o lls .

The controUed case study is also operated with disturbances in the reboiler heat supply.

The column is operated identically to the undisturbed case study but the reboiler heat duty

is stepped up and down (shown in Figure 4.10). The principle objective of using the same

case study as th a t used for the tray columns is so th a t the two modelling approaches can be

compared. Two im portan t issues are investigated. Firstly, it is necessary to estabUsh the

degree of discretisation acceptable for the simulations, i.e. the num ber of finite elements

for orthogonal coUocation to be employed. Secondly, so th a t the packed column model can

be compared to the rigorous and simplified tray column models, presented in C hapter 3

an equivalent num ber of trays must be estabUshed. This is done by analysing the packed

column model and establishing a suitable value for the HETP.

4 .2 .3 E ffe c t o f d i s c r e t i s a t io n

As mentioned earher. Orthogonal Collocation on Finite Elements M ethod (O CFEM ) is

used to discretise the axial dimension within the column packing. In this section, the

effect of the num ber of finite elements used during the sim ulation results is investigated.

The greater the num ber of finite elements adopted, the more accurate the simulation but

at the expense of greatly increased com putational time. It is therefore prudent to adopt

the smallest num ber of finite elements possible without compromising accuracy. How

ever, adopting a greater num ber of finite elements also gives more detailed information

about how quantities vary within the packing and when analysing HETP, it is necessary

to have this detailed inform ation, hence a larger number of finite elements may be appro

C H A P T E R 4. M O DE LLING OF RBD IN PACKED COLUMNS 72

priate. In Table 4.2 are displayed the batch times and accum ulated product for the two

operating policies (constant reflux ratio and controlled composition) modelled with third

order orthogonal coUocation on two and on four flnite elements. For the constant reflux

ratio poUcy the discrepancy between the predicted batch tim e and holdup of product in

the accum ulator is neghgible. However, for the controUed composition poUcy, there is a

1.27% discrepancy between the predicted batch times with neghgible difference between

the predicted accum ulator holdup. Due to the changing flowrate and therefore greater

dynamic change, a higher degree of discretisation may be needed. It is im portan t to note

th a t the 1.27% discrepancy in batch time may not justify employing the higher degree of

discretisation due to the vastly increased com putational tim es, 765 s to 4113 5, also shown

in Table 4.2.

Therefore, third order orthogonal coUocation on two finite elements wiU be adopted for

the simulations in this thesis. However, this level of discretisation yields only seven points

across the discrete space so four flnite elements where the num ber of points increases to

th irteen, wiU be used for the H ETP investigation.

2 Finite Elements 4 Finite Elements % DiscrepancyC onstant Reflux Ratio:

Batch Time [hr] 32.653 32.625 -0.09%Product Holdup [kmol] 12.824 12.835 -0.09%Simulation Time [s] 1523 7592 +398%

ControUed Composition:Batch Tim e [hr] 10.457 10.326 -1.27%Product Holdup [kmol] 10.296 10.283 -0.13%Simulation Tim e [s] 765 4113 +438%

Table 4.2: Comparison between level of discretisation for 8m packed column

4 .2 .4 D e t e r m i n a t i o n o f H E T P

As already m entioned, HETP, the Height Equivalent to a Theoretical P late, is a method

employed to reconcile column packing with equilibrium stage modeUing. Authors (Treybal

(1980) and Kreul et al. (1999)) have already concluded th a t H E TP varies, not only with

C H A P T E R 4. MODELLING OF R BD IN PACKED COLUM NS 73

type and size of packing, but also strongly with howrates of both liquid and vapour phases

and by component concentration. Thus, one would naturally expect a spatial variation

in a continuous column operating under steady state . In a batch column, one would in

addition also expect a strong variance with time. The difficulty with this approach hes

in the failure to account for, in a physically reahstic way, the behaviour of packing by

assuming a staged equihbrium model. In this section, it is considered how H ETP varies

within the packing section and with time. The m otivation is two fold, firstly to confirm

th a t H ETP variance is too great to justify equihbrium modelhng and secondly to examine

how changing conditions within the column affect the perform ance of the packing.

C alculation o f H E T P

In order to estim ate the number of trays equivalent to th a t of the packed column it is

necessary to assess how the “Height Equivalent to a Theoretical P la te” (H ETP) varies

within the packing. For this purpose, a column packing with a height of 8m was selected

and axially discretised using th ird order orthogonal coUocation on four finite elements,

this level of discretisation yields 13 discretisation points within the body of the packing.

Figure 4.2 shows the composition of ethyl acetate in the vapour phase as a function of

height together with the composition of the vapour at therm odynam ic equilibrium with the

liquid phase for the constant reflux ratio policy after 3 hrs of operation. The equihbrium

composition a t the bo ttom of the column is approxim ately 0.45. This is equivalent to the

vapour outflow composition of the first theoretical plate. This value is achieved on the

vapour composition curve at approxim ately 1 m. In o ther words, the first 1 m of packing

achieves the equivalent separation of one equihbrium stage. Therefore, the H ETP of the

packing at the bottom of the column is 1 m. The H ETP at any given location is simply

the horizontal distance between these curves (Furlonge, 2000).

The H ETP was analysed for the constant reflux ratio pohcy and for the controlled com

position case study. Figure 4.3 shows the profiles of composition of ethyl acetate in the

C H A P T E R 4. MODELLING OF R BD IN PA CKED COLUMNS 74

0.85

0.8

0.75

0.7

o 0.65

0.5

0.45 — Actual- - Equilibrium

HETP0.4

0.35

Height [m]

Figure 4.2: Vapour composition of Ethyl A cetate a t 3 hrs (constant reflux ratio)

reboiler and distillate and the distillate flow during the batch. The top two plots show the

profiles for the constant composition policy and the bottom two plots show the profiles

for the controlled composition policy. It is evident th a t the composition varies at the top

and the bo ttom of the column for the constant reflux ratio policy, albeit slowly for the

m ajority of the batch due to the high reflux ratio. There is a rapid drop in the distillate

composition after 25 hours. Here, the heavy components, predom inantly w ater, begin

to be removed. This rapid drop in distillate composition accompanies a shght drop in

distillate flow, which otherwise remains approxim ately constant at 0.12 m o l/s .

For the controlled composition pohcy, composition at the top of the column remains

constant past the to ta l reflux period due to the controller. The reboiler composition

increases to a maximum of about 0.2 at 3 hours and then decreases over the rem ainder

of the batch. The most dram atic variation in the column is th a t of the distiUate flow,

m anipulated to control the composition. The distillate increases to a peak at about 5

C H A P T E R 4. M ODELLING OF R BD IN PACKED COLUM NS 75

hours during the main reaction phase and then decreases towards the end of the batch

as the ra te of production of ethyl acetate begins to tail off. The consequence of this is

th a t the flow of liquid reflux back into the packing is initially high, decreases during the

middle portion of the batch and then increases again towards the end of the batch. This

is thought to have a profound effect on the efficiency of the column and consequently on

the HETP.

The actual and equilibrium vapour compositions (as shown in Figure 4.2) were extracted

at a num ber of tim e steps throughout the batch process. A program was w ritten within

M atlab (The M ath Works Inc., 1996) to interpolate the da ta using Lagrangian polyno

mials and to calculate the horizontal distance between the curves as a function of height

(Equivalent to H E TP). It should be noted th a t since the actual composition curve does

not extend past the top of the column it is not possible to calculate past the H ETP at the

top of the column.

For the constant reflux ratio policy a series of axial H ETP profiles for different times during

the batch are shown in Figure 4.4. The H ETP is approxim ately constant throughout the

packing between 0.78m and 0.91m for the first three tim e steps (6 hrs to 18 hrs). There

is a shght drop in H ETP at the bottom of the column which becomes greater as tim e

progresses. At 24hrs, this becomes most significant with the H ETP dropping to 0.62m at

the bottom and rising with height in the column. From Figure 4.3 it was shown th a t the

flowrates within the column remain approxim ately constant and therefore the variation in

H ETP is likely to be due principally to the change in concentration within the packing

during the batch . The m ean H ETP against tim e is shown in Figure 4.5. The mean H ETP

s ta rts a t about 0.9 m a t the s ta rt of the batch and decreases slightly to approxim ately

0.86 m for m ost of the batch up to 21 hrs where it dips shghtly down to 0.8 m at 21 hrs.

Generally, the H ETP falls as the batch proceeds which means the packing becomes more

efficient tow ards the end of the batch. The average H ETP with tim e and space for this

policy is 0.857 m

For the controlled composition case study a series of axial H E TP profiles for different

times during th e batch are shown in Figure 4.6. Up until the peak in distillate flow at 5

C H A P T E R 4. M ODELLING DE R B D IN PACKED COLUM NS 76

hours, the H ETP is approxim ately constant throughout the packing. As the reflux flow

falls, the H ETP falls. After the peak at 5 hours, greater variation in the packing occurs

w ith the bo ttom and top portions of the column decreasing in efficiency and the central

section between 2 m and 3 m continuing to increase in efficiency. The mean H ETP against

tim e is shown in Figure 4.7. The mean H ETP appears to approxim ately follow the change

in distillate flowrate, shown in Figure 4.3. The H ETP decreases from its maximum of

0.97; m , and hence efficiency increases as the flowrate begins to rise to its maximum at 5

hours. The H E TP reaches its minimum of 0.725 m at 6 hours and rises again towards the

end of the batch. This indicates th a t H ETP is a strong function of flowrate and increases

with increasing liquid reflux. The average H ETP with tim e and space is 0.825 m.

C H A P T E R 4. MODELLING OF R BD IN PACKED COLUMNS 77

DistillateReboiler

0.8

0.6

0.4

E 0.2

20 30 35

0 .14

0.12

S 0 .08

« 0 .06

S 0 .04

0.02

50 10 15 20 25 30 35Time [hrs]

0.8DistillateReboiler

0.6

B 0.2

050 1 2 3 4 6 7 8 9 10

0.7

0.6

o 0 .5

5 0 .4

® 0 .3

0.2

0 1 2 3 4 5 6 7 8 9 10Tim e [hrs]

Figure 4.3: Composition and Distillate Flowrate - (TO P C onstant reflux ratio - BOTTOM Controlled Composition)


0 .9 5

0.9

0 .8 5

0.8— 6 hrs

18 hrs 24 hrs^ 0 .75

0 .7

0 .65

0.6

Height [m]

Figure 4.4: H ETP packing profile against tim e for Reactive Case Study (C onstant reflux ratio)

0 .92

0.9

0.86

® 0 .84

0 .82

0.82520 30

Time [hr]

Figure 4.5: Mean H ETP profile for Reactive Case Study (C onstan t Reflux Ratio)


1 hr- - 3 hr

£CLtDX

0 .9

0.8

0.7

0.6

height [m]

Figure 4.6: H ETP packing profile against tim e for Reactive Case Study (Controlled)

0 .95

5 0 .9

0 .85

0 .75

0.71 3 4 5 7 8 92 6

Tim e [hr]

Figure 4.7: Mean H ETP profile for Reactive Case Study (Controlled)

C H A P T E R 4. MODELLING OF R BD IN PACKED COLUM NS 80

4.3 C om parison b etw een rate based and equ ilibrium m odels

The most im portant conclusion from the study in the previous section is th a t H ETP is

not constant within the column section or with time. This has potentially severe imphca-

tions when adopting the equihbrium modelling approach to model packed batch columns,

especially when hquid flowrate is varying during the batch such as during the controlled

composition policy. A rate-based approach is therefore expected to provide a more accu

rate representation of a packed column. To confirm this, the two approaches are compared.

In order to make this comparison it is necessary to ascertain the num ber of equilibrium

trays th a t corresponds to the 8 m of packing. This is difficult to do due to the variation

experienced in both the constant reflux ratio and especially in the controlled composition

policy. Using the average HETPs of the two operating policies, 0.857 m for the constant

reflux ratio and 0.825 m for the controlled composition gives columns of 9.3 trays and

9.7 trays respectively which both round up to 10 trays. It is also im portan t, due to the

high variations in HETP, to investigate the impfications of selecting values of H ETP at

the extremes. The H ETP varies most significantly for the controlled composition policy:

between 1.4 m and 0.6 m corresponding to a 6 tray and 14 tray column respectively.

Therefore, in this section the 8 m packed column is compared to the rigorous tray col

umn model (C hapter 3) with 6, 10 and 14 trays. The packed column is also compared to

the simplified tray model with 10 trays, to investigate the implications of this approach

to packed column modelling. Comparisons are made under the three different operating

policies: Constant reflux ratio [R = 0.95), constant composition of 0.6 mole fraction Ethyl

Acetate and the constant composition policy under disturbances.

In Table 4.3 are shown the batch times and to ta l product accum ulated with composition

0.6 mole fraction ethyl acetate. The table also shows the percentage discrepancy between

the different models as compared to the 8 m ra te based packed column model.

The constant reflux ratio, distillate composition and flow profiles are shown in Figure 4.8.

The performance of the rigorous and simplified tray column configurations are close to

th a t of the packed column model. The 10 tray rigorous model, derived from average

C H A P T E R 4. M O D E LLING OF RBD IN PACKED COLUM NS 81

H ETP corresponds m ost closely of the tray column models. It is noted th a t the simplified

10 tray model responds much quicker and has a similar initial response to the 14 tray

column. The distillate profiles are also close with the distillate flow m arginally lower for the

packed column model. The close profiles are reflected in the batch times and accum ulator

holdup values (Table 4.3). There is no discrepancy between the packed column model

and the rigorous 10 tray model for batch tim e with a slight overestim ate of 2.0% for the

accum ulator holdup. For the simplified 10 tray model, there is an overestim ate of 1.9%

in the batch tim e and an overestim ate of 3.5% in the accum ulator holdup. The smaller 6

tray rigorous column underestim ates both the batch tim e, —5.1% and holdup, —3.5% due

to its poorer separation performance. Similarly, due to its higher separation ability the

larger 14 tray rigorous column overestimates the batch tim e, +3.6% , and holdup, +5.9%.

The controlled composition profiles are shown in Figure 4.9. As with the constant reflux

ratio profiles the profiles of the packed column are close to those of the rigorous tray

columns. However it is interesting to note th a t, due to the controllers, there is oscillation

in the distillate flowrate in the region of 2 hours. The am plitude of the oscillation is larger

for the packed column th an for of the rigorous tray columns except th a t of the largest 14

tray rigorous tra y column. This suggest a slightly slower dynamic response of the packed

column and the larger tray column. This slower response is also reflected in the distillate

composition where the packed column and the larger tray column take slightly longer to

reach the controller set point. In term s of final product accum ulated and batch times

the variance between the models is larger than for the constant composition case study

(Table 4.3). The simplified 10 tray model has very significant 21.1% shorter batch tim e

and 2.8% larger product accum ulated. The rigorous 10 tray model is still close to the

packed column model, —0.8% batch tim e and +1.2% holdup . The larger and smaller

rigorous tray columns are more significantly different th an for the constant reflux ratio

policy. The discrepancies for the 14 tray column are 4.2% shorter batch tim e and 0.6%

larger holdup. For the smallest, 6 tray column the batch tim e is 15.3% longer and the

holdup is 2.0% larger.

The controlled composition with reboiler disturbance profiles are shown in Figure 4.10.

C H A P T E R 4. M ODELLING OF RBD IN PACKED COLUMNS 82

Due to the reboiler disturbances there is a larger difference between the packed column

profiles and the rigorous tray column profiles. However, the packed column profiles remain

much closer to those of the rigorous tray column models th an to those of the simpbhed

tray column model. The reboiler disturbances produce oscillations in the distillate flow

which is most pronounced for the packed column and the largest, 14 tray rigorous column.

The oscillations become more pronounced towards the end of the batch. The discrepancy

in accum ulator holdup is small at about (2%) for both columns. The discrepancy in batch

times and accum ulator holdups (Table 4.3) are generally larger than for the controlled

policy w ithout disturbances. The variance is greatest for the batch times. The simplified

10 tray model underestim ates the batch tim e by 20.1%, the rigorous 10 tray model overes

tim ates by 3.2%. The larger, 14 tray column underestim ates by 2.1%. The smaller, 6 tray

column overestim ates by a significant 25.8%. In term s of accum ulator holdup aU four tray

column models overestim ate the amount of holdup in the accum ulator. The simplified 10

tray model overestim ates by 3.0%, the rigorous 6 tray by 2.6%, the rigorous 10 tray by

1.6% and the rigorous 14 tray by 0.9%.

C H A P T E R 4. MODELLING OF R B D IN PACKED COLUMNS 83

Model Size Batch Time Acc. HoldupHours % discrepancy Holdup % discrepancy

C on stantR efluxPacked (NEQ) Simplified (EQ) Rigorous (EQ) Rigorous (EQ) Rigorous (EQ)

8 m10 Trays 6 Trays 10 Trays 14 Trays

32.65 33.2730.9932.65 33.84

+1.9 % -5.1

0 % +3.6 %

12.8213.2712.3713.0813.58

+3.5 % -3.5% +2.0 % +5.9 %

C ontrolledPacked (NEQ) Simplified (EQ) Rigorous (EQ) Rigorous (EQ) Rigorous (EQ)


10.158.0111.7010.07R72

-21.1 % 4-15.3 %o -0.8 %o -4.2 %

10.3010.5910.5110.4210.36

+2.8 % +2.0 % +1.2 % +0.6 %

C ontrolled + D is tu rb an cesPacked (NEQ) Simplified (EQ) Rigorous (EQ) Rigorous (EQ) Rigorous (EQ)


9.347M611.759.649.14

-20.1 % +25.8 % +3.2 % -2.1

10.2410.5510.5110.4010.33

+3.0 % +2.6 % +1.6 % +0.9 %

Table 4.3: Comparison between modeUing approaches, rate based model)

(EQ: equilibrium model, NEQ:


Packed (8m)

■ Rig 10 Tray ' Rig 14 Tray■ Simpl. 10 Tray

0.4

Q 0.2

15 20 25 350 5 10 30

0.15

fII 0.05

I

-0 .050 5 10 15 20 25 30 35

Time [hrs]

Figure 4.8: Distillate composition (top) and distillate flowrate (bottom ) for constant reflux ratio policy

0.7

■sO.5Q-E 0.4ÔS 031 0.2 b

0.1

0

" Packed (8m)' - ' Rig 6 Tray - - Rig 10 Tray Rig 14 Tray Simpi. 10 Tray

10 12

0.6

0.4

Q 0.2

Time [hrs]

Figure 4.9: Distillate composition (top) and distillate flowrate (bottom ) for controlled composition policy

C H A P T E R 4. M O DE LLING OF RBD IN PACKED COLUMNS 85

c 0.8

^06I 0.4

.= 0.2

wmm Packed (8m) Rig 6 Tray

- - Rig 10 Tray Rig 14 Tray— Simpl. 10 Tray

10 12

iZ 0.5

CFIIOTime [hrs]

S 105

<D 100

6Time [hrs]

Figure 4.10: D istillate composition (top), distillate flowrate (middle) and reboiler d isturbance profile (bo ttom ) for controlled composition policy

C H A P T E R 4. M ODELLING OF R B D IN PA CKED COLUM NS 86

4 .4 C onclusion

In this chapter, the production of ethyl acetate in a packed reactive batch distillation

column has been considered. Packed columns can either be modelled as an equivalent tray

column by determining a suitable value for the H ETP (Height Equivalent to a Theoretical

P late) or more physically reahstically by a rate-based model. The objective of this chapter

was to establish whether or not, the less physically realistic, tray column model would be

suitable for modelling the packed column for reactive purposes.

A packed column model which extended the work of Furlonge (2000) to include chemical

reaction throughout the liquid phase was presented. This model was used to simulate the

behaviour of the constant reflux ratio and controlled composition case study presented in

the previous chapter, in a packed batch column. A m ethod was presented for establishing

the H ETP for the column and this was used on both case studies. It was concluded th a t

the H ETP is not constant, varying with tim e and packing height as a result of varying

composition and liquid flowrate. Consequently, the controlled composition case study was

found to vary more than the constant reflux ratio case study due to greater changes in

liquid flowrate.

A comparison was made between the packed column model and rigorous and simplified

tray column models with 10 trays, which corresponded to the average H ETP found for

both the constant reflux ratio policy and the controlled composition policy. The packed

column was also compared to rigorous tray columns with 14 and 6 trays which are the

columns corresponding to the maximum and minimum H E TP values in the controlled

composition study. The rigorous column model with 10 trays agreed most closely with

the packed column model while the simplified model and the larger and smaller rigorous

column models had poorer agreement. This suggest th a t if the equilibrium approach is

taken then it is im portant to use the average H ETP and to model the column using the

rigorous tray column model. However, for the controlled case study with and without

disturbances the constant H ETP assum ption begins to break down due to the higher

variance in H ETP experienced within the packing due to the varying flowrates. The

C H A P T E R 4. MODELLING DE R B D IN PAC KED COLUM NS 87

greater variance may also be due to the different dynamics within the columns leading to

the slower behaviour of the packed column.

The variation in H ETP was found to be too great to justify the adoption of the equilibrium

approach for modelhng packed columns especially where liquid flowrates vary considerably,

such as during control. Therefore, in order to study control and controllability of packed

columns it is necessary to adopt a modelling approach th a t accounts for the physically

different mechanisms of heat and mass transfer encountered in a packed column and the

rate-based packed column model will be adopted when investigating packed column control

in Chapter 5.

C hapter 5

C ontrol o f reactive batch colum ns

In this chapter, the control o f reactive hatch distillation in both tray and packed

columns is considered. It is important to have an appreciation o f how design

decisions affect column control as this insight is incorporated into column de

sign and operation. Optimal design and operation may not be achievable with

out considering control simultaneously. Both a simulation based approach and

a frequency response approach to controllability is considered. The simula

tion approach requires a control scheme to be implem ented and is more time

consuming. The frequency response approach requires linear approximations

to non-linear process models. A number o f linear models are needed to track

changes in controllability through the batch. A robust method o f linearisation

is presented to deal with numerical problems arising from the rigorous tray col

um n model. These methods are applied to tray and packed columns and are

used to identify how the choice o f column and column size affects controllabil

ity. The effects of batch time and reaction are also investigated. It is concluded

that packed columns are harder to control than tray columns, control becomes

more difficult for larger columns, particularly fo r packed. For this case study,

control is found to get harder towards the end o f the batch and reaction is found

to have negligible direct effect on control. However, reaction affects the reflux

ratio profile which is found to have a significant effect on controllability.

C H A P T E R 5. C O N T R O L OF R E A C T I V E B A T C H COLUMNS 89

Disturbances (d)

Inputs (u)

ProcessControlled Outputs (y)

Measured Outputs ( y j

Figure 5.1: Process for control

5.1 In trod u ction

A general process th a t is to be controlled is illustrated in Figure 5.1. Initially, the require

ments of a control system need to be developed. These control objectives are formulated

such th a t the operational requirements for the process can be achieved. The process’ oper

ational requirem ents may be to perform optimally, maximising profit or minimising batch

tim e, for instance, with reference to safety, environm ental regulations, product specifica

tions and operational constraints.

It is im portan t to consider how easily these control objectives can be met for the given

process. Skogestad and Postlethw aite (1996) defined input-output controllability as being

this ability to meet the control objectives. They formally defined it as:

Inpu t-ou tpu t controllability is the ability to achieve acceptable control perfor

mance; th a t is, to keep the outputs (y) within specified bounds or displace

ments from their references (r) , in spite of unknown but bounded variations,

such as disturbances (d) and plant changes, using available inputs (u) and

available measurem ents ( y ^ and dm)-

W ith this definition of controllability in mind, it is also im portan t to consider which

m easurem ents (y-m, dm) should be made and which inputs (u) should be m anipulated to

achieve the best possible performance. Seborg et al. (1989) presented a series of guidelines

for doing so:

C H A P T E R 5. C O N T R O L OF R E A C T I V E B A T C H COLUM NS 90

1. Select control ou tputs th a t are not self-regulating

2. Select control outputs th a t have favourable dynamic and sta tic characteristics, i,e.

there should exist an input with a significant, direct and rapid effect

3. Select inputs th a t have large effects on the outputs

4. Select inputs th a t rapidly affect the controlled variables

It is im portan t to stress th a t controllability is an inherent property of a process and

can only be affected by physical changes to the process. Therefore, it is im portant to

understand how any modifications made to the plant affects its controllabihty.

There are a num ber of approaches to a controllability analysis: The most common ap

proach is the sim ulation approach which involves a candidate control system being im

plemented and then tested against a range of disturbances and set point changes whilst

evaluating the performance.

Alternatively, a frequency response approach can be taken to assess the controllability

of a process. In order to use the frequency response approach, it is necessary to use

linear approxim ations of the non-linear process models. For continuous processes, this

would typically be linearisations about the steady-state. However, for batch processes

where the operating point changes with tim e, it is necessary to bnearise a t multiple points

throughout the batch. R ather than representing the deviation away from steady-state, the

linear models represent the deviation away from the operating trajectory . The frequency

response approach has the advantage th a t it can be applied to open-loop processes, thereby

eliminating the effect of the controller on the controllability analysis.

A simulation approach and a frequency response approach to controllabihty analysis will

be adopted in this chapter. Firstly, the m ethods for controllabihty analysis wiU be outlined

for both the tim e domain and the frequency domain. Then a m ethod for applying the

controllabihty tools will be outhned, together with a m ethod for deaUng with numerical

problems in the rigorous tray column model. These m ethods are then used to investigate

C H A P T E R 5. CO NTR O L OF R E A C T I V E B A T C H COLUMNS 91

CondenserRefluxDrum

Distillate DReflux L t

Vapour Boilup

Reboiler

Figure 5.2: Batch Column

the controllability of the tray column and the packed column presented in Chapters 3 and

4.

5.1.1 Control of batch distil lation colum ns

In both non-reactive and reactive batch distillation, shown in Figure 5.2, there are a

num ber of common control objectives:

• Condenser Pressure Control: Typically controlled through the m anipulation of con

denser duty

• Reflux Drum Level Control: Normally controlled through the m anipulation of reflux

or distillate flow

• P roduct/ distillate composition control: Normally controlled through the m anipula

tion of reflux or distillate flow

Speciflcally for some reactive batch distillation operations:


• Reboiler tem perature control

C on d en ser p ressu re contro l

Condenser pressure control is a requirement of all distillation columns. This is typically

achieved through the m anipulation of the condenser duty. Pressure dynamics are fast

compared to other distillation phenomena which is the justification for assuming perfect

pressure control taken by some authors as discussed in C hapter 3.

R eflu x drum level control

Reflux drum level control is necessary in all types of columns as this is not a self regulating

process. It is necessary to m aintain a “reasonable quan tity” of inventory in the reflux drum

to help dam pen disturbances in the vapour flowrate entering the condenser. Normally

either hquid distillate flowrate, D or the reflux flowrate, L t -, is used for this purpose.

D istilla te co m p o sitio n contro l

In non-reactive batch distillation, a number of open-loop pohcies can be employed to

achieve the desired product composition in the accum ulator. This would entail either a

fixed reflux ra tio or tim e varying reflux ratio, both im plem ented through the m anipulation

of whichever out of the reflux flowrate, L t or distillate flowrate, D is not being used for

level control. Sprensen and Skogestad (1994) concluded th a t, for reactive batch distillation,

it is not rehable to apply these open loop pohcies and th a t some form of feedback control

is required. Here the composition of the distiUate, is m aintained at a desired value

through the m anipulation of either the reflux or distiUate flowrates.

R eaction tem p e r a tu re control

Sprensen and Skogestad (1994) indicated th a t for some reactions it is necessary to control

the tem pera tu re of the reboiler, where reaction tends to occur, to within tigh t bounds to


ensure a consistent product between batches. Tem perature can be controlled by m anipu

lating the reboiler boilup ra te hg.

Two candidate control schemes (Table 5.1) for reactive batch distillation are considered

by Sprensen and Skogestad (1994) although they considered perfect level control in the

reflux drum. Skogestad (1997) indicated th a t for continuous distillation, there is greater

interaction w ith the level control loop when employing Scheme 1 (D V ) than with Scheme

2 (L V ) .

M x XD T b P cScheme 1 L t D QcScheme 2 D L t Vb Qc

Table 5.1: Control Schemes

E q u ip m en t m od ifica tion s

As indicated earlier, controllability is a property of the process only and is independent

of the control system . It can only be affected by modification of the process equipment.

Some modifications th a t can be made to the process to modify controllability are outhned

below.

• Modifying the size of the reflux drum or adding ex tra buffer tanks wiU dampen

disturbances.

• Changing the height of the column or type of column will affect controllabihty.

Larger distances between top and bottom of the column has a decoupling effect on

control loops at the bottom and top.

• The choice of measuring equipment can introduce significant delays and therefore

control difficulties into the system. Sneesby et al. (1997) discussed composition mea

surem ent and reported th a t these fah into three categories: directly, using one or

more online analysers; indirectly, using tem perature to infer composition; and exter

nally, by taking samples and measuring in the laboratory. Using analysers has many


advantages but is expensive, requires regular m aintenance, and often introduces sig

nificant tim e delay into the process. Inferential control is cheaper but can be less

accurate. External m easurem ent is useful for m onitoring the process but cannot be

used for closed loop control. Sneesby et al. (1997) also noted th a t inferential control

needs to be used w ith caution and placements need to be m ade where the change in

tem perature , accurately reflects changes in composition.

5.2 C on tro llab ility m eth od s

In this section, controllability m ethods for both the tim e domain and the frequency domain

as presented in the literatu re are reported.

5.2.1 S im ulation controllability analysis

The sim ulation or tim e domain approach requires th a t a candidate control system for the

process control system is designed and implemented. The controllers would be tuned by

some m ethod such as th a t presented by Ziegler and Nichols (1942).

The perform ance of the selected control system can then be evaluated against a set of

likely disturbances and set-point changes. The most controllable configuration would be

the one with the “best” performance.

W hich criteria are adopted for evaluating this “b est” perform ance will depend on the

apphcation. The tuning m ethod described by Ziegler and Nichols (1942) was designed to

give good overall performance. However, some applications, notably level control, require

th a t the disturbances to the m anipulated flow are kept to a minimum rather than tight

level control, so th a t they are not passed on downstream .

An essential requirem ent of aU control systems is the avoidance of both input and output

constraints. These constraints are requirements of both safety and quality. In the case

of m anipulated variables, the wear and tear of the valves m ust be considered and rapid

movement of the valves between fully open and fully shut should be avoided.

C H A P T E R 5. C O N TR O L OF R E A C T I V E B A T C H COLUM NS 95

Time domain performance indicators th a t can be applied to both the controlled output {y)

and the m anipulated variables (u) are reported by Skogestad and Postlethw aite (1996).

Those th a t relate to the speed of response:

• Rise time: Time taken to reach 90% of its final value should be small.

• Settling time: Time after which output remains within ±5% of its final value. Nor

mally required to be small.

Those th a t relate to the quality of response:

• Overshoot: Peak value divided by the final value should be less than 20%.

• Decay Ratio: Ratio of second and first peaks, which should be less than 0.3.

• Steady-State offset: Difference between the final value achieved and the desired value.

This is often required to be small.

Shinskey (1994) mentioned a num ber of performance indicators based on the integration

of the controller error. As well as indicators of performance, controller settings can be

optimised through minimisation of these functions.

In teg ra ted error (IE )

pt21 E = e d t (5.1)

Jti

The error can be either positive or negative and therefore an Integrated Error of zero could

be obtained w ith a continuously oscillating system. Consequently, this is not a measure

of stability.

C H A P T E R 5. CO NTR O L OF R E A C T I V E B A T C H COLUM NS 96

In teg ra ted a b so lu te error ( lA E )

r t 2= / |e| dt (5.2)

Jti

>t2lA E

This represents the to ta l area under the response curve on both sides of zero error. It is

an improvement over Integrated Error as it incorporates stability. The lA E will only tend

to a finite value for a stable loop which achieves its set point.

In teg ra ted square error (IS E )

ISE = / e^dt (5.3)J t i

This is similar in nature to the Integrated Absolute Error from a stability point of view since

the squaring of the error removes the negative sign. However, also due to the squaring,

larger errors gain a greater weight from smaller errors. Shinskey (1994) indicated th a t,

as a consequence of this, when optimising controller param eters by minimising the ISE,

longer settling times are encountered than for minimising lAE.

In tegra l o f t im e and ab so lu te error (IT A E )

f t 2I T A E = / t \e \d t (5.4)

J t 2

This m easure puts greater weight on the setthng tim e and as such has an opposite effect

on the optim al param eters than ISE, tending to penalise long term errors more than short

term errors. The minimum I T A E response wiU tend to have higher peak response and

shorter settling tim e than minimum l A E and ISE responses. Shortening the settling time

is particularly useful in batch processes where it is an economical advantage to achieve

changes in s ta tes as quickly as possible when set-point changes are made.

Su m m ary

The drawback with the simulation approach, is th a t it is impossible to guarantee th a t the

conclusions drawn are due to the inherent properties of the process, which is of prim ary


interest. The control system selected, or for th a t m atte r, the set of disturbances and

setpoint changes chosen, may well be responsible.

5.2.2 Frequency response controllability analysis

In order to overcome the lim itations of the simulation approach, Skogestad and Postleth

waite (1996) indicated th a t it may be more useful to assess controllability in the frequency

domain ra ther than the tim e domain. The frequency domain approach uses Frequency

Analysis of the process. From this, information about the process’ controllability can be

derived. This technique can be applied to an open loop process w ithout controllers, and

can also be usefully applied to open loop processes w ith controller for controller design

and also in closed loop processes. As mentioned earlier, one cannot use the simulation

approach on uncontrolled, unstable processes. Thus the frequency domain approach offers

the advantage of being controller independent, the m ain shortcom ing of the simulation

approach. A brief overview of the nature of frequency analysis will be presented, followed

by a description of controUabiHty measures th a t can typically be appbed.

F req u en cy an a lysis

The principle behind frequency analysis is th a t a sinusoidal input is appbed to a process,

after a long tim e, the output will oscibate with exactly the same frequency as the input

signal. However, the output wib be out of phase with the input [phase lag) and the

am pbtude wib change [amplitude ratio). As the frequency of the impinging sinusoid

changes so wib the values of the phase lag and amplitude ratio. These are ibustrated

for a first order process, e.g. buffer tank, in Figure 5.3 where as frequency increases the

am pbtude of the output signal decreases and phase shifts to being 90° behind the input

signal. C ontrobabibty results about the process can be in terpreted from this information.

It should be noted th a t although a sinusoidal signal is not a typical process input, Fourier

analysis shows th a t any discrete signal is a compound of sinusoids at different frequencies.


§5 ° 2 ,-2 0

M -60

É -80

1010 10 10 '

Frequency (rad/sec)

Figure 5.3: Frequency response of a first order system

C on tro llab ility in th e freq u en cy dom ain

In order to use frequency response controllability tools, it is necessary to use linearised

versions of the non-linear process models with the assum ption th a t the hnear model ac

curately describes the non-hnear model for a brief period after linearisation. It is vital

to confirm th a t this is indeed the case through comparison with non-hnear simulation.

It should be noted th a t, for highly non-hnear processes such as distiUation columns, the

hnear models wih not accurately describe the process s ta te a t large deviations. W hen

operating in batch mode, where the states are continuously changing, it is necessary to

hnearise the process model at multiple points to appreciate how controhabihty changes

during the process in order to design the controllers.

Thus in order to carry out a controhabihty analysis of a plant one requires a sufficiently

accurate, normaUy non-hnear, model describing the operation of the plant. A general

procedure for this would be:

• Simulate non-hnear plant model to desired point of hnearisation.

• G enerate hnearised approxim ation to model

Apply controhabihty measures

C H A P T E R 5. C O N TR O L OF R E A C T I V E B A T C H COLUMNS 99

e=r-v c(s) g(s)

Figure 5.4: Block diagram of feedback control

These are considered in more detail in Section 5.3.2.

S en sitiv ity , com p lem en tary se n s it iv ity and loop tran sfer fu n ction s

Consider a linear process model in term s of deviation variables as follows

y = g(s)u + gd(s)d (5.5)

where y denotes output variables, u the m anipulated input and d a disturbance. g{s) and

gd{s) are transfer function models which describe the effect on the ou tpu t of the input and

the disturbance. These are the basis of the controllability measures.

Controller error is defined as

e = r — y (5.6)

where r(s) denotes the reference value (setpoint) for the ou tpu t.

Typically, some form of feedback control will be used to control the process, defined as:

u = c(«)(r - y) (5.7)

c(s) is the transfer function of the controller. If these equations are combined together

and eliminate u the following closed loop transfer function is formed:

y = T r + Sgdd (5.8)


e = - S r + Sgdd (5.9)

Here the sensitivity function 5 = (1 + gc)~^ and the com plem entary sensitivity function

T = gc(l + gc)~^ = I — S. The transfer function around the open loop is L = gc.

Scaling

So th a t the controllability measures can be interpreted and meaningful comparisons made,

it is necessary to express process models in term s of scaled, or normalised, variables. Thus

all the input, disturbance and output variables should have a m agnitude less than 1 [i.e.

within the interval between -1 and 1). Skogestad (1996) indicated the following scahng

m ethod: u = I'^'max where u' denotes the unsealed and u the scaled variable, and

the largest allowable input change. For disturbances d = d'/d '^^^, for error, e = e'/e'max^

for ou tput y = 2/V^maa; for reference, r = r ' / wher e d'^^^ is the largest expected

disturbance and is the largest allowed control error. In most cases, it can be assumed

th a t the maximum values {u'âx-> 'max &re independent of frequency.

C on tro llab ility m easures

The following controllability rules are taken from Skogestad and Postlethw aite (1996)

where they identified th a t it would be useful to quantify these reasonable heuristics for

control system design.

Let u>B denotes the bandw idth of the system, defined as the highest frequency where

\L{j ‘ B)\ = 1 and ojd denotes the frequency at which \gd{j^d)\ first crosses 1 from above.

R u le 1 Ensure th a t speed of response is fast enough to reject disturbances. It is necessary

th a t ÜJB >

R u le 2 Ensure th a t speed of response is fast enough to follow setpoint changes. It is

necessary th a t ujb > where tOr is the frequency up to which tracking is required.


|L | Margin to stay within constraints (|u|<l)

Margin for performance (|y|<l)|Gd|

1

Control needed to reject disturbances

z/2

Margins for stability and performance

Figure 5.5: Controllability requirem ents

R u le 3 Ensure th a t input constraints are not violated when perfectly rejecting distur

bances. Must require th a t \g(juj)\ > at frequencies where \gd{jco)\ > 1.

R u le 4 Ensure th a t input constraints are not violated for perfect setpoint tracking. Must

require th a t \g{jco)\ > Rmax up to frequency where tracking is required.

R u le 5 Ensure th a t speed of response is slow enough to allow for tim e delays in process

and m easurem ent. So with tim e delay 6 it is required th a t wg < 1 /0 .

R u le 6 Ensure th a t speed of response is sufficiently slow to allow acceptable control

performance at low frequencies where a real RHP-zero, z, exists in the process. It

is necessary th a t wg < z /2 .

R u le 7 Phase lag constraint. Must require th a t wg < ujy,. Here the “u ltim ate” frequency,

uju is where the phase oi g[juj) is —180°. NB. Similar to Bode stability criterion.

R u le 8 Need sufficiently high feedback gain to stabilise the system where a real open-loop

unstable pole in g{s) exists a t s = p. The condition wg > 2p needs to be satisfied

for stabilisation.


5.3 M eth o d s for con tro llab ility an alysis

In this section, m ethods are described for the performance of both the simulation and fre

quency response controllabiUty analysis described earlier. A robust m ethod for generating

linear process information for numerically unstable models is also described.

5 .3 .1 M e th o d for s im u la t io n c o n tr o lla b il ity a n a ly s is

In order to perform the simulation approach a process model, typically non-hnear is re

quired. A candidate control system needs to be implemented and the controllers tuned by

some m ethod, e.g. Ziegler-Nichols. The following procedure would then be performed:

1. Simulate the process undisturbed to point of interest. If the process is batch then this

would be one of a series of times throughout the batch. If the process is continuous

it would typically be to the operating steady state .

2. Apply set-point change or process disturbance

3. M onitor the tracking error in the controllers until process restabilises

4. Calculate controllability measure such as an integrated error, described earher in

Section 5.2.1

The procedure would be repeated for further times for batch processes. For both batch

and continuous processes the procedure would be repeated for a range of disturbances and

set-point changes.

5 .3 .2 M e th o d for fr e q u e n c y r e sp o n s e c o n tr o l la b il i ty a n a ly s is

In order to use the frequency response techniques and other linear control tools it is

necessary to generate a linear approxim ation to the uncontrolled process at the desired


point of linearisation. The linear process model has the following form:

X — A x + 3 u

y = C x + T)u (5.10)

The m athem atics of how the state-space model is generated from the set of non-hnear

differential and algebraic equations (DAEs) th a t form the uncontrolled process model

( f { x , x , y , u ) = 0) is set out in Appendix B .l. The transfer function representation of the

linear model, used in Section 5.2.2, is generated by taking the Laplace transform of the

state-space model. The inputs (u) and outputs (y) expressed in the linear state-space

model are subsets of those in the non-hnear process model. The inputs (u) are selected

as the m anipulated variables and process disturbances. In this work, these would include

such quantities such as flowrates (distillate and reflux), condenser duty and reboiler duty.

The outputs (y) are selected as the controlled and m easured process variables, such as

composition, hquid level, and pressure.

The foUowing procedure would be followed to generate the hnear state-space models of

the process:

1. Simulate process undisturbed to predeterm ined hnearisation point with controhers

if necessary

2. Remove control if present. Then extract hnear process inform ation

3. Scale the hnear process model (Appendix B.2)

4. Apply controUabihty tools to hnear models.

5. Repeat for further hnearisation points if required

5 .3 .3 R o b u s t m e th o d for l in e a r is a t io n

The aforementioned m ethod has proved to be adequate for simple equihbrium stage models

and the packed column model. However, numerical difficulties have been encountered with


0.01 nonlinear - rigorous linear - rigorous— linear - simple

0.009

17 0.008

0.007

% 0.006

Q 0.005

0.004

< 0.003

ÜJ 0.002

0.001

80 1000 20 40 60Time [si

Figure 5.6: Comparison of robust method(Sim ple) to standard m ethod(Rigorous)

the rigorous equilibrium stage model. This section outlines those problems and proposes

a m ethod for dealing with these problems.

The standard m ethod outlined in the previous section was applied to the ethyl acetate case

study presented in Chapter 3 modelled using the rigorous equilibrium tray model. The lin

earised approxim ation to this model is compared to the original (non-linear, uncontrolled)

model. As already discussed, it is expected th a t following a step change disturbance, these

model responses should be approxim ately the same, for a reasonable period, which should

be at least long enough for control to be effective. In Figure 5.6 the distillate composi

tion response following a unit step increase in the reflux flow is shown. As can be seen

from the flgure the linearised model response is initially in the correct direction but after

20 seconds, the response inverts and travels in the opposite direction. Clearly numerical

problems in the linearised model result in this response.

In order to resolve these issues, a modiflcation to the procedure is proposed. T ha t is to


use the rigorous model to generate the sta te inform ation and then to introduce this into

the uncontrolled simphhed model, presented in C hapter 3 for the purpose of linearisation.

The simphhed model has two im portant simphhcations compared to the rigorous model.

1. The pressure is assumed to be constant.

2. The dynamic energy balances are converted to algebraic ones where the hquid en

thalpy is assumed locally constant.

These assum ptions are locally correct for short periods and are numerically stable. The

results of applying the m ethod are also shown in Figure 5.6. As can be seen a much closer

agreement is obtained over the tim e period shown.

In summary, this modihed hnearisation m ethod can now be described as:

1. Using the rigorous model, simulate until the desired point of hnearisation

2. In itiate the simphhed model with the process conditions of the rigorous model at

the point of hnearisation

3. Linearise the simphhed model at this point and then scale hnear model

4. Investigate the controhability of the process based on the hnearised version of the

simphhed model

5. Repeat steps 1-4 for further hnearisation points w ithin the batch

It m ay be necessary to include controhers in the rigorous model to reach the desired

process conditions prior to hnearisation. As long as these controhers are not included in

the simphhed model, their effect wiU not inhuence the controUabihty. (The conditions at

the point of hnearisation are, nevertheless, a result of the controhers.)

The justihcation for using the rigorous model at ah is th a t there cannot be any conhdence

th a t the simphhed model would be able to achieve the same process conditions as th a t of

the rigorous model which was dem onstrated in C hapter 3. Consequently, the hnearised

models generated could be quite different.


g 0 .95

■ 6 TRAYS 8 TRAYS

' 10TRAYS

0.9

0.8

0 .75

Tim e [hr]

Figure 5.7: Internal reflux ratio profiles

5.4 C ontro llab ility o f batch d istilla tio n co lum ns

In this section, the controllability of both tray and packed batch columns is considered.

Both the frequency response m ethods and simulation m ethods are used to investigate the

effect of column type, column height and batch tim e have on the controllability of the

column. The direct effect of the reaction in reactive batch distillation is also considered.

5 .4 .1 C a se s tu d ie s

For the controllability analysis, six different column configurations are considered: the tray

column, w ith 6, 8 and 10 trays and the packed column, w ith 4, 6 and 8 m etres of packing.

As indicated earher, due to the non-flnearities of the model and the changing process

conditions, it is necessary to generate linear models a t a num ber of tim e steps within the

batch. Four tim e steps were chosen: 3 hours, 4.5 hours, 6 hours and 7.5 hours. All of

these tim e steps are within the production phase when distillate is being w ithdrawn from

the column. The ethyl acetate case study as presented in C hapters 3 and 4 is considered.

The columns are aU operated under controlled composition, to 0.6 mole fraction of ethyl

acetate. The reflux profiles for the three tray columns is shown in Figure 5.7

For the tray column, the linear models are generated using the Robust linearisation proce

dure, described earlier. The packed column is linearised directly from an uncontrolled col

umn. The linearisation was performed using the LINEARISEtd^sk w ithin ^rPROMS which

C H A P T E R 5. CO NTR OL OF R E A C T I V E B A T C H COLUM NS 107

Input N om in a l U'^ m axUi L q 2.15 mol / s ±2.15 mol / s1/2 ^ 0.383 molf s ±0.383 mol / sU3 Qreb 8.85 X 10" J / s ±8.85 X 10^ J / s

O utp ut N om in a l y m axyi M t 100 mol ±10 mol

0.6 ± 0.022/3 Treb 346 K ± 5 K

Table 5.2: Inputs and outputs for linear model (6 Tray column at 3 hrs lin. point)

generates the state-space matrices (A,B,C and D). The state-space inform ation was im

ported into M atlab (The M athworks Inc., 1996) for analysis.

5 .4 .2 S c a lin g

The linear models are scaled. W hen scahng the Hnearised models, the reflux flow, To, the

distillate flow, D , and the reboiler heat supply Qj-eb are considered as the inpu ts/d istu rbances,

u for the linear model. The outputs, y, considered are the condenser holdup, M j, the dis

tillate composition, X£>, together with the reboiler tem peratu re , Treb- The distillate and

reflux flowrates which are to be used as m anipulated variables are allowed to vary by 100%,

actual values depend on the column type, size and batch tim e. The reboiler, normally the

principal source of disturbance, is expected to not vary by more th an 10%. For the out

puts: the condenser holdup, M j, is allowed to vary by 10 %, ±10 mol. A small variance

in the distillate composition is allowed ±0.02. The reboiler tem pera tu re is perm itted to

vary by ± 5 K. The results are summarised in Table 5.2 for the 6 Tray column at the first

Hnearisation point, 3 hours.

5 .4 .3 L in e a r m o d e ls

The Hnear models developed are first used to understand the effect of step changes in the

inputs to gain some general insights into control.


In Figure 5.8 is displayed the distillate composition response to 1 m ol/s step increase in

reflux flow, distillate flow and to a 1 X 10^ J /s , step increase in reboiler heat duty a t 3

hrs for the 6 tray column. The gain is positive for the reflux flow as more liquid returns

to the column, improving separation. The gain is negative for the reboiler as vapour flow

through the column increases, while the hquid reflux rem ains constant, hence a decrease

in the internal reflux ratio. Both of these inputs have a strong effect on the distillate

composition. The step increase in distillate flow has no effect on the composition, this is

due to the fact th a t this causes no change to the internal flows within the column, it only

increases the rate a t which the reflux drum drains. Therefore, the distillate flow can only

affect the composition by interaction with the reflux flow being used to control the level

in the reflux drum.

In Figure 5.9 is shown the effects of a step change in the reflux flow, distillate flow and

reboiler heat duty on the tem perature of the reboiler at 3 hrs for the 6 tray column. In

creasing the reflux flow causes the tem perature to decrease due to the increase of high

volatility components to the column. The step increase in reboiler heat supply causes an

increase in tem perature, due to the greater ra te of evaporation of high volatility compo

nents. Again, as with the composition responses, a step increase in distillate has no effect

on the reboiler tem perature as no internal changes occur. It is noted th a t all three inputs

have minimal effect on the reboiler tem perature.

In Figure 5.10 is displayed the responses of aU three of the tray column linear models (6,

8 and 10 trays) to 1 m ol/s step increase in reflux flow. The flgure shows the response of

the 6, 8 and 10 tray distillation columns on the top, middle and bottom plots respectively.

On each plot are the responses at each of the four hnearisation points. It is evident th a t

for each of the columns, the responses become progressively larger with time. The final

step response, 7.5 hours, is significantly larger than those a t the previous tim e steps and

this difference becomes larger with column height. It is thought th a t this is due, in part,

to the lower holdup of m aterial in the column towards the end of the batch. This explains

why the larger columns have a larger response at the final linearisation point since they

have been operating at lower reflux and hence at 7.5 hours, holdup in the column will be

C H A P T E R 5. C O N TR O L OF R E A C T I V E B A T C H CO LUMNS 109

lower.

For the two candidate control systems DV and LV discussed in Section 5.1, using distillate

to control the composition cannot be achieved without interaction with the level control

loop where reflux is m anipulated. Therefore the performance of the DV scheme depends

heavily on the performance of the level controller.

The DV configuration does have practical advantages over the LV configuration, partic

ularly during start-up . While the column composition is increasing during to ta l reflux,

no distillate is withdrawn so the distillate composition controller would sa tu ra te to zero.

Level in the reflux drum is controlled by m anipulating the reflux flow. However, this pe

riod is uncontrollable using the LV configuration, where reflux is used to control distillate

composition and the distillate to control the level in the condenser. During sta rtup , the

reflux valve would be fully open but the distillate valve would be fully shut to m aintain

the level in the reflux drum. Consequently level control cannot be achieved and the reflux

drum empties.

0.25

0.2 -

^ 0.

IŒ 0.1 h§0 srt 0.05

1I 0

-0.05

- 0.1

........ Heat Duty (+ 1 x 1 0 * J/s)Distillate (+ 1 moi/s) Reflux (+1 mol/s)

100 300 Time [sec]

Figure 5.8: Distillate composition response to step change in inputs

5. CO N TEO I OF F F A C T /V F BATCH COFFAfNS 110

0.1

- 0 2

- 0 3

-0 4

-0 5 Heat Duty (+ ^x 10 J/s) Distillate (+ 1 mol/s) Rellux (+ 1 mol/s)

100 200 300 Time [sec]

400 500 600

F ig u re 5.9: Reboi le r t e in [ )e ra t i i r e r e sp o n se t o s t e p c h a n g e in in p u t s

5 . 4 . 4 F i ' e q u e n c y r e s p o n s e b a s e d c o n t r o l l a b i l i t y

In F ig u re 5.11 is d i splayed t h e f recpiency r e sp o n se o f t h e t h r e e t r a y c o l u m n s a t t h e final

l i n e a r i s a t io n p o i n t (7.5 h o ur s ) . T h e cr i t ica l f r eq uen cy , w h e r e p h a s e is —180°, is r eached

a t 3 .77 r a d / s for all co lu m n s . T h e r e is a di f fe rence in p h a s e l ag profi le b e t w e e n 0.1 a n d 1

r a d / s b e t w e e n c o l u m n s which m a y hav e an i m p o r t a n t effect on c o n t r o l l a b i l i ty if t h e r e is

e x t r a p h a s e l ag a d d e d to t h e s y s t e m by d e a d t i m e o r f ro m a P I con t ro l l e r .

In F ig u r e 5 .12 is d i splayed t h e f r e q u e n c y r es po n se o f t h e t h r e e p a c k e d co l u m n h e i gh t s a t

t h e f inal l i n e a r i s a t i o n p o i n t (7.5 ho u r s ) . T h e p h a s e l ag is sm a l l e r in t h e low f r e q u en c y

reg ion , 10“ ^ t o 10“ r a d / s , t h a n for t h e t r a y co l u m n . H ow ev er , t h e p h a s e lag inc reases

m o r e r a p i d l y t h a n for t h e t r a y c o l u m n w i t h t h e cr i t i ca l f r e qu en cy , Wg, b e i n g rea ch ed a t t h e

lower f r e q u e n c y o f 0 .0545 r a d / s for t h e 8 m co l u m n , 0 .069 8 r a d / s for t h e 6rn c o l u m n a nd

0 .0711 for t h e 4 m c o lu m n. T h i s su g g e s t s t h a t c o n t r o l l ab i l i ty ge t s p o o r e r wi t h inc rea s i ng

c o l u m n he ig h t a s t h e cr i t ica l f r e q u en c y ge t s lower.

W h e n c o n s id e r in g th e c h a n g e in f r e q u en c y r e sp o n se w i t h t im e . F ig u r e 5 .13 sho w s t h e 10

T r a y c o l u m n a n d F ig u re 5.14 show s t h e 8 m p ac ked c o l u m n . T h e r e s p o n se s a re gene ra l ly

s im i la r for ea ch t i m e s t e p , however , t h e r e is a s l igh t ly l a rg e r p h a s e lag a n d s l ight ly l a rge r


gain for the final tim e step which suggests th a t both columns become harder to control at

the end of the batch.

5 .4 .5 C o n tr o lle r tu n in g

In this section, the tuning of composition controllers for both packed and tray columns is

considered using the linear models for the column with 10 trays and the 8 m packed column.

Proportional controllers with integral action (PI) are used and tuned using the Ziegler-

Nichols m ethod. The scaled models are closed using proportional only (P) controllers

and subjected to a unit (0.02 molfrac) decrease in composition setpoint. The gain is

increased until oscillation occurs. The ultim ate gain, Ky,, and u ltim ate period, Py, for the

oscillations, together with the corresponding controller param eters, ( K and r /) are shown

in Table 5.3. As expected from the frequency response analysis, the tray column can take

a much higher gain before destabihsing compared to the packed column.

Column Kn Pu K TITrayPacked

-9780- 3

71 s 110 5

-4401-1 .3 5

5992

Table 5.3: Controller tuning param eters

The step response of the controlled hnear models are shown in Figure 5.15 for the tray

column and in Figure 5.16 for the packed column. The consequence of the packed column

being harder to control is shown by the the longer setthng tim e, approxim ately 60 times

longer th an the tray column.

Having developed the controllers for both the tray and packed columns, it is possible to

investigate the open-loop controUabihty of these processes. In Figure 5.17 is shown the

frequency response of the tray column. The composition control by reflux process is shown

as |G |, the effect of the reboiler disturbance \ G d \ a-nd the effect of the open-loop is shown

as \L\. Here the bandw idth of the disturbance, Wj, is 0.0038 ra d /s and the controUer

bandw idth, 4.53 rad /s . The disturbance bandw idth is the frequency up to which

control is required to reject disturbances. This is very low compared to the bandw idth of


the controller, and therefore the controlled process can be expected to reject disturbances

effectively. However, the gain and hence bandw idth, may need to be reduced in a real

system to account for unmodelled dead time. The requirem ent is th a t the bandw idth is

smaller than 1/6 where 6 is the dead time. In this case, the controller would be destabilised

by more than 0.2 seconds dead time. In a real system, the bandw idth could be reduced to

allow for dead tim e of up to 262 seconds and still reject disturbances. It should be noted

th a t, due to the disparity between non-linear and linear system s, confidence in the low

frequency region is not high. Therefore, in practice, a real system m ay not be capable of

rejecting expected disturbances with this dead time.

In Figure 5.18 is shown the frequency response of the packed column. Here, the bandwidth

of the controller loop, u i is 0.034 rad /s . Control to reject disturbances is required up to

0.0046 rad /s so therefore the packed column controller will be able to reject expected

disturbances. The composition controller for the packed column wiU be capable of dealing

with a dead time of up to 29 seconds and by reducing the bandw idth it is possible to allow

for a dead time of up to 216 seconds and still reject disturbances.

5 .4 .6 N o n - l in e a r m o d e l s im u la t io n s

Having investigated the composition controllability using the frequency response technique

and developed controller settings for the columns, the controllers are tested by simulation

of the non-linear, rigorous models for tray and packed columns. The simulation approach

to controllability is also applied. Two aspects of controllability are investigated. Firstly,

how controllability is affected during the course of the batch and secondly how the choice

of the num ber of trays or how the choice of packing over trays affects the controllability.

In order to make these comparisons, four column configurations are employed: 6 , 8 and

10 trays and 8m packing. Each column is simulated, undisturbed to each of four points in

time: 3 hours, 4.5 hours, 6 hours and 7.5 hours. Then a 10% step increase in the reboiler

heat supply, the maximum expected, is applied. The responses of the composition con

trollers was m onitored until the composition settled. The Integrated Absolute Error (lAE)

and the Integrated Time Absolute Error (ITAE) (Shinskey, 1994) are used to investigate


the controllability. These values for the four column configurations are shown in Table 5.4,

the final value for the packed column is unavailable as it failed to settle. As is reflected in

the frequency response study, control is poor at the end of the batch for aU case studies

and the packed column has poorer performance than the tray column. However, earlier

in the batch the column appears to get harder to control during the high reflux period

(Figure 5.7). This suggests th a t the flow of reflux has some effect on controllability.

6 Trays 8 Trays 10 Trays 8m PackingTime [hrs] lAE ITAE lAE ITAE lAE ITAE lA E ITAE3 4.67 1600 6.81 2890 6.81 2890 17.3 146004.5 6.17 1840 7.79 2860 8.49 4360 18.53 141006 6.36 1870 8T4 3940 8.81 3700 18.96 156257.5 5.87 3610 12.1 17000 3120 8 x 106 - -

Table 5.4: Integrated controller errors

The controller error is shown in Figure 5.19 for the the 10 tray column and in Figure 5.20

for the 8 m packed column. These reflect a similar performance to the linear simulations,

with the tray column settflng in 60 secs and the packed in approxim ately 1 hour. However,

the frequency of the oscillations is much lower which m ay be due to m ism atch between

the linear and non-hnear models. Also shown in Figure 5.19 is an example of the poor

performance a t the end of the batch where at 7.5 hrs where the column does not settle

within 60 seconds.

5.5 E ffect o f reaction

In this section, the controUabihty of both reactive and non-reactive systems are investi

gated for the tray column model. However, in order to com pare the controUabihty of the

two system s, their process conditions m ust be as similar as possible for a comparison to

be inform ative. Hence the foUowing approach is taken:

• The rigorous tray model is simulated with control and reaction to the 3.5 hour

hnearisation point.


• The Robust M ethod for Linearisation (Section 5.3.3) is used to generate the reactive

linear model

• From the same linearisation point as above, the reaction term s in the mass and

energy balances are set to zero and the Robust M ethod for Linearisation is used to

generate the non-reactive linear model

For simulation controllability comparison, the rigorous tray model is sim ulated with control

and reaction for 3.5 hours, this is the reactive non-linear model. The reaction term s are

also set to zero in the non-linear model to create the non-reactive non-linear model.

5 .5 .1 S im u la t io n c o n tr o lla b il ity

Figures 5.21 and 5.22 show the controlled responses of the column to a set point change

of 0.01 in the setpoint of the composition controller for the reactive and non-reactive non

linear models. Figure 5.21 shows the response of the controlled ethyl acetate distillate

composition and Figure 5.22 shows the response of the uncontrolled reboiler tem perature.

The dashed One represents the response in the reactive system model and the solid line

represents the response in the non-reactive system model. It can be seen th a t there is

no significant difference between the reactive and non-reactive systems in term s of the

distillate composition. There is a slight difference in the tem pera tu re response which

reflects the fact th a t ethyl acetate is not being produced in the non-reactive system model

and therefore the tem perature rises faster as the ethyl acetate concentration falls.

5 .5 .2 F r e q u e n c y r e s p o n s e c o n tr o lla b il ity

The process gain as a function of frequency for the reactive and the non-reactive model

systems are shown in Figure 5.23. There appears to be very little difference between the

reacting and non-reacting systems. The two curves separate slightly at high frequencies

but the process gain is very small and therefore inconsequential. The phase is slightly

different at low frequencies bu t otherwise similar (not shown). As already mentioned.


the further one moves from the point of hnearisation, the less r eh able the hnear model

becomes. Therefore no meaningful conclusions can be drawn from this area of the response.

For this case study it is concluded th a t reaction has minimal direct effect on control.

However, as indicated earher, reaction does change the reflux ratio profile and therefore

m ay indirectly affect control. It is beheved th a t for other kinds of reactions, for example

highly exothermic or those with tight tem perature requirem ents, control wih be more

profoundly affected by reaction.

5.6 C onclu sion s

In this chapter, the controhability of reactive batch distiUation columns was investigated

using, both frequency response techniques and simulation based techniques. The frequency

response techniques require the generation of hnear models. It has been concluded th a t

packed columns are harder to control than tray columns. Increasing the height of a packed

column makes the column harder to control, but increasing the num ber of trays has a much

less significant effect on controhabihty. Control becomes harder for both types of column

towards the end of the batch as the holdup lowers and the reflux ratio increases. It has

also been concluded th a t, for this specific case study, reaction does not have a significant

effect on control. However, due the distinctive reflux ratio profile in a reactive column, as

compared to a non-reactive column, the reaction does indeed make the column harder to

control.

CTQrvCn

Ethyl A cetate C om postion Ax Ethyl a ce ta te com position A x Ethyl A cetate C om position A x

d3

%

5

OO

oo

og

to>HO

OCe:


10 ' 6 Trays 8 Trays— 10 Trays

10

10"

10"

-90

-136

-225

-270

Frequency [rad/s]

Figure 5.11: Tray column frequency response of distillate composition to reflux flow at 7.5 hrs

4m Packing 6m Packing 8m Packing

0

-9 0

i -3 6 0

-4 5 0

-5 4 010' 1010' 10 10'

Frequency [rad/s]

Figure 5.12: Packed column frequency response of distillate composition to reflux flow at 7.5 hrs


3 hrs 4.5 hrs— 6 hrs— 7.5 hrs

I.-90

-135

S’-180

£-225

-270 L - 10- 10®10' 10

Frequency [ratfs]

Figure 5.13: Frequency response of distillate composition to reflux flow for 10 tray column

(0 10

0

-9 0

0) -2 7 0

qI -360,

-4 5 0

10" 10' 10 10’ 10"Frequency [rad/s]

Figure 5.14: Frequency response of distillate composition to reflux flow for 8m packed column

.5. CONTZÔI OF F F A C T /V F BATC F COFFAfNS' 119

- 0.2

- 0 .4

- 0.6

I -0.1

Q.E<

- 1.2

- 1 .4

- 1.6

-260

Tim e [sec]

F ig u re 5.15: T r a y c o l u m n closed loop r e s p o n s e t o s e t p o i n t c h a n g e ( L i n e a r m o de l )

- 0 .5

< - 1 .5

-2

- 2 .51000 2000

Time [sec]3 0 0 0 4 0 0 0

F ig u re 5.16: P a c k e d co l u m n closed loop r e s p o n s e t o s e t p o i n t c h a n g e ( L i n e a r m od e l )

C H A P T E R 5. CO NTR OL OF R E A C T I V E B A T C H COLUMNS 120

Frequency [rad/s]

Figure 5.17: M agnitude composition response (10 tray column)

Frequency [rad/s]

Figure 5.18: M agnitude composition response (8m packed column)

.5. CO NTRO I OF F F A C T iV F BATCF COFFM NS 121

0 015 3 hrs

0.01

0 005

bLU

-0 00525

-0 0 1

- 0 015

- 0.02

- 0 02520 30

Time |sec |40

F ig ur e 5.19: C o m p o s i t i o n r o n t r o l l e r e r r o r in res])onse to 10% s t e p inc re as e in r eboi l er h e a t d u t y (R ig o r o u s T r a y C o l u m n )

001

0.005

-0.005

UJ- 0.012

§^ -0.015

-0 02

-0 025

-0 .03500 1000 1500 2000

Time [s]2500 35003000 4000

F ig ur e 5.20: C o m p o s i t i o n con t ro l l e r e r r o r in r e sp o n se to 10% s t e p in c re ase in r eboi l er he a t d u t y (8 m Pack ed C o l u m n ) a t 3 h o u r s


0.62

0.615

0.61

< 0.605

0.6

^ 0.595 non-reactive reactive

0.595 10 15 200

Time [min]

Figure 5.21: Distillate composition response of reactive and non-reactive tray columns

346

345.8

345.6

Ô 345.4

345.2 non-reactive reactive

3450 5 10 15 20

Time [min]

Figure 5.22: Reboiler tem perature response of reactive and non-reactive tray columns


,-2

- 40)

.•t;cO)(0

-8

-10

-2

Frequency [rad/s]

Figure 5.23: Composition frequency response of reactive and non-reactive tray columns

C h apter 6

M od elling and control o f

sh ort-p ath colum ns

This chapter considers the modelling of reactive distillation in short path evap

orators. From the literature it has been identified that although this process is

used industrially no work has been undertaken on modelling this process and it

is therefore important to develop a model so that the process can be more fully

understood. A dynamic model for reactive short path distillation is presented

which may also be used for control studies. The model is demonstrated using a

complex, industrially motivated example with thermally unstable reactants and

products. The sensitivity of the process to changes in operating conditions and

the control o f the process is investigated.

6.1 In trod u ction

Reactive distillation has been applied successfully in industry where large capital and en

ergy savings have been made through the integration of reaction and distillation into one

system. Reaction yields can be increased by the removal of volatile products from the

reaction zone, pushing the equihbrium towards the products. However, if the m aterials

124

C H A P T E R 6. MODELLING A N D C O N TR O L OF SHO RT-PATH COLUMNS 125

involved are tem perature sensitive, they will degrade when exposed to high tem peratures

for extended periods of time. Operating batch distillation under vacuum will decrease the

distillation tem peratu re but large residence times w ithin the still m ay nevertheless result

in therm al degradation. Furtherm ore, the degree of vacuum is lim ited by the pressure

drop across the column. Short-path distillation addresses these issues. In these columns

(Figure 6.1), a liquid feed is applied continuously or semi-continuously to the inside wall

of a single, externally heated, tube. A ro tating wiper spreads and moves the film perpen

dicular to the flow, thereby avoiding hot spots in the liquid. A condenser in the centre

of the evaporator ensures a short-distiUation path with minimal pressure drop, allowing a

high degree of vacuum and large evaporation rates.

Performing a chemical reaction within a short-path evaporator is an atypical use for the

equipm ent. However, it is being considered by industry as an alternative to reactive

batch distillation, such as described in Chapters 3 and 4, when dealing with reactions

involving highly tem perature sensitive m aterials. No work has been undertaken in the

open litera tu re addressing the modelling of reactive short-path distillation. Therefore it

is im portan t to develop a model so th a t the behaviour of these processes can be more

fully understood. Secondly, it is likely, as with chemical reactors where highly exothermic

reactions take place, th a t the process is difficult to control. Therefore it is im portant

th a t the model developed is dynamic so th a t the controllability of the process can be

investigated. In this chapter, a dynamic model for short pa th distillation is presented. The

model is dem onstrated using a complex, industrially m otivated example with therm ally

unstable reactan ts and products. The sensitivity of the process to changes in operating

conditions is investigated.

6.2 M od ellin g o f short-p ath evaporators

In this section, the modelling of short-path evaporators is investigated. As identified in the

literature survey in Chapter 2, the following general modelling issues have been identified

CVfAPTEff G. AfODEIIEVG AND CO NTRO I OF SDO FT-PATF COFDAFV^ 12G

FEED

CONDENSER

HEATING JACKET

WIPING SYSTEM

VACUUM

— COOLING

RESIDUE

Figure 6.1: Short-path evaporator

for iioii-reactive short-path evaporators:

1. Modelling of heat, mass and momentum transfer within the evaporator and con

denser films.

2. Modelling of the mass transfer from the evaporator film surface and through the

vapour gap onto the condensing surface.

6 .2 .1 M o d e l l i n g o f f i lm p h e n o m e n a

L iqu id h y d ro d y n a m ic s

The flow pattern on the evaporator surface is com])lex, resulting from a combination of the

gravity assisted flow axially and that created by the action of the wiper blades. McKenna

C JM PT E /f G. M O D E IIJN G AND CO N TA O I OF SD OFT-FA TF C0FFM N5' 127

Wall

5

Blade

Bow Wave

Direction of blade movement

F ig u re (J.2: T o p view of m ix i ng b o w wave ( a d a p t e d f ro m M c K e n n a 1995^

(199 5) in d ic a t ed t h a t a rol l ing b o w w a v e fo rm s in f ron t o f t h e b la de , sh o w n in F ig u re 6.2,

p r ov id in g t h e m ix i n g ac t ion .

C v e n g r o s e t al. (1995) mode l l ed t h e film b e h a v i o u r in a c o l u m n w i t h s e g m e n t e d wipe r s .

T h e c o l u m n was d iv ided in to a series o f l a m i n a r sect ions jo in e d by m ix e r s o f ze ro l e ng th

to r e p r e s e n t t h e co l u m n wiper s . T h i s c o m p a r e d well to e x p e r i m e n t a l r e s id ence t i m e dis

t r i b u t i o n d a t a .

Gene ra l ly , t h e flow is a s s u m e d to be e i th e r l a m i n a r , r e s u l t in g in r a d i a l t e m p e r a t u r e an d

c o n c e n t r a t i o n g r a d i e n t s , o r t u r b u l e n t in which case th e r e a re , no ra d i a l va r i a t io n s . Micov

et al. (1 997) n o t e d t h a t t h e well m ix e d , t u r b u l e n t , r eg im e ca n be a p p r o a c h e d in a wiped

film e v a p o r a t o r o p e r a t i n g a t h igh s p e ed . However , a t h igh e v a p o r a t i o n r a t e s , t h e r m a l

a n d c o n c e n t r a t i o n g r a d i e n t s do exi s t a n d c a n n o t be e l i m in a te d . I t was d e m o n s t r a t e d t h a t

d u r i n g t h e t u r b u l e n t r eg ime , s e p a r a t i o n p e r f o r m a n c e is b e t t e r t h a n in t h e l a m i n a r r eg ime.

In o r d e r to m o d e l t h e fluid d y n a m i c s o f t h e fi lm, t h e a p p r o p r i a t e fo rm of t h e N av ie r -S to k es

cnpiat ion nee ds to be inc luded in th e m o d e l . T h i s can be q u i t e c o m p le x b u t w h e n a s s u m i n g

a l a m i n a r film, t h e Ni issel t f o rm of t h e e q u a t i o n is suff i cient , Micov e t al. (1997) . T h i s is

th e a p p r o a c h a d o p t e d in th is tl iesis. I t is a s s u m e d t h a t t h e r ad ia l m ix i n g does s igni f ican t ly

affect t h e flow a lo n g th e c o l u m n in t h e axial d imen s io n .

C H A P T E R 6. MODELLING A N D C O N TR O L OF SHO RT-PATH CO LUMNS 128

evaporator d is t i l la t ion condenser

I W

11vapour

0

Figure 6.3: Tem perature profile in a wiped film evaporator (Lutisan et al 2002)

T em p era tu re and concentration profiles

Tem perature profiles can be quite pronounced within the column with large effects on

the perform ance of the column, necessitating an energy balance. Tem peratures will tend

to rise axially downwards as the higher volatility products are removed. Also, there are

radial tem pera tu re differences, particularly when flow is lam inar. Figure 6.3 shows a

typical radial tem perature profile Lutisan et al. (2002).

In the lam inar regime, the tem perature profile in the film is commonly assumed to be

linear radially, whereas in the turbulent regime it is assumed to be constant radially.

As the industrial column under consideration is mechanically wiped, it is assumed th a t

the m ixture is well mixed in all dimensions other th an axially. Therefore tem perature is

assumed to vary only axially and not radially.

6 .2 .2 M o d e l l in g o f e v a p o r a t io n p h e n o m e n a

Kawala and Dakiniewicz (2002) indicated th a t there are three m ajor regimes encountered

in high vacuum distillation: Molecular Distillation, Interm ediate Range and Equilibrium


distillation. Which regime prevails depends on the Knudsen num ber, K n = A/ho, which

expresses the ratio of the mean free path of vapour molecules. A, to the distance between

evaporator to condenser, L q . As pressure decreases the mean free pa th increases and so

does the Knudsen num ber, K n .

M olecu lar d istilla tion (K n > 10)

Here the vapour molecules travel the distance between the evaporator and the condenser

with practically no colhsions. Evaporation proceeds at the m axim al ra te given by the

Langmuir-Knudsen equation:

" ^ 2 i r M iR T

W here j i is the rate of evaporation of component i {rnol/w?s) and P° is the saturation

pressure of component i {Pa). This equation applies to both the evaporation ra te from

the evaporator surface and the re-evaporation from the condenser surface.

I n term ed ia te range (0.05 < K n < 10)

Here evaporation rate is reduced through vapour interaction, reducing the mean free path

of the vapour molecules. The apparent rate of distillation in this range, Je , is described

by the following:

3E - f y- j i (6 .2 )

where / = [1 — (1 — F){1 — A; and n are experim ental coefficients. F , is the surface

ratio defined as:

^ = 6 ^ ( - )

de is the diam eter of evaporation surface curvature.

A num ber of authors,e.g. Lutisan and Cvengros (1995a)(1995b) and Batistella et al. (2000),

have considered the treatm ent of the vapour phase in detail and have used M onte-Carlo


based simulations to determine the overall behaviour of the vapour phase during the

interm ediate range.

E quilibrium distillation (K n < 0 .05)

Here collisions are frequent and vapour molecules frequently retu rn to the liquid. Therm o

dynamic equilibrium exists a t the liquid-vapour interface. The param eter / tends towards

the value of the surface ratio, F, and the evaporation ra te is given by the following equa

tion:

Je = F X ji (6.4)

For apparatus with planar evaporation and condensation surfaces, the surface ratio, F,

is 0.5 (Kawala and Dakiniewicz, 2002). Thus, as a result of interm olecular collisions, the

probability of a gas molecule reaching the condenser is equal to the probability of its return

to the evaporation surface.

6.3 D yn am ic sh ort-p ath d istillation m od e l

Previous authors (Batistella and Maciel (1996), Nguyen and Le GofRc (1997) and Lutisan

et al. (2002)) have considered steady-state models of non-reactive short-path distillation.

In this work, a dynamic model is presented th a t also considers m ultiple reactions within

the evaporator liquid film. It is assumed th a t due to the mechanical wiping perpendicular

to the flow, variations in tem perature and concentration due to reaction, evaporation and

external heating occur only in the axial direction. Axial flow, driven by gravity, is assumed

to be lam inar and vacuum sufficiently high th a t molecular distillation occurs (Kn > 10).

The model is implemented in the gPROMS process modelling system (Process Systems

Enterprise Ltd., 1999) and is shown in detail in the following section.

C H A P T E R 6. MODELLING A N D C O N TR O L OF SHO RT-PATH CO LUMNS 131

EVAPORATOR

I

CONDENSER

WALL^ I

1

-FILM

Figure 6.4: Cross section of Evaporator

6 .3 .1 M o d e l l in g a s s u m p t io n s

As a result of the literature review, in Chapter 2 the following assum ptions have been

m ade in the development of a short path distillation model:

• Dynamic component molar balance

• Dynamic energy balance

• Lam inar liquid hydrodynamics

• M olecular diffusion

D y n a m ic m olar balance

Component m olar balances are used to describe the flow of m aterial through the Uquid film

on the evaporating surface of the column (shown in Figure 6.4 as given by Equation 6.5).

The film is considered to be well mixed perpendicular to the flow of m aterial downwards


therefore the m aterial balance is performed over the cross-sectional film annulus. The first

term on the right hand side of the equation represents the flow in and out of this annulus.

The second term represents the flow of m aterial out of the evaporator film towards the

condenser surface, governed by the appropriate rate expression. The th ird term is the

reaction term , modelled as homogeneous within the evaporator liquid film. The m aterial

balances are defined for each component and across the whole length of the column with

the exception of the boundary at the top of the column.

d M ^ d F ^= i = l , . . , N c , V z G (0 ,Z ] (6.5)

i= i

The condenser surface is modelled similarly except th a t the interfilm transport term Niz

is positive and the reaction term is om itted.

D y n a m ic e n erg y balance

The energy balance (Equation 6.6) considers the transport of energy through the film by

m aterial (first term on right hand side) and th a t removed by m aterial evaporating from

the film (second term ). The energy balance also considers the heat produced/ absorbed by

reaction (th ird term ) and th a t supphed by the heating medium through the walls of the

column (fourth term ).

f iu E— = - — - Y , N i , , h l , - 2 w T , S f ' £ T i A H f ~ Q V z e ( 0 ,Z ] (6.6)

i=l j=l

Lam inar liquid hydrodyn am ics

Extrem ely complex fiow patterns exist within a wiped film evaporator, a combination of

both gravity assisted axial fiow and radial and tangential flows due to wiping. However,

for the purposes of calculating the film thickness, it is assumed th a t the flow patterns in

both the evaporator and condenser films are lam inar. Therefore the film thickness can be

described by N usselt’s equation. For the evaporator film this is;

f#. = Z'y'gCff)' V. e [0. z] (6.7)

C H A P T E R 6. M ODELLING A N D C O N TR O L OF SH O RT-PATH COLUM NS 133

Film thickness definition,

,E _ ^ T , z^T,2 ^pTTTe

M olecu lar diffusion

V zG [0 ,Z ] (6.8)

It is assumed th a t the pressure in the short pa th column is sufficiently low to perm it the

ra te of diffusion to be characterised by the Langmuir-Knudsen equation. It is also assumed

th a t there is no re-evaporation from the condenser.

= 1.006 X 27rr^— = A â J = i = 1, N c Vz G [0, Z] (6.9)y 2 'K m iR T ^

A ncillary equations

In addition to the above equations the following constraints and definitions are included

in the model:

Mole fraction norm alisation, Vz G (0, Z]

Ncî,z — 1 (6.10)

t=i

Definitions, i = 1 ,.., A c Vz G (0, z]

(6 .11)

( 6 .12)

C H A P T E R 6. M ODELLING A N D CO NTR O L OF SHO RT-PATH COLUMNS 134

6 .3 .2 B o u n d a r y a n d in i t ia l c o n d i t io n s

The conditions a t the boundary at the top of the column, z = 0, for the evaporator surface

are specified as being equal to the feed of the column. Here to ta l m olar how rate, composi

tion and tem perature are specified. For the condenser surface the same specifications are

m ade, however the feed flowrate is specified as neghgible but non zero.

The column is operated from steady state . Therefore the initial condition is th a t all time

derivatives are zero, d X /d t = 0.

6 .3 .3 N u m e r ic a l s o lu t io n

The model, a set of partial, differential and algebraic equations (PD A Es), is implemented

within the gPROMS modelling language. Spatial variations are only considered in the axial

direction. The m ethod used for spatial discretisation is the Backward Finite Difference

M ethod (BFDM ). Orthogonal Collocation on Finite Elements M ethod (O CFEM ), which

is typically more com putationally efficient for a given accuracy, proved to be numerically

unstable. The effect of the num ber of intervals selected for both dynamic and steady-state

simulations is discussed in Section 6.4.1.

6.4 C ase s tu d y

An industrially m otivated complex reaction scheme is used to dem onstrate the applicability

of the model presented in the previous section. The scheme is, shown in Figure 6.5. Both

the reactan t. A, and the product, B, are therm ally unstable, degrading to form products, D

and E. The volatile by-product, V, wiU react further with the desired product B, producing

C, thereby lowering yields of B. The forward reaction is strongly exothermic, causing the

film tem peratu re to rise down the length of the evaporator even in the absence of an

external heat source. However, the volatile by-product, V, removes some of the heat

produced, thereby hm iting the tem perature rise. In this case study, no external heating

C H A P T E R 6. MODELLING A N D C O N TR O L OF SH O RT-PATH CO LUMNS 135

is required, i.e. Q = 0 in Equation 6.6. The column configuration and properties of the

feed, pure A, is outlined in Table 6.1.

O perating under these conditions, the composition profiles of components A and B and

the tem perature profile within the column are as shown in Figure 6.6. The composition of

the desired components B rises steadily along the length of the column as the composition

of reactan t A decreases. Under these conditions, there is minimal production of the side

product C as the volatile component, V, is evaporated from the film. The tem perature of

the film decreases slightly along the column, from an initial tem pera tu re of 353 K to 350 K

as heat is being removed by the evaporating m ethanol. This reduction in tem perature also

ensures minimal therm al degradation of the reactant and main product.

This base case will be used initially to investigate the degree of discretisation required for

both steady-sate simulation and during dynamic changes as a result of disturbances. It

is then used to explore how the way in which the column is operated, affects the yields

of the desired product. Specifically, the effects of feed flowrate, feed tem perature, heat of

reaction and evaporation efficiency are considered.

A B + V c

N/

D EFigure 6.5: Reaction Scheme

Column Length 0.4 mEvaporator Diameter 0.2 mCondenser Diameter 0.04 mFeed Flowrate 1 mol / hrFeed Tem perature 353 KFeed Viscosity 0.1Feed Density 1000 k g /m ^

Table 6.1; Short path column configuration

CffAPTE7( 6. M O D EIM N G AND CO N TEO I OF 5VfOPT-PATE COEDAfNS' 136

Axial d istance [m]

— C om ponent A • • " C om ponent B

£ 0.6

0.4

O 0.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

353

352 5

^ 352

3 351 5

351

H 350 5

350

349 50 0 05 0.1 0 15 0 2 0 2 5 0.3 0.35 0.4Axial distance [m]

Figure 6.6: Base case profiles: compositions (top), tem perature (bottom )

6 .4 .1 E ffe c t o f d i s c r e t i s a t io n

As indicated previously, for stability reasons, the method of discretisation employed in the

conversion of the partial differential equations (PDEs) into differential equations is the

backward finite difference method (BFDM). In this section, the degree of discretisation

required for accurate representation is investigated. As the process is continuous, the

performance under steady sta te is im portant. However, as the model is also required for

control studies, it is im portant that the degree of discretisation is sufficient to represent

the model during dynamic responses to disturbances.

In Figure 6.7 (Top), the effect of a step increase of feed tem perature of 5 K on the column

is displayed. The resulting tem perature profile at the outlet in shown for two levels of

discretisation, 3 intervals and 20 intervals. It is clear th a t the two plots remain close

throughout the transition from one steady state to another. The discrepancy between

the two plots remains at less than 0.02%. Figure 6.7 (Bottom ) shows the steady-state

CH APTE R 6. MODELLING AND CONTROL OE SHORT-PATH COLUMNS 137

355

354

20 Elem ents 3 Elem ents353

352

351

35016 18 206 8 10 12 140 2 4

Time [tirs]

352 5 20 Elem ents - w - 10 Elem ents - B - 5 Elem ents - A - 3 Elem ents

? 352

3 351 5

351

H 350.5

350

349 ,50 5 0 6 0 70 0 1 0.2 0.3 0.4

Length [m]

Figure 6.7: Outlet Tem pérature profile resulting from disturbance (TO P) Steady-state tem perature profile (Bottom )

tem perature profile across the column before the step increase of feed tem perature. The

tem perature profile is displayed for 3, 5, 10 and 20 intervals. It is clear from the plot that

the 3 and 5 interval discretisations do not give sufficient coverage of the interior of the

column and consequently give different profiles. However, the 10 and 20 interval profiles

are close with negligible difference and therefore it can be assumed th a t 10 intervals are

sufficient to give good accuracy. This degree of discretisation will be used when assessing

the sensitivity of the column to changes in operating conditions in the following.

6 .4 .2 E ffec t o f fee d H ow rate

The liquid film on the evaporator surface, as it is assumed to be well mixed perpendicular

to the flow, shares many properties with a plug flow reactor, PER. However, there is an

im portant difference between the two. The plug flow reactor has a fixed volume due to the

C H A P T E R 6. MODELLING A N D CO NTR O L OF SHO RT-PATH COLUMNS 138

geom etry of the reactor. The volume of a short-path evaporator film on the other hand

depends on the depth of the film, 6, and the geometry of the evaporator. The film depth

depends on the fiowrate of the liquid feed and its density and viscosity within the lam inar

regime. The residence tim e in a reactor is defined as r = F /jP , where V is the volume of

the reactor and F is the volumetric flowrate of the feed. For the liquid film, the volume

V is the product of the surface area of the evaporator cylinder, 27rr’eZ, and the depth of

liquid, 6. Therefore the definition of space time becomes:

2'KTpZbT =

F(6.14)

where the film thickness, 6, is described by Bird et al. (1960) as:

6 =' S/^Fpg27rr.

(6.15)

Therefore for a PER, r oc F~^ but for the film reactor, r oc F ~ ^ . This is illustrated in

Figure 6.8. The film reacto r’s residence time decreases less rapidly with increasing fiowrate

due to the increase in cross sectional area as the film gets thicker.

1600 PFR FILM1400

1200

800

600

400

200 1 1.5 2 .52 3 3 .5 4Flowrate [mol/hr]

Figure 6.8: Effect of feed flowrate on residence tim e

The effect of the feed fiowrate is investigated for the case study for a range of fiowrates

from 1 m ol/h r to 4 m ol/h r as shown in Figure 6.9. As expected, the yield decreases, from

79% at 1 m ol/h r to 38% at 4 m ol/hr, due to the fall in residence tim e. It should be noted

th a t the decrease in yield would be expected to be larger in a P F R due to the more rapid

drop in residence time.

C H A P T E R 6. M ODELLING A N D C O N TR O L OF SHO RT-PATH COLUMNS 139

80

70

>: 50

40

302.5

Flow [mol/hr]3 .5

Figure 6.9: Effect of feed flowrate on reactor yield

6 .4 .3 E f fe c t o f v i s c o s i t y

The reacting m ixture in the evaporator is assumed to have a constant viscosity of 0.1 P aS .

Although in reality, the m ixture viscosity would be tem perature dependent. It is, however,

im portan t to investigate how the yield is affected by viscosity. In Figure 6.10 is shown

how the yield varies as viscosity varies between 0.0001 P a S and 10 P a S . At the lowest

viscosity, the yield of B, is only 33%. At the highest viscosity, the yield approaches 100%.

As viscosity increases, the film thickness increases (see Equation 6.15) and so does the

residence tim e, shown in Figure 6.11. At very high viscosities, outside the range of the

figure, the yield begins to fall as the therm al degradation reactions become significant.

100

9 0

80

70

I 6 0

50

40

3010' 10''

V iscosity [PaS]

Figure 6.10: Effect of film viscosity on reactor yield

C H A P T E R 6. MODELLING A N D CO NTR O L OF SHO RT-PATH CO LUMNS 140

800 0

6 0 0 0

400 0

2000

10'Viscosity [PaS]

Figure 6.11: Effect of viscosity on residence tim e

6 .4 .4 E f fe c t o f f e e d t e m p e r a t u r e

The effect of feed tem perature on the yield of the product is investigated and shown in

Figure 6.12. If the feed tem perature is too low then there is insufficient tim e for the

product to be formed due to the slower ra te of reaction. As the tem peratu re rises, the

rate of reaction rises and therefore the yield of product is higher. However, if the feed

tem peratu re is too high, then therm al degradation of both the product and the reactant

wiU occur, thus lowering the yield again. This is observed in Figure 6.12 as a maximum

yield of 99.6% is found at a feed tem perature of 368 K, however, as the tem perature

increases beyond this, the yield is reduced and at 383 K, has dropped to 94.8%. This is

due to the therm al degradation of the reactant A to D, and of the desired product, B to

E.

100

95

85

80

7535 0 355 360 365 370 375 380 385

Feed Tem p [K]

Figure 6.12: Effect of feed tem perature on reactor yield

C H A P T E R 6. M ODELLING A N D C O N TR O L OF SHO RT-PATH COLUMNS 141

6 .4 .5 E ffec t o f h e a t o f r e a c t io n

The effect of uncertainty in the heat of reaction is found to be quite dram atic. For the base

case, the heat of reaction is initially assumed to be identical to the heat of vaporisation of

the volatile by-product. The heat removed by the evaporation of the volatile by-product

balances the heat produced by the main reaction. A 1% increase in the heat of reaction

results in a slight warming of the reaction m ixture over the length of the evaporator and

consequently the yield increases (from 79% to 87%). Lowering the heat of reaction by 1%

causes a significant drop in tem perature across the reactor resulting in a decrease in yield

(from 79% to 70%). It is interesting to observe th a t the process is extremely sensitive to

the heat of reaction and th a t for simulations, small uncertainties wiU have a large effect

on the predicted yield of the reactor.

6 .4 .6 E ffe c t o f e f f ic ie n c y

So far, it has been assumed th a t evaporation takes place w ithout resistance in the vapour

phase and w ithout re-evaporation from the condenser. The impUcations of lower mass

transfer rates are now considered. For non-reactive evaporation, the lowering of evapo

ration efficiency, compared to molecular distiUation, significantly reduces the yield of the

desired product (Batistella et ah, 2000). In the reactive case considered here, the effects

are more subtle. The reduction in efficiency as a result of resistance in the vapour phase,

for example at higher pressures, has a number of effects. Firstly, the equilibrium con

centration of the volatile by-product, V rises in the Uquid establishing a new equiUbrium

concentration. The higher volatile by-product concentration will increase the ra te at which

the desired product is reacted further in the side reaction to C. The degree to which this

occurs depends largely on the ra te constant of the side reaction as weU as on the efficiency

of the evaporation. However, for this case study, these phenom ena are significant for only

very low efficiencies (< 1%) and therefore the column m ay be operated over a large range

of efficiencies w ithout significant reduction in yield.

In Figure 6.13 is shown the effect when efficiency drops to zero and separation of the volatile

CJfA rTER G. M O D E II/N G AND CONTJÔI OF SZfORT-FATN COFFM NS 142

component, V, stops. Here the mole fraction of the desired product, B, is almost negligible,

0.0068, whereas the mole fraction of the by-product, C has risen to 0.69 at the outlet. The

reason is th a t the desired product is consumed by reaction with the volatile component, V,

in the film by forming by-product, C. This last example provides an im portant motivation

for operating this reaction in an evaporator as it would be impossible to produce the

desired product, B, without separating off component V.

Vf o . 8 -

l o . e -

I 0 .4 -

EO 0 . 2 -

■ - • Com ponent A — Com ponent B Com ponent C

0.05 0.1 0 .15 0 .2 0.25Axial d istance [m]

0.3 0 .35 0.4

Figure 6.13: Evaporator composition profile with no separation of V

6.5 Control o f short-path distillation

In this section, the control of short-path distillation is considered. The case study (Sec

tion 6.4) established that the composition of the desired product, B, is particularly sen

sitive to feed flowrate, heat of reaction and to the tem perature of the feed. In order to

control the outlet concentration of the evaporator, the feed flowrate is the only variable

that can be manipulated. The other two variables: heat of reaction and feed tem perature

are considered as disturbances to the system. In this case study, where no external heat

ing is applied, the heat supply is discounted as a source of disturbance or as a potential

m anipulated variable. The frequency response approach, outlined in Chapter 5 is used to

investigate the controllability of the process.


6 .5 .1 L in e a r is a t io n

As the column operates continuously, the operating base case steady-sta te shall be used

as the linearisation point. Figure 6.14 (T O P) shows the dynamic response of the non

linear process model to step changes in the input variables. The step changes are a 0.5%

increase in the heat of reaction, a 10% increase in the feed flowrate of A and a, -\-l K

change in the feed tem perature. New steady-states are achieved, about 12 minutes after

the step changes are apphed. As reflected in the case study, the increases in the heat of

reaction and the feed tem perature both cause an increase in the composition of the desired

product, B. The increase in feed flowrate, as expected, causes a decrease in the product

composition as residence tim e falls. In Figure 6.14 (BO TTO M ) is shown the same step

changes for the linear version of the model. It is clear th a t the linear model provides a good

approxim ation to the non-linear process model and it can be used w ith confidence in the

control study. However, it should be noted th a t the linear model will not account for the

decrease in composition of the desired product B, encountered for large feed tem peratures

when therm al decomposition becomes significant.

6 .5 .2 F r e q u e n c y a n a ly s is

The linear model is scaled with the output composition allowed to vary by, ±0.01 mole

fraction. The m anipulated variable, feed flowrate of A, is allowed to vary by, ±10%. The

maximum expected disturbances are ±0.5% in the heat of reaction and ±1 K for the

tem perature of the feed. The scaled model is used for the frequency analysis.

In Figure 6.15 is shown the frequency response of the composition of product, B, to the

feed flowrate. It is clear th a t there is a very large phase lag in the process due to the

high order of the process. This will have a destabilising effect and limit the gain of any

implemented controller.


6 .5 .3 C o n tr o l le d r e s p o n s e

For the control, the feed flowrate is m anipulated to m aintain the column composition of

the desired product, B. Feed tem perature and heat of reaction are considered as distur

bances. A proportional integral controller is implemented and tuned using the approach

of Ziegler and Nichols (1942). The controller gain, K c is 0.25 and the integral tim e, tj is

12.5 min.

The frequency response of the gain of the open loop (process and controller) is shown

in Figure 6.16. The response of the feed tem perature disturbance is also shown on the

figure. The difficulty in controlling this process is indicated by the fact th a t the distur

bance bandw idth, Wj at 0.54 rad /m in is above th a t of the controller bandw idth, a t 0.05

rad /m in . Therefore this controller is not effective at meeting the requirem ents of rejecting

disturbances while m aintaining the product within the required limits during its response.

The linear models are simulated with control for a set-point change in Figure 6.17 and a

feed tem perature disturbance in Figure 6.18. The response to a disturbance in heat of re

action is approxim ately the same as th a t for the tem perature disturbance. The controller

response to both the set-point change and the disturbance case is very slow, taking over

100 minutes to settle. The conclusion from Figure 6.16, th a t the controller is ineffective

against disturbances, is shown by the fact th a t the ou tpu t concentration deviation be

comes 3.3 times the perm itted maximum during the response. Some form of feedforward

control m ay be necessary to meet the product tolerances specified.

It can be concluded th a t the control of these processes is difficult due to the large process

time lag along the length of the column and the sensitivity of the system to small dis

turbances in the heat of reaction and column feed tem perature. The column will become

uncontrollable at high tem peratures with the feedback control scheme described when the

composition of desired product, B, decreases due to therm al degredation (Figure 6.12).


The action of the controller would be to decrease flowrate, which would increase the de

gree of therm al degredation. Therefore some form of tem pera tu re control, in addition to

composition control may be necessary.

6.6 C onclusion

The objective of this chapter was to develop a dynamic model th a t describes the behaviour

of reactive distillation in a short-path distillation column and use th a t model to investigate

the control of this process. The model was dem onstrated using an industrially motivated

reaction. The reaction contained a therm ally unstable reactan t and desired product,

m aking it suitable for treatm ent in a short-path distillation column under vacuum where

both tem peratures and contact times are low.

The effect of feed flowrate, feed tem perature, heat of reaction and evaporation efficiency

were considered. The impact of these variables on product yield is significant except for

evaporation efficiency, where a significant effect is only encountered for very low efficien

cies. Therm al degradation effects were minimised by operating a t low tem perature and

low residence times, and yields enhanced by the removal of volatile by-products, which

successfully dem onstrated the suitability of short-path distillation for combined reaction

and separation. However, the control of this process is difficult. The sensitivity of the pro

cess to operational changes, especially in feed tem peratu re and heat of reaction, and the

long tim e lag in the column contribute to this difficulty. The composition control of this

process is m ade more difficult by the reactions as the trend of increasing desired product

composition w ith tem perature reverses at high tem peratu re due to therm al degradation.

Therefore good composition and tem perature control of this process is critical.

C H A P r E R 6. MODELLING AND CONTROL OE SHORT-PATH COLUMNS 146

0.05

0,04

0,03

0,02

0,01Q. A,IN

- 0,01

- 0,02

-0 ,0 310

T im e [m in]

step Rœponse

0,05

0,04

0,03

0,02

A,INg 0,01

- 0,01

- 0,02

-0 ,0 32 6 8 10 120 4

T im e [min]

Fi gu re 6 .14: P r o d u c t , B, co i np os i t i on s t e p r e sp on se s . N o n - l in e a r m o d e l ( T O P ) , l inear mod e l ( B O T T O M )

CVfAf TEA 6. AfODEIITNG AND CONTAOI OF ÊOAT-TATN COIDAfNS 147

10'

10'

10 '

720

ID -7 2 0 -

-1 4 4 0

-2 1 6 010

Frequency [rad/'min|

F ig ur e 6.15: S h o r t - p a t h coluinii f r e tpiency re sp o n se o f p r o d u c t B c o m p o s i t i o n to feed fiow

-2

-3

10Frequency [rad/min]

Fi g ur e 6.16: S h o r t - p a t h co l u m n f re r iuency r e sp o n se o f co n t ro l l e r loop a n d t e m p e r a t u r e d i s t u r b a n c e

C H A P T E R 6. MODELLING A N D CO NTR O L OF SHO RT-PATH COLUMNS 148

0.8

0.6

■S.2Q.E<

0.2

- 0.210080 90

Tim e [min]

Figure 6.17: Column composition response to a set-point change (controlled scaled linear model)

3 .5

2 .5

IQ.

I

0 .5

30 100Tim e [min]

Figure 6.18: Column composition response to a feed tem peratu re disturbance (controlled scaled linear model)

C hapter 7

C onclusions and d irections for

future work

This chapter summarises the main findings o f this thesis regarding the mod

elling and control of reactive batch distillation in tray and packed columns and

the modelling of reactive distillation in short-path evaporators. Some directions

for future work are also given.

7.1 C onclusions

From a survey of the literature (C hapter 2) it was established th a t the m ajority of the

previous work on reactive batch distillation has focused on modelling with some work

on optim isation of operating policies and some Umited work on control. Some authors

have considered control, for example, Reuter et al. (1989), Wilson and M artinez (1997a),

Wilson and M artinez (1997b) and M onroy-Lopereba and Alvarez-Ramirez (2000) but they

have tended to use simple models. It is also noted th a t with, the exception of Sprensen

and Skogestad (1994) and Sprensen et al. (1996), no work has been undertaken on the

controllability of reactive batch distillation. Analysis of the controllability forms the basis

for understanding the features th a t make control of these processes difRcult. Adequate

149

C H A P T E R 7. CONCLUSIONS A N D D IREC TIO N S FOR F U T U R E W O R K 150

control is essential for m aintaining consistent operation. The flexibility which is offered by

batch operations can be further enhanced by the knowledge of how to modify the process

design, its operation or control structure to yield b e tte r controller performance.

In the literature survey, it was also concluded th a t no work has been undertaken on the

modelling of chemical reaction in short-path columns or on the control of these processes.

The m ain objectives of this thesis were to study the controUablity of reactive distillation in

tray and packed batch columns and in short-path distillation columns. In order to study

control, it is essential to develop rigorous dynamic models th a t accurately capture the

process behaviour. In Chapters 3 and 4, the modelling of tray and packed columns were

considered. In C hapter 5, the control of these columns was investigated and in C hapter 6

the modelling and control of reactive short-path distillation columns were considered.

7 .1 .1 M o d e l l in g o f r e a c t iv e b a tc h d i s t i l l a t io n in t r a y c o lu m n s

The objective of C hapter 3 was to present a rigorous dynamic model for the simulation

of reactive batch distillation. A simplified, but more numerically robust, model was also

presented for use in the control studies in C hapter 5. In order to justify the adoption of the

rigorous model over the simplified model for simulations, the two modelling approaches

were compared for the production of ethyl acetate. Two operating policies: constant reflux

ratio and controlled distillate composition, were considered for the production of a 0.6 mole

fraction m ixture of ethyl acetate and sim ultated using both models. The controlled policy

was also considered with reboiler disturbances.

In comparing the two modelling approaches, it appears th a t, although the predicted batch

holdups are reasonably close, less than 5% discrepancy between the m ethods, the dis

crepancies in the predicted batch times are significant and they increase as the column

is operated under disturbances. For the constant reflux ratio policy, the reflux flow re

mains approxim ately constant throughout. For the controlled case study, the reflux flow

is changed continuously. For the disturbance case study, the reboiler disturbances in tro

duce greater variances in the reflux flow. This large difference in the performance of the

simplified model compared to the rigorous under control would justify the adoption of the

C H A P T E R 7. CONCLUSIONS A N D D IREC TIO N S F O R F U T U R E W O R K 151

rigorous model for simulating reactive batch distillation. As reported by Sprensen and

Skogestad (1994), reactive batch distillation cannot be effectively operated w ithout some

form of feedback control.

The simulation times for the two modelling approaches were m arkedly different. The

simplified model took just 36 s but the rigorous took 574 s to solve, approxim ately 16

times longer. The sixteen fold increase in com putational tim e is particularly significant

if the rigorous model were to be employed for optim isation. W hen using a feasible path

approach for dynamic optim isation, a complete simulation is required to evaluate the ob

jective function at each step in the optim isation m ethod. However, M onroy-Lopereba and

Alvarez-Ramirez (2000) indicated th a t the controlled composition study can be thought

of as being equivalent to the optim al reflux ratio policy. It is im portan t to note th a t the

“optim al” reflux ratio profile for the rigorous model m ay be very different to th a t predicted

for the simple model.

7 .1 .2 M o d e l l in g o f r e a c t iv e b a tc h d i s t i l l a t io n in p a c k e d c o lu m n s

In C hapter 4, the production of ethyl acetate in a packed reactive batch distillation column

was considered. Packed columns can either be modelled as an equivalent tray column by

determining a suitable value for the HETF (Height Equivalent to a Theoretical Plate) or

more physically realistically by a rate-based model. The objective of this chapter was to

establish whether or not the less physically realistic tray column model would be suitable

for modelling the packed column for reactive purposes.

A packed column model which extended the work of Furlonge (2000) to include chemical

reaction throughout the liquid phase, was presented. This model was used to simulate the

behaviour of the constant reflux ratio and controlled composition case study presented in

the previous chapter but now, in a packed batch column. A m ethod was presented for

establishing the H ETP for a batch column and this was used on both operating policies:

constant reflux ratio and controlled composition. It was concluded th a t the H ETP is not

constant, varying with both tim e and packing height as a result of varying composition

C H A P T E R ? . CONCLUSIONS A N D DIRECTIONS FO R F U T U R E W O R K 152

and liquid flowrate. Consequently, the controlled composition policy was found to vary

m ore th an the constant reflux ratio pohcy due to greater changes in liquid flowrate.

A comparison was made between the packed column model and the rigorous and simpflfied

tray column models with 10 trays, which corresponded to the average H ETP found for

bo th the constant reflux ratio policy and the controlled composition policy. The packed

column was also compared to rigorous tray columns with 14 and 6 trays which correspond

to the maximum and minimum HETP values for the controlled composition policy. The

rigorous column model with 10 trays agreed most closely with the packed column model

while the simplified model and the larger and smaller rigorous column models had poorer

agreem ent. This suggest th a t if the equilibrium approach is taken, then it is vital to use

a correct H ETP value and to model the column using the rigorous and not the simplified,

tray column model. However, for the controlled case study with and w ithout disturbances,

the constant H ETP assum ption begins to break down due to the higher variance in HETP

experienced within the packing due to the varying flowrates. The greater variance may

also be due to the different dynamics within the columns leading to a slower behaviour of

the packed column.

The variation in H ETP was found to be too great to justify the adoption of the equilibrium

approach for modelling packed columns especially where liquid flowrates vary considerably,

such as during control. Therefore, in order to study control and controllability of packed

columns it is necessary to adopt a modelling approach th a t accounts for the physically

different mechanisms of heat and mass transfer encountered in a packed column and the

rate-based packed column model will therefore be adopted when investigating packed

column control.

7 .1 .3 C o n tr o l o f r e a c t iv e b a tc h d i s t i l la t io n

In this chapter, the controllability of reactive batch distillation columns was investigated

using, bo th frequency response techniques and simulation based techniques. The frequency

response techniques require the generation of hnear models. It was concluded th a t packed

column are harder to control than tray columns. Increasing the height of a packed column

C H A P T E R ? . CONCLUSIONS A N D DIREC TIONS FO R F U T U R E W O R K 153

makes the column harder to control, but increasing the num ber of trays has a much less

significant effect on controllability. Control becomes harder for both types of column

towards the end of the batch as the holdup lowers and the reflux ratio increases. It has

also been concluded th a t, for this specific case study, reaction does not have a significant

effect on control. However, due the distinctive reflux ratio profile in a reactive column, as

compared to a non-reactive column, the reaction does indeed make the column harder to

control.

7 .1 .4 M o d e l l in g a n d c o n tr o l o f r e a c t iv e s h o r t - p a t h e v a p o r a to r s

The objective of this chapter was to develop a dynamic model th a t describes the behaviour

of reactive distillation in a short-path distillation column and use th a t model to investigate

the control of this process. The model was dem onstrated using an industrially m otivated

reaction. The reaction contained a therm ally unstable reactan t and desired product,

m aking it suitable for treatm ent in a short-path distillation column under vacuum where

both tem peratures and contact times are low.

The effect of feed flowrate, feed tem perature, heat of reaction and evaporation efficiency

were considered. The im pact of these variables on product yield is significant except for

evaporation efficiency, where a significant effect is only encountered for very low efficien

cies. Therm al degradation effects were minimised by operating at low tem peratu re and

low residence times, and yields enhanced by the removal of volatile by-products, which

successfully dem onstrated the suitabifity of short-path distillation for combined reaction

and separation. However, the control of this process is difficult. The sensitivity of the pro

cess to operational changes, especially in feed tem peratu re and heat of reaction, and the

long tim e lag in the column contribute to this difficulty. The composition control of this

process is made more difficult by the reactions as the trend of increasing desired product

composition with tem perature reverses at high tem peratu re due to therm al degradation.

Therefore good composition and tem perature control of this process is critical.

C H A P T E R 7. CONCLUSIONS A N D D IREC TIO N S F O R F U T U R E W O R K 154

7.2 D irection s for fu ture work

In this section, some of the lim itations of this work are highhghted and recommendations

for fu ture work are outlined.

7 .2 .1 M o d e l v a l id a t io n

This thesis has employed detailed models to explore reactive distillation in three unit op

erations. The more detailed models are expected to have closer agreement to experimental

da ta than simpler models used previously in the h terature . In order to verify the accuracy

of the tray, packed and short-path column models presented in this work, experimental

results are required. A close m atch between experim ental results and model predictions

would further justify the use of such a high level of modelling detail despite the associated

large com putational cost.

7 .2 .2 M o d e l l in g d e ta i l

Experim ental work such as th a t described in Section 7.2.1 m ay reveal the need for even

more accurate process models, reaction models and physical property models.

P r o c ess m odels

The rigour of the tray column model may be enhanced by taking the following affects into

account:

• Tray Efficiency - For non-reactive distillation it m ay be possible to introduce Mur-

phree tray efficiencies, however, as was indicated by Ruiz et al. (1995), no methods

have been reported for the effect of reaction on plate efficiencies.

• Entrainm ent Effects

• Downcomer dynamics

C H A P T E R 7. CONCLUSIONS A N D DIREC TIONS FO R F U T U R E W O R K 155

For the packed column, the following effects may be introduced:

• Mass transfer in the liquid phase

• Dispersion effects

R ea ct io n m odels

In this thesis, the reactions have been assumed homogenous. To gain further insight

into the affect of chemical reaction, rigorous kinetics should be used. In the case of het

erogeneously catalysed reactions, a thorough understanding of the tran sp o rt mechanisms

involved, is essential to model the reaction properly.

P h ysica l properties

In this thesis, the physical properties have been assumed to be ideal. Real systems are

typically non-ideal. To properly characterise the system it m ay be necessary to use more

rigorous physical properties. The presence of azeotropes affects the feasibility of certain

separations and may have implications for control.

For the short-path column, it is assumed th a t pure molecular distillation occurs. It is

also assumed th a t no resistance occurs in the vapour phase. At anything other than

high vacuum, some resistance wiU occur. Some authors have considered modelling the

affect of vapour phase resistance using M onte-Carlo simulations (Lutisan and Cvengros

(1995a)(1995b) and Batistella et al. (2000))

7 .2 .3 F u r th e r c o n tr o l s tu d ie s

Effect o f reaction

It was concluded th a t the particular reaction used in this thesis, did not have a significant

affect on controllability. It would be useful to explore this fu rther by considering a number

of different reactions to establish whether or not this is a general conclusion or specific to

the reaction chosen.

C H A P T E R 7. CONCLUSIONS A N D DIREC TIONS F O R F U T U R E W O R K 156

S y stem a tic approach to m od e l reduction

In this thesis, an approach was presented for avoiding num erical problems while generat

ing linear models. The procedure used a simplified model. The degree of simplification

necessary to ensure a numerically stable linear model, was established by tria l and error.

In this particular case, the energy balance was assumed to be algebraic and the pressure

constant. However, it would be useful to develop a system atic approach for this model re

duction to ensure th a t the minimum degree of simplification is applied while guaranteeing

numerical stabihty.

N om enclature

tte effective interfacial area per unit volume rn? jvn?

A tray area rr?

Ab cross-sectional area of packed bed m ?

Ah to ta l area of holes on a tray w ?

Dout distillate flowrate m o lls

Fyj francis weir formula wall interaction param eter -

g acceleration due to gravity m / 5^

h specific molar enthalpy J jm o l

hyjeir weir height m

hliquid level of liquid on tray m

H enthalpy flowrate W

heat of reaction j J jm o l

k' heat transfer coefficient W jK m ^

K°'^ overall mass transfer coefficient m js

Ki liquid-vapour equilibrium constant for component i -

L liquid flowrate m o ljs

Mb holdup per length of packed bed m o ljm

Mb,i holdup of component i per length of packed bed m o ljm

Mi m olar holdup of component i mol

Mtot to ta l molar holdup mol

N{ ra te of transfer of component j from vapour phase to liquid phase m o ljsm

P pressure Fa

157

N O M E N C L A T U R E 158

Q reb oiler heat duty J /s

Q c condenser heat duty J /s

Rout reflux flow m o l/s

Rint internal reflux ratio -

Te radius of condenser m

Te radius of evaporator m

f'j rate of reaction j mol / m ^s

T tem perature K

r* tem perature at interface K

n normal boiling tem perature K

Utot internal energy J

V vapour flowrate m o l/s

^tray volume available between trays w?

^vessel vessel volume m?

y vapour mole fraction mol jm o l

y* vapour mole fraction at the liquid/vapour interface mol / mol

z packing height m

Z to ta l length of packed bed m

Greek Letters

a pressure drop vapour flowrate param eter -

6 film thickness m

€ packing void fraction -

l'ij stoichiom etry of component i in reaction j -

P average molar density m o l/m ^

P average mass density kg jm ^

fugacity coefficient

N O M E N C L A T U R E 159

Superscripts and Subscripts

C condenser

E evaporator

N O num ber of components

N R num ber of reactions

L liquid phase

V vapour phase

in inlet stream

out outlet stream

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A ppendix A

Process M odels

In th is a p p e n d i x , t h e m o d e l e q u a t i o n s for t h e r igorous t r a y m o d e l , t h e r a t e - b a s e p ac k in g

sec t i on a n d t h e anc i l l a ry ecpj ipment a r e de ta i l ed .

A .l Equilibrium tray m odel

T h i s is t h e r i g o ro u s e qu i l ib r iu m t r a y co l u m n mo d e l , as i n t r o d u c e d in C h a p t e r 3.

Lin . VoutII Vin

F ig u re A . l : Sieve t r a y

Lout

A ssumptions:

No e n t r a i n m e n t effects

• No d o w n c o m e r d y n a m i c s

• A d i a b a t i c o[derat ion

166

A P P E N D I X A. PRO CESS MODELS 167

• Phase equilibrium

• Perfect mixing

• Im m ediate heat input

• Negligible holdup in the condenser

Molar balance on component, i:

— Linî,in "P Vinyi,in Loutî ôutVi "P ^ ^ ^ ] îj^ j — 1; Nq(Â.1^

Liquid and vapour contributions to component holdup:

M ; = + M ^ o tV i i = 1, -/Va (A .2)

Energy balance:

rITJ f \ N R= Li„hi + Vir,hJ„ - + ( -— ^ in ^ fn + ^ in ^ Y n ~ ~ Voutf^^ + ( ( A . 3 )

i= i

Vapour and liquid contributions to internal energy:

Utot = M tth ^ + M Yoth^ - P V tray (A.4)

Total volume constraint:

V t r o y (A.5)

Equilibrium relationship:

y I — R{X{ z = 1, N Q (A. 6)


Norm alisation equations:

N C N C

Y Xi = J2yi = ' ( A . 7)î=l i=\

Liquid Hydrodynamics:

hliquid = (A .8)

= 1 .7 7 6 4 8 8 L » ./ (A.9)

Vapour Inflow Characterisation:

A P - P in - P - p ghiiquid (A .10)

V^n = sqrt I j pYnAh (A .l l)\P in ^ /


A .2 R ate-b ased m od el o f packing sectio n

This is the rate-base model for the packing section of the column. This model extends the

work of Furlonge (2000) to include reaction in the hquid phase. Reaction term s are added

to the liquid phase component mole balance and energy balance.

L(z=Z)

mm

z = Z

PackedColumnSection

z = 0

V(z=0)

mm

Figure A .2: Packing section

Assumptions:

• Film theory for heat and mass transfer

• Negligible hquid phase mass transfer resistance

• Neghgible axial and radial dispersion in both hquid and vapour phases

• A diabatic operation

• Liquid phase chemical reaction

A P P E N D I X A. P R O C E SS MODELS 170

B o u n d a ry C o n d it io n s

Liquid Inlet:

Lin = L{Z) (A.12)

Xi,in = î {Z) * = 1> ", (A .13)

= H ^ { Z ) (A.14)

Vapour Inlet:

Vin = y (0 ) (A.15)

ÿi.TO = 2/i(0) 1 = 1 , . . , A c (A .16)

V in h l = A'^(O) (A.17)

L iq u id p h a s e Vz G [0, Z )

Molar balance on component i:

^ = ^ + ^•' + ( ^ ) g i = 1 ,.., A c (A.18)

Energy balance:

- + k '^ a M { T * - T ^) + E ^ ^ ( - A i 7 f )r,- (A.i = l \ P / 7 = 1

19)

Mo/e fraction normalisation:

NcY ^ X i ^ l (A.20)i = l

A P P E N D I X A. PRO CESS M ODELS 171

V a p o u r p h a s e Vz G (0, Z]

Molar balance on component i:

^ = i = (A.21)

Energy balance:

f ) T j V r ) T T ^

= - — - - T") - ' £ N i h Y ( P , T ’ ) (A.22)i=l

Mole fraction normalisation:

Nç= 1 (A .23)

i=i

L iq u id / v a p o u r in te r fa c e Vz G [0, Z] Mass transfer rate o f component i

IZj ~ ~ ~ (a .24 )

p= a ^ A }) j^ rj^ y {V i ~ V i ) * — I 5 ••? ~ 1

Energy balance:

N ç N çk '^ a ,A i ( T ' - T ^) + ^ A ./ if (P ,T ^ ) = k ' ' 'a ,A t{ T ' ' - T ) N ih J { P ,T ^ ) (A.25)

i=l Î=1

Phase equilibrium:

4 { P , T \ x ) x i = < t , Y ( P , T \ f ) y t i = l , . . , N c (A.26)

Mo/e fraction normalisation:

NcE ÿ .* = l (A.27)i=l

A P P E N D I X A . P R O C E S S M O D E L S 172

A u x il ia ry e q u a tio n s Vz G [0, Z]

D efinition o f enthalpy flowrate:

= L h ^ (A.28)

= VhX (A.29)

Definition o f component holdup:

= M^ Xi i = 1 , Ac (A .30)

Pi i = 1, N c (A .31)

Internal energy:

u t = M th ^ - P v ^ 'u t (A.32)

h}' - P v ^ M ^ (A.33)

Geometry constraint:

= (A t (A.34)

Total mass transfer:

NcA, = ^ JV < (A.35)

i = l

In addition to the equations given in the previous section, relationships for pressure drop,

liquid holdup, interfacial area, binary mass transfer coefficients and heat transfer coeffi

cients are required. These are detailed in Furlonge (2000).

A r r E N U /X A . PROCESS MODEIS 173

A .3 R eboiler m odel

VoutLin

Fill

F ig u re A . 3: Reboi le r

A s s u m p t io n s :

• i n s t a n t a n e o u s h e a t in p u t

• liipiid p h a s e r ea c t i on

• pe r fec t m ix i ng a n d e ipi i l ibr ium

M o la r balance on co m p o n e n t i:

(It

L iqu id a n d vapour con t r i bu t io ns to co m p o n e n t holdup:

Ml = + M y, i = I , . . , N c

E n e n jy Ixilance:

i = 1 , N c ( A . 36)

(It J = 1

Vapour a n d l iquid con t r ibu t ions to in ter na l energy:

(A .3 7)

(A .3 8)

V t o t — ~ t r a y A.39)

A P P E N D I X A. PROCESS M ODELS 174

Total volume constraint:

Vt.ay = ^ ^ (A.40)

Equilibrium relationship:

Pi = KiXi 2 = 1, N c (A.41)

Norm alisation equations:

N C N C

= = (A .42)4 = 1 4 = 1

A .4 C ondenser m odel

The condenser model is either operated as a perfect condenser with no subcooling or

sub cooling and partial condensing are perm itted. W hen operating as a perfect condenser

the equilibrium relationship is appUed. W hen this assum ption is relaxed, the equilibrium

relationship is removed and the condenser cooling duty Q c is specified.

Vin Lout

Figure A.4: Condenser

Assum ptions:

• Negligible m aterial holdup

• Negligible pressure drop

A P P E N D IX A. PRO CESS M ODELS 175

M ass balance

Lout = Vin (A.43)

= Vi,in i (A.44)

Energy balance

0 = — Louth^ — Q c (A.45)

Equilibrium relationship (perfect condenser only):

yi = KiXi i = 1 , N c (A.46)

Normalisation equation (perfect condenser only:

N CY , y i = l (A.47)i = l

Pressure:

Pin = P (A.48)

A .5 R eflu x drum m od el

Assum ptions:

• Variable holdup

• Liquid and vapour phase considered

• Adiabatic operation

• Liquid phase chemical reaction

APfENDTXA. PEOCES'ÂfODElS' 176

Figure A.5: Reflux Drum

C o m p o n e n t mole balance:

( C k ] Y u r ^

Liquid a n d vapour con t r ib u t ion s to co m p o n e n t holdup:

M, = + MtotVi i = 1, --AVa

En ergy balance:

^ = kn h i - E(-A//«)r,

Vapour a n d l iquid con t r ibu t ion s to in te rn a l energy:

:A.49)

A.50)

(A.51

U t o t — — P V v e s s e l (A.52)

Tota l vo lu me const raint :

E q u i I i b ri u m re I a t i o n s h ip :

y, = Kpx, i = l , . . , N c

:A.53)

(A.54)

A P P E N D I X A . P R O C E S S M O D E L S 177

Normalisation equations:

N C NO' ^ X i = ' ^ V i = 1 (A.55)i = l i = l

A .6 A ccu m u lator m odel

The model of the accum ulator is similar to th a t of the reflux drum , except th a t there is

no liquid flow leaving the unit, i.e. Lout = 0.

A p p en d ix B

Linearisation and scaling m ethods

In this appendix, the procedure used in Chapter 5 to generate the linear models

from the non-linear process models is outlined together with the method used

to scale the linear models.

B . l L inearisation o f m od el eq u ation s

In this section, the m ethod used by gPROMS (Process Systems Enterprise Ltd., 1999) to

generate the linear model in the “LINEARISE” task is outlined. The non-linear process

model is expressed as a set of nonlinear differential and algebraic equations of the following

form:

f ( x , x , y , u ) = 0 (B .l)

x{ t ) and y{ t ) represent the differential and algebraic variables, respectively, which are

determ ined by the ÿPROM Ssimulation. x( t ) is the derivative of x{ t ) with tim e and u( t )

are the input variables, given functions of time.

If we consider a point on the solution trad jectory (z*(<), x*( t ) , y*{ t ) , u*( t ) ) . If we linearise

the nonlinear equations at this point we can obtain a linear model of the following from:

^ S x + ^ ê x + ^ ë y + ^ S u = 0 (B.2)OX Ox Oy o u

178

A P P E N D I X B. L IN E A R ISA T IO N A N D SCALIN G M E T H O D S 179

This can be rearranged to the following:

6x+

Sx_dx du. 6u _dx dy. Sy

(B.3)

Assuming th a t d x d y is not singular, true for most systems of index 1 or less, then:

Sx -1 Sx

Sy ,dx dy. ,dx du. Su(B.4)

this is the linear form of the original non-linear form of equation (B .l) which is separated

into the two sta te space equations to give:

X = A x + B u

y = C x + T>u (B.5)

The LINEARISE task in gPROMS generates the constant m atrices (A, B, C and D) for

specified subsets of the input variables(u) and the ou tpu t variables(?/)

B .2 Scaling o f th e linear m odels

The general, unsealed, statespace representation of the linear process model is:

X = A x + B 'u '

y' = C x -f C u '

(B.6)

(B.7)

where x contains the process states, u' contains the unsealed inpu ts/ disturbances, y ' con

tains the unsealed output variables and A , B ', C and D ' are the unsealed constant m atri

ces generated, in this work, by the ^PROM S LIN E A R ISE id .sk . As indicated in C hapter 5

A P P E N D I X B. L I N E A R I S A T I O N A N D S C A L I N G M E T H O D S 180

it is necessary to scale the model. W hen considering the scaling of the model, the vector

u[nax contains both the maximum allowable values for the inputs and the expected m axi

mum values for the disturbances. Vector y'âx contains the maximum allowable values for

the outputs.

The diagonal ou tpu t scaling m atrix Sy with elements syij is defined as:

^ ya — ymax,i ~ I : - - f Ny (B-8)

syij = 0 i J = l , . . , N y i ^ j (B.9)

The diagonal inpu t/ disturbance scaling m atrix Su w ith elements suij is defined as:

SUii = '^max,i ~ (B.IO)

suij = 0 = (B .l l)

The scaled state-space model becomes:

X = A x + B u (B.12)

y — C x -f D u (B.13)

where

B = B'5,

C = 5 - ’ C'

D =

with u and y being the scaled input and output vectors.

Date post:	11-Feb-2022
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Modelling and Control of Reactive Distillation Processes

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