Modeling, Simulation and Control of a multicomponent

Batch Distillation Processes

Thesis submitted in partial fulfillment of the requirements for the award of

the degree of

Master of Technology

in

Chemical Engineering

by

P.V.Radha Krishna Adari

(06CH6027)

under the guidance

of

Prof. Amiya K. Jana

DDEEPPAARRTTMMEENNTT OOFF CCHHEEMMIICCAALL EENNGGIINNEEEERRIINNGG IINNDDIIAANN IINNSSTTIITTUUTTEE OOFF TTEECCHHNNOOLLOOGGYY

KKHHAARRAAGGPPUURR,, IINNDDIIAA

2008

CCEERRTTIIFFIICCAATTEE

This is to certify that the thesis entitled “Modeling, Simulation and

Control of a multicomponent Batch Distillation Processes” submitted by

P.V.Radha Krishna Adari (Roll no. 06CH6027), in the partial fulfillment for the

requirement for the award of degree of Master of Technology in Chemical

Engineering of Indian Institute of Technology, Kharagpur during the academic

session 2007-2008 is a bonafide record of the project work carried out by him under my

supervision and guidance. The approval does not necessarily endorse or accept every

statement made, opinion expressed or conclusion drawn as recorded in the thesis. It only

signifies the acceptance of the thesis for the purpose for which it is submitted.

____________________ Dr. Amiya K. Jana

Date: _______________ Department of Chemical Engineering Place: IIT Kharagpur Indian Institute of Technology

Kharagpur-721302

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IInnddiiaann IInnssttiittuuttee ooff TTeecchhnnoollooggyy

KKhhaarraaggppuurr--772211330022

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Prof. Amiya K. Jana for his

guidance and assistance in this project. The technical discussions with Prof. Amiya K.

Jana were always been very insightful, and I will always be indebted to him for all the

knowledge he shared with me. His prompt responses and availability despite his

constantly busy schedule were truly appreciated. The reality is that Prof. Amiya K. Jana

was much more than an advisor for me. He always helped me in all the technical and

non-technical issues during the production of this work. His encouragement and efforts

led this project to successful completion in a timely fashion.

I would like to extend my sincere thanks to my classmates. Finally, I express my

deep sincere thanks to my Parents who motivated and encouraged me for higher studies,

without which wouldn’t have been possible.

P.V.Radha Krishna Adari

International Journal Publications

Adari, P. V. Radha Krishna and Jana, Amiya K. (2008) "Comparative

Control Study of a High-Purity Ternary Batch Distillation," Chemical Product

and Process Modeling: Vol. 3: Iss. 1, Article 26.

Avialable at http://www.bepress.com/cppm/vol3/iss1/26

P.V. Radha Krishna Adari and Amiya K.Jana (2009) “Nonlinear Adaptive

Control of a Batch Reactive Distillation Column,” Chemical Engineering

Journal: Vol 150: Iss. 2-3, pages 516-526.

Available at doi:10.1016/j.cej.2009.03.015

CONTENTS Page. No List of Tables………………………………………………………. iii List of Figures……………………………………………………... iv Abstract ………………………………………………………… vi Chapter 1: Introduction……………………………………………. 1 Chapter 2: Literature review………………………………………. 5

Chapter 3: Modeling, Simulation and Control of a multicomponent

Batch Distillation Column

3.1. Process Description……………………………………. 8

3.2. Assumptions…………………………………………… 10

3.3. Modeling Equations ……………………….…………... 11

3.4. Simulation Algorithm………………………………….. 18

3.5. Synthesis of Control Scheme…………………………... 19

3.6. Simulation Results and Discussion…………....……….. 20

Chapter 4: Modeling, Simulation and Control of a multicomponent

Reactive Batch Distillation Column

4.1. Process Description……………………………………. 34

4.2. Assumptions…………………………………………… 35

4.3. Modeling Equations ………………………….….……. 35

4.4. Simulation Algorithm………………………………….. 40

i

4.5. Synthesis of nonlinear Control scheme……………….. 41

4.6. Simulation Results and Discussion……………………. 48

Chapter5: Conclusions and Scope of future work………………….. 63

Notation…………………………………………………………….. 64

References…………………………………………………………... 65

Appendix……………………………………………………………. 67

ii

List of Tables Table page

1. Batch distillation column specifications …………………………. 17

2. Tuning parameters and ISE values for servo performance with respect

to component (XD,1)……....…………..…………………...……… 31

3. Tuning parameters and ISE values for servo performance with respect

to component (XD,2)…………………………………………….….. 31

4. Tuning parameters and ISE values for regulatory performance with

respect to component (XD,1) (for heat input)..…………………....... 32

5. Tuning parameters and ISE values for regulatory performance with

respect to component (XD,2) (for heat input )..…………...…..…….. 32

6. Tuning parameters and ISE values for regulatory performance with

respect to component (XD,1) (for XB )..….……………..……….........33

7. Tuning parameters and ISE values for regulatory performance with

respect to component (XD,2) (for XB )..………….…………..……... 33 B

8. Reactive batch distillation column specifications...………..………. 39

9. Tuning parameters for ASE 1 and ASE 2 ……………..….….……..62

10. GSPI and GMC controller tuning parameters….....……….………..62

11. ISE values for comparison of ASE 1 and ASE 2 …………..…...….62

12. ISE values for comparison of GSPI and GMC ..………..….……….62

iii

List of Figures Figure page

1. Schematic diagram of multicomponent batch distillation process...…… 18

2. Bubble point calculations…………………………………………….... 23

3. Open-loop process dynamics at stat-up phase ……….……………….. 31

4. Open-loop process dynamics at production phase………………….….. 32

5a.Comparative closed loop performance of PI, NLPI and GSPI control

algorithms with a set point change in XD,1.............................................. 34

5b.Comparative closed loop performance of PI, NLPI and GSPI control

algorithms with a set point change in XD,2............................................. 35

5c.Third component in the bottom after second slop cut ............................ 36

6a.Comparative Regulatory performance of PI, NLPI and GSPI control

algorithms with step change in reboiler heat duty for XD,1………….….37

6b. Comparative Regulatory performance of PI, NLPI and GSPI control

algorithms with step change in reboiler heat duty for XD,2………….….38

7a. Comparative Regulatory performance of PI, NLPI and GSPI control

algorithms with step change in XB,1…………………………………….39

7b. Comparative Regulatory performance of PI, NLPI and GSPI control

algorithms with step change in XB,2………………………………….....40

8. Block diagram of an adaptive state estimator control strategy………...51

9. Open loop performance of reactive batch distillation without reaction.. 51

10. a. Open-loop response of the batch reactive distillation column under

total reflux ratio………………………………..………...................... 52

10. b. Open-loop response of the batch reactive distillation column under

reflux ratio 0.8……………………………………….…….…………. 53

iv

11. Comparison of the estimated outputs (ASE 1, ASE 2) and process

outputs with out any change for production stage…………………….. 54

12. Comparison of the estimated outputs (ASE 1, ASE 2) and process

outputs with two consecutive step changes in heat input to reboiler…..55

13. Comparison of the estimated outputs (ASE 1, ASE 2) and process

outputs with two consecutive step changes in Tray efficiency……...…56

14. Comparison of the estimated outputs (ASE 1, ASE 2) and process

outputs with initialization error in VnT ynT,3 ..……………………..….…57

15. Comparison of the estimated outputs (ASE 2) and process outputs

with initialization error in both VnT ynT,3 and r3…………………..……..58

16. Comparative closed loop performance of GSPI and GMC control

algorithms ………………………………………………………………59

17. Comparative Regulatory performance of GSPI and GMC control

algorithms with step change in reboiler heat duty for XD,3......................60

18. Comparative closed loop performance of GSPI and GMC control

algorithms with a set point change in XD,3 ..............................................61

v

ABSTRACT

Modeling, Simulation and Control of a multicomponent Batch Distillation Processes

BY P.V.Radha Krishna Adari

The present work is devoted to study the comparative closed-loop performance of

multicomponent non-reactive batch distillation followed by a multi-component reactive

batch distillation column. First, the dynamic model of the non-reactive batch column has

been developed and then simulated. In the next, three different control schemes, namely

proportional-integral (PI), Nonlinear Proportional-Integral (NLPI), and Gain Scheduled

Proportional-Integral (GSPI), have been designed. Then the control structures have been

employed on the simulated non-reactive batch distillation column for comparative control

study. Several simulation experiments have been conducted to examine the servo as well

as regulatory performance.

In the subsequent part, the model structure of a multi component reactive batch

distillation column has been constructed. Then dynamic simulation has been performed to

observe the column performance at start-up and production phase. In the next, two non-

linear adaptive state estimators have been designed for the concerned process.

Interestingly, the state predictor includes only the component mass balance equation

around condenser-reflux drum system. Subsequently, an adaptive control scheme has

been synthesized for the example reactive column. Comparative performance study has

been conducted between the non-linear adaptive controller and the GSPI using the

simulative rectifier.

vi

1

Chapter-1 Introduction

______________________________________________________

Distillation is the oldest separation process and the most widely used unit

operation in the industry. The process of distillation has been around since 4000 BC,

where it was first discovered by the Babylonians. It wasn’t until an Arabian alchemist

named Jabir Ibn Hayyan (721-815 AD) invented the alembic still that distillation became

used for beverage purposes. The process of distillation quickly spread after the Persian

physician and scientist Rhazes (865-925 AD) published a book on the process and theory

of distilling alcohol. In the early 19th century the basics of modern techniques including

pre-heating and reflux were developed, particularly by the French, then in 1830 a British

Patent was issued to Aeneas Coffey for a whiskey distillation column, which worked

continuously and may be regarded as the archetype of modern petrochemical units. In

1877, Ernest Solvay was granted a U.S. Patent for a tray column for ammonia distillation

and the same and subsequent years saw developments of this theme for oil and spirits.

Distillation separates two or more liquid components in a mixture using the

principle of relative volatility or boiling points. The greater the difference in relative

volatility the greater the nonlinearity and the easier it is to separate the mixture using

distillation. The process involves production of vapour by boiling the liquid mixture in a

still and removal of the vapour from the still by condensation. Due to differences in

relative volatility or boiling points, the vapour is rich in light components and the liquid

is rich in heavy components.

Often a part of the condensate is returned (reflux) back to the still and is mixed

with the outgoing vapour. This allows further transfer of lighter components to the

vapour phase from the liquid phase and transfer of heavier components to the liquid

phase from the vapour phase. Consequently, the vapour stream becomes richer in light

components and the liquid stream becomes richer in heavy components.

2

Different types of devices called plates, trays or packing are used to bring the

vapour and liquid phases into intimate contact to enhance the mass transfer. Depending

on the relative volatility and the separation task (i.e. purity of the separated components)

more trays (or more packing materials) are stacked one above the other in a cylindrical

shell to form a column. The distillation process can be carried out in continuous, batch or

in semi-batch (or semi-continuous) mode.

In continuous distillation column feed is fed to the column at one or more points

along the column. Liquid runs down the column due to gravity while the vapour runs up

the column. The vapour is produced by partial vaporisation of the liquid reaching the

bottom of the column. The remaining liquid is withdrawn from the column as bottom

product rich in heavy components. The vapour reaching the top of the column is partially

or fully condensed. Part of the condensed liquid is refluxed to the column while the

remainder is withdrawn as the distillate product. The column section above the feed tray

rectifies the vapour stream with light components and therefore is termed as rectifying

section. The column section below the feed tray strips heavy components from the vapour

stream to the liquid stream and is termed as stripping section.

Batch distillation is the oldest operation used for separation of liquid mixtures.

For centuries and also today, batch distillation is widely used for the production of fine

chemicals and specialised products such as alcoholic beverages, essential oils, perfume,

pharmaceutical and petroleum products. It is the most frequent separation method in

batch processes (Lucet et al., 1992). The essential features of a batch distillation column

are:

• A bottom receiver/reboiler which is charged with the feed to be processed and

which provides the heat transfer surface.

• A rectifying column (either a tray or packed column) superimposed on the

reboiler, coupled with either a total condenser or a partial condenser system.

• A series of product accumulator tanks connected to the product streams to

collect the main and or the intermediate distillate fractions.

3

Operation of such a column involves carrying out the fractionation until a desired

amount has been distilled off. The overhead composition varies during the operation and

usually a number of cuts are made. Some of the cuts are desired products (main-cuts)

while others are intermediate fractions (off-cuts) that can be recycled to subsequent

batches to obtain further separation. A residual bottom fraction may or may not be

recovered as product (Mujtaba, 1989).

The batch processing is a very attractive separation operation mainly for the

following reasons (Barolo and Berto, 1998a)

(i) A single batch column can separate a multicomponent feed mixture into

several products within a single operation,

(ii) The flexibility of a batch process is such that the frequent change of market

demands and strict product purity requirements can be accommodated,

(iii) The production amounts in a batch process are usually small with minimum

raw material inventories; this often results in an economic incentive.

The basic difference between a batch column and continuous column is that in

continuous column feed is continuously entering the column, while in batch distillation

the reboiler is normally fed at the beginning of the operation. Also, while the top products

are removed continuously in both batch and continuous operation, there is no bottom

product in the batch distillation. Since in a continuous operation the total product flow

rate equals that of incoming feed, the process reach steady state. In batch distillation, on

the other hand, the reboiler gets depleted over time, so the process is unsteady.

Batch distillation requires the least amount of capital for separating relatively pure

components. Continuous distillation generally requires a separate column for each

component. The same operation could be obtained with one batch distillation column

with product cuts.

Reactive distillation is one major step in the history of distillation in achieving

these goals. Reactive distillation processes couple chemical reactions and physical

separations into a single unit operation. These processes as a whole are not a new concept

as the first patent dates back to the 1920s. The initial publications dealt with

homogeneous self catalyzed reactions such as esterifications and hydrolysis.

Heterogeneous catalysis in reactive distillation is a more recent development and was

4

first described by Spes. While the concept existed much earlier, the first real world

implementation of reactive distillation took place in 1980s.

Batch distillation with or without chemical reaction is used in industry for the

production of small amounts of products with high added value and for processes where

flexibility is required. Distillation with chemical reaction is well suited for processes

where one of the products has a lower boiling point than other products and reactants.

The higher volatility of this product induces a decrease of its concentration in the liquid

phase, thus leading to higher reactant conversions than with reaction alone. The higher

volatility of this product results in a decrease in its concentration in the liquid phase,

therefore increasing the liquid temperature and hence reaction rate, in the case of an

irreversible reaction. With reversible reactions, elimination of products by distillation

favors the forward reaction. In both cases higher conversion of the reactants is expected

than by reaction alone. Therefore, in both cases, a higher amount of distillate

(proportional to the increase in conversion of the reactant) with desired purity is expected

than that obtained by distillation alone (as in traditional approach).

5

Chapter-2

Literature Review The flexibility provided by a batch rectifier, however, gives rise to challenging

control problems that are basically owing to the nonlinear, nonstationary, and finite time

duration nature of the underlying dynamics. The batch distillation is inherently an

unsteady state process and therefore, there is no normal condition at which the

conventional input/output linear model can be formulated and the employed controller

can be tuned. So, it is really a tough job to achieve satisfactory closed-loop process

response (Quintero-Marmol et al., 1991; Kim, 1999). Finefrock et al. (1994) mentioned

that if a PI composition controller is implemented, the controller gain should be varied

during the operation in order to enable control to be maintained. They suggested using a

gain-scheduled PI (GSPI) controller to accomplish the control objective. Frattini-Fileti

and Rocha-Pereira (1997) applied the GSPI control scheme on a binary batch distillation

column. In their strategy, the controller gain is increased during the batch operation in

order to meet the control objective. Predictive and adaptive control approaches have been

used to tackle the nonstationary nature of the batch rectifier. Li and Wozny (1997)

showed that the optimal profiles cannot be tracked with conventional linear controllers. To realize the optimum of multiple-fraction batch distillation, subsequently they

(Li and Wozny, 2001) presented a predefined optimal policy that can be tracked with an

adaptive control law. To follow the rapidly changing process dynamics, they have applied

a recursive least-square estimation technique with a variable forgetting factor for the

online plant identification. In recent years, various approaches for designing the nonlinear

control methodologies have been reported in the literature. Unless they are using “ad

hoc” designs, most of the nonlinear controllers, such as globally linearizing controller

(GLC) (Hunt et al., 1983; Kravaris and Chung, 1987; Isidori, 1989; Jana et al., 2005),

generic model controller (GMC) (Lee and Sullivan, 1988; Signal and Lee, 1992), internal

model controller (IMC) (Henson and Seborg, 1991; Barolo and Berto, 1998a) and many

others, require the feedback of state variables to implement the control laws.

Although direct integration of the process model, i.e., open-loop observer

(Kravaris and Chung, 1987), may be employed to estimate the required state feedback

6

and performs satisfactorily for a limited number of applications, a closed-loop

observation scheme with feedback correction is still desirable, especially, when the

system consists of modeling error(s) or pure integrator(s). The development of closed-

loop observers for nonlinear systems still provides an open area for research. Many

observation approaches have been developed so far and few of them are applied with

good results on the batch distillation processes (Quintero-Marmol et al., 1991; Quintero-

Marmol and Luyben, 1992; Barolo et al., 2000; Oisiovici and Cruz, 2000; Venkateswarlu

and Avantika, 2001; Zamprogna et al., 2004). It is recognized that the performance of a

nonlinear modelbased control system largely depends on the convergence ability as well

as robustness of the state estimator. Barolo and Berto (1998a) constructed a control

strategy within the framework of nonlinear internal model control (Henson and Seborg,

1991). To estimate the distillate composition from the selected tray temperature

measurements, the authors have used the extended Luenberger observer (ELO)

(Quintero-Marmol et al., 1991). The derived control law is then successfully applied on a

binary and a ternary batch rectifier. The authors reported the significant improvement

provided by a nonlinear controller over a linear control law.

The nonlinear model-based controller provided very fast process response

towards the desired operating point, and the reflux rate profile was relatively smooth.

Barolo and Berto recommended the use of a stochastic estimator (like an extended

Kalman filter (EKF)) when a large degree of noise is expected. Dechechi et al. (1998)

developed a nonlinear model predictive controller for overhead composition regulation of

a batch column. In their approach, an ELO coupled with an optimization problem has

been solved online and this approach can lead to a very complex controller (Barolo and

Berto, 1998a).

Alvarez-Ramirez et al. (2000) have presented a strategy that comprises of a

controller (classical PID with antireset windup) and an observer to estimate the modeling

error. A drawback of this control approach (Monroy-Loperena and Alvarez-Ramirez,

2003) is that they strongly rely on an accurate model of the batch distillation process.

Subsequently, Oisiovici and Cruz (2001) proposed an inferential control structure that

consisted of the GLC and the EKF (Baratti et al., 1995). They have shown that the

inferential control scheme yields good results for a high-purity multicomponent batch

7

distillation column. They pointed out that the performance of the EKF is greatly

influenced by the vapor–liquid equilibrium (VLE) data. The authors also found that the

tracking performance of the filter usually improved when larger sets of secondary

measurements were used, and/or when the sampling frequency was increased. But both

the options increase the design complexity and the computational load. Han and Park

(2001) have used the quasi dynamic estimator (QDE) (Quintero-Marmol et al., 1991) to

estimate the distillate composition in the closed-loop batch rectifier. But Quintero-

Marmol et al. (1991) found that the ELO provided better performance than the QDE for a

ternary batch column. Then, Monroy-Loperena and Alvarez-Ramirez (2003) developed a

technique for the identification and control of a batch distillation process. The feedback

controller is derived in the framework of robust nonlinear control (Alvarez-Ramirez,

1999) with modeling error compensation technique for the control of distillate

composition via manipulations of the reflux ratio.

Cuille and Reklaitis (1986) considered the simulation of reactive batch

distillation, with reaction occurring on the plates, in the condenser, and in the reboiler.

They considered the esterification of 1-propanol with acetic acid, but the example was

not suitable for use in batch distillation. Since 1-propanol (one of the reactants) is the

more volatile component in the system, the removal of species by distillation causes the

removal of reactant from the column thus decreasing conversion..

Wilson discussed the optimal design of batch distillation processes using a

simplified column model involving chemical reaction and using repeated simulation. For

a commercially used complex parallel reaction scheme and using a simple economic

model, he showed the benefit of integrating reaction and distillation. He generated a

number of plots of process efficiency for a range of alternative process and design

variable choices and suggested an optimal design and operation of reactive batch

distillation. (Wilson, J. A., 1987)

Albet et al. (1991) presented a method for obtaining operational policies for both

reactive and non reactive batch distillation systems using repeated simulation techniques.

Mujtaba and Macchietto, (1991, 1993) discussed reactive batch distillation is

presented as a proper dynamic optimization problem incorporating a detailed dynamic

model.

8

Chapter-3

Modeling, Simulation and Control of a multicomponent Batch Distillation Column

3.1. Process Description In batch distillation, a liquid mixture is charged into a vessel and heat is added to

produce vapor that is fed into a rectifying column. The liquid mixture can be a fresh feed

and also with any recycled slop cuts. During the initial startup period, the column

operates under total reflux condition in which vapor from the top of the column is

condensed and returned to the column. The operation of batch distillation described here

corresponds to a ternary system. During the column operation under total reflux

condition, the concentration of the lightest component buildup on the upper trays in the

column and the concentrations of the intermediate component and heaviest component

decreases in the top of the column but increases in the still pot. When the concentration

of the lightest component in the distillate reaches its specified purity level, then the

distillate product withdrawal is begun. During the withdrawal of the first product, there is

a composition front located in the lower part of the column that separates the lightest and

intermediate components. This front moves up the column as light product is removed.

When this front nears the top of the column, the distillate stream is diverted to another

tank as the 1rst slop cut. When the concentration of the intermediate component in the

distillate reaches its speci1ed purity level, the distillate is diverted to another tank in

which second product is collected. When the purity of the material in this tank drops to

the speci1ed purity level, the distillate stream is diverted into another tank, and the

second slop cut is collected until the average composition of the material remaining in the

still pot and on the trays in the column meets the purity speci1cation of the heavy

product.

In order to represent realistic operation of actual batch distillation column, a

rigorous nonlinear model that considers simultaneous effect of heat and mass transfer

operations and fluid flow on the plates is needed. Such batch distillation model is derived

from first principles involving dynamic material and component, and algebraic energy

9

equations supported by vapor–liquid equilibrium and physical properties. The

multicomponent batch distillation dynamics simulator has major computation functions

like vapor flow, liquid flow and tray holdup calculations, enthalpy calculations, average

molecular weight and density calculations, and vapor–liquid equilibrium calculations.

As assumed, the production phase the reflux drum holdup is kept constant

employing Proportional controller.

The operation of batch distillation described here corresponds to a ternary system

of cyclohexane–toluene–chlorobenzene. Among these constituent feed components,

cyclohexane is the lightest component, toluene is the intermediate component, and

chlorobenzene is the heaviest component. The model structure of the ternary distillation.

Fig. 1. schematic representation of the multicomponent Batch Distillation Process

10

3.2. Assumptions

Process is developed based on the following assumptions

• Staged batch distillation column with trays numbered from the bottom and top

(total 14 trays including still pot and reflux drum).

• Perfect mixing and equilibrium on all trays.

• Constant stage pressures (atmospheric) and tray efficiencies

(Vapor-phase Murphree efficiency = 75%).

• Negligible tray vapor holdups.

• Total condensation with no sub cooling in the condenser.

• Nonlinear Francis weir formula (Luyben, 1990) for tray Hydraulics calculations.

• Variable liquid holdup in each tray.

• Constant liquid holdup in the reflux drum (perfectly controlled by a conventional

proportional (P) controller with Proportional gain = −0.0005).

• Raoult’s law for the vapor–liquid equilibrium.

Batch distillation is inherently an unsteady state process. Consider the distillation

column as shown in Fig.1. Normally in batch processes the feed is charged in the still at

the bottom of the column. For this study feed is a cyclohexane-toluene-chlorobenzene

mixture, which has composition 0.4-0.4-0.2 respectively. This mixture is to be separated

by this batch process. The column shown in Fig.1. has 12 plates excluding reboiler and

reflux drum. It has divided into five sections for doing the material and energy balances

easily. MB is the amount of holdup in the still. QR amount of heat is supplied to still. Due

to that heat feed in the still changes its phase according relative volatility and VB amount

of vapors are formed which tries to go up through the first plate and so on. Now on first

plate there are four streams two inlet and outlet each. The suffix in the stream name

shows the stream coming from that plate. For example L1 is the amount of liquid stream

coming from first plate. V is used for vapor rate, M for hold up Hl for liquid enthalpy, Hv

for vapor enthalpy, x is liquid composition and y is vapor composition.

11

3.3. Modeling Equations Material balance, component balance and enthalpy balance equations can be

written accordingly,

The change in the heat energy for a very small amount of time can be considered

negligible i.e. the change is very less. So d(M Hl)/dt is very small, d(M Hl)/dt = 0; on

rearrangement, we get

3.3.1. Reboiler Section Total mass balance:

BB VL

dtdM

−= 1 (3.1)

Component mass balance:

,1 1. ,

( )B B jj B B j

d M xL x V y

dt= − (3.2)

Where j = 1,2….c (c = No. of components)

, 11. , , ,( ) ( )B j B

j B j B j B jB B

dx L Vx x y xdt M M

= − − − (3.3)

Energy balance:

RvBB

llBB QHVHL

dtHMd

+−= 11)( (3.4)

1 1( )l

RB v

B

L H QVH+

= (3.5)

3.3.2. First Plate

1121 LVLV

dtdM

B −−+= (3.6)

1 1,

, 2 2 1 1, 1 1,

( )jB B j , j j j

d M xV y L x V y L x

dt= + − − (3.7)

Where j = 1,2….c (c = No. of components)

12

1, 2 1, 1, 2 1 1, 1,

1 1 1( ) ( ) ( )j B

B j j j j j jdx V L Vy x x x y xdt M M M

= − + − − − (3.8)

lvlv

BB

l

HLHVHLHVdt

HMd111122

11 )(−−+= (3.9)

v

llvBB

HHLHLHVV

1

11221

−+= (3.10)

3.3.3. nth Plate

nnnnn LVLV

dtdM

−−+= +− 11 (3.11)

,

1 1, 1 1, , ,

( )n n jn n j n n j n n j n n j

d M xV y L x V y L x

dt − − + += + − − (3.12)

Where j = 1,2….c (c = No. of components)

, 1 11, , 1, , , ,( ) ( ) ( )n j n n n

n j n j n j n j n j n jn n n

dx V L Vy x x x y xdt M M M

− +− += − + − − − (3.13)

lnn

vnn

lnn

vnn

lnn HLHVHLHV

dtHMd

−−+= ++−− 1111)(

(3.14)

vn

lnn

lnn

vnn

n HHLHLHVV −+

= ++−− 1111 (3.15)

3.3.4. Top Plate (nT

th)

TTTT

nVnLnVRdt

ndM−−+= −1 (3.16)

,

, 1 1, , ,

( )T T jD j T T j T T j T T j

d M xn n Rx V y L x V yn n n n n ndt − −= + − − (3.17)

Where j = 1, 2….c (c = No. of component

1,1, , , , , ,( ) ( ) ( )nTnT i nT

nT i nT i D i nT i nT i nT inT nT nT

dx V R Vy x x x y xdt M M M

−−= − + − − − (3.18)

13

vTT

lTT

vTT

lD

lTT

nHnVnHnLnHnVRHdt

nHnMd−−+= −− 11

)( (3.19)

For start-up phase, R = VnT

)(11

lD

vT

lTT

vTT

T HnHnHnLnHnV

nV−−

= −− (3.20)

For production phase, for any R

vT

lTT

vTT

lD

TnH

nHnLnHnVRHnV

−+= −− 11 (3.21)

3.3.5. Reflux Drum

DRnVdt

dMT

D −−= (3.22)

,

, ,,

( )D D jD j D jT T j

d M xV y Rx Dxn ndt

= − − (3.23)

Where j = 1,2….c (c = No. of components)

14

3.3.6. Bubble point calculations For solving the vapor rates i.e. the energy balance equation, one requires the

enthalpy data. And to calculate the enthalpy; temperature should be known. Therefore, it

is necessary to have the temperature-composition correlation. The vapor-phase

composition in equilibrium with the liquid-phase is given by ,

( ) iiiiii xPTxky ,,* = (3.24)

iii xky =* (3.25)

Where k is equilibrium ratio, for this study k is calculated as follows,

t

si

i PPk = (3.26)

Where siP the vapor pressure was calculated by using Antoine equation, and tP

was the total pressure.

The Antoine equation is given as,

⎥⎦⎤

⎢⎣⎡

+−=

CTBAPs

i exp (3.27)

Where the A, B, C are the constants and T is the temperature. Here the pressure is

in mmHg and temperature is in K

So now the equilibrium ratio becomes,

t

i PCT

BAk

⎥⎦⎤

⎢⎣⎡

+−

=exp

(3.28)

15

Fig.2. Calculation of Temperature (T) and vapor composition (yi)

3.3.7. Enthalpy Calculations

The liquid and vapor enthalpies were calculated from the following equations.

Where the temperature is in (K) and enthalpy is in (kJ/kmol) 432 TeTdTcTbaH l ++++= (3.29) 432 TeTdTcTbaH v ++++= (3.30)

Calculate equilibrium ratio Ki from equation (28)

Calculate yi from equation (25) using given xi

Check f(T)=∑(yi) -1=0.00001

Obtain yi and T

Yes

No

Assume temperature T

Tnew=Told - 1

( )( )

f Tf T

16

3.3.8. Liquid flow rate Calculations

For the calculation of the liquid flow rates the Francis weir formula was used,

which is given by the following equation. Liquid flow rate is function of density,

molecular weight and tray parameters which includes the tray holdup, height, length and

column diameter. The change in the liquid rate is accounted by changes in the density and

indirectly the liquid composition.

( )

m

WD

mMWρL

H

cL

5.1

2 122.183999 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=ρ (3.31)

Where L liquid rate (lb-mol/hr)

ρ Liquid density (lb/ft3)

WL Weir Length (in)

WH Weir Height (in)

M Hold up (lb-mol)

m Average molecular weight (lbm/lb-mol)

Dc Column Diameter (in)

Density and molecular weight of liquid on the plate is calculated by the sum of

product of liquid composition & density, and liquid composition & molecular weight of

all the components present in the system respectively, is given by the following

mathematical relation.

iix ρρ ∑= (3.32)

i im x m= ∑ (3.33)

17

Table. 1. Batch Distillation Column specification

System Cyclohexane / Toluene / Chlorobenzene

Feed (kmol) 30

Feed composition 0.40 / 0.40 / 0.20

Number of trays (excluding reboiler) 12

Tray hold-up (kmol) 0.03

Heat input to still (kJ/min) 25,000

Distillate composition 0.99

Reflux drum hold-up (kmol) 1.0

Production rate (kmol/min) 0.10

Time step (min) 0.005

Column diameter (inch) 18

Weir Length (inch) 12

Weir Height (inch) 0.30

Murphee Tray Efficiency 0.75

18

3.4. Simulation algorithm

Modeling equations are solve by a stepwise procedure is given below.

Step1: Declare all the variables and initialize the start up phase liquid composition (xi)

and liquid hold up (Mn) on each plate.

Step2: Calculate vapor phase composition (yi) and temperature (T) on each plate

as shown in fig.2 bubble point calculation.

Step3: Calculate actual vapor phase composition (yi) by multiplying the Murphee

tray efficiency.

Step4: Find out the vapor and liquid enthalpy using equations (29) and (30) for all

the component on each plate and then total liquid and vapor phase enthalpies

by multiplying the respective compositions on each plate.

Step5: Then estimates the liquid flow rates using the Francis weir equation (31) In

section with the help of density, molecular weight, holdup and tray specifications.

Step6: Find out the vapor flow rates using the energy balance equations.

Step7: Calculate the liquid hold up on each plate by equations as modeled in section 2

Use the total reflux for start up phase.

Step8: Finally calculate the liquid phase composition. And then check for sum.

The sum of it should be equal to zero. If it is not then normalize it using

current and previous value.

19

3.5. Controller synthesis In closed-loop simulation study, the control objective is to recover the light

component and intermediate component at a constant purity. The manipulated input is the

reflux flow rate. Distillate light component and intermediate components are controlled

with any one of Proportional integral (PI) controller, Nonlinear Proportional integral

(NLPI) controller and Gain Scheduling Proportional Integral (GSPI) controller.

3.5.1. Proportional Integral controller

A PI Controller has two terms, one proportional to the error and other

proportional to the integral of the error. PI controller equation (process systems Analysis

and Control, Coughanor) is given as

0

tc

s ci

ku u k e edtt

= + + ∫ (3.34)

kc =gain

ti =integral time (min)

us =constant

This equation can be written as

0

1( )t

s ci

R R k e edtτ

= + + ∫ (3.35)

e =xDsp, j-xD

3.5.2. Nonlinear Proportional Integral Controller (NLPI)

The idea of NLPI controller is to modify the controller action in some way to

compensate for the nonlinearity of the process. The equation (Bequette, 2006) is given as

0

tc

s ci

ku u k e edtt

= + + ∫ (3.36)

(1 . )c cok k a e= + (3.37)

This means that controller output is effectively proportional to square of the error

kco=controller gain with zero error

e =absolute magnitude of error

20

a=adjustable constant

0

1( )t

s ci

R R k e edtτ

= + + ∫ (3.38)

e =xDsp, j-xD

3.5.3 Gain Scheduling Proportional Controller (GSPI)

A GSPI law (Bequette, 2006) is given as

0

tc

s ci

ku u k e edtt

= + + ∫ (3.39)

,

1 ,1

DSP

D

jc co

j

xk kx

−=

− for xD,j>xDSP,j

kc = kco for xD,j<xDSP,j

This equation can be written as

0

1( )t

s ci

R R k e edtτ

= + + ∫ (3.40)

e =xDsp, j-xD

All the above cases liquid holdup in reflux drum controlled by proportional

controller. Generally, a level controller is used to maintain the liquid holdup in the reflux

drum at a desired value. The variation of liquid holdup so small (± 1.0%) that is

reasonable to assume constant mD. This mD is controlled by manipulating the distillate

rate D. Proportional controller equation (process systems Analysis and Control,

Coughanor), given as

s cu u k e= + (3.41)

D=Ds+kcm*e (3.42) e=mDsp-mD

kcm = proportional gain

21

3.6. Simulation Results and Discussion Several simulation experiments have been carried out on the multi component

batch distillation column in open-loop as well as closed loop mode. The column is

employed for the fractionation of a hydrocarbon system, cyclohexane–toluene–

chlorobenzene. In this system, cyclohexane and toluene are separated as distillate

products and chlorobenzene is separated as a product in the still pot.

In the batch distillation operation, first the column may be brought to the steady

state by considering the total reflux startup procedure. Then the production phase is

started and the controller switched on to maintain the specified product quality.

Sometimes, the product is withdrawn as soon as the distillate composition reaches its

desired value, without waiting for the steady state to be attained. Notice that immediately

after the production phase is started, controller responses may be very aggressive. It

happens because of the following two reasons: (i) immediate withdrawal of the distillate

product, and (ii) the change of distillation composition from the steady state value to the

set point value.

Fig. 3 presents the dynamics of the uncontrolled distillate composition of the

example batch column at the start-up phase. The steady state composition of the lightest

component in the distillate product is 0.999 (very close to 1) and the column can be

considered as a high-purity distillation process. In the following, the open-loop as well as

closed-loop dynamics of this multicomponent batch rectifier are discussed in the

production phase only.

Fig. 4 illustrates the uncontrolled process dynamics at the production phase. The

production phase has been started from the steady state and with the withdrawal of

distillate. In the present simulation-based experiment, the distillate is discharged with a

constant flow rate of 0.1 kmol/min.

22

3.6.1. Constant Composition Control

In the present study, the set point composition has been fixed at a value of 0.985

for lightest and 0.97 for intermediate component. The column is started up as usual and

the lightest component withdrawal is begun as soon as this component met the

composition specification ,then set point step change is maintained from 0.995 to 0.985

at time= 30 min. The constant composition control is continued until the distillate

decreases to small (almost zero) value. The time taken to withdrawal the lightest

component is 91 min with purity of 0.985.At this moment the controller is switched off

and the intermediate component starts to reach the top of the column, time period 77 min

(91 min to 168 min). When this component has reached the desired purity (xDsp,2 =0.97),

constant composition control is started again. In between the withdrawal of lightest and

intermediate components, the distillate is collected as the slop cut. At this moment we

must note that in the simulation experiment, the distillate flow rate is manipulated by a

proportional level controller to maintain constant (nearly) liquid holdup in the reflux

accumulator. The time taken to withdrawal the second lightest component is 66 min with

purity of 0.97.At this moment the controller is switched off, during this second slop cut

third component starts to reach the bottom of the column up to purity of 0.938 (234 min

to 294 min).

Withdrawals of lightest and intermediate components are shown in Fig.7.a and 7.b

with servo performance (+5% step change in set point) at time 30 min and 190 min.

Regulatory performances are shown for (+5%) step change in heat input, feed change for

light component and intermediate component at time 30 min and 190 min(Fig.8 &Fig 9).

Comparative study among the PI, NLPI, and GSPI controllers based on the ISE

values.

ISE= 2

0

t

d te∫ (3.42)

e = error

23

Open loop performance

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (min)

X D,j

(mol

e fra

ctio

n)

CyclohexaneTolueneChlorobenzene

11.2 min

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 10 11 12

Time (min)

X B,j

(mol

e fra

ctio

n)

CyclohexaneTolueneChlorobenzene

Fig. 3. open-loop process dynamics at start-up phase

24

00.10.20.30.40.50.60.70.80.9

1

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Time (min)

X D,j (m

ole

fract

ion)

CyclohexaneTolueneChlorobenzene

147 min

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Time (min)

X B,j

(mol

e fra

ctio

n)

CyclohexaneTolueneChlorobenzene

Fig .4. open-loop process dynamics at the production phase

25

Servo Performance

0.97

0.975

0.98

0.985

0.99

0.995

1

0 10 20 30 40 50 60 70 80 90 100

Time (min)

X DSP

,1 /

XD

,1 (

mol

e fra

ctio

n)

Set PointGSPINLPIPI

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Time (min)

R (k

mol

/ m

in)

GSPINLPIPI

Fig.5a. Comparative closed loop performance of PI, NLPI and GSPI control algorithms

with a set point change in XD,1 (0.995 to 0.985 at time=30 min)

26

0.930.94

0.950.96

0.970.98

0.991

140 150 160 170 180 190 200 210 220 230 240

Time (min)

X DSP

,2 /

XD

,2 (

mol

e fra

ctio

n)Set PointGSPINLPIPI

00.10.20.30.40.50.60.70.8

140 150 160 170 180 190 200 210 220 230 240

Time (min)

R (k

mol

/min

)

GSPINLPIPI

Fig.5b.Comparative closed loop performance of PI, NLPI and GSPI control algorithms

with a set point change in XD, 2 (0.975 to 0.97 at time=190 min)

27

0.7

0.75

0.8

0.85

0.9

0.95

1

234 244 254 264 274 284 294

Time (min)

X B,3 (

mol

e fra

ctio

n)

Fig.5c.Third component in the bottom after second slop cut Regulatory performance

0.97

0.975

0.98

0.985

0.99

0.995

1

0 10 20 30 40 50 60 70 80 90

Time (min)

X DSP

,1 /

XD

,1 (m

ole

fract

ion)

Set PointGSPINLPIPI

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

Time (min)

R (k

mol

/min

)

GSPINLPIPI

Fig.6a.Comparative Regulatory performance of PI, NLPI and GSPI control algorithms

with +5% (25000 → 26250 kJ/min ) step change in reboiler heat duty at time =30min.

28

0.93

0.94

0.95

0.96

0.97

0.98

0.99

140 150 160 170 180 190 200 210 220 230 240

Time (min)

X DSP

,2 /

XD

,2 (m

ole

fract

ion)

Set PointGSPINLPIPI

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

140 150 160 170 180 190 200 210 220 230 240

Time (min)

R (k

mol

/min

)

GSPINLPIPI

Fig.6b.Comparative Regulatory performance of PI, NLPI and GSPI control algorithms

with +5% (26250 → 27562 kJ/min ) step change in reboiler heat duty at time =190 min.

29

0

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 60 70 80 90 100

Time (min)

X B,1

(mol

e fra

ctio

n)

0.97

0.975

0.98

0.985

0.99

0.995

1

0 10 20 30 40 50 60 70 80 90 100

Time (min)

X DSP

,1/ X

D,1

(mol

e fra

ctio

n)

Set PointGSPINLPIPI

0.3

0.40.5

0.60.7

0.80.9

1

0 10 20 30 40 50 60 70 80 90 100

Time (min)

R (k

mol

/min

)

GSPINLPIPI

Fig.7a. Comparative Regulatory performance of PI, NLPI and GSPI control algorithms with +5 % (0.17128, 0.56771, 0.26110 → 0.22128, 0.53318, 0.24553) step change in XB,1 at time =30min.

30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

140 150 160 170 180 190 200 210 220 230 240 250

Time (min)

X B,2

(mol

e fra

ctio

n)

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

140 150 160 170 180 190 200 210 220 230 240 250

Time (min)

R (k

mol

/min

)

Set PointGSPINLPIPI

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

140 150 160 170 180 190 200 210 220 230 240 250Time (min)

R (k

mol

/min

)

GSPINLPIPI

Fig.7b. Comparative Regulatory performance of PI, NLPI and GSPI control algorithms with +5 % ( 0.000095.48899,0.51067 → 0.000084,0.53899,0.46080) step change in XB,jat time =190 min.

31

Tuning Parameters Table2. (Servo Performance) For component X D,1

Controller Kci iτ a ISE

PI 35.65 6.5 ---- 0.000193259

NLPI 35.65 6.5 12.5 0.000182704

GSPI 35.65 6.5 ---- 0.000126476

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.0005 Table3. (Servo Performance) For component X D,2

Controller Kci iτ a ISE

PI 38.68 7.5 ---- 0.00000491465

NLPI 38.68 7.5 12.1 0.00000421346

GSPI 38.68 7.5 ---- 0.00000394175

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.00048

32

Table4. (Regulatory Performance) For component X D,1

Controller Kci iτ a ISE

PI 35.65 6.5 ---- 0.000221949

NLPI 35.65 6.5 12.5 0.000214235

GSPI 35.65 6.5 ---- 0.000126805

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.0005 Table5. (Regulatory Performance) For component X D,2

Controller Kci iτ a ISE

PI 38.68 7.5 ---- 0.000068712

NLPI 38.68 7.5 12.1 0.000062481

GSPI 38.68 7.5 ---- 0.000048687

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.00048

33

Table6. (Regulatory Performance) for XB change For component X D,1

Controller Kci iτ a ISE

PI 35.65 6.5 ---- 0.000019565

NLPI 35.65 6.5 12.5 0.000019283

GSPI 35.65 6.5 ---- 0.0000152386

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.0005 Table7. (Regulatory Performance) for XB change For component X D,2

Controller Kci iτ a ISE

PI 38.68 7.5 ---- 0.000058637

NLPI 38.68 7.5 12.1 0.000055245

GSPI 38.68 7.5 ---- 0.000043618

Liquid level in the reflux drum controlled by proportional controller. Tuning parameter for proportional controller kcm= -0.00052

34

Chapter- 4

Modeling, Simulation and Nonlinear Adaptive Control of a multicomponent Reactive Batch Distillation Column

4.1. Process Description Initially batch column is filled with reactants. In This work 4% of the total feed

charge as the total column holdup. Half of this holdup is taken as condenser holdup and

the rest is equally divided for the plate holdups. Plate compositions (M. Mujtaba and

Macchietto, 1997).In this present work the esterification of ethanol and acetic acid

considered. The reaction products are ethyl acetate (main product) and water. Sulfuric

acid in liquid form is used as catalyst.

The reversible reaction scheme is the following:

Acetic Acid (1) + Ethanol (2) Ethyl Acetate (3) + Water (4)

Rate of reaction r = k1C1C2-k2C3C4 (4.0)

Where rate constants are k1 = 4.76 x10-4 and k2 =1.63x10-4

The boiling temperatures are respectively 391.1, 351.5, 350.3, and 373.2 K. Ethyl

acetate, the main product has the lowest boiling temperature in the mixture and

consequently has the highest volatility. The continuous removal of this product by

distillation will shift the chemical equilibrium further to the right and will improve

conversion of reactants.

35

4.2. Assumptions 1. The vapor-phase holdup is assumed to be negligible compared to the liquid-phase

holdup on each phase.

2. Chemical reactions in the vapor phase are neglected.

3. The initial state of the column is the steady-state total reflux condition with no

reactions.

4. The liquid volumetric holdups on the plates will be assumed to be constant. Thus, the

model is directed at simulating the dynamics of the main production period during which

the hydrodynamic conditions are not widely varying.

5. The pressure drops and the plate efficiencies are constant during the operation

4.3. Modeling Equations Material balance, component balance and enthalpy balance equations can be written

accordingly,

The change in the heat energy for a very small amount of time can be considered

negligible i.e. the change is very less. So d(M Hl)/dt is very small, d(M Hl)/dt = 0; on

rearrangement, we get

4.3.1. Reboiler Section Total mass balance:

1 , ,

1 1

. .r c

BB f i m B i B

m i

dM L V r rdt

γ ε= =

= − + ∑∑ (4.1)

Component mass balance:

.,

, . ,1 1, ,1

( ) rB B i

f i m B i Bi B B im

d M xL x V y r r

dtγ ε

=

= − + ∑ (4.2)

Where i = 1,2….c (c = No. of components)

, 1 , . , . . , . , .1. , , ,1 1 1

( ) ( )r r c

B i B f i m B i B B f i m B i Bi B i B i B im m iB B

dx L Vx x y x r r x r rdt M M

γ ε γ ε= = =

= − − − + −∑ ∑∑ (4.3)

Energy balance:

, . , .1 11

( ) ( )l r

l vB B f i m B i BB B Rm

d M H L H V H Q r rdt

γ ε=

= − + + ∑ (4.4)

,1 11

rll

f B i BR Bm

B vB

L H Q r rHV

H

ε=

+ +=

∑ (4.5)

36

4.3.2. First Plate

1 , 1, 12 1 11 1

. .r c

f i m iBm i

dM V L V L r rdt

γ ε= =

= + − − + ∑∑ (4.6)

1 1,, . 1, . 1, 2 2 1 1, 1 1,

1

( ) ri

f i m iB B i ,i i im

d M xV y L x V y L x r r

dtγ ε

=

= + − − + ∑ (4.7)

Where i = 1,2….c (c = No. of components) 1, 2 1

, 1, 2, 1, 1, 1, , . 1, . 1 . , . 1, . 11 1 1 1 1 1

( ) ( ) ( )r r c

i BB i i i i i i f i m i B f i m i

m m i

dx V L Vy x x x y x r r x r rdt M M M

γ ε γ ε= = =

= − + − − − + −∑ ∑∑ (4.8)

1 1 , . 1, . 12 2 1 1 1 11

( ) ( )l r

v l v lf i m iB B

m

d M H V H L H V H L H r rdt

γ ε=

= + − − + ∑ (4.9)

1, . 112 2 1 11

11

rv l l l

f iB Bm

v

V H L H L H r rHV

H

ε=

+ − +=

∑ (4.10)

4.3.3. nth Plate

, ,1 11 1

. .r c

n f i m n i nn n n nm i

dM V L V L r rdt

γ ε− += =

= + − − + ∑∑ (4.11)

,

, ,1 1, 1 1, , ,1

( ) rn n i

f i m n i nn n i n n i n n i n n im

d M xV y L x V y L x r r

dtγ ε− − + +

=

= + − − + ∑ (4.12)

Where i = 1,2….c (c = No. of components)

1

, 1 1 , . , .1, , 1, , , ,( ) ( ) ( )r

m

n i n n n f i m n i nn i n i n i n i n i n in n n

dx V L Vy x x x y x r rdt M M M

γ ε=

− +− += − + − − − + ∑

. , . , .

1 1

r c

n f i m n i n

m i

x r rγ ε= =

− ∑∑ (4.13)

, . , .1 1 1 11

( ) ( )l r

v l v ln n f i m n i nn n n n n n n nm

d M H V H L H V H L H r rdt

γ ε− − + +=

= + − − ++ ∑ (4.14)

, .1 1 1 11

rv l l l

f n i nnn n n n n nm

n vn

V H L H L H r rHV

H

ε− − + +=

+ − +=

∑ (4.15)

37

4.3.4. Top Plate (nT

th)

, ,11 1

. .r c

nT f i m nT i nTnT nT nTm i

dM R V L V r rdt

γ ε−= =

= + − − + ∑∑ (4.16)

,

, . , ., 1 1, , ,1

( ) rnT nT i

f i m nT i nTD i nT nT i nT nT i nT nT im

d M xRx V y L x V y r r

dtγ ε− −

=

= + − − + ∑ (4.17)

Where i = 1,2….c (c = No. of components)

1

1,, . , .1, , , , , ,( ) ( ) ( )

r

m

nTnT i nT f i m nT i nTnT i nT i D i nT i nT i nT inT nT nT

dx V R Vy x x x y x r rdt M M M

γ ε=

−−= − + − − − + ∑

. , . , .

1 1

r c

nT f i m nT i nT

m i

x r rγ ε= =

− ∑∑ (4.18)

, . , .1 11

( ) ( )l r

l v l vnT nT f i m nT i nTD nT nT nT nT nT nTm

d M H RH V H L H V H r rdt

γ ε− −=

= + − − ++ ∑ (4.19)

For start-up phase, R = VnT

, .1 11

( )

rv l l

f nT i nTnTnT nT nT nTm

nT v lnT D

V H L H r rHV

H H

ε− −=

− +=

−

∑ (4.20)

For production phase, for any R

, .1 11

rl v l l

f nT i nTnTD nT nT nT nTm

nT vnT

RH V H L H r rHV

H

ε− −=

+ − +=

∑ (4.21)

4.3.5. Reflux Drum

, ,

1 1

. .r c

D f i m D i DTm i

dM V R D r rndtγ ε

= =

= − − + ∑∑ (4.22)

,

, . , ., ,,1

( ) rD D i

f i m D i DD i D iT T im

d M xV y Rx Dx r rn ndt

γ ε=

= − − + ∑ (4.23)

Where i = 1, 2….c (c = No. of components)

38

4.3.6. Calculation of tray volume

d=diameter of column

OA=d/2

AB=weir length

AC=BC

Sin θ= ACOA

θ=sin-1 ACOA

tan θ= ACOC

OC=tan θ *AC

Area of ADB (down comer area) = Area of OADB – Area of OAB

Area of OADB =22θ πd*

360 4

Area of OAB = 1 (AB)*(OC)2

Area of ABE = (Area of ADBE) – (Area of ADB)

Volume of tray = (Area of ABE) * (weir height)

O

A

B

C

θD

E

39

Table. 8. Reactive Batch Distillation Column specification

System Acetic Acid + Ethanol Ethyl Acetate + Water (1) (2) (3) (4)

Feed (kmol) 30

Feed composition 0.45 / 0.45 / 0.0 / 0.1

Number of trays (excluding reboiler & reflux drum) 8

Tray hold-up (kmol) 0.075

Heat input to still (kJ/min) 3200

Distillate composition 0.936

Reflux drum hold-up (kmol) 0.6

Production rate (kmol/min) 0.06

Time step (min) 0.005

Column diameter (inch) 18

Weir Length (inch) 12

Weir Height (inch) 0.30

Murphee Tray Efficiency 0.75

40

4.4. Simulation algorithm

Modeling equations are solve by a stepwise procedure is given below.

Step1: Declare all the variables and initialize the start up phase liquid composition (xi)

and liquid hold up (Mn) on each plate.

Step2: Calculate vapor phase composition (yi) and temperature (T) on each plate

as shown in fig 2; bubble point calculation.

Step3: Calculate actual vapor phase composition (yi) by multiplying the Murphee

tray efficiency.

Step4: Find out the vapor and liquid enthalpy using equations (3.29) and (3.30) for all

the component on each plate and then total liquid and vapor phase enthalpies

by multiplying the respective compositions on each plate.

Step5: Then estimates the liquid flow rates using the Francis weir equation (3.31) In

section with the help of density, molecular weight, holdup and tray specifications.

Step6: Find out the vapor flow rates using the energy balance equations.

Step7: Calculate the liquid hold up on each plate by equations as modeled in section 2

Use the total reflux for start up phase.

Step8: Find out volume of tray (v), the rate of reaction using equation (4.0)

Step9: Finally calculate the liquid phase composition. And then check for sum.

The sum of it should be equal to zero. If it is not then normalize it using

current and previous value.

41

4.5. Nonlinear adaptive control algorithm As stated, the adaptive control structure consisted of the nonlinear GMC and

an ASE. The closed-loop system having different controller elements and the process is

shown in Fig. 3. In the following, the detailed synthesis of the adaptive controller is

presented in generalized form.

Fig. 8. Block diagram for the adaptive control algorithm

4.5.1. Generic model control

In nonlinear modeling of dynamic processes (Guo et al., 2001), it may be considered that

the system is nonlinear in the states, disturbances, and control variables but linear in the

model parameters such that

dxdt

= f (x,d)θ + g1(u, x, d), (4.34.a)

y = cx, (4.34.b)

where the state x∈Rn, the model parameter θ∈ nℜ , the measurable disturbance

d ∈ qℜ , and the input u ∈ mℜ . Moreover, f and g1 are matrices of nonlinear

functions. In this study, it is assumed that all states are measurable and c (coefficient

matrix) is a unity matrix. From the basic principle of GMC (Lee and Sullivan, 1988),

the following control law can be derived (Guo et al., 2001)

Nonlinear GMC

Process

Output Map

Nonlinear ASE

ex yu

^x

ysp

42

f (x,d) θ + g1(u, x, d) − 1 2

0

0t

K e K edt− =∫ (4.35)

where e is the error (=ysp − y) to the controller, ysp is the set point value of the output y,

and K1 and K2 are diagonal n×n tuning parameter matrices. Eq. (4.35) implies that the

GMC algorithm comprises of dynamic process model, proportional action term, and

integral action term. If g1 is linear with respect to u, then one can write

g1 (u, x, d) = b(x, d) u. Accordingly, Eq. (35) yields

u = (b(x, d))−11 2

0

( , )t

K e K edt f x d θ⎡ ⎤

− −⎢ ⎥⎣ ⎦

∫ =0 (4.36)

The values of the elements of tuning parameter matrices can be found out based on the

following relationships given by Signal and Lee (1992)

11( , )

2

2 ii i

iK τ

τ= (4.37.a)

2( , ) 2

2

1i i

i

Kτ

= (4.37.b)

where τ1i and τ2i determine the shape and speed of the desired closed-loop trajectory (the

reference trajectory), respectively. The reference trajectory gives pseudo-second order

response for a step change in the set point. However, Yamuna and Gangiah (1991)

confirmed that the above relationships could be applied to compute the specified response

accurately. Once the values of τ1i and τ 2i are obtained, then K1 and K2 can be calculated

from Equations (4.37.a) and (4.37b).

4.5.2. Adaptive state estimation

In practice, there are two types of mismatch, structural mismatch and parameter

mismatch. The structural mismatch exists when there is a difference between the actual

plant model and the predictor model. The parameter mismatch occurs when the numerical

values of parameters in the predictor model differ with the true values. The effects of the

structural discrepancy on the closed-loop performance can be reduced if the imprecisely

known parameters are continuously updated.

In the present study, the model parameters in Eq. (4.34.a and 4.34.b) are supposed

to be time varying. Here, a nonlinear observer proposed by Farza et al. (1998, 1999) is

43

designed to estimate the poorly known parameters of the batch rectifier. It is also

assumed that the parameter dynamics in the nonlinear system (Eq. (4.34)) obey the

following general first-order equation: ddtθ = g2(u, x, d) +ε (4.38)

where g2 is a nonlinear function and ε is an unknown function that may depend on x, θ, u,

d, noise, and so on. The assumptions that have been made are: ε is an unknown but

bounded function and the disturbance d with its time derivative are also bounded.

The nonlinear system equations ((4.34) and (4.38)) can be expressed in the following

condensed form

( , ) ( , , )dZ F x d Z G u x ddy

ε−

= + + (4.39.a)

y = CZ, (4.39.b)

Where0x

Z ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 0 ( , )

( , )0 0

f x dF x d ⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 1( , , )

( , , )2( , , )

g u x dG u x d

g u x d⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 0

εε

− ⎡ ⎤= ⎢ ⎥⎣ ⎦

, C =[In, 0],

with In the n×n identity matrix. f is an n × n matrix which is differentiable and the

corresponding partial derivative is continuous. According to Farza et al. (1998), the

nonlinear adaptive observer can be used to track the vector Z as follows: ^

^ ^11( , ) ( , , ) ( , ) ( )Td Z F y d Z G u y d y d C Z ydt S C−−= + − −Γ (4.40)

(i)^ 2

^n

yZ

θ

⎡ ⎤= ∈⎢ ⎥⎢ ⎥⎣ ⎦

ℜ ,and ^ mθ ∈ℜ

(ii) 0

( , )0 ( , )nI

y df y d

⎡ ⎤Γ = ⎢ ⎥

⎣ ⎦, and

(iii) S is the unique symmetric positive-definite matrix which satisfies the algebraic

Lyapunov equation.

The gain of the estimator is obtained as

1112

2( , )

( , )

nT

Iy d

y dS C fα

α−−

−

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

Γ (4.41)

Where α >0 is a design parameter (Gauthier et al., 1992).

44

It is obvious from Eq. (41) that only a single tuning parameter α is involved in the

estimator. When ε = 0, the convergence of the observer error is an exponential one. In the

case where ε ≠ 0, the asymptotic error can be made arbitrarily small by choosing a

sufficiently large value of α. However, a very large value α may make the observer

sensitive to noise. Thus, the choice α of is a compromise between fast convergence and sensitivity to noise.

4.5.3. Controller synthesis

A nonlinear adaptive controller has been designed for a multicomponent reactive

batch distillation column. This model will be referred to as “the process”. The schematic

representation of the batch rectifier is shown in Fig. 1. In the closed-loop simulation

study, the control objective is to recover the more volatile at a constant purity. The

manipulated input is the reflux flow rate. At the beginning of the operation, it is assumed

that the reboiler, all the trays, and the reflux drum are filled with the liquid feed. During

the startup period, the liquid holdup in the reflux drum remains constant due to the total

reflux condition. As assumed, at the production phase the reflux drum holdup is kept

almost constant employing a traditional proportional controller.

4.5.4. Adaptive control strategy

As stated previously, the adaptive GMC–ASE controller consists of the GMC and

an ASE. The important features of its application to the batch column are:

• The nonlinear batch distillation model can be directly inserted into the control structure,

permitting for the inherent process nonlinearity to be taken into account.

• The relationship between feed forward and feedback control is explicitly accounted for

in the GMC controller.

• This control algorithm allows us to regularly update the parameters in the predictor

model. As a consequence, the effects of the structural mismatch on the controller

performance can be minimized.

• The mathematical formulation and tuning of the adaptive GMC–ASE controller, even

for this large multicomponent distillation system, are relatively simple.

45

4.5.5. Generic model controller

The component mass balance equation for the condenser accumulator system is

given as

,, , ,

( ) ( )D D jnT nT j D j j D j D

d M x V y R D x rdt

γ ε= − + + (4.42)

Actually, a level controller is employed to maintain the liquid holdup in the reflux

drum at a desired value. The variation of holdup is so small (±1.0%) that it is reasonable

to assume constant MD. Accordingly, the above equation becomes

, , ,( ) ,D j nT nT j D j i D j D

D

dx V y R D x rdt m

γ ε+− += (4.43)

Using Eq. (4.42) and simplifying, the following form of GMC controller equation

can be obtained for the concerned process:

, 1 20

,

( )t

nT nT j D j D

D j

V y r M K e K edtR D

x

γ ε+ − += −∫ (4.44)

where e = xDsp,j − xD,j , and xDsp,j is the set point value of xD,j . It is obvious from the above

controller Eq. (4.44) that the component vapor flow rate leaving top tray (VnT ynT, j), a

poorly known parameter, is required to estimate for the implementation of the GMC

algorithm. Notice that to obtain satisfactory controller tuning parameters for the closed-

loop batch distillation operation, we need to follow the conventional startup procedure

(Barolo and Berto, 1998b).

4.5.6. Adaptive state estimator

In the present study, the product composition (xD,j) is assumed as measured

variables (true state), whereas the component vapor flow rate leaving top tray (VnT ynT ,j)

and rate of reaction rD are the extra states having no dynamics. It is important to mention

here that although xD,j is obtained through direct measurement, that composition is also

estimated in the ASE to compute the residual, ˆx − x (=estimated value − measured

value).

The predictor model, which is required to design the ASE 1, consisted of the

component mass balance equation around the condenser–reflux drum system and the

extra state equation with no dynamics. The mathematical representation of the predictor

model is

46

, , , , . , .D i nT nT i D i D i i D i D

D

dx V y Rx Dx rdt M

γ ε+− −= (4.45.a)

( ) 0nT nTd V ydt

= (4.45.b)

The final structure of the ASE estimator can be obtained in matrix form by combining

Eqs. (4.40), (4.41) , (4.45.a) and (4.45.b) as ^

,

^ ^

D i

nT nT

d xdt

d V ydt

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

=10

0 0Dm

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

^,

^ ^

D i

nT nT

x

V y

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

+, , , ..

0

D i D i i D i D

D

Rx Dx rM

γ ε− − +⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

2

2

2 1

DM

α

α

⎡ ⎤− ⎢ ⎥⎢ ⎥⎣ ⎦

^, ,D i D ix x⎡ ⎤−⎢ ⎥⎣ ⎦

(4.47)

where ^

,D ix and ^ ^

nT nTV y are estimates of xD,i and VnT ynT,i , respectively. In the above

estimator structure, α1 and α2 are the tuning parameters. It is true that the dynamics of xD,i

and VnT ynT ,i are not same. As a consequence, it is better to estimate xD,j and VnT ynT ,i ,

using different tuning parameters, respectively. The values of these parameters have been

determined based on the guidelines suggested by Farza et al. (1999).

In this study, another Estimator ASE 2 is designed with known parameter, rate

of raction (rD), along with VnT ynT . Such attempts have been made in order to provide a

better test scenario for the proposed procedure. It is worth mentioning that xD is directly

obtained from the sensor model, composition of distillate is estimated in the ASE 2 to

compute the residual ( ˆx − x). The predictor model, which is required to design an

adaptive state estimator, consists of only two balance equations around the reflux-

condenser system. The mathematical representation of the predictor model is

47

Sub system 1:

^

,

^ ^,

D i

nT nT i

d xdt

d V ydt

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

=10

0 0Dm

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

^,

^ ^,

D i

nT nT i

x

V y

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

+, , , ..

0

D i D i i D i D

D

Rx Dx rM

γ ε− − +⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

1

2

2

2

DM

α

α

⎡ ⎤− ⎢ ⎥⎢ ⎥⎣ ⎦

^, ,D i D ix x⎡ ⎤−⎢ ⎥⎣ ⎦

(4.48)

Sub system 2:

^

,

^

D id xdt

r

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= .0

0 0

i D

Dmγ ε⎡ ⎤

⎢ ⎥⎢ ⎥⎣ ⎦

^,

^

D ix

r

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

+

^ ^, ,

0

nT nT D i D i

D

V y Rx DxM

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

.

2

D

j D

M

α

αγ ε

⎡ ⎤⎢ ⎥− ⎢ ⎥⎢ ⎥⎣ ⎦

^, ,D i D ix x⎡ ⎤−⎢ ⎥⎣ ⎦

(4.49)

In this estimator α1, α2, α3 are tuning parameters

48

4.6. Simulation Results and Discussion In the batch reactive distillation operation, first the column may be brought to the

steady state by considering the total reflux startup procedure with no reaction. Ethanol is

the lightest component in system reactants, ethanol reaches to top of the column and in

distillate ethanol mole fraction reaches to steady state 0.99385 at top are shown in the

Fig. 10a. Once reactants in the column reaches to steady state then reaction will take

place in the column. We have carried out several numerical simulations with the example presented by

Mujtaba and Macchietto. Fig.10b and Fig. 10c. Shows that open loop performance of the

composition for total reflux ratio and internal reflux ratio rf (R/VnT). It is noted that the

concentration of the reactant acetic acid in the distillate flow goes to almost zero

immediately.

Reaction mechanism becomes more important than the distillate separation

mechanism. For total reflux operation, the maximum achievable ethyl acetate

concentration is 0.935 mole fraction. This value imposes a limit in the achievable product

purity under batch operation. In fact, for batch operation (0 < rf < 1), the ethyl acetate

mole fraction increases in the first part of the batch time, achieves a maximum value, and

then decreases until the end of the operation. The first part of the batch operation where

the ethyl acetate mole fraction increases can be called the reaction phase because the

chemical reaction is the main drive of ethyl acetate in the distillate product. Compared

with nonreactive batch distillation where the mole fraction of the more volatile

component decreases along the operation of the uncontrolled batch distillation, the last

part of the batch operation can be called the separation phase. In fact, the process is

controlled by the separation during the second phase. It is also noted that the smaller the

reflux ratio, the smaller the operating time where the maximum ethyl acetate mole

fraction is achieved. Hence, inefficient batch operation is obtained with smaller values of

the reflux ratio.

49

4.6.1. Open Loop Performance of Estimators

Comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with

out any change for production stage are shown in Fig 11.

Disturbance in Heat input to reboiler

The performance of the designed state observer has been tested considering +10%

and then -10% step change in the heat input to the reboiler (from 3200 kJ/min to 3520

kJ/min at time = 2000 min and then from 3520 kJ/min to 3200 kJ/min at time=3000 min).

In Fig.12 shown good agreement between the process outputs and estimator outputs.

Uncertain tray efficiency

Fig.13. compares the estimated outputs and true process outputs with two

consecutive step changes, +10% and −10%, in the Murphree e tray efficiency (from 0.75

to 0.825 at time 2000 min and then from 0.825 to 0.75 at time 3000 min). In the present

situation, the ASE estimator again confirmed its convergence ability.

Initialization error

The initialization error performance of the ASE scheme has been shown in Fig 14

comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with 10%

initialization error in VnT ynT,3 (from 0.8692 to 0.9561 ). It is obvious from the figures that

good convergence against the initialization error is achieved by the proposed observation

approach.

The initialization error performance of the ASE scheme has been shown in Fig.15

comparison of the estimated outputs (ASE 2) and process outputs with 10% initialization

error in both VnT ynT,3 (from 0.8692 kmol/min to 0.9561 kmol/min) and rD (from

0.00000417 kmol/lit.min to -0.00000458 kmol/lit.min).

50

4.6.2. Constant composition control In the present study, the set point composition has been fixed at a value of 0.934

for product ethyl acetate (lightest component). The column is started up as usual and

product withdrawal is begun as soon as this component met the composition

specification. The time taken to withdrawal the product is 5400 min with purity of 0.934.

shown in fig.17. Comparative performance between the GSPI and GMC controllers based

on the ISE values.

Closed-loop servo performance of GSPI and GMC control algorithms with two

consecutive step changes in XD,3 from 0.934 to 0.9 at time 2000 min and then from 0.9

to0.934 at time 3000 min shown in fig. 18.

Closed-loop with regulatory performance (+10% step change in set point) at time

2000 min and 3000 min. shown in fig.19.

51

Open-Loop Performance

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0 1 2 3 4 5 6 7 8 9 10 11

Time (min)

XD

, i (

mol

e fra

ctio

n)

Acetic Acid EthanolEthyl AcetateWater

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 9 10 11

Time (min)

XB

, i(m

ole

fract

ion) Acetic Acid

EthanolEthyl AcetateWater

Fig. 9. Steady state response of batch reactive distillation without reaction.

52

00.10.20.30.40.50.60.70.8

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Time (min)

XB

,I (m

ole

fract

ion)

Acetic AcidEthanolEthyl AcetateWater

Fig. 10a. open-loop response of the batch reactive distillation column with total reflux ratio. (First 10 min without reaction)

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

Time (min)

XD

, i(m

ole

fract

ion)

Acetic AcidEthanolEthyl AcetateWater

53

-0.20

0.20.40.60.8

11.2

0 50 100 150 200 250 300 350 400 450 500 550 600 650

Time (min)

XD

, i (

mol

e fra

ctio

n)

Acetic AcidEthanolEthyl AcetateWater

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400 450 500 550 600 650Time (min)

XB

,I (m

ole

fract

ion)

Acetic AcidEthanolEthyl AcetateWater

Fig. 10b. open-loop performance of the batch reactive distillation column with start-up phase and production phase (RR=0.8 with chemical reaction)

54

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

X D,3

(m

ole

fract

ion) Process

ASE 1ASE 2

0

0.1

0.2

0.3

0.4

0.5

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

V nT

y nT,

3 (k

mol

/min

)

ProcessASE 1ASE 2

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

r D (k

mol

/lit.m

in)

ProcessASE 2

Fig. 11. Comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with out any change for production stage.

55

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

X D,3 (

mol

e fra

ctio

n)ProcessASE 1ASE 2

0

0.1

0.2

0.3

0.4

0.5

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

V nT

y nT,

3 (k

mol

/min

) ProcessASE 1ASE 2

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

r D (k

mol

/lit.m

in)

ProcessASE 2

Fig. 12. comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with two consecutive step changes in heat input to reboiler (from 3200 kJ/min to 3520 kJ/min at time 2000 min and then from 3520 kJ/min to 3200 kJ/min at time 3000 min).

56

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

X D,3

(mol

e fra

ctio

n)ProcessASE 1ASE 2

0

0.1

0.2

0.3

0.4

0.5

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

V nT

y nT,

3 (k

mol

/min

)

ProcessASE 1ASE 2

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

r D (k

mol

/lit.m

in)

ProcessASE 2

Fig. 13. comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with two consecutive step changes in tray efficiency (from 0.75 to 0.825 at time 2000 min and then from 0.825 to 0.75 at time 3000 min).

57

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

X D,3 (

mol

e fra

ctio

n)ProcessASE 1ASE 2

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

V nT

y nT,

3 (k

mol

/min

)

ProcessASE 1ASE 2

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

r D (k

mol

/lit.m

in)

ProcessASE 2

Fig. 14. comparison of the estimated outputs (ASE 1, ASE 2) and process outputs with 10% initialization error in VnT ynT,3 (from 0.8692 to 0.9561 ).

58

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

X D,3

(mol

e fra

ctio

n) ProcessASE 2

0

0.2

0.4

0.6

0.8

1

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

V nT

y nT,

3 (k

mol

/min

)

ProcessASE 2

-1.00E-04

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

r D (k

mol

/lit.m

in)

ProcessASE 2

Fig. 15. comparison of the estimated outputs (ASE 2) and process outputs with 10% initialization error in both VnT ynT,3 (from 0.8692 kmol/min to 0.9561 kmol/min) and rD (from -0.00000417 kmol/lit.min to -0.00000458 kmol/lit.min).

59

Closed-loop performance

0.932

0.9325

0.933

0.9335

0.934

0.9345

0.935

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

XD

,3/X

DSP

,3 (m

ole

fract

ion)

Set PointGSPIGMC

1.041.051.061.071.081.09

1.11.111.121.13

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

Ref

lux

Rat

e (k

mol

/min

)

GSPIGMC

Fig. 16. Comparative closed loop performance of GSPI and GMC control algorithms for production stage.

60

Servo performance

0.88

0.89

0.9

0.91

0.92

0.93

0.94

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

XD

,3 /X

DSP

,3 (m

ole

fract

ion)

Set PointGSPIGMC

00.20.40.60.8

11.21.41.6

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

Ref

lux

Rat

e (k

mol

/min

)

GSPIGMC

Fig. 17. Comparative closed loop performance of GSPI and GMC control algorithms with two consecutive step changes in XD,3 (from 0.934 to 0.9 at time 2000 min and then from 0.9 to0.934 at time 3000 min).

61

Regulatory Performance

0.930.9310.9320.9330.9340.9350.9360.9370.9380.939

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

XD

,3/X

DS

P,3

(m

ole

fract

ion) Set Point

GSPIGMC

1

1.05

1.1

1.15

1.2

1.25

1400 1900 2400 2900 3400 3900 4400 4900 5400

Time (min)

Ref

lux

Rat

e (k

mol

/min

)

GSPIGMC

Fig. 18. Comparative Regulatory performance of GSPI and GMC control algorithms with two consecutive step changes in reboiler heat duty (from 3200 kJ/min to 3520 kJ/min at time 2000 min and then from 3520 kJ/min to 3200 kJ/min at time 3000 min).

62

Tuning parameters Table. 9. Tuning parameters for ASE 1 and ASE 2

Estimator

α1

α2

α3

ASE 1

110

20

----

ASE 2

110

20

0.0055

Table. 10. GSPI and GMC controller tuning parameters

Controller kco τi K1 K2

GSPI 1.16 0.15 ---- ---- GMC ---- ---- -20 0.000012

Table. 11. ISE values for comparison of ASE 1 and ASE 2

ASE 1 ASE 2

XD,3 VnT ynT,3 XD,3 VnT ynT,3 rD Closed loop 0.00000626

8 0.0001088 0.00000121

9 0.000129 0.00000731

Q change 0.00004418 0.00011275 0.0000389 0.000258 0.00001371

Efficiency change

0.00004395 0.00011126 0.0000369 0.000241 0.00001385

Initial change in

VnT ynT

0.000043936

0.00011124 0.0000353 0.000235 0.00001412

Initial change in

VnT ynT & rD

____ _____ 0.00003873 0.0002805 0.00001416

Table. 12. ISE values for comparison of GSPI and GMC

GSPI

GMC

Closed loop

0.000013578

0.0000022574

Regulatory Performance

0.00010325

0.00002173

Servo Performance

0.00986812

0.00122158

63

Chapter -5

Conclusions and Scope of future work

Conclusion GSPI controller strategy is proposed for the control of constant composition

operations of a multicomponent batch distillation column compare to other two NLPI and

PI controllers. An adaptive control strategy is proposed for the control constant.

Composition operations of multi component reactive batch distillation column. Structural

and parametric mismatches were considered between the actual process and the predictor

model in order to provide a realistic test scenario for the proposed strategy. In this study,

the open-loop performance of the nonlinear adaptive estimator was inspected. Despite

structural discrepancy, disturbance, and parametric uncertainty, the observation scheme

provided sufficiently fast convergence of the estimation error towards zero. The GMC

control strategy showed relatively better performance than the GSPI controller for

constant composition control in reactive batch distillation.

Future Directions

1) Testing of the proposed adaptive (GMC-ASE) controller on a real time batch

reactive distillation process.

2) In corporation of dead-time and feed forward disturbance comparison.

3) Comparison with a widely used nonlinear model predictive controller ( NLMPC)

64

Notation

MB =liquid holdup in still pot (kmol)

MD =liquid holdup in reflux drum (kmol)

Mn =liquid holdup in the nth tray (kmol)

nT =total number of trays

QR =Heat input to the still pot(kJ/min)

R=Reflux flow rate (kmol/min)

RS =steady state value of R, (kmol/min)

VB =vapor boil-up rate (kmol/min)

Vn=vapor flow rate of vapor leaving nth tray (kmol/min)

VnT =vapor flow rate of vapor leaving top tray (kmol/min)

D=distillate flow rate (kmol/min)

Ln=liquid flow rate of liquid leaving the nth tray (kmol/min)

xB,i=composition of component i in the still

xD,i=composition of component i in the Distillate

xn,i=composition of component i in Liquid stream leaving the nth tray

r,i=rate of reaction of component (kmol/lit.min)

ε=volume of catalyst (lit)

k1, k2 =rate constants

K1, K2= GMC controller tuning parameters

vb =volume of reboiler (lit)

vd=volume of reflux drum (lit)

v=volume of tray (lit)

rf=multiplication factor (rf=1 for reactive section , rf=0 for non reactive section)

γi,m=stoichiometric coefficient of i th component of m th reaction.

α1, α2, α3=estimator tuning parameters.

rD,i=rate of reaction of component i in distillate.

rB,i= rate of reaction of component i in bottom.

rn,i = rate of reaction of component i in nth tray.

65

References

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67

Appendix. A. Data for Batch Distillation system

Table 1: Constants of enthalpy equations (29&30)

Constants Cyclohexane Toluene Chlorobenzene

Liquid Enthalpy

a 0 0 0

b -220.6 140.1 -1307

c 1.559 -7.615×10-2 7.669

d -3.141×10-3 2.317×10-4 -1.799×10-2

e 2.672×10-6 -2.769×10-13 1.587×10-5

Vapor Enthalpy

a 4.126×104 4.852×104 5.118×104

b -203.0 132.7 -1332

c 1.284 -0.2389 7.623

d -2.55×10-3 5.907×10-4 -1.791×10-2

e 2.084×10-6 -3.914×10-7 1.575×10-5

Table .2: Density, molecular weight and boiling point data

Property Cyclohexane Toluene Chlorobenzene

B.P. (oC) 80 110.8 132.1

ρ (gm/cc) 0.779 0.866 1.107

ρ (lb/ft3) 48.63 54.06 69.11

M.W 84.16 92.13 112.56

Table .3: Antoine’s constants Constant Cyclohexane Toluene Chlorobenzene

A 15.7527 16.0137 16.0676

B 2766.63 3096.52 3295.12

C -50.50 -53.67 -55.60

68

Appendix. B. Data for Reactive Batch Distillation system

Table .1: Antoine’s constant Constant Ethanol Acetic Acid Ethyl Acetate Water

A 16.80 18.912 16.1516 18.3086

B 3405.57 3803.98 2790.5 3816.44

C -56.34 -41.68 -57.15 -46.19

Table .2: Density, molecular weight and boiling point data

Property Ethanol Acetic Acid Ethyl Acetate Water

B.P. (K) 351.5 391.1 350.3 373.2

ρ (gm/cc) 0.789 1.049 0.897 1

ρ (lb/ft3) 65.486 49.255 62.4278 55.997

M.W 46.069 60.05 88.107 18

Table .3: specific heat of vapor constant

Constant Ethanol Acetic Acid Ethyl Acetate Water

a 14.6934 14.048 24.9801 7.9857

b 0.22987x10-1 0.21531x10-1 0.33297x10-1 0.46331x10-3

c -0.102199x10-4 -0.21534x10-4 -0.7316x10-6 -0.14028x10-5

d 0.2589x10-8 -0.4607x10-8 -0.1247x10-8 -0.65783x10-9

e -0.8044x10-12 0.1893x10-11 0.48242x10-11 0.9895x10-13

Cp=a+bT+cT2+dT3+eT4

HV=mcpΔT

HL=HV-λ

Latent heat λ= ( )

22

BRTC T

⎡ ⎤⎢ ⎥

+⎢ ⎥⎣ ⎦