Modeling and Optimization of Polymerization Reactors

7/28/2019 Modeling and Optimization of Polymerization Reactors

1/103

MODELING AND OPTIMIZATION OF

POLYMERIZATION REACTORS

SUBMITTED BY:

FIAZ AHMED TAHIR

(2002-POLY-1062)

MOHSIN ABBAS

(2002-POLY-1052)

SUBMITTED TO:

DR.JAVED RABBANI KHAN

DEPARTMENT OF CHEMICAL ENGINEERING

UNIVERSITY OF ENGINEERING & TECHNOLOGY

LAHORE.


2/103

Preface

Materials are more than mere components in technology; rather, the basic properties of

materials frequently define the capabilities, potential, reliability, and limitations oftechnology itself. Improved materials and processes will play an ever increasing role in

efforts to improve energy efficiency, promote environmental protection, develop an

information infrastructure, and provide modern and reliable transportation and civil

infrastructure systems. Advances in materials science and engineering, therefore, enableprogress across and broad range of scientific disciplines and technological areas with

dramatic impacts on society.

Among these materials which have grown tremendously during last few decades aresynthetic polymers. Today, polymers are found in a large variety of products e.g.,

automobiles, paints, and clothing, to name a few. Polymers have replaced metals in many

instances, and with the development of polymers alloys, applications in specialty areasare certain grow. The new and highly specialized application of polymers, along with the

trend toward totally quality management and global competitiveness, has served to driveup the quality expectations of the customer. These developments make it imperative to

operate the polymerization processes efficiently, which underscores the importance ofmodeling and optimization of polymerization reactors.

In a polymerization reactor, raw materials are mixed at specified operating

conditions to produce polymer(s) having desired properties. The end-use properties ofinterest include color, viscoelasticity , thermal properties, and mechanical properties

among others. To produce a polymer with such desired properties means that process

variables such as temperature, molecular weight, molecular weight distribution must betightly controlled. The manipulated variables available for controlling the variables of

interest at setpoints include the flow rates of raw materials and catalyst, temperature of

feed streams and temperature, and/or flow rated of heating/cooling mediums. Thusmathematical modeling of polymerization reactors which relate molecular weight and

molecular weight distribution of polymers with manipulated variables is very important.

Being undergraduate students of polymer engineering we cant model and optimize

the polymerization reactors in details because it is very difficult to model and optimizethe polymerization reactors specially on this level.

Now here is brief overview of our project.

We begin in chapter 1 with a brief overview of modeling, optimization, polymerizationtechniques and polymerization reactors.

In chapter 2, we started with brief concepts of NACL, WACL, NAMW, WAMW and

MWD. We followed this with the discussion of chemistry and kinetics of various

polymerization reactions.Chapter 3 gives the details of modeling, how to build a model, use of modeling, modeling

principles and how to model a polymerization reactor.

Chapter 4 is devoted to optimization in which we define objective function, variables,constraints, mathematical relationships between these, definition of optimization

problems and optimization solution methodologies.

2


3/103

In Chapter 5 we shifted our attention from chemistry and kinetics of polymerization to

modeling of batch polymerization reactors. We developed models for anionic, free

radical and step growth polymerization and solved these using MATLAB programming.Chapter 6 throw light on modeling of continuous stirred tank polymerization reactors.

The techniques used were anionic, free radical and step growth polymerization. Then we

solved these using MATLAB.Chapter 7 is about optimization of polymerization reactors. First we have discussed what

is multi objective optimization and then we have written multi objective optimization of

polyester reactor.We thank the Department of chemical engineering UET Lahore for their support in

this endeavor.

We pay special homage to our respective teacherDr. Javed Rabbani Khan, who really

paid their special attention in completion of our project.

FIAZ AHMED TAHIR

MOHSIN ABBAS

3


4/103

TABLE OF CONTENTS

CHAPTERS PARTICULARS PAGE NO.

CHAPTER 1 INTRODUCTION

1.1: Modeling 6

1.2: Optimization 7

1.3: Classification of polymerization reactions 71.4: Polymerization reactors 8

CHAPTER 2 POLYMER REACTION ENGINEERING

2.1: Molecular weight and molecular weight distribution 92.2: Kinetics of anionic polymerization 10

2.3: Kinetics of cationic polymerization 13

2.4: Kinetics of free radical polymerization 14

2.5: Kinetics of step growth polymerization 16

2.6: Kinetics of copolymerization 17

2.7: Polymerization reactors 19

2.8: Reactor selection

CHAPTER 3 MODELING3.1: What are models? 20

3.2: Reasons for developing models 20

3.3: General modeling principles 21

3.4: Classification of models 21

3.5: How to build a model 223.6: Modeling of Polymerization reactors 23

CHAPTER 4 OPTIMIZATION 4.1: How to define a model for optimization? 27

4.2: What makes a model hard to solve? 294.3: Mathematical relationships 30

4.4: Optimization solution methodologies 32

4.5: Algorithm solutions to optimization problems 32

4


5/103

CHAPTER 5 MODELING OF BATCH POLYMERIZATION REACTORS

5.1: Anionic polymerization 36

5.2: Free radical polymerization 38

5.2.1:Model for free radical batch polymerization

reactor 39

5.2.2:MATLAB solution of model 40

5.3: Step growth polymerization 45

5.3.1:Step growth polymerization in absence of catalyst 46

5.3.2:Model for step growth polymerization reactor

in absence of catalyst 47

5.3.2.1:MATLAB solution of model 48

5.3.3:Step growth polymerization in presence of catalyst 53

5.3.4:Model for step growth polymerization reactorin presence of catalyst 54

5.3.4.1:MATLAB solution of model 55

5.3.5:Comparison of catalyzed and non catalyzed reactions

56

CHAPTER 6 MODELING OF STIRRED TANK POLYMERIZATIONREACTORS

6.1: Anionic polymerization 586.1.1:Model for stirred tank anionic polymerization

reactor 59

6.2: Free radical polymerization 60

6.2.1: Model for continuous stirred tank free radical

polymerization reactor 61

6.2.2: MATLAB solution of model 626.3: Step growth polymerization 71

6.3.1: Model for continuous stirred tank step growth

polymerization reactor 72

6.3.2:MATLAB solution of model 73

6.4:Reactor dynamics 77

CHAPTER 7 OPTIMIZATION OF POLYMERIZATION REACTORS

7.1: What is multiobjective optimization? 817.2: Optimization of polyester reactor 82

5


6/103

CHAPTER 1

INTRODUCTIONSynthetic polymers have grown tremendously during last few decades. Today, polymers

are found in a large variety of products ranging from common to very specialized

applications. The new and highly specialized application of polymers, along with the

trend toward totally quality management and global competitiveness, has served to driveup the quality expectations of the customer. These developments make it imperative to

operate the polymerization processes efficiently, which underscores the importance of

modeling and optimization of polymerization reactors.In a polymerization reactor, raw materials are mixed at specified operating

conditions to produce polymer(s) having desired properties. The end-use properties ofinterest include color, viscoelasticity , thermal properties, and mechanical propertiesamong others. To produce a polymer with such desired properties means that process

variables such as temperature, molecular weight, molecular weight distribution must be

tightly controlled. The manipulated variables available for controlling the variables of

interest at setpoints include the flow rates of raw materials and catalyst, temperature offeed streams and temperature, and/or flow rated of heating/cooling mediums. Thus

mathematical modeling of polymerization reactors which relate molecular weight and

molecular weight distribution of polymers with manipulated variables is very important.

1.1 Modeling

Modeling is The representation of a physical system by a set ofmathematical relationships that allow the response of the system to various

alternative inputs to be predicted.

Reasons for developing process models are that we can improve or understand

chemical process operations, improve the quality of produced products, increasethe productivity of existing and new processes, for operator training and process

design etc.

General modeling principles are Steady state modeling, Dynamic modeling andConstitution relationships.

Models can be classified as. Theoretically based vs. empirical, Linear vs.

nonlinear, Steady state vs. unsteady state, Lumped parameter vs. distributed

parameter and Continuous vs. discrete variables

6


7/103

1.2 OPTIMIZATION

Reactors are often the critical stage in a polymerization process. Recently, demands onthe design and operation of chemical processes are increasingly required to comply

with the safety, cost and environmental concerns. All these necessitate accurate

modeling and optimization of reactors and processes. To optimize a reactor, first wedefine an objective function, constraints, variables. Then we develop a mathematical

relationship between these. Then we solve these by different optimization solution

methodologies.

1.3 Classification of Polymerization reactions

Polymerization reactions are classified as homogeneous Polymerization and heterogeneous

Polymerization.

In homogeneous polymerization common techniques are bulk polymerization and solution

polymerization. And in heterogeneous polymerization are emulsion polymerization,suspension polymerization, precipitation polymerization and solid-phase polymerization.

We will focus our attention mostly on bulk and solution polymerization techniques.

On the basis of kinetics main classification of polymerization reactions are cationic,

anionic, free radical and step growth or condensation polymerization.

1.4 Polymerization Reactors

Polymerization reactors can be classified by the phase involved in the reaction.

Classification of Polymerization Reactors

Continuous Phase Dispersed Phase Type of

Polymerization

Polymer solution None Homogeneous bulk

or solution pzn

Polymer solution Any (e.g.

condensation

product)

Heterogeneous bulk

on solution pzn

Water or other non

solvent

Polymer or polymer

solution

Suspension,

dispersion or

7


8/103

emulsion pzn

Liquid monomer Polymer (swollenwith monomer)

Precipitation orslurry pzn

Gaseous monomer Polymer Gas phase pzn

A polymerizer reactor will be heterogeneous whenever the polymer is insoluble in

the monomer mixture from which it was formed. If the polymer is soluble in its

own monomers, a dispersed phase polymerization requires the addition of a nonsolvent (typically water) together with appropriate interfacial agents. For high

volume polymers like high volume chemicals continuous operation is generally

preferred over batch.

In a batch reactor feed is entered and product is removed in batches.

In a semi batch reactor initiator or monomer is added continuously and product isremoved in batches.

Tubular reactors are occasionally used for bulk, continuous polymerizations. A

monomer or monomer mixture is introduced at one end of the tube and if all goes

well, a high molecular weight polymer emerges at the other.

Continuous stirred tank reactors are widely used for bulk, free radical

polymerizations. The details for polymerization reactors and their kinetics are

discussed in the next chapters.

8


9/103

CHAPTER 2

POLYMER REACTION ENGINEERING

While polymerization and the reactions of polymers are in many respects similar to ordinary

chemical reactions, there are some significant differences that make the former unique in thesense of reactor and reaction engineering. These are high viscosities and low diffusion rates

associated with concentrated polymer solutions and polymer melts. In polymerization reactors

we have to control the molecular weight and molecular weight distribution to achieve good enduse product properties. These properties include color, viscoelasticity, thermal properties and

mechanical properties. To produce a polymer with such properties means that process variable

such as temperature, molecular weight, molecular weight distribution and mooney viscosity

must be tightly controlled. In this chapter, we will briefly explain about degree ofpolymerization, molecular weight and molecular weight distribution and also will explain the

kinetics of various types of polymerization reactions.

2.1: Degree of polymerization, molecular weight and molecular weight distribution

Degree of polymerization:

The no of repeat units per chain is known as degree of

polymerization and it is denoted by x, it is also known as length of polymer chain.

Molecular weight:

Molecular weight of given chain is defined as degree of

polymerization times the molecular weight of repeat unit.

We have defined the degree of polymerization for a single polymer molecule. But not all

polymer molecules within a reactor have the same degree of polymerization. Rather, apolymer produced in a single reaction exhibits a distribution of chain lengths (degree of

polymerization). The distribution of chain lengths within a polymeric material may will be the

most important factor in determining its end-use properties. Therefore, it will be necessary to

develop a method of describing the distribution of chain lengths in a polymeric material.

As a single reaction exhibits a distribution of chain lengths,the mean of this distribution is the

number average chain length (NACL), this is the weight chain length distribution, and itsaverage is the weight average chain length (WACL). Respectively, there is the number

9


10/103

molecular weight distribution and the weight molecular weight distribution. Their averages are,respectively, the number average molecular weight (NAMW) and the weight average molecular

weight (WAMW). The values of NAMW and the WAMW will not be necessarily the same.This is because both are single number attempts to represent an entire distribution. It should be

noted that the end-use properties of a polymer are determined by the distribution of molecular

sizes which is independent of the average used to characterize it. IfPn(the concentration of

chains containing npolymer units) is known for all values ofn, the various averages may becalculated as follows

nPnNACL= n = (2.1)

Pn

n2PnWACL= w= (2.2)

nPn

(nw)PnNAMW= mn = (2.3)

Pn

(nw)2PnWAMW= mw = (2.4)

(nw)Pn

Molecular weight distribution :

Molecular weight distribution is defined bypolydispersity D which is given as mw w

D = = (2.5)

mn n

Inspection of Eqs. (2.1) through (2.5) reveals that the polydispersity takes a value of 1 for amonodisperse sample (one in which all of the chains are the exact same length). For any otherdistribution of chain lengths, the polydispersity will be greater than 1. On the other hand, thepolydispersity varies or slightly with average chain length.

Variations in degree of polymerization (and hence in molecular weight) occur for at least threereasons. The main mechanism by which the molecular weight distribution is broadened is

through the nature of the series-parallel reaction mechanisms leading to chain formation.Second mechanism is that of spatial or temporal variations in reaction conditions duringpolymerization. Variations in temperature, monomer concentration, etc in any reactor, and inresidence time in a continuous reactor, affect the individual chain lengths. The final mechanism ofvariation in degree of polymerization that of stochastic variations reaction rates on a molecularlevel. This however has been shown to be insignificant in relation to the previous two. The importconcept, then, is that a distribution of chain lengths will result due to the nature of the reaction

10


11/103

mechanisms, even when all environmental variables (temperature monomer concentration, etc.)are kept constant.

2.2: Kinetics of Anionic Polymerization

Addition polymerization can be carried out by a number of mechanisms. The free radicalmechanism is commercially predominant, but addition polymerization is often carried out byanionic and cationic mechanisms. Anionic polymerization takes place via the opening of a carbondouble bond on the monomer unit. Initiation takes place with the addition of a negative ion tothe monomer, resulting in the opening of a double bond and growth at the end bearing thenegative charge. Propagation proceeds by addition of monomer units with the carbanionremaining with the propagating chain end. Termination of a growing chain usually involvestransfer, and only results in the net loss of a growing chain if the new species is too weak topropagate. Because termination usually involves transfer to some impurity in the system, it ispossible, with carefully purified reagents, to carry out polymerization in which termination islacking. The resulting species are termed living polymers and may result in extremely narrow(essentially monodisperse) molecular weight distributions.

Anionic polymerization is employed with vinyl monomers containing electron-withdrawinggroups such as nitrile, carboxyl, phenyl, or vinyl in an aprotic non-polar solvent. It ischaracterized by high rates of polymerization and low polymerization temperatures. Strongbases such as alkyl metal amides, alkoxides, alkyls, hydroxides, and cyanides are often used toform the original carbanion.

(2.6)

AMn-+M AM-n+1 Propagation (2.7)

AMn-+B AMn+B

- Chain transfer (2.8)HereA-is the anion initiating the polymerization,B is the chain transfer molecule, andB-

is the new anion formed by chain transfer, which may or may not be capable of initiation of a

new chain. The mechanism can be written in a form which is more concise and consistent withthe subsequent treatment for free radical polymerization as

K

AC A- + C + Rate of initiation is given as(RI)

Initiationki RI= ki A

- M (2.9)

A- + M P1

Initiation

11


12/103

Rate ofpropagation is given as(RP)

kp

Pn + M Pn+1 Propagation RP = kp Pn M (2.10)

kf Rate of chain transfer

Pn + B Mn + B- Chain transfer RT = kfPnB (2.11)

HerePnis taken to meanAM- and MnrepresentsAMn. For very fast reactions, concentration

of reactive species (in this case, ionic chains) becomes essentially constant very early in thereaction. For this to happen here, the rates of initiation and chain transfer must reach steadystate quickly be equal. This is known as the quasi-steady-state approximation (QSSA).Based on the mechanism above and making the QSSA forPn, the rate of polymerizationcan be written as

At steady state

Rate of initiation = Rate of termination

ki A- M = kfPnB

ki A- M

Pn = (2.12)kf B

Putting the value of Pn in Eq.(2.10), so_

kp ki A- M 2

RP = (2.13)kf B

Rate of formation of ion A- is given as

-rA- = KAC - KA-C+ + kiA-M

At steady state (rA- =0)

Rearranging KAiA- = (2.14)

KC+-ki M

Putting value of A-

from Eq. 2.14 in Eq. 2.13 gives

Kkp ki AC M 2

RP = (2.15)K kf B C

+ -kikfBM

By putting K kf= kf and also kikf =0 , final eq. becomes

12


13/103

Kkp ki AC

M 2

RP =(2.16)

kf B C+

QSSA is only valid for significant chain transfer to an unreactive anion,B - . In theabsence of

rapid chain transfer (or of chain transfer to a nonpropagation anion), the rate of

polymerization will continue to rise as the total number ofliving chains increases untilinitiation is complete. Initiation is complete when all of the catalyst has been consumed;

from this point on, the number of live chains will remain constant.

The instantaneous degree of polymerization may be written as the rate of propagation divided

by the rate of chain transfer (rate of productionof dead chains).kp M

x = (2.17)

kf B

2.3 CATIONIC POLYMERIZATION

Cationic polymerization is another mechanism of addition polymerization. It proceedsthrough chain propagation via a carbonium ion with the opening of a double bond on themonomer unit as with anionic polymerization. The carbonium ion is formed by the reaction ofa strong Lewis acid (catalyst) with a weak Lewis base (co catalyst) followed by attack on thedouble-bonded monomer unit. Termination via terminal double-bond formation and chain

transfer to monomer and polymer are dominant.C'ationic polymerization is carried out with vinyl monomers containing electron-releasing groups such as alkoxy, phenyl, and vinyl. The system is characterized by very highrates of polymerization. The mechanism of cationic polymerization may be written as

Initiation(2.18)

Propagation (2.19)

Termination (2.20)

Chain transfer (2.21)

HereA is the catalyst andRHis the cocatalyst. These two species react to form thecatalyst-cocatalyst complex in Eq. (2.18). This complex donates a proton to the monomer,forming a carbonium ion. Because cationic polymerization is usually carried out in achlorinated hydrocarbon solvent of low dielectric constant, the anion ( A R -) cannot beseparated from the carbonium ion. Rather, the two form an intimate ion pair. Propagationtakes place by the addition of monomer to the growing chain end [Eq. 2.19]. Terminationoccurs with the formation of a terminal double bond and the regeneration of the catalyst-cocatalyst complex [Eq. (2.20)]. Chain transfer to monomer takes place as shown in Eq.

13

++

++

++

+

++

+

++

+

+

+

+

ARHMMMARHM

ARHMARHM

ARHMMARHM

ARHMMARH

ARHRHA

n

k

n

n

k

n

n

k

n

k

K

f

i

p

i

1


14/103

(2.21). The mechanism can be written in a form which is more concise and consistent with theprevious notation as

Initiation (2.18)

Propagation (2.19)

Termination (2.20)

Chain transfer (2.21)

HerePnis taken to meanHM+

nAR-

Based on the mechanism above, the rate of polymerization may be written as the product of

the propagation rate constant, the monomer concentration, and the concentration of live

chains (P). By applying quasi-steady-state approximation forPn and following the same

procedure as in anionic polymerization on equations (2.18) to (2.21) the rate ofpolymerization can be written as

Kkp ki A(AH) M 2

RP = (2.22)kt

If, as is often the case in ionic polymerization, termination is negligible, the QSSA is notapplicable and the term after the second equality in Eq. (2.22) cannot be used. In this case,the rate of polymerization will continue to rise as the total number of living chainsincreases until initiation is complete. Initiation is complete when all of the catalyst hasbeen consumed; from this point on, the number of live chains will remain constant. Theinstantaneous degree of polymerization may be written as the ratio of propagation to the sumof the rates of termination and transfer:

Mkk

Mk

PMkPk

PMkx

ft

p

ft

p

+=

+=

Thus, if transfer predominates, the degree of polymerization is a function only of temperature(through kp/kf}. If termination predominates and if the activation energy for termination isgreater than the sum of the activation energies for initiation and propagation (as is often thecase), both the rate of polymerization and the degree of polymerization increase withdecreasing temperature. These conditions are the reverse of those found in free radicalpolymerization and allow the attainment of high molecular weight.

14

1

1

1

PMMP

ARHMP

PMP

PMARH

ARHRHA

n

k

n

n

k

n

n

k

n

k

K

f

i

p

i

++

+

+

+

+

+

+

+

+


15/103

2.4 Free Radical Polymerization

Free radical polymerization is the most common of all addition polymerization mechanisms.When free radicals are generated in the presence of unsaturated monomers, the radical adds

to the double bond and the resultant unpaired electron generates another radical. This radicalis free to react with another monomer unit, and in this way the polymer molecule grows byadding monomer units while maintaining a free radical at the reactive end of the live(growing) chain. Chain growth continues until the radical is terminated or transferred toanother chain. The complete mechanism can be written as follows:

I dk 2R Initiation (2.23)M+R ki P1 (2.24)

Pn +M kp Pn+1 Propagation (2.25)

Pn+Pm ktc

Mn+m Termination by combination (2.26)

Pn+Pm ktd Mn+Mm Termination by disproportionation (2.27)

Pn+M dk Mn+P1 Chain transfer to monomer (2.28)

Pn+S dk Mn+S Chain transfer to solvent (2.29)

Pn+T dk Mn+T Chain transfer to transfer agent (2.30)

Pn+MmMn+Pm Chain transfer to polymer chain (2.31)

P+In Q Inhibition (2.32)

This complex set of reactions may be divided into initiation, propagation, termination,and chain transfer reactions. The rate of polymerization may be derived by applying mass-action kinetics to the elementary reactions in eqns (2.23)to(2.32).

PMR ki+ (2.33)1PMP kp+ (2.34)

2PPPkt+ (2.35)

By applying mass balance for R and P from equation yields

RMkfIkdtdR id = 2/ (2.36)

(2.37)

As in the free radical polymerization, the propagation step is very short as compared totermination and initiation so in above equation propagation term is neglected.

2/ PkRMkdtdP =

15

2

/ PkkpPMRMkdtdP ti =


16/103

At steady state, dP/dt=0 and also dR/dt=0

By applying this and solving above equations give

t

i

t

i

k

RMkP

k

RMkP

=

=2

(2.38)

Finally the rate of polymerization can be written as

t

ippP

k

RMkMkPMk

dt

dMR === (2.39)

In the case where inhibition is significant, P is given by

2/1

/1

12

+

=

MkInkk

IfkP

iint

d (2.40)

The instantaneous degree of polymerization can be defined as the rate of propagation divided bythe rate of production of dead chains (the sum of the rates of all reactions leading to deadchains):

PTkPSkMPkPkPk

MPkx

ftfsfmtdtc

p

++++=

222/1

2.5 Step-Growth Polymerization

Step-growth polymerization involves reaction of functional groups on adjacent monomermolecules with the evolution of water or other low-molecular-weight by-products. Thereaction is stepwise or step-growth in the sense that the reaction of each functional group isessentially independent of previous condensation reactions. There are no activated speciesas in addition polymerization.Condensation polymerizations are of two general types. A-B type and A-A/B-B type

Experimental observation of step-growth polymerization yields the following generalcharacteristics: early disappearance of the monomer, absence of any high polymer during theearly stages of reaction, and equilibrium between polymerization and depolymerizationreactions. These observations suggest a mechanism of linear condensation in which monomermolecules react to form dimers, the dimers react with each other to form tetramers (or withother oligomers to form larger oligomers), and the tetramers react with other oligomers to formlonger chains. Thus, in this step-growth mechanism, the monomer disappears rapidly as it isconverted to a dimer. A great deal of low-molecular-weight material is formed early in the

16


17/103

reaction, and the average chain length grows slowly as the polymer chains condense to formlonger chains. Because the condensation reaction is reversible, the polymerization is alwaysin equilibrium with the depolymerization reaction (hydrolysis). The depolymerization can becontrolled by continuously removing the water (or other by-product) of condensation, thus drivingthe polymerization to completion. The validity of this sort of polymerization mechanism has

been verified for a large number of linear condensation polymerizations.Rate expressions for step-growth polymerization can be written from mass-actionkinetics once the mechanism is understood. The condensation is catalyzed by acids. Thus,for a driven system, the rate of polymerization in the presence of an acid can be written as

A+B+H+ P+C (2.41)Where P is Polymer formed and C is a small molecule which is condensed out, A,B

monomers and H+ is acid which is used to catalyze the reaction.

Rate of Polymerization can be written as:

+== kABHdt

dARp (2.42)

For a stoichiometric ratio ofA andB, and assuming the acid concentration to be constantover the reaction, the rate of polymerization may be simplified to

RP =dt

dA = k' A 2 ( 2 . 4 3 )

In the absence of added strong acid, an acid functional group on the monomer can catalyzethe reaction. The kinetics then becomes

RP=dt

dA = k A 2B ( 2 .44)

Wher e A r ep r es en t s t he ac i d i c f unc t i ona l g r oup . For a s t o i ch i omet r i c

r a t i o o f f unc t i ona l g r oups t h i s becomes

RP=dt

dA = k A 3 ( 2 .45)

The progress of the polymerization reaction can be quantified by introducing the extent ofreaction, p, defined as the fraction ofA orB functional groups which have reacted at time t.The number average chain length is given by the total number of monomer molecules initiallypresent divided by the total number of molecules present at time t, which can be related tothe extent of reaction as follows:

PP)(NN

NNn === 1

110

00 ( 2 .46)

Inspection of eq. (2.46) will indicate that to obtain the necessary high number average

chain length, the extent of reaction must be well above 0.99.

2.6 COPOLYMERIZATION

One of the most important polymerization techniques is co polymerization. In this type of

polymerization two monomers with different functional groups are reacted. The

17


18/103

mechanism of co polymerization is same as that of polymerization. The steps in co-

polymerization are initiation propagation and termination, as for other polymerization

techniques. The mechanism is as under

A.

I.

B.

C.

The I. Symbol is for radical. The rate of decomposition is given by (RI = kII).The freeradicals (/.) react with monomer molecules to form chain radicals. The system to be

studied consists of three types of monomers. A, B, and C; hence the in it ia tion stage

can be symbolized. The primary chain radicals A., B., and C. can now react wi th

monomers and thereby create a growing chain. This growth phase is symbolized, whereA ., B., and C. now represent polymer chains ending wi th a radical attached to anA,B. orCmonomer. Since there are three monomers, there will be nine possiblereactions. The KJKare the"propagation" rate constants.

A. A.

B. B. P

.

C.

C.

The fourth step in the sequence is the t ermina tion reaction where two chain radicals

react to form a "dead" polymer molecule. There are six possible reactions in this last

step. Th is sequence co nt in ue s t h ro ug h these four steps unt il all t he monomer

present is converted to polymer or the rate of radical formation from th e in i t ia tor

decreases to zero.

18

Radical generation

.II ik

Initiation

..

.

..

CCI

BBI

AAI

+

+

+

Growth

..

..

..

..

..

..

..

..

..

CCC

BCB

ACA

CBC

BBB

ABA

CAC

BAB

AAA

CC

CB

CA

BC

BB

BA

AC

AB

AA

k

k

k

k

k

k

k

k

k

+

+

+

+

+

+

+

+

+ Termination

PBC

PCA

PBA

PCC

PBB

PAA

TCB

TAC

TAB

TCC

TBB

TAA

k

k

k

k

k

k

+

+

+

+

+

+

..

...

...

...

...

...


19/103

Kinetics of the co polymerization is as underRadical generation RI = kII (2.47)Monomer reaction rates

For monomer A ( )... CkBkAkVAR CABAAAA ++= (2.48)For monomer B ( )... CkAkBkVBR CBABBBB ++= (2.49)For monomer C ( )... CkAkCkVCR BCACCCc ++= (2.50)

2.7 POLYMERIZATION REACTORS

There are different types of Polymerization reactors. Here three types will be considered.

1. Batch (or semi batch) reactor

2. Plug flow reactor3. Continuous stirred tank reactor

1. Batch (or Semi batch) Reactors

The most common polymerization reactor on a numerical basis is the batch kettle. Batchkettle may range in size from a 5-gal pilot plant kettle, to a 30,000-gal production kettle.

They are generally constructed of stainless steel or glass lined.

If all reactants are added at the beginning of the polymerization, the kettle is said to beoperating in the batch mode. If a reactant is added during the course of polymerization.

The kettle is said to be operating in a semi batch mode.

2. Plug flow reactors

In a plug flow reactor, each element of the reaction mixture can be viewed as an

individual batch reactor. The batch time is the residence time in tubular reactor, which is

easily calculated as the total volume of the tube divided by the volumetric flow rate.Because no material enters or leaves the fluid element during the reaction time, all of the

kinetic relationships derived thus far for the batch reactor are directly applicable to theplug flow reactor.Tubular reactors (approximating plug flow characteristics) are

applicable in high volume polymerizations.

3.Continuous stirred tank reactors

19


20/103

The use of continuous stirred tank polymerization (in a single CSTR or train multiple

CSTRs in series) may be warranted for high volume products. The nature of the reactor

system results in low processing costs, high throughput and in most cases a highlyuniform product. The fact that the polymerization rate is constant will contribute to

product homogeneity. Large residence time CSTR systems are not particularly flexible

and are therefore best suited to extended production runs of a small number of products.In low residence time CSTRs (as in olefin polymerization), grade changes can be made

rapidly and low volume products can be made effectively.

CHAPTER 3

MODELING

In this chapter, we will study about use of modeling, modeling principles, how to developa model and at the last we will discus that how a polymerization reactor is modeled and

the techniques for solution of difference equations used for modeling purpose.

3.1 What are Models?

Models may be defined in many ways

The process of creating a depiction of reality, such as a graph, picture,

or mathematical representation. OR

The representation of a physical system by a set of mathematicalrelationships that allow the response of the system to various alternative

inputs to be predicted.

The Webster dictionary defines the term model as a small representation

of a planned or existing object .

From Mc Graw Hill dictionary we find the following definition, amathematical or physical system obeying certain specified conditions,

whose behavior is used to understand a physical, biological or social

system to which it is analogous in some way .

20


21/103

Here we define a process model as a set of equations (including

necessary input data to solve the equations) that allows us to predict the

behavior of a chemical process system.

3.2: Reasons for Developing Models

Many reasons for developing process models:

a) Improving or understanding chemical process operations

b) Improve the quality of produced products

c) Increase the productivity of existing and new processes

d) Operator training

e) Process design

f) Safety system analysis

g) Design control system design

h) To simulate and predict real events and processes

3.3: General modeling principles

3.3.1: Steady state modeling

For steady state modeling

[Mass or energy entering in system] - [Mass or energy leaving a system] = 0The in and out terms would then include the generation and conversion of species

by chemical reaction, respectively.

3.3.2 Dynamic modeling

In this class, we are interested in dynamic balances that have the form:

=

systemtheleaving

energyormassofRate

systemtheentering

energyormassofRate

systemwithindaccumulate

energyormassofRate

21


22/103

3.3.3: Constitution relationships: Sometimes, we need more relationships than

simple mass or energy balance, such as

Gas law e.g., ideal gas law PV=nRT

Arrhenius rate law K(T)=Aexp(-E/RT)

Equilibrium relationships yi=Kixi

Heat transfer TUAQ =

Flow through valvesgs

PxfCF vv

.)(

=

3.4: Classification of models

Models can be classified as

a) Theoretically based vs. empiricalb) Linear vs. nonlinear

c) Steady state vs. unsteady state

d) Lumped parameter vs. distributed parameter

e) Continuous vs. discrete variables

3.5: How to build a modelFor convenience of presentation, model building can be divided into four phases:

1) problem definition and formulation, 2) preliminary and detailed analysis, 3) evaluation

and 4) interpretation. Keep in mind that model building is an iterative procedure.

22

Experience,reality

Formulate model objectives,

Evaluation criteria, costs

of development

ManagementObjectives

Select key variables,

Physical principles to beapplied, test plan to be use

DevelopModel

Observations,data

Computer simulation,Software development

EstimateParametersEvaluate andverify modelApply model


23/103

3.6: Modeling of Polymerization Reactors

For modeling of polymerization reactors, the mass and energy balances incorporating the kinetics andreactor type should be combined with a description of the molecular weight development toproduce a model of a polymerization reactor which can accurately describe the production rateand character of the product. Often, however, the proper kinetic constants may be unknown. Insome cases, even the exact form of the reaction mechanism is not understood. In these cases, itmay be necessary to use parameter estimation to fit the model to experimental data. The

number of adjustable parameters should be kept low, however, or the model becomesdescriptive of particular data sets rather than mechanistic.

Any model, even if based on established kinetics, should be validated by simulation of data setswhich were not used in the estimation of parameters within the model. These data shouldpreferably cover operating regimes far from those of the data used for parameter estimation.Poor agreement suggests incorrect mechanisms or a tendency of the model to simplycorrelate the data from which its parameters were estimated. Validated polymerization reactormodels may be extremely valuable for design of the reactor system, optimization of the operatingparameters, simulation of potential new modes of operation, and even for the development of newproducts through the modification of the product via modification of the process. In addition,simulation studies can lead to the identification of potential stability problems such as

steady-state multiplicity and can be used to evaluate potential control schemes.

3.6.1: Mass and Energy Balances

Polymerization reactors can be modeled using the classical techniques of chemical reactordesign. Primary to this approach are mass balances over various chemical species as well

as an energy balance. In all cases, the knowledge of the chemical kinetics is used to

describe the rates of formation of various species or, in the case of the energy balance,

23


24/103

to describe the rate of heat generation via reaction. These terms are combined with

flow terms specific to the reactor in question. In the case of heterogeneous systems,

terms describing interphase mass or heat transfer may also be included. Some processesmay occur so rapidly that the species involved are assumed to be at equilibrium at all

times. By applying the principles of mass and energy balances

+

=

systemwithinconsumedor

generatedenergyormassofrate

systemtheleaving

energyormassofRate

systemtheentering

energyormassofRate

systemwithindaccumulate

energyormassofRate

We can derive the equations for mass and energy balance. A mass balance over amonomer may be written as

=dt

dMQfMf - QM - VkpPM, M(0) = M0 (3.1)

Where Vis the reactor volume, Qfand Q are the inlet and outlet volumetric flow rates,

respectively, andPand M are taken to be concentrations of polymer and monomerrespectively.

If P is assumed constant (rapid initiation and no chain transfer), a mass balance over the

live polymer may be written as

QPPQdt

dPV ff = , P(0) = P0 (3.2)

where the generation term may be negative or zero as above.

The energy balance may be written as

0)0(

)()(

TT

TTUAPMkHVTQCTQCdt

dTVC jpppffpp

=

+= (3.3)

The heat of reaction (propagation), and UandA are the overall heat transfer coefficient and heattransfer area for the cooling jacket. In view of the viscosity of polymerizing solutions and theeffect of micro mixing on molecular weight development, it may be desirable in some instances toincorporate a more complex mixing model for the reactor.

3.6.2 Molecular Weight Distribution

In modeling of polymerization reactors it is also important to control the molecular

weight and molecular weight distribution. Therefore it is necessary to develop themathematical models which relate the molecular weight and molecular weight

distribution with the process variables. So in the next chapter we will explain the

molecular weight distribution.

24


25/103

Models of the form of Eqs. (3.1), (3.2), and (3.3) contains derivative. To solve these we willhave to integrate these equations. These equations can be integrated numerically (or occasionally,analytically) from specific initial conditions to determine the transient behavior of the system.For design purposes, it may be acceptable to consider only the steady-state solution. This maybe obtained by setting the time derivatives to zero and solving forM, P, and T. Two methodsare discussed here.

The Method of Moments

If the kth moment of the NCLD is defined as

=

=1n

n

k

k Pn k=0,1 ,2 The NACL

may be written as the ratio of the zeroth to the first moments:

0

1

=n (3.5)

The variance of the NCLD is the second moment about the mean or

2

0

1

0

22

=

n (3.6)

Similarly, the kth moment of the WCLD may be written as

n

n

k

k Pnw

=

+=1

)1( k=1,2, (3.7)

The WACL may be written as

w =0

1

=

1

2

The variance of the WCLD may be written as2

1

2

1

3

2

0

1

1

22

=

=

w (3.8)

The means of the NMWD and WMWD may be calculated from the NACL and WACL viaEqs. (2.1) and (2.2). The variances of the NMWD and WMWD are functions of the variances

of the NCLD and WCLD, respectively, as follows:

222

nmn w = (3.9)222

wmw w = (3.10)

Thus it may be seen that if the leading moments of the NCLD (or WCLD) are known, the meanand the variance of any of the distributions (NCLD, WCLD, NMWD, or WMWD) characterizingthe product may be calculated directly. Once the NAMW and WAMW have been determined. It

(3.4)

25


26/103

is possible to reconstruct the entire distribution from an infinite set of moments. If thedistribution is not complex, a good approximation may be made with a finite (even small)number of moments. In cases of simple kinetics, it may be possible to derive equations forthe development of the moments directly from the mass balances. Consider batch anionicsolution polymerization. Equation dPn/dt = Mkp )( 1 nn PP may be multiplied by n

k

and then summed for all values ofn, resulting in

=

=

=

+=1 2

1

1

,n n

n

k

pn

k

n

pnk PnMkPnMk

dt

dPn (3.11)

=

=1

10

n

n PP

Here

= =1 ,nkn

k

Pn dt

d

dt

dPn knk

n

=

=1 (3.12)

The final term in Eq.3.11 may be evaluated for k=1,2,3 as

k=0, 02

1

= =

n

n

kPN (3.13)

k=1, 102

1 +=

=

n

n

kPn (3.14)

k=2,

210

2

1 2 ++=

=

n

n

kPn

(3.15)

the equations for the first three moments becomes

,00 =dt

d 100 )0( P= (3.16)

,01 Mkdtd p= 101 )0( P= (3.17)

,2 102

MkMk

dt

dpp += 102 )0( P= (3.18)

Integration of eqs. (3.16) through (3.18) together with the monomer balance will yield an

adequate characterization of the molecular weight development under isothermal

conditions.

26


27/103

z-TransformsAnother way of dealing with the differential difference equations describing the chain

length distribution is through the use ofz-transforms. The use of z-transforms is common

in digital process control, but can be used to solve systems of difference equations arisingfrom any source. The z-transform ofPnis defined as

),(),(0

tPztzF nn

n

=

= 0)( =tPn for n


28/103

CHAPTER 4

OPTIMIZATION

Reactors are often the critical stage in a polymerization process. Recently, demands on

the design and operation of chemical processes are increasingly required to comply withthe safety, cost and environmental concerns. All these necessitate accurate modeling and

optimization of reactors and processes. To optimize the reactors, in this chapter we have

discussed about optimization.

How to Define a Model for Optimization?

We need to quantify the various elements of our model: decision variables, constraints,

and the objective function and their relationships.

Decision Variables

Start with the decision variables. They usually measure the amounts of resources, such asmoney, to be allocated to some purpose, or the level of some activity, such as the number

of products to be manufactured, the number of pounds or gallons of a chemical to be

blended, etc.

Objective Function

Once we've defined the decision variables, the next step is to define the objective, which

is normally some function that depends on the variables. For example profit, productionrate, Conversion, yield, various costs, etc. We usually maximize or minimize objectivefunction such as we maximize profit, production rate, Conversion ,yield and minimize

various costs.

Constraints

You'd be finished at this point, if the model did not require any constraints. For example,

in a curve-fitting application, the objective is to minimize the sum of squared differencesbetween each actual data value, or observation, and the corresponding predicted value.

This sum has a minimum value of zero, which occurs only when the actual and predicted

values are all identical. If you asked a solver to minimize this objective function, youwould not need any constraints.

In most models, however, constraints play a key role in determining what values can be

assumed by the decision variables, and what sort of objective value can be attained.

28


29/103

Constraints reflect real-world limits on production capacity, market demand, available

funds, and so on. To define a constraint, you first compute a value based on the decision

variables. Then you place a limit (=) on this computed value.

General Constraints. For example, if A1:A5 contains the percentage of funds to be

invested in each of 5 stocks, you might use B1 to calculate =SUM(A1:A5), and thendefine a constraint B1 = 1 to say that the percentages allocated must sum up to 100%.

Bounds on Variables. Of course, you can also place a limit directly on a decisionvariable, such as A1


30/103

calculate a fixed lease cost per month, but also a lower cost per item processed with the

machine, if it is used.

Feasible Solution.

A solution (set of values for the decision variables) for which all of the constraints in the

model are satisfied is called a feasible solution. Most solution algorithms first try to find a

feasible solution, and then try to improve it by finding another feasible solution that

increases the value of the objective function (when maximizing, or decreases it whenminimizing).

Optimal Solution

An optimal solution is a feasible solution where the objective function reaches a

maximum (or minimum) value.

Globally Optimal Solution

A globally optimal solution is one where there are no other feasible solutions with better

objective function values.

Locally Optimal Solution

A locally optimal solution is one where there are no other feasible solutions "in thevicinity" with better objective function values -- you can picture this as a point at the top

of a "peak" or at the bottom of a "valley" which may be formed by the objective function

and/or the constraints. The Solver is designed to find optimal solutions -- ideally theglobal optimum -- but this is not always possible.

Whether we can find a globally optimal solution, a locally optimal solution, or a good

solution depends on the nature of the mathematical relationship between the variables and

the objective function and constraints (and the solution algorithm used).

What Makes a Model Hard to Solve?

Solver models can be easy or hard to solve. "Hard" models may require a great deal ofCPU time and random-access memory (RAM) to solve -- if they can be solved at all. The

good news is that, with today's very fast PCs and advanced optimization software fromFrontline Systems, a very broad range of models can be solved.

Three major factors interact to determine how difficult it will be to find an optimal

solution to a solver model:

a) The mathematical relationships between the objective and constraints, and the

decision variables

30
http://www.solver.com/tutorial6.htm#Mathematical%20Relationshipshttp://www.solver.com/tutorial6.htm#Mathematical%20Relationships


31/103

b) The size of the model (number of decision variables and constraints) and its

sparsity

c) The use ofinteger variables- memory and solution time may rise exponentially asyou add more integer variables

Mathematical Relationships

a) Linear programming problems

b) Smooth nonlinear optimization problemsc) Global optimization problems

d) Nonsmooth optimization problems

The types of mathematical relationships in a model (for example, linear or nonlinear, and

especially convex or non-convex) determine how hard it is to solve, and the confidenceyou can have that the solution is truly optimal. These relationships also have a direct

bearing on the maximum size of models that can be realistically solved.

A model that consists of mostly linear relationships but a few nonlinear relationships

generally must be solved with more "expensive" nonlinear optimization methods. Thesame is true of models with mostly linear or smooth nonlinear relationships, but a few

nonsmooth relationships. Hence, a single IF or CHOOSE function that depends on the

variables can turn a simple linear model into an extremely difficult or even unsolvablenonsmooth model.

A few advanced solvers can break down a problem into linear, smooth nonlinear and

nonsmooth parts and apply the most appropriate method to each part -- but in general,

you should try to keep the mathematical relationships in a model as simple (i.e. close to

linear) as possible.

Below are some general statements about solution times on modern Windows PCs (with,

say, 2GHz CPUs and 512MB of RAM), forproblems without integer variables

Linear Programming Problems -- where all of the relationships are linear, and henceconvex -- can be solved up to hundreds of thousands of variables and constraints, given

enough memory and time. Models with tens of thousands of variables and constraints

can be solved in minutes (sometimes in seconds) on modern PCs. You can have very

high confidence that the solutions obtained are globally optimal.

Smooth Nonlinear Optimization Problems -- where all of the relationships aresmoothfunctions (i.e. functions whose derivatives are continuous) -- can be solved up to tens of

thousands of variables and constraints, given enough memory and time. Models withthousands of variables and constraints can often be solved in minutes on modern PCs.

If the problem is convex, you can have very high confidence that the solutions obtained

are globally optimal. If the problem is non-convex, you can have reasonable confidence

that the solutions obtained are locally optimal, but not globally optimal.

31
http://www.solver.com/tutorial7.htmhttp://www.solver.com/tutorial7.htm#Sparsityhttp://www.solver.com/tutorial7.htm#Integer%20Variableshttp://www.solver.com/tutorial7.htm#Integer%20Variableshttp://www.solver.com/tutorial6.htm#Linear%20programming%20problemshttp://www.solver.com/tutorial6.htm#Smooth%20nonlinear%20optimization%20problemshttp://www.solver.com/tutorial6.htm#Global%20optimization%20problemshttp://www.solver.com/tutorial6.htm#Nonsmooth%20optimization%20problemshttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/tutorial7.htmhttp://www.solver.com/tutorial7.htm#Sparsityhttp://www.solver.com/tutorial7.htm#Integer%20Variableshttp://www.solver.com/tutorial6.htm#Linear%20programming%20problemshttp://www.solver.com/tutorial6.htm#Smooth%20nonlinear%20optimization%20problemshttp://www.solver.com/tutorial6.htm#Global%20optimization%20problemshttp://www.solver.com/tutorial6.htm#Nonsmooth%20optimization%20problemshttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htm


32/103

Global Optimization Problems -- smooth nonlinear, non-convex optimization problems

where a globally optimal solution is sought -- can often be solved up to a few hundred

variables and constraints, given enough memory and time. Depending on the solutionmethod, you can have reasonably high confidence that the solutions obtained are globally

optimal.

Nonsmooth Optimization Problems -- where the relationships may include functions

like IF, CHOOSE, LOOKUP and the like -- can be solved up to scores, and occasionallyup to hundreds of variables and constraints, given enough memory and time. You can

only have confidence that the solutions obtained are "good" (i.e. better than many

alternative solutions) -- they are not guaranteed to be globally or even locally optimal.

Model Size

The size of a solver model is measured by the number of decision variables and the

number of constraints it contains.

Most optimization software algorithms have a practical upper limit on the size of modelsthey can handle, due to either memory requirements or numerical stability.

Sparsity

Most large solver models aresparse. This means that, while there may be thousands of

decision variables and thousands of constraints, the typical constraint depends on only a

small subset of the variables. For example, a linear programming model with 10,000variables and 10,000 constraints could have a "coefficient matrix" with 10,000 x 10,000 =

100 million elements, but in practice, the number of nonzero elements is likely to be

closer to 2 million.

Integer Variables

Integer variables (i.e. decision variables that are constrained to have integer values at the

solution) in a model make that model far more difficult to solve. Memory and solution

time may rise exponentially as you add more integer variables. Even with highlysophisticated algorithms and modern supercomputers, there are models of just a few

hundred integer variables that have never been solvedto optimality.

This is because many combinations of specific integer values for the variables must be

tested, and each test requires the solution of a "normal" linear or nonlinear optimizationproblem. The number of combinations can rise exponentially with the size of the

problem. "Branch and bound" and "branch and cut" strategies help to cut down on this

exponential growth, but even with these strategies, solutions for even moderately largemixed-integer programming (MIP) problems can require a great deal of time.

32
http://www.solver.com/probconvex.htmhttp://www.solver.com/probconvex.htm


33/103

Optimization Solution Methodologies

Solving and obtaining the optimum values is the last phase in design optimization.

A number of general methods for solving the programmed optimization problems relateVarious relations and constraints that describe the process to their effect on objective

function. these examine the effect of variables on the objective function using analytic,graphical, and algorithmic techniques based on the principles of optimization and programmingmethods.

Procedure with One Variable

There are many cases in which the factor being minimized (or maximized) is an analytic function of

a single variable. The procedure then becomes very simple. Consider the example, where it is

necessary to obtain the insulation thickness that gives the least total cost. The primary variable

involved is the thickness of the insulation, and relationships can be developed showing how thisvariable affects all costs.

Cost data for the purchase and installation of the insulation are available, and the length of

service life can be estimated. Therefore, a relationship giving the effect of insulation thickness

on fixed charges can be developed. Similarly, a relationship showing the cost of heat lost as a

function ofinsulation thickness can be obtained from data on the thermal properties of steam,

properties of the insulation, and heat-transfer con-siderati6"ns. All other costs, such as

maintenance and plant expenses, can be assumed to be independent of the insulation thickness.

Procedure with Two or More Variables

When two or more independent variables affect the objective function, the procedure for

determining the optimum conditions may become rather tedious; however, the general approach is

the same as when only one variable is involved. Consider the case in which the total cost for a

given operation is a function of the two independent variablesx and y, or

CT= f(x,y)

By analyzing all the costs involved and reducing the resulting relationships to a simple form, the

following function might be found.

CT= ax +b/ xy + cy+d

Where a, b, c, and dare positive constants.

33


34/103

Graphical Procedure: The relationship among CT, X, and y could be shown as a curvedsurface in a three-dimensional plot, with a minimum value of CT occurring at the optimum

values ofx and v. However, the use of a three-dimensional plot is not practical for most

engineering determinations.

Analytical Procedure:The optimum value ofx is found at the point where (dCT/dx)y=yi, is equal tozero. Similarly, the same results would be obtained ify were used as the abscissa instead ofx. Ifthis

were done, the optimum value ofy (that isyi) would be found at the point where (dCr/dy)x=xi., is equal

to zero. This immediately indicates an analytical procedure for determining optimum values.

Algorithm Solutions to Optimization Problems

An algorithm is simply an objective mathematical method for solving a problem and is purely

mechanical so that it can be programmed for a computer. Solution of programming problems

generally requires a series of actions that are iterated to a solution, based on a programming

method and various numerical calculation methods. The input to the algorithm can be manual,

where the relations governing the design behavior are added to the algorithm. They can also beintegrated or set to interface with rigorous computer simulations that describe the design.

Use of algorithms thus requires the selection of an appropriate programming method,

methods, or combination of methods as the basic principals for the algorithm function. It also

requires the provision of the objective functions and constraints, either as directly provided relations

or from computer simulation models. The algorithm then uses the basic programming approach to

solve the optimization problem set by the objective function and constraints.

Linear Programming Algorithm Development

To develop this form of approach for linear programming solutions, a set of linear inequalities

which form the constraints are written in the form of "equal to or less than" equations as

nn

nn

bvavava

bvavava

+++

+++

...

...

22222121

11212111

................................................

mnmnmm bvavava +++ ...2211

or in general summation form

=

=jv j=1,2,,nwhere irefers to rows (or equation number) in the set of inequalities and j refers to

columns (or variable number).

34


35/103

As indicated earlier, these inequalities can be changed to equalities by adding

a set of slack variables mnn vv ++ ,....,1 (here v is used in place of S to simplify the

generalized expressions), so that

111212111 ... bvvavava nnn =++++ + 222222121 ... bvvavava nnn =++++ + .......................................................

mmnnmnmm bvvavava =++++ +...2211

or in general summation form

=

+ =+n

j

iinjij bvva1

)( i= 1,2,,m

for jv 0 j=1,2,..,n+m

In addition to the constraining equations, there is an objective function for the linearprogram which is expressed in the form of

z=maximum(or minimum) of nnjj vcvcvcvc ........2211 ++++

where the variables jv are subject to jv 0(j=1,2,.,n+m). Note that, in this case, all

the variables above nv are slack variables and provide no direct contribution to the value

of the objective function.

Simplex Algorithm:

The basis for the simplex method is the generation of extreme-point solutions by starting at anyone extreme point for which a feasible solution is known and then proceeding to a neighboring

extreme point. Special rules are followed that cause the generation of each new extreme, point to

be an improvement toward the desired objective function. When the extreme point is reachedwhere no further improvement is possible, this will represent the desired optimum feasible

solution. Thus, the simplex algorithm is an iterative process that starts at one extreme-point

feasible solution, tests this point for optimality, and proceeds toward an improved solution. If an

optimal solution exists, this algorithm can be shown to lead ultimately and efficiently to theoptimal solution.

The stepwise procedure for the simplex algorithm is as follows (based on the

optimum being a maximum):

1. State the linear programming problem in standard equality form.

2. Establish the initial feasible solution from which further iterations can proceed. A common

method to establish this initial solution is to base it on the values of the slack variables,

where all other variables are assumed to be zero. With this assumption, the initial matrix for

the simplex algorithm can be set up with a column showing those variables that will be

35


36/103

involved in the first solution. The coefficient for these variables appearing in the matrix table

should be 1 with the rest of column being 0.

3. Test the initial feasible solution for optimality. The optimality test is accomplished by the

addition of rows to the matrix which give a value ofZj for each column, whereZj is defined

as the sum of the objective function coefficient for each solution variable.

4. Iteration toward the optimal solution is accomplished as follows: Assuming that the

optimality test indicates that the optimal program has not been found, the following

iteration procedure can be used:

a. Find the column in the matrix with the maximum value of cj -Zj anddesignate this column

as k. The incoming variable for the new test will be the variable at the head of this column.

b. For the matrix applying to the initial feasible solution, add a column showing the

ratio bi /aik.

c. . Find the minimumpositive value of this ratio, and designate the variable in the

corresponding row as the outgoing variable.c. Set up a new matrix with the incoming variable, as determined under (a), substituted for the

outgoing variable, as determined under(b). The modification of the table accomplished by

matrix operations so that the entering variable will have a 1 in the row of the departing

variable and 0s in the rest of that column. The matrix operations involve row manipulations

of multiplying rows by constants and subtracting from or adding to other rows until the

necessary 1 and 0 values are reached.

d. Apply the optimality test to the new matrix.

e. Continue the iterations until the optimality test indicates that the optimum objective

function has been attained.

5. Special cases:

a. If the initial solution obtained by use of the method given in the preceding is not feasible, a

feasible solution can be obtained by adding more artificial variables which must then be

forced out of the final solution.

b. Degeneracy may occur in the simplex method when the outgoing variable is selected. If

there are two or more minimal values of the same size, the problem is degenerate, and a poor

choice of the outgoing variable may result in cycling, although cycling almost never occurs

in real problems. This can be eliminated , by a method of multiplying each element in the

rows in question by the positive coefficients of the kth column and choosing the row for the

outgoing variable as the one first containing the smallest algebraic ratio.

6. The preceding method for obtaining a maximum as the objective function can be applied to the

case when the objective function is a minimum by recognizing that maximizing the negative of

a function is equivalent to minimizing the function.

36


37/103

CHAPTER No. 5

MODELING OF BATCH POLYMERIZATIONREACTORS

In this chapter, the focus will be shifted from chemistry and kinetics of polymerization to

the mathematical description i.e. modeling of polymerization process. In this chapter, we

will apply mass and energy balances combined with the kinetics of polymerizationreactions for batch reactor. We will focus our main attention to the molecular weight and

molecular weight distribution for different polymerization processes using batch reactors.

We will model the equations to determine the effects of operating conditions on the meanchain length distributions and the breadth of distribution.

5.1 Anionic Polymerization

Let us consider batch anionic polymerization. Now we apply mass and energy balances to

batch reactor. As there are no inflow and outflow terms for batch polymerizer so theequation becomes

PMVkdt

dMV P=

which can be written as

(5.1)

0

0

=

=

dt

dP

dt

dPV

( ) 00 PP = (5.2)

The equation (5.1) and (5.2) can be solved to determine the time dependent behavior of

concentration of monomer and polymer.

To investigate molecular weight distribution it is necessary to develop mass balance over

the concentration of live chains of length n.The balance for P1 is simply

32

21

PMP

PPM

P

P

k

k

+

+

(5.3)

37

PMkdt

dMP=

( ) 00 MM =

11 MPk

dt

dPP=


38/103

similarly balance for P2 is

)( 122

212

PPMkdt

dP

MPkMPkdt

dP

P

PP

=

=

from this the balance for Pn can be written analogous to P2

(5.4)

we define variance as under

(5.5)

from this equations (5.1) (5.3) and (5.4) can be written as under

11 P

d

dP=

P1(0)=P10 (5.6)

)( 1= nnn PP

d

dP

Pn(0)=0, n>=2 (5.7)

M(0)=M0 (5.8)

taking the z-transform of equatins (5.6)and (5.7) one obtains

( ) ( ) ( )

,1

, 1 zFzd

zdF= ( ) 10

10, PzzF = (5.9)

equation(5.9) is separable and can be solved as

)exp()exp(),(

1

10

1

= zPzzFexpanding power series in z-1

(5.10)

now by comparing with definition of z-

transform the eq.(5.10) can be written as

)!1(

)()exp()(

1

10 =

nPtP

n

n

(5.11 )

this is poisson distribution with mean (1+ ) and variance .Polydispersity can be written as

2

1

20

===n

w

n

w

m

m

D (5.12)

the variance about mean NACL denoted by2

n can be written in term of moments as2

0

1

0

22

=

n (5.13)

38

)( 1= nnPn PPMk

dt

dP

( ) ''0

dttMk

t

P=

Pd

dM=

n

n

n

zn

PzF

=

=1

1

10)!1(

)()exp(),(


39/103

we know that NACL is defined as in term of moments as0

1

=n

2

0

12

=

n (5.14)

Dividing (5.14) on (5.13) gives

2

0

12

2

0

1

0

22

=

=

n

n

12

1

02

2

2

=

n

n 2

1

02

2

2

1

=+

n

n

from the definition of Polydispersity given by equation (5.12) it can be written as

2

2

1n

nD

+= (5.15)

from eq.(5.5) += 1nfor an ionic polymerization >>1 so one can be neglected in the above eq.

n =2

n

from eq(5.15) 2

2

1n

nD

+= =1+

n

1(5.16)

Hence for high degree of polymerization D approaches unity meaning NCLD approachesmonodispersity hence ionic polymerization in absence of termination or chain transfer isuseful for creating narrow molecular weight distribution.

The above discussion was for anionic polymerization without chain transfer or

termination. How ever for anionic polymerization where chain transfer takes place.The instantaneous degree of polymerization may be calculated as the rate of propagation

divided by the rate of chain transfer (rate of productionof dead chains).

kp Mx = (5.17)

kf B

5.2. Free Radical Polymerization

Consider the free radical polymerization mechanism Ignoring inhibition and considering batchsolution polymerization, the proper mass balances may be written assuming constant reactorvolume and isothermal operation.:

MPkdt

dMp=

0)0( MM = (5.18)

39


40/103

Ikdt

dId= (5.19)

The time derivatives ofR andPare set to zero, andR is eliminated from the two equations

2/1

2

+=

tdtc

d

kk

Ifk

P (5.20)

Equations (5.18), (5.19), and (5.20) define the conversion-time behavior of the reactor. Equations

(5.18) and (5.19) can be solved.Integrating Eq. (5.19)

=I

I

t

d

o

dtkI

dI

0

tkI

Id=

0

ln

tkdeII= 0 (5.21)

Now, integrating eq. (5.18)

=M

M

t

p

o

dtPkM

dM

0

PtkM

Mp=

0

ln

PtkpeMM= 0 (5.22)

Rate of Propagation is given by

PMkR pp =2/1

2

+= tdtc

dppkk

IfkMkR (5.23)

and

t

pn

R

RX =

which can be written as

2/1)(2

][

Iktfk

MkX

d

pn =

40


41/103

5.2.1 Model for Free radical batch polymerization reactors

a Et a Ed a Ep

Time(t)kd

I0 I

f

(Monomer concentration at any time (t))M

Rp(Rate of polymerization at any time t)

41

RTEtt aek

/=RTEd

d aek/

=RTEp

p aek/=

tkdeII= 0

=tIk

fk

kt

dp

eMM

2/1

2/1

0

fIkR di=

MIk

fkkR

t

dpp

2/1

2/1

=

( ) 2/12 fIkk

Mk

dt

p

n =


42/103

In model given above:

a=frequency factor

E=activation energy

5.2.2 MATLAB SOLUTIONOFMODEL

Knowing the initial concentrations of radical and monomer we can solve this model.

Values of activation energies , different initiator efficiencies and frequency factor valuesare available . we can solve this model using MATLAB,

EXAMPLE

Styrene is polymerized by free radical mechanism in a batch reactor. The initialconcentrations of monomer and initiator are 1 M and .001 M.

Model the reactor to determine (a) rate of initiation (b)initiator concentration

(c)rate of polymerization (d)number average degree of polymerization at any time?At 60 degree C initiator efficiency is0.30 and all other cnstants are as under

kd=1.2*10^-5 (1/sec),kp=176 (1/M s) and kt=7.2*10^7 (1/M s)

MATLAB SOLUTION

MODELM0=1;

I0=.001;

T=333;

f=.30;kd=1.2*10^(-5);

kp=176;

kt=7.2*10^7;t=input('enter time in seconds at which u want ur calculation=');

I=I0*exp(-kd*t); %concentration of free radicals at time t

Ri=-f*kd*I; % rate of initiationM=M0*exp(-kp*(f*kd/kt).^.5*t*I.^.5); %concentration of free radicals at time t

Rp=kp*(f*kd/kt).^.5*M*I.^.5; %rate of polymerization at any time t

mun=kp*M/(2*(f*kd*kt*I).^.5); %number average degree of polymerizationat time tLt=I/(kd*f*I); %free radical life time

disp('concentration of free radicals= ')

disp(I)

disp('concentration of monomers=')disp(M)

disp('rate of polymerization=')

disp(Rp)

disp('number average degree of polymerization=');disp(mun)

disp('free radical life time=')disp(Lt)

SOLUTION

enter time in seconds at which u want ur calculation=60concentration of free radicals=

42


43/103

9.9928e-004

concentration of monomers=0.9999

rate of polymerization=1.2440e-006

number average degree of polymerization=172.8977

free radical life time=

2.7778e+005

Effect of time on concentration of radical

I0=.001;

kd=1.2*10^(-5);

t=1:1800; I=I0*exp(-kd*t);

plot(t,I)

0 200 400 600 800 1000 1200 1400 1600 1800

9.7

9.8

9.9

10x 10

-4graph b/w time and concentration of radical

concentrationof radicals(M)

time(seconds)

43


44/103


45/103

Effect of time on degree of polymerization

M0=1; kp=176;

kt=7.2*10^7;

kd=1.2*10^(-5); I0=.001;

f=.30;

t=1:1800; I=I0*exp(-kd*t);

M=M0.*exp(-kp*(f*kd/kt).^.5.*t.*I.^.5);

Rp=kp*(f*kd/kt).^.5*M.*I.^.5;

mun=kp*M./(2*(f*kd*kt*I).^.5); plot(t,mun)

0 200 400 600 800 1000 1200 1400 1600 1800172.8

173

173.2

173.4

173.6

173.8

174

174.2

174.4graph b/w time and degree of polymerization

degree ofpolymerization

time(sec)

45


46/103

5.3:Step growth polymerizationThe polymerization in which polyfunctional reactants react to produce larger units in a

continuous stepwise manner.Step growth polymerization:

1. In absence of solvent or catalyst

2. In presence of catalyst5.3.1:step growth polymerization in absence of solvent or catalyst.

Assume the polyesterification is conducted in the absence of solvent or

catalyst and that the monomers are present in stoichiometric ratio.Then by applying the law of conservation of mass equation for batch

reactor. As there are no inflow and outflow terms so the eqn. can be written as

2kAdt

dA= (5.24)

integrating eq. (5.24)

=tA

A

dtk

A

dA

o 0

2 (5.25)

kdtA

=+

+

12

12

=A

A

t

o

dtkA

0

1

1

11

+=

=

=

ktAA

A

ktAA

A

A

A

ktAA

oo

o

o

oo

o

1)(

+=

ktA

AtA

o

o(5.26)

and P is given by

0

0 )(

A

tAAp

= (5.27)

and n

ppA

A

A

A

o

oon

=

==1

1

)1(

(5.28)

and w

p

pw

+=

1

1 (5.29)

Finally we get the value of D by dividing the above two equations. (5.28) and (5.29).

pDn

w +== 1

46


47/103

5.3.2 Model for batch polycondensation reactors(in absence of catalyst)

Activation energy of the reaction(E)

T(Temperature) Frequency factor(a)

k

Initial concentrationof monomer A(A0) A(t) Concentraion of

monomer at anytime(t) time (t)

A(t)

PMonomer conversion

P

P

Weight w n Number

Average Average

D.P D.P

Molecular WeightDistribution

47

1)(

0

0

+=

ktA

AtA

0

0 )(A

tAAP

=

Pn

=

1

1

n

wMWD

=

RTEaek

/=

P

Pw

+=

1

1


48/103

5.3.2.1 MATLAB SOLUTION OF MODELPolyesterification is conducted in the absence of solvent or catalyst and that the

monomers are present in stoichiometric ratios.Model the reactor .given is that the

dicarboxylic acid concentration is 3 mol L^-1 and polymerization rate constant is 10^-2Lmol^-1 S^-1

MODEL

k=10^-2;A0=3;

t=input('enter time at which u want to calculate parametersin sec=');

A=A0./(A0*k*t+1);p=(A0-A)/A0;

mun=1./(1-p);

muw=(1+p)./(1-p);D=muw./mun;

disp('monomer concentration=')

disp(A)disp('monomer conversion=')

disp(p)

disp('number average degree of polymerization is =')disp(mun)

disp('weight average degree of polymerization is =')

disp(muw)disp('mol weight distribution=')

disp(D)

SOLUTION enter time at which u want to calculate parametersin sec=1800

monomer concentration=

0.0545

monomer conversion=

0.9818

number average degree of polymerization is =

55.0000

weight average degree of polymerization is =

109.0000

mol weight distribution=

1.9818

48


49/103

Effect of time on monomer conversion

A0=3; t=[1:900];

k=10^-2;

A=A0./(A0*k*t+1); p=(A0-A)/A0;

plot(t,p)

0 100 200 300 400 500 600 700 800 9000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

conversion

time(sec)

49


50/103

Effect of time on NACL and WACL

t=[1:900];

k=10^-2;A0=3;

A=A0./(A0*k*t+1);

p=(A0-A)/A0; mun=1./(1-p);

muw=(1+p)./(1-p);

plot(t,muw,'--') hold on

plot(t,mun)

0 100 200 300 400 500 600 700 800 9000

10

20

30

40

50

60

muw

mun

numberavg.andweight avg.D.P

time(sec)

50


51/103

Effect of time on MWD

t=[1:900];

k=10^-2;A0=3;

A=A0./(A0*k*t+1);

p=(A0-A)/A0; mun=1./(1-p);

muw=(1+p)./(1-p);

D=muw./mun; plot(t,D)

0 100 200 300 400 500 600 700 800 9001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

MWD

time(sec)

51


52/103

Effect of stoichiometric ratio on degree of polymerization for different conversion

P=.996; r=[1:.01:1.25];

mun=(1+r)./(1+r-2*P);

plot(r,mun) hold on

P=.994;

mun=(1+r)./(1+r-2*P); plot(r,mun,'*')

P=.990;

mun=(1+r)./(1+r-2*P);

plot(r,mun,'^') P=.980;

mun=(1+r)./(1+r-2*P);

plot(r,mun,'--')

P=.970; mun=(1+r)./(1+r-2*P);

plot(r,mun,'^')

1 1.05 1.1 1.15 1.2 1.250

50

100

150

200

250

p=.996

p=.994

p=.990

p=.980p=.970

effect of stoichiometric ratio and conversion on number average degree of polymerization

number averagedegree ofpolymerization

stoichiometric ratio

52


53/103

5.3.3 Step growth polymerization in presence of catalyst

Let the catalyst is an acid functional group. For stoichiometric ratios of acidic

functional group. By applying Eq. of mass balance

3kA

dt

dA= (5.30)

Integrating the above eq.

ktA

=+

+

13

13

Applying the limits

ktAA

211

2

0

2=

ktAA

A 202

2

0 21 =

ktAA

A 202

2

0

21+= (5.31)

ktA

AA

2

0

2

02

21+=

ktA

AtA

2

0

0

21)(

+= (5.32)

P is given as

0

0 )(

A

tAAP

= (5.33)

nppA

A =

=

2

2

0

0

)1(1

)1(

2)1(

1

pn

= (5.34)

From the equations (5.30) to (5.34) we can develop a model of batch step growth polymerizationreactor in presence of catalyst.

53


54/103

5.3.4 Model for batch polycondensation reactor(in presence of catalyst in stoichiometric ratio )

Activation energy of the reaction(E)

T(Temperature) Frequency factor(a)

Modeling and Optimization of Polymerization Reactors

Documents

Transcript of Modeling and Optimization of Polymerization Reactors