## Chapter 4

## 4.1 Introduction

In this chapter, the derivation and the preparation of the theoretical account and equations for the constituents studied are shown in inside informations in the subdivisions. This chapter is divided into 5 chief classs which cover all from the constituent balances, population balances, polymerization rate, method of minute polymerization rate and inactive theoretical accounts used in developing this dynamic theoretical account.

## 4.2 Loop Reactor Modelling

As mentioned in Chapter 1 and Chapter 3, the cringle reactors in this survey are represented by a series of Continuous-Stirred-Tank Reactor ( CSTR ) . Two CSTRs in series are to stand for each cringle reactor with same capacity. The boundaries are set as shown in Figure 3.2 in Chapter 3.

## 4.2.1 Modelling Premises

In patterning the cringle reactors, several premises are made in order to cut down the complexness of the theoretical account developed. The premises are made based on sensible justification.

The slurry densenesss in several cringles are considered homogenous at all clip. The presence of a go arounding pump at the underside of loop reactors helps in blending the slurry medium.

There is no temperature gradient nowadays in the reactors. The reactors are ever maintained at 70 & A ; deg ; C with an first-class heat transportation between stuffs and equipped with an first-class temperature control system.

The accelerator is assumed to hold merely one active site. The accelerator is considered to hold been activated as it enters the reactors. The premise is made as the co-catalyst which is the TEAl is assorted with accelerator in the babe cringle before come ining the reactors. The map of TEAl is to trip the accelerator site.

Merely about eight to ten per centum of the accelerator sites are activated in this research survey. Sudden addition in temperature might do the active site of the accelerator to be deactivated.

All the polymerization kinetic parametric quantities are assumed to be changeless. This is supported by the fact that the reactor temperature is changeless.

The on the job volume of the mediums peers to the designed volume of the reactors. The loop reactors consists of merely two stages which are solid and liquid which are good assorted in the signifier of slurry which are wholly filled the reactors.

## 4.2.2 Overall Mass Balance

The overall mass balance equation used in this theoretical account derivation is obtained from Mohd-Zamry ( 2009 ) as it is derived from the rule of preservation of mass every bit shown as Equation 4.1

The cringle reactors are as assumed to be filled wholly with slurry. Therefore, the volume remains changeless. Besides, the mercantile establishment denseness of the reactor and the denseness inside the reactor are similar as justified in subdivision 4.2.1. This gives the overall mass balance equation for this theoretical account as in equation 4.2.

The recess and mercantile establishment slurry denseness ( kg/m3 ) in CSTR are represented as ?in and ?out severally. As volume of reactor ( M3 ) is VR while Qin and qout represent the recess and mercantile establishment volumetric flow ( m3/s ) .

From the industrial position, the end product of the flow rate go forthing the reactors are non controlled straight but the flow depends really much on the slurry densenesss of the reactor which greatly affected by the monomer and polymer profile every bit good as the polymerization rate. ( Mohd-Zamry, 2009 ) . For that intent, the equation for the mercantile establishment volumetric computation is obtained from Luo et Al. ( 2007 ) . The equation for the mercantile establishment volumetric computation can be referred from equation 4.3.

The denseness ( kg/m3 ) for propene and polypropene are represented as ?M and ?PP. CM, out is the mercantile establishment propene molar concentration ( kgmol/m3 ) .

## 4.2.3 Component Balances

There are several constituents taken into consideration for this theoretical account development. The general constituent balance is given by equation 4.4. ( Felder and Rousseau, 1999 ) . The equation can besides be represented into another similar signifier as shown in equation 4.5.

( 4.4 )

( 4.5 )

The constituents consist of active site of accelerator, monomer ( propene ) , the co-catalyst ( TEAl ) , H, the deactivated accelerator site. All the constituent balances for the above-named constituents are derived utilizing equation 4.5. The derivation of each of the constituents are summarised and tabulated in Table 4.1.

## 4.2.4 Population Balances

As for the minute of unrecorded polymer concatenation ( LPC ) and dead polymer concatenation ( DPC ) , the population balances are besides derived utilizing the equation 4.5. The primary use of this population balances in this theoretical account is to simplify the theoretical account by merely sing the concatenation length of the polymer formed which has more than 2 monomers.

Merely three orders of minutes as mentioned in Chapter 2 are used to develop the population balances. The derivation of each of the method of minutes for both the life and dead polymer concatenation are summarised and tabulated in Table 4.1.

Table 4.1 Balances for each constituent and overall system

Component

Symbol

Equation

Slurry denseness

?out

Outlet volumetric flow rate

qout

Active accelerator site

P0

Monomer

Meter

Hydrogen

Hydrogen

Co-catalyst

C

Dead accelerator site

Calciferol

0th minute of LPC

?0

1st minute of LPC

?1

2nd minute of LPC

?2

0th minute of DPC

?0

1st minute of DPC

?1

2nd minute of DPC

?2

From Table 4.1, a sum of 13 ordinary differential equations are developed from 3 balances. Out of 13 equations, two are derived from overall mass balances, 5 equations are derived from constituent balances and 6 equations are derived from population balances.

## 4.3 Polymerisation Rate Modelling

As in Chapter 2, the kinetic theoretical account from Lucca et Al ( 2008 ) is applied in this research survey except the site activation kinetic as activation of accelerator site is beyond the boundaries set for this research survey. The ground being is as false and justified in subdivision 4.2.1.

## 4.3.1 Component Polymerization Rate

For the derivation of polymerization rate, 6 constituents are considered. The constituents involved in this theoretical account are catalyst active sites, monomer, H, co-catalyst and accelerator dead sites.

## Catalyst Active Sites ( P0 )

## Monomer ( M )

## Hydrogen ( H )

## Co-Catalyst ( C )

## Catalyst Dead Sites ( D )

## 4.3.2 Population Polymerisation Rate

For the derivation of population polymerization rate, 6 methods of minute polymerization rate are considered. There are two chief classs of polymer concatenation which are unrecorded polymer concatenation and dead polymer concatenation. For each class, there will be zero, foremost and 2nd minute polymerization rate.

## 4.3.2.1 Live Polymer Chain ( PR )

In order to develop the complete unrecorded polymer concatenation for method of minute polymerization rate derivation, two chief parts are to be considered which are for the polymer concatenation peers to one and for polymer concatenation greater than 2. The summing up of both the polymer concatenation polymerization rate will give the complete unrecorded polymer concatenation polymerization rate.

For RP=1,

( 4.11 )

For RP?2,

For unrecorded polymer concatenation polymerization rate, , it equals to the summing up of equation 4.11 and 4.12.

## 4.3.2.2 Dead Polymer Chain ( SR )

Dead polymer concatenation polymerization rate is different from the unrecorded polymer concatenation polymerization rate. For a dead polymer concatenation, it merely starts for the concatenation length of the polymer with at least combination of two monomers ( R?2 ) . For the concatenation length of polymer peers to one ( R=1 ) , it is non considered to be a polymer as it is similar to monomer ( Mohd-Zamry, 2009 ) .

## 4.4 Method of Moment Polymerisation Rate

## 4.4.1 Method of Live Moment

A sum of three orders of method of minute are used in finding the physical belongingss of the polymer formed in the reactors. The most common minutes used are zero, foremost and 2nd order of minute. The general equation for method of minute for unrecorded polymer is as shown by equation 4.15.

## 4.4.1.1 Zero Moment of Live Polymer Chain

For zero minute of unrecorded polymer concatenation, ?0, the general equation is given by equation 4.15.

From equation 4.13, the term PR-1 exists. Rearrangement of the term is done as the method below:

Therefore, the zeroth minute of unrecorded polymer concatenation is formed by replacing equation 4.16 and equation 4.17 into equation 4.13.

## 4.4.1.2 First Moment of Live Polymer Chain

Using equation 4.15, the first minute of unrecorded polymer is formed as equation 4.19.

From equation 4.13, the term RPR-1 exists. Rearrangement of the term is done as the method below:

Therefore, the first minute of unrecorded polymer concatenation is formed by replacing equation 4.18 and equation 4.19 into equation 4.13.

## 4.4.1.3 Second Moment of Live Polymer Chain

Using equation 4.15, the 2nd minute of unrecorded polymer is formed as equation 4.21.

From equation 4.13, the term RPR-1 exists. Rearrangement of the term is done as the method below:

Therefore, the 2nd minute of unrecorded polymer concatenation is formed by replacing equation 4.21 and equation 4.22 into equation 4.13.

## 4.4.2 Method of Dead Moment

## 4.4.2.1 Zero Moment of Dead Polymer Chain

The general equation for method of minute for dead polymer concatenation is as shown by equation 4.22.

For zero minute of dead polymer concatenation, ?0, the general equation is given by equation 4.24.

Therefore, the zeroth minute of dead polymer concatenation is formed by replacing equation 4.25 into equation 4.14.

## 4.4.2.2 First Moment of Dead Polymer Chain

Using equation 4.24, the first minute of dead polymer is formed as equation 4.27.

Therefore, the first minute of dead polymer concatenation is formed by replacing equation 4.27 in equation 4.14.

## 4.4.2.3 Second Moment of Dead Polymer Chain

Using equation 4.24, the 2nd minute of dead polymer is formed as equation 4.29.

Therefore, the first minute of dead polymer concatenation is formed by replacing equation 4.29 into equation 4.14.

## 4.5 Finalised Polymerization Rate

Zeroth minute is used to find the entire polymer concentration in reactor. Therefore, due to this fact, zeroth minute is applied in the full polymerization rate for the constituents that being considered in this research survey.

## Catalyst Active Sites ( P0 )

## Monomer ( M )

## Hydrogen ( H )

## Co-Catalyst ( C )

## Catalyst Dead Sites ( D )

In drumhead, the full polymerization rates are tabulated as shown in Table 4.2.

Table 4.2 Polymerization Rate

Component

Polymerization Rate

Active accelerator site

Monomer

Hydrogen

Co-catalyst

Dead accelerator site

0th minute of LPC

1st minute of LPC

2nd minute of LPC

0th minute of DPC

1st minute of DPC

2nd minute of DPC

## 4.6 Inactive Models

In this research survey, a few inactive theoretical accounts are developed for the merchandise quality. The inactive theoretical accounts developed are the theoretical account for polymer weight norm molecular weight, Melt Flow Index ( MFI ) theoretical account and besides the production rate theoretical account.

## 4.6.1 Polymer Weight Average Molecular Weight ( WAMW ) Model

This theoretical account is developed utilizing the equation ( … ) as obtained from Asua ( 2007 ) and Mohd-Zamry ( 2009 ) . This theoretical account is used subsequently in developing the MFI theoretical account.

## 4.6.2 Melt Index Flow ( MFI ) Model

The MFI theoretical account is developed utilizing two values which are the fake value of the WAMW obtained from equation ( … ) and the MFI value from the works informations. The merchandise quality of MFI is known to hold reverse correlativity with the WAMW. Due to that, the MFI theoretical account developed in this research survey is considered to be inactive theoretical account as it is developed as conformity to the industrial informations and non utilizing the first rule method.

There are many classs produced in industry, in this instance, Titan Petchem ( M ) Sdn. Bhd. , but merely the homopolymer class with a MFI of 12.5±3.0 is studied and developed.

## 4.6.3 Production Rate Model

The production rate used for this survey is obtained from Titan Petchem ( M ) Sdn. Bhd.. This theoretical account is used in the Distribution Control System ( DCS ) in that company to foretell the production rate of the works. The production rate theoretical account used is seen to hold depends simply on the slurry densenesss of cringle reactor 2.

Combination of the equation 4.36, 4.37 and 4.38 give the concluding equation for production rate as equation 4.39.

## 4.7 Parameter Value

The value of accelerator active site is obtained through appraisal. As in subdivision 4.2.1, the accelerator is assumed to hold merely one active site but the activated sites are merely about 8 to10 per centum of the accelerator site available. The active site is obtained through the generation of 8 per centum from the mass of Titanium, Ti over the mass of accelerator, TiCl4. The expression for the active site, A is as shown in equation 4.40.

From equation 4.40, the equation can be farther simplified as equation 4.41 as the existent mass of Titanium is instead hard to be known.

The computation of the active site for the accelerator used in the industry as follows:

From computation, it can be seen that for every 1 kg of accelerator used, merely 0.02 kgmol of active site nowadays. As from the research carried out by Zacca and Ray ( 1993 ) , the active site value used is 0.01 gmol/g accelerator. Unfortunately, the method of computation and the premises in acquiring the value is unknown.