# Derivation And The Formulation Of The Model Biology Essay

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## 4.1 Introduction

In this chapter, the derivation and the formulation of the model and equations for the components studied are shown in details in the subsections. This chapter is divided into 5 main categories which cover all from the component balances, population balances, polymerisation rate, method of moment polymerisation rate and static models used in developing this dynamic model.

## 4.2 Loop Reactor Modelling

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

## 4.2.1 Modelling Assumptions

In modelling the loop reactors, several assumptions are made in order to reduce the complexity of the model developed. The assumptions are made based on reasonable justification.

The slurry densities in respective loops are considered homogeneous at all time. The presence of a circulating pump at the bottom of loop reactors helps in mixing the slurry medium.

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There is no temperature gradient present in the reactors. The reactors are always maintained at 70&deg;C with an excellent heat transfer between materials and equipped with an excellent temperature control system.

The catalyst is assumed to have only one active site. The catalyst is considered to have been activated as it enters the reactors. The assumption is made as the co-catalyst which is the TEAl is mixed with catalyst in the baby loop before entering the reactors. The function of TEAl is to activate the catalyst site.

Only around eight to ten percent of the catalyst sites are activated in this research study. Sudden increase in temperature might cause the active site of the catalyst to be deactivated.

All the polymerisation kinetic parameters are assumed to be constant. This is supported by the fact that the reactor temperature is constant.

The working volume of the mediums equals to the designed volume of the reactors. The loop reactors consists of only two phases which are solid and liquid which are well mixed in the form of slurry which are entirely filled the reactors.

## 4.2.2 Overall Mass Balance

The overall mass balance equation used in this model derivation is obtained from Mohd-Zamry (2009) as it is derived from the principle of conservation of mass as shown as Equation 4.1

The loop reactors are as assumed to be filled entirely with slurry. Therefore, the volume remains constant. Besides, the outlet density of the reactor and the density inside the reactor are similar as justified in section 4.2.1. This gives the overall mass balance equation for this model as in equation 4.2.

The inlet and outlet slurry density (kg/m3) in CSTR are represented as ρin and ρout respectively. As volume of reactor (m3) is VR while qin and qout represent the inlet and outlet volumetric flow (m3/s).

From the industrial perspective, the output of the flow rate leaving the reactors are not controlled directly but the flow depends very much on the slurry densities of the reactor which greatly affected by the monomer and polymer profile as well as the polymerisation rate. (Mohd-Zamry, 2009). For that purpose, the equation for the outlet volumetric calculation is obtained from Luo et al. (2007). The equation for the outlet volumetric calculation can be referred from equation 4.3.

The density (kg/m3) for propylene and polypropylene are represented as ρM and ρPP. CM,out is the outlet propylene molar concentration (kgmol/m3).

## 4.2.3 Component Balances

There are several components taken into consideration for this model development. The general component balance is given by equation 4.4. (Felder and Rousseau, 1999). The equation can also be represented into another similar form as shown in equation 4.5.

(4.4)

(4.5)

The components consist of active site of catalyst, monomer (propylene), the co-catalyst (TEAl), hydrogen, the deactivated catalyst site. All the component balances for the above-mentioned components are derived using equation 4.5. The derivation of each of the components are summarised and tabulated in Table 4.1.

## 4.2.4 Population Balances

As for the moment of live polymer chain (LPC) and dead polymer chain (DPC), the population balances are also derived using the equation 4.5. The primary usage of this population balances in this model is to simplify the model by just considering the chain length of the polymer formed which has more than 2 monomers.

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Only three orders of moments as mentioned in Chapter 2 are used to develop the population balances. The derivation of each of the method of moments for both the living and dead polymer chain are summarised and tabulated in Table 4.1.

Table 4.1 Balances for each component and overall system

Component

Symbol

Equation

Slurry density

ρout

Outlet volumetric flow rate

qout

Active catalyst site

P0

Monomer

M

Hydrogen

H

Co-catalyst

C

D

0th moment of LPC

μ0

1st moment of LPC

μ1

2nd moment of LPC

μ2

0th moment of DPC

γ0

1st moment of DPC

γ1

2nd moment of DPC

γ2

From Table 4.1, a total 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 component balances and 6 equations are derived from population balances.

## 4.3 Polymerisation Rate Modelling

As in Chapter 2, the kinetic model from Lucca et al (2008) is applied in this research study except the site activation kinetic as activation of catalyst site is beyond the boundaries set for this research study. The reason being is as assumed and justified in section 4.2.1.

## 4.3.1 Component Polymerisation Rate

For the derivation of polymerisation rate, 6 components are considered. The components involved in this model are catalyst active sites, monomer, hydrogen, co-catalyst and catalyst dead sites.

## 4.3.2 Population Polymerisation Rate

For the derivation of population polymerisation rate, 6 methods of moment polymerisation rate are considered. There are two main categories of polymer chain which are live polymer chain and dead polymer chain. For each category, there will be zero, first and second moment polymerisation rate.

## 4.3.2.1 Live Polymer Chain (PR)

In order to develop the complete live polymer chain for method of moment polymerisation rate derivation, two main parts are to be considered which are for the polymer chain equals to one and for polymer chain greater than 2. The summation of both the polymer chain polymerisation rate will give the complete live polymer chain polymerisation rate.

For RP=1,

(4.11)

For RP≥2,

For live polymer chain polymerisation rate, , it equals to the summation of equation 4.11 and 4.12.

## 4.3.2.2 Dead Polymer Chain (SR)

Dead polymer chain polymerisation rate is different from the live polymer chain polymerisation rate. For a dead polymer chain, it only starts for the chain length of the polymer with at least combination of two monomers (R≥2). For the chain length of polymer equals to one (R=1), it is not considered to be a polymer as it is similar to monomer (Mohd-Zamry, 2009).

## 4.4.1 Method of Live Moment

A total of three orders of method of moment are used in determining the physical properties of the polymer formed in the reactors. The most common moments used are zero, first and second order of moment. The general equation for method of moment for live polymer is as shown by equation 4.15.

## 4.4.1.1 Zero Moment of Live Polymer Chain

For zero moment of live polymer chain, μ0, 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 moment of live polymer chain is formed by substituting 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 moment of live 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 moment of live polymer chain is formed by substituting equation 4.18 and equation 4.19 into equation 4.13.

## 4.4.1.3 Second Moment of Live Polymer Chain

Applying equation 4.15, the second moment of live 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:

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Therefore, the second moment of live polymer chain is formed by substituting equation 4.21 and equation 4.22 into equation 4.13.

## 4.4.2.1 Zero Moment of Dead Polymer Chain

The general equation for method of moment for dead polymer chain is as shown by equation 4.22.

For zero moment of dead polymer chain, μ0, the general equation is given by equation 4.24.

Therefore, the zeroth moment of dead polymer chain is formed by substituting equation 4.25 into equation 4.14.

## 4.4.2.2 First Moment of Dead Polymer Chain

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

Therefore, the first moment of dead polymer chain is formed by substituting equation 4.27 in equation 4.14.

## 4.4.2.3 Second Moment of Dead Polymer Chain

Applying equation 4.24, the second moment of dead polymer is formed as equation 4.29.

Therefore, the first moment of dead polymer chain is formed by substituting equation 4.29 into equation 4.14.

## 4.5 Finalised Polymerisation Rate

Zeroth moment is used to determine the total polymer concentration in reactor. Therefore, due to this fact, zeroth moment is applied in the entire polymerisation rate for the components that being considered in this research study.

## Catalyst Dead Sites (D)

In summary, the entire polymerisation rates are tabulated as shown in Table 4.2.

Table 4.2 Polymerisation Rate

Component

Polymerisation Rate

Active catalyst site

Monomer

Hydrogen

Co-catalyst

0th moment of LPC

1st moment of LPC

2nd moment of LPC

0th moment of DPC

1st moment of DPC

2nd moment of DPC

## 4.6 Static Models

In this research study, a few static models are developed for the product quality. The static models developed are the model for polymer weight average molecular weight, Melt Flow Index (MFI) model and also the production rate model.

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

This model is developed using the equation (…) as obtained from Asua (2007) and Mohd-Zamry (2009). This model is used later in developing the MFI model.

## 4.6.2 Melt Index Flow (MFI) Model

The MFI model is developed using two values which are the simulated value of the WAMW obtained from equation (…) and the MFI value from the plant data. The product quality of MFI is known to have inverse correlation with the WAMW. Due to that, the MFI model developed in this research study is considered to be static model as it is developed as accordance to the industrial data and not using the first principle method.

There are many grades produced in industry, in this case, Titan Petchem (M) Sdn. Bhd., but only the homopolymer grade with a MFI of 12.5±3.0 is studied and developed.

## 4.6.3 Production Rate Model

The production rate used for this study is obtained from Titan Petchem (M) Sdn. Bhd.. This model is used in the Distribution Control System (DCS) in that company to predict the production rate of the plant. The production rate model used is seen to have depends merely on the slurry densities of loop reactor 2.

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

## 4.7 Parameter Value

The value of catalyst active site is obtained through estimation. As in section 4.2.1, the catalyst is assumed to have only one active site but the activated sites are only around 8 to10 percent of the catalyst site available. The active site is obtained through the multiplication of 8 percent from the mass of Titanium, Ti over the mass of catalyst, TiCl4. The formula for the active site, A is as shown in equation 4.40.

From equation 4.40, the equation can be further simplified as equation 4.41 as the actual mass of Titanium is rather difficult to be known.

The calculation of the active site for the catalyst used in the industry as follows:

From calculation, it can be seen that for every 1 kilogram of catalyst used, only 0.02 kgmol of active site present. As from the research carried out by Zacca and Ray (1993), the active site value used is 0.01 gmol/g catalyst. Unfortunately, the method of calculation and the assumptions in getting the value is unknown.