polimerisation reactor

7.5. POLYMERISATION REACTOR

The methyl methacrylate (MMA) polymerisation takes place in a CSTR (see

Figure 27), using azo-iso-butyronitrile (AIBN) as initiator and toluene as sol-

vent. The reaction is exothermic, and a cooling jacket is used to remove the heat

of reaction (Silva et al. 2001).

The reaction mechanism ofMMA free radical polymerisation consists of the

following steps:

Initiation:

I!k0 2RRM!kI P1

Propagation:

Pi M!kp

Pi1

Monomer transfer:

Pi M!kfm

Pi DiAddition termination:

Pi Pj!ktc DijDisproportionation termination:

Pi Pj!ktd

Di Djwhere I, P, M, R and D refer to initiator, polymer, monomer, radicals and dead

polymer respectively.

[(Figure_7)TD$FIG]

FIGURE 27 Polymerisation reactor.

Product and Process Modelling: A Case Study Approach206

7.5.1. Modelling Objective

The objective here is to develop a dynamic model so that the transient responses

of the compositions and the temperatures during the operation of the polymer-

isation reactor can be studied; existence of multiple solutions can be identified;

and closed-loop simulations performed by adding an appropriate control model.

7.5.2. Model Description

7.5.2.1 Model Assumptions

* The contents of the reactor are perfectly mixed.* Constant density and heat capacity of the reaction mixture.* Density and heat capacity of the cooling fluid stay constant.* Uniform cooling of fluid temperature.* The reactions only happen inside the reactor.* There is no gel effect (the conversion of monomer is low, and the proportion

of solvent in the reaction mixture is very high).* Constant volume of the reactor.* Polymerisation reactions occur according to the free radical mechanism.

7.5.2.2 Model Description

Based on the above assumptions, the mathematical model of the MMA poly-

merisation is given below.

Mass balance:

dCm

dt kp kfm

CmP0 F Cmin Cm

V42

dCI

dt kICI

FICIin FCI

V43

dD0

dt 0:5ktc ktd P20 kfmCmP0

FD0

V44

dD1

dt Mm kp kfm

CmP0 FD1

V45

Energy balance:

dT

dt DH kpCm

rCPP0 UA

rCPVT Tj F Tin T

V46

dTj

dt FCW TW0 Tj

V0

UArWCPWV0

T Tj 47

Chapter | 7 Models for Dynamic Applications 207

Defined (constitutive) relations:

P0 2f *CIkI

ktd ktc

s48

kr AreEr=RT ; r p; f m; I; td; tc 49Defined polymer average molecular weight (Pm):

Pm D1=D0 50The percentage monomer conversion (X):

X Cmin Cm=Cmin*100 51

7.5.3. Model Analysis

The above mathematical model consists of a total of 14 equations out of which 6

are ODEs and 8 are algebraic equations (AEs). The 6 state (dependent) variables

for the 6 ODEs are: Cm, CI, T, D0, D1, Tj. The 8 algebraic (unknown) variables

for the 8 AEs are: k, P0, Pm, X (note that k is a vector of 5 variables). The rest of

the variables must be specified. Table 22 lists the variables that need to be

specified together with their specified values. Table 23 shows the incidence

matrix, which is not ordered in a lower tridiagonal form but it can easily be seen

that the algebraic equations have a lower tridiagonal form.

Based on the incidence matrix given in Table 23, the calculation procedure is

as follows: at any time, t, the values of the state (dependent) variables are known

TABLE 22 List of Specified Variables and their Given Values

F = 1.0 m3/h Mm = 100.12 kg/kgmol

FI = 0.0032 m3/h f* = 0.58

FCW = 0.1588 m3/h R = 8.314 kJ/kgmol K)

Cmin = 6.4678 kgmol/m3 -DH = 57800 kJ/kgmol

CIin = 8.0 kgmol/m3 Ep = 1.8283 104 kJ/kgmol

Tin = 350 K EI = 1.2877 105 kJ/kgmolTW0 = 293.2 K Efm = 7.4478 104 kJ/kgmolU = 720 kJ/(hkm2) Etc = 2.9442 103 kJ/kgmolA = 2.0 m2 Etd = 2.9442 103 kJ/kgmolV = 0.1 m3 Ap = 1.77 109 m3/(kgmoh)V0 = 0.02 m

3 AI = 3.792 1018 1/hr = 866 kg/m3 Afm = 1.0067 1015 m3/(kgmoh)rW = 1000 kg/m

3 Atc = 3.8283 1010 m3/(kgmoh)Cp =2.0 kJ/(kgK) Atd = 3.1457 1011 m3/(kgmoh)CpW = 4.2 kJ/(kgK)


and so the AEs (Eq. 49 and then 48) are solved. Using these variables and the

known variables, the right-hand sides (RHSs) of ODES (Eqs. 4247) are deter-

mined, and based on these, new values of the dependent variables are obtained.

Equations 50-51 are used to calculate the polymer averagemolecular weight and

the percentage conversion at any time.

7.5.4. Numerical Solution

The model equations have been solved through ICAS-MoT (Sales-Cruz, 2006).

The calculated steady-state values for the state variables are given in Table 24. It

can be noted that this reactor has multiple steady states, which can be obtained

TABLE 23 Incidence Matrix (note that Eq 49 is actually five equations for the

vector k). The shaded Cells Indicate the ODEs and their Corresponding

Dependent Variables

Eq\Variable k P0 Cm CI D0 DI T Tj Pm X

Eq. 49 * *

Eq. 48 * * *

Eq. 42 * * *

Eq. 43 * *

Eq. 44 * * * *

Eq. 45 * * * *

Eq. 46 * * * * *

Eq. 47 * *

Eq. 50 * * *

Eq. 51 * *

TABLE 24 Calculated Steady-State Values for the MMA Polymerisation

Reactor

State 1 State 2 State 3

Cm, kgmol/m3 5.9651 5.8897 2.3636

CI, kgmol/m3 0.0249 0.0247 1.7661 10-04

T, K 351.41 353.40 436.20

D0, kgmol/m3 0.0020 0.0025 0.4213

D1, kgmol/m3 50.329 57.881 410.91

Tj, K 332.99 334.34 390.93

X, % 7.8 8.9 63.5

PM, kg/kgmol 25000 23000 975


by starting the dynamic simulation runs from different initial conditions but

keeping all the known variables at the listed values of Table 22. This is an

example of output multiplicity. The transient responses for the 6 state variables

are shown in Figures 28a-f. These results are obtained for the following initial

conditions of the state variables: Cm = 0 ; CI = 0.5 ;T = 200 ; D0 = 0 ; D1 = 0; Tj=0. This corresponds to steady state 1 in Table 24.

[(Figure_8b)TD$FIG]

FIGURE 28b Transient response of Cm (y-axis) versus time (x-axis).

[(Figure_8a)TD$FIG]

FIGURE 28a Transient response of CI (y-axis) versus time (x-axis).


[(Figure_8d)TD$FIG]

FIGURE 28d Transient response of DI (y-axis) versus time (x-axis).

[(Figure_8c)TD$FIG]

FIGURE 28c Transient response of D0 (y-axis) versus time (x-axis).

[(Figure_8e)TD$FIG]

FIGURE 28e Transient response of T (y-axis) versus time (x-axis).


The model equations as implemented in ICAS-MoTare given in Appendix 1

(see chapter-7-5-poly-reactor.mot file).

Discussion: Try to obtain all the three steady states and evaluate their

stability through calculating the eigenvalues. Add the gel-effect to evaluate

the changes in the transient responses. A more detailed polymerisation reactor

model is given by Lopez-Arenas et al. (2006) try to implement this model.

REFERENCES

Cooper, H., 2007. Fuel cells, the hydrogen economy and you. Chemical Engineering Progress.,

3443.

Fogler, S. H., 2005. Elements of Chemical Reaction Engineering, 4th Edition, Prentice Hall, USA.

Ingram, C.D., Cameron, I.T., Hangos, K.M., 2004. Chemical Engineering Science 59, 21712187.

Lopez-Arenas, T., Sales-Cruz, M., Gani, R., 2006. Chemical Engineering Research and Design 84,

911931.

Luss, D. and Amundson, N. R., 1968, Stability of Batch Catalytic Fluidized Beds, AIChE Journal,

14 (2), 211221.

Morales-Rodriguez, R., 2009. Computer-Aided Multi-scale Modelling for Chemical Product-

Process Design, PhD-thesis. Lyngby, Denmark: Technical University of Denmark.

Sales-Cruz, M., 2006. Development of a computer aided modelling system for bio and chemical

process and product design, PhD-thesis. Lyngby, Denmark: Technical University of Denmark.

Sundmacher, K., Schultz, T., Zhou, S., Scott, K., Ginkel, M., Gilles, G., 2001. Chemical Engineering

Science 56, 333341.

Xu, C., Follmann, P.M., Biegler, L.T., Jhon, M.S., 2005. Computers and Chemical Engineering 29,

18491860.

[(Figure_8f)TD$FIG]

FIGURE 28f Transient response of Tj (y-axis) versus time (x-axis).


polimerisation reactor

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