**2.3 The description of polymerization process and its mathematical representation**

The solution copolymerization of styrene and acrylonitrile taking place in a semibatch reactor is considered as a test bed for sequential implementation of the dynamic optimization strategy. The xylene is the solvent and AIBN is the initiator for the reaction. The feed which is a mixture of monomers, solvent, and initiator enters the reactor in semi-batch mode. The reactor initial volume is 1.01 L. The initially set design parameters are the solvent mole fraction *fs* = 0.25 and initiator concentration *I0* = 0.05 mol/L. The mole ratio of monomers in the feed, *M*1/*M*2, is 1.5, where *M*<sup>1</sup> and *M*<sup>2</sup> are the molar concentrations of the unreacted monomers (styrene and acrylonitrile). The homogeneous solution free-radical copolymerization of styrene with acrylonitrile is described by the following reaction mechanism [26].

Initiation:

$$\begin{aligned} I &\longrightarrow 2R\\ R + M\_1 &\xrightarrow{k\_{i1}} P\_{10} \\ R + M\_2 &\xrightarrow{k\_{i2}} Q\_{01} \end{aligned} \tag{13}$$

Propagation:

$$P\_{n,m} + M\_1 \xrightarrow{k\_{p11}} P\_{n+1,m}$$

$$P\_{n,m} + M\_2 \xrightarrow{k\_{p12}} Q\_{n,m+1} \tag{14}$$

$$Q\_{n,m} + M\_1 \xrightarrow{k\_{p21}} P\_{n+1,m}$$

$$Q\_{n,m} + M\_2 \xrightarrow{k\_{p22}} Q\_{n,m+1}$$

Combination termination:

$$P\_{n,m} + P\_{r,q} \xrightarrow{k\_{\rm t\,11}} M\_{n+r,m+q}$$

$$P\_{n,m} + Q\_{r,q} \xrightarrow{k\_{\rm t\,21}} M\_{n+r,m+q} \tag{15}$$

$$Q\_{n,m} + Q\_{r,q} \xrightarrow{k\_{\rm t\,22}} M\_{n+r,m+q}$$

Disproportionation termination:

$$P\_{n,m} + P\_{r,q} \xrightarrow{k\_{ld11}} M\_{n,m} + M\_{r,q}$$

$$P\_{n,m} + Q\_{r,q} \xrightarrow{k\_{ld2}} M\_{n,m} + M\_{r,q} \tag{16}$$

$$Q\_{n,m} + Q\_{r,q} \xrightarrow{k\_{ld2}} M\_{n,m} + M\_{r,q}$$

Chain Transfer:

$$\begin{aligned} P\_{n,m} + M\_1 &\xrightarrow{k\_{\,f11}} M\_{n,m} + P\_{10} \\ P\_{n,m} + M\_2 &\xrightarrow{k\_{\,f12}} M\_{n,m} + Q\_{01} \\ Q\_{n,m} + M\_1 &\xrightarrow{k\_{\,f21}} M\_{n,m} + P\_{10} \\ Q\_{n,m} + M\_2 &\xrightarrow{k\_{\,f22}} M\_{n,m} + Q\_{01} \end{aligned} \tag{17}$$

where *Pn*,*<sup>m</sup>* stand for a growing polymer chain with *n* units of monomer 1 (styrene) and *m* units of monomer 2 (acrylonitrile) with monomer 1 on the chain end. Similarly, *Qn*,*<sup>m</sup>* refers to a growing copolymer chain with monomer 2 on the end. The *Mn*,*<sup>m</sup>* denotes inactive or dead polymer.

The molecular weight (*MW*) and molecular weight distribution (*MWD*) of the copolymer are computed using three leading moments of the total number average copolymers. The instantaneous *kth* moment is given by:

$$\lambda\_k^{\;d} = \sum\_{n=1}^{\infty} \sum\_{m=1}^{\infty} \left( nw\_1 + mw\_2 \right)^k M\_{n,m} \quad , \; k = 0, 1, 2, \dots \quad \dots \tag{18}$$

where *w*<sup>1</sup> is the molecular weight of styrene and *w*<sup>2</sup> is the molecular weights of acrylonitrile. The total number average chain length (*Xn*), the total weight average chain length (*Xw*) and the polydispersity index (*PD*) are defined as:

*A Metaheuristic Tabu Search Optimization Algorithm: Applications to Chemical… DOI: http://dx.doi.org/10.5772/intechopen.98240*

$$\begin{aligned} X\_n &= \lambda\_1^d / \lambda\_0^d \\ X\_w &= \lambda\_2^d / \lambda\_1^d \\ \text{PD} &= X\_w / X\_n \end{aligned} \tag{19}$$

More details on modeling equations, reaction kinetics and numerical data pertaining to this system can be referred elsewhere [24].

The polymerization process model [24] is in the form of the general expression in Eq. (5). The set of state variables, *X* and the control vector *U* in the model are given by.

$$X(\mathbf{t}) = \left[M\_1(\mathbf{t}), M\_2(\mathbf{t}), I(\mathbf{t}), V(\mathbf{t}), \lambda\_0(\mathbf{t}), \lambda\_1(\mathbf{t}), \lambda\_2(\mathbf{t})\right]^\mathrm{T}$$

and

$$U(\mathbf{t}) = \left[T(\mathbf{t}), u(\mathbf{t})\right]^\mathrm{T} \tag{20}$$

In the above, *I* is the concentration of the unreacted initiator, *V* is the reaction mixture volume, *λ<sup>k</sup>* (*k* = 0, 1, … ) is the *kth* moment of the dead copolymer molecular weight distribution,*T* is the reaction mixture temperature, an *u* is the volumetric flow rate of the feed mixture to the reactor.

#### **2.4 Control objectives**

The desired values of copolymer composition (*F*1*D*) and number average molecular weight (*MWD*) are chosen to be 0.58 and 30000, respectively. Minimizing the deviations of copolymer composition, *F*<sup>1</sup> and molecular weight, *MW* from their respective desired values during the entire span of the reaction are specified as the desired objectives. In order to attain these objectives, the control variables are set as monomer addition rate, *u* and reactor temperature,*T*. If one manipulative variable is used to control one polymer quality parameter, the uncontrolled property parameters may deviate from their desired values as the reaction proceeds. The optimal control problem involves in optimizing the single objectives as well as both the objectives simultaneously. The objectives of SAN copolymerization are specified as.

$$J\_1 = \left[1 - MW(t) / \mathcal{M} \mathbf{M} \mathbf{D}\right]^2 \tag{21}$$

$$J\_2 = \left[1 - F\_1(t) / F\_1 D\right]^2 \tag{22}$$

$$J\_3 = \left[\mathbf{1} - F\_1(t)/F\_1D\right]^2 + \left[\mathbf{1} - M W(t)/M \mathbf{W} \mathbf{D}\right]^2\tag{23}$$

Here *MW* and *F*<sup>1</sup> are the molecular weight and copolymer composition, and *MWD* and *F*1*D* are their respective desired values. The notation *t* here refers discrete time. The hard constraints are set as.

$$\begin{aligned} \mathbf{0} \le \mathbf{u}(t) \le \mathbf{0}.\mathbf{0} \mathbf{7} \text{ (l/ min} \ )\\ \mathbf{320} \le T(t) \le \mathbf{368}(k) \\ V(t) \le \mathbf{4.0 l} \end{aligned} \tag{24}$$

These constraints on operating variables are selected by taking into consideration of reaction rate, heat transfer limitation and reactor safety. Determination of temperature policy,*T*, to satisfy *J*<sup>1</sup> (Eq. (21)) and monomer feed policy, *u*, to satisfy *J*<sup>2</sup> (Eq. (22)) are considered as single objective optimization problems.

Determination of both *u* and *T* policies that satisfy *J*<sup>3</sup> (Eq. (23)) is considered as a problem of simultaneous optimization.

#### **2.5 Design and implementation**

The design and implementation of tabu search for optimal control of copolymerization reactor is explained as follows. The total span of reaction time in SAN copolymerization reactor is split into finite number of time instants referred to discrete stages. The total time of reaction is fixed at 300 min. The duration of reaction time is divided into 19 stages with 20 time points, each stage having a duration of 15 min. Thus the discrete control sequences of feed flow rate, *u* and temperature,*T* are specified as.

$$\boldsymbol{u} = [\boldsymbol{u}\_1, \boldsymbol{u}\_2, \dots, \boldsymbol{u}\_{20}]^\mathrm{T} \tag{25}$$

$$T = \begin{bmatrix} T\_1, T\_2, \dots, T\_{20} \end{bmatrix}^T \tag{26}$$

The control sequences are located at equal distances. Initially, the control vector is specified as constant value at each of the discrete stages. The control input at the beginning of the first stage is chosen as the lower bound of the input space. The elements of tabu search for computing the optimal control policies are specified as follows. The sizes of both recency and frequency based tabu lists are set as 50. The sizes of these lists are chosen such that they prohibit revisiting of un-prosperous solutions in the search process. An intensification procedure in the form of a sine function is used. The oscillation period *α* of the sine function is controlled by a parameter (*θ*) whose value is 4.0001. A sigmoid function based aspiration criterion with the parameters as *kcenter* = 0.3 and *σ* = 7/*M* with *M* as specified number of iterations is used. For optimizing MW, ten neighbors are formed at the end of the first stage by considering random changes in the search space of *T* with an incremental change of �0.4 to 0.4. The number of neighbors specified for each control input of each stage is 10. The integration of process model is performed for a time period of 15 min with a time step of 1 min from the starting to the end of the first stage and the objective function values are calculated for all the generated neighbors at the end of this stage. The iterative convergence of TS leads to establish the best control input (*T*) along with its objective function. This provides the optimal control point for the first stage, which is then used as a starting point for the second stage solution. For second stage solution, random neighbors are created at the end of the second stage around the optimal *T* of the first stage. The integration of the model is performed from the beginning to the end of the first stage based on the initial control point (*T*) and from the starting to the end of the second stage for each of the neighbors generated at the end of second stage. The optimal control inputs for successive stages are computed until the end of last stage in a similar manner. The control input values that are computed at the end of each stage thus represents the optimal control policy for *T*. For *F1* optimization, ten neighbors are generated at the end of first stage with an incremental variation of �1.0 x 10�<sup>6</sup> to 1.0 x 10�<sup>6</sup> . In analogous manner, optimal control policy for *u*(*F*1) is found by following the similar TS procedure as in *T* policy. For determining the dual control policies of *T* and *u*, multistage dynamic optimization by TS is performed by accounting incremental variations in neighbors generation of *T* and *<sup>u</sup>* within the ranges of �0.4 to 0.4, and � 1.0 x 10�<sup>6</sup> to 1.0 x 10�<sup>6</sup> , respectively. This case of multistage dynamic optimization involves the computation of the objective function values for 100 neighbor combinations at each of the control point corresponding to *T* and *u.* **Figure 2** shows the implementation of TS strategy for optimal control of SAN copolymerization reactor.

*A Metaheuristic Tabu Search Optimization Algorithm: Applications to Chemical… DOI: http://dx.doi.org/10.5772/intechopen.98240*

**Figure 2.** *Multistage dynamic optimization of copolymerization reactor using tabu search.*
