Multiobjective optimization of long-chain branched propylene polymerization.
Polypropylene (PP) can be preferred to many other thermoplastics for the applications which need light weight and reasonable thermal resistant properties. Highly linear PPs can be produced by Ziegler-Natta and metallocene catalysts . Branching needs to be incorporated within the linear polymer chains to impart better melt strength properties, which then can be handled by processing techniques such as thermoforming, film blowing, blow molding, and so forth. Different experimental procedures, for example reactive extrusion  and electron beam irradiation , have been developed to produce long-chain branched polypropylene (LCBPP). Although various experimental techniques of synthesizing PP with branching are in progress, one of the theoretical approaches might be the modeling of LCBPP with a proposed mechanism which can validate the experimental results and then using that validated model to optimize and control the extent of branching.
However, it is not easy to produce LCBPP by direct synthesis method because the chemistry involved in this phenomenon is not very well understood. To bridge this gap, utilization of PP macromonomers has been studied  by several researchers by adapting different methods. As an example, isotactic LCBPP can be produced by the incorporation of vinyl-terminated macromonomers . In such case, long-chain branching density depends on macromonomer insertion rate in relative to the propylene monomer insertion rate. Shiono et al.  have used rac-[Me.sub.2]Si[(2-MeBenz[e]Ind).sub.2]Zr[Cl.sub.2] catalyst to copolymerize atactic polypropylene (aPP) macromonomer with propylene to produce LCBPP. LCBPP (with isotactic backbones and atactic side chains) has been produced by Ye and Zhu  by the tandem catalysis. By this method, second catalyst copolymerizes the propylene monomer with vinyl-terminated macromonomers, which is produced by the first catalyst prior to the copolymerization process. However, binary catalyst systems in a single reactor have shown to be effective for one-step production of long-chain branched polyethylene [6, 7]. Paavola et al.  have produced LCBPP by using the nonconjugated diene comonomers. Polymers obtained by this method exhibited broad molecular weight distribution (MWD) with a polydispersity index (PDI) value greater than 5. Langston et al.  produced LCBPP by the use of metallocene catalyst and T-reagent. Apart from the above experiments, some mathematical models are developed for the long-chain branched polyolefin. Mehdiabadi et al.  compared the performance of two continuous stirred-tank reactor (CSTRs) in series with semi-batch reactor performance by considering the general olefin polymerization system in which CSTRs were shown to have better performance in producing polymer with high long-chain branching density. Soares and Hamielec  explained the construction of molecular weight long-chain branching distribution with an analytical expression in a steady-state CSTR. Yiannoulakis et al.  calculated the molecular weight long chain branching distribution for ethylene polymerization reactors by fractionating the total polymer based on the number of long chain branches. However, building model that is validated with experimental data is a gap area found out of the literature survey.
Multiobjective optimization techniques are excellent candidates to find out optimal solutions that are conflicting in nature, for example simultaneous attainment of maximum molecular weight and grafting density in less polymerization time in a polymerization setup. The early efforts to solve multiobjective optimization problems (MOOPs) in polymer reaction engineering go back to the works of Tsoukas et al.  and Fan et al. . These studies are primarily based on Pontryagin's maximization principle to find solutions to the optimal control problems where various single objective optimization-based methods are used to transform the original MOOPs to obtain the Pareto optimal (PO) solutions. There are benefits of using evolutionary optimization methods for solving MOOPs. These population-based methods attack the MOOPs using a vector approach where all objectives are considered simultaneously as opposed to the single-objective optimization approaches for solving MOOPs. Multiple numbers of well spread PO solutions can be obtained in single optimization run using these evolutionary approaches. One such earlier effort is the multiobjective optimization study of optimal control of industrial nylon-6 semi-batch reactor , where evolutionary algorithms are shown to work better than conventional Pontryagin's maximization principle -based approaches to solve multiobjective optimal control problems. In another example, Raha et al.  investigated the effect of NaOH addition as a catalyst in semi-batch mode for epoxy polymerization, which was otherwise considered as a batch operation. Detailed optimal control studies have been performed by Mitra et al.  and Majumdar et al.  with relevant process constraints to find out optimal addition profiles for various other reactants that further support semi-batch operation as compared to conventional batch process. A thorough review of various such works from polymerization domain can be found in the literature [19-22]. Apart from this, multiobjective optimization studies related to polymerization and other chemical engineering applications can be found in recent review chapter .
In this study, an example of LCBPP that is produced by a binary catalyst system has been considered, experimental details of which can be found from the work of Ye and Zhu . The aim here is to develop a model with a mechanism which can validate the given experimental data  and then use the model to optimize and control the degree of branching of the polymer. Single-site coordination mechanism has been considered to model this system. Earlier, Zhu and Li  conducted modeling study of a binary catalyst system in steady-state CSTR to obtain a comb-branched polymer. They obtained a polymer with a maximum PD1 of 2.25 where the main chain and the side chains obtained with a theoretical PDI of 2. However, in the experimental findings  that are considered here, aPP macromonomers obtained are reported with a PDI of the order of 1.3 where the approach shown by Zhu and Li  does not lead to such lower PDI values. To obtain aPP polymer with very narrow MWD instead of Schulz-Flory distribution, reversible chain-transfer step  has been considered. From the above kinetic model, net rate of formation of live and dead polymers has been derived for aPP macromonomers and isotactic poplypropylene (iPP) copolymers. The zeroth-, first-, and second-order moment equations have been derived. The resultant equations are solved by the LIMEX DAE  solver which can predict weight average molecular weight (Mw) and PDI of aPP macromonomers and iPP copolymer and grafting density. Next, the nondominated sorting genetic algorithm (NSGA II) has been utilized to estimate the kinetic parameters, which are obtained by minimizing the sum of the square of the error between the experimental and the simulated values of some of the properties given in the experimental work  (e.g., [M.sub.w] and PDI of aPP macromonomers and iPP copolymer and grafting density, etc.). Though NSGA II has been used for multiobjective optimization, it can be used for the purpose of single-objective optimization by considering one of the objectives as constant. In the optimization exercise, the abovementioned validated model is extended to find the optimal values of the addition of catalysts and cocatalyst, second catalyst addition time that minimizes the total polymerization time while maximizing the iPP copolymer [M.sub.w] and grafting density, simultaneously. However, these three objectives are conflicting in nature. Hence, there is a need to find out the optimal process conditions to get the desired combination of various conflicting objectives. To cater this, a multiobjective optimization study has been performed using well-established nondominated sorting genetic algorithm II (NSGA II) . PO solutions are the set of solutions given by MOOP which are nondominating in nature. This study can be extremely beneficial for operating branched PP reactors that can lead to the desired results with optimal operating conditions.
MODEL AND OPTIMIZATION ASPECTS
The experiment for the production of branched PP starts with 200 mL of toulene and prescribed amount of cocatalyst inside a reactor, pressurized with propylene. The reaction starts with the addition of the first catalyst at zeroth time. The second catalyst is added later. In this binary catalyst system, first catalyst system [(2-ArN=C(Me)].sub.2][C.sub.5][H.sub.3]N}Fe[Cl.sub.2]/MMAO (1))  along with the cocatalyst produces macromonomers, whereas the second catalyst system (rac-[Me.sub.2]Si[(2-MeBenz[c]Ind).sub.2]Zr[Cl.sub.2]/MMAO (2))  copolymerizes aPP macromonomers with the propylene monomer to create long-chain branches (LCBPP). The proposed kinetic mechanism for the abovementioned LCBPP with twin catalyst system is summarized in Table 1. The rates of formation of live and dead polymers are derived from kinetic mechanism for aPP macromonomers and iPP copolymer and shown in Eqs. 1-4 in which, [P.sub.n] and [D.sup.=.sub.n] represent the aPP live and dead polymers of chain length n, whereas, [Q.sub.n] and [R.sub.n] represent the iPP live and dead polymers of chain length n. This method of generating equations for representing evolution of all species leads to a large number of equations. To handle such situation, moment balance modeling has been applied as given in Eqs. 5-16. Additionally, iPP copolymer live and dead polymer moments ([[mu].sub.x] and [v.sub.x]) are given in Eq. 17 and aPP live and dead polymer moments ([[lambda].sub.x] and [[mu].sup.=.sub.x]) are given in Eq. 18. Expressions for number average molecular weight ([M.sub.n]), [M.sub.w], and PDI are shown in Eq. 19 followed by the expression of grafting density in Eq. 20. In Eq. 19, [M.sub.n] and [M.sub.w] are represented in terms of iPP dead polymer because the concentration of the live polymer is almost negligible as compared to the concentration of dead polymer.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
d[R.sub.n]/dt = ([k.sub.al][cocat] + [k.sub.d2][[mu].sub.0)[Q.sub.n] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
d[v.sub.0]/dt = ([k.sub.d2][[mu].sub.0] + [k.sub.al][cocat])[[mu].sub.0] (14)
d[v.sub.0]/dt = ([k.sub.d2][[mu].sub.0] + [k.sub.al][cocat])[[mu].sub.1] (15)
d[v.sub.0]/dt = ([k.sub.d2][[mu].sub.0] + [k.sub.al][cocat])[[mu].sub.2] (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
[M.sub.n] = ([v.sub.1]/[v.sub.0])MW [M.sub.w] = ([v.sub.2]/[v.sub.1])MW PDI = ([M.sub.w]/[M.sub.n]) (19)
GD = 1000([alpha][k.sub.lcb][[mu].sub.0.sup.=]/[k.sub.p2]M) (20)
The grafting density depends on the time gap between the two catalyst additions  and the ratio of two catalyst concentrations . This means if both catalysts are added together in the beginning, the grafting density becomes zero and when more time is allowed between the additions of the two catalysts, more numbers of macromonomers get the chance to be grafted to the iPP back bone (owing to the accumulation of more number of macromonomers) before the addition of the second catalyst . One parameter, [alpha], designating the diffusional effect (i.e., all macromonomers will not be available to attack to the iPP back bone) has been introduced to the model apart from the kinetic constants appearing in the kinetic mechanism and estimated through the parameter estimation exercise. This parameter [alpha] which we have estimated as a time-averaged value is valid for the entire processing period. Parameter estimation exercise finds out those values of kinetic parameters and a that provide reasonable match between the model-predicted values and the experimental data available for certain polymer properties ([M.sub.w] and PDI of aPP macromonomers and iPP copolymer and grafting density). Predictions of grafting density are far away from the experimental data without considering the parameter [alpha]. Table 2 summarizes the results of experimental validation in the presence and absence of empirical parameter [alpha]. The results show the importance of considering [alpha] as a diffusion parameter and estimating that using parameter estimation exercise. However, parameter [alpha] is purely empirical in nature. Further, one may find 0-th, 1-st and 2nd moments variations of iPP dead polymer with polymerization time in Figure 1, 2, 3 of 'Additional Supporting Information' document provided. The error (e) expression used to estimate all these parameters is shown in Eq. 21. This error has been minimized through NSGA II by taking one of the objectives as constant. For this purpose, the model, which is embedded with LIMEX DAE solver, is integrated with the optimization routine of NSGA II . Based on the experimental conditions, different [alpha]-values are estimated depending on the different values of grafting density. Using these values, an empirical relationship has been developed for [alpha] which is a function of time gap between the two catalyst additions, Fe/Zr ratio (i.e., Catalyst 1 to Catalyst 2 ratio) and copolymerization time (time starting after the second catalyst is added till the completion of polymerization) and is shown in Eq. 22.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
[alpha] = [([t.sub.1]).sup.0.837] [([cat.sub.1]/[cat.sub.2]).sup.1.369] (3.51[t.sub.1] + 1.72[t.sub.2]) [10.sup.-6] (22)
where [t.sub.1]] represents second catalyst addition time, [cat.sub.1] and [cat.sub.2] represent the Catalyst 1 and Catalyst 2 concentrations, and [t.sub.2] represents the copolymerization time. The rationale behind building the expression 22 is to make use of it in the optimization exercise where the value of [alpha] can be predicted based on the supplied values of [t.sub.1], [t.sub.2], [cat.sub.1], and [cat.sub.2].
Problem Formulation and Optimization Procedure
Addition amounts for the two catalysts ([u.sub.1] and [u.sub.2]) and cocatalyst (MMAO) ([u.sub.3]), time of addition for the second catalyst ([u.sub.4]) and the total polymerization ([t.sub.p]) (70-180 min) are considered as decision variables for the optimization problem. These decision variables are to be decided by the optimization routine while attaining simultaneous minimization of total polymerization time, maximization of [M.sup.w], and maximization of GD. As these objectives are conflicting in nature, solving the MOOP helps in obtaining the PO or tradeoff solutions among various conflicting objectives. The abovementioned problem formulation with relevant constraints is summarized in Table 3. The constraint bounds in the optimization problem formulation have been chosen completely based on the experimental values  to avoid extrapolation errors of the model. For example, aPP PDI experimental values are 1.3 and 1.4, respectively. The constraint limit for this has been chosen [less
than or equal to] 1.45. In this, Case 1 represents the multiobjective optimization formulation based on the decision variable values of experimental run no. 1, whereas, Case 2 is based on the entire experimental range. This Case 2 multiobjective optimization study has been extended based on the process performance improvement in Case 1. To find a polymer in less processing time, one may get less [M.sub.w] and GD. A multiobjective optimization study for LCBPP is, therefore, performed here to obtain tradeoff solutions in the abovementioned conflicting scenario. To reduce the extrapolation errors, the bounds on the decision variables are fixed at the [+ or -] 10% of the experimental values  and additional constraints are posed (Table 3). All decision variables  are forced to lie within the lower and upper bounds to obtain a realistic final solution. MOOP has been performed by integrating the validated model with a well-established multi objective optimization routine, real-coded non-dominated sorting genetic algorithm (NSGA II) . This is a multidimensional search space and finding the globally optimum space of interest is always challenging. NSGA II has hopefully solved this complicated problem, which is coupled with highly nonlinear stiff ordinary differential equations with the decision variables bounded by the experimental observations.
RESULTS AND DISCUSSION
All experimental runs  were conducted at 1 atm. propylene pressure and 25[degrees]C in 200 mL of toluene solvent and various molecular properties such as [M.sub.w], PDI (of aPP, iPP-copolymer), and grafting density were determined. Propylene concentration in toluene is calculated by Soave-Redlich-Kwong equation of state for vapor-liquid phases in equilibrium [28, 29]. Kinetic parameters, those are estimated by the parameter estimation exercise as explained earlier, are listed in Table 4. Once the parameters are estimated, the model is ready for use in optimization of process-operating conditions within the experimental range. As the first catalyst produces only macromonomers, [beta]-hydride elimination mechanism has been considered as the chain-transfer step . The percentage of catalyst-active sites for the first catalyst system (1/MMAO) is calculated by [C.sub.1] = [eta][cat.sub.1] , where [cat.sub.1] represents the number of moles of catalyst introduced, [C.sub.1] is the moles of catalyst-active sites, and [eta] is the efficiency factor. The factor [eta] depends on the cocat/[cat.sub.1] (Al/Fe) ratio, for example as the ratio increases, the number of moles activated catalyst sites also increase. For the second catalyst system (2/MMAO), the mechanism of chain transfer to cocatalyst and bimolecular deactivation of live polymers has been considered . Compared to the second catalyst concentration, the cocatalyst (MMAO) is found to be present in much higher amount . Hence, the term "[k.sub.al][MMAO]" is considered to be constant  during the polymerization. Polymer chain length increases with the decrease in cocatalyst concentration, which is owing to lower chain transfer to cocatalyst. The effect of cocatalyst (MAO) on bimolecular deactivation has been considered in the literature . Decrease in bimolecular deactivation has been observed with increase in cocatalyst/catalyst ratio . In the present effort, the effect of MMAO/Zr (cocat/[cat.sub.2]) ratio on bimolecular deactivation has been taken into account. This rate constant is estimated by considering the related experimental values (i.e., cocat/[cat.sub.2] = 5000 and 7500) and listed in Table 4. With the decrease in this value, bimolecular deactivation increases that leads to produce polymers of lower chain length. In this study, the adopted model has been validated with the experimental  findings and the result is summarized in Tables (5-7). Model predictions for polymer properties for all experimental runs are found to corroborate experimental data reasonably well. Higher molecular weight polymers are obtained for the first two runs, which are owing to lower bimolecular deactivation (Table 4) as compared to the last three runs (Tables (5-7)). While comparing between the 3rd and the 5th runs (of same cocat/[cat.sub.2] = 5000), 5th run has provided higher chain length polymers. This is owing to high chain transfer to cocatalyst in case of the 3rd run as compared to the 5th run. Grafting density predictions are also summarized in Tables (5-7). As explained earlier, this value completely depends on the time gap between the two catalyst additions, two catalyst ratios, and copolymerization time. For Run 1, grafting density value is more as compared to the value in Run 2, which is owing to more time gap between two catalyst additions, whereas same [cat.sub.1]/[cat.sub.2] ratio has been maintained. Similarly, by comparing 3rd and 4th runs (of having similar catalyst concentrations), 4th run has lower long chain branching density. This is owing to less macromonomers present in the reactor because of less time gap between two catalyst additions. However, no experimental data for grafting density are available for the 4th and 5th runs. These values are predicted from the model. Lower branching density of 4th run as compared to the 3rd run indicates higher melting point , which is in line with the experimental observations. By comparing the 4th and 5th runs of having the same time gap between the two catalyst additions and the copolymerization time with various [cat.sub.1]/[cat.sub.2] ratio, 4th run shows more grafting density value compared to the 5th run (Tables (5-7)) owing to more [cat.sub.1]/[cat.sub.2] ratio. In Tables (5-7), validation results of 6th run have been provided, which is not included in the parameter estimation exercise. After the model is validated with the experimental data, it has been extended to investigate the optimal process-operating conditions to attain specific objectives as summarized in Table 3. First of all, one might be curious to see whether optimization result can give any better solution than the experimental results. First a targeted optimization search is done in a narrow decision variable space to figure out the performance similar or better than run 1 (Table 3, Case 1). Multiobjective optimization study is carried out for the run 1 experimental range (i.e., within [+ or -] 10% of the experimental process conditions; henceforth called as Case 1). The set of PO solutions for population of 100 is shown in Fig. 1. Polymerization time is arranged in ascending order in terms of an ordered chromosome number (Fig. 1a). Figure 1b and 1c represent the remaining two objective functions by using the same ordered chromosome numbers as shown in Fig. 1a. This way of representing the objectives helps to see the embedded tradeoff among objectives. These are multiple numbers of optimal solutions competing with each other. No single solution can be pointed as better than other solution in terms of all three objectives. While comparing two solutions (e.g., a and b), if one objective for a solution (solution a) looks better than another solution (solution b), this would definitely have some compromise in some other objective (i.e., other objective of solution a might be inferior to solution b). In that sense, all these solutions are equally important and none of them can be discarded right away. As shown in Figure 1, with ~8% increase in first catalyst concentration, grafting density is more as compared to the experimental Run 1 in less polymerization time (Table 8). This means that as the ratio of two catalysts increases ([cat.sub.1]/[cat.sub.2]), grafting density also increases. This grafting density also strongly depends on second catalyst addition time. One has to allow certain span of copolymerization time to get more iPP [M.sub.w]. Based on these results, we got certain process performance improvement. Hence, multiobjective optimization study has been extended to the entire range of experimental data (henceforth called as Case 2) to see more variety of process performances as well as improving the same strictly within the experimental limits.
Figure 2 shows the multiobjective PO solutions for the abovementioned three conflicting objectives for the entire range of experimental data. All decision variables are kept within the [+ or -]10% experimental range to control the model extrapolation errors because the estimated kinetic parameters are valid for a certain range of operating conditions. These PO solutions are projected into the individual 2D planes to have better understanding of the situation. Experimental points of Runs 1 and 3 (which are having GD > 8) represented in the same plot as filled points (circled points). A decent number of PO solutions are found better than the experimental points. The corresponding decision variables (amount of first catalyst, second catalyst, cocatalyst, and second catalyst addition time) are shown in Fig. 3a-d in different shades. Figure 3a represents PO solutions with the first catalyst concentration (Fe) as decision variable. Figure 3b-d is the same PO solutions where second catalyst concentration (Zr), cocatalyst concentration, and second catalyst addition time have been taken as decision variables, respectively. We can characterize these PO solutions and can find an interesting trend among the decision variables. If we concentrate on grafting density alone, we can see that higher grafting density can be achieved by maintaining operating conditions with higher values of [cat.sub.1]/[cat.sub.2] ratio (Fe/Zr) as well as higher second catalyst addition time. The amount of first catalyst addition (Fig. 3) spans across medium to higher range, whereas other decision variables are present accross the entire ranges. As told earlier, this is owing to the increase in the ratio of the first catalyst to second catalyst, which leads to more GD. Similarly, the time minimization occurs for higher values for [u.sub.1] (first catalyst) and moderate values for [u.sub.2], [u.sub.3], and [u.sub.4]. Of course, one has to see for a solution considering all three objectives in mind because settling for higher grafting density and iPP [M.sub.w] may lead to solutions with poor time productivity. From the PO set shown in Fig. 3, a particular solution can be chosen based on decision maker's preference and corresponding trends among the decision variables can be found out. This kind of optimal trend can be extremly useful for an operator to run a plant without much intervention of mere qualitative perceptions.
The PO solutions shown in Fig. 3, along with their corresponding values of ratio of Catalyst 1 to Catalyst 2, grafting density, and second catalyst addition time are shown in Fig. 4 with various copolymerization times (presented in shades). A primary look at the figure divides the points into two regions: (1) solutions with less copolymerization time and less catalysts ratio; (2) solutions with medium to high copolymerization time and high catalysts ratio. In the first region, their GD and second catalyst addition times are found to be quite varying. In the other region, the variation is found less for GD as well as for second catalyst addition time. In addition, with lower catalyst ratio, the optimizer has chosen more time gap between the two catalyst additions to achieve more GD at a less copolymerization time. However, lesser value of iPP [M.sub.w] is obtained in less copolymerization time as shown in Fig. 5. In Figure 5, the solutions which are grouped by an ellipse have almost similar copolymerization time. In case of these solutions, high [M.sub.w] points appear for low cocatalyst concentration. This is happening owing to low chain transfer to cocatalyst. Multiobjective optimization leads to multiple number of tradeoff solutions as opposed to a single solution in case of single-objective optimization. Tradeoff among solutions is clear as improvement in certain objective is obtained at the cost of deterioation in other objectives. However, at the end of the optimization study, one has to choose only one solution as the solution of choice and this selection needs decision maker's knowledge about how to prioritize among various objectives. The formulation provided in Table 3 could have been also presented by optimizing grafting density and polymerization time and constraining the [M.sub.w] to some higher value in the commercial range instead of considering Mw in the objectives. This is because in commercial operation one would be interested to produce polymer with same quality in terms of Mw. However, the formulation provided in Table 3 is more beneficial when the decision maker is not sure whether there exists a Pareto solution at a particular value of A/w (e.g., 500 kg/mol). In these cases, it is better to see at what different values of [M.sub.w] the solution exists and then decide which value of [M.sub.w] (may be 490 kg/mol) to be chosen. For a clear depiction, polymerization time versus grafting density has been plotted in Fig. 6 to show different polymers of almost similar molecular weight (e.g. [M.sub.w] = 670,000-676,000 g/mol).
As the MWD is of great interest owing to its direct relationship with various polymer properties, MWD of two PO points is calculated. One of the ways to calculate the MWD for branched polymer is numerical fractionation, where the whole polymer population is classified into number of classes based on the number of branches  (e.g., linear polymers belong to zeroth class, polymers with one LCB go to Class 1, and so on.). According to this method , the rates of moments have been derived for each class of live and dead polymer chains as shown in Eqs. 23 and 34. MWD of each class of polymer chains is calculated using a two-parameter model following Schultz-Flory distribution (Eq. 35). Once the individual distribution is achieved, the overall MWD is calculated by the weighed sum of all individual class distributions. In this method, the number of classes should be chosen properly to construct the complete MWD. The belowmentioned convergence criteria have been applied for accurate construction of MWD (Eq. 36). MWDs for the two Pareto points of having different Mws and GD are compared and shown in Fig. 7, and the corresponding MWDs of grafted side chains are shown in Fig. 8. As shown in Figure 8, it can be concluded that grafted side chains may exhibit very narrow MWD. It is evident from Figure 8 that high [M.sub.w] plot exhibits wider MWD and shifted toward higher chain length as compared to the curve with lower [M.sub.w].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
[dv.sub.0,0]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.0,0] (26)
[dv.sub.1,0]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.0,0] (27)
[dv.sub.2,0]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.2,0] (28)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)
[dv.sub.0,m]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.0,m] (32)
[dv.sub.1,m]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.1,m] (33)
[dv.sub.2,m]/dt = ([k.sub.a1][cocat] + [k.sub.d2][[mu].sub.0])[[mu].sub.2,m] (34)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)
([v.sub.1] - [[summation].sup.N.sub.m=0] [v.sub.1,m])/[v.sub.1] [less than or equal to] 0.02 (36)
where [w.sub.m](r) represents the weight fraction of the polymer chains of wth class with a degree of polymerization of r.
Multiobjective optimization has been formulated for various conflicting process objectives with relevant constraints using a model validated with the data collected from the literature. Maximization of iPP weight average molecular weight and grafting density have been attained along with the simultaneous minimization of total polymerization time without violating the process constraints. Real-coded NSGS II has been used to find the multiobjective PO solution and corresponding operating conditions. Optimization routine provided wide variety of solutions in the entire terrain of search for this twin catalyst system. Here, though we are maximizing the grafting density, several grafting density values are obtained by this multiobjective optimization approach. Two catalysts and one cocatalyst concentrations and time gap between the two catalyst additions are used as decision variables. One of the objective functions, grafting density, strongly depends on the time gap between the two catalyst additions, ratio of the two catalysts, and copolymerization time. Another objective function, iPP [M.sub.w], depends on cocatalyst which is owing to chain transfer to cocatalyst and cocat/[cat.sub.2] ratio (bimolecular deactivation). The optimization exercise not only leads to a variety of competitive process choices but also shows improvement in process objectives as compared to the existing literature data. Solutions originating from different regions of the PO set were considered and the possible reasons for their occurrence were analyzed in detail in terms of reaction mechanisms proposed.
NOMENCLATURE aPP Atactic polypropylene [C.sup.H.sub.I] Hydride-activated complex [C.sup.Me.sub.2] Methylated catalyst-activated complex [cat.sub.1] First catalyst concentration (mol/L) [cat.sub.2] Second catalyst concentration (mol/L) Cocat Cocatalyst concentration (mol/L) [D.sup.=.sub.n] aPP macromonomer concentration of chain length "n" (mol/L) [DP.sub.n,m] Number average degree of polymerization of mth class polymer chains [DP.sub.w,m] Weight average degree of polymerization of mth class polymer chains GD Grafting density iPP Isotactic polypropylene [ksub.i1] Initiation rate constant (L/(mol.min)) [k.sub.[beta]] [beta]-H elimination constant (1/min) [k.sub.[beta]r] Reversible chain transfer to metal rate constant (L/ (mol.min)) [k.sub.p1] Propagation constant for the first catalyst system (L/ mol.min)) [alpha][k.sub.1cb] Effective long-chain branching rate constant (L/ (mol.min)) [k.sub.a2] Activation rate constant for the second catalyst system (L/(mol.min)) [k.sub.i2] Initiation rate constant for the second catalyst system (L/(mol.min)) [k.sub.p2] Propagation rate constant for the second catalyst system (L/(mol.min)) [k.sub.d2] Deactivation rate constant for the second catalyst system (L/(mol.min)) [k.sub.a1] Chain transfer to cocatalyst for second catalyst system (L/(mol.min)) [k.sub.ri1] Reinitiation eith hydride metal complex (L/(mol.min)) [M] Monomer concentration (mol/L) [P.sub.n] aPP live polymer of chain length "n" (mol/L) [Q.sub.n,i] iPP copolymer of chain length "n" and "i" long-chain branches (mol/L) [R.sub.n,i] Dead iPP copolymer of chain length "n" and "i" long-chain branches (mol/L) [t.sub.2] Copolymerization time (min) [[lambda].sub.0] Zeroth moment of aPP live polymer (mol/L) [[lambda].sub.1] First moment of aPP live polymer (mol/L) [[lambda].sub.2] Second moment of aPP live polymer (mol/L) [[mu].sup.=.sub.0] Zeroth moment of aPP macromonomer (mol/L) [[mu].sup.=.sub.1] First moment of aPP macromonomer (mol/L) [[mu].sup.=.sub.2] Second moment of aPP macromonomer (mol/L) [[mu].sub.0] Zeroth moment of iPP live polymer (mol/L) [[mu].sub.1] First moment of iPP live polymer (mol/L) [[mu].sub.2] Second moment of iPP live polymer (mol/L) [v.sub.0] Zeroth moment of iPP dead polymer (mol/L) [v.sub.1] First moment of iPP dead polymer (mol/L) [v.sub.2] Second moment of iPP dead polymer (mol/L) [[mu].sub.0,m] Zeroth moment of iPP live polymer of mth class (mol/L) [[mu].sub.1,m] First moment of iPP live polymer of mth class (mol/L) [[mu].sub.2,m] Second moment of iPP live polymer mth class (mol/L) [v.sub.0,m] Zeroth moment of iPP dead polymer of mth class (mol/L) [v.sub.1,m] First moment of iPP dead polymer of mth class (mol/L) [v.sub.2,m] Second moment of iPP dead polymer mth class (mol/L)
[1.] D. Graebling, Macromolecules, 35, 4602 (2002).
[2.] D. Auhl, J. Stange, H. Munstedt, B. Krause, D. Voigt, A. Lederer, U. Lappan, and K. Lunkwitz, Macromolecules, 37, 9465 (2004).
[3.] W. Weng, W. Hu, A.H. Dekmerzian, and C.J. Ruff, Macromolecules, 35, 3838 (2002).
[4.] T. Shiono, S.M. Azad, and T. Ikeda, Macromolecules, 32, 5723 (1999).
[5.] Z. Ye and S. Zhu, J. Polym. Sci. Part A: Polym. Chem., 41, 1152 (2003).
[6.] Z.J.A. Komon and G.C. Bazan, Macromol. Rapid. Commun., 22, 467 (2001).
[7.] R.F. de Souza and O.L. Casagrande Jr., Macromol. Rapid. Commun., 22, 1293 (2001).
[8.] S. Paavola, T. Saarinena, B. Lofgren, and P. Pitkanen, Polymer, 45, 2099 (2004).
[9.] J.A. Langston, R.H. Colby, T.C. Mike Chung, F. Shimizu, T. Suzuki, and M. Aoki, Macromolecules, 40, 2712 (2007).
[10.] S. Mehdiabadi, J.B.P. Soares, and A.H. Dekmezian, Macromol. React. Eng., 2, 529 (2008).
[11.] J.B.P. Soares and A.E. Hamielec, Macromol. Theory Simul., 5, 547 (1996).
[12.] H. Yiannoulakis, A. Yiagopoulos, P. Pladis, and C. Kiparissides, Macromolecules, 33, 2757 (2000).
[13.] A. Tsoukas, M. Tirrel, and G. Stephanopoulos, Chem. Eng. Sci., 37, 1785 (1982).
[14.] L.T. Fan, C.S. Landis, and S.A. Patel, in Frontiers in Chemical Reaction Engineering, L.K. Dorais Wamy and R.A. Mashelkar, Eds., Wiley Eastern, New Delhi, 609 (1984).
[15.] K. Mitra, K. Deb, and S.K. Gupta, J. Appl. Polym. Sci., 69, 69 (1998).
[16.] S. Raha, S. Majumdar, and K. Mitra, Macromol. Theory Simul., 13, 152 (2004).
[17.] K. Mitra, S. Majumdar, and S. Raha, Ind. Eng. Chem. Res., 43, 6055 (2004).
[18.] S. Majumdar, K. Mitra, and S. Raha, Polymer, 46, 11858 (2005).
[19.] K. Mitra, Inter. Mat. Rev., 53, 275 (2008).
[20.] V. Bhaskar, S.K. Gupta, and A.K. Ray, Rev. Chem. Eng., 16, 1 (2000).
[21.] R.B. Kasat, A.K. Ray, and S.K. Gupta, Mater. Manufact. Process., 18, 523 (2003).
[22.] M. Ramteke and S.K. Gupta, Int. J. Chem. Rxn. Eng., 9, 1 (2011).
[23.] S. Sharma and G.P. Rangaiah, "Multi-Objective Optimization in Chemical Engineering," in Multi-Objective Optimization Applications in Chemical Engineering, John Wiley, Chichester (2013).
[24.] S. Zhu and D. Li, Macromol. Theory Simul., 6, 793 (1997).
[25.] P.D. Hustad, R.L. Kuhlman, E.M. Carnahan, T.T. Wenzel, and D.J. Arriola, Macromolecules, 41, 4081 (2008).
[26.] P. Deuflhardt, E. Hairer, and J. Zugk, Numer. Math., 51, 501 (1987).
[27.] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, Chichester (2001).
[28.] M. Atiqullah, H. Hammawa, and H. Hamid, Ear. Polym, J., 34, 1511 (1997).
[29.] S.I. Sandler, Chemical and Engineering Thermodynamics, John Wiley & Sons, New York (1989).
[30.] B.L. Small and M. Brookhart, Macromolecules, 32, 2120 (1999).
[31.] J.B.P. Soares and T.F.L. McKenna, Polyolefin Reaction Engineering, Wiley-VCH, Weinheim, Germany (2012).
[32.] E. Ochoteco, M. Vecino, M. Montes, and J.C. de la Cal, Chem. Eng. Sci., 56, 4169 (2001).
Anitha Mogilicharla, Saptarshi Majumdar, Kishalay Mitra
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502205, Andhra Pradesh, India
Correspondence to: Kishalay Mitra; e-mail: email@example.com Additional Supporting Information may be found in the online version of this article.
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 2. Prediction of grafting density values with/without considering the parameter [alpha]. Predicted without considering [alpha] with fresh Inclusion Run no. Experiment  parameter estimation of [alpha] 1 8.4 2.66 8.2 2 1.7 2.13 1.7 3 8.6 3.09 7.5 4 1.72 0.31 5 1.8 0.008 TABLE 3. MOOP formulation. Maximize [M.sub.w] Maximize GD Minimize [t.sub.p] 500,000 [less than or equal to] [M.sub.w] [less than or equal to] 700,000 4500 [less than or equal to] cocat/[cat.sub.2] [less than or equal to] 8000 [M.sub.w,app] [greater than or equal to] 2500 [PDI.sub.app] [less than or equal to] 1.45 GD [greater than or equal to] 7 Bounds on decision variables Case 1 [m.sup.min.sub.1] = 67.5E-06mol/L; [m.sup.max.sub.1] = 82.5E-06mol/L [m.sup.min.sub.2] = 9E-06mol/L; [m.sup.max.sub.2] = 11E-06mol/L [m.sup.min.sub.3] = 0.0675mol/L; [m.sup.max.sub.3] = 0.0825mol/L [m.sup.min.sub.4] = 0.2[t.sub.p]min; [m.sup.max.sub.4] = 0.8[t.sub.p]min Case 2 [m.sup.min.sub.1] = 14E-O6mol/L; [m.sup.max.sub.1] = 82.5E-06mol/L [m.sup.min.sub.2] = 9#-06mol/L; [m.sup.max.sub.2] = 16.5E-06mol/L [m.sup.min.sub.3] = 0.045mol/L; [m.sup.max.sub.3] = 0.0825mol/L [m.sup.min.sub.4] = 0.2[t.sub.p]min; [m.sup.max.sub.4] = 0.8[t.sub.p]min [t.sup.min.sub.p] = 70min; [t.sup.max.sub.p] = 180min TABLE 4. Kinetic rate constants for the binary catalyst system. [k.sub.i1] 4.7789 x [10.sup.3] (L/(mol.min)) [k.sub.p1] 1.0659 x [10.sup.6] (L/(mol.min)) [k.sub.[beta]] 8.9738 x [10.sup.7] (1/min) [k.sub.[beta]r] 8.3145 x [10.sup.6] (L/(mol.min)) [k.sub.ri1] 1.4799 (L/(mol.min)) [k.sub.a2] 8.8243 x [10.sup.2] (L/(mol.min)) [k.sub.i2] 6.5754 x [10.sup.3] (L/(mol.min)) [k.sub.p2] 9.4277 x [10.sup.7] (L/(mol.min)) [k.sub.lcb] 8.3375 x [10.sup.8] (L/(mol.min)) [k.sub.al] 8.5325 x [10.sup.4] (L/(mol.min)) [k.sub.ral] 13.9312 x [10.sup.4] (L/(mol.min)) [k.sub.d2] (Al/Zr = 7500) 12.6322 x [10.sup.10] (L/(mol.min)) [k.sub.d2] (Al/Zr = 5000) 56.8449 x [10.sup.10] (L/(mol.ntin)) TABLE 5. Comparison of model predicted values with experimental data  for aPP macromonomer. Second catalyst [cat.sub.1] [cat.sub.2] Al (M) addition time Runno. ([micro]M)  ([micro]M)   (min) 1 75 10 0.075 90 2 75 10 0.075 30 3 75 15 0.075 120 4 75 15 0.075 30 5 16.67 10 0.05 30 6 75 5 0.075 0 aPP experiment  aPP Predicted [M.sub.w] x [M.sub.w] x Runno. [10.sup.-3] (g/mol) PDI [10.sup.-3] (g/mol) PDI 1 3.6 1.3 4.4 1.4 2 3.6 1.4 3.2 1.34 3 3.3 1.3 4.5 1.4 4 3.1 1.3 2.5 1.34 5 3.0 1.3 2.6 1.33 6 3.4 1.3 2.5 1.34 TABLE 6. Comparison of model predicted values with the experimental  data for iPP copolymer. iPP experiment  iPP predicted [M.sub.w x [10.sup.-3] [M.sub.w x [10.sup.-3] Runno. (g/mol) PDI (g/mol) PDI 1 631.8 2.7 632 2.2 2 564.7 2.5 544 2.2 3 447.3 2.3 485 2.4 4 395.2 2.4 378 2.4 5 514.4 2.3 554 2.4 6 548.8 2.5 595 2.1 TABLE 7. Comparison of model predicted values with the experimental  data for grafting density. GD Run no. Experiment  Predicted Melting point  i 8.4 8.2 144.4 2 1.7 1.7 148.6 3 8.6 7.5 145.6 4 0.31 149.7 5 0.008 153.5 6 0 0 155.1 TABLE 8. Process performance with various decision variables for Case 1. Data set Fe ([micro]M) Zr ([micro]M) Cocatalyst (M) Experimental 75 10 0.075 MOOP 82.3 9.83 0.0751 MOOP 82.3 9.5 0.0716 Polymerization time [M.sub.w] [10.sup.-3] Data set (min) (g/mol) GD Experimental 180 631.8 8.4 MOOP 178 648 8.92 MOOP 178.4 666 8.9
|Printer friendly Cite/link Email Feedback|
|Author:||Mogilicharla, Anitha; Majumdar, Saptarshi; Mitra, Kishalay|
|Publication:||Polymer Engineering and Science|
|Date:||May 1, 2015|
|Previous Article:||Infrared melt temperature measurement of single screw extrusion.|
|Next Article:||New method for determining the optical rotatory dispersion of hydroxypropyl cellulose polymer solutions in water.|