Multiobjective dynamic optimization of an industrial nylon 6 semibatch reactor using genetic algorithm

The nondominated sorting genetic algorithm (NSGA) is adapted and used to obtain multiobjective Pareto optimal solutions for three grades of nylon 6 being produced in an industrial semibatch reactor. The total reaction time and the concentration of an undesirable cyclic dimer in the product are taken as two individual objectives for minimization, while simultaneously requiring the attainment of design values of the final monomer conversion and for the number-average chain length. Substantial improvements in the operation of the nylon 6 reactor are indicated by this study. The technique used is very general in nature and can be used for multiobjective optimization of other reactors. Good mathematical models accounting for all the physicochemical aspects operative in a reactor (and which have been preferably tested on industrial data) are a prerequisite for such optimization studies. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 69–87, 1998     

Multiobjective optimization of an industrial nylon‐6 semi batch reactor using the a‐jumping gene adaptations of genetic algorithm and simulated annealing

The elitist nondominated sorting genetic algorithm (NSGA‐II) and multiobjective simulated annealing (MOSA) with the robust fixed‐length jumping gene adaptation (aJG) are used to solve three computationally intensive multiobjective optimization problems for an industrial semi batch nylon‐6 reactor. In Problems 1 and 2, the batch time and the final concentration of the undesirable side‐product (cyclic dimer) are minimized while maintaining desired values of the degree of polymerization of the product and the monomer conversion (monomer conversion is maximized as a third objective in Problem 3). The histories of two decision variables, pressure [or vapor release rate] and jacket fluid temperature, are used to obtain the Pareto optimal fronts. The study predicts considerable improvement over earlier results when (i) a single‐stage steam jet ejector is used to create subatmospheric pressures in the reactor, (ii) when the jacket fluid temperature is taken as a function of time, and (iii) when some amino caproic acid (from the depolymerization of scrap nylon‐6) is added to the feed. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers     

Optimization of nonvaporizing nylon 6 reactors with stopping conditions and end-point constraints

In this study, optimal temperature profiles (or histories),T(t), are obtained for nonvaporizing plug-flow (or batch) Nylon 6 reactors using the minimum principle. Two objective functions are studied, one in which the monomer conversion, conv_tf, is maximized, and the other in which the undesirable side product (cyclic dimer) concentration in the output stream, [C₂]_tf, is minimized. The control variable, temperature, is constrained to lie between 220°C and 270°C in order to ensure single phase polymerization. The most significant difference between this study and earlier ones is that the residence (or reaction), time t_f, is not specified a priori, but is determined optimally by the use of a ‘stopping’ condition such that the polymer product has a number-average chain length, μ_n, equal to some desired value μ_n.d. Simultaneously, an end-point constraint is used, which, depending on the objective function used, forces either the cyclic dimer concentration or the monomer conversion at the end of the reactor to lie at a specified value, [C₂]_d or conv_d. Thus, this algorithm incorporates stopping conditions as well as end-point constraints and so is more complex than earlier ones, but the results are more meaningful. Different nonisothermal optimal temperature profiles are obtained for the two objective functions studied, depending on the values of μ_n.d, conv_d, [C₂]_d, and the feed water concentration, representing the complex interplay of several opposing factors.     

Study of parametric sensitivity in an autothermal nylon 6 reactor

The parametric sensitivity of an industrial autothermal nylon 6 reactor was studied. The sensitivities of the temperature maxima with respect to various parameters of the model are computed numerically. The sensitivity peaks were found to occur (almost) at the same value of the input parameter, thus confirming the generalized nature of the thermal parametric sensitivity criterion. It is shown that this criterion can easily be used to find safer regions of operation of the reactor. The variation of the number‐average chain length of the product, μ_nf, with the variation of input parameter, W₀, was also studied. A methodology was suggested to obtain the desired ranges of operation of the reactor which represent an optimal balance between the thermal sensitivity and the sensitivity of μ_nf. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 73: 333–343, 1999     

On‐line optimizing control of bulk polymerization of methyl methacrylate: Some experimental results for heater failure

An experimental unit has been assembled to carry out on‐line optimizing control of the bulk polymerization of methyl methacrylate (MMA). A rheometer‐reactor assembly is used. Temperature and viscosity measurements are used to describe the state of the system. The polymerization is carried out under an off‐line computed optimal temperature history, T_op(t). A planned disturbance (heating system failure) is introduced at time t₁. This disturbance leads to a fall in the temperature of the reaction mass. A new optimal temperature history, T_reop(t), is re‐computed on‐line and is implemented on the reaction mass at time t₂, when the heating is resumed. This procedure helps ‘save the batch’. A genetic algorithm is used to compute this reoptimized temperature history in a short period of ～2 min of real time. The feasibility of the on‐line optimizing control scheme has been demonstrated experimentally. Replicable results for the viscosity history, η(t), of the polymerizing mass under several non‐isothermal conditions have been obtained. These experimental results are quite trustworthy, even though the model predictions are only in approximate agreement with them, perhaps because of the extreme sensitivity of results to the values of the model parameters. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 85: 2350–2360, 2002     

Optimization of nylon 6 reactors with end-point constraints

Optimal temperature profiles for nylon 6 polymerization in plug–flow reactors have been obtained with end-point constraints involving the degree of polymerization and the cyclic dimer concentration, using the most recent kinetic information. Computations suggest that the temperature at the feed end of the reactor must be maintained close to the highest permissible level (determined by the boiling point of the ?-caprolactam). The temperatures in this region control the degree of polymerization more than other variables. Thereafter, the temperature should be reduced. This second zone controls the undesirable cyclic dimer concentration. The effect of a systematic change of values of the various design variables is studied. The profiles obtained herein are qualitatively similar to those obtained by earlier workers using similar formulations. However, they differ significantly from the profiles obtained by us earlier, using different objective functions which are more relevant to the design of new reactors. Attempts have also been made to obtain a global optimal scheme to produce polymer of a desired degree of polymerization and cyclic dimer content, using as short a reactor as possible, and using the water content and the modifier concentration in the feed as the independent variables.     

Multiobjective optimization of an industrial wiped film poly(ethylene terephthalate) reactor: some further insights

Multiobjective optimization of an industrial third-stage, wiped-film poly(ethylene terephthalate) reactor is carried out, using a pre-validated model. The two objective functions minimized are the acid and vinyl end group concentrations in the product. These are two of the undesirable side products produced in the reactor. The optimization problem incorporates an end-point constraint to produce polymer having a desired value of the degree of polymerization (DP). In addition, the concentration of the di-ethylene glycol end group in the product is constrained to lie within a certain range of values. The possible decision variables for the problem are the reactor pressure, temperature, catalyst concentration, residence time of the reaction mass in the reactor and the speed of rotation of the agitator. The nondominated sorting genetic algorithm (NSGA) is used to solve this multiobjective optimization problem. It is found that this algorithm is unable to converge to the correct solution(s) when two or more decision variables are used, and we need to run the code several times over (with different values of the computational variable, S_r, the seed for generating the random numbers) to obtain the solutions. In fact, this is an excellent test problem for future multiobjective optimization algorithms. It is found that when temperature is kept constant, Pareto optimal solutions are obtained, while, when the temperature is included as a decision variable, a global unique optimal point is obtained.     

Multiobjective optimization of the continuous casting process for poly (methyl methacrylate) using adapted genetic algorithm

The nondominated sorting genetic algorithm (NSGA) has been used to optimize the operation of the continuous casting of a film of poly (methyl methacrylate). This process involves two reactors, namely, an isothermal plug flow tubular reactor (PFTR) followed by a nonisothermal film reactor. Two objective functions have been used in this study: the cross‐section average value of the monomer conversion, x̄_mf , of the product is maximized, and the length, z_f , of the film reactor is minimized. Simultaneously, the cross‐section average value of the number‐average molecular weight of the product is forced to have a certain prescribed (desired) value. It is also ensured that the temperature at any location in the film being produced lies below a certain value, to avoid degradation reactions. Seven decision variables are used in this study: the temperature of the isothermal PFTR, the flow rate of the initiator in the feed to the PFTR (for a specified feed flow rate of the monomer), the film thickness, the monomer conversion at the output of the PFTR, and three coefficients describing the wall temperature to be used in the film reactor. Sets of nondominating (equally good) optimal solutions (Pareto sets) have been obtained due to the conflicting requirements for the several conditions studied. It is interesting to observe that under optimal conditions, the exothermicity of the reactions drives them to completion near the center of the film, while heat conduction and higher wall temperature help to achieve this in the outer regions. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 78: 1439–1458, 2000     

Optimization of nonvaporizing nylon 6 reactors with stopping conditions

In this study, optimal temperature profiles or histories T(t) are obtained for nonvaporizing plug-flow or batch Nylon 6 reactors using the minimum principle. Two objective functions are studied, one in which the monomer conversion is maximized, and the other in which the undesirable cyclic dimer concentration in the product stream is minimized. The control variable, temperature, is constrained to lie between 220 and 270°C in order to ensure single-phase polymerization. The most significant difference between this study and earlier ones is that the residence or reaction time t_f is not specified a priori, but is determined optimally by the use of a “stopping” condition such that the polymer product has a number-average chain length μ, equal to some desired value μ_n,d. This makes the algorithm considerably more complex, but the results are more meaningful. Qualitatively different optimal temperature profiles are obtained for the two objective functions studied, representing the complex interplay of several opposing factors in determining optimal conditions. This study also lays the foundation for even more complex, but relevant, optimization studies.     

Effect of the reaction bath temperature in a fixed-bed reactor for oxidation of oxylene over V2O5/TiO2 catalysts

The effects of the reaction variables in the operation of a fixed-bed reactor for oxidation ofo-xylene over V₂O₅/TiO₂ catalysts were studied experimentally using a bench reactor. Reaction temperature, feed flow rate and feed concentration ofo-xylene were found to have significant effects on the product distribution and the temperature profile in the reactor. Drastic enhancements ofo-xylene oxidation reaction were observed at some conditions, which was ascribed to the effect of heat accumulated in the bed and indicated a possible way to increase the productivity in the industrial condition. This paper was presented at the 8th APCChE (Asia Pacific Confederation of Chemical Engineering) Congress held at Seoul between August 16 and 19, 1999.     

Crystallization of cyclic oligomers on the surface of nylon 6 fibers

When scoured nylon 6 filament was exposed to either water vapor or n-alcohol vapor, a portion of the cyclic [NH(CH₂)₅CO]_x, with x = 2 through 7, and the ?-caprolactam present migrated to the filament surface and crystallized. A variety of crystal forms were observed. The variations depended partly on the vapor to which the sample had been exposed. During water vapor exposures, the cyclic dimer migrated more easily to the filament surface than the cyclic monomer and the cyclic tetramer migrated more easily than the cyclic trimer. It is postulated that the secondary valence forces acting between the dimer and the nylon 6 polymer are weaker than that between the ?-caprolactam and the polymer since the dimer forms intramolecular hydrogen bonds. Consequently, the dimer migrates more easily than the monomer, which can form hydrogen bonds with the polymer. Also the cyclic trimer can form hydrogen bonds with the nylon 6 polymer and thus migrates less easily than the cyclic tetramer, which can form intramolecular hydrogen bonds. In alcohol vapor exposures, the n-alcohol used influenced the concentration of cyclic monomer and oligomer, which migrated to the surface of the nylon 6 filament. The variation of the monomer and oligomer concentration on the fiber surface with the alcohol used in the exposure is discussed.     

On‐line optimization of free radical bulk polymerization reactors in the presence of equipment failure

An on‐line optimizing control scheme has been developed for bulk polymerization of free radical systems. The effects of random errors, as well as one kind of a major disturbance (heating system failure), have been studied. A model‐based, inferential state estimation scheme was incorporated to estimate, on‐line, the parameters of the model (and thereby, the monomer conversion and molecular weight of the polymer) using experimental data on temperature and viscosity. A sequential quadratic programming technique was used for this purpose. A major disturbance, such as heating system failure, leads to a deteriorated final product unless an on‐line optimal temperature trajectory (history) is recomputed and implemented on the reactor. Genetic algorithm was used for this purpose. It has been found that, if the “sensing” of the major temperature deviation from the optimal value and rectification of the heating system is achieved well in advance of the onset of the Trommsdroff effect, use of a reoptimized temperature history is sufficient to produce the desired product without significantly altering reaction time. However, if such a disturbance occurs late, a single‐shot intermediate addition of an optimal amount of initiator needs to be used in addition to changing the temperature history to produce polymers having the desired properties in the minimum reaction time. Other types of failures can similarly be handled using the methodology developed. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 71: 2101–2120, 1999     

Design stage optimization of an industrial low-density polyethylene tubular reactor for multiple objectives using NSGA-II and its jumping gene adaptations

Design stage optimization of an industrial low-density polyethylene (LDPE) tubular reactor is carried out for two simultaneous objectives: maximization of monomer conversion and minimization of normalized side products (methyl, vinyl, and vinylidene groups), both at the reactor end, with end-point constraint on number-average molecular weight (M_n,f) in the product. An inequality constraint is also imposed on reactor temperature to avoid run-away condition in the tubular reactor. The binary-coded elitist non-dominated sorting genetic algorithm (NSGA-II) and its jumping gene (JG) adaptations are used to solve the optimization problem. Both the equality and inequality constraints are handled by penalty functions. Only sub-optimal solutions are obtained when the equality end-point constraint on M_n,f is imposed. But, correct global optimal solutions can be assembled from among the Pareto-optimal sets of several problems involving a softer constraint on M_n,f. A systematic approach of constrained-dominance principle for handling constraints is applied for the first time in the binary-coded NSGA-II-aJG and NSGA-II-JG, and its performance is compared to the penalty function approach. A three-objective optimization problem with the compression power (associated with the compression cost) as the third objective along with the aforementioned two objectives, is also studied. The results of three-objective optimization are compared with two different combinations of two-objective problems.     

Multiobjective dynamic optimization of a nonvaporizing nylon 6 batch reactor

Multiobjective dynamic optimization has been carried out on a nonvaporizing nylon 6 batch reactor. Three objective functions have been identified, viz., minimization of the concentration of unreacted monomer in the product, minimization of the dimer concentration, and minimization of the reaction time, for producing polymer having a specified value, μ_{n, d}, of the number average chain length. Two problems have been studied in this paper, each consisting of two objective functions taken from the above set. Pareto solutions have been generated using an algorithm based on Pontryagin's minimum principle and the method of Lagrangian multipliers. The effects of various physical and computational parameters have been studied, and methods have been developed to overcome the numerical difficulties that arise during the solution. The Pareto sets so generated can be coupled with the surrogate worth trade-off (SWT) method, which facilitates interaction with a decision maker (DM). The optimal temperature histories obtained for the two problems studied are quite different and suggest that one must solve the three-dimensional problem in which the vector objective function incorporates all three objective functions. Results from the present study could be used as starting guesses to converge rapidly on the solution of the three-dimensional problem.     

Dynamic optimization of an industrial semi-batch nylon 6 reactor with end point constraints and stopping conditions

In this study, optimal vapor release rate (or pressure) histories have been generated for an industrial semi-batch nylon 6 reactor using Pontryagin's minimum principle. The batch time has been taken as the objective function, which is to be minimized. The pressure is constrained to lie between a lower and an upper limit. The temperature, a state variable, is also constrained to lie between 220°C and 270°C in order to ensure single-phase polymerization. Optimization has been carried out with a single end-point constraint (on monomer conversion) and a stopping condition (obtaining a product having a desired degree of polymerization, μ_n). Techniques have been developed to overcome the discontinuities present in the model, as well as to take care of state variable constraints. The effects of various physical and computational variables on the optimal pressure history and the corresponding batch time have been studied. It is found that the optimal batch time is almost 50% of the industrial value used currently. Interestingly, the optimal pressure history is quite similar qualitatively with the current practice though quantitatively there is a significant difference. Improvements in reactor operation along these lines have been reported. © 1996 John Wiley & Sons, Inc.     

Estimation of Kinetic Parameters for Hydrogenation Reactions Using a Genetic Algorithm

The kinetics of acetylene hydrogenation in a fixed‐bed reactor of a commercial Pd/Al₂O₃ catalyst has been studied. The hydrogenation reactor considered in this work is an essential part of a vinyl chloride monomer (VCM) plant. Three well‐known kinetic models were used to simulate the hydrogenation reactor under industrial operating conditions. Since none of the models provide appropriate prediction, the industrial data and calculated values were compared and optimum kinetic parameters were evaluated utilizing a genetic algorithm (GA) technique. The best kinetic parameters for the three models were determined under specified industrial operating conditions. The hydrogenation reactor was simulated using the estimated optimum kinetic parameters of the three models. Simulation results from the three models were compared to industrial data and the best kinetic model was found. This kinetic model with the evaluated optimum kinetic parameters can well predict the behavior of the industrial hydrogenation reactor to improve the performance of the process.     

Continuous enzymatic synthesis of Z-kyotorphin amide in an enzyme-immobilized fixed-bed reactor

α-Chymotrypsin immobilized on natural and inexpensive supports such as diatomaceous earth was used as catalyst for Z-Tyr-Arg-NH₂ (Z-kyotorphin amide) synthesis. In order to obtain the optimal reaction conditions, a 2² factorial experimental design was used. The factors considered were cosolvent (dimethylformamide) concentration and temperature; optimal product yield was achieved at 40% (v/v) dimethylformamide and 25°C. A sequential kinetic model was considered which generally gave good agreement between experimental and theoretical data for continuous synthesis of Z-kyotorphin amide in a packed-bed immobilized reactor system. The activation energy for the synthesis was determined to be 48.0 ± 2.3 kJ mol^?1.     

Prediction of molecular weight distribution of high-density polyethylene made in a continuous reactor from fixed ethylene concentration batch reactor data

The polydispersity index, M_w/M_n, of the product of an ideal continuous stirred tank reactor can be predicted from batch reactor data. A method for the prediction, in the case of high-density polyethylene synthesis, comes from finding the effect of catalyst age upon yield, M_w, and M_n in a batch reactor operated at constant pressure. These catalyst age effects are combined with the catalyst age distribution in the continuous reactor, and integration over all ages gives the yield and the polydispersity index for the continuous reactor product. This scheme for prediction was applied to a particular catalyst system, and the calculated values have been found to agree with observations.     

Mass Transfer Characteristics of Syngas Components in Slurry System at Industrial Conditions

The gas‐liquid mass transfer behavior of syngas components, H₂ and CO, has been studied in a three‐phase bubble column reactor at industrial conditions. The influences of the main operating conditions, such as temperature, pressure, superficial gas velocity and solid concentration, have been studied systematically. The volumetric liquid‐side mass transfer coefficient k_La is obtained by measuring the dissolution rate of H₂ and CO. The gas holdup and the bubble size distribution in the reactor are measured by an optical fiber technique, the specific gas‐liquid interfacial area aand the liquid‐side mass transfer coefficient k_L are calculated based on the experimental measurements. Empirical correlations are proposed to predict k_L and a values for H₂ and CO in liquid paraffin/solid particles slurry bubble column reactors.