Reduced-order methodologies for feedback control of particle size distribution in semi-batch emulsion copolymerization

Within-batch feedback control strategies are developed for the regulation of the particle size distribution (PSD) in a semibatch vinyl acetate (VAc)/butyl acrylate (BuA) emulsion copolymerization system. These strategies are also applicable for regulation of distributions in other particulate systems governed by population balances. In the first strategy, PID controllers are employed for regulating nucleation and growth events through tracking the nominal trajectories of total number of particles and the solids content by manipulating the feed-rates of the more reactive monomer, BuA, and the surfactant. The second control strategy is based on tracking nominal trajectories of the moments of the distribution with a quadratic dynamic matrix controller (QDMC). To determine the appropriate number of moments to describe the PSD during various stages of a nominal batch, a maximum-entropy approach is utilized. In the final and most complex approach, a nonlinear model predictive controller is designed utilizing the detailed population balance model of the system. The ill-conditioning resulting from the high-dimensionality of the resulting dynamical system is removed by principal component analysis (PCA)-based model order reduction and a multi-rate estimator is designed to compensate for the measurement delay associated with the PSD measurements.     

Feedback controllability assessment and control of particle size distribution in emulsion polymerisation

In this article, the importance of particle size distribution (PSD) control as a means for the inferential control of the rheology of emulsion polymers is illustrated. A controllability assessment is presented to illustrate the attainability or otherwise of bimodal PSD using feedback control through a consideration of the process mechanisms—measurement limitations and process constraints that prevent the implementation of feedback corrections. The suitability of a batch-to-batch iterative feedback PSD control is demonstrated, which could act in addition to any in-batch feedback control, the latter being less feasible in certain cases, as argued in this article. A detailed population balance model is used for the batch-to-batch control, which simplifies model update and feedback correction.     

Measurement methods of particle size distribution in emulsion polymerization

The particle size distribution of polymer always develops in emulsion polymerization systems, and certain key phenomena/mechanisms as well as properties of the final product are significantly affected by this distribution. This review mainly focuses on the measurement methods of particle size distribution rather than average particle size during the emulsion polymerization process, including the existing off-line, on-line, and in-line measurement methods. Moreover, the principle, resolution, performance, advantages, and drawbacks of various methods for evaluating particle size distribution are contrasted and illustrated. Besides, several possible development directions or solutions of the in-line measurement technology are explored     

On-line multi-variable predictive control of molar mass and particle size distributions in free-radical emulsion copolymerization

A multi-variable model predictive control (MPC) was formulated to solve control problems associated with a combination of regulation and targeting desired set-points. We investigated the simultaneous control of key polymer properties: the particle size (PSD) and molecular weight distribution (MWD) by manipulating the flow rates of the monomers (styrene, MMA), surfactant, initiator and the temperature of the reactor. A multi-input-multi-output (MIMO) formulation was constructed for the constrained optimal control problem to maximize the width of the PSD (with M_n at a constant set-point), and to maximize the average molar mass. The strategy developed within a gPROMS-API-DCS environment allowed real-time implementation of model-based control of the process. The optimal control problem was implemented via an interface to a dynamic optimization code. Major improvements in process operation and polymer property control resulted on the implementation of our multi-variable MPC algorithm. The manipulation of the four flow rates and the temperature increased the degree of freedom in the system and achieved tighter PSD and MWD control. The on-line performance of MPC for MWD and PSD control was found to be satisfactory.     

Nonlinear model order reduction and control of particle size distribution in a semibatch vinyl acetate/butyl acrylate emulsion copolymerization reactor

This paper addresses the control of the full particle size distribution (PSD) in a semibatch emulsion copolymerization reactor. The numerical approximation of a fundamental population balance model results in a high order system to accurately describe the distribution of particle size; therefore, model order reduction is required. Pseudo random input signals are input to the mechanistic model to generate a data set which covers the reachable region of the system, on the basis of which the transformation matrices are calculated by principal component analysis (PCA). A linear time varying model with reduced order obtained from the transformation matrices is augmented in the prediction equation of linear model predictive control. The performance of the controller is evaluated to drive the particle size distribution at the final time of the batch to the desired distribution in the presence of disturbances. This paper is dedicated to Professor Hyun-Ku Rhee on the occasion of his retirement from the Seoul National University.     

On the role of an unconventional rigid rodlike cationic surfactant on the styrene emulsion polymerization. Kinetics, particle size and particle size distribution

An unconventional amphiphile (1-[ω-(4′-methoxy-4-biphenylyloxy)octyl]pyridinium bromide, PC8) was used as surfactant in the emulsion polymerization of styrene. At low surfactant concentration (6, 12 or 36 mmol l⁻¹), curves of polymerization rate versus conversion obeyed the typical behavior characterized by intervals I, II and III. However, at high concentration (48 or 72 mmol l⁻¹) the interval II was not observed. The particle size distribution curves showed two families of polymer particles, indicating the participation of at least two mechanisms of particle formation, one being the simple micellar nucleation and the other probably the coagulative nucleation of precursor particles. The latter was considered to occur during the nucleation interval.     

Measurement based modeling and control of bimodal particle size distribution in batch emulsion polymerization

In this article, a novel modeling approach is proposed for bimodal Particle Size Distribution (PSD) control in batch emulsion polymerization. The modeling approach is based on a behavioral model structure that captures the dynamics of PSD. The parameters of the resulting model can be easily identified using a limited number of experiments. The resulting model can then be incorporated in a simple learning scheme to produce a desired bimodal PSD while compensating for model mismatch and/or physical parameters variations using very simple updating rules. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Inferential closed-loop control of particle size distribution for styrene emulsion polymerization

In this work, a new control strategy for controlling the particle size distribution (PSD) in emulsion polymerization has been proposed. It is shown that the desired PSD can be achieved by controlling the free surfactant concentration which in turn can be done by manipulating the surfactant feed rate. Simulation results show that the closed-loop control of free surfactant concentration results in a better control of PSD compared to open-loop control strategy, in presence of model mismatch and disturbances. Since the on-line measuring of ionic free surfactant concentration is difficult, conductivity which is related to it is measured instead and used for control purposes. The closed-loop control of conductivity also results in a better control of PSD compared to open-loop control strategy, but its performance is not as good as controlling free surfactant concentration in presence of model mismatch.     

Modelling and simulation of batch and semi-batch emulsion copolymerization of styrene and butyl acrylate

This work deals with modelling of emulsion polymerization processes using batch and semi-batch reactors. Specific attention is paid to copolymerization of the system styrene/butyl acrylate.The main key parameters of the model are identified on the basis of batch experimental data in order to describe the complete sketch of emulsion polymerization. The model is then used to simulate, under several operating conditions, the polymerization rate, the global monomers conversion, the number and weight average molecular weights as well as the particle size distribution and the glass transition temperature. Then, the model is generalized to the use of semi-batch processes and validated for this application.     

Particle size distribution control in emulsion polymerization

Effects of the operating policies—the initial initiator amount; the initial emulsifier amount; the monomer addition mode: batch or semibatch; and the monomer addition rate under “monomer‐starved conditions” for the control of particle size distribution (PSD)—were studied through a model that simulates batch and semibatch reactor operations in conventional emulsion polymerization. The population balance model incorporates both the nucleation stage and the growth stage. The full PSDs were reported, which have normally been omitted in earlier studies. It was shown through simulations that the broadness of the distributions, both initial (obtained after the end of nucleation) and final (after complete conversion of monomer), can be controlled by the initial initiator amount and the emulsifier amount. The higher initiator amounts and the lower emulsifier amounts favor narrower initial and final distributions. The shape of the initial PSDs and the trends in the average size and range were preserved with subsequent addition of monomer in the batch or in the semibatch mode, although the final PSD was always considerably narrower than that of the initial PSD. The addition of monomer in the semibatch mode gave narrower distribution compared to that of the batch mode, and also, lower monomer addition rates gave narrower distributions (larger average sizes), which was a new result. It was further shown through simulations that, under monomer‐starved conditions, the reaction rate closely matched the monomer feed rate. These conclusions are explained (1) qualitatively—the shorter the length of the nucleation stage and the larger the length of the growth stage (provided the number of particles remains the same), the narrower is the distribution; and (2) mathematically—in terms of the “self‐sharpening” effect. Experimental evidence in favor of the self‐sharpening effect was given by analyzing the experimental particle size distributions in detail. The practical significance of this work was proposed. © 2004 Wiley Periodicals, Inc. J Appl Polym Sci 92: 2884–2902, 2004     

Operation of semi-batch emulsion polymerisation reactors: Modelling, validation and effect of operating conditions

A detailed dynamic model was developed for a styrene emulsion polymerisation semi-batch reactor to predict the evolution of the product particle size distribution (PSD) and molecular weight distribution (MWD) over the entire range of monomer conversion. A system exhibiting zero-one kinetics was employed, with the model comprising a set of rigorously developed population balance equations to predict monomer conversion, PSD and MWD. The modelling equations included diffusion-controlled kinetics at high monomer conversion where the transition from the zero-one regime to a pseudo-bulk regime occurs. The model predictions were found to be in good agreement with experimental results. Both particle growth and the PSD were found to be strongly affected by the monomer feedrate. Reactor temperature had a major influence on the MWD which was, however, insensitive to changes in the monomer feedrate. These findings were confirmed experimentally. As a result, it seems reasonable to propose that the use of the monomer feedrate to control the PSD and the reactor temperature to control the MWD are appropriate in practical situations. Consequently, an optimal monomer feed trajectory was developed off-line (using the validated reactor simulation) and verified experimentally by producing a polymer with specific PSD characteristics.     

Modelling particle size distributions and secondary particle formation in emulsion polymerisation

An extensive model is given for the particle size distribution (PSD), particle number, particle size and amount of secondary nucleation in emulsion polymerisations. This incorporates what are thought to be all of the complex competing processes: aqueous phase kinetics for all radical species arising from both initiator and from exit (desorption), radical balance inside the particles, particle formation by both micellar and homogeneous nucleation mechanisms, and coagulation (the rate of which is obtained using the Healy–Hogg extension of DLVO theory). The predictions of the model are compared to extensive experimental results on rates, time evolution of the particle size distribution, and relative amounts of secondary nucleation, for styrene initiated by persulfate with sodium dodecyl sulfate and with sodium dihexyl sulfosuccinate as surfactants. For this system values of almost all of the many parameters needed for the model are available from independent measurements, and thus no significant parameter adjustment is plausible. Accord with experiment is imperfect but quite acceptable, supporting the validity of the various mechanisms in the model. Effects such as the experimental variation of particle number with ionic strength, as well as calculated coagulation rate coefficients as functions of particle size, suggest that coagulation of precursor (i.e., newly-formed) particles is a significant effect, even above the cmc. The modelling also suggests why secondary nucleation occurs readily in systems stabilised with polymeric surfactant.     

Investigation into the effect of carboxylic acid monomer on particle nucleation and growth in emulsifier-free emulsion copolymerization of styrene-butadiene-acrylic acid

Latexes of carboxylated styrene-butadiene rubber were prepared via batch emulsion copolymerization with different amounts of acrylic acid in the absence of emulsifier. The effect of acid monomer was investigated in the particle formation and growth. It was observed that the amount of acrylic acid strongly affected the particle formation. The number of particles and thus polymerization rate increased with increasing of the acid content. There was no significant difference in the polymerization rate per particle in all experiments. The results show that in this case particle growth process is less dependent on the acrylic acid amount in comparison with its influence on nucleation stage and then particle number. Several parameters such as polymerization rate and number of latex particle per unit volume of the aqueous phase were calculated. Attempt was made to evaluate the average number of growing chain per particle. Also average particle diameter of the above carboxylated SBR latexes was obtained through some calculations from the direct measurement of average particle diameter in the swollen state by light scattering technique for the first time.     

Integrated run-to-run and on-line model-based control of particle size distribution for a semi-batch precipitation reactor

The dynamics of particle size distribution (PSD) in a precipitation process are represented by a population balance (PB) and related differential–algebraic equations. The control of PSD is studied by using a closed-form solution of the PB. Batch-to-batch control and on-line single batch control strategies are investigated for controlling a semi-batch reactor. A systematic integration of the two strategies is shown to have a complementary effect on the control performance.     

乳液聚合中乳胶粒粒径大小及分布的影响因素

在乳液聚合中,乳胶粒的大小及分布对乳液的性能及其应用有很大的影响,同时也反映了乳液聚合反应进行的过程。本文综述了影响乳胶粒粒径大小及分布的各种因素,如聚合工艺、乳化剂、单体种类、聚合温度、引发剂等,并介绍了不同粒径乳液的性能及其应用。     

Measurement of particle size distribution of tripalmitin crystals in a model solution using a laser diffraction method

The form and size distribution of tripalmitin (PPP) crystals formed by quenching a solution in a batch crystallizer have been studied using scanning electron microscopy and a laser diffraction technique. Although the accuracy of the laser technique for particle sizing was affected by the nonsphericity of the crystals, the technique proved to be effective and very reproducible. Crystals recovered by filtration and sonication for laser sizing exhibited markedly different particle size distributions (PSD) and shapes compared to those prepared using another standard route, namely, suspension in butanol and centrifugation. Solutions of 2–10 wt% PPP in a paraffin solvent were crystallized at 25–40°C and displayed two ranges of behavior: (i) rapid growth under strongly supersaturated conditions, yielding narrow PSD of plate and needle-like crystals; and (ii) slow growth in the metastable regime, yielding spherulitic aggregates of platelets that broke down under sonication during preparation for laser sizing. Suspension in alcohol followed by centrifugation yielded aggregates that did not break down under sonication, indicating that the sample preparation route affected the result of the analysis.     

Study of the solution and aqueous emulsion copolymerization of vinylidene chloride with methyl acrylate in the presence a poly(ethylene oxide) macromolecular RAFT agent

The reversible addition-fragmentation chain transfer (RAFT) copolymerization of vinylidene chloride (VDC) with methyl acrylate (MeA) was studied in the presence of poly(ethylene oxide)-based macromolecular RAFT (macroRAFT) agents of the trithiocarbonate type (PEO-TTC) in solution and in aqueous emulsion. Firstly the formation of PEO-b-P(VDC-co-MeA) diblock copolymers was performed in toluene solution at 30 °C and a good control over the polymerization with high chain-end functionality was shown. A first aqueous emulsion copolymerization of VDC with MeA was performed using one of the amphiphilic PEO-b-P(VDC-co-MeA) diblock copolymers as macromolecular stabilizer. Then, in a series of experiments the PEO-TTC macroRAFT agents were directly tested as both chain transfer agents and stabilizing agents in similar conditions (aqueous batch emulsion copolymerization of VDC with MeA at 70 °C). The influence of the nature and concentration of the initiating system and the presence or not of a buffer were studied. We demonstrated that in simple conditions, nanometric latex particles composed of amphiphilic PEO-b-P(VDC-co-MeA) diblock copolymers formed by polymerization-induced self-assembly (PISA). It can thus be concluded that PEO-TTC macroRAFT agents are valuable non-ionic macromolecular stabilizers in the emulsion copolymerization of VDC and MeA and allow the formation of core–shell diblock copolymer particles in the absence of free surfactant. However, when rather high molar masses of the hydrophobic PVDC-based block were targeted, the determined molar masses deviated from the theoretical values.     

Population balance modeling of particle size distribution in monomer‐starved semibatch emulsion polymerization

The evolution of particle size distribution (PSD) in the monomer‐starved semibatch emulsion polymerization of styrene with a neat monomer feed is investigated using a population balance model. The system under study ranges from conventional batch emulsion to semicontinuous (micro)emulsion polymerization depending on the rate of monomer addition. It is shown that, contrary to what is often believed, the broadness of PSD is not necessarily associated with the length of nucleation period. The PSDs at the end of nucleation are found to be independent of surfactant concentration. Simulation results indicate that at the completion of nucleation the particle size is reduced and the PSD narrows with decreasing rate of monomer addition despite nucleation time increasing. The broad distribution of particles frequently encountered in semibatch emulsion polymerizations is therefore attributed to stochastic broadening during the growth stage. The zero‐one‐two‐three model developed in this article allows perceiving that the dominant kinetic mechanism may be different for particles with different sizes. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Calculation of the particle size distribution in suspension polymerization using a compartment-mixing model

A modelling strategy for effective estimation of the particle size distribution (PSD) in suspension polymerization is presented. The strategy consists of coupling a population balance equation (PBE) and a compartment-mixing (CM) model to account for the non-homogeneous mixing in the tank reactor. The values for the rate of energy dissipation of each compartment are estimated from Computational Fluid Dynamics (CFD) calculations and experimental reports on systems with the same agitator and geometric characteristics. Model predictions using the CM model are compared with predictions that assume homogeneous mixing and experimental data on PSD from styrene and divinylbenzene pilot-plant suspension polymerization reactors of 1 and 5 L with Rushton and PBT impellers.     

Predictive control of particle size distribution in particulate processes

In this work, we focus on the development and application of predictive-based strategies for control of particle size distribution (PSD) in continuous and batch particulate processes described by population balance models (PBMs). The control algorithms are designed on the basis of reduced-order models, utilize measurements of principle moments of the PSD, and are tailored to address different control objectives for the continuous and batch processes. For continuous particulate processes, we develop a hybrid predictive control strategy to stabilize a continuous crystallizer at an open-loop unstable steady-state. The hybrid predictive control strategy employs logic-based switching between model predictive control (MPC) and a fall-back bounded controller with a well-defined stability region. The strategy is shown to provide a safety net for the implementation of MPC algorithms with guaranteed stability closed-loop region. For batch particulate processes, the control objective is to achieve a final PSD with desired characteristics subject to both manipulated input and product quality constraints. An optimization-based predictive control strategy that incorporates these constraints explicitly in the controller design is formulated and applied to a seeded batch crystallizer. The strategy is shown to be able to reduce the total volume of the fines by 13.4% compared to a linear cooling strategy, and is shown to be robust with respect to modeling errors.