Control of Distributions in an Emulsion Polymerization Reactor

Emulsion polymerization is an important process for the polymer industry that has significant advantages over bulk and solution polymerization processes. These advantages result mostly from the multiphase and compartmentalized nature of the emulsion polymerization, delivering a high versatility to product qualities. Despite these advantages, the control of emulsion polymerization reactors is very challenging due to the complexity of emulsion process and the lack of adequate on-line measurement instrumentation. The control of particle size distribution (PSD), which is critical for desired end-product performance, suffers from both of these aspects. The resulting control problem is high-dimensional due to the complexity of the process and lacks fast high-resolution size distribution measurements. Although the control of the emulsion polymerization systems is a well studied topic typically involving the regulation of one or more of the lumped properties (e.g. moments of the distributions), the control of the distribution shape is largely an open problem. The indirect control of PSD through moments reduces the high dimensionality of the problem but regulation to complex distributions is not always possible. However, the end-use properties (mechanical, rheological, optical) of the polymer products may depend on obtaining the full distributions, particularly when the target distribution is complex and/or multimodal. For example, bimodal distributions in paint applications may be desirable for rheopectic behavior. The population balance equation (PBE) model previously developed by Immanuel et al. and the experimental system in the Doyle group are utilized for developing and testing the various strategies.

The Experimental System

With the goal of the on-line control of the particle size distribution (PSD), the first aspect of the research was to construct an experimental facility incorporating instrumentation for the measurement of PSD and other important process variables, and to evaluate the feasibility of on-line measurement and estimation of important process variables. This aspect was a collaborative effort involving Dr. Timothy J Crowley and Dr. Edward S Meadows. The approach was based on first-principle model-based process calculation strategy that enabled the inference of the maximum possible process information with measurements of the latex density alone (besides the feed rates). This information combined with the PSD measurements using the state-of-the-art instrument for PSD measurement (Capillary Hydro-DynamicFractionator), enables the estimation of all process variables of interest for control purposes. The pictures show the experimental facility.

Figure 1. Experimental System

Figure 2. Capillary Hydro-Dynamic Fractionator (CHDF)

Population Balance Model

The evolution of PSD in emulsion polymerization is determined by three major phenomena: nucleation, growth and coagulation. Nucleation accounts for the formation of a new particle, which occurs either by the entry of polymeric radicals generated from the aqueous phase into surfactant micelles (micellar nucleation), or in the absense of any external surfactants (homogenous nucleation). Growth account for the increase in the size of these polymeric particles due to the polymerization of the radicals housed within them with the monomers absorbed within. Coagulation (agglomeration) accounts for the merging of particles with each other due to insufficient stabilization. Particle stabilization is through amphoteric surfactant molecules, which are endowed with both hydrophilic and lyophilic characteristics. These phenomena of nucleation, growth and coagulation can be cast in a population balance framework. While population balance equations are material balances performed on particles in the various size ranges, they also account for the internal re-alignment of the particles due to the growth and coagulation processes.

There is a considerable literature on the modeling of PSD, starting from the model of Min and Ray (1974), and including the models of Rawlings and Ray; Storti and Morbidelli; and Gilbert, Richards and Congalides, among others. A model was developed in our group for Styrene homopolymerization, based on the Zero-One approach, and was recently extended to encompass non-isothermal conditions.

The model for the vinyl acetate (VAc)-butyl acrylate(BuA) emulsion co-polymer system stems from the more general and comprehensive model of Saldivar, Dafniotis and Ray. The model is based on the pseudo-homopolymerization approach, which in turn employs the Long-chain hypothesis and a pseudo-steady state approximation. The novel contributions in this model compared to earlier models are:

(1) A mechanistic formulation is employed in modeling the distribution of the radical concentration inside the particles (average number of radicals/particle) (as opposed to using empirical formulations which stem from the traditional Smith-Ewart formulations). This approach captures the right size dependence, as seen in the experimental results, and also avoids the prediction of sharp fronts that are contrary to reality.

(2) Modifications are proposed in the partitioning calculations for the surfactants, to account for the tendency of the non-ionic surfactants to partition into the dispersed phases (be absorbed by them). This tendency of the non-ionic surfactants introduces interesting modifications to the nucleation phenomena (as seen in several experimental studies), which is captured well by the model.

(3) A model is proposed for the particle stabilization mechanism. Unlike their ionic counterparts, (wherein the stabilization mechanism has been well established through the DLVO theory), studies addressing the stabilization mechanism of non-ionic surfactants in the domain of emulsion polymerization are sparse. The current model accounts for particle stabilization mainly through the forces imparted by the bulky surfactant chains, though ad-hoc allowances have been made to account for the inevitable shear effects.

The complete model shows very good qualitative match, and a reasonable quantitative match, to the experimental results. A computationally-efficient solution technique, based on an effective decomposition of the fast and slow modes of the system, have been developed. This new technique gives faster solutions, which puts model-based control within the feasibility realm, from a computational-standpoint.

Control Studies

A relatively simple controller to serve as a benchmark was designed before considering advanced control strategies for the semi-batch VAc/BuA emulsion copolymerization system. The proposed simple strategy consists of regulating the three major phenomena (nucleation, growth, and coagulation) that govern the particle size distribution. To control these events, the monomer and surfactant flowrates were chosen as inputs whereas the average particle size, total number of particles, and the solids content were chosen as the outputs. Series of step tests introduced to the system at different times with various magnitudes were used to obtain the transfer functions relating the output deviations from their nominal trajectories to the inputs. Nonlinear regression was then used to obtain the transfer functions of the input/output pairs suggested by the Relative Gain Array (RGA) analysis. The control studies showed that the regulation of the average particle size was not necessary but the regulation of the total number of particles and the solids content was sufficient. Proportional Integral Derivative (PID) controllers were used for the regulation of the outputs. The proposed control strategy was proven to be successful in updating the input trajectories to regulate the whole PSD to the desired distribution.

The model-based control studies by Doyle and coworkers revealing that the delayed infrequent measurement of PSD, the primary controlled variable, deteriorates the quality of in-batch feedback control motivated a multi-rate control approach. Availability of frequent and reliable on-line density measurements for emulsion polymerization systems encouraged the use of multi-rate estimation schemes as a remedy for the delayed infrequent PSD measurements. A multi-rate estimator is used in which a periodically time-varying Riccati equation is solved to obtain the optimal multi-rate filter. This Kalman filter utilizes the density measurements with high sampling rates when PSD measurements are not available, detecting possible disturbances on PSD through density measurements. This filter is embedded in a model predictive controller (MPC) to regulate the endpoint PSD. The MPC controller uses the nonlinear PBE model for open loop multi-step ahead predictions and the effect of the future manipulated changes is modeled linearly so that the resulting optimization problem at each time step (that is solved by the controller) is a quadratic program. Principal component analysis (PCA) based model order reduction was employed to remove the high dimensionality of the control problem, where the principal directions of variability were calculated by PCA and the original states that represent the PSD and the conditions of the reactor were projected to the principal components space. The designed controller was able to act upon the frequent secondary measurements to detect the disturbances and take corrective action on the primary mode of the distribution as well as the secondary mode in the case of a target bimodal distribution.