A Sensitivity Approach to Reachability Analysis for Particle Size

Ind. Eng. Chem. Res. 2004, 43, 327-339

327

A Sensitivity Approach to Reachability Analysis for Particle Size Distribution in Semibatch Emulsion Polymerization Charles David Immanuel and Francis Joseph Doyle, III* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716

Experimental and simulation-based sensitivity studies on the process of evolution of particle size distribution (PSD) in semibatch emulsion polymerization are presented. The experimental studies identify an effective control strategy for PSD, and the appropriate manipulated variables. The complementary simulation studies identify the potentially reachable PSDs in the complex operating space. The study also examines the effect of uncertainties and disturbances on the reachable distributions, to evaluate the benefits of in-batch feedback control. 1. Introduction The control of distributed chemical processes is gaining increased attention due to the potential process and product improvement that is obtained through direct optimization of quality variables.1-4 For example, the particle size distribution (PSD) in emulsion polymerization is critical in determining the end products properties of adhesion, rheological properties, and mechanical strength. The target distribution in most cases is multimodal and non-Gaussian. The control of distributions and profiles motivates a detailed analysis of the controllability of these systems. The mechanism of formation of distributions and profiles are intricate and are influenced in complex and in some cases nonintuitive ways by the inputs available in the process, hence the need for the sensitivity analysis, to evaluate the feasibility of control of distributions and identify an effective control strategy and a suitable combination of manipulated variables. In addition to these sensitivity studies, a very pertinent issue is the identification of classes of distributions that can be produced in these processes. In the highly multivariate character that these distributions represent, the attainable regions of the different variables could be correlated, thereby limiting the attainable classes of distributions. While it is easier to ascertain these controllability issues for linear lumped parameter systems, it is not so in the case of the nonlinear distributed parameter systems. This paper first presents experimental sensitivity studies on the process, to demonstrate the feasibility of control of distributions. The second objective of the paper is to analyze the reachability of the system, namely, the ability to steer an output from an initial state to a final state in a finite time, using the available inputs. Controllability includes the ability to maintain the output at this final state beyond the end point. The reachability problem is approached by employing a detailed population balance model of the system. The focus of the present study is on emulsion polymerization, in which the particle size distribution (PSD) is determined by the three major phenomena of nucleation, growth, and interparticle coagulation. One has a * To whom correspondence should be addressed. Present address: Department of Chemical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106. Tel.: (805) 893-8133. Fax: (805) 893-4731. E-mail: doyle@ engineering.ucsb.edu.

variety of potential inputs to be employed for the control of PSD. These include the feeds of the surfactants (emulsifiers), monomers, and initiators, the concentration of the reagents, and the temperature. It is important to identify the best inputs and the best control configuration. While most of the (few) studies on the control of PSD have focused on surfactant alone, or surfactant and initiator as manipulated variables, Crowley et al.5 examined the effectiveness of control with surfactant feed directly as a manipulated variable, and also with surfactant concentration in the aqueous phase as a manipulated variable. Meadows et al.6 examined the suitability of temperature as a manipulated variable for PSD control. Liotta et al.7 considered the weight fraction polymer within the particles as the manipulated variable, which in turn was regulated using monomer feed, for the control of diameter ratio of a seeded bidisperse population (lumped variable). Distributed parameter systems (DPS) are characterized by partial differential equations, in the case under consideration the population balance equation, resulting in infinite-dimensional systems. These systems can be approximated as finite-dimensional multivariable systems using suitable discretization techniques, with a very high ratio of the number of correlated outputs (controlled variables) to the number of inputs (manipulated variables). As stated above, the correlation and interaction between the nucleation, growth, and coagulation events sets certain constraints on the types of distributions that can be produced in the emulsion. Even in the case of a monodisperse population, there is an upper limit on the particle size to yield a dispersion of solids in the aqueous phase with an appreciable solids content in the latex. As the particle size of the monodisperse population increases, the achievable solids content is reduced. Further, the relative rates of nucleation and growth limits the attainable polydispersity of monodisperse populations. Allowing for a distribution of particle sizes in the population expands the attainable domain of solids content. However, the attainable distributions are dictated by the achievable rates of particle nucleation and growth and by the particle stability, in addition to the strong interaction among these. Although there are process inputs to manipulate some of these subprocesses independently, there is also a high degree of coupling with certain inputs. Considering all these issues and the constraints on the inputs,

10.1021/ie030145p CCC: $27.50 © 2004 American Chemical Society Published on Web 08/19/2003

328

Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004

the actual attainable distributionssboth unimodal and multimodalsare limited. Formal controllability studies on particulate systems, aimed at determining the controllable distributions, are rather limited. To avoid unattainable set points, some studies in the past on the control of distributions have utilized a partial control-like strategy in which a few of the outputs are controlled while the rest are allowed to evolve in an open-loop manner.8,3 Semino and Ray9 addressed the issue of controllability of various particulate systems including emulsion polymerization. They addressed the control of PSD in emulsion polymerization by manipulating the nucleation phenomenon. Considering that the micellar nucleation phenomenon is influenced by the feeds of the surfactant and that of the initiator and/or the inhibitor, they found that the nucleation phenomenon can be completely controlled when the feeds of either the surfactant or the initiator/ inhibitor is unconstrained. A strategy for determining the attainable bounds was developed by Glasser et al.,10 and demonstrated by Smith and Malone,11 for more simple problems. Liotta et al.7 employed a similar strategy in their study on the reachability of diameter ratio (lumped) of bidisperse populations by manipulating the competitive growth phenomenon in seeded emulsion polymerization of styrene. They identified the reachable regions of diameter ratio and examined its sensitivity to the initial conditions. The upper and the practical lower limits on the monomer concentration within the particles were used to identify the reachable regions. Other studies employ a similar philosophy, but rely on simple simulations in determining the reachable regions and in analyzing the processes.12 Recently, there have been studies that address the development of the reachable regions through the solution of optimization problems.13,14 The latter study deals with determining the reachable regions of PSD in styrene emulsion polymerization. In this study, Wang and Doyle III have identified the reachable domains in terms of the “-reachability”, defined to be the domain of distributions that are reached subject to a tolerance in the defined objective function. In the current study, a combination of experiments and simulations using population balance models are employed to analyze the evolution of PSD in the semibatch emulsion copolymerization of vinyl acetate (VAc) and butyl acrylate (BuA), using nonionic surfactants. The experimental studies are mainly used to identify a suitable control configuration and the manipulated variables. The analysis addresses the hierarchy of the individual subprocesses of nucleation, growth, and coagulation, where possible. The simulation studies extend the analysis to identify the reachable bimodal distributions, with the objective of extracting information on the capabilities and limitations of the process with regard to the control of distributions. 2. Experimental Sensitivity Studies A state-of-the-art experimental facility was utilized to study the sensitivities in the process of evolution of PSD. The facility includes a 3-L stirred reactor, equipped for semibatch operation and provided with instrumentation for on-line measurement of latex density and PSD. See ref 15 for a detailed description of the facility and for the reproducibility characteristics of the equipment and the overall process. The system under investigation is vinyl acetate (VAc)-butyl acrylate (BuA)

emulsion copolymerization. A nonionic surfactant is used as the emulsifier, with a redox initiation mechanism (tert-butyl hydrogen peroxide, t-BHP, and sodium formaldehyde sulfoxylate, SFS). In interpreting the experimental results, it is important to understand the underlying mechanisms of the evolution of PSD. In emulsion polymerization, the polymer is produced as a colloidal dispersion of particles in the aqueous phase. Although the polymerization is primarily initiated in the aqueous phase (by watersoluble initiators), the predominant locus of polymerization lies within the particles. Thus, each of these particles constitutes a microscale bulk polymerization reactor, comprising active and terminated polymer chains and the monomers. The monomers are usually sparingly solubilized in the aqueous phase and are usually in a thermodynamic equilibrium between the aqueous phase and the particles. When these phases are saturated with the monomers, separate monomer droplets result as additional dispersed phases. The dispersed phases are stabilized by surfactants, which, by virtue of their amphiphilic character, adsorb onto the dispersed phases, and thereby keep them apart in the aqueous phase. The PSD evolves according to the interplay between the phenomena of particle nucleation, growth, and coagulation. When the concentration of surfactant in the aqueous phase exceeds the critical micelle concentration value of the surfactant, the surfactants form micelles that serve as nucleii for particles. Particles are formed by the entry of a polymer radical from the aqueous phase into the micelles. Particle nucleation can also occur under micelle-free conditions by the precipitation of a polymer radical initiated in the aqueous phase upon the attainment of a critical chain length, corresponding to the solubility of the radicals in water. The nucleated particles grow by the polymerization of the radicals within them with the monomers absorbed into the particles. The particles can also grow in discrete leaps by coagulating with each other, which occurs due to the instability of the particles. The nonionic surfactants employed in this study have the tendency to partition into the dispersed phases, unlike the ionic surfactants which restrict themselves to the aqueous phase and the interfaces. This aspect results in complications mainly in the nucleation pattern, as was demonstrated previously.16,17,15 Thus, the identification of the best control inputs assumes greater intricacy while employing nonionic surfactants. The series of experiments aimed at analyzing the process sensitivities are described next. Figure 1a shows the base case feed profiles pertaining to the two monomers VAc and BuA (pure components), surfactant solution (22.7 wt % in water), t-BHP solution (3.33 wt % in water), and SFS solution (3.41 wt % in water). The initial mixture consists of 1-L of DI water, 52 g of VAc monomer, and 0.1 g of ferrous ammonium sulfate for the coordination of the redox initiation. Figure 1b shows the evolution of the PSD during the course of the batch, and Figure 1c shows the corresponding profile of total particles. Figure 1c indicates a gradual and prolonged nucleation event, which coupled with the size-dependent growth phenomenon and particle coagulation results in a broad and interconnected distribution at the end of the batch (Figure 1b). Figure 1d shows the profile of solids content, as obtained from gravimetry and from the densimetric calculations pre-

Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 329

Figure 1. Results corresponding to the base case ab initio experiment.

sented in ref 15. It records a modest 21% solids content at the end of the batch. This base case recipe is subjected to various perturbations, and their effects on the process are studied. These studies are used to infer the ability to steer the entire distribution in certain directions and to identify effective control configurations. The experiments and the inferences from them are described below. 2.1. Effects of Surfactant. The experiment described in this subsection was designed to determine the effect of changing the surfactant feed profile on the PSD and the conversion of the monomers. The surfactant feed rate was perturbed as shown in Figure 2a. Figure 2c compares the profiles of total particles in the current and the base case experiments. The total number of particles nucleated in this experiment is much lower (∼1016), and also nucleation is not prolonged, unlike in the previous experiment. This is mainly due to the decrease in the surfactant feed rate between 10 and 15 min (compared to the base case experiment). The decrease in the number of particles in the latex results in larger growth rates, thereby causing the particles to grow to a larger size (Figures 2b). This figure also shows some large size particles (larger than 600 nm), suggesting a coagulation event that could have occurred during the reduced surfactant feed rate. The end-point solids content increases to about 25% at a final overall conversion of about 95% (almost 20% higher than in the base case). This aspect clearly demonstrates the complex

and nonintuitive behavior of the system. Increasing the surfactant feed rate to effect secondary nucleation and a bimodal end-point distribution could cascade into lower growth rates and particle sizes, and hence a lower conversion. This necessitates using other inputs in the process, such as the feed rates of initiators and monomers, to correct this situation. Another aspect of the process that is evident from this experiment is the competitive particle growth phenomenon. This is seen in Figure 2d, which depicts the evolution of the bimodal distribution (number-averaged plot) along the course of the batch. It shows a more interconnected and diffuse distribution at the intermediate time range, which becomes separated into a more clear bimodal distribution toward the end of the batch. This is due to the strongly size-dependent growth, with the larger particles growing faster than the smaller ones. The sizedependent growth and the lack of inputs to manipulate this phenomenon demonstrate the importance of the timings and durations of the various nucleation events. In producing a multimodal distribution, the polydispersity of each mode and the separation between the modes is mainly influenced by the duration and the timings of the nucleation events. 2.2. Effects of MonomersVinyl Acetate. The next experiment was designed to investigate the influence of the feed rate of one of the monomers on the evolution of the PSD and other pertinent outputs. Vinyl acetate feed rate was perturbed as shown in Figure 3a. Figure

330


Figure 2. Effect of a perturbation in the surfactant feed rate relative to the base case experiment.

3c,d compares the profiles of total particles and solids content between the two experiments. Despite the reduced feed rate of the VAc monomer at the start of the reaction, there is no appreciable difference between the two cases at early times (through 30 min) of the batches. A plausible explanation is that the system still remains saturated with the monomers. But unlike in the base case, there is a steep increase in the number of particles at approximately 37.5 min, coinciding with the drop in the VAc feed rate at this time (Figure 3a). One plausible explanation is that all the monomer droplets disappear rapidly from the system, coinciding with the decrease in the feed of VAc at 37.5 min. This results in releasing all the absorbed surfactants back into the aqueous phase and causing rapid nucleation. The end-point distribution (Figure 3b) shows appreciable mass of very small particles, due to the larger number of particles nucleated at the later times and also due to the prolonged monomer addition in this experiment. The solids content in this experiment increases by about 3%. These results clearly show that the monomer feed affects the PSD by influencing not only the rate of growth but also the rate of nucleation. While a monomer-starved condition might decouple the effect of monomer feed on the nucleation process, it would result in reduced growth rates.

2.3. Effects of MonomersButyl Acrylate. Figure 4 depicts results corresponding to a perturbation in the feed rate of BuA monomer (shown in Figure 4a). There is a much reduced nucleation rate in this case, as is seen from the profile of the total particles (Figure 4c), and the nucleation process continues through the course of the batch. The end-point PSD (Figure 4b) shows a signature of coagulation in the form of the large particles (larger than 600 nm). The results suggest a complex dependence of the surfactant partitioning on the monomer composition. One explanation is that the surfactant solubility in the monomer droplets is larger at higher VAc composition in the droplets. 2.4. Effects of Initiator. Figure 5 shows sensitivity results from an experiment in which the concentration of the redox initiator pair (t-BHP and SFS) were doubled relative to the base case, while the feed rates were maintained the same as in Figure 1a. Figure 5b compares the profiles of total particles, which shows that in this case there is a larger nucleation event at the initial times (up to 15-20 min). Thereafter, the total number of particles remains relatively constant until approximately 40 min, at which time there is a second nucleation event. Thus, the end-point distribution is a more prominent bimodal distribution (as seen in Figure 5a). This could be attributed to a larger nucleation event at the early times (by either of micellar or homogeneous


Figure 3. Effect of a perturbation in the VAc feed rate relative to the base case experiment.

mechanisms or both), which depletes the micelles and thereby prevents further nucleation events until approximately 40 min when more micelles are formed. The new batch of micelles could be due to either the increase in the surfactant feed or the drop in the VAc feed (and the associated depletion of the droplets). Thus, there are two clearly distinct nucleation events, contrary to the prolonged micellar nucleation event that characterizes the base case experiment. This observation clearly demonstrates the strong ability to influence the nucleation events through initiator feed rates or composition. Thus, the rate-limiting step in this case is clearly the formation of micelles. However, there is no appreciable effect on growth, as seen by the comparable particle sizes. 2.5. General Perturbation. A combined perturbation of the variables was performed to see its effect on the process. Specifically, the feed of the surfactant solution and the monomers were delayed relative to that shown in Figure 1a, while the VAc monomer in the initial mixture was left unchanged. This result was presented in ref 15 in a related context. Its implication under the present context is the relatively strong sensitivity of the process to even small and inevitable process variations, such as the delays in the pumps, etc. and the inability to correct these errors using feedback control. This result highlights the irreversible nature of the process, and hence the need for very stringent operating practice.

2.6. Implications of the Sensitivity Results for the Control of PSD. In the experiments described above, the effects of the manipulative variables on the PSD and solids content were studied experimentally. Each of the feeds of the surfactant, monomers, and initiators have profound influence on the evolution of the distribution. However, the observed influences are complex and nonintuitive. It is natural to expect the surfactants to affect the nucleation phenomenon and particle stability and the initiators and monomers to affect mainly the growth phenomenon. Thus, it is logical to utilize the feed rates of surfactant and initiator as manipulated variables for the control of PSD, as was done with ionic surfactants by other researchers.5,18 However, in the current case, it is seen that all the reagentsssurfactant, monomer, and initiatorsinfluence the nucleation phenomena, albeit in different ways. Monomer affects the growth phenomenon, while the effect of the initiator on growth is seen to be minimal. Surfactants, whose major role is particle stabilization (emulsification), do affect the coagulation phenomenon. In addition to these direct effects, the variables also have secondary effects, which come into play because of the interaction among nucleation, growth, and coagulation. For instance, the surfactants can affect the growth process indirectly, by affecting the number of radicals/particle and the monomer concentration inside the particles (by varying the particle number). Also, the behavior observed is quite complex and cannot be

332


Figure 4. Effect of a perturbation in the BuA feed rate relative to the base case experiment.

adequately represented in terms of any simple dynamics, even for continuous processes (for example, first order, time delays, and inverse responses). Thus, a detailed model-based optimization and control strategy is appropriate for this process. Another aspect that is revealed in the current study is the strong size dependence seen in the growth phenomenon, which results in a relative broadening of the distribution with growth. Note that this is in contrast to the observations made by Liotta et al.,7 who observed a relative narrowing of seeded bidisperse populations with growth. Modeling studies also support the observation on the broadening of the distribution.19 This is a clear illustration of the ineffectiveness of lumped approaches in certain cases and lends strong support to the distributed parameter route pursued by researchers in recent years.4,3,1,2 This strong size dependence necessitates tight control on the timings, durations, and magnitudes of the nucleation events in producing the desired distributions with particular mean sizes and standard deviations of the modes. Once the nucleation event has occurred, there is very little latitude to alter the competitive growth. A third aspect that is evident from these experiments is the irreversibility that is characteristic of these processes. Each of the nucleation, growth, and coagulation processes exhibit a certain irreversible character. For example, when a unimodal distribution is being produced, if the actual nucleation rate deviated from

the desired rate, the effect of this error on the particle sizes can be corrected by suitably modifying the growth rate (by recruiting multiple process inputs). However, the effect of this error on the breadth of the distribution cannot be rectified (the import of this irreversibility on the breadth of the distributions being dependent on the end applications). Similarly, in producing multimodal distributions, if the nucleation rate for the first nucleation event is erroneous in implementation, this can be corrected (in a relative sense, and bearing with the skewness of the distributions) by correcting not only the growth but also all the subsequent nucleation rates. On the other hand, if the primary nucleation event is implemented as planned, but the secondary nucleation event is erroneously implemented, it might leave an uncorrectable effect on the distribution. Similarly, a larger growth rate might result in larger sizes, which again cannot be corrected (no shrinkage possible with respect to the polymer mass in the particles). A strong coagulation event would also leave an indelible mark on the distribution (see ref 20 for examples of these). The interactive nature between nucleation, growth, and coagulation again advises caution in the control of distributions. From the preceding analysis, it is evident that one must employ multiple inputs for the control of distributions, with the surfactant and VAc monomer being the most suitable inputs. Initiator feed could also be recruited, but mainly to control the nucleation phenom-


subprocesses are controlled. Such a configuration was actually employed in our control studies21 and is akin to the idea of partial control.22 A similar idea is also implicit in the studies of Semino and Ray.9 3. Simulation-Based Reachability Analysis In the previous section, experimental sensitivity results were detailed, based on a perturbation in either the timing of a step change in input or the magnitude of a step change. A more exhaustive sensitivity study would examine the effect of all possible such perturbations on the distribution. This analysis becomes almost impossible experimentally (although the results presented above are some of the most relevant perturbations). In addition to these open-loop studies, it is also of interest to study the effectiveness of feedback control in eliminating the effects of uncertainties and disturbances in the processsa robustness analysis for the open-loop reachable distributions. A comprehensive population-balance model has been developed for this process, both without coagulation and incorporating coagulation events, and validated against experiments.19,23 It models the PSD in terms of a particle density F(r,t) dr, which is defined as the moles of particles of size within a small interval r and r + dr. The PBE is given by

∂ ∂ dr F(r,t) + F(r,t) ) ∂t ∂r dt Rcoag(r,t) + δ(r - rnuc)Rnuc(t) (1)

(

Figure 5. Effect of a perturbation in the initiator concentration relative to the base case experiment.

enon. A monomer-starved (droplet-free) condition might be preferable to render the nucleation event a prerogative of the surfactant feed alone (and the initiator feed, if utilized). However, too low concentrations would result in suboptimal growth rates. Although coagulation could potentially be used to shape the distributions, the best strategy is to minimize coagulation events to the extent possible. Also, the irreversibility considerations and the interactions suggest an hierarchical strategy in which the subprocesses (nucleation, growth, and coagulation) are controlled individually, thereby producing the target distribution. This hierarchical strategy can also be employed within a partial control configuration in which only a subset of these three major

)

where the partial derivative with respect to r accounts for the particle growth and Rcoag(r,t) accounts for the coagulation events. Particle nucleation events, which are restricted to the smallest particle size in the distribution, enter through a boundary condition. A computationally efficient solution method has been developed, which enables rapid simulations of the model.24 This model can be utilized to perform more comprehensive sensitivity studies. First, the open-loop analysis is presented, followed by a robustness analysis to determine the effects of common disturbances and uncertainties. 3.1. Open-Loop Reachability. A typical semibatch recipe is considered, which is divided into 11 intervals of a fixed duration of 11 min each. The feed rates of the reagents are held constant within each interval (piecewise constant input profiles). The surfactant feed rate and the VAc feed rate in the early intervals are varied, and the sensitivities with respect to each of these variables are analyzed. The surfactant feed rate in the first interval is constrained to lie between 0.83 and 4.16 mL/min of 22.7 wt % aqueous solution. While the lower limit is fixed to enable considerable micellar nucleation at early times (there is no surfactant in the intial mixture), the upper limit is based on constraints that allow secondary nucleation. A feed rate above this upper limit would result in the nucleation of a very large number of particles initially, which upon growth would render it impossible to cause the surfactant concentration in the aqueous phase to exceed the critical micelle concentration (which is a prerequisite for causing a micellar nucleation event), subject to the upper constraints on the pumps. The feed rates in the remaining four intervals are constrained only by the limits of the pumps. The VAc feed rate is likewise constrained to lie between set limits. The feed rates of the BuA monomer

334


and the initiator components are fixed at nominal values. Different values of the feed rates of surfactant solution and the VAc monomer in each of these intervals are considered. Simulations are performed for all combinations of the reagent feed rate values. Thus, a total of 7776 simulations are performed in this combinatorial study. The end-point distributions produced in each of these cases are analyzed, by approximating as combinations of Gaussian distributions for the purposes of characterization. Thus, each mode is characterized by a mean size, standard deviation in size, and the total particles in the mode. An analysis of the ratio of the mean diameters of the bimodal distributions (among all the distributions produced) shows a monotonic yet nonlinear dependence on the feed rate of the surfactant in each of these intervals. This suggests that the distributions produced envelope all the attainable distributions (subject to the discretization of the input profiles, the constraints imposed, and the modeling errors). Thus, one can draw an envelope of the reachable domains around these points, which characterize the exactly reachable distributions. Due to several reasons including model uncertainties and measurement noise, it is advisable to allow a tolerance value on the attainable proximity to the target distribution. Wang and Doyle III14 factor this aspect into their study by defining an -reachability, which is the reachability of distributions within a defined tolerance (contrasted against the exact reachability). These -reachable domains can be easily inferred from the exactly reachable distributions identified in the present approach (by drawing a circle around each of the data points, the radius of which represents the margin ). Figures 6 and 7 depict results pertaining to these open-loop simulations. Figure 6 characterizes the bimodal distributions produced in these simulations in terms of the means and the standard deviations of the two modes. This figure depicts the types of reachable bimodal distributions. It shows that as the size of the smaller (secondary) mode increases, the standard deviation of this mode increases, resulting in a broad distribution (also with a reduced total particles in this mode). However, one can produce a larger average size in the larger mode and still obtain a relatively lower standard deviation. This case actually corresponds to clearly separated bimodal distributions (as seen in the right end of the plot in Figure 7a). Figure 7a shows a plot of the diameter ratio of the bimodal distribution, which shows restrictions on the diameter of the primary mode for a given diameter of the secondary mode. In general, as the diameter of the smaller mode increases, the range of reachable diameters of the larger mode increases. A suitable strategy for the control of distributions, as was highlighted in the Experimental Section, is to control the individual processes of nucleation, growth, and coagulation separately. Also, as was pointed out earlier, in emulsion polymerization, it is desirable to minimize the problematic coagulation events, utilizing the nucleation and growth events as manipulated variables for the control of PSD. Thus, it is of interest to analyze the attainable distributions in the absence of coagulation events. Figure 7b shows a plot of the diameter ratio of the attainable bimodal distributions under coagulation-free conditions, generated utilizing a coagulation-free model.19 In the absence of the coagulation events which contribute to discrete growth in the particle size, the attainable size of the smaller mode is

Figure 6. Bimodal distributions produced with surfactant and VAc monomer as the manipulated variables.

much reduced. Also, the attainable range of size for the larger mode is less sensitive to the size of the smaller mode, and the size of the larger mode itself is also lower. A direct parallel to the ideas of attainability region determination described by other authors10,11,7 does not exist for this complex and intricate process. However, simulations were used to determine the lower and upper limits in the attainable profiles of total particles and solids content (see ref 25 for plots). These show that the profile of solids content that can potentially be followed is relatively narrow, and the attainable end-point solids content lies in the range of 19-23.5% for the input profiles considered. These plots also show that the rates diminish toward the end of the batch, suggesting that these reachable distributions are retained beyond the end point of the distribution. The reachable distributions are useful in setting proper targets for the optimization problem involved in designing a recipe to achieve a target end-point distribution. Even though the target PSD is dictated by the end-use application of the emulsion latex, the reachability results aid in identifying if this target PSD is achievable and, if not, to identify the achievable distribution that is closest to the target (the point that is closest to the target in these figures, based on a Euclidean distance, for example). Similarly, the profiles of the total particles and solids content aid in setting proper targets for the optimization problem involved in designing a recipe that tracks a particular PSD trajectory in leading to a target end-


Figure 7. Effect of coagulation on the diameter ratio of the attainable distributions with surfactant and VAc monomer as the manipulated variables.

point PSD. Such a problem is presented in ref 21, in which the optimization problem was formulated in a multiobjective framework. A hierarchical control idea, based on the control of the individual nucleation, growth, and coagulation events, was employed. Also, the partial control idea, based on the elimination or minimization of the coagulation events, was utilized. The distribution trajectory was re-cast as equivalent trajectories of nucleation and growth rates, which in turn were converted into equivalent trajectories of total particles and solids content, employing the idea of the control of instantaneous properties. When the piecewise constant feed rates of surfactant alone is employed as design variables, the attainable limits for the end-point solids content experiences an upward shift in the lower limit. However, the attainable region of the distributions retains its shape (see ref 25 for plots). This result suggests a potential for mid-course correction, wherein an error introduced by the uncertainty in the feed rate of surfactant can be corrected by manipulating the monomer feed rate and vice versa. This result is in perfect agreement with the findings from the experimental studies presented previously, on the interaction between nucleation and growth, and the effects of surfactant and VAc on the process. To re-visit the scenario that was presented among the experimental results, a lower nucleation rate (caused by a lower than intended feed rate of the surfactant) would necessitate a decrease in the growth rate by reducing the monomer feed (to offset the interaction between nucleation and growth). Thus, one could prevent the particles

from growing to a larger size than the target, which would have occurred had no correction been made in the monomer feed. But this correction in the size by a decrease in the monomer feed would result in a lower solids content. The sensitivity of these results to the input parametrization (in particular, the duration of the zero-order holds) was examined. It was seen that a larger interval at early times results in very large primary nucleation rates, particularly at larger surfactant feed rates in these intervals. This deprives the system of its ability to cause a secondary nucleation event (for which it needs to breech the critical micelle concentration barrier). This clearly highlights the importance of allowing small enough intervals in the process, particularly for openloop optimization, in producing multimodal distributions. One can also pose the problem of allowing the duration of the intervals as additional optimization variables. It is pertinent to reiterate that, in this study, a uniform gridding of the inputs was adopted, and every possible combination of inputs were simulated, to identify the reachable distributions. These distributions identified are exactly reachable, from which reachable regions based on different definitions can be deduced. However, one could instead use a statistical design of experiments approach26 to sample the reachable space (and identify similar exactly reachable distributions). This might lead to a considerable reduction in the number of simulations needed to be performed in arriving at the reachable distributions. 3.2. Robustness to Uncertainties and Disturbances. The simulation studies presented above examine the open-loop attainability of distributions in semibatch processes. However, in addition to the inevitable model uncertainties, the process is characterized by several disturbances. This section examines the alterations to the attainable distributions introduced by three most common disturbances: first, the latex carryover, second, the seeded polymerization (large latex carry-over), and third, a model/parametric uncertainty which is rectified mid-course during the batch. Batch-to-batch latex carryover and the associated uncertainty in the initial condition is a prevalent issue in industrial practice, motivating the analysis of its effect on the reachable distributions. Figure 8 examines the effect of an uncertainty in the initial conditions on the reachable distributions, based on an arbitrary initial distribution (obtained from one of the experiments) with an initial solids content of less than 0.5%. Both VAc and surfactant feeds were considered as manipulated variables, and the coagulation effects were considered. Though most of the nominal reachable regions are covered in this perturbed case, there are other possible distributions that can be reached in the face of this uncertainty in the initial condition (particularly with respect to the secondary (smaller) mode, as seen in Figure 8c,d). The reason for the strip of large sizes in Figure 8b is that, at low surfactant feed rates at the early intervals, the initial particles prevent any primary nucleation event by taking up most of the surfactants, and also grow rapidly with the high concentration of the monomers, until the nucleation of the secondary mode. The nucleation of the secondary mode is advanced in these cases, accounting for the larger particles (larger than 150 nm) in the secondary mode (Figure 8c compared with Figure 8d). At higher feed rates of surfac-

336


Figure 8. Effect of the initial distribution (of small mass) on the reachable bimodal distributions.

tants in the initial intervals, the new particles nucleated dominate the particles in the initial batch, thereby preventing them from being evident in the end-point distribution. The implication of these results is that running a batch in open loop with pre-optimized inputs, in the face of latex carryover, might result in considerable difference in the resultant distribution compared to the target. Thus, one needs to employ in-batch feedback control, which can correct these errors and drive the distribution toward the target. However, even though feedback can be used to bring the distribution back to the target (as seen by the nominal region lying entirely within the perturbed region in Figure 8), the large particles are still present (in negligible quantity). In the perturbed case, the entire lower limit on the attainable profile of solids content is reduced relative to the nominal case (Figure not shown). However, the upper limit essentially follows the nominal case except at the early times (where both limits are above the nominal case due to the initial particles). This again is a result that indicates that multiple manipulated variables can bring the distributions back to the target (in a relative sense), although one might have to sacrifice performance on the solids content tracking. Figure 9 presents results to examine the effect of a larger mass of initial particles, but with a much reduced particle sizes and a narrower distribution. This case can be considered either as a larger initial disturbance with over 1% solids content or as seeded emulsion polymerization. Figure 9 shows the reachable bimodal distributions in the perturbed case, while the reachable distributions corresponding to the nominal case are shown

in Figure 6. These results indicate an inverse effect from what was seen in the previous case with a lower mass of initial particles (latex carryover). In the current case, the perturbed distributions cover a smaller domain than the nominal case. Thus, in-batch feedback has a limited utility in this case. For example, very large sizes of the larger mode is not possible (Figure 9b compared with Figure 6b), as the seed particles add to the nucleated particles, thereby causing reduced growth rates and reduced particle sizes. The distributions with very large sizes in the larger mode (larger than 250 nm, Figure 6b) seen in the nominal case correspond to low surfactant feed rates in the early intervals and large enough feed rates at the later intervals to cause a secondary nucleation event. The particles with the larger size range in the smaller mode (above 80 nm in Figure 6a) are also not possible due to the cascaded effect of the seed and growth on the nucleation event. Figure 10 presents results to examine the effect of an early disturbance in the process that is removed midcourse, on the reachable PSDs. The perturbation introduced could be due to several reasons, including model and parameter uncertainties, implementation errors, and shear-induced coagulation, which could be a stochastic effect. This analysis is complementary to the latex carryover-related initial condition disturbance analysis presented in the previous two cases, which cannot be rectified mid-course (the particles carried over are always there). Figure 10a shows the perturbed distribution relative to the nominal one, at 22 min in the batch. This corresponds to a time when a nucleation event is underway (as seen by the large peak of particles


Figure 9. Comparison of the reachable bimodal distributions in the seeded case with those in the nominal (ab initio) case (Figure 6).

in the smallest end). Figure 10b shows the end-point distributions that are reached in the nominal case. Assuming that after 22 min there are no further stochastic effects, or that the source of the uncertainty that caused the deviation in the distribution at 22 min has been removed, the resultant distributions at the end of the batch are shown in Figure 10c, which shows a very profound difference when compared to Figure 10b, even for such a small variation. (Note that all the other process states are assumed to be unaltered between the nominal and perturbed cases). Figure 10d characterizes the attainable bimodal distributions in terms of the mean and the diameter ratio. The nominal reachable region is much more concentrated than the reachable region corresponding to the perturbed distribution. Given the monotonicity with respect to the inputs’, one could conclude that at least part of the nominal region lies within the perturbed region. This region of intersection is amenable to feedback correction, while a target in the complementary region (to the perturbed region) cannot be attained after this mid-course disturbance. In the latter case, if the source of this uncertainty is a deterministic disturbance, then the best recourse is to embark on a batch-to-batch control strategy.27,28 On the other hand, if it is caused by stochastic effects, one has to perform robust design, say based on worst-case considerations.29 4. Summary The sensitivities in the process of the evolution of PSD in semibatch emulsion copolymerization using nonionic

surfactants were studied via experiments and simulations. The experimental studies indicate the following: (1) The need for multiple process inputs for the control of PSD. (2) Surfactant and a monomer (in this case the primary monomer VAc) as the inputs, particularly when employing nonionic surfactants. (3) The suitability of a hierarchical control strategy in which the individual rates of nucleation, growth, and coagulation are controlled to produce the desired complete distribution. (4) Irreversibility in several aspects of the processs while some of these such as the skewness of the distributions may not be critical depending upon the end-use applications, others such as the effect of a larger growth rate or a stronger coagulation rate might leave an intolerable deviation in the distribution. (5) Inherent limitations within the process that restrict the type of distributions that can be produced. (6) The limitations in the process that in some cases translate into lower solids content (and hence have a bearing on the economy of the process), in employing feedback to correct the correctable errors in the distribution. (6) The need for a detailed first-principle model to be employed for the open-loop and closed-loop control of PSD. The simulation studies were used to gain further insights into the potentials and limitations in the process and to determine the type of distributions that can actually be produced considering system and external limitations. A controllability analysis based on a linearized model is unsuitable due to the discontinuity in the process and its highly nonlinear character. A rigorous mathematical analysis of the reachability and controllability is also beyond the reach of this process, due to the underlying complexity. Thus, a simple simulation-based analysis was performed, bearing in mind the practicality or the operability of the process. The study gave insight into the restrictions on the types of distributions that can be produced and also revealed the effect on these restrictions under different control configurations. The effect of a partial control configuration (with coagulation not being recruited as a manipulated variable for control of PSD) was examined. The total control configuration, which controls all the three major subprocesses, was seen to produce a much wider class of distributions, even though this configuration is subject to questions about the feasibility of enabling tight control on the coagulation events. The study identified the types of exactly reachable distributions (currently restricted to a low solids regime). Key aspects of the distributions (such as the mean diameter ratio of bimodal distributions) were found to have a monotonic (though nonlinear) dependence on the inputs. On the basis of this observation, reachability domains are determined by envelopes around these reachable points. A particular discretization of the inputs along the batch and a fixed reaction time were considered in this study. Relaxation of these restrictions were examined, which gave further insight into the formulation of the problem of optimization and control of PSD. Further, the effect of uncertainties and disturbances on the reachable distributions were analyzed. The results reveal two different scenarios: (1) One in which in-batch feedback control is feasible despite the internal limitations (irreversibility) and

338


Figure 10. Characterization of the distributions that result from the nominal case, and from the case wherein the source of an early disturbance has been removed mid-course (at 22 min into the batch spanning 121 min).

external limitations (constraints, sparse and delayed measurements, etc.). (2) Another in which in-batch feedback control might not be effective in all cases in correcting the errors, thereby advocating batch-to-batch control. These situations suggest a combination of in-batch and batch-to-batch feedback control strategy for the control of PSD in semibatch emulsion polymerization. There is also enormous potential for robust optimization formulations, which take explicit account of potential uncertainties and disturbances in performing a conservative open-loop optimal design of the process. Acknowledgment The authors acknowledge collaboration and support from Dr. Cajetan F. Cordeiro of Air Products and Chemicals Inc. and Dr. Yang Wang at the University of California, Santa Barbara. Financial support from the University of Delaware Competitive Fellowship, the Office of Naval Research, and the University of Delaware Process Control and Monitoring Consortium are also gratefully acknowledged. Literature Cited (1) Braatz, R. D.; Hasebe, S. Particle Size and Shape Control in Crystallization Processes. In AIChE Symposium Series: Chemical Process Control-VI; Rawlings, J. B., Ogunnaike, B. A., Eaton,

J. W., Eds.; AIChE: New York, 2001; Vol. 98 of 326, pp 307-327. (2) Christofides, P. D. Control of Nonlinear Distributed Parameter Systems: An Overview and New Research Directions. AIChE J. 2002, 341-346. (3) Daoutidis, P.; Henson, M. A. Dynamics and Control of Cell Populations in Continuous Biorroeact. In AIChE Symposium Series: Chemical Process Control: VI; Rawlings, J. B., Ogunnaike, B. A., Eaton, J. W., Eds.; AIChE: New York, 2001; Vol. 98 of 326, pp 274-289. (4) Doyle, F. J., III; Soroush, M.; Cordeiro, C. Control of Product Quality in Polymerization Processes. In AIChE Symposium Series: Chemical Process Control-VI; Rawlings, J. B., Ogunnaike, B. A., Eaton, J. W., Eds.; AIChE: New York, 2001; Vol. 98 of 326, pp 290-306. (5) Crowley, T. J.; Meadows, E. S.; Kostoulas, E.; Doyle, F. J., III. Control of Particle Size Distribution described by a Population Balance Model of Semibatch Emulsion Polymerization. J. Process Control 2000, 10, 419-432. (6) Meadows, E. S.; Crowley, T. J.; Immanuel, C. D.; Doyle, F. J., II. Non-isothermal Modeling and Sensitivity Studies for Batch and Semi-batch Emulsion Polymerization of Styrene. Ind. Eng. Chem. Res. 2003, 42 (3), 555-567. (7) Liotta, V.; Georgakis, C.; Sudol, E. D.; El-Aasser, M. S. Manipulation of Competitive Growth for Particle Size Control in Emulsion Polymerization. Ind. Eng. Chem. Res. 1997, 36, 32523263. (8) Gatzke, E. P.; Doyle, F. J., II. Model predictive control of a granulation system using soft output contstraints and prioritized control objectives. Powder Technol. 2001, 121, 149-158. (9) Semino, D.; Ray, W. H. Control of Systems Described by Population Balance Equations - I. Controllabity Analysis. Chem. Eng. Sci. 1995, 50 (11), 1805-1824.

Ind. Eng. Chem. Res., Vol. 43, No. 2, 2004 339 (10) Glasser, D.; Hildebrandt, D.; Crowe, C. A geometric approach to steady flow reactors: The attainable region and optimization in concentration space. Ind. Eng. Chem. Res. 1987, 26, 1803-1810. (11) Smith, R. L.; Malone, M. F. Attainable regions for polymerization reaction systems. Ind. Eng. Chem. Res. 1997, 36, 10761084. (12) Russel, B. M.; Henriksen, J. P.; Jorgensen, S. B.; Gani, R. Integration of design and control through model analysis. Comput. Chem. Eng. 2002, 26, 213-225. (13) Kauchali, S.; Rooney, W. C.; Biegler, L. T.; Glasser, D.; Hildebrandt, D. Linear programming formulations for attainable region analysis. Chem. Eng. Sci. 2002, 57, 2015-2028. (14) Wang, Y.; Doyle, F. J., III. Reachability Studies for Particle Size Distribution in Emulsion Polymerization. In preparation, 2003. (15) Immanuel, C. D.; Crowley, T. J.; Meadows, E. S.; Cordeiro, C. F.; Doyle, F. J., III. Evolution of Multi-Modal Particle Size Distribution in Vinyl Acetate-Butyl Acrylate Emulsion Co-Polymerization. J. Polym. Sci., Polym. Chem. 2003, 41 (14), 2232-2249. (16) Piirma, I.; Chang, M. Emulsion Polymerization of Styrene: Nucleation Studies with Nonionic Emulsifier. J. Polym. Sci.: Polym. Chem. Ed. 1982, 20, 489-498. (17) Ozdeger, E.; Sudol, E. D.; El-Aasser, M. S.; Klein, A. Role of Nonionic Surfactant Triton X-405 in Emulsion Polymerization. III. Copolymerization of Styrene and n-Butyl Acrylate. J. Polym. Sci., Part A: Polym. Chem. 1997, 35, 3837-3846. (18) Flores-Cerrillo, J.; MacGregor, J. F. Control of Particle Size Distribution in Emulsion Semi-batch Polymerization using Midcourse Correction Policies. Ind. Eng. Chem. Res. 2002, 41, 18051814. (19) Immanuel, C. D.; Cordeiro, C. F.; Sundaram, S. S.; Meadows, E. S.; Crowley, T. J.; Doyle, F. J., III. Modeling of Particle Size Distribution in Emulsion Co-Polymerization: Comparison with Experimental Data and Parametric Sensitivity Studies. Comput. Chem. Eng. 2002, 26 (7-8), 1133-1152. (20) Immanuel, C. D.; Doyle, F. J., III. Open-loop Control of Particle Size Distribution in Semi-batch Emulsion Co-Polymeri-

zation using a Genetic Algorithm. Chem. Eng. Sci. 2002, 57 (20), 4415-4427. (21) Immanuel, C. D.; Doyle, F. J., III. A Hierarchical Strategy for Control of Particle Size Distribution Using Multi-Objective Optimization. AIChE J., in press. (22) Kothare, M. V.; Shinnar, R.; Morari, M. On defining the partial control problem: Concepts and examples. AIChE J. 2000, 46, 2456-2474. (23) Immanuel, C. D.; Cordeiro, C. F.; Sundaram, S. S.; Doyle, F. J., III. Population Balance PSD Model for Emulsion Polymerization with Steric Stabilizers AIChE J. 2003, 49 (6), 1392-1404. (24) Immanuel, C. D.; Doyle, F. J., III. ComputationallyEfficient Solution of Population Balance Models Incorporating Nucleation, Growth and Coagulation. Chem. Eng. Sci. 2003, 58 (16), 3681-3698. (25) Immanuel, C. D. Modeling, Analysis and Control of Particle Size Distribution in Semi-batch Emulsion Polymerization. Ph.D. Thesis, Department of Chemical Engineering, University of Delaware, Newark, DE, 2002. (26) Asprey, S. P.; Macchietto, S. Defining robust optimal dynamic experiments. J. Process Control 2002, 12, 545-556. (27) Crowley, T. J.; Harrison, C. A.; Doyle, F. J., III. Batchto-batch Optimization of PSD in Emulsion Polymerization using a Hybrid Model. In Proceedings American Control Conference, Arlington, VA, 2001; pp 981-986. (28) Lee, K.; Lee, J. H.; Yang, D. R.; Mahoney, A. W. Integrated run-to-run and on-line model-based control of particle size distribution in semi-batch precipitation reactor. Comput. Chem. Eng. 2002, 26 (7-8), 1117-1131. (29) Ma, D. L.; Chung, S. H.; Braatz, R. D. Worst-Case Performance Analysis of Optimal Batch Control Trajectories. AIChE J. 1999, 45 (7), 1469-1476.

Received for review February 18, 2003 Revised manuscript received June 2, 2003 Accepted June 6, 2003 IE030145P

A Sensitivity Approach to Reachability Analysis for Particle Size

Recommend Documents