Elsevier

Computers & Chemical Engineering

Volume 127, 4 August 2019, Pages 11-24
Computers & Chemical Engineering

A reduced-order multiscale model of a free-radical semibatch emulsion polymerization process

https://doi.org/10.1016/j.compchemeng.2019.04.029Get rights and content

Abstract

Free-radical emulsion polymerization is a heterogeneous process where simultaneous and competitive physicochemical events occur over a wide range of time/length scales. Although a highly accurate representation of the process is possible by multiscale modeling, common approaches face several issues, including the stochastic nature of finer scales, the timely exchange of information between scales, and the high computational load for the model solution. In this work, a reduced-order computationally-tractable multiscale model is proposed while preserving a good predictive capability. The model integrates microscopic-scale calculations based on kinetic Monte Carlo simulations (stochastic), a mesoscopic-scale representation of the particle size distribution through a novel statistical approach and a deterministic description of the macroscopic-scale. The proposed model resulted in faster satisfactory predictions (compared to the model based on the Fokker Planck Equation) of traditional macro and mesoscopic variables, along with the average number of free radicals and secondary nucleation on a microscopic scale.

Introduction

Multiscale modeling is a term employed in several fields which commonly refers to a specific set of methods used for simultaneously describing the dynamics of a system at a different time and length scales (Keil, 2012). For lumped processes, Ordinary Differential Equations (ODEs) are adopted to describe the time evolution of the system. For distributed processes, hyperbolic/parabolic Partial Differential Equations (PDEs) are preferred to describe not only the temporal but also the spatial evolution of the system. For instance, Navier-Stokes and Population Balance Equations (PBEs) are the basis for the design and optimization of a wide variety of dynamic processes (Christofides, 2001). In contrast, for processes where important phenomena at microscopic scales take place, multiscale approaches coupling continuum-type lumped/distributed parameter models with Molecular Dynamics (MD) or kinetic Monte Carlo (kMC) simulations are implemented because of their ability to describe phenomena that are inaccessible using macroscopic continuum laws and equations (Vlachos, 2005).

All physical and chemical phenomena are actually multiscale processes, because the macroscopic behavior observed at human scale is the result of the interaction of elementary particles, either at the microscopic, molecular, atomic or sub-atomic scales. Modeling these types of processes requires understanding the behavior of the system at each scale having a relevant effect on the overall phenomenon. Accurate and efficient multiscale models are required for a better prediction of the behavior of the studied system, as well as for the design of better optimization/control strategies. Given all the details and specificity found for each multiscale system of interest, they must be considered as particular case studies (Christofides, Armaou, 2006, Crose, Kwon, Nayhouse, Ni, Christofides, 2015, Crose, Kwon, Tran, Christofides, 2017, Majumder, Broadbelt, 2006, Rasoulian, Ricardez-Sandoval, 2014, Rasoulian, Ricardez-Sandoval, 2015, Varshney, Armaou, 2008). In this paper, the modeling of a particular free-radical Emulsion Polymerization (EP) process for the synthesis of structured polymer particles is considered as a case study.

PDE/ODEs - kMC-based multiscale modeling approaches have been receiving a lot of attention in the international research community in the last few years in order to be used in optimization and control tasks (Christofides, Armaou, 2006, Crose, Kwon, Nayhouse, Ni, Christofides, 2015, Crose, Kwon, Tran, Christofides, 2017, Kimaev, Ricardez-Sandoval, 2017, Majumder, Broadbelt, 2006, Rasoulian, Ricardez-Sandoval, 2014, Rasoulian, Ricardez-Sandoval, 2015, Varshney, Armaou, 2008). There is a significant amount of publications on multiscale processes modeling, optimization, and control, including some recent reviews (Braatz, Alkire, Seebauer, Rusli, Gunawan, Drews, Li, He, 2006, Ricardez-Sandoval, 2011, Vlachos, 2012) in which the main tools available to address this kind of problems as well as some open research questions are well explained. It is important to remark that because of the curse of dimensionality associated to the multiscale models (Kimaev, Ricardez-Sandoval, 2017, Ricardez-Sandoval, 2011), model-order reduction techniques have been applied in several of the optimization/control PDE/ODEs - kMC-based approaches (Chaffart, Rasoulian, Ricardez-Sandoval, 2016, Chaffart, Ricardez-Sandoval, 2017, Chaffart, Ricardez-Sandoval, 2018, Kwon, Nayhouse, Christofides, Orkoulas, 2013, Rusli, Drews, Ma, Alkire, Braatz, 2006). Model-order reduction has been mainly focused on the development of a closed-form model for the microscopic-scale by using statistical modeling techniques in order to either reduce the model computational load or to quantify the associated uncertainty due to the stochastic nature of the kMC simulations. However, less attention has been paid to the dimensional reduction of the macro- and meso-scopic scales in order to reduce the computational load by requiring less kMC executions, which indeed would be another alternative to deal with the curse of dimensionality. The latter is the kind of approach pointed out in this work.

Different interesting processes from the chemical engineering point of view have been addressed from a multiscale modeling perspective, such as: the manufacture of on-chip copper inter- connections by electro-chemical deposition of copper into a trench (Braatz, Alkire, Seebauer, Rusli, Gunawan, Drews, Li, He, 2006, Li, Drews, Rusli, Xue, He, Braatz, Alkire, 2007, Rusli, Drews, Ma, Alkire, Braatz, 2006), the growth mode transitions that occur during physical vapor deposition of thin films from a fluid in a vertical, stagnation-flow geometry (Christofides, Armaou, 2006, Lam, Vlachos, 2001); concentration variations in the fluid in two dimensions for catalytic flow reactor (Majumder, Broadbelt, 2006, Ulissi, Prasad, Vlachos, 2011); batch crystallization process to produce tetragonal hen-egg-white lysozyme crystals (Kwon, Nayhouse, Christofides, Orkoulas, 2013, Kwon, Nayhouse, Christofides, Orkoulas, 2013, Kwon, Nayhouse, Christofides, Orkoulas, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014), among others.

In the chemical vapor deposition process, particularly, during the epitaxial thin film growth process in the stagnation point flow chamber, the evolution of the surface microstructure is captured through kMC simulations which represent the surface roughness trajectories as a combination of absorption, desorption and migration stochastic events (Christofides and Armaou, 2006). The multiscale nature of the process has been captured using nonlinear PDEs embedded with lattice-based kMC simulations. Moreover, the control problem of obtaining thin films with uniform surface roughness had been solved based on this PDE-kMC dynamical model (Rasoulian, Ricardez-Sandoval, 2015, Rasoulian, Ricardez-Sandoval, 2016).

On the other hand, stochastic fluctuations in the microscopic-scale has been handled in a crystallization process, particularly, in the crystal growth of the tetragonal form of hen-egg-white lysozyme. Kwon, Nayhouse, Christofides, Orkoulas, 2013, Kwon, Nayhouse, Christofides, Orkoulas, 2013 modeled and controlled this protein crystallization process taking into account both nucleation and growth rates. In that work, the growth of each crystal was simulated via kMC comprising adsorption, desorption and migration events on the (110) and (101) crystal faces. PBE were used to describe the evolution of the particle size and shape distribution inside the reactor. Kwon, Nayhouse, Christofides, Orkoulas, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014 coupled the PBE and the kMC simulation with the mass and energy balances obtaining a multiscale representation for the crystallizer. Then, the multiscale model was used for the design of a Model Predictive Control (MPC) scheme to regulate the average shape of the crystal population in a continuous operation mode.

In summary, well-defined approaches for modeling each scale of interest in a variety of processes have been developed (D’hooge, Steenberge, Reyniers, Marin, 2016, Gooneie, Schuschnigg, Holzer, 2017). Additionally, several algorithms to simulate multiscale systems have been reported (Keil, 2012, Kwon, Nayhouse, Christofides, 2015, Xie, Liu, Luo, 2018, Xie, Luo, 2017), where the latest developments are addressed to guarantee the numerical stability and reduction of the computational cost. However, despite the similarities between the crystallization and EP process for synthesizing structured polymer particles, the latter has not yet been tackled by means of a multiscale approach that simultaneously takes into account scales below the mesoscopic-scale, e.g., the microscopic-scale through a kMC simulation. One aspect of the crystallization processes which has received a lot of attention is the simultaneous modeling of the size and shape of the formed crystals (Kwon, Nayhouse, Christofides, Orkoulas, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014, Kwon, Nayhouse, Orkoulas, Christofides, 2014). However, it is important to mention that the crystals shape is different to the non-crystal polymers (which is the case of the polymers particles obtained from an EP process.) and that, as the result of surface Gibbs-free energy minimization, the polymer particle shape tends to be spherical or elliptical (Sundberg and Durant, 2003).

The most common approaches for modeling and simulating processes containing a dispersed media (as is the case for EP process) represent the studied system as a set of coupled ODEs-PDEs. However, such approximations are based on the assumption that the system behaves as a continuum entity, which does not represent closely the actual phenomenon. A suggested approximation for overcoming this is to combine continuum models with coarse models by assuming that the system is not a continuum but a collection of a large number of particles. Thus, allowing to observe the behavior of each of those particles separately, i.e., multiscale approaches (Matous et al., 2017). For the specific case of EP processes, the most recurrent approach employed for modeling a particular EP process couples mass and energy balances with PBE (Crowley, Meadows, Kostoulas, III, 2000, Hosseini, Bouaswaig, Engell, 2012, Kiparissides, 2006, Sweetman, Immanuel, Malik, Emmett, Williams, 2008). This type of approaches have shown to be very effective only in the simplest case of pure growth processes (Sheibat-Othman et al., 2017).

However, the real picture is more complicated because the particle size is not the only factor which distinguishes the particles from each other. Besides the size of the particle, the characteristic length, volume, mass, age, composition, and other characteristics of an entity in a distribution should be taken into account (Gunawan et al., 2004). This can be achieved by considering any of the above particle characteristics as internal coordinates to describe the evolution of the Particle Size Distribution (PSD). The main issue in doing so is that by taking into account additional internal coordinates leads to a multidimensional PBE system. Main drawbacks of this are: i) reaching a solution is very challenging, still an open research problem up to four internal variables or dimensions (Gunawan, Fusman, Braatz, 2004, Reinhold, Briesen, 2015), ii) the stochastic nature of the evolution of the internal variables is still neglected (Hosseini et al., 2012), and iii) it is still not clear how to validate multidimensional PBE-based while models from different scales can be separately validated and afterwards integrated in a coupled simulation scheme (D’hooge et al., 2016). For instance, an alternative approach for solving the problem could be extending the model by incorporating additional scales and representing the evolution of the system in those scales through stochastic simulation approaches as Brownian Dynamics or kMC simulations (Hernández and Tauer, 2008a).

Modeling EP processes is a challenging task because the rate of events that take place in this process range from about 100 to 109 s1 and involve entities of very different length scales, such as ions and molecules ( < 1nm), macromolecules (110nm), polymer particles (10nm1μm) and monomer droplets ( > 1μm) (Hernández and Tauer, 2008b). Those difficulties in the modeling of EP reactors coupled with the lack of robust online measurements for critical process parameters have forced practitioners to use several empirical or semi-empirical equations. Furthermore, industrial EP reactors suffer from process variability, that usually makes the process irreproducible from batch to batch (Dimitratos et al., 1994). This variability includes random disturbances in the process operating conditions, which in the EP case corresponds to stochastic fluctuations occurring at the finest scales (D’hooge et al., 2016). For example, fluctuations of the particle state around the mean size cause a stochastic broadness in the PSD (Hosseini et al., 2012). For all these reasons, a very precise representation of the process is only possible if different simulation techniques are integrated into a multiscale simulation approach (D’hooge et al., 2016).

The integration of different scales in emulsion polymerization was initially done by incorporating the meso-scale of particle size distribution in the macroscopic polymerization model (Dimitratos et al., 1994). Almost in parallel, the micro-scale of radical dynamics and chain growth was considered using Gillespieâs stochastic simulation algorithm (Gillespie, 1976), also known as kMC simulation. kMC was originally used for modeling the molecular weight distribution of polymers (Tobita et al., 1994), although without a multiscale integration. Integration of polymerization models at different scales, including the micro-scale was explored for both emulsion (Hernández, 2008) and mini-emulsion polymerization (Rawlston, 2010). However, integrating the micro-, meso- and macro-scopic scales in a single model is still challenging, particularly from the computational load point of view.

Up-to-date modeling approaches for describing the dynamic evolution of polymerization processes include sophisticated descriptions at the macro- and micro-scales (Xie, Liu, Luo, 2018, Xie, Luo, 2017, Yan, Luo, Guo, 2011, Yao, Su, Luo, 2015). Full Computational Fluid Dynamic (CFD) simulations have been also developed, which additionally include the turbulence characteristic of the fluid pattern (Xie, Liu, Luo, 2018, Xie, Luo, 2017). All these approaches result in an accurate representation of the studied phenomenon. However, they might be seen as impractical for carrying out model-based optimization and control tasks in real-time implementations. Furthermore, it must be noticed that selecting an adequate solution strategy for integrating continuum with coarse models (in order to exchange information among the scales) is a very important task(Gooneie et al., 2017). Such exchange of information could lead to numerical instability during the numerical solution of the problem (Ricardez-Sandoval, 2011). Additionally, numerical solution of the multiscale models imposes several restrictions as high computational load or the need for parallelizable numerical schemes, not to mention that those models can hardly be used for subsequent tasks such as process optimization and/or control because of the computational load and the lack of a closed-form of the scales below the mesoscopic-scale (Xie and Luo, 2017).

In this work, the micro-, meso- and macro-scales are combined into the framework of ODE/PDE - kMC multiscale models to simulate the synthesis of core-shell particles by emulsion polymerization. The multiscale model is proposed attempting to keep a good balance between model predictions accuracy and tractability. Therefore, the proposed model may be a proper tool for being used in model-based-optimization and control applications. The multiscale model integrates microscopic-scale calculations based on kMC simulations (stochastic). Mesoscopic-scale representation of dispersed media is carried out by using variance algebra concepts for describing the stochastic nature of particle growth. This is precisely the main step towards the construction of the reduced-order model. On the other hand, the macroscopic-scale is described based on mass and energy balances (deterministic). The developed model predicts the traditional macro- and mesoscopic variables, along with the average number of free radicals and the secondary nucleation rate at the microscopic scale, with a low computational load. In consequence, this work develops a reasonable computational model while preserving a good level of detail in the formulation. Although the macroscopic-scale description is kept simple, it must be noticed that the modelling effort is on the microscopic-scale representation (where an integration of chemical, colloidal and hydrodynamic events is performed using the kMC technique (Hernández and Tauer, 2008a).), as well as in the mesoscopic-scale (where a reduced-order model is proposed for describing the mean and standard deviation of the particle size distribution by means of only two ODEs), and in an efficient scales integration. Taking into account these key points, the model might be used for implementing a control strategy towards improving the process productivity while avoiding secondary particle nucleation.

Finally, it is important to remark that the usual approach for modeling the meso-scale based on PBE requires discretization of the particle radius for all the particles in the whole domain for which the equations system will be solved, which increases exponentially the required computational load for solving the problem. For example, let us suppose that it is decided to discretize the domain in N-points. Also, in order to reach a good approximation to the number of free radicals, the kMC (micro-scale) should be run X-times at each discretization point. Therefore, for solving the kMC, N × X executions must be performed. Assume that 50 points are selected for discretization and that the kMC will be run 50-times. In such case, the kMC would be executed 2500-times, increasing the computational load drastically. In contrast, by using the reduced-order model instead of the PBE, the kMC would be run only for one value of the particle size (i.e. just 50-times in our hypothetical example). This represents an important reduction on the computational time required for solving the problem, which is the main reason why it is suggested to use the proposed reduced-order model for optimization and control applications.

This paper is organized as follows. The process description and multiscale mathematical model deduction are presented in Sections 2 and 3, respectively. The algorithm for numerically solving the multiscale mathematical model is presented in Section 4. Simulation results are reported in Section 5. Finally, conclusions are presented in Section 6.

Section snippets

Process description

Free-radical Emulsion Polymerization (EP) processes are used for producing a wide range of products in an environmentally friendly manner due to the utilization of water as the dispersion medium in the process. In this type of processes, monomers polymerize in the form of emulsions (i.e. colloidal dispersion) using an inert medium in which the monomer is moderately soluble (not totally insoluble). Commercial polymerization of vinyl acetate, chloroprene, various acrylate copolymers and

Dynamical model description

The macroscopic scale is modeled with ODEs according to the laws of conservation of mass and energy. The macroscopic model represents the slowest reactor dynamics but serves as the interface for the lower scales. The macroscopic model allows relating the process inputs with the meso- and microscopic states that could be chosen as the process controlled outputs. At the microscopic scale, an integration of chemical, colloidal and hydrodynamic events is performed using the kMC technique (Hernández

Multiscale model solution

A very important aspect of multiscale model solution is the exchange of information among the different scales. At lower-scales the model requires information about the state of the system (temperature, pressure, composition, etc.) which is determined at upper-scales. At the same time, the larger-scales model requires parametric and structural information of the system obtained at lower-scales. Therefore, top-down and bottom-up information exchange procedures must be clearly defined. Fig. 2

Simulation results

Parameters values used in the simulation are reported in Tables 3 - 6. Table 3 summarizes the kinetic and thermodynamic parameters related with both macro- and meso-scales. Table 4 shows reactor input feed flows. Table 5 presents the initial conditions for the state variables at the macroscopic-scale. Table 6 shows values for the kinetic and thermodynamic parameters used in the kMC simulation. Notice that some of the kMC simulation parameters depend on the conditions at the macroscopic-scale.

In

Conclusions

A multiscale model for describing the dynamic evolution of free-radical polymerization was successfully implemented with reasonable computational complexity. The developed multiscale model includes the traditional macroscopic and mesoscopic dynamics of the free-radical polymerization process, and also the average number of free-radicals and secondary nucleation rate (which are important microscopic states). These two microscopic states are critical to satisfactorily obtaining structured polymer

Acknowledgments

Jorge-Humberto Urrea-Quintero gratefully acknowledges the financial support by COLCIENCIAS via a Doctoral Scholarship (Grant number 727–2015).

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