Dynamic optimization of an emulsion copolymerization process for product quality using a deterministic kinetic model with embedded Monte Carlo simulations

Abstract

We present the dynamic optimization of an emulsion copolymerization process described by a deterministic kinetic ordinary differential equation model including a stochastic Monte Carlo submodel, describing the growth of polymer chains within particles. For the considered semi-batch operation, the time dependent input trajectories for monomer and initiator flow rates are optimized along with the isothermal reactor temperature. We use the surrogate-model-based optimizer MATSuMoTo for the optimization to avoid the need to compute derivatives of the stochastic model. Radial basis functions with linear polynomial tails are selected as surrogate functions which are updated during optimization by the newly evaluated points. A relevant application problem formulation together with results for two case studies are presented. Qualitatively similar input trajectories are obtained for different optimization runs due to the stochastic process and the limited number of iterations. All solutions reduce the batch time significantly.

Introduction

Increasing market pressure and the demand for constant high quality polymers necessitates optimization of existing polymer production processes. Emulsion copolymers are widely used in adhesives, drug delivery or paints (Asua, 2007), mostly produced in semi-batch operating modes. Because emulsion copolymers are products-by-process, particular attention must be paid to the operating conditions. For the product quality, the final properties of the copolymer determined by the composition, various copolymer distributions and particle morphology are relevant. The molecular weight distribution (MWD) is an important polymer characteristic, as it determines many end-user quality properties.

Many methods are available for modeling the molecular weight, whose detailed descriptions are out of scope of this article. Nele et al. (1999) gave an overview of the advantages and disadvantages of the various methods for the computation of the MWD. A general classification can be made into statistical methods, population balance methods, Monte Carlo methods and moment equations. Monte Carlo methods, as used in this article, simulate specific chains and allow for describing quantities difficult to explain by macro-scale deterministic modeling.

Deterministic model equations can be readily used for optimization. Vicente et al. (2001) utilize chain transfer agent (CTA) to control the MWD described by cumulative Schulz-Flory distributions. Vicente et al. (2002) employed iterative dynamic programming for full MWD, including bi-modal distributions. The MWD was computed via an adaptive orthogonal collocation technique and experimental results were presented. Alhamad et al. (2005) modeled the particle size distribution and molecular weight distribution and optimized a semi-batch operation. Pontes et al. (2011) optimized the MWD of polyethylene described by orthogonal collocation for a steady-state continuous operation with different reactor configurations. Tjiam and Gomes (2014) developed a population balance model for particle size and molecular weight and discretized it to obtain a differential-algebraic equation system. This system of equations was used to minimize the number average molecular weight within bounds by manipulating the monomer flow rate and reactor temperature. Zhang et al. (2015) used a probablity density function to describe the MWD and control to process to obtain the desired MWD.

For the optimization of Monte Carlo models, no exact derivatives are available, motivating the use of derivative-free optimization. Conn et al. (1997) published an overview of derivative-free optimization methods, which can be grouped into direct search and model-based methods. The former rely on the evaluation of the objective function only, and the latter construct a surrogate model which is used for the optimization. As the surrogate model is cheap to evaluate and information about derivatives is usually available, it is efficient to use gradient-based optimization. The review of Amaran et al. (2014) was focused on simulation optimization, where costly simulations must be run to compute objective or constraint function values. Different optimization algorithms were presented and corresponding software solutions. Cozad et al. (2014) introduced an algorithm for the adaptive generation of algebraic models for simulation-based optimization. So far, only few articles have dealt with the optimization using embedded Monte Carlo simulations. Gao et al. (2018) applied three different derivative-free optimizers to a kinetic Monte Carlo simulation of a free radical polymerization. They minimized the reaction time and manipulated the polymeric microstructure. Their most complex case included the multi-objective minimization of batch time while keeping the prescribed number average molecular weight and dyad fraction of compolymers. The degrees of freedom consisted of reaction temperature, conversion, mole fraction of a monomer and initial initiator concentration, leading to four degrees of freedom. Ma et al. (2018) solved a problem with chemical composition distributions for a terpolymerization at steady state, where the composition is computed via a Monte Carlo approach. They proposed an error estimation and a successive boundary shrinkage formulation to deal with the uncertainties of the Monte Carlo model. The derivative-free solver BOBYQA was used as an optimizer. The seven degrees of freedom involved the flow rates of three monomers, hydrogen, catalyst, cocatalyst and temperature. Mohammadi et al. (2018) proposed what they call “Intelligent Monte Carlo” for polymerization. An artificial neural net (ANN) is trained with the outputs of a kinetic Monte Carlo simulation. This ANN was used in a genetic algorithm to find optimal operating conditions. The three degrees of freedom for the optimization were initial masses of reactants.

In this article, we use for the prediction of the molecular weight distribution for process optimization a hybrid dynamic model, which combines a deterministic kinetic model together and a stochastic Monte Carlo model. The Monte Carlo model is a good choice, as it correctly captures transfer to polymer reactions which can happen only within a single particle. A common problem of macro-scale deterministic modeling is that branched polymer chains are shifted between particles with is non-physical. However, this stochastic Monte Carlo part of the model does not allow for gradient-based optimization because no analytical derivatives can be computed and finite differences lead to inconsistent gradients for small perturbations. Methods like “common random numbers”, “common reaction path” (Rathinam et al., 2010) or “coupled finite differences” (Anderson, 2012) were suggested to reduce the variance of the estimate of the derivative. As shown by Wright and Ramsay (Wright and Ramsay, 1979), the method of common random numbers can also lead to an increased variance, if the stochastic outputs have a non-positive covariance.

The practical problem of minimizing the batch time as economic objective while meeting strict quality constraints on the MWD is addressed. It requires the use of a full dynamic model and time varying trajectories of input and state variables in contrast to often used steady-state models as the process conditions change considerably over time in the semi-batch operating mode.

In Section 2, the hybrid model of the semi-batch copolymerization is briefly introduced and the optimization problem is formulated in Section 3. The results of two case studies are presented and discussed in Section 4. Conclusions are drawn in Section 5.