(669d) Calorimetric Estimation Employing Unscented Kalman Filtering for a Batch Emulsion Polymerization Reactor

Reaction calorimetry is a very useful tool to monitor exothermic polymerization reactions as it is based on the measurement of heat generated by the reaction, which can be then associated to the polymerization rate. Compared to other techniques, reaction calorimetry has the advantages of being a noninvasive technique, robust, fast (when compared to typical off-line characterization techniques) and low cost.^[1-5]

Heat flow calorimetry is frequently used to monitor and control batch and semi-batch polymerization reactions.^[6-16] Nonetheless, this technique requires the determination of the overall heat transfer coefficient between the reaction medium and the jacket (UA), which is a function not only of the reactor setup (reactor geometry and materials of the wall) and operation conditions (stirring rate and cooling fluid flow rate), but also of the properties of the reaction medium that in polymerizations changes throughout the reaction as the viscosity of the reaction medium can increase drastically and fouling at the reactor wall can occur.

Nonlinear state estimation can be employed to calculate the variation of UA during batch and semi-batch polymerization reaction. The well-known Kalman Filter is only suitable for linear systems and the Extended Kalman Filter (EKF) has become a standard formulation for nonlinear state estimation being employed to calculate UA for several polymerization reactions ^{[2, 17, 18, 19]}. However, EKF may cause significant error for highly nonlinear systems because of the propagation of uncertainty through the nonlinear system.

The Unscented Kalman Filter (UKF) is an enhanced filtering technique that effectively addresses the aforementioned issues ^{[20, 21]} .The idea is to produce several sampling points (Sigma points) around the current state estimate based on its covariance. Then, propagating these points through the nonlinear map to get more accurate estimation of the mean and covariance of the mapping results.  In this way, in fact, it computes the Jacobian but without making use of a linear approximation, hence incurs only the similar computation load as the EKF.

Despite the apparent advantages of the UKF over the EKF, the former has not been yet used widely in polymer reaction engineering. Moreover, the analysis of the filter (UKF) has been done with simulated data^{[22, 23]} that add a known noisy to the system, in addition, the same model that produced the output was provided to the algorithm that estimate the state, this is inconvenient because in the real world, the estimation is provided with model mismatch or unknown-model phenomenon. Finally, the covariance plays an important role in the stochastic filters. Usually, these matrices (Q and R) are defined by default in literature, but with real data measurement from the process tuning the covariance results in a difficult task. Recently, an off-line estimation of the two matrix has been proposed for nonlinear system^[24], the method linear time-varying autocovariance least-squares (LTV-ALS) use a linear version of the model to obtain the full version of the matrices, also the technique can be limited to obtain the diagonal elements of the covariance matrices Q and R. 

The objective of this work is to implement UKF to estimate the variation of UA and of the total heat loss (Q_loss) in order to calculate the heat generated by the reaction during a batch emulsion polymerization reaction by heat flow calorimetry. To evaluate the performance of this procedure, in-line monitoring of batch vinyl acetate emulsion polymerization reactions were carried out in a jacketed stainless steel tank reactor with an internal volume of 5 liters instrumented to operate as a calorimetric reactor at different reaction conditions and variable cooling fluid flow rate (4.0 -  24 L/min).

Results show that the use of UKF is a practical alternative to the online calorimetric estimation without the use of intrinsic equation or privileged information (e.g. gravimetric conversion experimental data obtained during the reaction and used to correct the estimation parameters). It was observed that different operation mode (low or high cooling fluid flow rate through the jacket) that affects the overall heat transfer coefficient and influences the term that represented the diagonal covariance for UA, which make it necessary to tune the covariance for the different operation conditions. As the cooling fluid flow rate is a known variable, an strategy is proposed in which the covariance is changed according to this variable making the estimation of UA using UKF possible, in order to calculate the heat of polymerization.

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