Heterogeneous solid-liquid batch reactors are widely applied to produce fine chemicals in metallurgical, agrochemical and pharmaceutical processes. Compared with fundamental homogenous reactions, solid-liquid reactions are usually coupled with mass transfer, such as liquid phase diffusion and solid particle dissolution. However, a detailed reaction mechanism is typically inaccessible in a real industrial plant due to limited prior knowledge and difficulty of measurements. Understanding reaction mechanisms and constructing acceptable models based on obtainable information are desired to facilitate process optimization and control. In this study, we first introduce a model identification and parameter estimation procedure and implement it on our batch reactors with limited real plant/experimental data. Optimal control is then implemented on the estimated model to reduce total batch time while satisfying operation constraints and final concentration requirements.
Two particular mechanisms -- solid surface reaction models and dissolution controlled models are considered in the targeted solid-liquid reactions. A uniform kinetic model is developed with a model indicating factor to differentiate mechanisms. Particle morphology is considered similar to that described in [1] by introducing a shape factor. In addition, several other parameters are also included, such as activation energies, diffusion coefficients and heat transfer coefficients. A detailed procedure for parameter estimation [2] and model identification [3] is implemented, including pre-selection of estimable parameter subset, simultaneous estimation by using measured data, estimation quality evaluation, model simplification and model discrimination. In our case, only reactor temperature profiles and highly limited concentrations are measured. Small confidence intervals and good data fitting are obtained by the identified model.
The motivation behind the model identification effort is to develop an optimal control solution to minimize batch time and increase process capacity. The end of batch is defined by satisfying final conversion requirements. Feeding of liquid reactive components and reactor/jacket inlet temperatures are treated as decision variables. Optimal control is obtained by solving the optimality conditions of the Hamiltonian together with the corresponding adjoint system. A dynamic model is derived with multiple stages describing different batch operation steps. Orthogonal collocation on finite elements is applied to discretize the problem, resulting in a large scale nonlinear programming problem formulated and solved in AMPL/IPOPT. Finally, to address potential parametric uncertainties, disturbances and model mismatch, a robust optimization strategy is applied by adding back-off terms to ensure that constraints are satisfied in the worst-case scenario.
References
[1] Salmi, Tapio, Henrik Grénman, Johan Wärnå, and Dmitry Yu Murzin. "New modelling approach to liquid–solid reaction kinetics: From ideal particles to real particles." Chemical Engineering Research and Design 91.10 (2013): 1876-1889.
[2] Lin, Weijie, Lorenz T. Biegler, and Annette M. Jacobson. "Modeling and optimization of a seeded suspension polymerization process." Chemical Engineering Science 65.15 (2010): 4350-4362.
[3] Stewart, W. E., Y. Shon, and G. E. P. Box. "Discrimination and goodness of fit of multiresponse mechanistic models." AIChE journal 44.6 (1998): 1404-1412.
[4] Diehl, Moritz, Hans Georg Bock, and Ekaterina Kostina. "An approximation technique for robust nonlinear optimization." Mathematical Programming 107.1 (2006): 213-230.
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