463592 Parameter Estimation and Model Discrimination of Batch Solid-Liquid Reactors
Heterogeneous solid-liquid reactions are of great significance in metallurgical, agrochemical and pharmaceutical processes. This kind of industrial process is usually conducted in batch reactors, generating fine chemicals with high profits. Hence, understanding the mechanism and constructing qualified models are desired to facilitate process optimization and controller design. Motivated by this, we explore a model building exercise for a real industrial process that involves multiple solid-liquid reactions in a series of two batch reactors. Key activities are to elucidate the kinetic and transport models, estimate model parameters with limited plant data, and validate the models.
For our target application, the batch reactor operates sequentially through steps such as preparing, feeding, reacting, impurity removing, sampling and discharging. In the reacting step, solid particles react with liquid reactants and generate solid and/or liquid products. The kinetic mechanism is complicated through coupled mass and heat transfer and different models have been proposed by previous researchers, including two particular mechanisms -- solid surface reaction models  and dissolution models  with insoluble solid component reactions taking place on the interface and products sticking on or cracking away from the reaction surface. On the other hand, if solid components dissolve into the solvent, even in trace amounts, reactions may occur in liquid phase and the dissolution rate influences the production rate.
It is well known that model discrimination is difficult and requires carefully designed experiments. However, in this case only limited plant data are available to identify the model and uncertainties in the reaction process, such as solid particle sizes and shapes, must be resolved. Moreover, simple models with assumptions of uniform size and ideally nonporous spherical particles cannot fit data well. To deal with these issues, a generalized model is applied for both surface reaction and dissolution-controlled mechanisms. The particle morphology is considered similar to that described in  by introducing a shape factor. A particle size distribution is also implemented. This model requires estimation of several parameters, including shape factors, activation energies and diffusion coefficients. To enhance the model estimability, a number of parameters are lumped in the model.
Based on the above process characteristics, a dynamic model is derived with multiple stages. The parameter estimation is solved in an Errors-in-Variables-Measured (EVM) optimization formulation . Orthogonal collocation on finite elements is applied to discretize the problem, resulting in a large scale nonlinear programming problem. The problem is formulated in AMPL and solved by a nonlinear optimization solver IPOPT. The estimation performance is evaluated by calculating confidence intervals at the optimal solution. In our case, only reactor temperature profiles and end-point compositions are measured, with no concentration data available during reactions. Results show good fitting on process data and small confidence intervals, which indicates good model parameter selection and estimation. These results lead us to elucidate the likely mechanistic model for subsequent optimization studies.
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