A Systemic Approach for Pellet Reactor Modeling: Application to Water Treatment

The development is reported of a model for a fluidized-bed process for phosphate precipitation and removal from wastewater. The general framework of this study involves a two-step procedure. The first modeling level was previously presented elsewhere and leads to the development of a thermodynamic model for the computation of phosphate conversion for the Ca–PO4–H2O system. The second step of the modeling procedure is the core of this report and computes the reactor efficiency from the identification of the so-called pellet reactor model as a reactor network involving a combination of elementary systems representing basic ideal flow patterns (perfect mixed flow, plug flow, and so on). For solving the involved mixed-integer quadratic programming (MIQP) problem, a hybrid procedure based on simulated annealing (SA) and quadratic programming (QP) is implemented. The SA generates reactor network structures deduced from a superstructure and for each one, a QP is carried out. The goal is to find not only the most appropriate model, but also the simplest one (in terms of the smallest possible number of elementary units). Thus the objective function is augmented with an outer penalty quadratic function representing the number of elementary units. The methodology was first validated on an example previously treated with a classical MINLP method involving a limited number of variables. The main interest of the approach proposed here is that it can handle large-size problems with a limited exploration of the search space and can be consequently extended to combinatorial problems. A comparison with experimental results obtained in the pilot unit designed for this study shows the efficiency of the systemic approach