Simulation of a Hydrometallurgical Leaching Reactor modeled as a DAE system

An existing dynamic model of the main reactor in the Silgrain process for the production of Si from FeSi has been extended here in order to resemble more closely the behavior of the real reactor. The previous model was based on the application of macroscopic mass conservation law, the population balance equation and the assumptions of complete mixing and isothermic conditions. The major modifications are the inclusion of the condition governing the entrainment of particles in the outflow, and the formulation of the energy balance. The extended model consists of 1 integrodifferential equation, 4 implicit ordinary differential equations, 7 algebraic equations and 3 integral equations. After discretization in the particle size space, a system of differential and algebraic equations (DAE) is obtained. DAEs are not ODEs and they require analysis and characterization and may require reformulation. After such analysis, it was concluded that the system is implicit index-one for the usual range of operation and that a method based on the Backward Differentiation formulas (BDF) can be used for its solution. The model was implemented in Matlab and the ode15s code was used for solving the system of equations. The simulation results are satisfactory and seem to match qualitatively with the known operation of the reactor. The model is suitable for further use in designing a model-based control scheme.