(240d) Size Control and Parameter Estimation for Fluidized Bed Reaction

This paper presents a particle size distribution control scheme for a fluidized bed reactor. The controller incorporates passivity based inventory control theory as well as an observer-based parameter estimator. The method can be applied to a wide range of processes modeled with classical population balance techniques. We demonstrate the control and estimation with a model for fluidized bed production of solar-grade silicon particles through thermal decomposition of silane (SiH₄) gas. Our model tracks the creation, growth, and agglomeration of particles through discrete size intervals, and it has been calibrated against data obtained from a pilot-scale fluidized bed reactor.

Inventory control, as described by Farschman et al. (1998), uses passivity theory to ensure stability of averages of extensive variables, which are termed inventories. The discrete population balance model is composed of mass balances over discretized size intervals, and we consider the mass of particles of desired size to be process inventories. We then develop feedback-feedforward controls for the system, which can be used to control the size distribution of the particulate system.

Using a population balance framework to model and control particulate processes often requires information about unmeasured states or parameters. The observer-based estimator described by Dochain (2003) enables decoupling of state and parameter estimations, which can improve parameter estimation. Once estimation equations are developed for an unmeasured parameter, we choose a tuning method that is appropriate for the existing model and control dynamics.

The proposed control and estimation scheme has been tested on a population balance model of the fluidized bed production of silicon. Existing kinetic models have been used to determine the rate of silane decomposition, which occurs heterogeneously at the surface of solid silicon to produce gray, crystalline silicon and homogeneously in the gas phase to produce brown, amorphous silicon powder. The model of nucleation and condensation kinetics includes one adjustable parameter, which indicates how much powder is scavenged by existing particles and how much powder is exhausted, subsequently decreasing the process yield. In this application, the inventory controller is used to design product and seed flow rates to ensure the silicon hold-up is held constant and that the average particle size tracks its set point within the feasible range of operation. Also, the adjustable parameter, which indicates process yield, is estimated using an observer-based estimator with the closed loop model and control scheme. The overall stability of the control system including parameter estimation is analyzed using passivity and L₂ stability theory based on thermodynamic storage functions derived from statistical mechanics.

Farschman, C. A., Viswanath, K. P. & Ydstie, B. E. (1998), ?Process systems and inventory control', AIChE Journal 44(8), 1841-1857.

Dochain, D. (2003), ?State and parameter estimation in chemical and biochemical processes: a tutorial', Journal of Process Control 13, 801-818.