(498c) Passivity Based Inventory Control of Particulate Systems

In this paper we will describe the
application of passivity based inventory control for size distribution control in
particulate processes. The theory we present can be applied to a wide range of
processes modeled with classical population balance techniques.  In the paper
we will describe the development of a control system for an industrial fluidized
bed solar-grade silicon production process, which is presently at the pilot
plant evaluation stage.  Our model has been calibrated against data obtained
from the pilot-scale fluidized bed reactor. The control system has not yet been
tested industrially. Simulation studies and robustness results will be
presented in the paper.

During the production of solar
grade silicon in a fluidized bed reactor, silane gas (SiH₄)
thermally decomposes into hydrogen gas and two different types of solid
silicon.  The heterogeneous silane decomposition produces crystalline silicon
on the surface of existing silicon.  Homogeneous decomposition produces
amorphous silicon powder, which aggregates to form new particles, is scavenged
by existing particles, or is exhausted with gas flowing through the system. Existing
kinetic models have been used to determine relative amounts of heterogeneous
and homogeneous decomposition. These models cannot predict whether the powder
contributes to particle nucleation and growth or whether it escapes as loss. 
The loss of powder determines the process yield and is consequently an
important design parameter.  In fact the overall success of the process depends
on careful control of yield losses since the market price of silicon powder is
considerably lower than that of solid silicon. 

We have developed a population
balance model of the silicon process which tracks the creation, growth, and
agglomeration of particles. The nucleation and condensation kinetics have been
included, and these allow us to model how the yield depends on operating
conditions using one adjustable parameter. This parameter will be estimated
using on line parameter adaptation via a state observer. The model predictions
have been calibrated so that model predictions of size distributions match
industrial pilot plant data.

A feedback-feedforward control
system based on the inventory control method has been developed to stabilize
the process at different operating conditions. The inventory balance controller
is designed so that the total hold-up of silicon is kept constant. Seed and
recirculation rates control the size distribution so that it tracks its
setpoint within the feasible range of operation. The overall stability of the
control system including state and parameter estimation is analyzed using
passivity and L₂ stability theory based on thermodynamic storage
functions derived from statistical mechanics.