Control Issues Arising in Population Balance Models

This paper provides an overview of modelling, measurement, and control issues arising in systems modeUled by population balances. The population balance is a partial differential equation describing the dynamics of some general particle size distribution. The independent variables in the PDE are time and one or more internal particle coordinates, such as size, age, activity, etc., that fully characterize the state of the particle. Population balance models therefore can present a different set of issues than those arising in standard distributed parameter systems in which the independent variables are time and spatial location. The remaining process states, such as concentrations and temperature, are modelled -with integro-differential equations. The integrodifferential equations and the population balance's nonlocal boundary conditions are the sources of interesting and problematic dynamic behavior in continuous processes. This behavior includes open-loop instability and long period oscillations. The solution of optimal control profiles for batch processes is also difficult and computationally expensive. Accurate, on-line measurement of the particle size distribution for feedback control has been a long-standing hurdle, but has become possible in some situations due to improvements in measurement technologies such as laser light scattering and digital imaging. Crystallization from solution is used in this paper as an example of population balance models to illustrate each of these issues and demonstrate useful methods for model identification and process control.