(100c) Simplifying Population Balance Models to Promote More Widespread Use

Population balance models are used sparingly in industry, and one likely reason is because the mathematics appears daunting, coming in the form of partial-integrodifferential equations. By contrast, every undergraduate engineer has been exposed to (and most feel comfortable with) the interpretation of chemical rate data to obtain the kinetics for reactor design. Likewise, they are comfortable with the integration of ODEs, either singly or as a set. The aim of this paper is to outline a method for teaching interpretation of particle evolution rate data so that it is as accessible as reaction kinetics to the undergraduate chemical engineer.

Under a broadly applicable set of assumptions, population balance models for systems in which particles transform by accretional growth, collisional growth and/or breakage have the possibility of reaching similarity solutions or stationary states. This paper uses this recognition to simplify population balance models of one internal dimension into a single ODE for the evolution of an appropriate mean of the population. The assumption is sometimes made that the evolving population is of a single size, i.e, the "monodisperse approximation." The paper shows that the single ODE when at similarity solution or stationary state is identical in form to the monodisperse approximation with certain invariant moments connected to the rate kernel order embedded in the effective size-independent portion of the rate kernel. This makes data analysis analogous to the extraction of rate constant and reaction order from chemical reaction trajectory data, only this time what is extracted is the size-independent portion of the effective rate kernel and the order of the kernel in size. These particle transformation kinetics can be used for design of particle unit operations in the same way chemical kinetics are used for reactor design.