(47b) On a Class of Analytical Solutions for the Population Balance Equation with Aggregation and Breakage

The analysis of reactive, dispersed-phase systems relies almost entirely on the accurate representation of the behavior of particles, their chemistry, and hydrodynamic interaction. This type of analysis is generally approached by solving the population balance equation (PBE) coupled with species transport to track the progress of chemical reactions.

Population balances are ubiquitous in the many branches of scientific and engineering applications. Albeit the theoretical foundations of PBE models have long been established, the literature appears to be lagging in analytical solutions except for a handful of studies. Analytical solutions provide an avenue for closed form, simple evaluations that help in identifying the key complexities of a particular model as well as benchmarks for experimental and numerical investigations.

In this talk, we discuss two approaches for deriving exact and approximate solutions for the population balance equation. We first present a novel transformation that maps the non-homogeneous PBE into a simple advection equation with the internal coordinate growth rate acting as the advecting velocity. This applies to a special class of population balances, namely, those that track one internal coordinate r with birth and death rates dependent only on r. Our transformation allows us to derive exact solutions for an arbitrary initial population. These are documented and applied to a plug flow reactor model for a mineral carbonation process where the formation and precipition of calcium carbonate is tracked via the population balance equation. 

We then extend our analysis to more complex, nonlinear systems where exact, closed form solutions cannot be found. We apply the Homotopy Analysis Method (HAM) to an expanded version of the plug flow reactor example where the supersaturation ratio is made to be time dependent. Our results indicate that the required simplicity of exact analytical solutions for PBE models can be relaxed by using novel analytical techniques that aid in linearizing the complex birth and death rates.