Population Balance Equations (PBE) are used for modeling a variety of
particulate processes as well as various stochastic phenomena in science and engineering.
However PBEs are difficult to solve because they describe the evolution of a probability
density function (PDF) in high dimensional spaces. Due to their unique mathematical
structure and properties, these equations require special solution techniques. Moment
methods are a class of solution techniques that evolve only a few moments of the PDF.
While moment methods are simpler, they are known to have closure problems, i.e. a
finite set of moment equations do not fully describe the PDF or its evolution. The purpose
of this dissertation is to investigate a closure scheme for the moment equations that is
based on Gaussian quadrature. This approach, known as the Quadrature Method of
Moments (QMOM), is very general as it does not require any a priori assumptions on the
form of the PDF. In this study, I first evaluate the accuracy of the moment closure by applying QMOM to solve some well known problems in aerosol science, such as particle
nucleation and growth in well stirred reactors and size dependent transport of aerosol
particles. I find that results obtained using QMOM compare favorably with results
obtained using more expensive techniques. Moment methods are particularly suited for
implementation in CFD codes. As an example of a model for smoke detectors, I use
QMOM to simulate smoke entry and light scattering in a cylindrical cavity above a
uniform flow. As further examples, I describe the use of QMOM in applications such as
statistical uncertainty propagation and simulation of turbulent mixing and chemical
reaction using the PDF transport equation. While moment methods are widely applicable,
they have some limitations. I find that the solutions depend on the choice of moments
and that there may not be a globally optimal set of moments. This becomes more
problematic for solutions of multivariate PBEs using an extension called the Direct
Quadrature Method of Moments (DQMOM). The insights from this work can lead to a
greater appreciation of the benefits and limitations of moment methods for solving PBEs.Mechanical Engineerin