(356d) Detailed Population Balance Modelling Using a Hybrid Particle Model

Population balance modelling
provides a powerful tool to study formation and growth of particulate species
in a wide range of systems from crystallisation to combustion and atmospheric
processes. In many applications, the particle structure is an important
determinant of its properties; thus, it is necessary to study the relationship
between process conditions and the evolution of particle morphology. Detailed
particle models have been developed^[1] to track the structure of
complex particles formed in combustion synthesis, for example of soot, silica
and titania. These can be used to study materials synthesis, supplementing
experimental understanding. A Monte Carlo approach is required to resolve
particle structure in more than a few dimensions. This treats the formation and
growth processes as events that occur in proportion to their respective rates,
evolving a discrete ensemble of computational particles to resolve the particle
size distribution (PSD). Operator splitting can be used to couple gas-phase
kinetics and the particle dynamics. For industrially relevant process
conditions, the particle dynamics are incredibly rapid – which can lead to very
high number densities and broad particle size distributions^[2]. This
heterogenous mix of complicated aggregates and many newly-incepted, simple primary
particles is challenging to treat efficiently with fixed computational ensemble
sizes.

We recently developed a hybrid
particle-number and particle (PN/P) model^[3] to facilitate use of a stochastic
algorithm and a detailed particle model under aggressive particle process
rates. This model aims to resolve the particle size distribution more
efficiently under high rate conditions, using different levels of details for
different regions of the particle type space. Small particles are tracked using
a particle-number model which requires only a single internal coordinate and
complex particles are tracked using a discrete ensemble of computational
particles in which particle structure is described in up to hundreds or even
thousands of dimensions (i.e. primary particle connectivity and sizes). In the
current work, we present the new algorithm and demonstrate its use to study a
wide range of process conditions and track detailed variation in particle
product morphology with a significant improvement in computational efficiency
over the standard, single-ensemble approach. We also show that the algorithm is
exact; thus, the improved performance does not compromise prediction accuracy.

^[1]Shekar, S., Menz, W. J., Smith, A. J., Kraft, M.,
Wagner, W. On a multivariate population balance model to describe the structure
and composition of silica nanoparticles, Computers and Chemical Engineering 43
(2012) 130–147. doi: 10.1016/j.compchemeng.2012.04.010.

^[2]Boje, A., Akroyd, J., Sutcliffe, S., Edwards, J.,
Kraft, M, Detailed Population Balance Modelling of TiO₂ Synthesis in
an Industrial Reactor, Chemical Engineering Science 164 (2017) 219–231.
doi: 10.1016/j.ces.2017.02.019.

^[3]Boje, A., Akroyd, J., Kraft, M, A hybrid
particle-number and particle model for efficient solution of population balance
equations, Journal of Computational Physics, in press (2019). doi:
10.1016/j.jcp.2019.03.033.