INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

14 2. Mathematical Modeling
number of droplets within a range of droplet characteristics such as size, velocity,
temperature and solute mole fraction within the droplet, which constitutes the phasespace
at a spatial location. The solution of this equation coupled with the gas phase
flow field equations provides the average properties of the spray, for example mean
droplet diameter, Sauter mean droplet diameter, mean droplet velocity and many other
statistical properties.
Among the existing Eulerian models, the multi-fluid method allows the detailed
description of poly-disperse droplet size and velocity through correlations. Such an
approach has been shown to be derived from the Williams’ spray equation, Eq. (2.23),
by Laurent and Massot [45] under the mono-kinetic spray assumption, which states
that the velocity dispersion of the spray distribution function at a given time, spatial
location and droplet size has to be zero. This assumption is important since it defines
the validity limit of the multi-fluid model and also results in the ”pressure-less gas
dynamics” structure of the transport equations for conservation of mass and momentum
of droplets [96]. These conservation equations are derived by discretizing the droplet
distribution using a finite volume technique in fixed size intervals called sections or
fluids.
This approach has been extended to higher order of accuracy [46], but discretization
of droplet size phase-space is still a problem that needs to be addressed. The issues
related to the mono-kinetic assumption have to be relaxed if the coalescence of droplets
are to be considered, which is addressed by introducing a semi-kinetic equation and
the results are presented for the evaporation and coalescence in spray flows [47, 97].
However, the validation in multi-dimensional configurations and the evaluation of the
level of accuracy of such model versus the reference Lagrangian simulations as well
as the related issue of a detailed study of the effective computational cost of the two
approaches is not yet understood [98].
Next most notable method is the method of classes (CM) or discretized population
balances (DPB) which is based on the discretization of droplet internal co-ordinates
of the population balance equation [99–102] into a finite series of bins. The CM’s
compute the mean properties of the population such as droplets or particles within
these bins by solving the discretized population balance equation. CM’s are divided
into two categories namely, (1) zero-order methods, and (2) higher order methods. In
zero-order methods, the droplet size distribution (DSD) is considered to be constant
in each class, and they are ”extremely stable”. Recently, Vanni [103] reviewed and
compared the wide variety of zero-order CM’s. In higher order methods, the DSD
is defined in a specific functional form for every section of discretization, and these
methods are usually more accurate but less robust [101–103]. The CM’s present the
main disadvantage of requiring a large number of classes to work with good accuracy,