INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

2.1. State of the Art 13
the governing equations for the gas phase include appropriate source terms to compute
the effects of droplets.
The other important SF method for modeling sprays is the CFM, which employs
a continuum formulation of the conservation equations for both phases [81, 82]. The
motion of both droplets and gas are treated as interpenetrating continua. The work of
Faeth [72] gives an extensive review of all the Euler – Lagrangian models.
The Euler – Lagrangian approach is so far considered to be effective in many applications,
which gives detailed information at the micro-level, however it has significant
drawbacks as listed by Archambault [67]. For instance, inclusion of droplet–droplet
interactions such as coalescence and breakup, which occur quite frequently in spray
flows, increases the computational complexity. The computational cost could be very
expensive due to the large number of droplets needed to reach the statistical convergence,
and it may pose difficulties and numerical instabilities in coupling of Lagrangian
description of dispersed phase with the Eulerian equations of the gas phase. The computational
cost is also dependent on mass loading of the dispersed phase. According
to Archambault [67], the vertices of the droplet trajectory and numerical grid of the
gas phase never coincide, hence a sub-grid model is required in order to compute the
exchange rate between the phases [83]. Grid independent solutions are quite difficult to
obtain [84], which could be because of an insufficient number of droplets in a grid cell
leading to a significant error as can be observed in the regions of high droplet number
density.
The study of Garcia et al. [85] and Riber et al. [86] describe and analyze the
comparison of computational time between Euler – Euler and Euler – Lagrangian in
homogeneous and non-homogeneous flows.
There is a tremendous amount of literature available on the Eulerian – Lagrangian
approaches in spray flows and spray drying [76, 87–93], and references therein. As the
focus of the current work is about Euler – Euler approach to spray flows, this section
presents the review of available literature in this area.
A numerous Eulerian models have been recently developed where the disperse phase
described based on a kinetic equation and continuum phase is resolved using Navier–
Stokes equations. The basic idea in kinetic equation based Eulerian methods is that
instead of solving the usual Euler equations for the dispersed phase, the evolution of the
moment transform of the kinetic equation is solved, which resembles Navier – Stokes -
like equation, and this equation is coupled to the continuum phase with the appropriate
source terms. Such a kinetic equation is first derived by Williams [42], known as
Williams’ spray equation which is analogous to Boltzmann’s equation of molecules [94,
95]. The derivation of Williams’ spray equation is given by Archambault [67] and
Ramakrishna [41]. This equation describes the temporal evolution of the probable