INAUGURAL–DISSERTATION zur Erlangung der Doktorwürde der ...

2.2. Euler – Lagrangian Approach 17
applied to study multi-component (fuel mixture) droplet evaporation [126]. Gumprich
et al. [127, 128] analyzed the dense turbulent sprays using DQMOM, and DQMOM
coupled with Eulerian multi-size moment model [129]. Madsen [61] extended DQMOM
to include droplet coalescence in spray flows by neglecting the effects of evaporation,
whereas Fox et al. [60] further improved DQMOM to model evaporating and coalescing
spray flows but his study assumed simplified models for evaporation and coalescence.
So far DQMOM has not been considered to treat the process of spray drying.
In the present study, DQMOM is used to describe the disperse phase consisting of
poly-disperse liquid droplets, whereas the gas phase is not yet resolved but its inlet
flow properties are taken for computing the droplet motion and evaporation. In this
work, the DQMOM is implemented in two dimensions, which is done for the first time,
and applied to study bi-component evaporating spray flows.
2.2 Euler – Lagrangian Approach
In the Euler – Lagrangian approach, the mean field equations are used only for the
continuous gas phase. The droplet properties are defined along the path lines followed
by the droplet. The trajectories of droplets are tracked for each droplet group by using
a set of equations that describe their physical transport in flow field. In the current
study, the discrete droplet model is used to define the droplet phase whereas the gas
phase is modeled using the Navier – Stokes equations.
2.2.1 Gas Flow
The Euler – Lagrangian model DDM includes Euler equations for the gas phase with
source terms for the dilute spray, which is described in Lagrangian coordinates. The
instantaneous Navier – Stokes equations in an axisymmetric, two-dimensional configuration
with no swirl for a dilute spray yield [68, 130]
∂(ρu i )
∂t
+ ∂(ρu iu j )
∂x j
∂ρ
∂t + ∂(ρu j)
= S l,1 , (2.1)
∂x j
= − ∂p
∂x i
+ ∂τ ij
∂x j
+ ρg i + S l,ui , (2.2)
where ρ, u i and p are the density, velocity component and pressure of the gas flow,
respectively. g i is the acceleration due to gravity and the quantities S l,1 and S l,ui are
the source terms due to spray evaporation [130, 131]. τ ij is the viscous stress tensor
given by
( ∂ui
τ ij = µ + ∂u j
− 2 ∂x j ∂x i 3
)
∂u k
δ ij , (2.3)
∂x k