Biomedical Engineering Reference
In-Depth Information
gross features of the phasic fluid flows and heat transfers are solved, which provide
sufficient information for the behaviour of the mixture.
A detailed derivation of the Eulerian method is beyond the scope of this topic;
instead the final form of the governing equations is presented. Furthermore, there are
different approaches to link the Eulerian particle phase with the fluid phase, such as
the
mixture
, and
Volume of Fluid
(VOF) models, but we will limit our discussion to
the
two-fluid
Eulerian-Eulerian model. For a full derivation and discussion on these
issues, the reader may refer to the topics of Crowe et al. (1998), Brennan (2005),
and Yeoh and Tu (2009). The derivation of the multiphase equations may begin by
considering an arbitrary volume subject to two conditions: (i) is much smaller than
the distance over which the flow properties will vary significantly, and (ii) is large
enough so that it contains representative samples of each of the phases. In practice,
however, these two conditions are not always satisfied, and an averaging process,
which restricts the solutions to macroscopic flows, is applied. Generally averaging
may be performed in time, space, over an ensemble, or in some combination of these.
For each phase to be modelled there is a set of
n
conservation equations to represent
the
n
number of phases that is solved with the volume fractions being tracked within
every computational cell. The volume fraction of a component or phase can be
denoted by
α
N
. The sum of all phase volume fractions is equal to 1; thus for a
two-phase flow having components or phases
A
and
B
,wehave
α
A
+
α
B
=
1.
There are a few different approaches to link the Eulerian particle phase with the
fluid phase. For a 2D steady flow, the
continuity equation
for each phase is given
as
∂
(
α
N
u
)
∂x
∂
(
α
N
v
)
∂y
+
=
χ
N
(6.2)
where
χ
N
is the net rate of mass transfer to/from the nth phase.
The
u
and
v
momentum equations
are
∂
(
α
N
uu
)
∂x
(
ν
∂
(
α
N
vu
)
∂y
1
ρ
∂p
∂x
+
∂
∂x
ν
T
)
∂
(
α
N
u
)
∂x
+
=−
α
N
+
(
ν
+
ν
T
)
∂
(
α
N
u
)
∂y
F
N
,
u
∂
∂y
+
+
χ
N
,
u
+
(6.3)
(
ν
∂
(
α
N
uv
)
∂x
∂
(
α
N
vv
)
∂y
α
N
1
ρ
∂p
∂y
+
∂
∂x
ν
T
)
∂
(
α
N
v
)
∂x
+
=−
+
(
ν
F
N
,
v
∂
∂y
ν
T
)
∂
(
α
N
v
)
∂y
+
+
+
χ
N
,
v
+
(6.4)
χ
N
,
u
and
χ
N
,
v
account for both the net rate of momentum exchange between phase
N and other phases.
F
N
,
u
and
F
N
,
v
are the additional body forces that act on the
N
th
phase. These additional momentum exchanges include
drag, lift, buoyancy, pressure
gradient, thermophoretic,
and
diffusion forces
.
To summarise, Table
6.1
provides a comparison between the two modelling
approaches, their characteristics, and advantages/disadvantages.
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