Theory and simulation of dispersedphase multiphase flows

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Postgraduate Course in Multiphase Flows
Theory and simulation of dispersed-phase
multiphase flows
Organized by Center of Excellence in Multiphase
Flow Research Lappeenranta University of
Technology Lecturer Payman Jalali,
Docent Autumn 2007
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Continuous phase equations
Averaging procedures Basically, there are 3 ways
of averaging time, volume, and ensemble
averaging. -Time averaging The averaging time
must be large compared to the local fluctuation
time.
- Volume averaging It is carried out by
averaging properties at an instant in time over a
volume and ascribing the average value to a point
in the flow. The averaging volume must be much
larger than the molecular distance cube and much
smaller than the container volume
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Continuous phase equations
Phase average is the average over the volume
occupied by a phase

• Ensemble averaging
• It avoids the shortcomings of time and volume
  averaging but it is much more difficult to
  implement. It is based on the probability of the
  flow field being in a particular configuration at
  a given time. For example, if the density is
  measured many times over a region and found that
  there are N different configurations (realization
  or ensemble), and for a realization ? at a given
  time t we have the distribution function

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Continuous phase equations
Assume that this ensemble has occurred n(?)
times. The ensemble average is then
In the limit where we have infinite realizations,
we have
In other words, ensemble averaging considers all
statistical variation of data in space and time
for calculating the average values.
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Continuous phase equations
Boundary particles We aim at developing the
equations for the carrier phase using the shell
balance method. The conservation principles are
applied to a finite size volume with an arbitrary
boundary. The boundary may pass through several
particles namely boundary particles. These
particles are responsible for a blockage area of
the carrier flow. They also contribute to
coupling effects such as mass transfer to and
forces on the carrier phase fluid inside the
control volume.
Blockage effects Consider the following figure.
The surface with area A cuts through several
particles in the field. The area of the particles
intersected by the surface is the blockage area,
Ab. Now, we try to quantify the ratio of the
blockage area to the total area (Ab/A) as a
function of particle volume fraction.
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Continuous phase equations
Assume that particles are spherical and of the
same size. The cross-sectional area of the cut
where the surface passes through the particle is
given by
There is a uniform probability for a particle to
lie at a distance ? from the surface. So the
probability distribution function is
If there is a total of N particles intersected by
the surface, the blockage area is given by
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Continuous phase equations
With the above uniform distribution we get
Then we have
Thus the flow rate of continuous phase through a
cloud of particles will be
u is not the superficial velocity, but it is the
phase velocity of the continuous phase.
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Continuous phase equations
Property transfer The boundary droplets also
contribute to the mass, momentum and energy
transfer to the continuous phase. How should we
determine the contributions of particles which
are only partially inside the volume?
As an example, consider the mass transfer from
boundary particles shown in the figure. The
center of droplet A is located at ?A and the mass
transfer to the carrier phase inside the
computational volume is
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Continuous phase equations
The sum of mass transfer rate to the inside and
outside of the computational domain is equal to
the total mass transfer rate over the surface of
the droplet, so
The two droplets A and B can be chosen as
complementary droplets so that
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Continuous phase equations
Quasi-one-dimensional flow It is a flow in which
the variations of flow in one direction are
considered but the variations in the cross-stream
direction are neglected. An example is the flow
with gently sloping walls.
The control surface is broken into two parts The
surface through the continuous phase at stations
1 and 2 and along the wall and the surface
adjacent to the droplets inside the control
volume.
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Continuous phase equations
Continuity equation Rate of mass accumulation
Net efflux of mass 0
For models treating the particles as points, the
volume fraction of the continuous phase is 1
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Continuous phase equations
Divide by V
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Continuous phase equations
Momentum equation Rate of momentum accumulation
in control volume Net efflux of momentum from
control volume Force on fluid in control volume
F is the force on the continuous phase. We assume
uniform flux over the droplet surface and all the
droplets have the same speed and evaporation
rate. Then
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Continuous phase equations
The forces acting on the fluid are the pressure
forces on the boundary, the shear stress on the
wall, the drag forces due to the dispersed phase,
and the body forces on the fluid such as
gravitational force. The boundary droplets need
special consideration. The average pressure
across the control surface 1 is p1. The force on
the continuous phase due to pressure on the
portion of the droplet inside the control volume
is
The pressure on the droplet surface is
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Continuous phase equations
If the second integral is calculated on
complementary droplets it will result the
negative of the force due to pressure on a
droplet completely inside the domain. So, the
number of droplets are assumed such that
complementary pair of boundary droplets are
included in N. So, the second integral will be
taken into account in drag force. So, the total
force due to pressure on surface 1 is
Thus the net force due to pressure is
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Continuous phase equations
The force due to wall friction is
The force due to gravity is
Finally, the force on the fluid due to dispersed
phase drag, assuming all the dispersed phase
elements move at the same velocity, is
The unsteady forces have been neglected here. The
force due to pressure gradient on the droplet can
be expressed as
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Continuous phase equations
We can combine the pressure force and the drag
force due to pressure gradient and call the sum
of them as Fp, so
Summing all the forces and substituting them into
the momentum balance equation for the fluid phase
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Continuous phase equations
The differential form of the momentum equation is
Question If the dispersed phase is treated as
points, how the momentum equation will change?
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Continuous phase equations
Practice Evaluate ?V for a mixture of 100 ?m
coal particles in air at standard conditions. The
particle volume fraction is 0.01, the material
density of the coal is 1300 kg/m3 and the
velocity difference is 1 m/s. Assume the coal
particles can be treated as spherical.