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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Basic properties of dispersed phase flows
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Basic properties of dispersed phase flows
So, both coupling parameters are unimportant.
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Basic properties of dispersed phase flows
Problem 2- Droplets are released in a hot air
stream and evaporate as they are convected with
the flow as shown. The mass coupling parameter is
small while the latent heat coupling parameter is
large. Sketch the variation of the gas
temperature, gas velocity, gas density and
droplet velocity for both one-way and two-way
coupling in the duct.
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Basic properties of dispersed phase flows
One-way coupling
Two-way coupling
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Size distribution of dispersed phase
Droplet or particle size can be a very important
parameter governing the flow of a dispersed
two-phase mixture. So, it is useful to have a
basic knowledge of the statistical parameters
relating to particle size distributions. For
spherical particles or droplets, a measure of the
size is the diameter. For nonspherical particles,
an equivalent diameter mest be selected to
quantify the size. Readers are urged to have the
basic background on statistics, therefore, you
may refer to the lecture notes concerning
statistical definitions such as probability
density function (pdf), common types of
distributions etc http//www2.et.lut.fi/ttd/Nano
Course/Lecture4.pdf http//www2.et.lut.fi/ttd/Nan
oCourse/Exercise4.pdf
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Size distribution of dispersed phase

• Discrete size distributions
• Practically, size measurement of discrete phase
  (particles or droplets) is so that we only
  express their sizes in finite number of groups.
  For example, the size of solid particles are
  measured using different sieves in the lab. The
  number of size groups will be equal to the number
  of sieves of different mesh sizes.

Average particle diameter among n size groups
The rest of discussion is left for students,
which are about different distribution functions,
variance calculations etc.
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Particle-Fluid Interaction
Particle-fluid interaction refers to the
exchange of properties between phases and is
responsible for coupling in dispersed phase
flows. The conservation equations for single
particles or droplets are introduced. The
phenomena responsible for mass, momentum and
energy transfer between phases are presented.
Single particle equations The governing
equations for a single particle of mass M are
given as
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Particle-Fluid Interaction
Consider the droplet subject to a loss of mass
with the velocity of gases through the control
surface with velocity w. - Continuity
equation The droplet continuity equation simply
states that the rate of change of droplet mass
is the negative value of the mass efflux
through the droplet surface,
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Particle-Fluid Interaction
- Translational momentum equation The momentum
equation of a droplet with losing mass states
that the rate of change of momentum within the
control volume plus the net efflux momentum
through the control surface is equal to the
forces acting on the system (surface and body).
This is comparable to a rocket which ejects
some mass per unit time that creates thrust.
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Particle-Fluid Interaction
- Angular momentum equation The angular
momentum equation is responsible for the rotation
of the droplet based on the moments of body and
surface forces. For uniformly evaporating
(combusting) particle with spherical shape the
role of the mass efflux from the surface on the
rotation of particle is zero, so
I is the instantaneous momentum of inertia about
an axis of symmetry. The torque T is directly
resulted from the surface shear forces.
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Particle-Fluid Interaction
- Energy equation The energy of a droplet
consists of the external (kinetic energy) and
internal energy as well as the energy associated
with surface tension. Heat transfer term
includes both convective and radiative heat
transfer. The work term includes the flow work
due to the efflux of fluid through the control
surface as well as the work associated with the
forces on the droplet.
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Particle-Fluid Interaction
The kinetic energy of the evaporated mass
(efflux velocity) is very small compared to the
enthalpy change. The final form of the energy
equation reduces to
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Particle-Fluid Interaction
Mass coupling Mass coupling can occur through a
variety of mechanisms such as evaporation,
condensation or chemical reactions. a)
Evaporation or condensation It is the transport
of the droplet vapor to or from the droplet
surface and the change of phase at the surface.
The driving force is the concentration
difference of the droplet vapor between the
surface and the free stream. The mixture is then
a binary mixture (carrier gasdroplet vapor). By
Ficks law
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Particle-Fluid Interaction
This equation can be written in the following
form
We can continue approximating this equation as
The average properties between the surface and
the freestream are called the film conditions.
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Particle-Fluid Interaction
The constant of proportionality is the Sherwood
number. (Why?)
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Particle-Fluid Interaction
The Sherwood number is 2 for a droplet which is
evaporating with radial symmetry with no forced
or free convection effects. The mass fraction of
the vapor at the droplet surface can be evaluated
if the droplet temperature is known. The partial
pressure of the vapor at the surface is the
saturation pressure corresponding to the droplet
temperature. The mole fraction of the droplet
vapor at the surface is the ratio of the partial
pressure to the local pressure
The corresponding vapor mass fraction is
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Particle-Fluid Interaction
The effect of a relative velocity between the
droplet and the conveying gas is to increase the
evaporation or condensation rate. This effect is
usually represented by the Ranz-Marshall
correlation for Sherwood number
The evaporation of a droplet can be represented
by the D2-law which states that the square of the
droplet diameter varies linearly with time.
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Particle-Fluid Interaction
The lifetime of evaporating droplet can be
obtained by setting D0
Note that models have been investigating
evaporation in clusters (where a group of
evaporating droplets are forming clusters). It is
found that in dense clusters, evaporation occurs
primarily due to diffusion effects (Sh2), while
convection plays the dominant role in very dilute
flows.
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Particle-Fluid Interaction
b) Mass transfer from slurry droplets This is
important in the technology of drying such as
making powdered milk. Slurries are atomized and
sprayed into a hot gas stream where the water is
driven off and dried product is collected. The
droplet material is assumed as a porous medium
formed by the solids. During drying, the size of
the droplet may not change considerably, but the
mass decreases as the moisture is gone. The
drying process has generally two stages - the
constant rate stage - the falling rate periods
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Particle-Fluid Interaction
In the constant rate stage, the drying rate
proceeds as if the slurry droplet were a liquid
droplet so that a liquid layer forms on the
slurry droplet
The diameter of droplet is almost constant
during the constant rate period as well as other
parameters, so dm/dt remains constant. The
amount of moisture is identified by the moisture
ratio or wetness
When the wetness reaches the critical moisture
ratio, the drying enters the falling rate period.
Within the falling rate period the primary
resistance to mass transfer is the transfer of
liquid through the pores of the solidphase. Mass
transfer rate is considerably slowerand it
progressively slows down.
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Particle-Fluid Interaction
The following picture shows the qualitative
variation of droplet temperature, moisture
content, and the evaporation rate versus time
indicating the two stages of drying.
C. Crowe, M. Sommerfeld, Y. Tsuji. Multiphase
flows with droplets and particles. CRC Press
(1998).
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Particle-Fluid Interaction
c) Combustion The combustion of a single
droplet is modeled as a liquid fuel droplet
surrounded by a flame. The simplest model is
based on a spherically symmetric flow of vapor
from the droplet, and oxidizer from the
surroundings. A flame front occurs where the
fuel vapor and oxidizer meet and react. The
position of the flame front is established by the
heat transfer necessary to evaporate the droplet
and supply fuel to support the flame. In the
basic model, the convective and conductive heat
transfer are equated
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Particle-Fluid Interaction
Integration of this equation between the flame
radius and the droplet radius yields
D is the droplet diameter (2rf). The linear
proportionality between the burning rate and
droplet diameter leads to a D2-law for burning
rate where the burning rate constant is
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Particle-Fluid Interaction
Example Calculate the burning time of a 200
micron diesel oil droplet. We had the total
burning time of a droplet as
From the given table, the burning rate constant
is 7.9x 10-7 m2/s, so the burning time is
The effect of a relative velocity between the
droplet and the oxidizer is to increase the
burning rate. One way to estimate this effect is
to assume that the Ranz-Marshall correlation is
valid in the form of
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Particle-Fluid Interaction
The combustion of a coal particle is completely
different. It consists of 4 different
components - volatiles combustible gases such
as methane - moisture -char - ash 1) volatile
and moisture are driven off. It is quick. 2)
combustible volatiles contribute to a gas-phase
flame. 3) Char burns very slowly