Theory and modeling of multiphase flows

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Theory and modeling of multiphase flows
Payman Jalali Department of Energy and
Environmental Technology Lappeenranta University
of Technology Lappeenranta, Finland Fall 2006
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CFD modeling of multiphase flows

• Different approaches can be applied in modeling
  two-phase flows
• Lagrangian approach Fluid phase is a continuum
  and the time average is taken by following a
  certain solid particle and observing it at some
  time interval. Particle trajectories are
  calculated from the equation of particle motion.
  Lagrangian averages are popular especially in
  modeling the dynamics of a single particle or a
  dilute suspension.
• Eulerian approach Particle phase is also treated
  as a continuum. The formulation consists of three
  essential parts the derivation of field
  equations, constitutive equations and interfacial
  conditions. The field equations state the
  conservation equations for, e.g., the momentum
  and mass. The constitutive equations close the
  equation system by taking into account the
  structure of the flow field and material
  properties by experimental correlations. The
  Euleriam approach has been the most widely used
  group of averaging for multiphase flows, because
  of its close relation to measuring techniques.

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CFD modeling of multiphase flows Mixture model

• An alternative method to the Eulerian method is
  applied in the classical mixture theory in which
  the principles of continuum mechanics for a
  single phase are generalized to several
  interpenetrable continua. The basic assumption is
  that, at any instant of time, all phases are
  present at every material point. The balance
  equations are assumed for mass and momentum
  conservation. It also requires constitutive
  relations to close the system of equations.
• Here, we will focus on basic equations, field
  equations of mixture model, the relative
  velocity, constitutive equations, and
  applications of the mixture model.

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CFD modeling of multiphase flows
CFD modeling of multiphase flows Mixture model
Basic equations The average velocity of phase k
is
The average is taken inside some averaging
domain such as volume, time-step, a set of
experiments or a group of particles. Hence,
balance equations of mass and momentum can be
written for the averaged quantities given for
each phase.
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CFD modeling of multiphase flows Mixture model
Continuity equation for phase k
Momentum equation for phase k
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CFD modeling of multiphase flows Mixture model
Before solving mass and momentum balance
equations, constitutive equations for the average
stress terms, turbulent stress and the
interaction forces between phases have to be
formulated. The type of the constitutive
equations depends on the averaging approach. The
existence of several types of multiphase flows
makes derivation of constitutive relations very
complex. In practice, multiphase modeling
commonly employs single-phase closure relations
extended to multiphase situations. A common
simplification in multiphase calculations is that
the dispersed phase is assumed to consist of
spherical particles of a single, average particle
size. The interactions between different
dispersed phases are frequently neglected.
Simplified assumptions are used at the walls. No
model generally valid for all possible multiphase
multiphase situations exists. Due to complexity,
there is little hope such models can be ever
developed. A large portion of the work in the
derivation of closure relations is based on
empirical information.
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CFD modeling of multiphase flows Mixture model
Field equations
Consider a mixture with n phases. Assume that one
of the phases is a continuous fluid indicated
with a subscript c. The dispersed phases can
comprise of particles, bubbles or droplets. The
dynamics of the system is comprehensively
described by Mass and momentum balance equations
together with the appropriate constitutive
equations.
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CFD modeling of multiphase flows Mixture model
From continuity equation for phase k, we sum them
over all phasesfor the mixture
We know that the total mass is conserved,
therefore
So the continuity equation for the mixture will
be
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CFD modeling of multiphase flows Mixture model
Continuity equation will be summarized as the one
for single-phase flow if we consider it in terms
of average volumetric flux
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CFD modeling of multiphase flows Mixture model
Momentum equation for the whole mixture is the
sum over all momentum equations of phases. Note
that in the following formula you can write only
one component of velocity at a time
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CFD modeling of multiphase flows Mixture model
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CFD modeling of multiphase flows Mixture model
The influence of the surface tension force on the
mixture is
It is dependent of the geometry of interface.
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CFD modeling of multiphase flows Mixture model
Continuity equation for phase k can be written
after eliminating the phase velocity
This is usually called diffusion equation and the
mixture model is the diffusion model.
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CFD modeling of multiphase flows Mixture model
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CFD modeling of multiphase flows Mixture model
Question Why do we have the following relation?
Question Why do we have?
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CFD modeling of multiphase flows Mixture model
Drift velocity and alternative formulation of the
phase continuity equation If the phase densities
are constants and the interphase mass transfer
can be neglected, the phase continuity equation
can be written in terms of the volumetric flux jk
Continuity of a phase
The approach based on this formula is called the
drif-flux model
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CFD modeling of multiphase flows Mixture model
The continuity equation will be written then as
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CFD modeling of multiphase flows Mixture model
It is worth to mention that so far in formulating
the mixture model equations from the full
multiphase model we have not made any further
assumptions. The field equations for the mixture
and the continuity equation for phase k in terms
of mixture velocity were obtained from the
original phase equations using only algebraic
manipulations. However, the closure of the field
equations requires some assumptions as in full
multiphase models. The most critical
approximation of the mixture model will be made
in replacing the phase momentum equations with
algebraic equations for the diffusion velocity
uMk.
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Reference
Manninen M., Taivassalo V., Kallio S., On the
mixture model for multiphase flow, VTT
Publications (1996)