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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Separated two-phase flows
First model Steady flow in which phases are
considered together but their velocities are
allowed to differ Continuity
Momentum
Energy
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Separated two-phase flows
Second model Phases are considered
separately Instead of using correlations, we can
use more equations to relate the variables. This
can be accomplished by writing separate
conservation laws for the components, rather than
using only the equations for the entire
mixture. Continuity We write the mass balance
equation for each phase in an element of pipe as
follows Rate of mass increaseInput rate-Output
rate-Mass loss due to phase changeMass generation
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Separated two-phase flows
Mathematical equation for the mass balance of
phase 2 will be
For the liquid phase (phase 1) we have
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Separated two-phase flows
In the form of divergence, we can write
continuity equations as
If the area of the pipe is allowed to vary and
the continuity equations are integrated across
the duct
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Separated two-phase flows
Momentum We can write the momentum balance
equation for each phase in an element of pipe as
follows Rate of momentum increaseInput
rate-Output ratebody forcesurface force
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Separated two-phase flows
Momentum eq. of phase 2
Momentum eq. of phase 1
The term of surface forces is a source term which
contains all involved surface forces. The surface
force term includes Pressure gradient, viscous
forces, hydrodynamic drag, apparent mass effects
during relative acceleration, particle-particle
forces, and forces due to momentum changes during
evaporation or condensation (phase change) and so
on.
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Separated two-phase flows
In the continuity equation, if we neglect the
phase change term and generation source
If the density of phase 2 is constant, the
momentum balance equation for phase 2 will be as
follows
We can exclude the pressure gradient term from
the surface forces
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Separated two-phase flows
If we have incompressible Newtonian fluid having
only one component with no phase change, the
usual results of viscous flow are obtained
Example In annular flow, liquid flows as a film
on the wall of a pipe while gas flows down a
central cylindrical core. For a pipe of diameter
D let the interfacial and wall shear stresses be
?i and ?w. Assuming symmetrical vertical flow
with the positive direction measured upward,
derive the values of Ff, Fg, ff, fg, bf, bg and
hence the equations of motion.
The force exerted by the fluid on the gas is
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Separated two-phase flows
Note that the diameter of the gas flow is
Similarly, for the fluid phase we have
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Separated two-phase flows
The values of ff and fg are
The body forces bf and bg are
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Separated two-phase flows
The equations of motion can be then written in
the following form
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Separated two-phase flows

• Flow with phase change
• If phase change happens in the system, the
  following terms must be considered in the
  analysis
• Drag force from the duct wall on components 1
  and 2, namely Fw1 and Fw2 per unit volume of the
  flow.
• Drag force between the components is F12 acting
  on component 1 in the direction of motion and in
  the opposite direction on component 2.
• Since the two components have different
  velocities, any phase change will result in a
  change of momentum. The mass rate of phase change
  per unit length is dM2/dzd(Mx)/dzM.dx/dz and
  the velocity change is v2-v1. Thus the force due
  to momentum increase in phase change per unit
  volume of the pipe is

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Separated two-phase flows
It is not clear how much of this force is to be
charged to stream 1 and stream 2. In general, we
can describe a fraction ? of the phase change
force to stream 2, and the rest 1- ? to stream 1.
The choice of ? may be different in various
systems. Therefore, the two momentum equations of
steady flow with gravity as the only body force
will be
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Separated two-phase flows
These equations can be combined to result the
momentum balance for the mixture.
In another form (relative motion equation) which
does not include the pressure gradient term, we
have
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Separated two-phase flows
Evaluation of phase change parameter ? The
concept of entropy generation can help us to
evaluate this parameter. The entropy source per
unit length caused by irreversibilities and heat
transfer is
Using the combined equation for the mixture
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Separated two-phase flows
The energy equation for the two phases together
in steady state is
From the thermodynamics we know that
So, one has
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Separated two-phase flows
Combining three framed equations above, we can
get
Wall shear on the liquid
Heat transfer
Wall shear on the gas
Relative motion of phases
The entropy generation due to relative motion is
something new which could not be seen in
one-phase flow. In the cases ?0 (single-phase
liquid) or ?1 (single-phase gas) or v1v2
(homogeneous flow) the relative motion term is
zero (trivial solution). The only nontrivial
solution is obtained when the term inside
brackets vanishes.
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Separated two-phase flows
As we can say that
Combined equation was
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Separated two-phase flows
Now, we have two expressions for ?, which are
equivalent to the following two equations for
phases
Compare it to momentum equations for each phase
including ?
Therefore, with the condition that there is no
entropy generation due to relative motion term we
conclude that for isentropic process of phase
change
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References

• Wallis G.B., One-dimensional two-phase flow,
  McGraw-Hill Book Company, New York (1969)