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
Steady flow in which phases are considered
together but their velocities are allowed to
differ In this simpler version of separated flow
model, the single velocity assumption for the two
phases is relaxed and the two phases are allowed
to have different velocities. Continuity
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Separated two-phase flows
Momentum
The equation is similar to what we wrote in the
homogeneous flow model, but the mean velocity is
related to phases velocities through quality,
x. Similar formula also applies to the enthalpy
of the mixture Density is determined by the
volume fraction
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Separated two-phase flows
Energy
Empirical correlations are needed for wall shear
stress and volume fraction of the second phase in
terms of the flow rates, fluid properties and
geometry. In gas-liquid flow, a common way of
correlating the wall shear stress is made by
Martinelli, in which the actual two-phase shear
stress is expressed as a factor ?2 times the wall
shear stress in a related single-phase flow of
liquid.
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Separated two-phase flows
For a vapor-liquid mixture, the void fraction
correlation is often expressed as a function of
pressure and quality
The equation for pressure drop in homogeneous
flow will be changed to the following form in
separated flows
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Separated two-phase flows
Maximum possible flow rate per unit area in a
nozzle containing separated phases of liquid and
vapor Consider a saturated liquid enters a
nozzle at high pressure with negligible kinetic
energy. It is discharged through the nozzle by
flashing (vapor formation due to pressure drop).
At the downstream with low pressure, the quality
is x. If the flow is in equilibrium and adiabatic
with negligible gravitational effects, what is
the maximum possible flow rate per unit area at
the downstream point?
One can use the energy equation for the nozzle
where there is no heat transfer (adiabatic),
mechanical work and gravitational effects
?h is the enthalpy change between the inlet and
outlet conditions. v is the velocity of phases.
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Separated two-phase flows
This equation can be derived from the energy
equation as following
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Separated two-phase flows
If the velocity ratio is nvg/vf , equations for
the enthalpy change and mass flow rate G can be
combined to result the following expression
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Separated two-phase flows
If thermodynamic path is known, ?h and G are
fixed. In order to maximize ? (and hence G) we
must have
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Separated two-phase flows
Then the maximum value of the mass flow rate is
Remark The value of x does not change much with
thermodynamic path, but ?h is a maximum for an
isentropic process. Thus the maximum value of
mass flow rate should also occur if the flashing
process is reversible and isentropic. Also,
choking is regarded as a condition of maximum
possible discharge through a nozzle exit area.
The current equation can be then argued that
corresponds to choking condition, where the Mach
number (M) is 1. Moody has some experimental
measurements for choked flow rates of steam-water
mixture which shows the trend of data are matched
with the lines given by this equation in the
range of 0.2ltxlt0.4.
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Separated two-phase flows
Diagram plotted by Moody for steam-water mixture
in different ranges of steam quality
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Separated two-phase flows
Wall shear stress and void fraction In order to
solve mass, momentum and energy equations we need
to have additional expressions for the wall shear
stress (frictional pressure drop) and void
fraction. We assume that the gas and liquid
phases have their own shares for the frictional
pressure drop. Thus we may define two-phase
Martinelli parameters as
Completely liquid flow Completely gas flow
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Separated two-phase flows
At the critical point where the phases are
indistinguishable the relationships between
two-phase multipliers of liquid and gas phases
are Laminar flow Turbulent flow
(1)
(2)
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Separated two-phase flows
A simple model of separated flow can be developed
by assuming that the two phases flow, without
interaction, in two-dimensional separate
cylinders and the sum of the areas is equal to
the real area of pipe. The pressure drop in each
of the imaginary cylinders is the same as in the
actual flow due to frictional effects only, and
that is calculated from single-phase flow theory.
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Separated two-phase flows
The results of this analysis are
(3)
In laminar flows n2, for turbulent flows
analyzed on a basis of friction factor n2.375 to
2.5, for turbulent flows calculated on a basis of
mixing-length n2.5 to 3.5.
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Separated two-phase flows
We can define a new variable X to avoid having
the unknown two-phase pressure drop in both the
correlating parameters
X2 gives a measure of the degree to which the
two-phase mixture behaves as the liquid rather
than as the gas. Gas and liquid can be either in
different regimes (laminar-turbulent) or the same
regime. Martinelli has given correlations for
different combinations of regimes for gas and
liquid phases.
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Separated two-phase flows
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Separated two-phase flows
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Separated two-phase flows
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Separated two-phase flows
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Separated two-phase flows
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References

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