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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Homogeneous flow theory
Problem 1 What is the frictional pressure drop
in a pipe carrying a homogeneous mixture of steam
and water with the following data? Compare the
results with a prediction with constant friction
factor. Pipe length L1 m, saturated water at
entrance xi0, vertical tube with diameter
d1.2cm. Flow rate is Gf 270 kg/s.m2. The heat
flux to the tube is ?6.5x105 W/m2. ( Assume that
Cf00.015 represents the constant value of
friction factor). a) 24 atm, and b) 68 atm.
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Homogeneous flow theory
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Homogeneous flow theory
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Homogeneous flow theory
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Homogeneous flow theory
Problem 2 Write the governing equations of flow
in a transient problem for homogeneous two-phase
flow.
Continuity
Momentum
Energy
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Homogeneous flow theory
We use continuity and momentum equation in this
equation
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Homogeneous flow theory
We use this equation now for a flow in a
high-pressure straight-tube evaporator in which
pressure changes and viscous dissipation are
small compared with the other energy terms during
a transient state. The tube is heated by a
uniform flux and there is no shaft work.
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Homogeneous flow theory
This equation is called propagation equation for
quality changes. It gives the dynamic response of
a boiler channel. If a given particle is
identified by the time t0 at which it starts to
evaporate, this equation can be integrated to
give
WHY?....
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Homogeneous flow theory
Consider the fluid particle moving with the fluid
and during which its quality changes. If we
attach our eyes to the particle (Lagrangian point
of view) we will observe the substantial time
rate of quality change for the particle
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Homogeneous flow theory
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Homogeneous flow theory
This figure shows how x varies with time as heat
flux increases. Use the data of the solved
problem.
vfg0.08153 phi6.5e5 d0.012 hfg1.85e6
Omega4vfgphi/d/hfg vf0.00119 t00.0011
x(vf/vfg)(exp(Omegat)-1) plot(t,x)
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Separated flow model
Problem 3 Using separate-cylinders model, show
that for a constant friction factor, the value of
the exponent for two-phase multipliers equation
is n2.5.
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Separated flow model
If the gas flows in a separate cylinder as well
as liquid in a separate one, we have
In imaginary gas cylinder
In constant Cf this is pressure drop of the
equivalent single-phase flow through the main pipe
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Separated flow model
This gives us
References

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