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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• Primary Contents of the Course
• Fundamentals of Continuum Mechanics -
  Vectors in a Cartesian coordinate system -
  Tensor algebra, summation convention and tensor
  notations
• Problems in Continuum Mechanics -
  Navier-Stokes equations for 1-phase fluid
  flows - Basic principles and definitions in
  two-phase flows - Conservation equations in
  two-phase flows Homogeneous Flows
  Separated Flows

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Basic definitions in two phase flows

• Mass flow rate (mass per unit time) MM1M2
• Volumetric rate of flow (volume per unit
  time) QQ1Q2
• Q1M1/?1, Q2M2/?2.
• Volume fraction of phases ?phaseVphase/Vtot

Note In different systems the name of ? could be
different. For instance, it is called void
fraction in gas-liquid flows as it represents the
gas phase. In porous media, it is called porosity
representing the gas phase filling the pore space
between solid boundaries. Then, it is solid
volume fraction for the solid phase.
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Basic definitions in two phase flows

• Quality (mass fraction of phases) x1M1/M
• Mass flux (mass flow rate per unit
  area) GG1G2M1/AM2/A
• Volumetric flux (volumetric flow rate per unit
  area) jj1j2G1/?1G2/?2
• Volumetric flux and the volume fraction are
  related by jphase?phase vphase

By convention, the volume fraction of the second
phase may be used in notation.
v Velocity of phases
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Basic definitions in two phase flows

• Relative velocity of phases v12v1-v2-v21
• Drift velocities of phases v1jv1 j, v2jv2
  j
• j is the average volumetric flux, which is a mean
  velocity
• Drift flux is volumetric flux of a phase
  relative to a surface moving at the average
  velocity j21?(v2-j), j12(1-?)(v1-j)
• Using the above-mentioned equations, we obtain

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Basic definitions in two phase flows
The plot of j1 and j2 is important as it can
represent what regime of flow appears in the
system.
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Homogeneous flow theory

• Suitable average properties are determined and
  the mixture is treated as a pseudofluid that
  obeys the usual equations of single-phase flow.
  All standard methods of fluid mechanics can then
  be applied.
• Weigthed average quantities required velocity,
  temperature, density, transport properties
  (viscosity, diffusion coefficient, conductivity
  etc.)
• Differences in velocity, temperature, and
  chemical potential between the phases will
  promote mutual momentum, heat and mass transfer.
• If one phase is finely dispersed in the other
  (in equilibrium), the average values of velocity,
  temperature and chemical potential are the same
  as the values for each phase, and it is
  homogeneous equilibrium flow.

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Homogeneous flow theory
One-dimensional steady homogeneous equilibrium
flow in a duct Continuity Momentum Energy
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Homogeneous flow theory
In addition to continuity, momentum and energy
equations we have some knowledge about the
equation of state. For example, for a steam-water
mixture, the steam tables are used. xM2/M.
Mean density For steady homogeneous flow with
velocity equilibrium
Momentum equation may be rearranged for pressure
gradient
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Homogeneous flow theory

• Here, we further develop momentum equation by
  expressing different terms contributing in
  pressure drop.
• Frictional pressure gradient
• The average wall shear stress can be expressed in
  terms of friction factor Cf

The ratio of A/P is expressed as D/4 in fluid
mechanics where D is called hydraulic diameter.
Then we will have
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Homogeneous flow theory
Since we have homogeneous flow vj Also
So, we will have
b) Accelerational pressure drop
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Homogeneous flow theory
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Homogeneous flow theory
For the two phases in equilibrium, ?f and ?g, or
likewise, ?1 and ?2 are only functions of
pressure, therefore
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Homogeneous flow theory
c) Gravitational pressure drop
Combining with
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Homogeneous flow theory
Phase change
Area change
Gravity
friction
Similar to Mach number
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References

• Segel L.A., Mathematics applied to continuum
  mechanics, Dover Publications Inc., New York
  (1987)
• Wallis G.B., One-dimensional two-phase flow,
  McGraw-Hill Book Company, New York (1969)
• Anderson J.D., Computational fluid dynamics,
  McGraw-Hill, Inc., (1995)
• Hjertager B.H., Basic numerical analysis of
  multiphase flows, Lecture notes, HUT, ESPOO (2006)