Governing equations for multiphase flows. Continuum hypothesis. - PowerPoint PPT Presentation

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Governing equations for multiphase flows. Continuum hypothesis.

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Lecture 3 Governing equations for multiphase flows. Continuum hypothesis. Fragmentation mechanisms. Models of conduit flows during explosive eruptions and results. – PowerPoint PPT presentation

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Title: Governing equations for multiphase flows. Continuum hypothesis.


1
Lecture 3
  • Governing equations for multiphase flows.
    Continuum hypothesis.
  • Fragmentation mechanisms.
  • Models of conduit flows during explosive
    eruptions and results.
  • Volcanic plume dynamics in the atmosphere.

2
Dynamics of dispersed systems
Bubbles
Mixture properties
Particles
3
Mixture properties (continue)
Continuity equations ?Mass fluxes Momentum
equations ?Momentum exchange Energy
equations ?Heat fluxes
4
Conduit flow during explosive eruption
  • Schematic view of the system
  • Flow regimes and boundaries.
  • Homogeneous from magma chamber until pressure gt
    saturation pressure.
  • Constant density, viscosity and velocity,
    laminar.
  • Vesiculated magma from homogeneous till magma
    fragmentation.
  • Bubbles grow due to exsolution of the gas and
    decompression.
  • Velocity and viscosity increases.
  • Flow is laminar with sharp gradients before
    fragmentation due to viscous friction.
  • Fragmentation zone or surface (?).
  • Fragmentation criteria.
  • Gas-particle dispersion from fragmentation till
    the vent.
  • Turbulent, high, nonequilibrium velocities.
  • subsonic in steady case, supersonic in transient.



x
t
5
Modelling strategy
  • Equations
  • Mass conservation for liquid and gas phases
  • intensity of mass transfer, bubble nucleation and
    diffusive growth
  • Momentum equations
  • gravity forces, conduit resistance, inertia
  • momentum transfer between phases
  • Energy equations
  • energy transfer between phases
  • dissipation of energy by viscous forces
  • Bubble growth equation - nonequilibrium pressure
    distribution
  • Physical properties of magma - density, gas
    solubility, viscosity
  • Fragmentation mechanism
  • Boundary conditions - chamber, atmosphere,
    between flow zones

6
Models of fragmentation
4
FP - fragmentation at fixed porosity.

4
OP- critical

R

-
p
p

m
4
g
m
overpressure in a
R


growing bubble
p
ö
æ
s
3





2
r

ç
R
R
R
2
g
p
2
R
ø
è
m
small
4
SR - critical
elongation strain-
rate
7
Hydrostatic vs. Lithostatic pressure gradient
8
Chocked flows
High pressure
Low pressure
Flow

9
Boundary conditions
  • Magma chamber
  • pressure, temperature
  • initial concentration of dissolved gas -
    calculate volume fraction of bubbles
  • Atmosphere
  • Pressure is equal to atmospheric if flow is
    subsonic
  • Chocked flow conditions - velocity equal to
    velocity of sound
  • Need to calculate discharge rate

10
Slezin (1982,1983,1992)
  • Main assumptions
  • Conduit has constant cross-section area
  • Magma - Newtonian viscous liquid, mconst
  • Bubbles do not rise in magma
  • When a 0.7 - fragmentation, porous foam
  • After fragmentation a 0.7, all extra gas goes
    to interconnected voids.
  • When concentration of gas in voids 0.4 -
    transition to gas particle dispersion.
  • Particles are suspended (drag forceweight)



11
Slezin (results)
12
Woods, Koyaguchi (1994)
  • Gas escape from ascending magma through the
    conduit walls.
  • Fragmentation criteria a a.
  • Magma ascends slowly - looses its gas - no
    fragmentation - lava dome extrusion.
  • Magma ascends rapidly - no gas loss -
    fragmentation - explosive eruption.
  • Contra arguments
  • Magma permeability should be gt rock permeability.
  • Vertical pressure gradient to gas escape through
    the magma.

13
Barmin, Melnik (2002)
  • Magma - 3-phase system - melt, crystals and gas.
  • Viscous liquid m (concentrations of dissolved gas
    and crystals).
  • Account for pressure disequilibria between melt
    and bubbles.
  • Permeable flow through the magma.
  • Fragmentation in fragmentation wave.
  • 2 particle sizes - small and big.

14
Mass conservation equations (bubbly zone)
a - volume concentration of gas (1-a) - of
condensed phase b - volume concentration of
crystals in condensed phase r - densities, m-
melt, c- crystals, g - gas c - mass fraction
of dissolved gas k pg1/2 V - velocities, Q -
discharge rates for m- magma, g - gas n -
number density of bubbles
15
Momentum equations in bubbly zone
r - mixture density l - resistance coefficient
(32 - pipe, 12 -dyke) k(a) - permeability mg-
gas viscosity p- pressure s- mixture, m-
condensed phase, g-gas
16
Rayleigh equation for bubble growth
Additional relationships
17
Equations in gas-particle dispersion
F - interaction forcessb - between small and
big particles
gb - between gas and big particles
18
Fragmentation wave
19
Steady discharge vs. chamber pressure
20
Pressure profiles in the conduit
21
Model of vulcanian explosion generated by lava
dome collapse
22
Assumptions
  • Flow is 1D, transient
  • Velocity of gas and condensed phase are equal
  • Initial condition - V 0, pressure at the top of
    the conduit gt patm, drops down to patm at t 0
  • Two cases of mass transfer equilibrium (fast
    diffusion), no mass transfer (slow diffusion)
  • Pressure disequilibria between bubbles and magma
  • No bubble additional nucleation

23
Mechanical model
24
Results of calculation (eq case)
25
Discharge rate and fragmentation depth(eq case)
26
Pulsing fragmentation
27
Seismic record of eruption
28
Results of simulations (no mt case)
  • Discharge rate and fragmentation depth

29
Volcanic plumes
Plinian Collapsing
High - comes to stratosphere Ash fallout, climate
change Acid rains, aviation hazards
Pyroclastic flow generation
30
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31
(No Transcript)
32
Unsolved problems
  • Physical properties of magma
  • Magma rheology for high strain-rates and high
    bubble and crystal content
  • Bubbly flow regime
  • Incorporation of bubble growth model into the
    conduit model
  • Understanding bubble interaction for high bubble
    concentrations
  • Understanding of bubble coalescence dynamics,
    permeability development
  • Thermal effects during magma ascent - viscous
    dissipation, gas exsolution

33
Unsolved problems (cont)
  • Fragmentation
  • Fragmentation in the system of partly
    interconnected bubbles
  • Partial fragmentation, structure of fragmentation
    zone, particle size distribution
  • Gas-particle dispersion
  • Momentum and thermal interaction in highly
    concentrated gas-particle dispersions

34
Unsolved problems (cont.)!
  • General
  • Coupling of conduit flow model with a model of
    magma chamber and atmospheric dispersal model
  • Deformation of the conduit walls during explosive
    eruption
  • Visco-elastic deformation
  • Erosion
  • Interaction of magma conduit flow with permeable
    water saturated layers - phreato-magmatic
    eruptions
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