lecture 3 governing equations for multiphase flows. continuum hypothesis. fragmentation mechanisms....
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Lecture 3Lecture 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.
Dynamics of dispersed systemsDynamics of dispersed systems
Mixture properties:
0
mass of componentBulk density =
volume of mixture
mass of componentPhase density =
volyme of component
volume of componentVolume fraction =
volume of mixture
mass frac
ii
mixture
ii
i
ii
mixture
m
m
mass of component
tion = mass of mixture
;
ii
mixture
mixture i mixture i
mX
m
m m
Bubbles
Particles
Mixture properties (continue)Mixture properties (continue)
component velocity =
mixture velocity =
ij iji
i
i i
mixture
mixture i
m VV
m
V
Continuity equations Mass fluxes
Momentum equations Momentum exchange
Energy equations Heat fluxes
Conduit flow during explosive eruptionConduit flow during explosive eruption
Schematic view of the system
xt
Flow regimes and boundaries. Homogeneous from magma chamber until
pressure > 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.
Modelling strategyModelling 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
Models of fragmentationModels of fragmentation FP - fragmentation at fixed porosity.
SR - critical elongation strain-rate
OP- critical overpressure in a growing bubble p
gpm
pp mg
RR
4
3
RRRR
22
2
small
Hydrostatic vs. Lithostatic Hydrostatic vs. Lithostatic pressure gradientpressure gradient
Chocked flowsChocked flows
FlowHigh pressure
Low pressure
Q Chocked
low highp p
Boundary conditionsBoundary 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
Slezin (1982,1983,1992)Slezin (1982,1983,1992)Main assumptions:Conduit has constant cross-section areaMagma - Newtonian viscous liquid, =constBubbles do not rise in magmaWhen = 0.7 - fragmentation, porous foamAfter fragmentation = 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 force=weight)
Slezin (results)Slezin (results)
WoodsWoods,, Koyaguchi Koyaguchi (1994) (1994)• Gas escape from ascending magma through the
conduit walls.
• Fragmentation criteria = *.
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 > rock permeability. Vertical pressure gradient to gas escape through the magma.
Barmin, Melnik (2002)Barmin, Melnik (2002)
• Magma - 3-phase system - melt, crystals and gas.
• Viscous liquid (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.”
0 0
0 0
00
1 1 1
1 1
m c m m
g g m m g
m m
c V Q
V cV Q
nV n V
Mass conservation equations (bubbly zone)Mass conservation equations (bubbly zone)
- volume concentration of gas (1-) - of condensed phase
- volume concentration of crystals in condensed phase
- 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
Momentum equations in bubbly zoneMomentum equations in bubbly zone
2
3.50
,
1
ms
gg m
g
s m g
c Vdpg
dx Dd pk
V Vdx
p p p
k k
- mixture density
- resistance coefficient
(32 - pipe, 12 -dyke)
k() - permeability
g- gas viscosity
p- pressure “s”- mixture, “m”- condensed phase, “g”-gas
Rayleigh equation for bubble growthRayleigh equation for bubble growth
4m g m
m
dR RV p p
dx c
Additional relationships:
3 04, R
3 g gR n p T
0 0
1
g g g b m g
m b m b
m s g s
gas
big particles
V V Q
V Q
V Q small particles
Equations in gas-particle dispersionEquations in gas-particle dispersion
0 0
0 0 0
1 1
1 ; 1;
mm b m m b gb sb
gg m s g g m s gb sb
m m c s b g
dVV g F F
dxdV dp
V g F Fdx dx
p RT
F - interaction forces:”sb” - between small and big particles
“gb” - between gas and big particles
Fragmentation waveFragmentation wave
0 0 0
2 0 2
0 2 0 2
Conservation la
1 1
1 1
ws
1
g g g g g b m
m s g b m
m g m m g g
b m g m g m s g
m m
V V V
V V V
p p
gas phase
condensed phase
mixturemomentum
big particle
V V
p V V
s mV V mo
*
Additional relat
; ;
1 1
ions
g m
s b
entu
p p p
m m
m
Steady discharge vs. chamber pressureSteady discharge vs. chamber pressure
Pressure profiles in the conduitPressure profiles in the conduit
Model ofModel of vulcanian vulcanian
explosion generated explosion generated by lava dome by lava dome
collapse collapse
AssumptionsAssumptions
• Flow is 1D, transient
• Velocity of gas and condensed phase are equal
• Initial condition - V = 0, pressure at the top of the conduit > 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
Mechanical modelMechanical model
0
2
Conservation of mass and number density of bubbles:
No mass transfer:
Equilibrium mt:
Moment
0, 0, 0,
(
um
1 ) 1 ,
,
32
:
g g l l
l m c g g
m
V V n nV
t x t x t x
p p c k p
V pV Vg
t x D
2
0 3
*
Rayleigh equation
Fragmentation condition
, 1 ,
4, ,
4 3
:
m l g
g l g gm
g l
p p p c
a a aV p p p RT a n
t x
p p p
Results of calculation (eq case)Results of calculation (eq case)
Discharge rate and fragmentation depthDischarge rate and fragmentation depth
(eq case)(eq case)
Pulsing fragmentationPulsing fragmentation
Seismic record of eruptionSeismic record of eruption
Results of simulations (no mt case)Results of simulations (no mt case)Discharge rate and fragmentation depth
Parameter Calculated Observed*
Duration 100 –600 s 60-300 s Max velocity 118-142 m/s 120-130 m/s Fragmentation depth 200-1400 m 200-1000 m Volume of material 105- 106 m3 2 105- 106 m3
*Druitt et. al. (2001)
Volcanic plumesVolcanic plumesPlinian Collapsing
High - comes to stratosphere
Ash fallout, climate change
Acid rains, aviation hazards
Pyroclastic flow generation
Unsolved problemsUnsolved 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
Unsolved problems (cont)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
Unsolved problems (cont.)!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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