Liquid Droplet Vaporization

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Liquid Droplet Vaporization

• References
• Combustion and Mass Transfer, by D.B. Spalding, I
  edition (1979, Pergamon Press).
• Recent advances in droplet vaporization and
  combustion, C.K. Law, Progress in Energy and
  Combustion Science, Vol. 8, pp. 171-201, 1982.
• Fluid Dynamics of Droplets and Sprays, by W.A.
  Sirignano, I edition (1999, Cambridge University
  Press).
• The Properties of Gases and Liquids, by R.C.
  Reid, J.M. Prausnitz and B.E. Poling, IV edition
  (1958, McGraw Hill Inc).
• Molecular Theory of Gases and Liquids, by J.O.
  Hirschfelder et al, II edition (1954,John Wiley
  and Sons, Inc.)

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Mass Transfer I
DEFINITIONS IN USE

• density mass of mixture per unit volume ?
  kg/m3
• species - chemically distinct substances, H2O,
  H2, H, O2, etc.
• partial density of A mass of chemical compound
  (species) A per unit
• volume ?A kg/m3
• mass fraction of A ?A/? mA
• note
• ?A ?B ?C ?
• mA mB mC 1

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DEFINITIONS IN USE

• total mass velocity of mixture in the specified
  direction (mass flux) mass of
• mixture crossing unit area normal to this
  direction in unit time
• GTOT kg/m2s, GTOT ru (density x velocity)
• total mass velocity of A in the specified
  direction GTOT,A kg/m2s
• note GTOT,A GTOT,B GTOT,C GTOT
• convective mass velocity of A in the specified
  direction
• 
  mAGTOT GCONV,A
• note GCONV,A GCONV,B GCONV,C GTOT
• but generally, GCONV,A ? GTOT,A
• diffusive mass velocity of A in the specified
  direction
• GTOT,A GCONV,A GDIFF,A
• note GDIFF,A GDIFF,B GDIFF,C 0

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DEFINITIONS IN USE

• velocity of mixture in the specified direction
  GTOT/r m/s
• concentration a word used loosely for partial
  density or for mass fraction
• (or for mole fraction, partial pressure, etc.)
• composition of mixture set of mass fractions

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mass flux in
-
mass flux out
mass accumulated

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The d2 Law - assumptions

• Spherical symmetry forced and natural convection
  are neglected. This reduces the analysis to
  one-dimension.
• No spray effect the droplet is an isolated one
  immersed in an infinite environment.
• Diffusion being rate controlling. The liquid
  does not move relative to the droplet center.
  Rather, the surface regresses into the liquid as
  vaporization occurs. Therefore heat and mass
  transfer in the liquid occur only because of
  diffusion with a moving boundary (droplet
  surface) but without convection.
• Isobaric processes.
• Constant gas-phase transport properties. This
  causes the major uncertainty in estimation the
  evaporation rate (can vary by a factor of two to
  three by using different, but reasonable,
  averaged property value specific heats, thermal
  conductivity, diffusion coefficient, vapour
  density, etc).
• Gas-phase quasi-steadiness. Because of the
  significant density disparity between liquid and
  gas. Liquid properties at the droplet surface
  (regression rate, temperature, species
  concentration) changes at rates much slower than
  those of gas phase transport processes. This
  assumption breaks down far away from the droplet
  surface where the characteristic diffusion time
  is of the same order as the surface regression
  time.

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Gas-phase QUASI-steadiness characteristic times
analysis.
In standard environment the gas-phase heat and
mass diffusivities, ag and dg are of the same
order of 100 cm2s-1, whereas the droplet surface
regression rate, K -d(D02)/dt is of the order
of 10-3cm2s-1 for conventional hydrocarbon
droplet vaporizing in standard atmosphere. Thus,
there ratio is of the same order as the ratio of
the liquid-to-gas densities, . If
we further assume that properties of the
environment also change very slowly, then during
the characteristic gas-phase diffusion time the
boundary locations and conditions can be
considered to be constant. Thus the gas-phase
processes can be treated as steady, with the
boundary variations occurring at longer time
scales.
When (at which value of D8) this assumption
breaks down, i.e. when the diffusion time is
equal to the surface regression time? D82/ dg
D02/K, but . So, the steady
assumption breaks down at such a distance that
For standard atmospheric conditions it breaks
down at For near- or super-critical conditions,
where its invalid everywhere.
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The d2 Law assumptions
(vii) Single fuel species. Thus it is unnecessary
to analyze liquid-phase mass transport. (viii)
Constant and uniform droplet temperature. This
implies that there is no droplet heating.
Combined with (vii), we see that liquid phase
heat and mass transport processes are completely
neglected. Therefore the d2 Law is essentially a
gas-phase model. (ix) Saturation vapour pressure
at droplet surface. This is based on the
assumption that the phase-change process between
liquid and vapour occurs at a rate much faster
than those for gas-phase transport. Thus,
evaporation at the surface is at thermodynamic
equilibrium, producing fuel vapour which is at
its saturation pressure corresponding to the
droplet surface temperature. (x) No Soret, Dufour
and radiation effects.
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Heat and mass diffusion from kinetic theory
Soret term
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Heat and mass diffusion from kinetic theory
Dufour term
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The Stefan flow problem

• Steady state
• Vapour diffuses upwards and escapes
• Air does not dissolve in liquid
• Gj is uniform
• There is no reaction
• Known
• x0 mVAPmVAP,0mVAP,SAT
• xx1 mVAPmVAP,1
• Find
• GTOT
• mVAP(x)

Stefan flow Molecules of
the evaporating liquid are moving upwards. They
push the air out of the tank, thus no air is
present in the tank. Therefore, only the vapour
of the liquid is moving (diffuses). Where (for
which values of x) do you think the expressions
for mVAP(x) and GTOT will be valid?
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The Stefan flow problem - solution
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The Stefan flow problem - solution
region of validity
almost linear behavior
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Droplet evaporation I (no energy concerns)

• The phenomenon considered
• A small sphere of liquid in an infinite gaseous
  atmosphere vaporizes and
• finally disappears.
• What is to be predicted?
• Time of vaporization as a function of the
  properties of liquid, vapor and environment.
• Assumptions
• spherical symmetry (non-radial motion is
  neglected)
• (quasi-) steady state in gas
• GVAP independent of radius
• large distance between droplets
• no chemical reaction

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Vapor concentration distribution mVAP in the gas.
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1
mVAP,0
mVAP
mVAP,8
r
r0
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Limitations

• mVAP,0 has a strong influence, but is not
  usually known, it depends on temperature.
• relative motion of droplet and air augments the
  evaporation rate (inner circulation of the
  liquid) by causing departures from spherical
  symmetry.
• the vapour field of neighbouring droplets
  interact
• mVAP,0 and mVAP,8 may both vary with time.
• GVAP usually depends on temperature and
  composition.

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The Energy Flux
DEFINITIONS IN USE
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S
E dE
E
Dx
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0
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0, for the case of Stefan flow
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Droplet evaporation II
Go
G GTOT,VAP
ro
E
r
Qo
heat flow to gas phase close to liquid surface
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y -x
1
0
So, a positive G0 reduces the rate of heat
transfer at the liquid surface. It means that if
the heat is transferred to some let us say solid
surface, that we want to prevent from heating up,
we should eject the liquid to the thermal
boundary layer (possibly through little holes).
This liquid jets will accommodate a great part of
the heat on vaporization of the liquid. Thus,
well prevent the surface from heating
transpiration cooling. The smaller the holes the
smaller a part of heat towards the liquid
interior and, subsequently towards the solid
surface.
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Clausius-Clayperon equation for
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Linkage of equations
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Equilibrium vaporization droplet is at such a
temperature that the heat transfer to its surface
from the gas is exactly equals the evaporation
rate times the latent heat of vaporization This
implies
See slide A for Q0?G0L
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slide A
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• Cases of interest
• When T8 is much greater than the boiling-point
  temperature TBOILING, mVAP,0 is close to 1 and T0
  is close to TBOILING. Then the vaporization rate
  is best calculated from
• When T8 is low, and mVAP,8 is close to zero, T0
  is close to T8. This implies T0T8. Thus, mVAP,0
  is approximately equal to the value given by
  setting T0T8 in
  and the vaporization rate can
  be calculated by

As in example with water droplet evaporating at
100C
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The choice depends on whether T0 or mVAP,0 is
easier to estimate
Evaporation rate m2/s
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Qualitative results for D2-Law
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Droplet heat up effect on temperature and lifetime
Fastest limit
Slowest limit
(r0/r0,INITIAL)2
T
1
Diffusion limit
Distillation limit
380
D2 Law
Surface Temperature
Center Temperature
0
300
(aLIQ/r0,INITIAL2)t
(aLIQ/r0,INITIAL2)t
0.2
0.1
0.1