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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.)
Mass Transfer I
DEFINITIONS IN USE:
• density – mass of mixture per unit volume ρ [kg/m3]
• species - chemically distinct substances, H2O, H2, H, O2, etc.
• partial density of A – mass of chemical compound (species) A per unit volume ρA [kg/m3]
• mass fraction of A – ρA/ρ = mA
note: ρA + ρB + ρC + … = ρ mA + mB + mC + … = 1
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 = u (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
DEFINITIONS IN USE:
• velocity of mixture in the specified direction = GTOT/ [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
s
mD
ms
kgΓ
jD
DΓ
Gx
jΓ
x
mG
2
j
j
j
jj
j
jjj,DIFF
mixture the in oft coefficien diffusion
measured is which in direction in distance
mixture the in oft coefficien exchange
:where
law alexperimentdiffusion of Law sFick'
t
ρ
x
GVρG
zwyvxuV
t
ρVρ0
z
ρw
y
ρv
x
ρu
t
ρ
0Δt0Δz
ρwΔ
Δy
ρvΔ
Δx
ρuΔ
Δt
Δρ
ΔxΔyΔzΔt
ΔρΔxΔyΔz
ΔxΔyΔwwΔxΔzΔvvΔyΔzΔuuΔρρ
ρwΔxΔyρvΔxΔzzyu
:case ldimensiona-onefor then ,but
where
or
while
thus , : volume in daccumulate mass
out flux mass
influx mass
u,x
v,y
w,z
xy
z
mass flux in
mass flux out
mass accumulated
-
=
The d2 Law - assumptions(i) Spherical symmetry: forced and natural convection are neglected. This
reduces the analysis to one-dimension.
(ii) No spray effect: the droplet is an isolated one immersed in an infinite environment.
(iii) 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.
(iv) Isobaric processes.
(v) 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).
(vi) 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.
Gas-phase QUASI-steadiness – characteristic times analysis.
In standard environment the gas-phase heat and mass diffusivities, g and g
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.
g
liqg
K
When (at which value of D∞) this assumption breaks down, i.e. when the
diffusion time is equal to the surface regression time? D∞2/ g ≈ D0
2/K, but
. So, the steady assumption breaks down at such a distance that g
liq
0D
D
g
liqg
K
For standard atmospheric conditions it breaks down at
For near- or super-critical conditions, where its invalid everywhere.
03
0g
liq0 D3210DDD
10g
liq
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.
Heat and mass diffusion from kinetic theory
tcoefficien diffusion thermal
tcoefficien diffusion binary
species of fraction mass
velosity average with moving
scoordinate torespect with species of molecule of velosity
scoordinate fixed torespect with species of molecule of velosity
volume)unit per species of molecules of(number densitynumber
Law) s(Fick' termgradient ionconcentrat
term force external
termgradient pressure
T1
12
2,12,1
22112221110
0
011
1
2,1
1
221111
11
111
221122
2222222
T121221
2
1111
D
D
2,1mn
mnmn;VmnVmn1
v
v
1vvV
1v
2,1n
n
n
XnXnXmp
mn
plnmn
n
n
XnXnXmp
mnpln
mn
n
n
n
nd
TlnDdDmmn
VmnJ
Soret term
x
Tλ
xt
TρC
Dλ
VVD
D
mmn
kTTq
x
nD
xt
n
TlnDn
nDmm
nJ
TlnDdDmmn
VmnJ
0k
0k
k
D
D
mmnk
TlnkdDnn
nVV
DD;DDdd0JJ
v
12
2112
T1
21
112
1
T1
11221
2
1
T121221
2
1111
T
T
T
12
T1
212T
T11221
2
21
2112T2
T12121
:allyexperiment i.e. , asmanner same the in determined is
:effect radiation and reaction chemicalwithout flux Heat
law second sFick'
where vessel, closed stationaryconcider llwe' determine To
:becomes forces external andgradient pressure of abscence the in
regionhot the towards moving is 1component
region cold the towards moving is 1component
diffusion )(molecular ordinary and diffusion thermal the of
importance relative the of measure a is ratio, diffusion thermal -
and since
Dufour term
Heat and mass diffusion from kinetic theory
jjj
jj
TOT
jjj
jTOTjjj
j,DIFFj,CONVj,TOT
j
j
jjj,TOT
Rt
m
x
m
xx
mG
Rmtx
m
xGm
xm
GGG
m
jR
Rtx
G
j
:for equation aldifferenti
reaction chemical by volume,unit per species of production of rate -
condition, continuity to similarly
species chemical of onconservati of LawjR
Rate of accumulation of
mass of component j
Mass flow rate of component j into
the system
Mass flow rate of component j out of
system
Rate of generation of mass of
component j from reaction
Rate of depletion of mass of
component j from reaction
constdxRdx
dmmG
0dx
md
dx
dmG
0dx
dm
dx
d
dx
dmG
Rdx
dm
dx
d
dx
dmG
j
jj
jjTOT
2
j2
jj
TOT
j
jj
jTOT
jj
jj
TOT
:equation state-steady of form integral the
:constant is also if
:conditions reaction no steady
:conditions steady
:cases particular , species chemical of onconservati of Law
jjj
jj
TOT Rt
m
x
m
xx
mG
jjj
jj
TOT Rt
m
x
m
xx
mG
jjj
jj
TOT Rt
m
x
m
xx
mG
00
0
00
The Stefan flow problem• Steady state
• Vapour diffuses upwards and escapes
• Air does not dissolve in liquid
• j is uniform
• There is no reaction
Known:
x=0 mVAP=mVAP,0=mVAP,SAT
x=x1 mVAP=mVAP,1
Find:
• GTOT
• mVAP(x)
air
liquid
x
0
x1
Vapour diffuses
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?
The Stefan flow problem - solution
constxG
1ml
1mGdx
dm0GGG
Gconstdx
dmGm
mm x x
mm m 0 x
0dx
md
dx
dmG
VAPVAP
VAPVAP
VAPAIR,TOTVAP,TOT
VAP,TOTVAP
VAPVAP
1VAP,VAP1
VAP,SAT0VAP,VAP
2VAP
2
VAPVAP
j
n
:yields nintegratio second
:thus ; ,
:so moves,vapour the onlythat note
:yields nintegratiofirst
:conditions boundary
:uniform is plus ,conditions rection no steady,
The Stefan flow problem - solution
xG
expm11m
m1
mm1lnx
G
constxG
1ml
const1ml
VAP0,VAPVAP
0,VAP
1,VAP0,VAP1
VAP
1VAP
1,VAP
0,VAP
and
hence
n
n
:conditions boundary inserting
1VAP
xG
0,VAP
1,VAP0,VAP
m1
mm
1
0,VAPm
1,VAPm
1x
0
1
1xx0
almost linear behavior
region of validity
Droplet evaporation I (no energy concerns)
The phenomenon considered:A small sphere of liquid in an infinite gaseous atmosphere vaporizes andfinally 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
• ΓVAP independent of radius
• large distance between droplets
• no chemical reaction
1mrGrdr
dm
rGrdr
dmGm
rGrGrG
r
G
rGGr
VAP2
002VAP
VAP
200
2VAPVAPVAP
200
200,VAP,TOT
2VAP,TOT
0
0
200
2
ie
:law sFick' from therefore
vapour of onconservati (ii)
surfacedroplet of radius -
area surfaceunit per
liquid of change phase of rate -
where
onconservati mass (i)
Vapor concentration distribution mVAP in the gas.
ro
r
GoG = GTOT,VAP
0VAP,
VAP,0VAP,
VAP
200
VAP,VAP
0VAP,VAP0
VAP
200
VAP
2VAP
200
VAP
VAP
m1
mm1ln
rG
mm: r
mm:rr
constr
1rG1mln
r
drrG
1m
dm
:is rate nevaporatiofor solution the
:are conditions boundary
gintegratinafter
:equation aldifferenti separable to leadsent rearrangem
:Solution
follows as is ondistributi the of form equalitativ the
that note ,
:ondistributivapor for solution the
0r
r
0VAP,0mVAP
r
r
VAP,
0VAP,VAP,VAP
r
r
VAP,
0VAP,
VAP,
VAP
rrm11rm
m1
m1m11rm
m1
m1
m1
m1
0
VAP,
00
mVAP
mVAP,0
mVAP,∞
r0r
1
dtm1
mm1ln4dDD
D,m1
mm1ln
rG
G2
dt
dD
D
G
dt
drr4G
dt
drr4
GGG
dt
drρπr4πr
3
4ρ
dt
dVρ
dt
d
dt
dm
t
0t 0,VAP
,VAP0,VAP
LIQ
VAP
D
D
00
00,VAP
,VAP0,VAP
0
VAP0
LIQ
00
0
LIQ
0020LIQ
0LIQ
20
0VAPOURLIQ
0LIQ
20
30LIQLIQLIQ
droplet
0
initial,0
:for DE the
rate onvaporizatifor solution the reminding hence,
: diameter, of terms inor
:thus ,
so direction, outwards the in moving is
vapour the while center, its towards moving is (liquid) surfacedroplet
:droplet liquid theFor
0VAP,
VAP,0VAP,VAP
LIQ2
initial,0VAP
VAP20
VAP
initial,00
0VAP,
VAP,0VAP,
LIQ
VAP20
2initial,0
,VAP0,VAPVAP
m1
mm1lnΓ8
ρDt
0tD
t
0tDD
m1
mm1ln
ρ
Γt8tDD
mm
: setting
by given therefore is , zero, to fallsdiameter the whichat time the
when where
:to leads
variables separating then time, oft independen all are , , If
:Solution
0VAP,
VAP,0VAP,VAP
LIQ2
initial,0VAP
3LIQ
5OH
OH
m1
mm1lnΓ8
ρDt
m
kg1000ρ
m s
kg 106.2Γ
0075.0m
2
2atm 1 and C10at air saturatedfor
:Data
on.vaporizati of time the Calculate mm. 0.1 isdiameter initial Its
.atmosphere 1at air dry into vaporates C10at droplet water A
0
0
s43.6
0075.010075.0
1ln106.28
1000101.0t
5
23
VAP
0
100
200
300
400
500
600
700
0 0.2 0.4 0.6 0.8 1
Initial droplet diameter [mm]
Eva
po
rati
on
tim
e [s
]
• 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,∞ may both vary with time.
• ΓVAP usually depends on temperature and composition.
Limitations
The Energy FluxDEFINITIONS IN USE:
velocity mixture is where
mixture of enthalpy stagnation
mixture of enthalpy specific
at of value
etemperaturbut anything of
tindependen pressure,constant at ofheat specific :where
:gases ideal of mixture afor that note
mixture a in component of enthalpy partial
V2
Vhh
h
hmh
h
Thh
jc
hdTch
jh
2
all jjj
0j0j,
j
0j,
T
T
jj
j
0
heat flux per unit area (caused by temperature gradient)
where thermal conductivity of mixture (Fourier's Law)
The total energy flux across a surface, per unit area, c
Q
dTQ λ
dxλ
E
2
an be regarded
as the algebraic sum of individual components as follows:
, heat flux
, shear work
, ki2
S
E Q
W
V G
,
netic energy
, enthalpy flux
The I Law of thermodynamics stands:
where S is source from outside the control volume (eg by radiation)
j TOT jall j
h G
dES
dx
E
E +dE
S
x
2
,
2
here G
and use has already been made of 0.
The genaration rate of species per unit volume is . The
TOT, j jSj TOT, j
all j
TOT jall j
j
dG dhdWdE d dT d VS λ G h G
dx dx dx dx dx dx dx
G
dG
dxj R
0
, ,
0
,0
refore:
and from definition of 0
Because all the in the definition of are constant:
H
TOT j TOT jj j j j
all j all j
j jall j
j, j
Tj
TOT, j TOT, j j j TOT, j j
T
dG dGR h h R
dx dx
R R
h h
dh d dTG G c dT h G c
dx dx dx
2
ence, the D.E. for T:
2S
j j TOT, j jall j all j
dWd dT d V dTS λ G h R G c
dx dx dx dx dx
Note, that a flow-rate-average specific heat could be defined as:
then the last term of the D.E. , but in general,
TOT, j jall j
TOT, j jall j
j jall j
G c
cG
dT dTG c Gc
dx dx
c c c m c
To receive another form of D.E. for temperature, note that:
or
but 0, and
TOT, j j DIFF, j j TOT, j j j j DIFF, j
j TOT, j j DIFF, j DIFF, jall j all j all j
j DIFF, j DIFFall j
G m G G c G c m G c G
c G cG c G G
c G cG
Thus:
, j j DIFF, jall j all j
j TOT, j j DIFF, jall j all j
c c G
c G cG c c G
0
fuel the of combustion ofheat the is where
:made be can onsubstituti following the species,
single a ofthat to linked be can species the all of rates reaction the if
vanishes termlast the , allfor or if obviously,
:apply now can we
etemperatur thefor D.E. the of formfirst the of Instead
FU
FUFUall j
jj
DIFF,jj
all jDIFF,jj
all jjj
2S
all j
jTOT,j
TOT,jj
2S
H
RHRh
j0G0cc
dx
dTGcc
dx
dTcGRh
2
V
dx
dG
dx
dW
dx
dTλ
dx
dS
dx
dhG
dx
dGh
2
V
dx
dG
dx
dW
dx
dTλ
dx
d
dx
dES
0, for the case of Stefan flow
Droplet evaporation II
ro
r
Go G = GTOT,VAP
E
Qo
2 20 0
(i) mass conservation
Gr G r
0
where
- rate of phase change of liquid
per u
G
0
2
,
nit surface area
- radius of droplet surface
(ii) Energy 2
S j TOT jALL j
r
VE Q W G h G
0
0
2 20 0
2 2, 0 ,0 0 0 ,0
2 20 0 0
2 00 0
0
together with and
TOT VAP VAP VAP VAPr r
r r
VAP
Er E r
dT dTr G c T T h r G h
dr dr
dTGr G r λ Q
dr
QdTr G c T T
dr G
2
0r
heat flow to gas phase close to liquid surface
0VAP0
0
0
0VAP
00VAP0
00
20VAP0
VAP0
00
rcG
GQ
TTc1ln
rrTTcG
QTTlnconst
rTT
constr
1rcG
cG
QTTln
to leads
at putting , so
at :is condition boundary
:Solution
20
0
00VAP0
2 rG
QTTcG
dr
dTr
:follows as ygraphicall presented be may this
or ,
:found be to is when needful is solution of form eAlternativ
1rcG
exp
rcG
TT
rQ
1rcG
exp
TTcGQ
Q
0VAP0
0VAP0
0
00
0VAP0
0VAP00
0
0
00
TT
rQ
0VAP0 rcG
y = -x
0
1
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, we’ll 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.
dt
dT
dt
dm
,constT,m
pressure,Tfm
LGQcr
3
dt
dT
r4LGdt
dTcr
3
4r4Q
1rGc
exp
TTGcQ
G
dt
dr
m1
mm1ln
rG
VAP,
,VAP
00,VAP
00LIQLIQ0
0
200
0LIQLIQ
30
200
00VAP
00VAP0
LIQ
00
0,VAP
,VAP0,VAP
0
VAP0
and for needed be will equations additional two not, if
:atmosphere theFor
i.e.
nevaporatiodroplet upheat droplet droplet the toheat overall
:surfacedroplet on balanceheat a From
onconservati energy from -
radiusdroplet the reduces onvaporizati -
onconservativapour from -
Clausius-Clayperon equation for pressure,Tfm 00,VAP
T
1
T
1
R
LMexpPTP
T
P
atm1PC100TPT
RT
dTLM
P
dP
MR
vP
PM
RTv
Tv
L
dT
dP
boiling
VnV
V
0nboiling
T
T2
V
P
P V
V
V
VV
VVV
V
V
0
boiling
V
n
: etemperatur
arbitraryat pressurevapour thefor expression following
the to leeds water)for at as ,at (say
point known some from gintegratin
vapour the ofweight molecular - constant, gas universal -
vapour the of volume specific - surface, theat pressurevapour -
,
GASGASVAP0,VAP
VAP0,VAP0,VAP
0nboiling
VAPn00,VAP0,VAP
GG
V
GG
VV
MM
Mm
T
1
PatT
1
R
LMexp
P
P
P
TP
TP
PP
TP
P
TP
fraction massvapour while
:caseour In
gas of pressure partial is gas, of fractionmolar is
pressure prevailing is andvapour of fractionmolar is where
and
:law sRault' to according
ambience) the(for gas andvapour namely gases ideal of mixture assuming
0,VAP
,VAP0,VAP
0
VAP0 m1
mm1ln
rG
LIQ
00 G
dt
dr
1
rGcexp
TTGcQ
00VAP
00VAP0
LGQcr
3
dt
dT00
LIQLIQ0
0
pressure,Tfm 00,VAP
0,VAPm
dt
dT00T
0Q
0G
dt
dr0
Linkage of equations
dt
dT
dt
dmVAP, and
:equations additional two be may There
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:
LGQ 00
time withinvariant is i.e. 000LIQLIQ0
0 T0LGQcr
3
dt
dT
VAPVAPc0VAP
0,VAP
,VAP0,VAP
00,VAP0
0,VAP
,VAP0,VAP
0
VAP0
0VAP
0VAP0
0VAP0
00
0VAP00
L
TTc1
m1
mm
TmG
m1
mm1ln
rG
L
TTc1ln
rcG
rcG
G/Q
TTc1lnLG-Q
1
and between relation following the to leads of neliminatio
vapour of onconservati from
to leads
into ngsubstituti
See slide A for –Q0≠G0L
LIQ
COMB0VAP
COMB
LIQ
0VAP
0VAP0
LIQ
0VAP
LIQ
LIQ00
HL
QTTcB
Q
HL
TTcB
rcG
HL
TTc1ln
H
L
.HLG-Q
:surface)droplet the towards sourceheat l(additiona
numerator the in sourse chemical called-so - combustion of
heat the includesnumber transfer Spalding of version" full" The
on.vaporizatifor number transfer Spalding -
:thus vaporized, liquid of mass
unitper heatingdroplet for needed energy -
onvaporizati ofheat latent specific -
:where
and surface,droplet theat mequilibriu
micthermodyna no is there speaking, Generally
slide A
:from calculated be
should prevail actually are which and of values The
unity to tends
zero, to tends or infinity to tends when obviously,
grearrangin
,pressureTfm
LTTc
1
m11m
mT
m
LT
LTTc
1
m11m
L
TTc1
m1
mm1
00VAP,
Le
0VAP
VAP,0VAP,
0VAP,0
0VAP,
Le
0VAP
VAP,0VAP,
Le
0VAP
0VAP,
VAP,0VAP,
massfor ydiffusivit
heatfor ydiffusivit
VAP
VAP
VAP
VAPVAP
VAPVAP DD
c/
cLe
0VAP,m
0T
1
BOILINGT
VAPVAPc0VAP
VAP,0VAP,
LTTc
1
m11m
Eq. Clapeyron-Clausius
,pressureTfm 00VAP,
VAP,m
T
0VAP,m
0T
of tionrepresanta Graphical
,pressureTfm
LTTc
1
m11m
00VAP,
c0VAP
VAP,0VAP,
VAPVAP
0 along this lineT T
Cases of interest:
(i) When T∞ 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:
(ii) When T∞ is low, and mVAP,∞ is close to zero, T0 is close to T∞. This implies T0≈T∞. Thus, mVAP,0 is approximately equal to the value given by setting T0=T∞ in and the vaporization rate can be calculated by:
,0 0 ,VAPm f T T pressure
VAPVAPc0VAP
VAP,0VAP,
LTTc
1
m11m
,0 0 ,VAPm f T T pressure
VAP,m
T
0VAP,m
0T
00 0
0 0
ln 1 ln 1VAP VAP BOILING
VAP VAP
c T T c T TG G
c r L c r L
,0 ,0
0 ,0
, ,0
0 ,
ln 11
ln 11
VAP VAPVAP
VAP
VAPVAP
m mG
r m
f T pressure mG
r f T pressure
As in example with water
dropletevaporating at 100C
KtDtD
L
TTc1ln
ct8DtD
m1
mm1lnt8DtD
G
dt
dr
m1
mm1ln
rG
L
TTc1ln
rcG
mT
2initial,0
20
0VAP
LIQVAP
2initial,0
20
0,VAP
,VAP0,VAP
LIQ
VAP2initial,0
20
LIQ
00
0,VAP
,VAP0,VAP
0
VAP0
0VAP
0VAP0
0,VAP0
:that reminding
fromor
fromeither
obtained be may onvaporizati the during time anddiameter
droplet between relation the and of ncalculatioAfter
Evaporation rate [m2/s]
The choice depends on whether T0 or mVAP,0 is easier to estimate
0,VAPm
0GD
LIQT
time
Qualitative results for D2-Law
Droplet heat up effect on temperature and lifetime
r
Tr
rrt
T
rr0
t,rfTTT
INTERIORLIQ
2
LIQINTERIORLIQ
0
INTERIORLIQ0INTERIORLIQ
,
limit" diffusion"
model heatingdroplet Transient
LGQcr
3
dt
dT
TTtfT
00LIQLIQ0
0
ORLIQ INTERI00
while
limit" ondistillati"
model tyconductivi liquid InfiniteSlowest limitFastest limit
Distillation limit
Diffusion limit
D2 Law
Center Temperature
Surface Temperature
T
(LIQ/r0,INITIAL2)t
(r/r0,INITIAL)2
(LIQ/r0,INITIAL2)t
Distillation limit
Diffusion limit
D2 Law
models) two
between difference the illustrate toorder
in (here conditions standard
in burning 300KT etemperatur
initial ofdroplet octane anfor results
VAPLIQ λλ
300
380
0.20.1
1
0
0.1