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