1B ReitzEPD9636Spray Submodels

Nozzle flow, atomizationdrop drag, dispersion,breakup, collision,vaporization

130 deg.

150 deg.

180 deg.

Start of Injection=120 degrees

Han et al. SAE970625 Early InjectionHomogeneous Charge

2B ReitzEPD9636Discrete Drop Spray Model

• Drop injected with specified size, velocity (spray angle), temperature, distortion,…

Standard KIVA – DDM

Stochastic parcel model

• Low pressure, single component fuel vaporization model• O’Rourke collision/coalescence model

• Drop break up modeled with Taylor Analogy Breakup (TAB) model

• Solid sphere drop drag correlations

3B ReitzEPD9636

• Pump-line-nozzle system

Describe flows in chambers, high pressure pipe, moving parts

pumping chamber

delivery chamber

nozzle chamber

high pressure pipe

sac chamber

delivery valve

needle valve

pump plunger

feed/spill port

Injected drop spraycharacteristics - drop size, velocitytemperature, ….

Fuel System Modeling

Bosch Injection Rate Shape

-5

0

5

10

15

20

25

30

35

40

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

Tim e (s)

Mas

s Fl

ow (m

g/m

s)

Bosch rate-of-Injection data

4B ReitzEPD9636

R/D

L/D

InitialSMD

Cavitationregion

1 vena 2UmeanC

c

C r dc = − −[(.

) . / ] /10 62

11 42 1 2

Contraction coefficient (Nurick (1976)

0.00 0.04 0.08 0.12 0.160.6

0.7

0.8

0.9

1.0

r/d

c c sharp inletnozzle

C c

Cavitation Inception

Sarre et al. SAE 1999-01-0912

• Account for effects of nozzle geometry

Cavitating flow

Yes No

Non-cavitating flow

P < PvCavitation if

12 2( )C Cc c

P P2 1/

5B ReitzEPD9636

21

1

pppp

CC vcd −

−=

uC P P C P

C P Peffc c v

c v=

− + −−

2 1 22

1 2

1

( )( )ρ

A C P PC P P C P

Aeffc v

c c v=

−− + −

22 1 2

21

1 2

( )( )

C ldd = −0 827 0 0085. .

u CP P

eff d=−2 1 2( )ρ

A Aeff =

Cavitating flowYes No

P P2 1/ Non-cavitating flow

Nozzle discharge coefficient

Effective injection velocity

Effective nozzle area

Nozzle discharge coefficient

Effective injection velocity

Effective nozzle area

Lichtarowicz (1965)

ERC Nozzle Flow Model

6B ReitzEPD9636Jet Atomization Regimes

a.) Rayleigh breakup. Ddrop > Djet

b.) 1st wind-induced Ddrop ~ Djet

c.) 2nd wind-induced Ddrop < Djet d.) Atomization Ddrop << Djet Breakup at nozzle exit.

Jet velocity (Weber Number)

2 /gWe U a

We>40We~1

7B ReitzEPD9636‘Blob’ Injection model

• Inject ‘blobs’ at nozzlewith characteristic size equal to effective nozzlediameter

• Allow ‘blobs’ to breakupfollowing drop/jet breakup model

L

Blobs

InjectedBroken up

8B ReitzEPD9636Liquid Jet Atomization Models

Tan θ = v/Vrel

• Provide breakup drop size• Provide drop velocity

η = R η 0e ikz +ωt

‘Blob’

Kelvin-Helmholtz Instability Model

KH Wave Reitz Atom.Spray TechVol. 3, 309-337,1987Wave+FIPA Habchi SAE970881Wave+TAB Beatrice SAE950086

λ

r = B λo

t = B τ1

Vrel

break

Liquid

Gas

Wave breakup model

θ

9B ReitzEPD9636Linear Stability Analysis

= σ kρ1a2

1 - k 2a2 l2 - k 2

l2 + k 2 I1 kaI0 ka

+ ρ2ρ1

U - iω /k 2 k 2 l2 - k 2

l2 + k 2 I1 ka K0 kaI0 ka K1 ka

ω2 + 2v1k 2ω I1' ka

I0 ka - 2kl

k 2+l2 I1 kaI0 ka

I1' la

I0 la

Dispersion relationship:

Λa = 9.02 1 + 0.45 Z 0.5 1 + 0.4 T 0.7

1 + 0.87 We21.67 0.6

Curve fits:

Ω ρ1a3

σ0.5

= 0.34 + 0.38 We21.5

1 + Z 1 + 1.4T 0.6

We1=ρ1U2aσ ; We2=ρ2U2a

σ ; Re 1=Uav1

Z=We10.5

Re 1 ; T=ZWe2

0.5

where

10B ReitzEPD9636

R/DL/D

Breakup lengthNozzle flowNozzle flowmodelmodel

ERC Jet Breakup Model‘Blob’ injection size ‘a’

tan( ) ( )θ π ρ

ρ24

= ⋅A

f Tg

lv/U =

Drop initial velocity

L = C aρ1

ρ2

/ f(T )

Breakup lengthΛ

η=η0eΩt

r=B0Λ

KH Model

KHKHKH

aBΛΩ

= 1726.3τ

ΚΗΛ= 0BrKH

Drop/Blob breakup da/dt = - (a -r) / τ

11B ReitzEPD9636

s

Λ

LdL

d = 1.89 dL

• Sheet breakup length and resulting ligament diameter:

• Maximum growth rate ΩS and wave number KS determinedfrom dispersion relation for liquid sheets

SS

bS

UULΩ

=Ω

= 12ln

0ηη

S

bL K

hd 16=

LISA Model - Schmidt et al. SAE 1999-01-0496

Liquid Sheet Breakup Modeling

1

32242

12

1 42ρ

σννω kkQUkkr −++−=

12B ReitzEPD9636

0.7 msec 1.7 msec 2.7 msec

Injector hole diameter 560 µmInjection pressure 4.76 MPaFuel mass 0.0437 gAmbient conditions 1 atm, 298 C

SMD

( µµ µµ

m )

Time (ms)

MeasuredPredicted

80

60

40

20

0 0 1 2 3 4 5 6

Gasoline Hollow Cone Sprays

0

2

4

6

8

10

12

0 1 2 3 4 5 6

Measured Pre-sprayMeasured Main SprayComputed Pre-sprayComputed Main Spray

Pene

tratio

n (c

m)

Time (ms)

13B ReitzEPD9636Droplet Drag Modeling

• Steady-state Stokes viscous drag, added-mass andBasset history integral

ρLVd dv / dt =CDAf

ρgU2

2U / U

F = 6πrµ g v + 12 ( 4

3 πr3ρg )dvdt

+ 6r2 πµρg

dvdt'

t − t '0

t

dt 'dv/dt =

• General form

dvdt

=9µ

2ρlr2 (u − v) = (u − v) / τ m

τ m = 2ρl r2 / 9µ

Stokes limit – low Reynolds number flow: CD = 24/Re

gMomentum Relaxation time

v = v 0 exp(−t / τ m)

14B ReitzEPD9636Form Drag & Distortion

y

CD =CD,sphere(1+2.632y)

• Drop distortion – Liu et al. SAE 930072

CD = 24Re d

1 + 16

Re d2/3 Re d ≤ 1000

0.424 Re d > 1000CD =

• Corrections to Stokes Drag

y – from TAB Breakup model

Af = π a 2

• Magnus lift, Saffman lift, thermophoretic forces, Stefan flow effects usually neglected

15B ReitzEPD9636Turbulence & Drop Dispersion

G( ′ u ) = 4 / 3πk( )−3/ 2 exp(−3 ′ u 2 / 4k)

• Monte Carlo method (Gosman 1981)

u = u + ′ u

Vortex structure

St >>1

St ~1

St <<1

δ

Drop-eddy interaction time Eddy life time Residence time

l = Cµ3/ 4k 3/ 2 / ε

te = l / 2k / 3 tp = l / u − v

t int = min(te ,tp )

δ = l

16B ReitzEPD9636

• Breakup due to capillary surface wavesHinze (1955) and Engel (1958)

Drop Breakup• Mechanisms of drop breakup at high velocities poorly understood - Conflicting theories

• Bag, 'Shear' and 'Catastrophic' breakup regimes

• Boundary Layer Stripping due to Shear at the interfaceRanger and Nicolls (1969) Reinecke and Waldman (1970)

• Stretching and thinning – dropdistortion - Liu and Reitz (1997)

δ(x)

Delphanque & Sirignano (1994)

17B ReitzEPD9636

Nozzle

1.27

Gas

Liquid drop

Liquidinjectionorifice

Low velocity drop breakup

Drop distortion

18B ReitzEPD9636

air-jet

Diesel Water

Stretching and Thinning breakup mechanism Liu & Reitz (1997)

High velocity drop breakup

We = 260

19B ReitzEPD9636

Breakupstages

Deformation orbreakup regimes Breakup process Weber number References

First breakup stage

(1) Deformationand flattening We 12<

(b) Bag breakup≤12 We 100≤

(including theBag-and-Stamenbreakup)

Pilch and Erdman[6]

(c) Shear breakup We 80< Ranger andNicolls[10]

(d) Stretching and thinning breakup

≤100 We 350≤ Liu and Reitz [24]

Second breakup stage

(e) Catastrophic breakup

≤350 We Hwang et al.[3]

Air

Air

Bag growth Bag burst Rim burst

Air

Air

Flatteningand thinning

Air

l

RTwaves KH waves

Drop Breakup Review

20B ReitzEPD9636Drop Breakup Models

t1 = D1ρlr

3

σ

Lifetimes of unstable drops:

Bag breakup

t2 = D2rU

ρl

ρg

Stripping

Reitz and Diwakar SAE 860469

• Check We inequalities for each drop parcel each timestep

• If criteria met for a time equal to life time then new drop size is specified using equalities

nf r f3 = niri

3

21B ReitzEPD9636Drop Distortion Modeling

y

Taylor Analogy Breakup Model (TAB)

y = 2 x/r

if y> 1 droplet breaks up:

We = Wecrit > 6.0For low speed drops

For high speed drops

tbu =π2

ρl r3

2σtbu = 3

rU

ρl

ρg

TAB ModelO’Rourke SAE 872089Pelloni & Bianchi SAE99 Tanner SAE 970050

2

2 3 2

52 83

g l

l l l

Uy y y

r r r

22B ReitzEPD9636Wave Breakup Theory

τ = 0.82B1ρa3

σ

• Jet stability theory

low speed (inviscid) jets

τ = (B1a/U) ρ1/ ρ2high speed (inviscid) jets

t t+dt t = tbu

'Wave' Model

TAB Model

λ

r = B λo

t = B τ1

Vrel

break

Liquid

Gas

Wave breakup model

23B ReitzEPD9636

Air jet

DropsRT waves

KH waves

λ

Λ

Product drops

• High Speed Drop Breakup Mechanism

Hwang et al. Atom. & Sprays, 1996

Catastrophic Drop Breakup

• Rayleigh Taylor Breakup

gt = accelerationK =

−gt ρl − ρg( )3 σ

Ωt =2

3 σ

−gt ρl − ρg( )[ ]3

2

ρl + ρg

24B ReitzEPD9636

R/DL/D

Breakup length

Λ

η=η0eΩt

r=B0Λ

Jet/drop breakupKH Model

Nozzle flowNozzle flowmodelmodel

ERC KH-RT Atomization Model‘Blob’ injection

KHKHKH

aBΛΩ

= 1726.3τ

Drop size (KH)

ΚΗΛ= 0BrKH

Drop breakupda/dt = - (a -r) / τ

Drop size (RT)

Drop breakupRT Model

2 πB2 KrRT =

1 Ω tτRT =

25B ReitzEPD963675(10)25 split injection

26B ReitzEPD9636Comparison with Engine Sprays

-12 -10 -8 -6 -4 -2 0 205

10152025303540455055

Measured KH-RT (Lb) Model KH Model

Spra

y Ti

p Pe

netra

tion

(mm

)

CAD ATDC

Spray Tip Penetration

Sandia Engine (Dec, 1997)

Cummins optical-access engineCELECT systemL/D=4.1, Dnozzle=0.194 mmSharp-edge inlet

27B ReitzEPD9636

Equivalence RatioL = 0.5 H = 4.5

KH KH-RTSpray drops Ricart, Reitz, Dec - ASME 1997

KH-RT & Breakup Length Model

9 btdc

7 btdc

5 btdc

• Limited liquid penetration length

28B ReitzEPD9636Drop Collision & Coalescence

∆ =rsmallrl arg e

0

0,1

0,2

0,3

0,4

0,5

0 20 40 60 80 100 120

2*Wec

Impa

ct p

aram

eter

x

Coalescence

Reflexive separation

'grazing'Stretching separation

present study:satellite

formationor

shattering collisionpossible

∆ 0

x = 1 grazing

x = 0 head on σρ 2UrWe smallc =

• Small dropcolliding withbig drop ismore likely tocoalesce

29B ReitzEPD9636Collision Probability

ν12 = N2 π(r1 + r2 )2 E12 |v1 − v2 |/Vol

• Collision frequency – O’Rourke and Bracco 1980

1

2

• Collision efficiency

E12 =K

K +1 / 2

2

~ 1 K =29

ρl v1 − v2 r22

µ g r1

Number of collisions fromPoisson process

p(n) = e -ν12∆t ν12∆t n/n!

0 < p <1 random number

30B ReitzEPD9636Drop Coalescence

x =12

5 We1+ ∆3( )116

1 + ∆( ) ∆3 1+ ∆2 − 1 + ∆3( )23

12

• Grazing-coalescence boundary – Ashgriz and Poo JFM 1990

Drops fly apart if rotational energy of colliding pair exceedssurface energy of combined pair

0 < x <1random number

31B ReitzEPD9636Grazing - Stretching Separation

• Collision dynamicsEnergy and angular momentum conservation:

• Grazing – drops move in same direction but at reduced velocity• Coalescence – mass average properties of colliding drops

32B ReitzEPD9636Drop Reflexive Separation

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70 80 90 1002*We

²=1²=0.75²=0.5

Coalescence

Reflexive separation

2 We

∆ 1 + ∆3( )2∆6 η1 + η2( )+ 3 4 1 + ∆2( )− 7 1+ ∆3( )2

3

≥ 0

η1 = 2 1− ξ( )2 1− ξ2( )12 −1

η2 = 2 ∆ − ξ( )2 ∆2 − ξ 2( )12 − ∆3

with ξ =12

x 1+ ∆( )

Tennison et al. SAE 980810

33B ReitzEPD9636Shattering Collisions

ur

r0

r

2δ

t=tbreakuprc

t=0

λ

θr1 r2

• Model basedon thestabilityanalysis ofcombineddroplets thatelongate intoa ligamentafter acollision

Georjon & Reitz, Atom. & Sprays, 1999

34B ReitzEPD9636Drop Vaporization

• Vaporization in a non-convective environment– well understood for single component, low pressure– D2 Law

Drop

Liquid-Vapor Interface: Equilibrium or

Non-equilibrium

Heat transfer to drop: convection (conduction), radiation

Mass transfer with surroundings: vaporization, condensation, gas solubility

Internal circulation and profiles: temperature, concentration, velocity

Relative Drop Motion

r

TR

Tinf

T YR

Y Yinf R

35B ReitzEPD9636KIVA Vaporization Models

Frossling correlation - Lefebvre, Atomization & Sprays 1989

Mass transfer number

Sherwood number

Fuel mass fraction at drop surface

R = dr / dt = −ρ DBSh / (2ρ1r )

B = (Y1* − Y1 ) / (1− Y1

* )

Sh = (2.0 + 0.6 Re d1/ 2 Sc1/ 3 )

ln(1+ B)B

Y1* = W1 / W1 + W0 (

ppv(Td )

−1)

Vapor pressure Pv from thermodynamic tables

36B ReitzEPD9636Drop Heat-up Modeling

Change in drop temperature from energy balance

Rate of heat conduction to drop from Ranz-Marshall correlation

Qd = α (T2 − T1)Nu / (2ρ r)

Nu = (2.0 + 0.6Red1/ 2 Pr1/ 3 )

ln(1+ B)B

d l d d d dr c T r RL T r Q43

4 43 2 2 ( )

37B ReitzEPD9636Other Effects

• High pressure effects (N2 solubility)• Drop distortion• Drop internal flow

– effective diffusivity

• Multicomponent fuels

0 20 40 60 80 100160

200

240

280

320

360

Chevron - Summer Chevron - Winter

Tem

pera

ture

(deg

C)

% Recovered

• Fuel effects:– Cetane number (auto-ignition)– Volatility (10%, 50% boiling point)

38B ReitzEPD9636Continuous Thermodynamics

f I I I( ) ( )( )

exp ( )

;

= − − −FHG

IKJ

= + =

−γβ α

γβ

θ αβ γ σ αβ

α

α

1

2

Γ2

Fuel composition represented by:• Γ-Distribution function• α, β shape parameters; γ origin shift

Fuel composition represented by:• Γ-Distribution function• α, β shape parameters; γ origin shift

0

0.005

0.01

0.015

0.02

0 100 200 300 400

DieselGasolineKerosene

Dis

tribu

tion

Func

tion

f(I)

Molecular Weight I

Fuel Diesel Gasoline Keroseneαβγ

18.510.00.0

5.715.00.0

50.03.5250.0

θσ

18543

85.535.8

176.2524.9

C14H30

Lippert and Reitz SAE 972882

39B ReitzEPD9636Drop Vaporization Processes

Gasoline Droplet Diesel Droplet

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160

180

200

220

240

0

20

40

60

80

100

120

140

160

180

200

220

240 Vapor mass fraction @ surface [%] Droplet Temperature [deg C]

Diameter 2 [10 4mm 2] Mean of Liquid Composition [MW] Width of Liquid Composition [MW] Boiling Temperature [deg C]

Time [ms]0 10 20 30 40

0

50

100

150

200

250

300

350

400

0

50

100

150

200

250

300

350

400 Vapor mass fraction @ surface [%] Droplet Temperature [deg C]

Diameter 2 [10 4mm 2] Mean of Liquid Composition [MW] Width of Liquid Composition [MW] Boiling Temperature [deg C]

Time [ms]

Han et al. SAE 970625