spray submodels epd9636 1b reitz - uw-madison submodels.pdf · 20b reitz drop breakup models...
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![Page 1: Spray Submodels EPD9636 1B Reitz - UW-Madison Submodels.pdf · 20B Reitz Drop Breakup Models EPD9636 t1 =D1 ρlr 3 σ Lifetimes of unstable drops: Bag breakup t2 =D2 r U ρl ρg Stripping](https://reader031.vdocuments.mx/reader031/viewer/2022013000/5c1103a209d3f2b60f8b6b1b/html5/thumbnails/1.jpg)
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
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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
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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
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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/
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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
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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
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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
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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
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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
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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
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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 −++−=
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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)
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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)
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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
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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
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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)
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17B ReitzEPD9636
Nozzle
1.27
Gas
Liquid drop
Liquidinjectionorifice
Low velocity drop breakup
Drop distortion
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18B ReitzEPD9636
air-jet
Diesel Water
Stretching and Thinning breakup mechanism Liu & Reitz (1997)
High velocity drop breakup
We = 260
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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
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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
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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
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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
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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
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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 =
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25B ReitzEPD963675(10)25 split injection
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ( )
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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)
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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
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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