Spray Submodels EPD9636 1B Reitz - UW-Madison Submodels.pdf · 20B Reitz Drop Breakup Models...
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Transcript of Spray Submodels EPD9636 1B Reitz - UW-Madison Submodels.pdf · 20B Reitz Drop Breakup Models...
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