Application of the Direct Quadrature MethodApplication of the Direct Quadrature Method of Moments to a Hollow-Cone Water Spray
Jesper Madsen, Tron Solberg and Bjørn H. HjertagerJespe adse , o So be g a d jø je tageAalborg University Esbjerg, Denmark
Homepage: hugin.aue.auc.dkHomepage: hugin.aue.auc.dk
Danfoss Pressure-Swirl Atomizer
ApplicationsD ti h ti b• Domestic heating burners
Daily production• About 30 000 nozzlesAbout 30 000 nozzles
Characteristics• Air-cored vortex• Hollow conical sheet• Hollow-cone spray
L it• Low capacity– 1.46 – 6.55 kg/h
• Spray anglesp y g– 2θ = 30° – 90°
Slide 2CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
DQMOM-Multi-Fluid Model
Eulerian multi-fluid model in Fluent 6.2O h• One gas phase
• N distinct droplet phases– One phase for each size classOne phase for each size class– Coupled to population balance equations
Di t Q d t M th d f M t (DQMOM)Direct Quadrature Method of Moments (DQMOM) • Droplet size distribution (DSD)
( ) ⎡ ⎤∑N
• Transport equations for weight and
( ) ω δ=
⎡ ⎤= −⎣ ⎦∑1
q qq
n d d d
ωq
weighted abscissa • DSD evolves due to breakup and coalescence
δ ω=q q qd
Slide 3CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
DQMOM Transport Equations
The DQMOM representation of the DSD involves the solution of
[ ]
( ) ( ) δ ω
α
π πα ρ α ρ ρ ρ
∈
∂ ∂+ = −2 3
Volume fraction, , 1,
2 3
q
q l q l i q l q l q
q N
U d S d St ( ) ( )
[ ]
δ ωρ ρ ρ ρ∂ ∂ , 2 3
Diameter 1
q qq l q l i q l q l qit x
d q N[ ]
( ) ( ) δ ωπ πα ρ α ρ ρ ρ
∈
∂ ∂+ = −
∂ ∂3 4
,
Diameter, , 1,
23 2q q
q
q l q q l i q q l q l q
d q N
d U d d S d St x
ω δ
∂ ∂ 3 2
Source terms and : From first 2N integer moments
it x
S S
( )
ω δ
ω=
− ∑1
g
1
q q
q
Nkq
qk d S [ ]δ
−
=
+ = ∈ −∑ 1
1; 0,2 1
q k
Nkq m
qk d S S k N
Slide 4CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Breakup Kernel
The WAVE atomization model breakup kernel1 d d( )( )τ−
= <−2 2
1,
1
where breakup time
st qq st q
q q bu
d da d d
b d
τ =
=
where breakup time stable diameter
bu
std
Breakup daughter distribution function
( )−⎧ =⎪= ⎨1 3
Symmetric breakup:
2 if 2 qd db d d( )
( ) ( )−
= ⎨⎪⎩
= 3 3
0 otherwise
2
q
k k kq q
b d d
b d
Slide 5CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Outcome of Collisions
2
We l rel smallU dρ=
1 3
Wecoll σ=
⎡ ⎤⎛ ⎞
( )
1 3We
min 1,4.8
collbounE
f γ
⎡ ⎤⎛ ⎞⎢ ⎥= ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( )4.8min 1,
Wecoalcoll
fE
γ⎡ ⎤= ⎢ ⎥
⎣ ⎦
( ) 3 22.4 2.7f γ γ γ γ= − +
large
small
dd
γ =
Slide 6CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Collision Kernels
β π+
= =2Collision coefficient: ;2
p qpq pq rel pq
d dd U d
( )
β
β
;2
C l k l i
pq pq rel pq
E E( )
( )
β=Coalescence kernel: min ,pq boun coal pqc E E
( ) β= −Collision-induced fragmentation kernel: 1pq coal pqe E
Diameter of droplet fragments (Post and Abraham, 2002):
( )+1 33 31.89 p qd d
=fragd( )
( )γ+ +2 212 7 32.81We 1 1
p q
coll
Slide 7CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
LISA Model
Linearized Instability Sheet Atomization (LISA) model (Schmidt et al. 1999) Fil f ti• Film formation
• Sheet breakup and atomization
Exit diameter D0 436 μm Injection pressure ΔP 850 kPa Mass flow rate m 3 2 kg/hMass flow rate lm 3.2 kg/h Cone angle 2θ 80° Injection velocity Ul 28.9 m/s Film thickness δ0 31.6 μm Sheet thickness parameter K 3809 μm2
Sheet breakup time τ 227 μsSheet breakup time τbu 227 μs Sheet wave length ΛS 884 μm Droplet diameter dD 48.6 μm p D μ
Slide 8CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Initial Droplet Size Distribution
DDM• Selected randomly
DQMOM• Rosin Rammler (RR)• Selected randomly
• Rosin-Rammler volume distribution
• Rosin-Rammler (RR)• Li & Tankin (LT) model• Nodes are calculated from 2N
moments using the PD algorithm
Slide 9CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Computational Grid
Domain (r, x > 4 mm)• Axisymmetric• Axisymmetric• Unstructured• Fine resolution close to orifice
DDM Grid• Orifice is comprised of one cellOrifice is comprised of one cell• Size: 200×200 μm• Cells: 3,944
DQMOM Grid• Sheet is fitted into one cell• Size: 30×30 μm• Cells: 5,424
Slide 10CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
Sauter Mean Diameter (SMD)
Evolution of Sauter Mean Diameter (SMD), d32 [μm]Evolution liquid volume fraction (iso-lines: αl = 10-6 and αl = 10-5)Evolution liquid volume fraction (iso lines: αl 10 and αl 10 )
DQMOM (RR)
The grid superimposed has a spacing of 50×50 mm
Slide 11CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008
SMD
Good agreement
Result of initial DSD
Effect of DDM collisions model
DQMOM (RR)DQMOM (RR)• Correct trend• Slope changes
SMD li d di d• SMD at centerline underpredicted• SMD at periphery overpredicted
Slide 12
Axial Velocity
Reasonable accurate• Correct trend• Correct trend
DQMOM (RR) most accurate• Centerline values underpredicted• Periphery values overpredicted• Finer grid resolution compared toFiner grid resolution compared to
DDM
Slide 13
DSD
Axial location x = 80 mm
Reasonable good agreement
Experimental results: wider range
DQMOM-multi-fluid modelDQMOM multi fluid model• Nearly monodisperse DSD’s
– No droplet breakupFew collisions– Few collisions
– Change due to convection• Agreement at shorter distances
Slide 14
Conclusions
DQMOM-multi-fluid model applied to a low-speed hollow-cone spray• Low relative velocitiesLow relative velocities
– No secondary droplet breakup• Wide angle spray
Few collisions– Few collisions• DSD determined by primary breakup of the liquid sheet
Predictions compared to • Experimental Phase Doppler Anemometry (PDA) data• Fluent Discrete Droplet Model (DDM)
The model for the primary breakup of the liquid sheet is importantThe model for the primary breakup of the liquid sheet is important• LISA model appears to be valid
DQMOM with Rosin-Rammler initial DSD shows reasonable results for• Axial droplet velocities• Sauter mean droplet diameters• Droplet size distributionsDroplet size distributions
Slide 15CFD in Chemical Reaction Engineering V, Whistler, BC, Canada, 15-20 June 2008