Application of the DirectApplication of the Direct Quadrature ...kdS δ [] − = +=∈−∑ 1 1;0,21 qk N k qm q kdS S k N Slide 4 CFD in Chemical Reaction Engineering V, Whistler,

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


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


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


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 =


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


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


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


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


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


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


Documents

Application of the DirectApplication of the Direct Quadrature ...kdS δ [] − = +=∈−∑ 1 1;0,21 qk N k qm q kdS S k N Slide 4 CFD in Chemical Reaction Engineering V, Whistler,