on the combustion of hollow cone sprays generated by...

On the Combustion of Hollow Cone SpraysGenerated by Pressure-Swirl Injectors

A. Linan1 J. Urzay2 J. Arrieta-Sanagustın3 A.L. Sanchez3

1E.T.S.I.AeronauticosUniversidad Politectnica de Madrid

2Center for Turbulence ResearchStanford University

3Departamento de Ingenierıa Termica y de FluidosUniversidad Carlos III de Madrid

5th Meeting of the Spanish Section of the Combustion Institute,May 23–25, 2011 Santiago de Compostela

Liquid-Fuel Injection in Gas-Turbine CombustorsLIQUID−FUEL INJECTION IN GAS TURBINE COMBUSTORS

RQL COMBUSTOR CONCEPT (NASA) APTE & MOIN (2006), LES − LAGRANGE

COMPLEX FUEL−ATOMIZATION PHYSICS (THIN FILMS, SWIRL, TURBULENT COFLOWS)

QUANTITATIVE EXPERIMENTS ARE SCARCE BECAUSE OF HIGH LIQUID−VOLUME FRACTION

NEED ANALYTICAL MODELS OF INJECTION BASED ON CONSERVATION LAWS

KULKARNI et al. (2010)

NUMERICAL SIMULATIONS OF THESE PROCESSES ARE COSTLY AND INACCURATE

Hollow Cone Sprays Generated by Pressure-Swirl Injectors

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Taylor Analysis of the inviscid swirl atomizer (1948)

The liquid fuel is forced through tangential slots to generate a strongswirling flow in the atomizer chamber.

i

a

pa!

paha

he uer

pQ volumetric flow rateΓ : circulation of the azimuthal velocityri : injector radius

Inviscid Axisymmetric Flow

v · ∇(rw) = 0→ w = Γ2πr

Conservation of Head: pa

ρ+ 1

2

(Γ/(2π)ri−ha

)2

= pa

ρ+ 1

2

(Γ/(2π)ri−he

)2

+ u2e

2

ha ∼ he ri → Q = 2πriheue = Γ

r1/2i

[2h2e(ha − he)]1/2

Q is maximum if he = 23ha, that is if ue = Γ

2πri

√he

ri= we

√he

ri

he =(QΓ

)2/3

r1/3i we = Γ

2πriue = Q1/3Γ2/3

2πr4/3i

Steady Axisymmetric Thin Films

l

a

paue

wr

z

r

hu

u!

elp

S = we

ue=(

Γri

Q

)1/3

=√

ri

he

>∼ 1

Conservation of circulation

wl = Γ2πrl→ wl

we= ri

rl

Conservation of total pressure head:

u2l + w2

l = u2e + w2

e →(ul

ue

)2

= 1 + S2

[1−

(ri

rl

)2]

Conservation of axial momemtum flux⇒ u = ul cos θ = ue

ul

ue= 1

cos θ=√(

drl

dz

)2+ 1→ drl

dz= S

√1−

(ri

rl

)2

As rl →∞ the swirling motion decays (i.e., wl → 0), so thatul

ue→ ul∞

ue=√

1 + S2 and tan θ = drl

dz→ tan θ∞ = S

Steady Axisymmetric Thin Films

0 2 31 4

5

4

3

2

1

0

z/re

rl

re

S = 1.5 S = 1.0

S = 0.5

!!he/re = 0.1

drl

dz= S

√1−

(ri

rl

)2

, rl(0) = ri

rl

ri=

√1 +

(Szri

)2

ul

ue=

√1 + S2

[1−

(ri

rl

)2]→ ul

ue=

r1+(S2+1)

“Szri

”2

r1+“

Szri

”2

Q = 2πriheue = 2πrlhul → hhe

= riue

rlul= 1r

1+(S2+1)“

Szri

”2

Liquid-Film Break-Up and Atomization

SPRAY

AIR

AIRLIQUID

SWIRLER

FILM

(a)

(b)

(c)

The Two-Continua Description of Spray Combustion

d2a

L

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2a

l

Typical spray-combustion applications

L ld ∼ n−1/3

Under local stoichiometric conditionsρla

3 ∼ Sρgl3d

ald∼(ρg

Sρl

)1/3

∼ 0.1 1

A continuum description of the liquid phase can be carried out for allcomputational cells with size δ such that

L δ ld a


The chemistry is described in terms of an overall irreversible reaction

F + sO2 → (1 + s)P + q S = s/YO2air' 15

Gas-Phase Equations for a monodisperse spray (YO = YO2

YO2air

):

∂ρ

∂t+∇ (ρv) = nm

∂

∂t(ρv) +∇ · (ρvv) = ∇ · ¯τ −∇p+ nmvd − nf

∂

∂t(ρYF) +∇ · (ρvYF)−∇ ·

(ρDT

LF

∇YF

)= ωF + nm

∂

∂t

(ρYO

)+∇ ·

(ρvYO

)−∇ ·

(ρDT

LO

∇YO

)= SωF

∂

∂t(ρYP) +∇ · (ρvYP)−∇ ·

(ρDT

LP

∇YP

)= (1 + S)ωF

∂

∂t(ρcpT ) +∇ · (ρvcpT )−∇ · (ρDT cp∇T ) = −qωF

−n [m (Lv − cpTd) + qd] +∂p

∂t


Liquid-Phase Equations:

∂n

∂t+∇ · (nvd) = 0

4

3πρla

3cl

(∂Td

∂t+ vd · ∇Td

)= qd

∂

∂t

(4

3πρla

3

)+ vd · ∇

(4

3πρla

3

)= −m

4

3πρla

3

(∂vd

∂t+ vd · ∇vd

)= f

Equation of State: pρ

= R0T∑iYi

Mi


Source terms f , m, and qd

Under local stoichiometric conditions

ρla3 ∼ Sρgl3d →

a

ld∼(ρg

Sρl

)1/3

∼ 0.1 1

Isolated droplet response in the local gas environment, which iscreated collectively by all droplets.

The droplet response depends on the local Reynolds numberRe = 2aUr/νg based on the relative velocity Ur = |v − vd|.

For Re 1:f = 6πµa(v − vd)

If Lv

RFTB' 15 1 and tc,d = a2

αl tv,d ∼ a2

αg

ρl

ρg:

Td < TB : qd = 4πκa(T − Td), m = 0

Td = TB : qd = 0, m = 4πρDTa ln[1 + cp(T−TB)+YOq/S

Lv

]


In the limit of infinitely fast chemistry, the reaction terms involving ωF

become Dirac-delta sinks, and can be eliminated by using linearcombinations of the conservation equations to generate chemistry-freecoupling functions. If unity Lewis numbers are assumed for all species(LF = LO = LP = 1)

∂

∂t(ρYF) +∇ · (ρvYF)−∇ · (ρDT∇YF) = ωF + nm

and 1/S times

∂

∂t

(ρYO

)+∇ ·

(ρvYO

)−∇ ·

(ρDT∇YO

)= SωF

yields

∂

∂t

[ρ

(YF −

YO

S

)]+∇ ·

[ρv

(YF −

YO

S

)]

−∇ ·[ρDT∇

(YF −

YO

S

)]= nm


This chemistry-free conservation equation is conveniently written interms of the mixture fraction variable

Z = SYF−YO+11+S

∂

∂t(ρZ) +∇ · (ρvZ)−∇ · (ρDT∇Z) = nm

The solution of this equation can be used together with thefast-chemistry assumption YFYO = 0 to compute YO and YF in theregion ΩF, where YO = 0, and in the region ΩO, where YF = 0, both

regions being separated by the flame surface, where Z = ZS = 11+S

.

This determines the reactant composition from the piecewise linearrelationships

YF = Z−ZS

1−ZSin ΩF where Z ≥ Zs

YO = 1− ZZS

in ΩO where Z ≤ Zs


Similarly, we eliminate the reaction term from the energy conservationequation by introducing the total enthalpy

H = cp(T − TA) + (YO − 1)q/S

where TA is the air temperature in its feed stream.

∂

∂t(ρH) + ∇ · (ρvH)−∇ · (ρDT∇H) =

− n m [q/S + Lv − cp (TB − TA)] + qd

If needed, the product composition can be obtained from

∂ (ρP )

∂t+∇ · (ρvP )−∇ · (ρDT∇P ) = 0

where

P = YP + (YO − 1)(1 + S)/S + Z(1 + S)/S

Combustion of Hollow Cone Sprays

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Cone Spray Characteristics:

Cone semiangle θ∞

Initial droplet radius ao

Initial droplet velocity ul∞

Droplet injection rate n = Q43πa

3o

Droplet acceleration (evaporation) time:

43πρla

3 dvd

dt= 6πµga(v − vd)→ td = 2

9

ρla2o

µg

Characteristic length scales: lc = ul∞td and yc = (νgtd)1/2

Characteristic droplet density:

n = 2πlc sin(θ∞)ycul∞nc → nc = n

2π sin(θ∞)ν1/2g u2

l∞t3/2d


Dimensionless variables: l = l′

lc, y = y′

yc, (u, ud) = (u′,u′

d)

ul∞,

(v, vd) = (v′,v′l)√

νg/td,(T, Td) = (T ′,T ′

d)

TB, a = a′

ao, n = n′

nc, H = H′

cpTB

Steady B-L Gas-Phase Equations λ = (4/3)πa3oρlnc

ρB∼ 1

S:

1

l

∂

∂l(ρlu) +

∂

∂y(ρv) = λnm

1

l

∂

∂l(ρlu2) +

∂

∂y(ρvu) =

∂2u

∂y2+ λnmud + λan(ud − u)

1

l

∂

∂l(ρluZ) +

∂

∂y(ρvZ) =

1

Pr

∂2Z

∂y2+ λnm

1

l

∂

∂l(ρluH) +

∂

∂y(ρvH) =

1

Pr

∂2H

∂y2− λn(γm+ qd)

with γ = q/S+Lv−cp(TB−TA)

cpTBand m = 0 and qd = 2a

3Pr(T − Td) if

Td < 1 and m = 2a3Pr

ln(1 + T−1+YOq/S

β

)and qd = 0 if Td = 1.


Liquid-Phase Equations:

1

l

∂

∂l(nlud) +

∂

∂y(nvd) = 0

ud∂

∂l(a3Td) + vd

∂

∂y(a3Td) =

cp

clqd

= 2a

3Pr(T − Td) if Td < 1

= 0 if Td = 1

ud∂a3

∂l+ vd

∂a3

∂y= −m

= 0 if Td < 1

= 2a3Pr

ln(1 + T−1+YOq/S

β

)if Td = 1

ud∂ud

∂l+ vd

∂ud

∂y=

1

a2(u− ud)

ud∂vd

∂l+ vd

∂vd

∂y=

1

a2(v − vd)


Boundary conditions Z = SYF−YO+11+S

, H = (T − TA) + q(YO−1)S

Y <<1

A

YP 8

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ly

TP

8F

TOxidizer side (y → +∞):u = Z = H = 0

Product side (y → −∞):u = Z − ZP = H −HP = 0

where

ZP = SYF∞+1

1+S> Zs

andHP = TP − TA − q

S

Integral conservation equations:

l

∫ +∞

−∞(nud)dy = 1 and l

∫ +∞

−∞(λ−1ρu2 + na3u2

d)dy = 1

Note that as l→ 0: ud → 1⇒ n ∼ 1yl→∞


As l→ 0 there exists a self-similar solution with two regions

Inner spray (y ∼ l):

ξ =y

l

a→ 1ud → 1Td → Td0

, n =N(ξ)

l2, vd = ξ,

T = Tdu = 1

Surrounding gas flow with n = 0 (y ∼ l1/2):

η = y√l, ρu = Fη, ρv = (1

2ηFη − 3

2F )

(Fη

ρ

)ηη

+3

2F

(Fη

ρ

)η

= Zηη +3

2PrFZη = Hηη +

3

2PrFHη = 0

η →∞ : Fη = Z = H = 0η → −∞ : Fη = Z − ZP = H −HP = 0

η = 0 :F = 0 (v = 0)Fη = To (u = ud = 1)H = To − TA + (YO − 1)q/S (T = Td = Td0)


Initial profiles

TP = 1.5, YF∞ = 0.2, TA = 0.8, Td0 = 0.6, Zs = 1/15.3

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