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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
!8
"
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 Two-Continua Description of Spray Combustion
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
The Two-Continua Description of Spray Combustion
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
The Two-Continua Description of Spray Combustion
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
]
The Two-Continua Description of Spray Combustion
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
The Two-Continua Description of Spray Combustion
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
The Two-Continua Description of Spray Combustion
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
Combustion of Hollow Cone Sprays
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.
Combustion of Hollow Cone Sprays
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)
Combustion of Hollow Cone Sprays
Boundary conditions Z = SYF−YO+11+S
, H = (T − TA) + q(YO−1)S
Y <<1
A
YP 8
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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→∞
Combustion of Hollow Cone Sprays
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)
Combustion of Hollow Cone Sprays
Initial profiles
TP = 1.5, YF∞ = 0.2, TA = 0.8, Td0 = 0.6, Zs = 1/15.3
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