594
ISSN 1392 - 1207. MECHANIKA. 2011. 17(6): 594-600
Simulation of evaporating isopropyl alcohol droplets injected into
a turbulent flow
H. Merouane*, A. Bounif**, M. Abidat***
*University of Science and Technology, Oran, Algeria, E-mail: [email protected]
**University of Science and Technology, Oran, Algeria, E-mail: [email protected]
***University of Science and Technology, Oran, Algeria, E-mail: [email protected]
http://dx.doi.org/10.5755/j01.mech.17.6.1001
Nomenclature
B M - mass transfer number; BT - thermal transfer number;
C d - drag coefficient; C P - specific heat, J/kgK; D - drop-
let diameter, m; D m - binary mass diffusivity coefficient;
h - enthalpy, J/kg; J - mass flux, kg/m2s; Lv - latent heat
of vaporization, J/kg; Lv e - effective latent heat of vaporization, J/kg; m - mass flow rate, kg/s; Sc - Schmidt number; ScT -Schmidt turbulent number; Pr - Prandtl number;
PrT - Prandtl turbulent number; Re - Reynolds number;
~
S m - mass source term; S ε - energy dissipation source
term; T g - gas temperature, K; TS - droplet temperature
surface, K; U d - droplets velocity, m/s; U g - gas velocity,
m/s; Y - mass fraction.
Greek symbols.
ε - turbulent energy dissipation rate; φ - heat transfer rate;
λ - thermal conductivity, W/mk; ν - cinematic viscosity,
m2/s; μ - dynamic viscosity, kg/ms; μ T - turbulent dynamic viscosity; ρ L - droplet density; ρ g - gas density.
Subscripts.
d - droplet, F - fuel, g - gas, L - liquid, s - surface.
1. Introduction
Spray vaporization and combustion studies are of
primary importance in the prediction and improvement of
systems utilizing spray injection, in liquid fuelled combustion systems such as industrial boilers, gas turbines, direct
ignition diesel engines, rocket and air-breathing engine
applications. In these systems the vaporization is the dominant process, the fuel is injected into the combustion
chamber. The spray is made of fine fuel droplets of different sizes. The fuel vapour mixes with the oxidizer gas to
form the reactive mixture is ignited, which mixes to produce the spray combustion [1]. Usually the injection of the
spray is in a turbulent flow, this accelerating the molecular
diffusion of heat and species, the dominant factor for improved combustion efficiency and reducing the formation
of pollutants such as unburned carbon monoxide and NOx.
Many excellent reviews already exist on the various phenomena related to droplet motion and vaporization. The
work on droplet motion is reviewed by Clift and Gauvinv
[2], Clift and al [3], Leal [4], The spray modeling work is
reviewed by Williams [5], and Faeth [6], and the droplet
vaporization models by Law [7], Sirignano [8], and Ag-
garwal and al [9]. The basic droplet vaporization model for
an isolated single component droplet in a stagnant environment was formulated by Godsave [10], Spalding [11].
The aim of the present paper is to extend our
model to turbulent jet applications, and to account for the
two-way coupling between the fluid turbulent and the
droplet heat transfers.
2. Assumptions and mathematical model
The following assumptions are considered in the
present model:
- the droplet is assumed to be spherical;
- the spray is assumed to be dilute, under this assumption droplet collisions are ignored and the effect of adjacent droplets on droplet transport rates is neglected;
- the pressure is assumed to be constant and equal to
the local mean ambient;
- equilibrium conditions at the droplet/gas interface
are assumed;
- uniform physical properties of the surrounding
fluid;
- the gas phase Lewis number is assumed to be unity
in the droplet mode;
- radiation between the droplets and their surroundings is neglected;
- temperature inside the droplet is uniform.
3. Droplets vaporization
Under steady state vaporization, solutions of the
species conservation and energy equations are obtained by
analyzing the interface flux conditions between the liquid
and gas phases (Fig. 1).
3.1. Mass conservation
With the assumption of quasisteady vaporisation,
the mass flaw rate is constant, independent of radius.
G
1 d (r 2 ρ vr )
=0
div( ρ v ) = 2
dr
r
4 π rs2 ρ vr = m = constant
(1)
(2)
where m is the mass transfer rate from the droplet to the
air; vr is the radial velocity.
595
3.2. Mass conservation of the fuel vapor
(
The presence of the oxidant around the droplet
during the evaporation process is not necessary. Therefore,
only the fuel vapor is considered
(
)
dY ⎞
d
d ⎛
4 π ρ g r 2 vr YF = ⎜ 4 π ρ g Dm r 2 F ⎟
dr
dr ⎝
dr ⎠
or
F − 4 π r 2 ρ F Dm
mY
d (YF )
dr
= m = const
(3)
(4)
with YF is the fuel mass fraction; Dm is the binary mass
diffusivity.
Heat flux
)
(11)
By integration of Eq. (11) using the boundary
conditions:
r = rS , T = TS
r = ∞ , T = T∞
where we obtain the temperature profile in the gas phase
and the vaporized mass flow
T = TS −
Lv ⎛
Lv ⎞
+ ⎜T∞ −TS + ⎟ exp
CP ⎝
CP ⎠
⎛
⎞
m
⎜−
⎟
⎜ 4πρg Dmr ⎟
⎝
⎠
(11)
m = 4 π ρ g Dm rs ln(1 + BT )
BT =
Droplet
T∞ ,Y∞
(12)
where
Mass flux
TS ,YS
d
d ⎛
dT ⎞
4 π ρ g r 2 vr CPT = ⎜ 4 π r 2 λ
dr
dr ⎝
dr ⎟⎠
C P ( T∞ − TS )
Lve
(13)
On taking into account the stationary system
B = BM = BT
Fig. 1 Droplet vaporisation processes
By integration of equation (3) using the boundary
conditions
r = rS , YF = YFs
r = ∞ , YF = 0
where we obtain the profile of the fuel mass fraction and
mass flow of spray
(14)
The expression for a vaporization process in a
quiescent surrounding air
ρg
dD 2
D ln(1 + B)
= −8
dt
ρL m
(15)
In the present of significant convection, the
change of droplet diameter is represented as follows, Ranz
and Marshall [12]
ρg
dD2
= −8 Dmln (1+ B) 1+ 0.3Re1/2 Pr1/3
dt
ρL
(
)
⎛
⎞
m
YF = 1 − exp ⎜ −
⎟
D
r
πρ
4
F m ⎠
⎝
(5)
m = 4 π ρ F Dm r ln(1 + BM )
(6)
3.4. Droplet dynamics
(7)
The droplet dynamics are simulated using a Lagrangian point-particle model and the equation governing
the droplet motions can be written [13]
where
Y −Y
BM = Fs ∞
1 − YFs
(
G⎞
G
⎛
div ( ρ v h ) = div ⎜Φ + ∑ hi ji ⎟
i
⎝
⎠
(8)
where
G
Φ = −λ grad (T )
G
G
J i = − ρ Di grad (Yi )
(
ρd − ρ g
dU d
3 ρ g Cd
Ud −U g Ud −U g +
=−
ρd
dt
4 ρd D
3.3. Energy conservation
(Fourier’s law)
(9)
(Fick's Law)
(10)
Because the transfer of heat from the environment
to the droplet is only by conduction, the equation of energy
conservation is given by
)
(16)
)
(17)
where C d is the drag coefficient that depends on the Reynolds number for a spherical configuration of the droplet
24
⎧
⎪Cd = Re
⎪
24
⎪
1 + 0.15Re0.687
⎨Cd =
Re
⎪
⎪Cd = 0.438
⎪
⎩
(
)
,
Re < 1
,
Re ≤ 1000
,
Re > 1000
596
Re =
Ud − U g .D
νr
where
.
(18)
4. Gas phase
The vapour produced by the droplets is a mass
source for the fluid, moreover the vaporization process
generates modification in the momentum and energy balances both phases. Fluid phase equations then contain
many extra source terms. Assuming that the vapour production does not modify the liquid phase density, the governing equations read as follows Berlemont et al [14]
ξ k = G − CD ρε + Sk
⎛ ∂U ∂U j
G = μT ⎜ i +
⎜ ∂xj
∂xi
⎝
(
∂x j
(19)
∂xj
) = − ∂ ⎡P +
∂xi ⎣
k
2
3
ρ k ⎤⎦ −
∂
∂xi
⎡2
∂U j ⎤
⎢ ( μ + μT )
⎥+
∂xj ⎦⎥
⎣⎢ 3
⎞ ~
⎟+S
⎟ u
⎠
(
~ ~
∂ ρ U jθ ′ 2
~
⎛ μ
μ ⎞⎛ ∂T
⎜⎜
+ T ⎟⎟⎜
⎜
⎝ Pr PrT ⎠⎝ ∂x j
ξθ = Cθ 1 μ t
⎞ S~H
⎟+
⎟ Cp
⎠
(21)
4.4. Mean vapor mass fraction
(
)
~
⎛μ
μ ⎞⎛ ∂Y
⎜⎜ + T ⎟⎟⎜
⎜
⎝ Sc ScT ⎠⎝ ∂x j
⎞ ~
⎟ + Sm
⎟
⎠
(22)
4.5. Turbulent equations
Fluid turbulence is still described using a (k-ε),
but the modification of the continuity equation leads to
extra terms in the k-equation, ε-equation and for energy
and vapor mass fraction equations.
4.5.1. The kinetic energy equation
(
∂x j
)=
∂
∂x j
⎛ μT
⎜
⎜σ
⎝ k
(26)
(27)
k
⎞⎛ ∂k
⎟⎜
⎟⎜ ∂x
⎠⎝ j
)=
∂
∂x j
⎛ μT
⎜
⎜σ
⎝ θ
~
⎞⎛ ∂ θ ′ 2
⎟⎜
⎟⎜ ∂ x
j
⎠⎝
⎞
⎟ + ξθ
⎟
⎠
(28)
~ ~
∂θ ∂θ
ε ~ ~
− Cθ 2 ρ θ ′ + Sθ
∂x j ∂x j
k
(29)
The classic constants corrections for turbulent
round jet predictions are given
CD = 1, σ k = 1, σ ε = 1.3, σ θ = 1.92,
Cε 1 = 1.44, Cε 2 = 1.92, Cθ 1 = 2.8, Cθ 2 = 1.92.
5. Computational details
where Sc and ScT are the Schmidt and Schmidt turbulent
number, respectively.
~
∂ ρU j k
⎞
⎟ + ξε
⎟
⎠
with
~
where T is the mean temperature; S H is the enthalpy
source term.
~~
∂ ρU iY
∂
=
∂x j
∂x j
∂x j
(20)
4.3. Energy equation
)
⎞ ⎛ ∂ε
⎟ ⎜⎜
⎠ ⎝ ∂x j
4.5.3. Scalar fluctuation transport equation
where Su is the momentum source term.
(
⎛ μT
⎜
⎝ σε
Sε is the energy dissipation source term.
⎛ ~ ∂U~
∂
(μ + μ t )⎜⎜ ∂U i + j
+
∂xi
∂x j
⎝ ∂x j
~~
∂ ρU iT
∂
=
∂x j
∂x j
∂
∂x j
ε
ε2
ξε = Cε 1 G − Cε 2 ρ + Sε
4.2. Momentum equation
(
)=
where
~
where S m is the mass source term.
∂ ρUU
i j
(25)
4.5.2. Energy dissipation rate equation
∂ ρU j ε
( )
⎞ ∂U
⎟ i
⎟ ∂xj
⎠
~
where G is the turbulent energy production term; S k is
the extra source term.
4.1. Continuity equation
∂
~
~
ρU i = S m
∂xi
(24)
⎞
⎟ + ξk
⎟
⎠
(23)
Fig. 2 shows a schematic of the computational
domain used for the model coaxial combustor investigated
by Sommerfeld and Qiu [15]. The chamber consists of an
annular section discharging hot air into a cylindrical test
section. The central section consists of a nozzle through
which isopropyl alcohol at 313 K is sprayed into the test
section. Hot air at constant temperature of 373 K enters the
annulus. The boiling temperature of isopropyl alcohol at
atmospheric pressures is 355 K and thus the evaporation is
dominated by mass transfer effects. The air and liquid
mass flow rates are 28.3 and 0.44 g/s, respectively. The
over all mass loading is small resulting in a dilute spray.
The outer radius of the annulus r = 32 mm. The mean axial
inlet velocity in the annular section is 18 m/s
The numerical simulation is performed by using
the industry code "FLUENT" which uses a numerical
method to finite volume coupled with a multigrid resolution scheme.The flow governing equations are solved by
using a performed SIMPLE algorithm and using the stan-
597
dard k-ε model of turbulence.
Fig. 3 shows the computational grid used. The
configuration of the jet is considered to be axisymmetric.
The definitions of the geometry and mesh generation were
performed by using the Gambit mesh with a quadrilateral
mesh of shape. The area is modelled for dimensions of
radius 100 mm and 350 mm in length that corresponds to a
half cylinder.
The mesh consists of 100 nodes in the radial direction and 80 nodes in the axial direction. The stitches are
tightened in the area near the injector
merfeld.
The annular flow is also characterized by a peak
velocity of 18 m/s between 20 and 30 mm in Fig. 4.
The results of Figs. 6 and 7 show the evolution of
the mean radial velocity as a function of radial distance to
the plane of the injector at distances 25 and 50 mm. In
these figures the radial mean velocity is negative near the
jet axis; this is due to internal circulation of the flow at the
entrance of the jet.
x = 25 mm
Simulation
Exp. Sommerfeld
20
Air
-1
Axial mean velocity (m.s )
15
Liquid supply
10
5
0
-5
Spray nozzle
0
20
40
60
80
100
Radius (mm)
Fig. 4 Axial mean velocity profile at 25 mm
20
40
64
x = 50 mm
Simulation
Exp. Sommerfeld
-1
Axial mean velocity (m.s )
15
20
Fig. 2 Schematic diagram of the computational domain for
model combustor of Sommerfeld and Qiu [15]
10
5
0
Wall
Air inlet
0
20
40
60
80
100
Radius (mm)
Outlet
Fig. 5 Axial mean velocity profile at 50 mm
x = 25 mm
Simulation
Exp. Sommerfeld
3
Symmetry
1
-1
Fig. 3 Schematic of the computational grid
2
Radial mean velocity (m.s )
Fuel injection
6. Results and discussion
6.1. Flow without injection
First, the single-phase air flow through the model
combustion chamber is simulated (without injecting any
liquid droplets).The radial variations of the mean and RMS
axial velocity field at different axial locations are compared with the experimental data
The results of Figs. 4 and 5 show the evolution of
the mean axial velocity as a function of the distance to the
plane of the injector at distances 25 and 50 mm.
The profiles obtained by simulation are in good
agreement with measurements results obtained by Som-
0
-1
-2
-3
-4
-5
0
20
40
60
80
100
Radius (mm)
Fig. 6 Radial mean velocity profile at 25 mm
Figs. 8-10 and 11 show the results of the RMS axial and radial velocity as a function of the distance to the
598
x = 50 mm
Simulation
Exp Sommerfeld
3
5,0
x = 25 mm
Simulation
Exp. Sommerfeld
4,5
4,0
3,5
3,0
-1
VRMS(m.s )
plane of the injector at distances 25 and 50. We observe in
these figures that these results are in good agreement with
experimental results with a small overestimate; this overestimation is expected because the axial and radial fluctuations are almost identical in the prediction while a strong
anisotropy is observed in the experience and this anisotropy is difficult to reproduce by simulation.
2,5
2,0
1,5
1,0
0,5
2
0,0
-0,5
-1
Radial mean velocity (m.s )
1
0
0
20
40
60
80
100
Radius (mm)
-1
Fig. 10 Radial mean velocity RMS profile at 25 mm
-2
5,0
-3
x = 50 mm
Simulation
Exp.Sommerfeld
4,5
-4
4,0
3,5
-5
40
60
80
100
Radius (mm)
Fig. 7 Radial mean velocity profile at 50 mm
-1
20
VRMS(m.s )
3,0
0
2,5
2,0
1,5
1,0
x = 25 mm
Simulation
Exp.Sommerfeld
5,0
4,5
0,5
0,0
-0,5
4,0
0
40
60
80
100
Fig. 11 Radial mean velocity RMS profile at 50 mm
3,0
URMS(m.s )
20
Radius (mm)
3,5
-1
2,5
We also note that apart from the axial zone that is
to say from 40 mm tube annular turbulence remains very
low in our predictions and this may be due to low coverage.
2,0
1,5
1,0
0,5
6.2. Flow with droplets injection
0,0
-0,5
0
20
40
60
80
100
Radius (mm)
Fig. 8 Axial mean velocity RMS profile at 25 mm
5,0
x = 50 mm
Simulation
Exp.Sommerfeld
4,5
4,0
3,5
-1
URMS(m.s )
3,0
2,5
2,0
1,5
1,0
0,5
0,0
-0,5
0
20
40
60
80
Radius (mm)
Fig. 9 Axial mean velocity RMS profile at 50 mm
100
The study of the flow without evaporation gave a
good agreement with the experimental configuration of
Sommerfeld and Qiu [15]. We will resume this configuration by injecting a spray of isopropyl alcohol in the center
of the flow to fully test our model and compare our results
with the measured results. The flow rate of the liquid injected is 0.443 g/s, and the outlet temperature of injector is
313 K. The ambient temperature is 373 K. The mean axial
velocity of the gas flow is 18 m/s. The initial droplet diameter in the inlet is 20 to 50 μm.
The results of axial mean velocities of the droplets
are shown in Figs. 12 and 13 for two different distances to
the plane of the injector. These results are in a good
agreement with experimental results with a small overestimate.
Figs. 14 and 15 shows the distributions of the
mean diameter as a function of the distance to the plane of
the injector at distances of 25 and 50 mm.
We observed in these figures that the distribution
is at a pace backward, and although this profile due to initial injection.
599
The results have a good overall agreement with
experimental results measured at the entrance plane of the
tube with a small overestimation of the plans from the
tube.
fined in the vicinity of the injection and inside the jet. This
is due to holding of the circulation droplets near the nozzle
where they evaporate
x = 25 mm
Simulation
Exp.Sommerfeld
18
16
45
Droplets diameter (µm)
14
Droplets mean velocity (m/s)
x = 50 mm
Simulation
Exp.Sommerfeld
50
20
12
10
8
6
4
40
35
30
25
2
20
0
-2
15
-4
5
10
-6
15
20
25
30
35
40
Radius(mm)
-8
0
10
20
30
40
50
60
70
80
90
100
Fig. 15 Droplets diameter at 50 mm
Radius (mm)
Fig. 17 chows the size droplets distributions, we
observe in this figure that the recirculation gas flow trap all
droplets or becomes towards the path of the flow is also
noted that though these droplets follow the flow and differ
very little from the gas jet
Fig. 12 Droplets velocity at 25 mm
20
x = 50 mm
Simulation
Exp.Sommerfeld
18
Droplets mean velocity (m/s)
16
14
12
10
8
6
4
2
Fig. 16 Mass fraction contours
0
0
10
20
30
40
50
60
70
80
90
100
Radius (mm)
Fig. 13 Droplets velocity at 50 mm
50
x = 25 mm
Simulation
Exp.Sommerfeld
Droplets diameter (µm)
45
40
Fig. 17 Droplets size distribution
35
7. Conclusion
30
25
20
15
20
25
30
35
40
Radius(mm)
Fig. 14 Droplets diameter at 25 mm
Fig. 16 chows the vapour mass fraction contours,
we observe in this figure that almost all the steam is con-
The study we have presented is intended to properly compare our results with numerical experiments.
Simulating the flow without injection gives a better approach for axial and radial mean velocity, The RMS axial
and radial velocities are in good agreement with experimental results with a small overestimate, this overestimation is expected because the axial and radial fluctuations
are almost identical in the model of simulation while a
strong anisotropy is observed in the experience and this
anisotropy is difficult to reproduce by simulation. We also
note that apart from the axial zone that is to say from
600
40 mm, the turbulence is very low due to poor circulation.
We also note that the droplets follow the flow and differ
very little from the gas jet.
The results of the axial mean velocities of the
droplets are in a good agreement with experimental results
with a small overestimate.
References
1. Birouk, M.; Chauveau, C.; Sarh, B.; Quilbars, A.;
Gökalp, I. 1996. Turbulence effects on the vaporization of monocomponent single droplets, Combustion
Science and Technique 111: 413-428.
2. Clift, R.; Grauvin, W.H. 1971. Motion of entrained
particles in gas stream, Can. J. Chem. Eng. 49: 439.
3. Clift, R.; Grace, J.R.; Weber, M.E. 1978. Bubbles,
Drops and Particles, New York. Academic Press.
4. Leal, L.G. 1980. Particle motions in a viscous fluid,
Annual Review of Fluid Mechanics 12: 435-176.
5. Williams, A. 1973. Combustion of droplets of liquid
fuels, A Review, Combust. Flame 21: 1-31.
6. Faeth, G.M. 1983. Evaporation and combustion of
sprays, Energy Combustion Science 9: 1-76.
7. Law, C.; Binark, M. 1979. Fuel sprays vaporization
humid environnement, Journal of Heat and Mass Transfer 22: 1009-1020.
8. Sirignano, W.A. 1986. The formulation of spray combustion models, resolution compared to droplet spacing, Journal of Heat and Mass Transfer 8: 633-639.
9. Aggarwal, S.K.; Chitre, S. 1992. On the structure of
unconfined spray flames, Combust. Sci. Technol. 81:
97.
10. Godsave, GA. 1953. Studies of the combustion of
drops in a fuel spray, the burning of single drops of
fuel, Fourth International Symposium on Combustion,
Cambridge, Massachusetts: 818-830.
11. Spalding, D.B. 1953. The combustion of liquid fuels,
In 4th International Symposium in Combustion, the
combustion Institute: 847-864.
12. Ranz, W.E.; Marshall, W.R. 1952. Evaporation from
drops, Journal of Chemical Engineering Progress 48:
173-180.
13. Sirignano, W.A. 1983. Fuel droplet vaporization and
spray combustion, Proc. Energy Combust. Sci. 9: 291322.
14. Berlemont, A. Grancher, M.S.; Gouesbet, G. 1995.
Heat and mass transfer coupling between vaporizing
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H. Merouane, A. Bounif, M. Abidat
IZOPROPILO ALKOHOLIO LAŠELIŲ, ĮPURKŠTŲ Į
TURBULENTINĘ SROVĘ, IŠGARAVIMO
IMITAVIMAS
Reziumė
Darbe pristatomas izopropilo alkoholio lašelių
skaitinis imitavimas turbulentinėje dviejų fazių srovėje.
Lašeliai įpurškiami į žiedinį įkaitinto oro vamzdį –
200 mm vidinio skersmens ir 1500 mm ilgio aliumininę
cilindrinę kamerą. Sommerfeldo eksperimente panaudotas
64 mm skersmens įpurškimo vamzdis. Skaitinis imitavimas
atliktas naudojant laisvąjį pramoninį kodą, naudojant skaitinį baigtinių tūrių metodą ir daugiatinklę dalijimo schemą.
Pagrindinės tekėjimo lygtys sprendžiamos naudojant žinomą SIMPLE algoritmą ir struktūrinį turbulencijos modelį. Gauti rezultatai gerai sutapo su Sommerfeldo ir Oiu
eksperimentų duomenimis.
H. Merouane, A. Bounif, M. Abidat
SIMULATION OF EVAPORATING ISOPROPYL
ALCOHOL DROPLETS INJECTED INTO A
TURBULENT FLOW
Summary
In this work we present a numerical simulation of
the isopropyl alcohol droplets in a turbulent, two-phase
flow. The droplets are injected into the annular heated air
tube which is an aluminium cylindrical chamber of internal
diameter 200 mm and 1500 mm of length. The diameter of
the injection tube is 64 mm in the Sommerfeld experimental configuration. The numerical simulation is performed
by using the Fluent industry code, which uses a numerical
finite volume method coupled with a multigrid resolution
scheme. The flow governing equations are solved by using
a performed SIMPLE algorithm and using the standard k-ε
model of turbulence. The results are compared with the
experimental data obtained by Sommerfeld and Qiu to
show good agreement.
Received January 24, 2011
Accepted November 10, 2011