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A Numerical Investigation of Thermal Diffusion Influence on Soot Formation in Ethylene/Air Diffusion Flames

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A Numerical Investigation of Thermal DiffusionInﬂuence on Soot Formation in Ethylene/AirDiffusion Flames
HONGSHENG GUO
a,
*, FENGSHAN LIU
a
, GREGORY J. SMALLWOOD
a
and O¨MER L. GU¨LDER
b
a
Combustion Research Group, Institute for Chemical Process and Environmental Technology, National Research Council of Canada,1200 Montreal Road, Building M-9, Ottawa, Ont., Canada K1A 0R6;
b
Institute for Aerospace Studies, University of Toronto,4925 Dufferin Street, Toronto, Ont., Canada M3H 5T6
Thermal diffusion, caused by temperature gradients, tends to draw lighter molecules towarmer regions and to drive heavier molecules to cooler regions of a mixture. The inﬂuenceof thermal diffusion on soot formation in coﬂow laminar ethylene/air diffusion ﬂames isnumerically investigated in this paper. Detailed reaction mechanisms and complex thermal andtransport properties are employed. The fully elliptic governing equations are solved.Radiation heat transfer from the ﬂames is calculated by the discrete-ordinates method coupledto an SNBCK-based wide band model. A simpliﬁed two-equation soot model is used.The interactions between soot and gas-phase chemistry are taken into account. The thermaldiffusion velocities are calculated according to the thermal diffusion coefﬁcients evaluated basedon multicomponent properties.The results show that thermal diffusion does affect soot formation in ethylene/air diffusionﬂames. Although the effect on soot formation in pure ethylene/air ﬂame is not signiﬁcant, theinﬂuence is enhanced if lighter species, such as helium, are added to the fuel or the air stream.The peak integrated soot volume fraction doubles if thermal diffusion is not taken into account inthe simulation of the ﬂame with helium addition to the air stream.
Keywords
: Combustion; Thermal diffusion; Laminar ﬂame; Soot; Simulation; Radiation
INTRODUCTION
Heat and mass transport processes have been shown to beof great importance for soot formation (McLintock, 1968;Schug
et al.
, 1980; Axelbaum
et al.
, 1988; Axelbaum &Law,1990;Gu¨lder
etal.
,1996;Glassman,1998;Guo
etal.
,2002a). Thermal diffusion (Soret effect), caused bytemperature gradients in a mixture, gives an additionalterm in the diffusion velocity of a chemical species.It tends to draw lighter molecules to hot regions and todrive heavier molecules to cold regions of the mixture.
In a pioneering work, Dixon-Lewis (1968) observedthat the thermal diffusion ﬂux of hydrogen could be of the same order of magnitude as the ordinary diffusionﬂux, caused by concentration gradients, in a hydrogen/airﬂame. The same observation was made byGreenberg (1980) in the study of one-dimensionalhydrogen/air ﬂames using a one-step chemistry modeland phenomenological expressions for the thermaldiffusion coefﬁcients. Later it was found by Warnatz(1982) that the laminar ﬂame speeds of both lean and richhydrogen/air ﬂames were lower when thermal diffusionwas taken into account, although only the thermaldiffusion ﬂuxes of atomic and molecular hydrogen wereconsidered in the simulation. In the study of vortex-ﬂameinteractions in hydrogen jet diffusion ﬂames, Hancock
et al.
(1996) showed that the thermal diffusion effectcouldn’t be neglected in the numerical simulation. Recentstudies of Ern and Giovangigli (1998, 1999) furtherindicated the importance of thermal diffusion in variousreactive ﬂows. In the paper of Ern and Giovangigli (1998),it was shown that thermaldiffusion was important notonlyfor the prediction of structures of hydrogen/air andmethane/air Bunsen ﬂames, but also for the prediction of NO in a counterﬂow methane/air ﬂame. Being differentfrom the result of Warnatz (1982), the study of Ern and
ISSN 1061-8562 print/ISSN 1029-0257 online
q
2004 Taylor & Francis LtdDOI: 10.1080/10618560310001634203
*Corresponding author. Tel.:
þ
1-613-991-0869. Fax:
þ
1-613-957-7869. E-mail: hongsheng.guo@nrc-cnrc.gc.ca
International Journal of Computational Fluid Dynamics
, February 2004
Vol.
18 (2), pp. 139–151
Giovangigli (1999) indicated that for hydrogen/airﬂames, while the speeds of lean ﬂames were lower,those of rich ﬂames were higher when thermal diffusionof all species was considered. The inﬂuence of thermaldiffusion on the speeds of methane/air ﬂameswas negligible. More recently, Williams (2001) revealedthat thermal diffusion caused an increase in the predictedextinction strain rate of methane/air counterﬂow ﬂames.
In spite of the importance of thermal diffusion,little attention has been paid to the inﬂuence of thermal diffusion on soot formation processes.It was totally neglected in some studies (McEnally
et al.
, 1998; Smooke
et al.
, 1999; Bennett
et al.
, 2001),while only the thermal diffusion of light species(such as H
2
and H) was taken into account inother studies, such as our two recent papers (Guo
et al.
(2002a,b) and Kennedy
et al.
(1996)). To ourknowledge, the relative inﬂuence of thermal diffusionon soot formation has not been reported previously inthe literature.
In the present paper, soot formation processes in apure ethylene/air and four helium or argon dilutedethylene/air coﬂow laminar diffusion ﬂames aresimulated. The objective is to investigate the relativeinﬂuence of thermal diffusion on soot formation. Weemploy the primitive variable method in which the fullyelliptic governing equations are solved with detailedgas-phase chemistry and complex thermal and transportproperties. The effects of soot inception, growth andoxidation on gas-phase chemistry are taken intoaccount. For the soot kinetics process, a simpliﬁedtwo-equation soot model is used. Radiation heat transferfrom CO
2
, CO, H
2
O and soot is calculated using thediscrete-ordinates method coupled to an SNBCK-basedwide band model.
NUMERICAL MODEL
The ﬂame conﬁguration studied is a coﬂowaxisymmetric laminar diffusion ﬂame. The fuelstream ﬂows from the centre pipe, and the oxidantstream ﬂows from the annular concentric space. Exceptfor the pure ethylene/air diffusion ﬂame, ﬂames withhelium or argon addition to the fuel or air stream arealso studied.
Gas-phase Governing Equations
The numerical model solves the fully elliptic governingequations for the conservation of mass, momentum,energy, gas species mass fractions, soot mass fractionand soot number density. In cylindrical coordinates(
r
,
z
), the governing equations for the gas-phase are(Kuo, 1986):
Continuity:
››
r
ð
r
r
v
Þ þ
››
z
ð
r
r
u
Þ ¼
0
:
ð
1
Þ
Axial momentum:
r
v
›
u
›
r
þ
r
u
›
u
›
z
¼
2
›
p
›
z
þ
1
r
››
r r
m
›
u
›
r
þ
2
››
z
m
›
u
›
z
2
23
››
z
m
r
››
r
ð
rv
Þ
2
23
››
z
m
›
u
›
z
þ
1
r
››
r r
m
›
v
›
z
þ
r
g
z
:
ð
2
Þ
Radial momentum:
r
v
›
v
›
r
þ
r
u
›
v
›
z
¼
2
›
p
›
r
þ
››
z
m
›
v
›
z
þ
2
r
››
r r
m
›
v
›
r
2
231
r
››
r
m
››
r
ð
rv
Þ
2
231
r
››
r r
m
›
u
›
z
þ
››
z
m
›
u
›
r
2
2
m
vr
2
þ
23
m
r
2
››
r
ð
rv
Þ þ
23
m
r
›
u
›
z
:
ð
3
Þ
Energy:
c
p
r
v
›
T
›
r
þ
r
u
›
T
›
z
¼
1
r
››
r r
l
›
T
›
r
þ
››
z
l
›
T
›
z
2
X
KK
þ
1
k
¼
1
r
c
pk
Y
k
V
kr
›
T
›
r
þ
V
kz
›
T
›
z
2
X
KK
þ
1
k
¼
1
h
k
W
k
v
k
þ
q
r
:
ð
4
Þ
Gas species:
r
v
›
Y
k
›
r
þ
r
u
›
Y
k
›
z
¼
2
1
r
››
r
ð
r
r
Y
k
V
kr
Þ
2
››
z
ð
r
Y
k
V
kz
Þþ
W
k
v
k
;
k
¼
1
;
2
;
. . .
;
KK
;
ð
5
Þ
where
u
and
v
are the velocities in axial (
z
) and radial (
r
)directions, respectively;
T
the temperature of the mixture;
r
the density of the mixture (soot and gas);
W
k
themolecular weight of the
k
th gas species;
l
the mixturethermal conductivity;
c
p
the speciﬁc heat of the mixtureunder constant pressure;
c
pk
the speciﬁc heat of the
k
th gas
H. GUO
et al.
140
species under constant pressure;
v
k
the mole productionrate of the
k
th gas species per unit volume;
p
the pressure.It should be pointed out that the production rates of gasspecies include the contribution due to the soot inception,surface growth and oxidation (see the next section).Quantity
h
k
denotes the speciﬁc enthalpy of the
k
th gas species;
g
z
the gravitational acceleration in the
z
direction;
m
the viscosity of the mixture;
Y
k
the massfraction of the
k
th gas species;
V
kr
and
V
kz
the diffusionvelocities of the
k
th gas species in
r
and
z
directions;and
KK
the total gas-phase species number. The quantitieswith subscript
KK
þ
1 correspond to those of soot. As anapproximation, the thermal properties, obtained fromJANAF thermochemical tables (Chase
et al.
, 1985), of graphite are used to represent those of soot.
The last term
q
r
on the right-hand side of Eq. (4) is thesourcetermduetoradiationheattransfer.Itisobtainedbythediscrete-ordinates method coupled to a statistical narrow-band correlated-K (SNBCK)-based wide bandmodel fortheproperties of CO, CO
2
, H
2
O and soot (Liu
et al.
, 1999).The spectral absorption coefﬁcient of soot is obtained byRayleigh’stheoryforsmallparticlesandtherefractiveindexof soot due to Dalzell & Saroﬁm (1969) is
5
:
5
f
v
=
l
, with
f
v
being the soot volume fraction and
l
the wavelength.
The diffusion velocity consists of three terms:ordinary diffusion, thermal diffusion and correctiondiffusion velocities. Therefore:
V
kx
i
¼
V
okx
i
þ
V
Tkx
i
þ
V
cx
i
;
k
¼
1
;
2
;
. . .
;
KK
;
x
i
¼
r
;
z
:
ð
6
Þ
Both the ordinary and thermal diffusion velocities canbe obtained by the detailed multicomponent method, orthe approximate mixture-average method. The former isaccurate, but computationally expensive. The latter iscorrect asymptotically in some special cases, such as in abinary mixture, in diffusion of trace amounts of speciesinto a nearly pure species, or systems in which all speciesexcept one move with nearly the same diffusion velocity.Since the current study concentrates on the relativeinﬂuence of thermal diffusion, the ordinary diffusionvelocity
V
okx
i
, caused by concentration gradient, isobtained by the approximate mixture-average formulation(Kee
et al.
, 1986), i.e.
V
okx
i
¼
2
1
Y
k
D
k
›
Y
k
›
x
i
k
¼
1
;
2
;
. . .
;
KK
;
x
i
¼
r
;
z
:
ð
7
Þ
Quantity
V
Tkx
i
is the thermal diffusion velocity,whose inﬂuence will be investigated in the presentpaper, in
x
i
(
r
or
z
) direction for the
k
th gas species.It is obtained by the detailed multicomponent formulation(Kee
et al.
, 1986):
V
Tkx
i
¼
2
D
T
k
r
Y
k
1
T
›
T
›
x
i
k
¼
1
;
2
;
. . .
;
KK
;
x
i
¼
r
;
z
ð
8
Þ
where
D
T
k
is the thermal diffusion coefﬁcient obtained bythe method given by Kee
et al.
(1986).
The correction diffusion velocity
V
cx
i
is used to ensurethat the net diffusive ﬂux of all gas species and soot is zero(Kee
et al.
, 1986).
Quantity
D
k
in Eq. (7) is related to the binary diffusioncoefﬁcients through the expression:
D
k
¼
1
2
X
k
X
KK j
–
k
X
j
D
jk
;
k
¼
1
;
2
;
. . .
;
KK
ð
9
Þ
where
X
k
is the mole fraction of the
k
th species, and
D
jk
is the binary diffusion coefﬁcient.
Soot Model
Although some detailed kinetic models of soot inception,growth and oxidation have been derived, such as those byFrenklach
et al.
(1984) and Frenklach and Wang (1990,1994), they are too complex and computationallyexpensive to be implemented in simulations of multi-dimensional combustion systems. Conversely, the appli-cability of purely empirical soot models is questionableunder conditions different from those under which theywere originally formulated. Based on some semi-empirical assumptions, McEnally
et al.
(1998) andSmooke
et al.
(1999) used the sectional model to simulatethe soot formation processes. In addition to themomentum, energy and gas species conservationequations, several soot section equations (usually morethan 10) need to be solved. The model developed byLeung
et al.
(1991) and Fairwhether
et al.
(1992)has been successfully used in our previous studies(Guo
et al.
, 2002a,b) for the simulations of ethylene/airdiffusion ﬂames. It has been shown that this modelcan capture the features of the effects of inertspecies dilution on soot formation in ethylene/air diffusionﬂames. As only two additional equations need to be solvedfor soot processes in this model, it is used again in thepresent paper.
Two transport equations are solved for soot massfraction and number density. They are
r
v
›
Y
s
›
r
þ
r
u
›
Y
s
›
z
¼
2
1
r
››
r
ð
r
r
V
T
;
r
Y
s
Þ
2
››
z
ð
r
V
T
;
z
Y
s
Þ þ
S
m
ð
10
Þ
r
v
›
N
›
r
þ
r
u
›
N
›
z
¼
2
1
r
››
r
ð
r
r
V
T
;
r
N
Þ
2
››
z
ð
r
V
T
;
z
N
Þ þ
S
N
ð
11
Þ
where
Y
s
is the soot mass fraction,
N
is the sootnumber density deﬁned as the particle number per unitmass of mixture. Quantities
V
T
;
r
and
V
T
;
z
are theparticle thermophoretic velocities. They are obtained bythe expression for a free molecular aerosol
NUMERICAL INVESTIGATION OF THERMAL DIFFUSION 141
(Talbot
et al.
, 1980):
V
T
;
x
i
¼
2
0
:
55
mr
T
›
T
›
x
i
x
i
¼
r
;
z
:
ð
12
Þ
Although the particle thermophoretic motion is alsoa kind of thermal diffusion, its effect was not studiedin this paper, since the emphasis in this paper is thethermal diffusion of gas species (Eq. 8).The source term
S
m
in Eq. (10) accounts for thecontributions of soot nucleation (
v
n
), surface growth(
v
g
) and oxidation (
v
O
). Therefore,
S
m
¼
v
n
þ
v
g
2
v
O
:
ð
13
Þ
The model developed by Leung
et al.
(1991) andFairwhether
et al.
(1992) is used to obtain the threeterms on the right-hand side of Eq. (13). The modelassumes the chemical reactions for nucleation andsurface growth, respectively, as:
C
2
H
2
!
2C
ð
S
Þ þ
H
2
ð
R1
Þ
C
2
H
2
þ
n
C
ð
S
Þ
!
ð
n
þ
2
Þ
C
ð
S
Þ þ
H
2
ð
R2
Þ
with the reaction rates given by the expressions:
r
1
¼
k
1
ð
T
Þ½
C
2
H
2
ð
14
Þ
r
2
¼
k
2
ð
T
Þ
f
ð
A
s
Þ½
C
2
H
2
ð
15
Þ
where
f
(
A
s
) denotes the functional dependence on sootsurface area per unit volume. Similar to our previousstudies (Guo
et al.
, 2002a,b), a simple linear functionaldependence is used, i.e.
f
(
A
s
)
¼
A
s
.
Neoh
etal.
(1981)investigatedthesootoxidationprocessin ﬂames, and found that the oxidation due to both O
2
and OH is important, depending on the local equivalenceratio. The radical O also contributes to soot oxidationin some regions. Therefore the soot oxidation by O
2
, OHand O are accounted for by the following reactions:
0
:
5O
2
þ
C
ð
S
Þ
!
CO
ð
R3
Þ
OH
þ
C
ð
S
Þ
!
CO
þ
H
ð
R4
Þ
O
þ
C
ð
S
Þ
!
CO
:
ð
R5
Þ
The reaction rates for these three reactions wereobtained by:
r
3
¼
k
3
ð
T
Þ
T
1
=
2
A
s
½
O
2
ð
16
Þ
r
4
¼
w
OH
k
4
ð
T
Þ
T
2
1
=
2
A
s
X
OH
ð
17
Þ
r
5
¼
w
O
k
5
ð
T
Þ
T
2
1
=
2
A
s
X
O
ð
18
Þ
where
X
OH
and
X
O
denote the mole fractions of OHand O, and
w
OH
and
w
O
are the collision efﬁciencies forOH and O attack on soot particles, respectively.The collision efﬁciency of OH is treated as thatdescribed by Kennedy
et al.
(1996), who accounted forthe variation of the collision efﬁciency of OH with timeby assuming a linear relation between the collisionefﬁciency and a dimensionless distance from thefuel nozzle exit. A collision efﬁciency of 0.5 forradical O attack on the particles is used in this study(Bradley
et al.
, 1984).
All the reaction rate constants,
k
i
ð
i
¼
1
;
. . .
;
5
Þ
;
aresummarized in Table I.
The source term
S
N
in Eq. (11) accounts for the sootnucleation and agglomeration, and is calculated as:
S
N
¼
2
C
min
N
A
r
1
2
2
C
a
6
M
C
ð
S
Þ
pr
C
ð
S
Þ
1
=
6
£
6
k
T
r
C
ð
S
Þ
1
=
2
C
ð
s
Þ½
1
=
6
r
N
11
=
6
ð
19
Þ
where
N
A
is Avogadro’s number (
6
:
022
£
10
26
particles
=
kmol
Þ
;
C
min
is the number of carbon atoms inthe incipient carbon particle
ð
9
£
10
4
Þ
(Fairwhether
et al.
,1992),
k
is the Boltzman constant
ð
1
:
38
£
10
2
23
J
=
K
Þ
;
r
C
(
S
)
is the soot density (1800kg/m
3
),
C
(s) is the moleconcentration of soot (kmol/m
3
),
M
C
(s)
is the molar massof soot (12.011kg/kmol), and
C
a
is the agglomeration rateconstant for which a value of 3.0 (Fairwhether
et al.
, 1992)is used.
Numerical Scheme
The ﬂames modeled in this study are generated with aburner (Gu¨lder
et al.
, 1996) in which the fuel streamﬂows from a 10.9mm inner diameter vertical tube, andthe oxidant stream ﬂows from the annular regionbetween the fuel tube and a 100mm inner diameterconcentric tube. The wall thickness of the fuel tube is0.95mm.
The computational domain covers an area from0 to 3.0cm in the radial direction and 0 to 11.0cm inthe axial direction. The inﬂow boundary
ð
z
¼
0cm
Þ
corresponds to the region immediately above thefuel nozzle exit. This computational domain hasbeen shown to be large enough by a sensitivity calculation.Totally, 104
£
71 non-uniform grids are used. Finergrids are placed in the reaction zone and near the fuelnozzle exit region by a grid adaptive reﬁnement method.
TABLE I Rate constants, as
k
i
¼
A
exp
ð
2
E
=
RT
Þ
(units are kg, m, s,kcal, kmol and K)
k
i
A E Reference
k
1
1.35E
þ
06 41 Fairwhether
et al.
(1992)
k
2
5.00E
þ
02 24 Fairwhether
et al.
(1992)
k
3
1.78E
þ
04 39 Fairwhether
et al.
(1992)
k
4
1.06E
þ
02 0 Neoh
et al.
(1981)
k
5
5.54E
þ
01 0 Bradley
et al.
(1984)
H. GUO
et al.
142
FIGURE 1 Comparison of predicted and measured ﬂame temperature and soot volume fraction.FIGURE 2 Comparison of soot volume fraction for pure ethylene/airﬂame. a. Integrated soot volume fraction; b. Radial soot volume fractionat axial height of
z
¼
1
:
0cm
:
TD represents
Thermal Diffusion
.
FIGURE 3 Thermal diffusion factors of some main species and radicalsat the axial height of
z
¼
1
:
0cm
for pure ethylene/air ﬂame.NUMERICAL INVESTIGATION OF THERMAL DIFFUSION 143

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