ACTA MECHANICA SINICA, Vo1.19, No.3, June 2003
The Chinese Society of Theoretical and Applied Mechanics
Chinese Journal of Mechanics Press, Beijing, China
Allerton Press, INC., New York, U.S.A.
SIMULATION
COMBUSTION
ISSN 0567-7718
OF NOx FORMATION
IN TURBULENT
USING
A USM TURBULENCE-CHEMISTRY
ZHOU Lixing ( ~ 2 q ~ ) t
QIAO Li ( ~
Sg)
Z H A N G Jian ( ~
SWIRLING
MODEL*
f~)
(Department of Engineering Mechanics, State Key Laboratory of Clean Coal Combustion,
Tsinghua University, Beijing 100084, China)
A B S T R A C T : A unified second-order moment (USM) turbulence-chemistry model for simulating
NOx formation in turbulent combustion is proposed. All of correlations, including the correlation of
the reaction-rate coefficient fluctuation with the concentration fluctuation, are closed by the transport
equations in the same form. This model discards the approximation of series expansion of the exponential function or the approximation of using the product of several 1-D PDF's instead of a joint PDF. It
is much simpler than other refined models, such as the PDF transport equation model and the conditional moment closure model. The proposed model is used to simulate methane-air swirling turbulent
combustion and NOx formation. The prediction results are in good agreement with the experimental
results.
K E Y W O R D S : swirling turbulent combustion, NOx formation, second-order moment model
1 INTRODUCTION
The numerical simulation of NOx formation
in turbulent combustion is frequently used in the
optimization design of low-NOx burners and furnaces.
Although the direct numerical simulation
(DNS)[ 1], large-eddy simulation (LES)[ 2], probability density distribution function ( P D F ) transport
equation model[ 3] and the conditional moment closure (CMC) model [4], developed in recent years, can
well simulate the interaction of turbulence with detailed chemistry, these refined models need rather
large computer m e m o r y and computation time. T h e y
can be used only in simulating very simple flows
in fundamental studies.
For engineering combustion modeling, the widely used models, including
those adopted in the commercial software, are the
Eddy-Break-Up (EBU)-Arrhenius (denoted as E-A in
the following text) model, the p r e s u m e d - P D F fastchemistry model for turbulent combustion [5] and the
p r e s u m e d - P D F finite-rate chemistry model for NOx
formation [6]. However, the PDF-fast-chemistry model
cannot account for the finite-rate kinetics and the EA model is also not handy to account for the finiterate kinetics. The P D F finite-chemistry model using
a product of several single-variable P D F ' s instead of
a joint P D F leads to an under-prediction over the averaged reaction rate. Alternatively, the second-order
moment turbulence-chemistry models, based on the
idea of second-order m o m e n t turbulence models, are
more reasonable t h a n the E-A model and the presumed P D F models, while they are more economical
t h a n other refined models.
It is well known t h a t the difficulty in developing
the second-order moment turbulence-chemistry models lies in the treatment of the exponential function of
t e m p e r a t u r e in the time averaging procedure. Previously, two versions of the second-order moment models were developed. In the first version a series expansion of the exponential t e r m with an approximation of E / R T << 1 is made[ 5]. This model is used
to simulate methane-air and hydrogen-air turbulent
diffusion combustion Iv's]. T h e predicted t e m p e r a t u r e
and main species concentration are in good agreement
with experimental results, but the NOx concentration
is under-predicted. The reason is that for NOx formation the activation energy is large and E / R T is much
larger t h a n unity, so the approximation of E / R T << 1
leads to a significant under-prediction over the averaged reaction rate. The second version is the second-
Received 27 October 2001, revised 24 May 2002
* The project supported by the Special Funds for Major State Basic Research of China (G1999-0222-07)
t E-mail: [email protected]
Vol.19, No.3
Zhou Lixing et al.: Simulation of NOx Formation in Swirl Combustion
order-moment P D F model, in which the concentration
correlation is closed using the second-order m o m e n t
equations, while the temperature-concentration correlation is closed using the presumed PDF[ 9,1~ Simulation of methane-air jet diffusion combustion shows
t h a t this version of the second-order moment model,
not using the series expansion approximation, is much
better than the first version of the second-order moment model and the E-A model. However, it still does
not entirely get rid of the approximation of using the
product of two single-variable P D F ' s instead a joint
PDF.
In this paper, a third version of second-order
m o m e n t model, i.e. a unified second-order m o m e n t
(USM) model is proposed. The feature of this new
model is that when solving the time-averaged reaction rate, the correlation of the reaction-rate coefficient fluctuation with the concentration fluctuation is
closed using the t r a n s p o r t equation in the same form,
as t h a t of the t r a n s p o r t equations for all other correlations, assuming that the production t e r m of this
correlation is proportional to that of the temperatureconcentration correlation. The validity of this closure
can be verified only by experiments. Besides, the
effect of reaction on the dissipation of correlations
is taken into account. T h e proposed model is used
to simulate methane-air swirling turbulent combustion and NOx formation. The prediction results are
validated using the experimental results taken from
Ref.[ll].
2
THE
USM
MODEL
Eq.(1) becomes
Ws --
+
rfuk'ro'x + Lxk'Yf']
For an elementary reaction of any two species or
a global one-step reaction with two species, for exampie, fuel and oxygen, the instantaneous reaction rate
is
(2)
where k = f exp(-E/RT)p(T)dT, p(T) is the temperature PDF. The correlations MY' and Y~Yorx are
closed using a unified form of transport equations.
The generalized form of t r a n s p o r t equations of these
correlations when accounting for the effect of chemical reaction on the dissipation using two time scales
is
0
05
-Sgx5
] +
0~0~
Cgl"T
OXj OXj
(ab)
Cg2
~
~-
<
/9r
(3)
where
--1
k
T T Z -C
Cg2 = 2.0
CgI = 2.8
a+ b = 1
For the correlation of the fluctuation of the reactionrate coefficient with the concentration k'Y', it is
very difficult strictly deriving the transport equation. Assuming that its t r a n s p o r t equation takes the
same form as t h a t for the transport of temperatureconcentration correlation and its production and dissipation are proportional to those of T'Y', we have
0
TURBULENCE-CHEMISTRY
209
0 (,;0k Y,)
Cgl/AT ~
Cg2p
+
(4)
Ws = Bp2YfuYox exp(- e/RT)
where Yfu, Yox, T express the instantaneous values of
fuel mass fraction, oxygen mass fraction and temperature, respectively, E denotes the activation energy
and R the universal gas constant. After taking the
Reynolds averaging the time-averaged reaction rate
takes the following form
9T
Ws = Bp2kYfuYox
= Bp2( + #)(?fu + U3( ox + Y'x)
where, k =
When
(1)
exp(-e/RT).
neglecting the
When taking the top-hat P D F of temperature, the
time-averaged reaction-rate coefficient is
third-order
correlation
=
Tt2
For the chemical kinetics of methane-air combustion a global reaction kinetics of Arrhenius type is
taken as
Wfu = 1.0 X 101~
e x p ( - 1 . 8 4 x 104/T)
(5)
ACTA MECHANICA SINICA
210
The time-averaged reaction rate is
~uk'Yo'x + ?o~k'y~']
(6)
where B1 = 6MNo/4MNHa, B2 = M N O / J ~ f N H a , M is
the species "molecular weight.
Therefore the laminar reaction rate of NO formation is
WNO ~- WNO,fuel -]- Wz -F Wp
The transport equation of Yf~Yot~is
_
ot
o (~ oyi'Yg,x ~
~
Cg2P
a
o~
Ofox
~Xj ] ~- ggl#t OXj OXj
i-a)
-~ TA
y, y,
fu o x
WNO,fue I = WNO,fue 1 X (1 + Z1)
ENHa
l Y,'O2
k'1Y/NHa ~_ k'1Y~02
Zl -- YNH3Yo2 + ~IYNH3 kl?O2
(7)
where cgl = 2.8, Cg2 = 2.0, a = 0.5. The transport
equation of k ' Y ' is Eq.(4).
For NO formation the Zeldovich mechanism of
thermal NO formation and Finemore mechanism of
prompt NO formation are used as
Wz = 9 x 1012TO.3exp ( 3 8 4 4 0 )
[N2][O2]
(8)
-+
Z2=-
([M]- [N2])([M]- [N2]- [CO2])]}.
(9)
-
[02]
G--l+40exp(-~-)+2.6xlO13T
exp
For fuel NO
--
4NH3
+ 6NO
-4.
is used
~ 5N2 + 6H20
(10)
4NH3
The fuel NO
+ 502 ~ 4NO
1 =
13 588
T= + T , ) + e x p
(L3588~ ]
~ - T'Jj
(
+exp
3 S I M U L A T I O N OF NOx F O R M A T I O N I N
METHANE-AIR SWIRLING TURBULENT COMBUSTION
]4
L~
reaction rate is determined by
4.0
•
IOspYNHayb2B2 exp ( \
16105~]
T-T']
by Eq.(3).
+ 6H20
WNO,fueI = 1.8 x IOspYNHaYNoB1 exp ( - - - W ~ o+, f u e
~1 [exp (
The geometrical configuration and sizes of the
swirl combustor to be predicted are shown in Fig.1
and Table 1. The fuel--methane is supplied from
the central tube and the swirling air is supplied from
the annular tube with a swirler. The air flow rate is
8.gm3/h and the methane flow rate is 0.8932m3/h.
The swirl number is s = 0.43. A small amount of
[H20]
the following mechanism
(14)
The correlations in Eqs.(13), (14) are determined also
where T is the temperature (K), [M] is the mole concentration (mole/cm3s), F and G are given by
x 10-Z~ 3exp
Y/NH3Y/O2 k2I Y/NHa k2I E02
I
- +- +- YNHaYO2 k2YNHa k2Yo2
1 [
(16105)
k 2 = ~ exp
'~+T'
F3/2G[N2][CH4][021 a/2 [H20]1/2/
[1 + 3 000 exp(-15 185/T)]
+
where
=
Wp = {3 x 1010exp(-2 900/T).
(13)
W;o,f.,ol = W;o,f.o 1 x (1 + 12)
f~l
F=I-I.1
(12)
For the turbulent reaction rate of NO formation,
the time-averaged reaction rate using the USM model,
for example, for fuel NO is
~xj
aXj
2003
)
1T5)
- -
(11)
methane+ammonia
Fig.1 The swirl combustor
Zhou Lixing et al.: Simulation of NOx Formation in Swirl Combustion
Vol.19, No.3
4
Table 1 The geometrical sizes of the combustor
PREDICTION
T h e m o d i f i e d k-e t u r b u l e n c e m o d e l w i t h a corr e c t i o n a c c o u n t i n g t h e effect of swirl a n d unified
s e c o n d - o r d e r m o m e n t t u r b u l e n c e - c h e m i s t r y m o d e l are
used t o s i m u l a t e t u r b u l e n t swirling flows, combustion and NO formation. The computation domain
is 0.9 m x 0.08 m. 80 x 45 s t a g e r r e d g r i d n o d e s are
a d o p t e d , as shown in Fig.2. T h e differential e q u a t i o n s
are d i s c r e t i z e d into finite difference e q u a t i o n s a n d t h e
F D E ' s a r e solved using t h e S I M P L E C a l g o r i t h m . For
b o u n d a r y c o n d i t i o n s , u n i f o r m d i s t r i b u t i o n of different
v a r i a b l e s is t a k e n at t h e inlet; s y m m e t r i c a l c o n d i t i o n s
are t a k e n a t t h e axis a n d f u l l y - d e v e l o p e d flow cond i t i o n s are t a k e n at t h e exit. No-slip c o n d i t i o n s are
t a k e n a t t h e wails. For n e a r - w a l l grid n o d e s t h e wall
f u n c t i o n a p p r o x i m a t i o n is used. T h e c o m p u t e r codes
consists of a b o u t 4 000 s t a t e m e n t s . R u n n i n g a case in
a P e n t i n u m - 3 - 5 5 0 P C t a k e s a b o u t 10 h.
1.0
0.8
o~
0.6
0.4
0.2
0.0
I
0
5
0
.
06
IIllllUlllllfi]LEklllllllllllllllllllllilLIIIII]lll
i
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t
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i
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I [ I I
i i i i
I I I I
nell
I I I [ [ I I k I I
Ullllml[]l]B[llllllllllllllllllllllll[llll]]lll
0.05
i
i
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[i
i k [ [ i i i i
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .I. .I. .I. .I. .I. I I I I I I
IIIIIIIIIlIIIlIlilllllllllllllllllllllllllll[lllllll
J I I I I I IIII
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I I
i i
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In
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~ i i ~
I I I i
~l
I J
I]
[I
II
.I .I .I . I . [ . I . I . . . .
I I
I I
I
"~- 0.04 ...........................................................
m,~,nnnnnnnnnnnmnmnl.ullllllnunnn
ii i i n n n n n n n n n n ~ n n
ii
0.03 ...........................................................
0.02
mllullmHIIIIIIIIIIIIII]lllllll]llllll
II
] I I I I I I I I I I I I I I I I
~[F~IIIIIIIIIIIIIIIIII]IIII]HIIIIII]I]II
II
I L [ ~ I I I I I I I I I I I I I
I::I:,~:::::::::::II:::ET, I:P,:IIIIIII
W~ llll::::llll
~:::::::::::::::I:['??,II,[F,I::2J,
V,I N Ill III:::::I
:::::::::::::::::::::::::::::::::::::::::::::
:::::
I
I
I
[
I
I
I
I
i
I
I
I
I
i
I
I
I
I
i
I
I
I
[
i
I
I
I
I
i
I
I
I
Fig.3 Streamlines
I I
i r
I I
IF
I
0.08
I
I I [ I I I I II
0.06
i]
"--- 0.04
1
: I I I I I 1
I I I I I I I I I l
: : : : : : : : : : : :
0.01
0
/~
0.2
0.4
1339
1274
1218
0.6
0.00
0.0
0.8
0.2
0.4
x/m
Fig.2 Computation
1173
0.02
':':':': ': ':
0
10
x/R
0.08~!!!!!!!!!!!!!!!![!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]!!!!]!!![!!
!!
' !: !', !', !', !: !', !', !', !', !!
':'''''''''''''']E[`'''''''''''''''''''':'::`''':'•'::::::::
',: ', ',
:',
',',
i i i
i
I I I
I
DISCUS-
Figure 3 gives the streamlines. It can be seen
that under the effect of swirl there is a large-size corner recirculation zone and a small-size central recirculation zone. Figure 4 shows the temperature maps.
The high temperature region developed in the corner
recirculation zone and the central recirculation zone
plays the role of flame stabilization. Lower temperature exists at the inlet and near-wall region. Figure
5 gives the comparison of predicted temperature profiles at 7 cross sections with the experimental results
in Ref.[ll]. The agreement is good. The comparison
of the predicted NO concentration with the experimental results is shown in Fig.6. In general, the prediction results are near to those measured. Quantitatively, the model over-predicts the NO concentration
a m m o n i a (4.91%) is a d d e d to t h e fuel t o s i m u l a t e fuel
N O in case of gas c o m b u s t i o n . T h e e x p e r i m e n t a l resuits for t u r b u l e n t c o m b u s t i o n a n d N O f o r m a t i o n in
this c o m b u s t o r are t a k e n from Ref.[11].
IHIIIIIIIIIIII]II[IIIIIIIIIIIIIIIIIII[II[II[Illll
m......i,i,m..mmmmu,Hllll,,lll
HIIIIIIIIIIIIII/II}IIIIIIIIIIIIIIIIIIIII[IIIIIIIll
mIIIIIIImI~ImUIIUUUUlUUIIUHIIIII~IIII
AND
SION
D1/mm D2/mm Da/mm Df/mm Dout/mm Lf/mm
8
i0
30
160
180
900
0.07
RESULTS
211
0.6
0.8
x/m
Fig.4 Temperature maps (unit: K)
domain and grid arrangement
)
-%.
4
t
~4
O ~
x=17.5
r
600
'
x----52.6
x=27.5
i
,
1 200
600
1 200
600
1 200
600
1 200
600
,
T/K
predictions
9
experiments
Fig.5 Temperature profiles
,
vl
1 200
600
x=702
~
1200
,',
600
1200
ACTA MECHANICA SINICA
212
2003
-5
i
@
I
I
I
x~5
10
-
7
"~
~=40
x=70
a----52.~
i
,
i
v
600 1 200
iv
,
600 1 200
600 1 200
600 1 200
600 1 200
600 1 200
I
.l
600 1 200
[NO]/ppm
predictions
9
experiments
Fig.6 NO concentration profiles
in t h e first 4 cross sections. T h i s d i s c r e p a n c y m a y b e
c a u s e d b y t h e a d o p t e d over-simplified r e a c t i o n kinetics. It can b e seen t h a t d u e t o t h e effect of swirl t h e
N O c o n c e n t r a t i o n in t h e c o r n e r region is higher.
5
CONCLUSIONS
(1) T h e N O f o r m a t i o n in swirling t u r b u l e n t combust i o n c a n b e r e a s o n a b l y s i m u l a t e d using t h e unified s e c o n d - o r d e r m o m e n t t u r b u l e n c e - c h e m i s t r y
model
(2) M o r e d e t a i l e d kinetics s h o u l d b e t a k e n into account to improve the predictions.
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823~830
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