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.
ISSN 05677718
SI MULATI ON OF NOx FORMATI ON I N TURBULENT SWI RLI NG
COMBUSTI ON USI NG A USM TURBULENCE CHEMI STRY MODEL*
ZHOU Lixing ( ~2q~) t QIAO Li ( ~ Sg) ZHANG Jian ( ~ f~)
(Department of Engineering Mechanics, State Key Laboratory of Clean Coal Combustion,
Tsinghua University, Beijing 100084, China)
ABSTRACT: A unified secondorder moment (USM) turbulencechemistry model for simulating
NOx formation in turbulent combustion is proposed. All of correlations, including the correlation of
the reactionrate 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 exponen
tial function or the approximation of using the product of several 1D PDF's instead of a joint PDF. It
is much simpler than other refined models, such as the PDF transport equation model and the condi
tional moment closure model. The proposed model is used to simulate methaneair swirling turbulent
combustion and NOx formation. The prediction results are in good agreement with the experimental
results.
KEY WORDS: swirling turbulent combustion, NOx formation, secondorder moment model
1 I NTRODUCTI ON
The numerical simulation of NOx formation
in turbulent combustion is frequently used in the
optimization design of lowNOx burners and fur
naces. Although the direct numerical simulation
(DNS)[ 1], largeeddy simulation (LES)[ 2], probabil
ity density distribution function ( PDF) t ransport
equation model[ 3] and the conditional moment clo
sure (CMC) model [4], developed in recent years, can
well simulate the interaction of turbulence with de
tailed chemistry, these refined models need rat her
large computer memory and comput at i on time. They
can be used only in simulating very simple flows
in fundamental studies. For engineering combus
tion modeling, the widely used models, including
those adopted in the commercial software, are the
EddyBreakUp (EBU)Arrhenius (denoted as EA in
the following text) model, the presumedPDF fast
chemistry model for turbulent combustion [5] and the
presumedPDF finiterate chemistry model for NOx
formation [6] . However, the PDFfast chemi st ry model
cannot account for the finiterate kinetics and the E
A model is also not handy to account for the finite
rate kinetics. The PDF finitechemistry model using
Received 27 October 2001, revised 24 May 2002
a product of several singlevariable PDF's instead of
a joint PDF leads to an underprediction over the av
eraged reaction rate. Alternatively, the secondorder
moment turbulencechemistry models, based on the
idea of secondorder moment turbulence models, are
more reasonable t han the EA model and the pre
sumed PDF models, while they are more economical
t han other refined models.
It is well known t hat the difficulty in developing
the secondorder moment turbulencechemistry mod
els lies in the t reat ment of the exponential function of
t emperat ure in the time averaging procedure. Previ
ously, two versions of the secondorder moment mod
els were developed. In the first version a series ex
pansion of the exponential t erm with an approxima
tion of E/RT << 1 is made[ 5]. This model is used
to simulate methaneair and hydrogenair turbulent
diffusion combustion Iv's]. The predicted t emperat ure
and mai n species concentration are in good agreement
with experimental results, but the NOx concentration
is underpredicted. The reason is t hat for NOx forma
tion the activation energy is large and E/RT is much
larger t han unity, so the approxi mat i on of E/RT << 1
leads to a significant underprediction over the aver
aged reaction rate. The second version is the second
* The project supported by the Special Funds for Major State Basic Research of China (G1999022207)
t Email: zhoulx@mail.tsinghua.edu.cn
Vol.19, No.3
Zhou Lixing et al.: Simulation of NOx Formation in Swirl Combustion
209
ordermoment PDF model, in which the concentration
correlation is closed using the secondorder moment
equations, while the t emperat ureconcent rat i on cor
relation is closed using the presumed PDF[ 9,1~ Sim
ulation of methaneair jet diffusion combustion shows
t hat this version of the secondorder moment model,
not using the series expansion approximation, is much
bet t er than the first version of the secondorder mo
ment model and the EA model. However, it still does
not entirely get rid of the approximation of using the
product of two singlevariable PDF's instead a joint
PDF.
In this paper, a third version of secondorder
moment model, i.e. a unified secondorder moment
(USM) model is proposed. The feature of this new
model is t hat when solving the timeaveraged reac
tion rate, the correlation of the reactionrate coeffi
cient fluctuation with the concentration fluctuation is
closed using the t ransport equation in the same form,
as t hat of the t ransport equations for all other cor
relations, assuming t hat the production t erm of this
correlation is proportional to t hat of the t emperat ure
concentration 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. The proposed model is used
to simulate methaneair swirling turbulent combus
tion and NOx formation. The prediction results are
validated using the experimental results taken from
Ref.[ll].
Eq.(1) becomes
Ws  +
rfuk'ro'x + Lxk'Yf'] (2)
where k = f exp(E/RT)p(T)dT, p(T) is the tem
perat ure PDF. The correlations MY' and Y~Yorx are
closed using a unified form of t ransport equations.
The generalized form of t ransport equations of these
correlations when accounting for the effect of chemi
cal reaction on the dissipation using two time scales
is
0
05 Sgx5 ] +
0~0~ ( ab)
Cgl"T OXj OXj Cg2 ~ ~ < /9r (3)
where
k
TTZ 
C
Cg I = 2.8
1
Cg2 = 2.0 a+ b = 1
For the correlation of the fluctuation of the reaction
rate coefficient with the concentration k'Y', it is
very difficult strictly deriving the t ransport equa
tion. Assuming t hat its t ransport equation takes the
same form as t hat for the t ransport of t emperat ure
concentration correlation and its production and dis
sipation are proportional to those of T'Y', we have
2 THE USM TURBULENCECHEMISTRY
MODEL
For an elementary reaction of any two species or
a global onestep reaction with two species, for exam
pie, fuel and oxygen, the instantaneous reaction rate
is
Ws = Bp2YfuYox exp( e/RT)
where Yfu, Yox, T express the instantaneous values of
fuel mass fraction, oxygen mass fraction and t emper
ature, respectively, E denotes the activation energy
and R the universal gas constant. After taking the
Reynolds averaging the timeaveraged reaction rate
takes the following form
Ws = Bp2kYfuYox
= Bp2( + #)(?fu + U3( ox + Y'x) (1)
where, k = exp(e/RT).
When neglecting the thirdorder correlation
0 0 (,;0k Y,)
Cgl/AT ~
Cg2p + (4)
When taking the t ophat PDF of t emperat ure, the
timeaveraged reactionrate coefficient is
9T = Tt2
For the chemical kinetics of methaneair com
bustion a global reaction kinetics of Arrhenius type is
taken as
Wfu = 1.0 X 101~ exp(1.84 x 104/T) (5)
210 ACTA MECHANICA SINICA 2003
The timeaveraged reaction rate is
~uk'Yo'x + ?o~k'y~'] (6)
The transport equation of Yf~Yot~ is
_
ot ~xj
o (~ oyi'Yg, x ~ o~ Ofox
aXj ~ ~Xj ] ~ ggl#t OXj OXj
ia)
a y, y,
Cg2P ~ fu ox (7)
TA
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 ( 38440) [N2][O2] (8)
Wp = {3 x 1010 exp(2 900/T).
F3/2G[N2][CH4][021 a/2 [H20]1/2/
([M] [N2])([M] [N2] [CO2])]}.
[1 + 3 000 exp(15 185/T)] (9)
where B1 = 6MNo/4MNHa, B2 = MNO/J ~f NHa, M is
the species "molecular weight.
Therefore the laminar reaction rate of NO for
mation is
WNO ~ WNO,fuel ] Wz F Wp (12)
For the turbulent reaction rate of NO formation,
the timeaveraged reaction rate using the USM model,
for example, for fuel NO is
WNO,fue I = WNO,fue 1 X (1 + Z1)
E l Y,' k' Y/ k' Y~
NHa O2 1 NHa ~_ 1 02
Zl  YNH3Yo2 + ~IYNH3 kl?O2
(13)
 + +
W;o,f.,ol x + = W;o,f.o 1 (1 12)
Y/ Y/ k I Y/ k I E I
NH3 O2 2 NHa 2 02
Z2=  +  + 
YNHa YO2 k2YNHa k2Yo2
where
13 588
1 [exp ( T+
f~l ~ = T,) +exp ( L3588~
]
= ~  T'Jj
1 [ ( 16105) ( 16105~]
k2= ~ exp '~+T' +exp T T']
(14)
The correlations in Eqs.(13), (14) are determined also
by Eq.(3).
where T is
centration (mole/cm3s), F and G are given by
F=I  I.1 x 10Z~ 3exp  [02]
G  l +40exp(  ~ ) +2.6xl O13T 4.
exp  [H20]
For fuel NO the following mechanism is used
4NH3 + 6NO ~ 5N2 + 6H20
4NH3 + 502 ~ 4NO + 6H20
The fuel NO reaction rate is determined by
WNO,fue I = 1.8 x IOspYNHaYNoB1 exp (    
+ IOspYNHayb2B2 exp (
W~o,f ue 1 = 4.0   
\
the temperature (K), [M] is the mole con
(10)
)
1T5 )
(11)
3 SIMULATION OF NOx FORMATION IN
METHANE AIR SWIRLING TURBU
LENT COMBUSTION
The geometrical configuration and sizes of the
swirl combustor to be predicted are shown in Fig.1
and Table 1. The fuelmethane 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
]4 L~
methane+ammonia
Fig.1 The swirl combustor
Vol.19, No.3
Zhou Lixing et al.: Simulation of NOx Formation in Swirl Combustion
211
Tabl e 1 The geomet r i cal si zes of t he combus t or
D1/mm D2/mm Da/mm Df/mm Dout/mm Lf/mm
8 i0 30 160 180 900
ammoni a (4.91%) is added to t he fuel t o si mul at e fuel
NO in case of gas combust i on. The exper i ment al re
suits for t ur bul ent combust i on and NO f or mat i on in
this combust or are t aken from Ref.[11].
The modi fi ed ke t ur bul ence model wi t h a cor
rect i on account i ng t he effect of swirl and unified
secondorder moment t ur bul ence chemi st r y model are
used t o si mul at e t ur bul ent swi rl i ng flows, combus
t i on and NO format i on. The comput at i on domai n
is 0.9 m x 0.08 m. 80 x 45 st ager r ed gri d nodes are
adopt ed, as shown in Fig.2. The di fferent i al equat i ons
are di scret i zed i nt o finite difference equat i ons and t he
FDE's are sol ved using t he SI MPLEC al gori t hm. For
boundar y condi t i ons, uni form di st r i but i on of different
vari abl es is t aken at t he inlet; symmet r i cal condi t i ons
are t aken at t he axis and ful l ydevel oped flow con
di t i ons are t aken at t he exit. Nosl i p condi t i ons are
t aken at t he wails. For nearwal l gri d nodes t he wall
funct i on appr oxi mat i on is used. The comput er codes
consists of about 4 000 st at ement s. Runni ng a case in
a Pent i num 3 550PC t akes about 10 h.
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IIIIIIIIII/ILl[lllllllllllllllllllllllllllllllllll[I J I I I I I IIII I I I I
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::::::::::::::::::::::::::::::::::::::::::::: ::::: : : : : : : : : : : : :
0.01 ':':':': ': ':
0
0 0.2 0.4 0.6 0.8
x/m
Fig.2 Computation domain and grid arrangement
4 PREDI CTI ON RESULTS AND DI SCUS
SI ON
Figure 3 gives the streamlines. It can be seen
that under the effect of swirl there is a largesize cor
ner recirculation zone and a smallsize central recir
culation 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 tempera
ture exists at the inlet and nearwall region. Figure
5 gives the comparison of predicted temperature pro
files at 7 cross sections with the experimental results
in Ref.[ll]. The agreement is good. The comparison
of the predicted NO concentration with the experi
mental results is shown in Fig.6. In general, the pre
diction results are near to those measured. Quantita
tively, the model overpredicts the NO concentration
1.0
0.8
0.6
0.4
0.2
0.0
o~
0 5
x/R
Fig.3 Streamlines
I
10
0.08
0.06
" 0.04
0.02
/~ 1339 1274 1218 1173
0.00
0.0 0.2 0.4 0.6 0.8
x/m
Fig.4 Temperature maps (unit: K)
~4
O ~
r '
600 1 200
600 1 200
%.
x=17.5
,
600 1 200
x=27.5
600 1 200
T/K
predictions
i , vl
600 1 200
experiments
4
)
t
x52.6 x=702
, ~ ,',
600 1200 600 1200
Fig.5 Temperature profiles
212
i
x~5
i , i v ,
600 1 200
ACTA MECHANICA SINICA
10  7 "~
@
600 1 200
~=40
i v
600 1 200 600 1 200 600 1 200
[NO]/ppm
predictions experiments
Fig.6 NO concentration profiles
I
5
I
I
x=70
a52 .~ I
.l
600 1 200 600 1 200
2003
in t he first 4 cross sections. Thi s di scr epancy may be
caused by t he adopt ed oversi mpl i fi ed r eact i on ki net 
ics. It can be seen t hat due t o t he effect of swirl t he
NO concent r at i on in t he corner regi on is higher.
5 CONCLUSI ONS
(1) The NO f or mat i on in swi rl i ng t ur bul ent combus
t i on can be r easonabl y si mul at ed using t he uni
fied secondorder moment t ur bul ence chemi st r y
model
(2) More det ai l ed ki net i cs shoul d be t aken into ac
count t o i mprove t he predi ct i ons.
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