(Department of Engineering Mechanics, State Key Laboratory of Clean Coal Combustion, Tsinghua University, Beijing China)

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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 0567-7718
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 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 exponen-
tial 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 condi-
tional 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.
KEY WORDS: swirling turbulent combustion, NOx formation, second-order moment model
1 I NTRODUCTI ON
The numerical simulation of NOx formation
in turbulent combustion is frequently used in the
optimization design of low-NOx burners and fur-
naces. Although the direct numerical simulation
(DNS)[ 1], large-eddy 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
Eddy-Break-Up (EBU)-Arrhenius (denoted as E-A in
the following text) model, the presumed-PDF fast-
chemistry model for turbulent combustion [5] and the
presumed-PDF finite-rate chemistry model for NOx
formation [6] . However, the PDF-fast -chemi st ry model
cannot account for the finite-rate kinetics and the E-
A model is also not handy to account for the finite-
rate kinetics. The PDF finite-chemistry model using
Received 27 October 2001, revised 24 May 2002
a product of several single-variable PDF's instead of
a joint PDF leads to an under-prediction over the av-
eraged reaction rate. Alternatively, the second-order
moment turbulence-chemistry models, based on the
idea of second-order moment turbulence models, are
more reasonable t han the E-A 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 second-order moment turbulence-chemistry 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 second-order 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 methane-air and hydrogen-air 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 under-predicted. 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 under-prediction 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 (G1999-0222-07)
t E-mail: zhoulx@mail.tsinghua.edu.cn
Vol.19, No.3
Zhou Lixing et al.: Simulation of NOx Formation in Swirl Combustion
209
order-moment PDF model, in which the concentration
correlation is closed using the second-order moment
equations, while the t emperat ure-concent rat i on cor-
relation is closed using the presumed PDF[ 9,1~ Sim-
ulation of methane-air jet diffusion combustion shows
t hat this version of the second-order moment model,
not using the series expansion approximation, is much
bet t er than the first version of the second-order mo-
ment 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 PDF's instead a joint
PDF.
In this paper, a third version of second-order
moment model, i.e. a unified second-order moment
(USM) model is proposed. The feature of this new
model is t hat when solving the time-averaged reac-
tion rate, the correlation of the reaction-rate 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 methane-air 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 TURBULENCE-CHEMISTRY
MODEL
For an elementary reaction of any two species or
a global one-step 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 time-averaged reaction rate
takes the following form
Ws = Bp2kYfuYox
= Bp2( + #)(?fu + U3( ox + Y'x) (1)
where, k = exp(-e/RT).
When neglecting the third-order correlation
0 0 (,;0k Y,)
Cgl/AT ~
Cg2p + (4)
When taking the t op-hat PDF of t emperat ure, the
time-averaged reaction-rate coefficient is
9T = Tt2
For the chemical kinetics of methane-air 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 time-averaged 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
i-a)
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 time-averaged 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 10-Z~ 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 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
]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 k-e t ur bul ence model wi t h a cor-
rect i on account i ng t he effect of swirl and unified
second-order 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 y-devel oped flow con-
di t i ons are t aken at t he exit. No-sl i p condi t i ons are
t aken at t he wails. For near-wal 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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::::::::::::::::::::::::::::::::::::::::::::: ::::: : : : : : : : : : : : :
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 large-size cor-
ner recirculation zone and a small-size 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 near-wall 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 over-predicts 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
x----52.6 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
a----52 .~ 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 over-si 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 second-order 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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