2.8.10 -1. 2.8. i0 -2. 6.0.10 -13. 6.g.10-I 1. l.l.10 -5. 7.r 0 -9. 8.6.10 -8. 1.0.10 8. 5.6.10 -1. 5.9. ..... CH3. H. O. C2H6. CH30. C2H3. C2H. C2H5. CH2. CH. 3. 1.361E+03 .... -2, 54E-07. CO. 4. OH. 5. I , 03E-04. 9, 10E-05. -i, 20E-05. CH3(]H +. OH. 6.
Twenty-first Symposium (International) on Combustion/The Combustion Institute, 1986/pp. 795-807
DEVELOPMENT OF A VARIATIONAL METHOD FOR CHEMICAL KINETIC SENSITIVITY ANALYSIS D. GROUSET, P. PLION, E. ZNATY Division Energetique SociOtO Bertin & Cie---40220 Tarnos/France
S. GALANT Division Informatique SociOt~ Bertin & Cie--13290 Les Milles/France
A new variational sensitivity analysis method has been developed and applied to chemical kinetic schemes in aeronautic combustion chambers. The method has been validated by comparison with the brute torce method for hydrogenoxygen combustion in a well stirred reactor assembly. Both methods agree very well as long as the nondimensional sensitivity coefficients are not too small. The variational method requires much less computational time than the brute force one. The method is then applied to complete kinetic schemes for methanol/oxygen and ethane/air combustion in a real well stirred reactor assembly. The reactions re-arrangement provided h~ the method is found to be a good tool for kinetic studies: the method may be used as a pre-processor for mechanism reductions. Results with reduced schemes for methanol (eleven reactions among the eighty-eight originaI ones) seems fairly good. For ethane, forty six reactions are needed for correct predictions, but the reduction could be carried out, using other chemical concepts. These results, together with the low computational times, are encouraging and the variational method may now be used to enhance kinetic studies.
Introduction A classical way to m o d e l a e r o n a u t i c combustion c h a m b e r s is to describe t h e m as a set o f well stirred reactors (W.S.R.). As already shown by a n u m b e r o f aircraft e n g i n e m a n f a c t u r e r s 1, accurate predictions o f c o m b u s t i o n efficiency, stability o r pollutant emissions b e c o m e available w h e n c o m p l e t e detailed kinetic schemes are used. T h e use o f such kinetic schemes in W.S.R. codes is e x p e n s i v e in t e r m s o f c o m p u t a t i o n a l time, a n d is unrealistic for 3D codes. So, e q u i v a l e n t kinetic schemes are c o m m o n l y developed, such as that o f H a u t m a n et al. 2, later a d a p t e d by Bellet et al. 3, A l t h o u g h such kinetic schemes give correct results for n o m i n a l conditions, they usually c a n n o t predict limiting p h e n o m e n a such as extinction. T o automatically r e d u c e c o m p l e t e kinetic schemes to different, locally e q u i v a l e n t schemes (for a given purpose), the first step is to classify reactions using a sensitivity analysis. T h e aim o f this study is to d e v e l o p a new m e t h o d for fast sensitivity analysis--a variational m e t h o d i a n d to show how it can h e l p in kinetic studies. Sensitivity analysis can be d o n e by m a n y
m e t h o d s . T h e o r i e s are well established (see Cacuci 4 for g e n e r a l systems o f n o n l i n e a r equations). T h e first m e t h o d is the brute force m e t h o d , it is simple but r a t h e r lengthy. T h e F o u r i e r m e t h o d , used by C u k i e r et al., 5'6'7, is in fact an extension o f the b r u t e force m e t h o d which may give second o r d e r sensitivity coefficients. T h e A.I.M. m e t h o d o f K r a m m e r et al. 8 also leads to second o r d e r coefficients. T h e G r e e n ' s f u n c t i o n m e t h o d a p p l i e d by H w a n g et al. 9 is suitable for differential e q u a t i o n systems i.e. time e v o l v i n g chemical problems. T h e direct m e t h o d used by Dickinson and Gelinas ~~ is simpler and also well suited for differential equations. T h e F o u r i e r m e t h o d seems to be the most c o m m o n l y a p p l i e d to chemical kinetic studies 5-7"it-It. O n the contrary, the variational m e t h o d , which has great m a t h e m a t i c a l advantages 13, has only been sparsely used. S e i g n e u r 14 a p p l i e d it to the differential e q u a t i o n system describing o z o n e reactions in the a t o m s p h e r e . We have applied the variational m e t h o d to a set o f n o n l i n e a r equations d e s c r i b i n g a W.S.R. network. We start with a b r i e f review o f the equations to be treated a n d o f the principle of the sensitivity m e t h o d . T h e n we p r e s e n t results
795
796
REACTION KINETICS
for a validation test and for real cases which may lead to reductions o f chemical kinetic schemes.
thus, Eqs. (1), (2) and (3) yield a set of algebraic equations [J (M + 1) in number] for which a general expression can be writen:
Combustion chamber simulation Let us consider a combustion chamber divided i n t o J coupled well stirred reactors where given mass flow rates of oxidizer and fuel are distributed. T h e concentrations (or mass fractions) of M molecular species built of K different atoms must be calculated as functions of the residence time and temperature. These species are coupled by a chemical kinetic scheme of R reactions. The t e m p e r a t u r e level is a result of chemical reactions and must also be calculated. Only the steady state is sought and an evolution equation may be written for each species m in each reactor i:
F (Y*,p) = 0
where Y is the array of variables Cjm and Tj; its value at the solution is Y*. p is the array of parameters, including the preexponential factors k and activation t e m p e r a t u r e s Ta o f the Arrhenius law (4 R in total in n u m b e r for a set of R reversible reactions). In each reactor, the search for the steady state uses the classical iterative Newton m e t h o d : ~F -1 Y~+~ = Y~ - ( - ~ ) F(Y",p) (6)
,c,,,, =0 r -J'" (c,,,, c,,,, / p, /
Mathematical background of the variational method
I
I
I
evolution
J
convection o,,~ ~ , ,~ reaction
0,/, external feed
(1) I
When a converged solution has been obtained in all reactors, a n o t h e r convergence loop ensures the coupling between reactors.
A solution Y* of Eq. 5 can be obtained for any physical value of the parameters p. T h e sensitivity analysis goal is to d e t e r m i n e the variations of Y* when the p a r a m e t e r s are changed. A sensitivity coefficient is defined: - - - ~dY* - - along F (Y*,p) = 0
_
Due to the A r r h e n i u s terms I/V, these equations are nonlinear and remain nonlinear for minor species. On the other hand, for major species, K in number, equations (1) are replaced by conservation equations (2) for atoms k in reactor i:
,,
p, / J
\ Pj
P,
=o
(s)
with
h,~(T) = horn+
I
AV* Ap
VS+~p)-- Y& -
Ap
OF
OF dY* dp+O(dp2) = 0
Cp(T) dT
(4)
(9)
The direct m e t h o d identifies the function F with the h y p e r p l a n tangential to the solution manifold and hence:
T
0
(8)
This method has been extended to Fourier's which gives second o r d e r coefficients and allows large p a r a m e t e r perturbations. But these methods are computationally time consuming. We will focus on the direct method and on the variational one. A first o r d e r d e v e l o p m e n t of F leads to:
dF
/
+qo(~h,,(To'-C~p~h,,(To))]
-
(2)
= 0
T e m p e r a t u r e is calculated by the enthalpy conservation equation which is nonlinear in T, even if it does not include the Arrhenius terms because of Cp variations with temperature:
,, L j,1
(7)
T h e first sensitivity analysis method is the brute force method. After the solution is found for the parameters' real values, complete sets of solutions associated with small perturbations of the parameters are sought and the sensitivity coefficients are then calculated using finite differences:
o
+,o,(;r
(5)
~
d~ =
\Or/
Op
(10)
V A R I A T I O N M E T H O D FOR S E N S I T I V I T Y ANALYSIS
The numerical calculation of~r does not require any stationary solution other than the central one, but simply a solution of linear systems, It is therefore much less time consuming than the brute force method. Due to the n u m b e r of species, reactions and parameters, the result--~-p v* will actually be a very large array [size:J 9 (M + 1)' 4 R]. If the topic of interest can be abstracted in an "objective" function L (Y) of the variables Y only, we will then calculate the scalar product:
dL
dL dY*
dp
dr
(11)
dp
in order to reduce the n u m b e r of data. T h e "obje&ive" function L may be the temperature, or a linear combination of u n b u r n t hydrocarbon concentrations or of other pollutant species... The variational analysis makes use of this information reduction possibility before solving the systems. For all F a n d L, there are Lagrange multipliers )t such that H = KF + L, the system Hamiltonian, would be m i n i m u m for the correct values of k:
OH ay = 0
dL ( O F ~ -1 so
=-Tf\a-f/
(12)
797
then:
eL
el. dr
dL(OF' - OF-xOF
Working with "objective" functions ensures a direct weighting of species. For L scalar, only one linear system must he solved--that one that yields X. A more powerful tool is provided by using a vector of several L functions which could summarize the effects of parameters on combustion efficiency, pollutant emissions or radiative properties of exhaust gases, fbr example. In this last case both X and ~L will be arrays.
Qualification tests of the method for a simple case The method has been tested on a simple case consisting of 2 reactors in series (V1 = 2.46. l0 '~ m 3 and V2 = 2 V0. A lean mixture of hydrogen and oxygen (H2 + 10 O2) is fed into the first reactor. T h e chemical scheme consists of 17 reversible reactions extracted from the Westbrook e t a l . scheme for ethylene 15, shown in Table I. Table II displays the steady solution: the temperature and composition (8 species) have been calculated. T h e chosen "objective" function is the mean formation enthalpy per
TABLE 1 Kinetic s c h e m e for the qualification test (from [ 15]) k *
r
T & + (K)
n+
k -
T a- (K)
n-
1
.186.109
8 395
0
.148.108
300
0
H+O 2
O*OH
2
.182.105
0 050
0
.832.100
3 075
1
H2+O
H*OH
3
.339.108
9 175
0
.316.107
550
0
H20 * O
2 OH
4
10 150
0
.219.108
2 575
0
H20 + H
H 2 * OH
900 52 500
0
.282.108 .101.1012
16 395
0
H202 -~ OH
H20 + HO 2
0
H20 + M
H*OH*M
500
0
.229.1010
22 950
0
H*O2*M~
HO 2 9 M
OH + 02
5
.955.108 .1 .108
6
.219.1011
7 8
.166.10 ~ .501.108
500
0
.606.108
28 505
0
HO 2 * O
9
.251.109
950
0
.120.10 $
20 050
0
HO 2 * H
2 OH
10
.251.108
350
0
.550.108
28 900
0
HO 2 + H
H 2 * 02
11
.501.108
H20 * 02
12 13
.398.108 .120.1012
1~ 15 16
.513.1010
57
.219.109
17
(13)
a#=d---f'~=-drkar/ at,- ap
0
-2
500
0
.631.109
36 930
0
HO 2 + OH
21 320
0
.1 .108
500
0
H202 * 02
2 HO 2
22 750
0
.922.103
- 2 503
0
H202 + M
2OH+M
.17 .107
1 775
0
.720.106
9 350
.1 .105
0
0
.729.1010
51 860
500
0
.068.100
0
~g 000
0
-302.100
0
0 -1 -
0,28 0
H202 * H
HO 2 * H 2
O*H+M~
OH*M
O2+M H2+M
=
20*M
=
2H*M
798
REACTION KINET1CS
TABLE II Data and stead}" solution for the qualification test Reactor
Volume
1
(10 -3 m 3)
2
2,t+6
TABLE iII Comparison of the brute force method with the variational method on the qualification test: arrangement of the first 20 reactions according to their sensitivity coefficient
4,92 Brute force method
Feedin 8
(mole/s)
H2
3.1,10 -2
0
02
3.1.10 -I
0
Temperature
Composition
(K)
966
Coefficient
-
971
(mole)
2.927
10-3
2.855
10-3
~ c-I co
riational method
~
7
]
1+
- 2.853
10-3
7+
I
11 +
- 2,324
10-3
11 +
1.255
IO-3
12-
12-
2.962 2.879
1+
10-3 10-3
2.873
10-3
- 2.348
10-3
1.263
10-3 I0 -#
-
[
t
H
7.1,10 "11
6.0.10 -13
O
4.5.10 -9
6.g.10-I 1
H2 OH
2.4,10 -/4
l.l.10 -5
6
13+
6.101
10-0
6.157
2.0.10 "8
7.r
0-9
7
5+
'
- 5.958
10-4
- 6.084
10-4
HO 2
7,4.10 -7
8.6.10 -8
8
'
8+
{ - 4.185
10-t~
8+
I - 4.292
10 -4
]
2.397
10 -~
3+
2.310
10-4
2,019
10-4
2+
2.026
10-4
5
1.g64
10-~
t . . . . . 4.677 .....
10-5
H202 02 H20
1.4.10 -6
1.0.10 8
9
[
3+
2.8.10 -1
5.6.10 -1
0
I
2+
2.8. i0 -2
5.9. I0 -2
.1 L4
2
7-
3
unit mass of c o m b u s t i o n products in both reactors. This is a reasonable m e a s u r e of combustion c o m p l e t i o n :
L,
E,,, C,,,,~,,,
10-5
17t
12+
I
1"5/~1
10-5
[ - /~.926 10 .6
5
]
3-
[
1.303
10-5
i
6
I
9-
1.240
10-5
- 4.289
10-6
6
1.126
10-5
2.338
10-6
7 I
10
15+
9
04) i 16+ [
T h e sensitivity of this objective to the pre-exponential factors is calculated bv the b r u t e force m e t h o d a n d the variational one. T h e n o n dimensional sensitivity coefficients (} ~ or rk ~dL\ lead to a r r a n g e the 20 first reactions as shown on Table III. T h e two m e t h o d s are in very good agreernent as soon as the sensitivity coetficients have more than a m i n i m u m value of 10 -5, However, the brute force m e t h o d is not an absolute reference. Its coefficients are only first-order a p p r o x i m a t i o n s because no calculation have been made for c e n t e r e d variations of k (see Eq. 8). Note that i n c l u d i n g t e m p e r a t u r e a m o n g the variables a n d consequently in the J a c o b i a n system dF/dY is the main condition for the accuracy of direct or variational m e t h o d s a n d the a g r e e m e n t with the brute force m e t h o d . T h e effects of t e m p e r a t u r e p e r t u r b a t i o n d u e to p r e - e x p o n e n t i a l factor p e r t u r b a t i o n are well taken into a c c o u n t in the sensitivity coefficients. T h e use of the p r e - e x p o n e n t i a l factors as parameters is numerically convenient. T h e sensitivity coefficients to the activation t e m p e r a -
10-5
H992
[
8
~],,, C,,,, h .....
5.646
I
t
4.592
10-6
1.081
10 -5
- 2.193
10-6
- 1.060
10-5
- 1.627
10-6
8.63I .__
10-6
8.423
lO-7
I
ture or t e m p e r a t u r e e x p o n e n t of a generalized A r r h e n i u s law can easily be derived, viz:
dL _ dT~
k dL T dk
dL dn
k L~ T) dLdk
(15)
Results for real cases As the variational m e t h o d gives the same results as the brute force m e t h o d , we have adapted it to m o r e realistic cases. Let us consider an aeronautic type c o m b u s t i o n chamber divided into 3 W.S.R. (fig. 1). T w o kinds o f feed have b e e n tested. First, a stoichiometric mixture o f m e t h a n o l and o x y g e n is injected at the c h a m b e r head and the products are progressively diluted by carbon dioxide. T h e thermal power o f 4 0 0 k W , the v o l u m e o f 0.186 10 -3 m 3, the total mass flow rate is 0.314 kg/s and the pressure is 60 bar. T h e second case is nearer to aeronautical conditions. Ethane is injected at the c h a m b e r head and air is provided along the flame tube. T h e r m a l power, v o l u m e and total mass flow rate are identical but the pressure is only 20 bar. In these conditions, the stoichiome-
VARIATION METHOD FOR SENSITIVITY ANALYSIS
i---i. CO .
.
.
.
.
.
.
.
,~o~/~ .
'//5...,;,,..0T-6~
- -
~
--
-"-~a~--'da"20 -is CH~OH+O. or C H 2 6
Injector ~
--
i'll
x~
.
,,.e~'qm, 2~
II/I//I
.
.
1st r e a c t o r V 1 : 40 c m '
.
" ~._~_~. ..........
---~l
~
~ _%_
.. "ill ~
s3 g/s
~
.
II
~-
- - ~
~0 s/~
13t4 sis
.
.
I
"~
~ - -
221
l[
2nd reactor
'I
I[
V 2 : 36 c m '
'
t,
.
I
799
~ . 2 ~ ~
.
"~ff-ls~ ~
I
sis
__~
-.__ 3rd reactor
I
V3 : 90 cm s
I
I I I / 1 I I I 7 7 / 7 ; 1I 1I1 1I 1 x < 1 r
~7 si~
-
44 g / s
3I, g/~
/
.
.
.
.
FIG. 1. The combustion chamber as a well stirred reactor assembly try evolves from a slightly rich mixture in reactor 1 (0 = 1.08) to a lean mixture in reactor 3 (qs = 0.44). The chemical scheme is again Westbrook's ethylene scheme 15, but C4 species have been suppressed. We then have to deal with 88 reversible reactions involving 25 or 26 species d e p e n d i n g on the presence of nitrogen. T h e objective function is again the mean massic formation enthalpy but 3 such functions have been defined---one in each reactor. T h e steady solutions are presented in Table IVa (for CHaOH + O2) and IVb (for C 2 H 6 + air). T h e species are arranged in decreasing order of concentration. T h e nondimensional sensitivity coefficients to pre-exponential factors lead to a classification of the reactions with respect to the sum of the absolute values of both coefficients for direct and reverse reactions. Due to the limited space available in this paper, only results with respect to the third objective, i.e. the formation enthalpy of the exhaust gases, are presented in Table Va (for methanol) and Vb (for ethane). T h e symbols < = > , < - > or - > > reactions indicate the relative weight of the forward and backward reactions. As we perturbed the 2 factors independently, equilibria have been displaced and reactions near equilibrium appear to be important, at the top of the classification. Due to the moderate temperature and the relatively high pressure, the main oxidant species are OH and HO2; reactions including O radicals appear very low in the classification. Comparison of sensitivity coefficients in the case of ethane-air for 20 a n d 60 bar shows that the lower the pressure, the more important the
three body reactions are. These reactions may become bottle necks for carbon oxidation at 20 bar. At this point, we are able to conclude as to the computational time requirements of the different methods. A Newton iteration modifying the 27 variables of a reactor, according to Eq. 6, needs about 2.6 s CPU on our PRIME 850 computer and 100 to 180 iterations are required for convergence, d e p e n d i n g on the initial solution. The influence of 176 parameters has been studied in 3 reactors. The brute force method would require about 176 perturbations • 4 iterations x 3 reactors = 2 112 iterations, or 5 500 s CPU time, fbr the Ay*/Ap determination. For the variational method, the gradients calculauon 9 ~, or ~p, 0F (~ 0L.is obtained . 9 125 s in computation time (equivalent to 48 iterations) and the sensitivity to the 3 objectives within 2.1 s (1 iteration). T h u s the variational method takes about 45 times less than the brute force method and, in any case, a complete sensitivity picture can be obtained in less time than needed for the the solution calculation.
Use of sensitivity analysis for chemical scheme reduction Without any chemical consideration except completion (to avoid dead-end species), we attempted to reduce the chemical schemes by using only the u p p e r part of the reaction arrangements of Tables V. Results are presented on Table VI for the state variables steady solution and Table VII for the chemical kinetic schemes and sensitivity coefficients.
TABLE IV Steady solution obtained in the 3 reactors with the complete kinetic scheme for: a) methanol/oxygen combustion at 60 bar b) ethane/air combustion at 20 bar a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2l 22 23 24 25
b
1 2 3 4 5 6 7 8 9 i0 ll I2 13 14 15 16 17 18 19 20 21 22 23 24 25 26
T
2
2.333E+03
T
CO2 H20 02 CO H202 CH3OH OH HO2 CH20 CH2OH H2 HCO CH4 CH3 H O CH30 C2H2 ('2H4 C2H6 C2H3 C2H C2H5 CH2 CH
6.614E-01 3.273E-01 7.946E-03 1.331E-03 8.861E-04 8.695E-04 8.259E-05 7.795E-05 5.580E-05 3.170E-06 1.210E-06 3.245E-07 2.574E-07 9.020E-09 1.791E-09 7 . 1 4 7 E - 10 2 . 8 9 0 E - 10 4 . 0 3 2 E - i1 1.269E- 11 1.338E-12 6 . 8 3 7 E - 14 6 . 3 9 5 E - I4 2 . 3 7 4 E - 16 2 . 0 7 0 E - 16 7.637E-20
CO2 H20 02 CO H202 CH3OH HO2 CH20 OH CH2OH H2 CH4 HCO CH3 H C2H2 C2H4 O CH30 C2H6 C2H C2H3 C2H5 CH2 CH
T
2.174E+03
T
7.I88E-01 1.559E-01 9.338E-02 1.462E-02 6.800E-03 3,250E-03 2.515E-03 2.251E-03 1.872E-03 2.562E-04 1.819E-04 1.465E-04 3.761E-05 1.181E-05 7.395E-06 4.464E-06 1.748E-06 1.564E-06 1.397E-06 2.814E-07 4.092E-08 3.406E-09 1.696E-09 1.379E- 11 1.683E- 12 8 . 2 3 1 E - 15
N2 H20 02 CO2 CO CH3OH C2H4 H202 C2H2 C2H6 HO2 CH20 OH H2 CH2OH C2H3 HCO CH4 C2H5 C2H CH3 H CH30 O CH2 CH
N2 H20 C()2 02 CO (;2I-t4 CH3OH C2H6 C2H2 (;I-120 HO2 H202 OH H2 CH2OH C2H3 C2H HCO CH4 C2H5 CH3 H CH30 O CH2 CH
i
1
2
800
3
1.699E+03
T
7.740E-01 2.169E-01 8.596E-03 3.227E-04 1,201E-04 4,203E-05 6,440E-06 4 , 9 0 0 E - 06 3,387E-06 4,823E-08 3.346E-08 2.539E-09 9 , 3 0 1 E - 10 7 , 0 2 8 E - 12 2 , 3 1 9 E - 12 1.297E- 12 3.121E-13 2 . 7 8 5 E - 13 2 . 0 0 0 E - 13 2 . 6 0 0 E - 14 1.677E- 16 1,256E- I6 1.728E-18 1.650E-21 1.650E-21
CO2 H20 02 CO H202 CH3OH CH20 HO2 OH H2 CH2OH CH4 H CO C2H2 C2H4 CH3 H O C2H6 CH30 C2H3 C2H C2H5 CH2 CH
1.712E+03
T
7.458E-01 1.053E-01 7.702E-02 6.565E-02 4,432E-03 6,417E-04 4,220E-04 2.831E-04 1.983E-04 1.608E-04 6,527E-05 4.766E-05 9,547E-06 1,133E-06 7.838E-07 1.842E-07 1.017E-07 3.607E-08 2.703E-08 1.421E-08 1,485E- 10 1.190E-11 6 . 4 4 1 E - 12 1.743E-12 4 . 4 1 6 E - 15 1,659E- 18
N2 02 H20 CO2 CO CH3OH H202 C2H4 C2H6 C2H2 CH20 HO2 OH H2 CH2OH C2H3 H CO CH4 C2H5 C2H CH3 O CH30 H CH2 CH
3
1.361E+03 8.311E-01 1.591E-01 9,562E-03 1.990E-04 3.304E-05 1 ,2 1 3 E- 0 5 1,897E-06 1.295E-06 2.471E-07 6.757E-09 3.546E-09 2,719E-10 5 , 0 8 8 E - 11 1 .7 3 3 E- 1 3 5,991E-14 3 , 4 4 0 E - 14 1 .1 6 3 E- 1 4 4.956E-15 4 , 0 7 4 E - 15 2 . 0 5 9 E - 15 6 . 5 3 5 E - 18 2.112E-18 1 .4 5 5 E- 1 9 1.642E- 21 1.642E- 21
1.372E+03 7 .5 8 6 E- 0 1 1.147E-01 7.549E-02 4.717E-02 3,203E-03 3,436E-04 1.750E-04 1 .6 6 0 E- 0 4 5.050E-05 4.943E-05 3.523E-05 1 .7 7 1 E- 0 5 1,064E-06 1 .1 8 6 E- 0 7 5,494E-08 4.962E-08 1.525E-08 7,986E-09 4.823E-09 2.734E-10 2.314E-12 1.634E- 13 1.314E- 13 1,075E- 13 7 . 9 5 2 E - 17 5,014E-21
V A R I A T I O N M E T H O D FOR S E N S I T I V I T Y A N A L Y S I S
801
TABLE V A r r a n g e m e n t o f the 88 complete kinetic scheme reactions in decreasing order: a) for m e t h a n o l / o x y g e n c o m b u s t i o n at 60 bar b) for ethane/air c o m b u s t i o n at 20 bar ~+ and ~ - are the n o n d i m e n s i o n a l sensitivity coefficients to k + and k = I ~+ I + I ~ - I is used for the classification Reaction f o r m u l a symbols mean: < = > 0.98 < I ~+/~I < 1.02 < > 0.5 < I ~+1~I < 2. - - > 2.0 < l ~ I < 200 -
->> a
200
o
r
I
2
3 4 5
I, 3 1 E - 0 4 I, 2 4 E - 0 4 I, 1 7 E - 0 4 1. 0 3 E - 0 4 I , 03E-04
o
I
- - > - >>
4M
) >
+
+ ~ 4
0
-i
0
0
+
- >> < "- > -- >
0 HCO CH3 CH3 CO CH3
+ 4 + * 4 +
OH H202 H H2[i HD2 OH
+ + 4 + * + 4
- )) -)) -- ) -> >
HCO
CH20H HCO CH20
02 02 H20 H202 H HD2
CH3
+
H202
t
53 54 55 56 57 58
(< - } > > > ) >
R E A C T I O N KINETICS
802 +
r
o
o
]4E
"02
1
2,
2!
1 , 90E-0~_~ ~. 2 6 E - E L ? I . 22E--02
3 4
!
1, 151-07
1. 9 0 E - 0 2 -4, 5 3 f ] - 0 3 - i , 10E-02
o
Reaction
" 9, 91E
EI~
.... I . 2 4 E - ~ 7 8. 0 6 E - 0 3 ~ . 2]E-03
H
4
[70 H'!02 H02
I ~ ~
formula
0;! H02 OH
"*
-~P
OH
*
+M
>> ) )
+
- '* > - } }
C02 CH3
"~ +
H CH20
+
(
H02
~
I-,14.)2
9 -i
}
CH?O HCO
+
-t
02 H20
C2H2 C2H2
OH OH 02
6, 9 1 E - 0 3 6, 1 3 E - 0 3 5, 7 7 E - 8 3 5, 61E-03 4. 1 8 E - 0 3 3, 8 8 E - 0 3 2. 08E -03 i, 6 3 E . - 0 3 I, 5 1 E - 0 ~ i. 17E-03
3. 4 9 E - 0 4 -q, 59E.-05 -I, 6 2 E - 0 3 - ? . 76E'-12 I . 16E>-03
20 21 22 23 24
8. 6 2 E - 0 4 5, 3 9 E - 0 4 I. 9 3 E - 0 4 I, 5 7 E - 0 4 1,08E-04
- B , &2E- 04 3, 4 9 E - 0 4 I, 9 3 E - 0 4 1, 0 8 E - 0 4 1 0EE-04
- 2 . 2 ] E - ~2 '~I . 9 0 E 0 4 - 3 . 00E'" 1 3
25 26 27 28 29 30 31
9, 4 s 7. 62E-05 6, 4 4 E - 0 5 4, 8 7 E - 0 5 4, 7 8 E - 0 5 2, 79E" 05
-9. 3 3 E - 0 5 7, 5 0 E - 0 5 5, 0 0 E - O b 4, 66E-05 4, 7 8 E - 0 5
0, 5 2 E 0 7 .-1, 22E- 06 -3 . 4 4 E - 0 5 -2, 0 3 E - 0 6 -4, 04E-08
I
32
I
2, 67E-05 -7. B4E-O 7 I, 751-05
2, 2 1 E 0 5 6, 81E-10
12
12 13 14 15 ]6 17 18 19
'
i
33 34 35
I, 3 5 E ' - 0 5 ], 1 5 E - 0 5 I, I I E - 0 5
36 :37
7. 05E-06 7. 03E-06 5, 2] E-06
38 39 40 4] 42 4L~ 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
77E--05 2, 26E-05 2. 21E-05 2,
5, 59E-0~ 2, 51E'-E]6 3, 67E-09
751i-06 73E-06 66E-06 46E-06 10E-06
3, 4 9 E - 0 6 5, 2 ] E - 0 6 4, 96E'-06 4, 2 6 E " 0 6 3, 46E-06 - I, 1 9 E - N . 5 2, 0 2 E - 0 6 i, 7 ~ E - ' 0 6 -0. 73E-07 I. 641--10 --2, 3 ) E - 0 9 i, 03C-06
9, 1 5 E - 0 7 5, 1 7 E - 0 7 4, B B E * 0 7 4. 45U?-07 3 , OE)E-"0 7 2, 9 0 E - - 0 7
15E-07 82E-07 91E:- 0 7 3bE'-09 - 2 , 9].~:-'07 "-3, 00E-10
2, 45E-07 2, 44E-0 :z
6, 96E-'08 2, 4IE-07 I, 13E-07
4, 9 9 E - 0 6 4, 2 6 E - 0 6 : 3. 46E-06 2, ] 81--06 2, 02E--06
i, I. I. I. I,
l
i, 0 2 E - 0 3
5, 61E-03 4, 1 0 E - 0 3
i, 75E-07 i . 67E-07 1, 00E- 07
9, -I. -2. -6,
~ ] . 401~-07 - 7 . 63 E - 0 8
OH 02
-4, 5 4 E - 0 , 5 -3. 08E-13 "-i, 57E-04 - 2 . 32E-'04 -3, I I E - 0 3 -3. 6 0 E - 0 3 -4, 75E> 03 "07
-] , 8 2 E - 0 9 -3. 54E 03 ":!. 0 4 E - 0 3 O, 9 6 E - ( ~ 6 ] 5 ] E 01~ - J . 55E 05
I I
+ + +
-4.
851E..-05
C2H HC()
-2
18E-.07
[H2CH
+
[2
HCI) CH2(]H C2H3 CH20 Ci?H5
4 + + 4 4
OH
-I, 261 -06 ~.!, 7 l , ! l i - 0 ~ : ~ "5, 04E 06
H[:(] H':!O C;;!H4
+ 4 +
O? H
7, 74E '09 "'-] , i~SE -0'5 -5, 9 5 E - 0 6 -8, 67E -06
CH20 CH30H C2H4 CH4
4
FIEf2 (]
" - 7 , 0',5E-06 - 3 , [;4E 0 6 -2, 84E 10
H24] C2H5
-iL 3 3 E - 0 0 -] , ] 51-- I 1 - 6 . 41!E 9 , (~JS[-07 ] , ILXE I E I "-7, 4 7 [ - 1 2 ' l!. ~.~gE 0 7 ~ I , 661!i-06 1, 46E> 06 -6. 3 [ ' ] E ' - 0 8 - 4 . 57L
I~
3, 351-07 1, 9 8 4 - 0 7 4, :~8;-07 I. 73E- 00
2, 9 0 E - 0 7 -J , 75E -07 '- 3 , 1 6 F - - O ~ P
-6.,~7E'00 2. 7 1 E - 0 8 2. 4E'IE:- 0 0
:
+
+ +
02
01-4
H C2H6 CH30H C2H2 HE):! CH4
I .t + + *
02 CH,3
+
CH3 H02 [:2H4
~ ~ ~
H202
"~
CH20 0
4 4 4 + ~
~
H20;; EO
CH,3 C2H
4 +
- > >
HCO
*
.IM *
9" > - >>
H
~
C0
CH20
4
HO"!
-I. . . . . > +M -- ) -- >
CO CH20 C2H2
-t
H20
+M
HCO
4
...... >
->>
C2H4 )
H20 CO
+
H
4
F-IC)2 H
4
-
>
riCO
4
9t
<
4M ~- ) 9M < "- )
[)
[]H
~M 9t
( < -
C21t
~
OH
0
<
[IH3
+
H202
02 H
4 9*
9 >>
OH
-~
[II I
H207
+
OH
+
< "" >
H?(J
+
HIIV
4
lt{)2
+
C1"12(]
~
[)2
4
4M
< < ..... >
[ Hi:'O
~
H
+M
,:
H
4
C[!
+M + 9 + +M
(=>
HOT C02 H2(] H2[) OH
+ 4 + + +
OH HO2 02
+ + + +
-'-> --> ->) -->
C2H5 CH20H HCO C2H3 C2H2 C2H3 C02 H H02 CH3
+ + + *
-7.73E-05 1,04E-04 4,57E-05
-2,07E-07
C[)
-5,69E-05
H20
-8,91E--06 --I,34s I, IIE..-05 9,49E-'06
H02
6 7 8
7.59E-05 2,68E-05 2.07E-05
6.70E-05 1,34E-05 -9,59E-06
9 10
9.52E'-'06
-3,031E-OB
9, I I E - . 0 6
B. 9 7 E - 0 6
-I, 35E- 0 7
HC[i CH2[!H
II
7, 40E-08
4, 2 6 E - - 0 8
-3. 1 4 1 - 0 8
HFO
+
1 2 3 4
02 H02 OH
2.12E-02 1,96E-02 1,24E-02 1.21E-02 1,04E-02
1,14E-'02 1.96E-02 -'4,53E-03 -I,10E-02 5.19E-03
-9. B I E - 0 3 -1.27E-07 7,88E'-03 1,19E-03 -5.19E-03
H CO H202 H02 H202
+ ~ + +
B,44E-83 7.70E-03 7.60E-03
8,40E-03 7,53E-03
-4,38E-05 -I,69E--04
7.60E-03
-2.76E-14
7.05E-03
6. B 2 E - 0 3
C2H6 CH30H C2H2 C2H4
+ + + +
6,80E-03 6,03E-03 5.93E-03 5,75E-03 4,07E-03
3.73E-03
-3.07E-03
C2H3
+
+M
2,~9E-03 5,93E-03 1.01E-03 3,74E-04 4,02E-03
C2H4 CO H2O H202 C2H4 HCO CH20 CH30H C02
+ + + + + + + +
4M + *M 9
4,02E-03
-3,55E-03 -3,79E-07 -4,74E-03 -3,69E-.03 -1.69E-09 2,03E-03
CH20
+
+
-)) > C2H5 * H2 C2H6 + ~ - >} CH3 + CH3 H + 02 4 0 4 OH C2H6 + CH3 * "- ) ) C2H5 + CH4 ~ .................................. _ ................................... CH4
+
011
+
'- )
CH3
4
H2O
VARIATION METHOD FOR SENSITIVITY ANALYSIS An 11 reaction scheme could be deduced for methanol-oxygen combustion, that brings only 12 species in action. T h e temperatures and concentrations are nearly u n c h a n g e d with regard to those of the complete scheme, even in the first reactor (a 2 K difference in temperature), although the r e d u c e d scheme was ded u c e d from sensitivity coefficients in the third reactor. For ethane-air combustion, 46 reactions involving 24 species had to be kept to get a comparable agreement. Several parallel ways subsists in the reaction steps. A more reduced scheme (only 19 reactions o f the 88 original ones), which is not presented here, has also been tested. It works quite well in the third reactor but some discrepancies a p p e a r in the first and second reactors where the reduction decreases the overall reaction rate and temp e r a t u r e (2078 K and 1696 K instead of 2174 and 1711 k respectively). This fact is mainly due to the choice o f the objective function. T h e mean massic enthalpy in the third reactor as an objective ensures correct fuel consumption and t e m p e r a t u r e rise at the c h a m b e r exit but may alter intermediate steps. Conclusion
A variational m e t h o d has been developed and a d a p t e d to WSR assemblies commonly used to predict aeronautical combustion chamber performance. O u r study gives a clear indication of the a g r e e m e n t between the brute force m e t h o d and the variational one and o f the large CPU time advantage o f the latter. A complete sensitivity analysis (for all parameters for a given objective) may be obtained in less time than n e e d e d for the solution calculation. T h e results o f this analysis lead to arranging the reactions in decreasing o r d e r of importance. In kinetic studies, the reactions are more commonly a r r a n g e d with respect to the relative weight of the different reaction paths (see Warnatz et al.16 for instance). O u r classification may differ from these, but it gives additional information. It can be used as the first step for a scheme reduction to be completed by conventional chemical reduction techniques. T h e use of rules such that quasi-steady state for some intermediate species, equilibrium for some reversible reactions and the cancelling o f secondary parallel paths o r nonoccuring reverse reactions will then lead to drastic but efficient reductions such as those proposed by Cathonnet et al. 17 or Peters is. Such equivalent schemes may also include pseudo-species which a p p e a r for sets o f similar species.
805
However, the conditions within a real combustion chamber are quite variable. The stoichiometry and t e m p e r a t u r e evolve extensively from the chamber head to the exit. Several locally equivalent schemes may be required. T h e objective functions of the sensitivity analysis are also multiple (efficiency, pollution . . . . ). W o r k is now in progress to develop an automatic method for scheme reduction u n d e r such conditions. T h e variational sensitivity analysis appears to be an efficient pre-processor for this purpose.
Symbols
a C Cp F h H J K k L M n p q R T V W Y k P cr
atomic composition of molecular species species concentration (mole/m 3) heat capacity generic name o f functions enthalpy Hamihonian n u m b e r of reactors n u m b e r of different atoms pre-exponential factor objective function n u m b e r of molecular species t e m p e r a t u r e e x p o n e n t in the Arrhenius law generic name o f p a r a m e t e r s mass flow rate between reactors n u m b e r o f reactions temperature reactor volume reaction rate generic name of variables array stoichiometric coefficient o f reactions Lagrange multiplier specific mass sensitivity coefficient molar mass
Subscripts a
i,j k m r 0
activation reactors atom species reaction feed value
Superscripts + o -
direct sense formation reverse sense
806
REACTION KINETICS
Acknowledgments This study has been supported by the Direction des Recherches, Etudes et Techniques of the French D616gation G~n6rale de l'Armement. We want to thank MM. J. BESNAULT, P. CLAVIN (DRET) and MM. G. BAYLE-LABOURE, M. DESAULTY, Y. CHAUVEAU (SNECMA) for fruitful discussions during the course of this study.
9. 10. i1. 12.
REFERENCES 13. 1. CHAUVEAU,Y., DESAULTY, M. & BAYLE-LABOURE, G.: "Pr6diction des performances de rendement et de stabilit6 d'une chambre de combustion". 62nd AGARD PEP Symposium, Paper 34, Cesme 1983 2. HAUTMAN, D.J., DRYER, F.L., SCHUG, K.P. & GLASSMAN,I.: Combust. Sci. Technol., 25, p. 219 (1981) 3. BELLET, J.C., CAMBRAY, P., CHAMPION, M. & KARMED, D.: "Etude de probl~mes fondamentaux de la combustion dans les foyers de turbor6acteurs au moyen d'un r6acteur tubulaire". 62nd AGARD PEP Symposium, Paper 34, Cesme 1983 4. CACUCI, D.G., WEBER, C.F., OBLOW, E.M., MAP.ABLE,J.H.: Nuclear Sci. and Engen., 75, 88110 (1980) 5. CUKIER, R.I., FORTUIN, C.M., SHULER, K.E., PETSGHEK, A.G. AND SCHAIBLY,J.H.: J. Chem. Phys. 59, 8 (1973) 6, CUKIER, R.I., SCHAIBLY,J.H. & SHULER, K.E.: J. Chem. Phys. 63,3 (1975) 7, CUKIER, R.I., LEVlNE, H.B., SHULER, K.E.: J. Computational Physics, 26, 1-42 (1978) 8. KRAMER, M.A., CALO, J.M., RABITZ, H. & KEE,
14. 15.
16.
17.
18.
R.F.: "AIM: the analytically integrated Magnus method for linear and second order sensitivity coefficients". Sandia Report. SAND 82-8231 (1982) HWANG, J.T., DOUGHERTY, E.P., RABITZ, S. & RABITZ, H.: J. Chem. Phys., 69, 11 (1978) DIGKSON, R.P., GELINAS, R.J.: J. Computational Physics, 21, 123-143 (1976) BONI, A.A. AND PENNER, R.C.: Combust. Sci. Technol., 15, 99-106 (1977) TEETS, R.E., BECHTEL, H.H.: Eighteenth Symposium (International) on Combustion, p. 425-432, The Combustion Institute (1981) HENRYSON, H., HUMMEL, H.H., HWANG, R.N., STAGEY, W.M. ANn TOPPEL, B.J.: "Variational sensitivity analysis. Theory and application". Inspec Conference paper 76A29412. Advanced reactors: physics, design and economics, p. 568577, Atlanta 1974. Pergamon 75 SEIGNEUR, C., STEPHANOPOULOUS, G., CARR, R.W.: Chem. Engen. Sci., 37, 6 (1982) WESTBROOK,C.K., DRYER, F.U AND SHUG, K.P.: Nineteenth Symposium (International) on Combustion, p. 153-166, The Combustion Institute (1983) WARNATZ, J., BOCKHORN, H., MOSER, A. AND WENZ, H.W.: Nineteenth Symposium (International) on Combustion, p. 197-209, The Combustion Institute (1983) CATHONNET, M., GAILLARO, F., BOETTNER,J.C. AND JAMES, H.: "Etude exp6rimentale et mod61isation de la cin6tique de combustion des hydrocarbures". 62nd AGARD PEP Symposium, Paper 22, Cesme 1983 PETERS, N.: "Numerical and asymptotic analysis of systematically reduced reaction schemes for hydrocarbons flames". Invited Paper at the Numerical Simulation of Combustion Phenomena Symposium, Lecture Note in Physics 241, Springer Verlag, 1985.
COMMENTS
M. Frenklach, Pennsylvania State University, USA. l) Have you compared the computational efficiency of your technique with methods other than the rather slow brute-force method? 2) Your proposal of using sensitivity analysis for reduction of reaction mechanisms has also been expressed by other researchers from time to time. Regardless of the technique, sensitivity analysis cannot in principle achieve this purpose.1 For example, a sensitivity value of zeor does not necessarily identify an "unimportant" reaction; it may identify an extremely fast, not rate-limiting and yet crucial reaction in the mechanism.
REFERENCE 1. FRENKLACrt,M.: in Combustion Chemistry, ed. W. C. Gardiner, Chap. 7 Springer-Verlag, New York, 1984.
Author's Reply. 1) The Fourier Method for sensitivity analysis requires more parameters perturbation than the brute force method to ensure the nondependence of excitation modes. So that method is still longer than the brute force one. The direct method has some analogy with the variational method. The CPU time
V A R I A T I O N METHOD FOR SENSITIVITY ANALYSIS advantage of the variational method over the direct method is small (about 2%); but the variational method gives more synthetic information. Its advantages should be greater when applied to differential equations. The Green method shows good performance and is rapid; but, if that method is well suited for sets of differential equations, it seems difficult, or impossible, to apply it to sets of algebraic equations such as those governing W.S.R.
807
2) It is perfectly true that the sensitivity analysis results are not sufficient to proceed to an optimized and efficient scheme reduction. However, a good sensitivity analysis seems to be a first necessary step. T h e combined use of sensitivity analysis and of usual chemical rules (quasisteady state, equilibrium, scheme completion...) will then lead to correct reductions.
