kinetic equations for a CO/HZ/air oxidation model have been constructed by a special perturbation method (SPM) developed recently by the authors. The SPM is.
Pergamon
Chemd
Enyrnwriny
PII: SOOOS-2509(97)00176-O
Scmce, Vol. 52, No. 23, pp. 43 I7 -4321, 1991 (’ 1997 Elsevier Science Ltd All rights reserved Printed m Greal Britam 0009%2509197 817.co + 0 00
Reduced kinetic equations of a CO/Hz/air oxidation model by a special perturbation method Genyuan Li and Herschel Rabitz* Department
of Chemistry, Princeton
University, Princeton, NJ 08540, U.S.A.
(Received 7 June 1996; accepted 5 February
1997)
Abstract-Reduced kinetic equations for a CO/HZ/air oxidation model have been constructed by a special perturbation method (SPM) developed recently by the authors. The SPM is a modification of the constrained nonlinear lumping approach to the simplification of kinetic models. The reduced kinetic equations are only composed of the variables for the temperature and the concentrations of main species (CO, Hz, 02, CO2 and HzO), and retain high accuracy in the solutions. All the eliminated variables, the concentrations of radicals and other unimportant species, can be also accurately calculated from the solutions of the reduced kinetic equations by employing some algebraic expressions. The SPM gives significantly better results than the traditional quasi-steady-state approximation when the latter breaks down. NC 1997 Elsevier Science Ltd
Keywords: Lumping; model reduction; combustion;
1. INTRODUCTION
Physico-chemical processes in typical combustion and other kinetic environments can necessitate reaction models composed of a very large number of strongly coupled chemical species as well as spatial transport processes. The needs of engineering modeling in this domain call for simplifying the models, and much work on model reduction has been done in recent years (Ho, 1995, 1996; Li and Rabitz, 1995; Yannacopoulos et al., 1995). Recently, the authors developed a combined symbolic and numerical approach of constrained nonlinear lumping for the dimension reduction of complex reaction systems (Li and Rabitz, 1996a). The application of this method to an Hz/O2 oxidation system shows very good accuracy. The drawback of this method is that the formula contains complicated second-order terms. To reduce the complexity, a special perturbation method (SPM) has been developed (Li and Rabitz, 3996b). The first-order approximation of the regular perturbation expansion for an eliminated variable in a singular perturbation system is first modified by introducing a correction term, and then transformed by the Shanks transformation (Shanks, 1955). The resultant expression has a simple form, and the solu-
*Corresponding
author. 4317
perturbation.
tion retains high accuracy. Although the SPM is a modification of constrained nonlinear lumping, it was developed from a different perspective. To understand this method, one only needs to know common perturbation methods without the knowledge of some advanced mathematical concepts used in constrained nonlinear lumping (Li, et al., 1994; Tomlin et al., 1994). The SPM has been successfully applied to an Hz/O2 oxidation model. A general reduced model without any radicals has been obtained, which can be applied to a wide range of initial conditions and has very good accuracy for the temperature and all the species (Li and Rabitz, 1996b). As Hz/O2 and CO/O, systems are two basic components of hydrocarbon oxidation models, in this paper, we apply the SPM to a CO/Hz/air oxidation system to construct a reduced model with only the main species CO, HZ, Or, CO2 and HzO. The solutions of the resultant reduced model retains high accuracy for the temperature and all species, including the minor ones. The paper is organized as follows. In Section 2 we briefly summarize the SPM. In Section 3, the operating formulas based on the SPM for first- and/or second-order reaction systems are provided. In Section 4, the reduced differential equations for the CO/Hz/air oxidation model are constructed by the SPM and a comparison between the results of the original and reduced models is presented. Finally. Section 5 contains conclusions and a general discussion.
G. Li and H. Rabitz
4318
+ E2[ gz(y, z(O)) Z(L)
2. METHODOLOGY OF SPM We first briefly introduce the basic principles of the SPM. We refer the reader to our earlier paper for additional details (Li and Rabitz, 1996b). The kinetics of a homogeneous reaction can be described by an n-dimensional ordinary differential equation system
dy
x
JJEP
= f(Y),
+ : gL,L(y, z(“))(z(1’)2]+
/ dy,ldt
\
dyz/dt
dzldt
where Y = 1.~1 YZ
\
fl(Y) .MY)
(2)
fn-I(Y)
dyn- ,ldt \
I =
I
. . Y,-~),
\ 4dY, 4 I z =Y,,
g(y,z) =.MY,z),
and we symbolically introduce a parameter E and will finally set c = 1. To eliminate the variable z, perturbation theory is commonly used, i.e. the fast variable z is expanded in a regular perturbation series: z = z(o’ + EZ(l) + E2z(2’+ .
(3)
where E is a small positive number and the zck”s are only functions of yj’s. In complex systems, E may only be implicit and eq. (3) may be understood with E set to unity. Substituting this expansion into the differential equations of Yj and eliminating the equation for z gives a reduced system describing the observable slow motion. The successive terms of eq. (3) account for increasingly more detail on the influence of the fast variable z upon the slow ones yj’s (Lomov, 1992). The partial sum Ci=, Ekz (” of eq. (3) is referred to as the nth order approximation for z. In order to determine the terms on the right-hand side of eq. (3), first expand g(y, z) about z(O), i.e. g(y, z) = g(y, z(O)) + EgZ(y 1z@‘)z(”
(4)
partial where gZ and g__,=are first- and second-order derivatives of g with respect to z, respectively. Substituting eqs (3) and (4) into the last equation of eq. (2) and requiring that the various orders of E vanish separately gives the formulas for the zck)‘s.
z(O) = cp(Y)> where y is an n-dimensional vector that specifies the state of the n species constituting the kinetics mode1 under isothermal conditions, or the state of n - 1 species plus the temperature T in the nonisothermal case; f(y) is an n-dimensional function vector, that describes the physical and/or chemical processes of the model. Chemical systems typically evolve over vastly different time scales. For instance, the concentrations of the major or readily measurable species, often evolve slowly in comparison with the concentrations of short-lived intermediates. The kinetic equations of such systems can be simplified by eliminating the fast variables by representing them approximately as functions of the slow ones. Without loss of generality, suppose y. in y is a fast variable we wish to eliminate. In order to deduce the formulas of y., eq. (1) is rewritten as
.”
which is obtained for z(O).
by explicitly
solving
(5) g(y, z(O)) = 0
1 dz”’ z(I) = ___ ~ gz(y, z(O)) dt
(6) where q,,(y) is the partial derivative of cp with respect to yj. If the z on the right-hand side of eq. (6) is approximated by z(O) = q(y), then we have z(1’ _ - g
(y
(7) lp(y)) Y;%(Y) dyj(ydtq(y))
Z 3
which is only a function of the elements of y. This is a standard procedure to construct the first-order approximation of z in a regular perturbation expansion (Van Kampen, 1985). As the third term z(‘) in eq. (3) involves second-order derivatives, which are quite complicated, the second-order approximation is rarely used. Now we set c: = 1 so that the formulas of zCk)(k = 0, 1) given above can still be used for the singular perturbation system where the small parameter E is contained implicitly. In the remainder of the paper, we will only consider the perturbation with E = 1 in eq. (3). From eq. (5) one can see that the zeroth-order approximation z(O)is just the quasi-steady-state (QSS) approximation for z. If z(O) is not a good approximation, one may use the higher-order approximations in eq. (3). However, the eliminated oariables in singular perturbation systems cannot be expressed as convergent regular perturbation series because their expansions are only asymptotically convergent or even divergent. In this circumstance, an eliminated variable cannot be computed to any degree of accuracy by increasing the number of terms of the corresponding partial sum of the perturbation expansion. The reason for this deviation seems apparent for the first-order approximation. When z(O)is close to z, the approximation of z by z(O) in the expression of dy,(y, z)/dt in eq. (6) will not introduce a large error. However, when z(O) is quite different from z, this approximation may cause a large error. A natural choice for possibly improving the accuracy of the first-order approximation of the perturbation series is to employ z(O) + z(l) instead of
Reduced kinetic equations
q(l) _
_ -
gl(y,;(rli
1:; Q,(Y) dyj(y’ ‘1; + ‘(“).
Here we use P(l) to distinguish have
(8)
it from z(l). Then, we
; z $0) + 5”’ = $Jl + $1) + (S(l) _ $1)) = .,(O)+ z(1) + A,-“‘,
Most practical reaction mechanisms are composed of first- and/or second-order reactions. In this case. sj(y, z) and g(y, z) are polynomials of Jj and 2, and then z(O),z(l), AZ”’ can be readily determined. In the following the formulas for first- andior second-order reactions are provided. The deduction of these formulas can be found in our previous paper (Li and Rabitz, 1996b).
(9)
When we used this modified first-order approximation to an Hz/O1 oxidation model (Li and Rabitz, 1996b), it did not improve the accuracy. This suggests that the divergence of the regular perturbation seriesfor singular perturbation systems is intrinsic. To circumvent this problem, we employ the Shanks transformation which can accelerate the convergence of some slowly convergent series and induce convergence in some divergent series (Shanks, 1955). This transformation is also named as Aitken’s 6’-process (Press et a[., 1992). Let ,(r, represent the nth partial sum of the series given by eq. (3) and define the simplest Shanks transformation by T (9 n,’ =
4319
3. SPECIFIC FORMULAS FOR z”“,i(” AND AZ”’
z”) to replace z in dJ>j (y, z)/dt, i.e.
rYnscl”_, - .sp.‘_
3.1. g(y, z) is u linear,function ofz If a reaction system does not contain the secondorder reaction between z and itself, then ~(y. -_) is a linear function of :, i.e. giy, 2) = a: + h
where a and b are only functions of yj‘s. For a chemical reaction system. a is always negative because (1~ represents the sum of the rates for the reactions leading to the disappearance of species Z. Conversely. h is always positive because it represents the sum of the rates for the reactions forming species 2. Solving g(y, z(O)) = 0 gives
1
.‘r, + .Yn_, - 2Y”_,
,((‘I = _ h/c,
(10) and the partial
For II = 2 we have
119)
derivative ,/,(Y> 40)) = a.
(‘0)
For z(l) and A-_“’, we need to consider two different conditions: isothermal and nonisothermal.
Setting
the Shanks
(18)
,Yo = $0)
(12)
,Y, = z(O) + =(I)
(13)
Jy2 = $0) + z(1) + AZ”’
(14)
transformation
3.1.1. Isothermal condition. Under the isothermal condition, the kinetic equations do not contain the differential equation for the temperature T and they have the form as follows:
gives /
dy,;dt dp2.dr
\
; .fi(Y) \ .fZ(Y)
(71) (15) The resultant formula, eq. (15), is only a modification of the first-order approximation by putting a parameter p in front of z(l): ;=z
(01+
/gz”’
dJ,” _ , ldt
.f;z- I(Y)
dzjdt
\ &Y> --I /
For this system we have
(16)
where
and fl= 1 _
l
Az(I)/Z(‘)
(17)
We used eq. (15) to calculate the concentrations of all radicals in an H2/02 oxidation model. For all the initial conditions we used, the accuracy was very good. The formula of AZ(l) is quite simple whenfj(y, z) and g(y, z) are polynomials or some other simple functions. These conditions make the SPM useful in practical applications.
(23) where Lj represents the sum of the linear terms of 2 in fj(y, z) after replacing z by 1; When f;(y, z) does not contain z, L, = 0. 3.1.2. Nonisothermal condition. Under thermal condition, the kinetic equations
the nonisocontain the
G. Li and H. Rabitz
4320 differential equation the following form: /
T and have
for the temperature
dylldt
fl(Y)
dyzldt
fz(Y)
\
=
\
dT/dt
G(Y, z, T)
dz/dt
g(y, z)
z(o’ = -b-d=
(24)
.fn-z(Y)
dy. - zldt
where a, b, c are only functions of yj’s. Similarly, a, b are negative, and c is positive. We put 2 in front of bz so that the expression of the square root in the following formulas has a simple form. In this case, we have
(29)
a g=(y, z(0)) = - 2JPZc.
I
(30)
For z(l) and AZ(~), we also need to consider isothermal and nonisothermal conditions, respectively.
where
3.2.1. Isothermal condition z(1’ =
(25) Here I signifies the reaction number. T, is the ambient temperature, R, and - AH, are the rate and heat of reaction 1, respectively, x the heat transfer coefficient, S the surface area, V the volume, IS the molar density, and C, is the specific heat at constant pressure. Under the nonisothermal condition, we have
4(@
_
ac)
j;l
h(y’
‘(O’)
2bz”’ + c aa
8b ac + 2z’O’ G’ Z& ayj >
a
(31)
where z(O) is given by eq. (29), and AZ”’ = _
(z(o’+E, $)
z(O))
z(1) = - ;;+,,,
n-1
1
-
n-l 1 4(b2 _ a~) .c {Liz”’ I 1
+ Qj[2z’o’z’1’ + (z”‘)‘]}
J
-
& P
2bz(O’+
[CRI(Y, z(~‘)(- AHI)
c
aa
a
G
I
P
+
ah ac 22’0’ &+ZjQ >
where Lj and Qj represent the sums of the terms for the linear and quadratic terms of z in fj(y, z) after replacing z by 1, respectively. When fj(y, z) does not contain z, Lj and Qj are zero.
1 x[~(z(~'~+~)(~+~)] -$(T-
T,)
(26)
where Rl(y, z(O))and $c, are just R, and oC, after z in these expressions is replaced by z(O).
3.2.2. Nonisothermal condition. Under thermal condition, we have
z(l’
=
-;li;
I
(34)
where LR, and QR, are the rate expressions of reaction I, which contain 7 linearly and quadratically, after replacing z by 1. When the rate expression of reaction [ does not contain 2, LK, and QR, are zero. If we wish to eliminate k variables (yi, i = 1, 2, . k), we choose each one of them as z and construct the corresponding expression of eq. (15). These k expressions are coupled algebraic equations and can be iteratively solved by computer. When the inner irrrution method is employed, only a few steps typically are necessary to reach converged values (Wang and Rogg. 1992). The resultant values of the yi’s are substituted into the remaining n - k differential equations of y,‘s for further calculation. 4. THE REDUCED
KINETIC
EQUATIONS
OXIDATION
FOR A COIHJAIR
MODEL
We will apply the SPM to a CO/HZ/air oxidation model which contains 13 species (N2, CO, Hz, 02, H20, COZ, OH. H, 0, H02, HCO, H202 and CH20) and 67 reactions, and is a homogeneous reaction in a continuously stirred tank reactor. The mechanism and the rate data for each reaction are given in Table 1 (Maas and Pope, 1992a). The differential equation of temperature T is given by eq. (25). A constant value for xS/V was chosen as 0.8 x 10m3 W cme3 K 1 according to the report of Leeds (Tomlin r’t u/., 1992). The thermochemical data are described by polynomial fits given by the NASA thermodynamic tables (Burcat, 1984), which may be different from those used by Maas and Pope. Hence, the solutions of the kinetic equations may not be the same as theirs. Here we only use this model to show the accuracy of the SPM. Including the variable temperature T, this system is a 14-dimensional model. In combustion problems, the QSS and partial equilibrium approximations are often used to eliminate the variables corresponding to the concentrations of the radicals and low concentration species. i.e. OH. H. 0, HO*, HCO, Hz02 and CH20. In some ranges of compositions and temperatures, these approximations have quite large errors, especially for the radicals OH, H and 0. In contrast, we will show that the SPM can always give good results. 4. I Elimination of’indiciduul radicals 4. I. I. Rudiculs OH, H and 0. In this model there are three second-order reactions between two OH radicals (Reactions 7, 18 and 28 in Table 1, and Reaction 28 involves a third body). Thus, the kinetic
332 I
equation for [OH] is a quadratic function of [OH]. The approximation formulas of z(‘)‘, z”‘, A:“’ for [OH] will be given by eqs (29), (33) and (34), respectively. The resultant eq. (15) is substituted into the other 13 equations for further calculation. The initial conditions of the calculation are as follows: ‘To = 1600 K, pressure 1 atm, 90 vol.% air. IO vol.% CO + Hz. initial ratios [CO] : [HJ = I : O.OO~).OOS : I One of the results for [OH] given by solving the reduced differential equation system is given in Fig. I, For all the initial conditions used, the SPM gives almost exact results. In contrast, the QSS approxunation has large errors. As with the OH radical, the kinetic equations for [H] and [0] are also quadratic functions of [H] and [O], respectively. The approximation formulas of z(O),z(‘). AZ”’ for [H] and [0] can be determined similarly as those for [OH]. Two of the results of [H] and [0] given by solving the reduced differential equation systems are also given in Fig. 1. For all the initial conditions used. the SPM gives almost exact results except for the ignition being delayed a little. In contrast. the QSS approximations for [H] and [O] have quite large errors. 4.1.2. Species HO?, HCO, H202 and CH20. The QSS approximations for [HO,], [HCO]. [H20J and [CH20] are quite satisfactory for the temperature and the concentrations of the main species for most of the initial conditions used, but they introduce quite large errors for the concentrations of other species. Since we are usually only concerned with the temperature and the main species, we will use the QSS approximation (i.e. z”‘) for these unimportant species to reduce the complexity of the reduced equations. 4.2. Elimination of’some speck sirnultuneousl~~ In combustion systems, a very small difference of initial conditions may yield totally different solutions. When we eliminate species simultaneously by the SPM and QSS approximations, a small error for one species may cause a large error for another. and the new error feeds back on the former causing an even larger error. Finally, the resultant error of the solution may not be acceptable. Therefore, even though each species has a very good accuracy as shown above. this does not guarantee that the simultaneous approximation will also have similar accuracy. Thus. we found that the error is not acceptable under some initial conditions. Moreover. the perturbation expansion (the QSS approximation is only the zeroth-order approximation, and the SPM is the modified first-order approximation of this expansion, respectively) is the outer solution for singular perturbation systems. It does not provide the inner solution (O’Malley. I99 1). Hence, the initial concentrations for some radicals given by the SPM and QSS approximations can be nonzero. This differs from the real initial condition we often use where the initial concentrations of radicals are zero. This difference may cause a large error. To overcome this difficulty we begin with the original
G. Li and H. Rabitz
4322 Table
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 31 38 39 40 41 42 43 44 45 46 41 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
1. Reaction
02+H-+OH+0 0H+O-r02+H H,+O+OH+H OH+H-tH,+O HZ+OH+H20+H H,0+H+H2+OH OH+OH-H,O+O H,O+O-+OH+OH H+H+M-+H,+M H2+M-tH+H+M H+OH+M+HZO+M HZO+M+H+OH+M O+O+M+O,+M Oz+M+O+O+M H+O,+M+HO,+M HOz+M+H+02+M HOz+H-rOH+OH OH+OH+HO,+H HOz+H+Hz+02 H,+Oz-+HOz+H HOz+H-+HzO+O H,O+O+HOz+H HOz+O-+OH+OZ OH+OpHOz+O HO,+OH+H,O+O, H,O+O,+HO,+OH HO2 + HO2 + H202 + O2 OH+OH+M-rHZOZ+M H202+M+OH+OH+M H,Oz+H+HZ+HOZ H,+H02+H202+H HzO,+H-H,O+OH H,0+OH-+H202+H H,Oz+O-+OH+HO, 0H+H02-+H,0,+0 H,Oz + OH --+ H,O + HOz H,O + HO, -+ H,Oz + OH CO+OH+COz+H CO,+H+CO+OH CO+HOz+COz+OH COz+OH-+CO+H02 CO+O+M+COZ+M CO,+M-CO+O+M co+02-+coz+o co~+o-+co+02 HCO+M+CO+H+M CO+H+M-+HCO+M HCO+H-rCO+HZ CO+Hz-+HCO+H HCO+O-+CO+OH CO+OH+HCO+O HCO+O+CO,+H CO,+H-+HCO+O HCO+OH-CO+H,O CO+H,O-HCO+OH HCO+OZ+CO+HOz CO+HO,-HCO+O, CH,O+M+HCO+H+M HCO+H+M-tCHzO+M CH20+H+HCO+HZ HCO+H,-+CH,O+H CH,O+O-+HCO+OH
mechanism
and rate constants
2.00 x lOI 1.47 x 1Ol3 5.06 x 10“ 2.24 x lo4 1.00 x lo8 4.46 x IO8 1.50 x lo9 1.51 x 1O’O 1.80 x 10” 6.98 x 10” 2.20 x lo= 3.80 x lo= 2.90 x 10” 6.78 x 10” 2.30 x 10” 2.66 x lOI* 1.50 x 1Ol4 1.63 x 1Ol3 2.50 x 10’3 8.39 x 1Ol3 3.00 x 1Ol3 3.29 x 1Ol3 1.80 x 1OL3 2.67 x lOI 6.00 x 1Ol3 8.97 x lOI 2.50 x 10” 3.25 x 10” 2.11 x lo= 1.70 x 1012 9.35 x 10” 1.00 x 10’3 2.66 x 1Ol2 2.80 x 10’ 3 6.80 x 1Ol2 5.40 x 10’2 1.32 x 1013 4.40 x 106 6.12 x lo8 1.50 x 1OL4 2.27 x 1OL5 7.10 x 10’” 1.69 x lOI 2.50 x 1Ol2 2.55 x 1Ol3 7.10 x 1014 1.07 x lOI 2.00 x lOI 1.17 x 1Ol5 3.00 x 1Ol3 7.72 x 1013 3.00 x 10’3 1.07 x 1Ol6 1.00 x 1014 2.60 x 1Ol5 3.00 x 10’2 5.21 x 1012 1.40 x IO” 2.62 x 1Ol5 2.50 x 1Ol3 1.82 x 1Ol2 3.50 x 10’3
-
-
0.00 0.00 2.67 2.67 1.60 1.60 1.14 1.14 1.00 1.00 2.00 2.00 1.00 1.00 0.80 0.80 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 2.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.50 1.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
70.3 2.1 26.3 18.4 13.8 71.1 0.4 71.6 0.0 436.0 0.0 499.4 0.0 496.4 0.0 206.2 4.2 158.0 2.9 232.8 7.2 232.2 - 1.7 220.3 0.0 293.2 - 5.2 0.0 206.8 15.7 91.2 15.0 307.6 26.8 94.4 4.2 143.0 - 3.1 94.1 98.7 349.7 - 19.0 506.4 200.0 229.0 70.3 8.6 0.0 374.4 0.0 366.5 0.0 463.7 0.0 437.7 0.0 144.5 320.0 - 56.8 16.7 76.0 14.6
Reduced kinetic equations
4323
Table 1. (cod.)
-
4: 63 64 65 66 67
HCO+OH+CH,O+O CHzO + OH + HCO + Hz0 HCO + HZ0 --t CHZO + OH CHzO + HO2 + HCO + HzOz HCO + HzOz + CHzO + HO2 *A, in cm
mol-’
bl = H2 + 0.350,
1x10*0
I.l?x 3.00 x 9.71 x 1.00 x 1.32 x
10’2 1O1” IO” 10” IO”
n,
E:
0.00 0.00 0.00 0.00 0.00
66.0 5.0 127.6 33.5 17.3
s-‘; El in kJ/mol; k, = AIT”’exp(- E,:RT). + 6.50H20 + 0.50N2 + 1.5OCO + l.50C02
’
* ‘I
*
’ ‘I
’
’ -1 ’
. 38 ’
’ ‘8 .
’ ‘1 7 -
Exact solution SPM approximation ------QSS approximation --..-.-.
SPM approximation ------QSS approximation ---.--.-
1x10-6
1x10”
0.0001
0.001
0.01
0.1
1
time, s Fig. I. Comparison between the results of [OH], [H] and [0] given by the exact solution, the SPM and QSS approximations for [OH], [H] and [0] individually.
differential equations for a very short period, and then switch to the perturbation expressions. This short period is different for different species. It is longer for the species whose initial rate is not zero. Also as more species that are treated simultaneously, then a longer time is needed. In the calculations this period is about lo-‘- 10-4s. 4.2.1. Eliminution of OH, H and 0 simultaneously by the SPM. To show the accuracy of the SPM, we eliminate the radicals OH, H and 0 simultaneously. The resultant expressions for [OH], [H] and [0] are three coupled algebraic equations, which can be iteratively solved. The resultant values for these radicals are substituted into the remaining differential equations for the other species and the temperature. This
calculation gives not only the evolution of the temperature T and the concentrations of other species by solving the remaining differential equations, but the concentrations of OH, H and 0 by substituting back into the algebraic expressions. The results are almost exact for the temperature and all species concentrations for all the initial conditions studied. Some illustrating results are given in Figs 2 and 3. 4.2.2. Simultaneous elimination of’OH, H, 0 by the SPA4 and H02, HCO, HzOz and CH20 hq’ the QSS approximation. After elimination of OH, H and 0 by the SPM, we further eliminated HOz, HCO. H202 and CHzO by the QSS approximation. The reduced equations have very good accuracy for the temperature and the main species. One of the results is shown
4324
G. Li and H. Rabitz
.
2800 2600
-
2400
-
p .,
1x10-5
.
. .I
I
8x10”-
u .I
. .,
.
-
. .,
. I,
.
m
a I,
.
Exact solution SPM approximation -------
0.1
0.01
0.001
0.0001
1.2x10’*
.
.
. .,
. .,
1
.
, , -
10
.
. ,,
Exact solution SPM approximation -------
I ,-
O2
4xioI’: 1x10-6 * 4x10’5
0.0001
1x10-5
q .,
r .,
.
0.01
0.001
*
I
I .,
1
0.1
*
*
I *,
Exact solution SPM approximation -------
3x10’5 _
. I(
- .
_
H:,
time, s Fig. 2. Comparison between the results for T and the concentrations of the main species given by the exact solution, and the simultaneous SPM approximation for [OH], [H] and [O]. Reaction condition: 1600 K, 1 atm, 90 vol.% air, [CO]:
[Hz] = 1: 0.005.
SPM approximation
-------
lxlo,o~
1 E .s !I! %
1x10-6 1x10*0 .
1x10-5 I
0.0001 ’
I
0.001 ’
I
.
0.01 0.1 n 1 ’ Exact solution SPM approximation -------
1
I
6 lxlolo~~ 1c * 1x10-6
I 1x10-5
.
0.0001
I 0.001
.
1 0.01
,
I 0.1
time, s Fig. 3. Comparison between the results for the concentrations solution, and the simultaneous SPM approximation for [OH],
of the rest species given by the exact [H] and CO]. Reaction condition is the
same as that in Fig. 2.
in Fig. 4. The solutions for the temperature and the concentrations of CO, 02, COz and HZ are almost exact. The solution for Hz0 has some error. For
comparison, the results of the temperature given by the QSS approximation for all eliminated species under the same time switch are also shown in Fig. 5.
Reduced
.
2800
y
2600
g
2400 -
. q,
a
- 8,
2200 -
;
2000 -
c
1800 -
.
- .,
.
4325
. .,
.
5 a,
Exact solution Combined SPM and QSS approximation
-
F
kinetic equations
ifinn . .___ 1x10-5 1.2x101*
0.0001
.
8x10”:
-
0.001
. =, -
- 0I
0.1
0.01
*
c -I
0
1
* a, *
10
m 0, 0
- .-
-------
-
O2
’
1xl o-6 4x10’5 . 3x10’5 r
, ,
- ~1 Exact solution Combined SPM and QSS approximation
0
, -------
s -’
’
1x10-5
. .I
’
~4~~~~
0.0001
I
I .,
0.001
.
. .,
.
’
0.01
. I,
’
I’
’
’ ’
.
.
1
0.1
.
. .,
Exact solution Combined SPM and QSS approximation
-------
_
Ha
time, s Fig. 4. Comparison
between the results for T and the concentrations of the main species given by the exact for [OH], [HI. [0] with QSS approximation for other eliminated species. Reaction condition is the same as that in Fig. 2.
solution. and the simultaneous SPM approximation
From mation
these figures one can see that the QSS approxihas quite
for some initial condiSPM and QSS approxima-
large errors
tions, but the combined tion is always accurate. The same treatment has been applied to another initial condition: a CO/H,/air system corresponding to a 6/10 mixture of syngas (40 vol.% CO, 30 vol.% HZ and 30 vol.% N2) and air at 1200 K and 1 bar (Maas and Pope, 1992b). The comparison of the results for temperature given by the exact solution and the combined SPM and QSS approximation is given in Fig. 6. The result given by the QSS approximation for all eliminated species is also shown in the figure. The results for main species given by the combined SPM and QSS approximation has similar accuracy to those in Fig. 4.
5. CONCLUSIONS AND DISCUSSION
The SPM for model dimension reduction is applied to a CO/HZ/air oxidation model with 13 species and 67 reactions. The reduced equations are only composed of the variables corresponding to the temperature and the concentrations of the main species: CO, HZ, 02, CO2 and H20. The radicals OH, H and 0 are approximated by the SPM, and the other species
eliminated by the QSS approximation. The reduced equations have very good accuracy for the temperature and all main species for a wide range of initial conditions. If we approximate all climinated species by the SPM, then the concentrations of all the species including the eliminated species can be almost exactly determined by the reduced equations.
As the SPM only contains the zeroth- and the tirstorder terms of the regular perturbation expansion, the formula is quite simple. For first- and/or secondorder reaction systems, the expressions Z(O) , 2”) and AZ(l) under isothermal and nonisothermal conditions have been given. The SPM reduces to a recipe for general application, making the SPM practical for applications to model dimension reduction. Hydrogen and carbon monoxide oxidation systems are two basic components of hydrocarbon combustion models. We have successfully constructed a reduced hydrogen oxidation model without any radicals. The present result for the CO/HZ/air model reduction shows that one can construct accurate reduced models for hydrocarbon combustion systems. The SPM may also be applicable to other scientific areas.
G. Li and H. Rabitz
4326 2800
.
2600 2400
-
2200
-
2000
-
. .,
.
. .I
.
. .,
*
- .,
8
- .,
-
. .,
.
. .,
.
I .,
a
. .,
I
, ,
.
. s
Exact solution ------Combined SPM and QSS approximation QSS approximation __-_-___ _
1800 1600 ’ 0.00001 2800 a 2600 2400
* *’ 0.0001
. .,
0.001
0.001
0.0001
. .,
. .,
s
10
0.1
0.01
8
c I,
5
I .,
. .,
0.001
0.0001
0.1
0.01
_
10
1
m
.
Combined SPM and QSS approximation Exact solution ------QSS approximation ---..-.-
2600 2400 -
0.00001
1
Combined SPM and QSS approximation Exact solution ------QSS approximation ....-.-.
-
0.00001 2800 8
0.1
0.01
a , _
10
1
time, s Fig. 5. Comparison between the results for T given by (1) the exact solution, (2) the simultaneous use of the SPM approximation for [OH], [HI, [0] along with the QSS approximation for other eliminated species, and (3) the QSS approximation for all eliminated species under different initial conditions.
1
7000
y
6000
-
0.00001
’
’ #I
’
8 -1
-
’ -1
’
“I
Exact solution Combined SPM and QSS approximation ------QSS approximation ._..--..
0.001
0.0001
0.01
0.1
’ _
1
time, s Fig. 6. Comparison between the results of T for syngas oxidation given by (1) the exact solution, (2) the simultaneous use of the SPM approximation for [OH], [HI, [0] along with the QSS approximation for
other eliminated species, and (3) the QSS approximation for all eliminated species.
Acknowledgements The authors acknowledge of Energy and the Petroleum can Chemical Society.
NOTATION
a
AI b C
CP
function of yj’s preexponential factor of reaction function of yj’s function of yj’s specific heat
defined as C, after z in the expression is replaced by z(O) E, activation energy of reaction I jth element of vector f(y) .6(Y) defined as f.(y, z) g(Y* 4 first-order partial derivative of g with reYz spect to z second-order partial derivative of g with Yz. I respect to z change WY, z, T) rate of temperature positive integer j G
support from the Department Research Fund of the Ameri-
I
Reduced
k
h 1
Lj LR,
kinetic equations
positive integer rate constant of reaction I positive integer sum of the linear terms of z inf;(y, z) after replacing z by 1; when fj(y, z) does not contain z, L, = 0 defined as RI, which contains z, after replacing = by 1; when RI does not contain 2, LR, = 0
n
Z;
positive integer power of T in the expression of kl sum of the quadratic terms of z in f,(y, 2) after replacing I”by 1; when,fj(y, z) does not contain -“. Q, = 0 universal gas constant rate of reaction I n-dimensional real space surface area partial sum of the first n + 1 terms for a series time temperature ambient temperature Shanks transformation volume jth element of vector y and the remained variable eliminated variable kth order term of the regular perturbation expansion for : modified first order term of the regular perturbation expansion for z
n-dimensional function vector n-dimensional function vector n- or (n - I)-dimensional variable Greek
B c 0 & cp(Y) (P,.,(Y) %
vector
letters defined as l/(1 - Az(“,‘z(“) small positive number molar density defined as 0 after z in the expression is replaced by :“” the expression of ?“I partial derivative of q(y) with respect to yj heat transfer coefficient REFERENCES
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1327
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