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The asymptotic structure of premixed methanol-air flames
382
COMBUSTION AND FLAME
91:382-398 (1992)
The Asymptotic Structure of Premixed Methanol-Air Flames B. YANG and K. SESHADRI* Center for Energy and Combustion Research, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093-0310, USA
N, PETERS Institut fiir Technische Mechanik, RWTH Aachen, W-5100 Aachen, Germany The asymptotic structure of methanol-air flames is analyzed using a reduced four-step mechanism, for values of equivalence ratio Ob from 0,52 to l.(I and values of pressure p from 1 to 10 arm. The reduced mechanism was deduced from a starting mechanism, containing 22 elementary chemical reactions. In the analysis, the overall flame structure is subdivided into four z o n e s - - a preheat zone with thickness of order unity, an inner layer with thickness of order 6, an oxidation layer with thickness of order e, and a postflame zone with thickness of order unity. The analysis is performed for 6 .~ e < 1.0. The inner layer is located between the preheat zone and the oxidation layer, and in this layer, finite-rate reactions related to the consumption of the fuel, to form primarily H 2 and CO and some H 2° and CO2, are considered. In the oxidation layer finite-rate reactions related to the oxidation of H 2 and CO to 1t20 and CO 2 are considered. Numerical integration of two coupled, nonlinear, second-order, ordinary differential equations is performed to resolve the structure of the oxidation layer. Analytical expressions are obtained for predicting the burning velocity, c,. At the stoichiometric conditions with p = 1 atm the model predicts values of t~u which are somewhat lower than those calculated from full numerical integration of the conservation equations. However, the model predicts the observed decrease in the value of c, with decrcasing values of ~b and increasing values of p.
INTRODUCTION
Asymptotic analysis, using reduced chemicalkinetic mechanisms, has been successfully employed previously to describe the structure and predict the burning velocities of laminar, premixed methane-air flames [1-4]. The analysis reported here attempts to provide a similar asymptotic description of the structure of laminar, premixed methanol-air flames, using reduced chemical-kinetic mechanisms. Numerical calculations using reduced chemical-kinetic mechanisms have been performed previously to describe the structure and to predict the burning velocities of laminar methanol-air flames [5-8]. These reduced mechanisms were deduced systematically from a detailed chemical-kinetic mechanism using procedures similar to those suggested by Peters [9]. The calculations show that the principal path of oxidation for methanol is CH3OH-CH2OH-CH20-HCO-CO,
*
Corresponding author.
0010-2180/92/$5.00
H 2-
C02, H 2 0 [5-8]. For premixed flames, a reduced four-step [5] and a reduced five-step [6] mechanism have been proposed. In the fivestep mechanism steady-state approximation is not introduced for the intermediate species C H 2 0 , whereas in deducing the four-step mechanism, steady-state approximation is introduced for CH20. It was found that the burning velocities, v, calculated at a value of pressure, p = 1 atm, using a detailed chemical-kinetic mechanism and a reduced five-step mechanism deduced from this detailed mechanism were in good agreement [6] over a wide range of values of the equivalence ratio, 4~. Similar comparisons were also made for values of p between 1 and 10 atm [6] and some discrepancies were found between the results of numerical calculations performed using the detailed chemical-kinetic mechanism and the five-step mechanism at high pressures. These discrepancies were attributed to inaccuracies associated with some of the approximations introduced in calculations using the reduced chemical-kinetic mechanism [6]. Also, the values of ~', calculated using a reduced four-step Copyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.
STRUCTURE
OF
PREMIXED
METHANOL-AIR
mechanism were found to agree reasonably w e l l w i t h t h e v a l u e s o f vu c a l c u l a t e d u s i n g a d e t a i l e d c h e m i c a l - k i n e t i c m e c h a n i s m [5]. Reduced five-step mechanisms and reduced four-step mechanisms have also been proposed for characterizing the structure of counterflow m e t h a n o l - a i r d i f f u s i o n f l a m e s [7, 8]. T h e c r i t i cal c o n d i t i o n s o f e x t i n c t i o n o f t h e c o u n t e r f l o w diffusion flame predicted by these reduced m e c h a n i s m s w e r e g e n e r a l l y in g o o d a g r e e m e n t with those calculated using a detailed chemical-kinetic mechanism. With the exception of the profile of CHzO calculated using the fourstep mechanism, the profiles of a number of major species, intermediate species and radicals c a l c u l a t e d u s i n g t h e r e d u c e d m e c h a n i s m s were found to agree well with those calculated using the detailed mechanism. In general the results calculated using the five-step mechanism agreed better with the results calculated
FLAMES
383
using the detailed mechanism than those calculated using the four-step mechanism. For simplicity, in the asymptotic analysis des c r i b e d h e r e , t h e f o u r - s t e p m e c h a n i s m is e m ployed. The analysis closely follows previous asymptotic analysis of premixed methane-air f l a m e s [2, 3].
REDUCED CHEMICAL-KINETIC MECHANISM A chemical-kinetic mechanism describing the o x i d a t i o n o f m e t h a n o l is s h o w n in T a b l e 1. This particular set of elementary reactions was c h o s e n as t h e s t a r t i n g p o i n t o f t h e a n a l y s i s , because the calculated values of vu for prem i x e d f l a m e s [6] a n d c r i t i c a l c o n d i t i o n s o f ext i n c t i o n o f d i f f u s i o n f l a m e s [8] u s i n g t h i s m e c h anism and a reduced mechanism deduced from this set of elementary reactions were found to
TABLE 1 Chemical Kinetic Mechanism a No. If lb 2f 2b 3f 3b 4f 4b 5f 5b 6 7 8 9 10 11 12f 12b 13 14 15 16 17 18 19 20 21 22
Reaction H ~ OH + O OH + O ~ O z + H H 2 + O ~ OH + H OH + H ~ H z + O H 2 + OH ---, H 2 0 + H H 2 0 + H ~ H 2 + OH OH + OH ~ H 2 0 + O H 2 0 + O ---, OH + OH 0 2 + H + M* --* HO z + M* HO z + M* ~ O z + H + M* H O 2 + H ~ OH + OH HO 2 + H ~ H 2 + 0 2 HO 2 + OH ~ H 2 0 + 0 2 HO 2 + H --, H 2 0 + O H + H + M* ~ H 2 + M* OH + H + M* ~ H 2 0 + M* CO + OH ~ CO 2 + H CO 2 + H + CO + OH CHO + H ~ CO + H 2 CHO + OH --, CO + H 2 0 CHO + M* ~ CO + H + M* CHeO + H ---, C H O + H 2 C H 2 0 + OH --9 CHO + H 2 0 CH2OH + H --, C H 2 0 + H 2 CH2OH + 0 2 ~ C H 2 0 + HO 2 CH2OH + M* "--, C H 2 0 + H + M* CH3OH + H ---, CH2OH + H 2 CH3OH + OH ~ CH2OH + H 2 0 0 2 +
A~ 2.000E 1.568E 5.060E 2.222E 1.000E 4.312E 1.500E 1.473E 2.300E 3.190E 1.500E 2.500E 6.000E 3.000E 1.800E 2.200E 4.400E 4.956E 2.000E 1.000E 7.100E 2.500E 3.000E 2.000E 1.000E 1.000E 4.000E 1.000E
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
14 13 04 04 08 08 09 10 18 18 14 13 13 13 18 22 06 08 14 14 14 13 13 13 13 14 13 13
n
En
0.00 0.00 2.67 2.67 1.60 1.60 1.14 1.14 -0.80 -0.80 0.00 0.00 0.00 0.00 -1.00 -2.00 1.50 1.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
70.30 3.52 26.30 18.29 13.80 76.46 0.42 71.09 0.00 195.39 4.20 2.90 0.00 7.20 0.00 0.00 -3.10 89.76 0.00 0.00 70.30 16.70 5.00 0.00 30.10 105.10 25.50 7.10
Rate constants are in the form k, = AnTnexp[-E,,/(l~T)]. Units are moles, cubic centimeters, seconds, Kelvins and kJ/mol. Third-body efficiencies are H20: 6.5, CO2: 1.5, CO: 0.75, 02: 0.4, N2: 0.4, all other species 1.0.
384
B. Y A N G E T AL.
agree reasonably well with measurements. A four-step mechanism can be deduced from the detailed mechanism shown in Table 1 by introducing steady-state approximation for C H 2 O H , C H 2 0 , H C O , OH, O, and H O 2 and eliminating these species from the detailed mechanism with the elementary chemical-kinetic steps 18, 16, 15, 3, 2, and 8, respectively [6, 8, 9]. The resulting mechanism can be written as I.
C H 3 O H + 2H ~ 3H 2 + CO,
//.
CO + H 2 0 ~ CO 2 + H2,
III.
H + H + M ~ H 2 + M,
IV.
3H 2 + O 2 ~ 2H + 2 H 2 0 .
kl21CFCH CHc o =
k'13C n + klsC M ' k5fCMCo:
CHO2
,
,
k6
[klfCo2 d- "~kl2bfH CO =
+ ( k k,iC./k'6 )Co ]C. yklbCr t + k'2fCH2
,
(3)
where k, is the rate constant of the elementary reaction n. The subscripts f and b refer to the forward and the backward reactions, respectively, and subscript F refers to the fuel. Other quantities appearing in Eqs. 3 are defined as
The net reaction rates wk, k = I, II, Ill, IV for the overall reactions I - I V can be related to the elementary reactions w~, n = 1, 2 . . . . 22 shown in Table 1 and are W I = W21 q- W 2 2 ,
k'2f = k2f + "yK3k4b ,
k'2b = k2b + "Yk4f,
k'6 = k 6 + k 7 + y k 8 + k 9, k'13 = k13
+ Tkl4,
k~l = k21 + yk22.
k]6
= k16 +
Ykl7, (4)
WII = W l 2 , Wil I :
W 5 q- W10 -k- WI1 q- W13 q'- W14 - - W20 ,
WIV = W 1 q'- W 6 + W 9 .
(1)
T o minimize algebraic complexities, as in previous analysis [2-4], the elementary reaction 3 is assumed to be in partial equilibrium, yielding the algebraic relation Con = yCH, where C~ is the molar concentration of species i and CH2O "y =
K3CH-----~,
(2)
in which K 3 = 0.23 e x p ( 7 5 3 7 / T ) is the equilibrium constant of elementary reaction 3. The steady-state concentration of CH2OH, C H 2 0 , HCO, H O 2, and O determined from setting the source terms in the species balance equation for these quantities to zero can be written as
k'21C~C, CCHzO H =
kl8C H + kl9Coz + k20C M ' k'21C F
CcHzo
r k16
,
The concentration of the third body CM can be written in terms of the Chaperon efficiency 'lc/i of species i(i = 1,2 . . . . . N ) as C M = N
[ p W / ( t ~ T ) ] ~_~ (rliYJWi), where Y/ and W, are i=l
the mass fraction and the molecular weight of species i, respectively, T is the temperature, W is the average molecular weight of the mixture, the gas constant R = 82.05 atm cm3/(molK), and the pressure p is in atmospheres. THE CONSERVATION EQUATIONS FOR A STEADY PREMIXED FLAME For a steady, planar, adiabatic deflagration at low Mach numbers, the equation of motion implies that the pressure is constant. The equation for mass conservation can be integrated to yield the result p c = puvu, where p is the gas density and v is the gas velocity. The index u denotes conditions in the unburned gas. Lewis numbers for species i are defined as L i = h / ( p c p D i ) , where h is the thermal conductivity and Cp is the mean specific heat; the diffusion coefficient O i is taken to be that of species i with respect to nitrogen, and the binary-diffusion approximation is employed. The values of the I_~wis number for all species
STRUCTURE OF PREMIXED M E T H A N O L - A I R FLAMES are assumed to be constant. The nondimensionalized species and energy balance equations can be written as [2-4] =
.~H(XH) = --2W I
-
2o)i11
+
2Wlv,
preheat zone O( 1 )
L ~ '~-]
CHaOH~'~
" ~ H 2 ( X H 2 ) = 3 W l + °911 + OAIII -- 3OAIV, -~°H20(XH20)
.~
385
-
oxidation layer O(Q
To ~
[/
inner layer 0(5)
~--" --OAII q'- 2OAIV'
= -oA,v, 0 ~co(Xco)
x
lb
= OAI -- OAII' Fig. 1. A schematic illustration of the overall flame structure.
co2(Xco) = OA., ,~'~('7") = QIOA l + QIIOAtl -t- QIIIOAIII
+ Qlv OAw.
(5)
The operators are defined as S ~ / - d / d x and . 9 ' = d / d x - d2/dx 2. The nondimensional independent variable x is related to the spatial coordinate x' as x = f f ( p V C p / A ) dx'. The quantities X/ (i = 1 . . . . . N), r, wk, and Qk, with k = I - I V , are defined as
(1/Li)d2/dx 2,
r/wp
T-
rFuW, '
Tb - L '
L
I~WFW k OAk
with thickness of order e, where H 2 and CO are oxidized to H 2 0 and C 0 2 , and a postflame zone with thickness of order unity. It is presumed that 6 ~ e-~ 1. The concentration of fuel is zero in the oxidation layer, and the H radicals are in steady state. In the postflame zone the products are presumed to be in chemical equilibrium and the values of T b and Y/b in this zone are calculated by assuming that only the species 0 2, H2, CO, CO 2, and H z O are present and that the enthalpy and the element mass fraction are equal to those in the unburned gas.
cpVFu(p.v.) 2'
YF,(-AHk)
Qk = cp(Tt,
-
Tu)W
F '
(6)
where Yvu is the mass fraction of fuel in the initial reactant stream, Tb is the adiabatic flame temperature and ( - A H k) is the heat release in that step. The subscript b denotes conditions in the postflame zone. A schematic illustration of the presumed structure of the premixed flame, consistent with the results of numerical calculations [6], is shown in Fig. 1 and is similar to the asymptotic structure of the premixed m e t h a n e - a i r flames analyzed previously [2, 3]. As shown in Fig. 1 the structure consists of four z o n e s - - a preheat zone with thickness of order unity, an inner layer with thickness of order 6 where fuel is consumed by radicals to form H e and CO as well as some H 2 0 and CO 2, an oxidation layer
ASYMPTOTIC ANALYSIS OF THE INNER LAYER Since the inner layer is asymptotically thin, the convective terms in the operators ~ / and S z can be neglected when compared with the diffusive terms. The influence of reaction 5b will be neglected in this layer. Following the development in Ref. 2, the inner layer is presumed to be located at the origin x = 0 and at this point T = T °, X i = Xi O, i = 0 2 , H2, CO, H 2 0 , CO2, and N2, in which the superscript 0 implies that the quantities are evaluated at x = 0. The quantities T O and Xi ° represent the characteristic temperature and scaled concentrations of species i in the inner layer and are the leading terms in an asymptotic expansion. The quantities r °, Xco2 °, XH20 °, a n d XN20 are of order unity, whereas XH: °, Xco °, and Xo2 ° are of order e. To analyze the structure of this
386
B. YANG ET AL.
layer, the following expansions are introduced: X = 61~,
O=
'yklbksfCM(k 7 + k 9 + 3'ks) kl fk'2[k'6.,¥H2
X F = 6LFYF,
X n = (2 6 L n / t o ) y H,
(klo + ykll)CM
z = r ° + 6t, +
klfXo2
R °,
XM
WF --. YFuW
X i = X i ° + 3Liy i,
(11) (7)
i = H 2 , C O , O 2 , H 2 0 , CO 2 , a n d N 2,
where the quantities YF, YH, Y~, and t are presumed to be of order unity. The small parameter 8, which characterizes the thickness of the inner layer, and the parameter to are defined as
6
kt?Xo2 0
2 6L n
to
k 'zqL F
R°
and
,
klyk,2yXoXn2
R =
]
1/2
yk~b[yk':b + (kl0 + Tkl,)CM]
(8) Introducing the expansions of Eq. 7 and Eq. 8 into the first and the second expression of Eq. 5 and using Eqs. 1-4 and 6, the following equations are obtained to the leading order in 6: d2yF
(9)
d~2 = LyFYH,
With the exception of /x, the temperature dependence of all other parameters shown in Eqs. 11 is neglected, because the relevant activation energies are not large. Hence, these parameters are treated as constants and their values are evaluated at the inner layer. It will be shown later that it is necessary to include the temperature dependence of /~ to suppress a logarithmic singularity that would otherwise appear for large negative values of (. Boundary conditions for Eqs. 9 and 10 obtained from matching with the preheat zone solution are
dy~ d,~
- -1,
YH = 0 ,
XYH
+
tr + YH
]
i x + ~r+yH
I
1 - K °_ - O°y___p_H--yH 2 ] (10)
1 + 0YH
J
where
L = Ak'2] 62LF R°, k tl°3R O
X
o , k15XM
6
k 20 X M
I~
k~R o ,
T°klb °R° k '2f° X H2o ,
- oX 0
O"
(k 7 + Tk8)k5fCM
K=
k~ik ~
k19
O,
klsORO ,
(12)
For large positive values of ~', solution to Eqs. 9 and 10 must match with the solution in the oxidation layer, where H radicals are in steady state and YF = 0. Hence,
dyF d~
-- 0,
-0°+
d2y' =-----~H ddd2 LtoYH[YF(-1 +Xyn
as ~'-~ --~.
YH =YH~ = as ( -~ ~.
i f 0 ° = + 4(1 - K °) 2 (13)
It follows from Eq. 13 that dyH/d ~ = 0 as ,~ --, ~. The parameter to can be roughly viewed as representing the ratio of thickness of the fuel-consumption layer, which is of order 6, to the thickness of the radical nonequilibrium layer, which is of order R {}. The structure of the fuel-consumption layer is governed by the overall reaction I, whereas the structure of the radical nonequilibrium layer is governed by the overall reactions III and IV. Following previous analysis [2], two limiting solutions to Eq. 9 and Eq. 10 can be obtained depending on the value of to. In the limit to -~ 0, fuel consumption reaction occurs in a thin layer of thickness of O(8to-2/3). Following previous analysis [2] of the structure of the radical nonequilibrium
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES layer, where YF = 0, which is described briefly in the appendix, it can be shown that for small values of the parameters K°, 0 °, and 0, the quantity L is given by the expression L 0 =2o9 1 +
+ 30 0 +-~qJ
.
387
parameters K°, 0 °, X, 0, o-, and /x are sought by writing I as I=
(14)
[I]00 + K° ~
+x
0o +
+q, oo
In the limit ro ~ ~, the fuel consumption reaction occurs in a relatively broad layer, and embedded in this layer is a thin layer of thickness of 0 ( 6 w - 1 / 3 ) , where reaction IV is not in equilibrium. In this limit, it follows from Eq. 10 that H radicals are in steady state in the fuelconsumption layer. Thus, setting the source term on the right-hand side of Eq. 10 to zero the following result is obtained: (1
Y F ="
- K ° O°ytt --yH 2) (1 + XYH)( /x + o" + y H )
(15)
(1 + ~0yH)[~r + yH(1 + X/X" + 2 o ' x ) + 2XYH2]
where (1 -- K° -- O°yn --yH2) 2 (1 + XYH)2( IZ + O" + y H ) 2 y H
(l q- ~yH)2[O" 4- (1 q- X/z + 2 o ' x ) y H + 2XYH2] 2 0 + 2y H
X
1 - K ° -- OOyH - - y H 2
1 + XYH
1
tx + o- + yH
+
+
+ too"®+~ -~ oo'
(18)
where the subscripts 00 imply that these functions of YH must be evaluated setting K° = 0 ° = X = 0 = ~ r = / x = 0 . The expression for [I]oo and the partial derivatives of I can be written as [1]00 = 2YH2 -- 2yH 4,
¢,
+ 5yH 5,
[ 0 I / 0 o " ]oo [aI/ot~]oo
]
= O,
= 2yn - 3yH3 + (YH)-'
(19)
Integration of the expression for [OI/OlZ]oo shown in Eq. 19 with respect to YH leads to a singularity at YH = 0. To avoid this singular behavior, as in previous analysis [2], the temperature variation of /z must be considered near the region where Yrl is small, so that the exponential decrease in the value o f / x is more rapid than the decrease in the value of YH as YH---' 0. Introducing the approximate expansion for I shown in Eq. 18 into Eq. 16 followed by the use of Eq. 19 and including the temperature variation o f / z near the region where YH is small [2], after some rearrangement the expression L_]
1 + ~by H
1 + Xtx + 2 o x + 4XYH
[01/00°]oo =YH -- 3yH 3, [c71/0X]oo = [0I/0~O]oo = YH -- 6yH 3
(16)
×
oo
[ OI/OK 0 ]00 = -- 2yH 2,
An expression for L can be obtained by introducing Eq. 15 into Eq. 9 and integrating once, using the boundary conditions Eqs. 12 and 13, which yields the expression
I =
ooI[ 00°]0o ,1
= - - 8 I t 1 + - - 5K o + - - 15 0 ° 15 [~ 2 16 5
5
+ --~-/x
~
+gx+ ~0
)-1
~r + (1 + X/x + 2crX)y H + 2XyH 2 ]" (17) To simplify the integration, only the leading terms in an expansion for small values of the
(~ - In/x ° - In E . (20)
388
B. Y A N G E T AL.
is obtained [2], where
E~, =
E20(T~ - Tu)a }}TO2 ,
and C = 0.5772 is the Euler's constant and E20 is the activation energy of the elementary reaction 20. The parameter o- does not appear in the expression for L~ * shown in Eq. 20 because [ M ~ ao" ]00 = 0. An ad hoc approximation to determine L for all values of ~o was proposed and tested previously for m e t h a n e - a i r flames [2]. Since the structure of the inner layer and the governing equations in this layer for m e t h a n o l - a i r premixed flames are similar to those for m e t h a n e - a i r premixed flames, the same ad hoe approximation is used here, which can be written as
L =L~ 1-
1 + 0.18L--
"
(21)
The definition of the parameter ~0 shown in Eq. 11 is identical to the definition of the parameter A appearing in the analysis of Seshadri and Peters [2] of the structure of the inner layer of m e t h a n e - a i r flames. If all the parameters except A appearing in the analysis of Seshadri and Peters [2] are set equal to zero, then the value of L calculated as a function of to from the ad hoc approximation represented by Eq. 21 for various values of A was found to agree reasonably well with the values of L calculated from direct numerical integration of the equations governing the structure of the inner layer. For simplicity, Seshadri and Peters [2] assumed that the influence of other parameters on the value of L is similar to that of A. This reasoning is extended here to m e t h a n o l air flames. If all the parameters except ~0 are set equal to zero, then Eqs. 9 and 10 are identical to the equations governing the structure of the inner layer of m e t h a n e - a i r flames [2], if all the parameters appearing in these equations except A are set equal to zero. Therefore values of L calculated as a function of oJ from Eq. 21 can be expected to agree well with the values of L calculated from direct numerical integration of Eqs. 9 and 10 if all the parameters appearing in this equation
except w and ~O are set equal to zero. It is presumed that the influence of all parameters appearing in Eq. 9, on the value of L is similar to that of ~O. Therefore Eq. 21 together with the complete expressions of L 0 and L~ shown in Eqs. 14 and 20 will be used to calculate the value of L as a function of to. The burning velocity v,, can be calculated by using the results obtained in this section and the rate data shown in Table 1, together with knowledge of YFu, T~,, Tb, T °, and Xi °. The following analyses are needed in determining T O and X/°. ANALYSIS OF T H E OXIDATION LAYER In the oxidation layer X F = O, and H radicals are in steady state. Hence, from the second expression in Eq. 5 it follows that to m = wlV, which, neglecting reaction 5b and for small values of 0 and K, yields the result
XH=R
1
2
2 "
The parameters K and 0 depend on T, Xo2, and XH2; hence variation in the value of these parameters must be considered in the oxidation layer. Since this leads to considerable algebraic complexities, these parameters are treated as constants and their values are evaluated at T = T O and X i = X i °. With these simplifications, Eq. 5 reduces to d2xco t~2
-- (.OII
d2 dx2 [XH2 +
XCO]=
2tOin ,
d2 dx 2 [XH2 nt- XCO -- 2Xo2 ] = 0,
d2 d,x2 [Xco q- Xco2] : 0, d2
~[x.:
+ X.2o] = 0,
d2 dx2 [(Olll + O i v ) ( X H 2 + X c o ) / 2 + Q l l x C O + -r] = 0,
(23)
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES where for convenience the definition x i = X J L i is introduced. For stoichiometric and near stoichiometric flames the values of X , 2,b, Xo2, b and Xco, b are presumed to be of the order of e. Hence, the following expansions are introduced 2qx = erl,
qXH2 = ELH~( b + 0.5ZH~),
qXco = e L c o ( b a + 0.5Zco), 2qXo2 = eLoz( a + z ), 2qXH20 = 2qXu,.O,b -- eLHzOZHz ,
389
2.28 e x p ( - 9 6 3 / T ) are the equilibrium constants of elementary reactions 1 and 2, respectively. In order to express the temperature dependence of the source terms in Eq. 23 in an exponential form, the temperature dependence of the preexponential factor of the rate constants k, were expanded around a reference temperature Tref as T" = (Trer)nexp(n)exp(-nTreJT). The value of Tret was chosen to be 1600 K. The quantity O)ul appearing in Eq. 23 can be written in terms of the expansions shown in Eq. 24 using the third expression of Eq. 1, and Eqs. 6, 25, 26, as
2qXco2 = 2qXco>b - eLco2Zco, r= 1-
et,
(24)
O.)ii I =
X ((Z + a)3/Z(z + b -
where Z = 0.5(ZH2
KI2
XH2oLco '
qXH2,b b - - - ,
+ S°G'nlexp( - m 2 t )
+ OllZco ,
K3Xco2LH2
0 . 5 Z c o ) 3/2
- a l / 2 b 3 / 2 ( z + a)
"{-ZCO),
2qt = ( O [ l l + Qlv) z a=
2q•3DlliGxll
× [ ( z + a ) ( z + b - 0.5Zco) 2 - ab2]}
a - 2qXo:,b eLo~
×exp(mlt),
(27)
where
and
eLH z
A (ksfCM Kll/2K21/2K3 )°
(Qll + Qlll + QIv) q=
2
(25)
(LH2Lo2)3/2(1
' Dii I :
where KI2 -= 0.0089 exp(11169/T) is the equilibrium constant of the elementary reaction 12. The quantity E is the small expansion parameter and is identified later in the analyses. With zn~ --, 0, Zco ---, 0, as r/---, ~, these forms automatically satisfy the boundary conditions in the postflame zone as well as the coupling relations in Eq. 23. Introducing the expansions Eqs. 24 into the expression for R shown in Eq. 8, and neglecting ( k l o + " Y k l l ) C M / ( ] t k ' 2 b ) in comparison to unity, the quantity X H can be written as
E2Kll/2K21/2KsLH23/2Lo21/2 21/2XH20q 2 ×
1
K0 2
0° ] ~ ,
(26)
where K 1 = 12.76 e x p ( - 8 0 3 2 / T ) and K 2 =
K ° / 2 - 00/2)
25/2XH200q 4 (2K1K2LH2)I/2(klo/Y + kll ) (1- K°/2- 0°/2)
S =
Gill
,
(28)
k5fLo21/2 X 2-----~° ° exp(-mlt°), XH2O SOOO~(Tb - L )
m 1 =
m2=
( z + b -- 0.5Zco)3/2(z q- a) l/2
XH=
--
Tb 2 2578e(T b Tb2
-
GPIII =
exp(m2t°),
and
7".)
Here the variation of the values of the quantities 3', and k l o / k u have been neglected, and ml, m2, which are presumed to be of order unity, were determined from the rate data shown in Table 1 and assuming as in methane-air flames that ( a / c p ) o t T °'7 [10].
390
B. YANG E T AL.
The t e r m s al/2b3/2(z + a) and ab 2 arising from reaction 5b, and the backward rates of reactions 10 and 11, although small, are included to ensure that to m = 0 as r l ~ oc. Hence, the parameters multiplying these terms were evaluated at T = T b. For large values of Dnl, the small parameter • is chosen to be E = Dul 1/4
(29)
Introducing the expansions of Eq. 24 into the second expression of Eq. 23, followed by the use of Eqs. 27-29 with X H 2 0 ° / X H 2 0 = 1, in the leading order the following differential equation is obtained:
d2z dr/2
where the parameter m3, which is presumed to be of order unity, was determined from the data shown in Table 1. For simplicity, the quantity a is treated as a constant and is calculated using Eq. 25 with K 3, and Kl2 evaluated at T = T o, and Xco ' and XH,o set equal to their respective values in the postflame zone. Introducing the expansions of Eq. 24 into the first expression of Eq. 23, followed by the use of Eq. 31 and 32, in the leading order the following differential equation is obtained: d2zco d~ 2
-- {(Z q- a)3/2(z q- b -
- D n ( z + a)X/2( z + b - 0.5Zco) 1/2
0 . 5 Z c o ) 3/2
-al/2b3/2( z + a) +S°exp(m2(t
×
×exp[rn,(t1
d~
2
dzco
t°)],
1+ a - -
Z
ate=O,
d~
-Oas~-~
~.
The boundary conditions at rl = 0 and 7/--' oo were obtained from matching with the inner layer and the postflame zone, respectively. The source term %1 appearing in Eq. 23 can be written in terms of the expansions shown in Eq. 24 and using the second expression of Eq. 1, and Eqs. 6, 25, 26, 28, and 29 as
"b a )
1/2
E?
- to)],
1 a t rl = 0,
-
d~
dz
2qDiiGll(Z
2°)
--
×exp[-m3(t
dzco
(30)
¢.011 =
Zc°
° - t))
× [(z + a ) ( z + b - 0.5Zco) 2 - ab2]}
dz
(
d~
-
0
as
7---'
~-
(33)
The boundary conditions at r / = 0 and rt were obtained from matching with the inner layer and the postflame zone respectively. Equations 30 and 33 can be integrated numeri0 cally to determine the quantities z ° and Zco as a function of a, b, m 1, m 2, m 3, S °, and D n. These quantities with the exception of S O depend on e, which may be expressed as a function of T o according to
X ( z + b - 0.5Zco) 1/2
(
-1 -+ za
× Zc°
e x p ( - m 3 t ),
(31)
•
Tb - T o t°(T0 - T,)
(34)
where The quantity L defined in Eq. 11 can be expressed in terms of the expansions shown in Eq. 24 as
q(1 + a ° ) k l z f X n 2 o ° L c o D n =
Gu
EksfCMOK3OLH2Lo2
(1 + ~) (1 + a ) °
exp(m3t°),
4 4 4 5 • ( T b - T,) m 3 =
'
Tb 2
L=
• x3/2T kl/'2(z° +a)5/2(z°-O.Szco° +O) Co2
ks~'CM°k'2°LF(1 - K°/2 - 0o/2)
(32)
'
(35)
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES where use has been made of Eqs. 8, 28, and 29. Equation 35 provides an expression for L in terms of T °, z °, and Zco°. Since Eq. 21 provides an independent expression for L in terms of these quantities, Eqs. 21, 30, 33, and 35 provide three equations that can be used to determine T °, z °, and Zco °. In view of Eq. 29 and 34, the burning velocity can be calculated by rewriting the first expression of Eq. 28 and using the second of Eq. 11. Thus
""=wFT
1.2, Lo2 = 1.1, L H 2 0 = 0.83, Lco 2 = 1.39, Ln2 = 0.3, L n = 0.18, and Lco = 1.11. Calculations were performed for values of 4) between 0.52 and 1.0, for values of p between 1 atm and 10 atm, and for T, = 300 K. Calculations were not performed for values of ~b greater than 1.0 because the simplified chemical-kinetic mechanism employed here is not valid for stoichiometrically rich flames [6]. Calculations for values of p greater than 10 atm were not performed because the approximations introduced in the asymptotic analysis become inaccurate at higher values of p. Figures 2-5 show results obtained for various values of 4) with p = 1 atm. Figure 2 shows results for Zco °, z °, a, and b obtained from simultaneous integration of Eqs. 30 and 33, demonstrating that the values of Zco° and z °, are nearly constant. Figure 3 shows that the value of w decreases with increasing values of ~b; however, L is a weak function of 4). Figure 4 shows the values of the parameters K°, 0 °, /z°, E,~, 0, and g and confirms that the current expansions treating K°, 0 °, and /x° as small quantities to be reasonably accurate. However, all values of ~0, and the values of X for values of 4~ approximately greater than 0.85, are slightly greater than unity. Hence, the quantity I cannot be accurately estimated using the expansions shown in Eq. 18. This may lead to inaccuracies in the values of L 0 and L , calculated from Eqs. 14 and 20, which in turn may L F =
V
(I%CMK~I/2K[/:K3)°(1-,~'}'2- 0y2) ( LH2Lo2)3/2( Tb - To) 4 2S/eq4XH2o°(T b - T~)4(t°) 4 (36) RESULTS AND D I S C U S S I O N
Equations 21, 30, 33, 35, and 36 were used to calculate the characteristic temperature at the inner layer T °, and the burning velocity v, as a function of the equivalence ratio ~b, and the pressure p. In these calculations the value of (A/Cp) appearing in Eq. 36 was estimated from the relation (A/cp) = 2.58 × 10-4(T/298) °7 g / ( c m s). The Lewis numbers for the various species were presumed to be constant, with
2
0.3 0.2
10
-0.1
,,(--
Zco
.(-
zo
391
°
-0.01 ~
~
-0.001
O. I -
0.01 0.5
0.6
.
.
.
.
i
0.7
.
.
.
.
i
0.8
.
.
.
.
i
0.9
0.0001
Fig. 2. Results of numerical integration of Eqs. 30 and 33, showing z °, Zco °, a, and b as functions of 4~ for p = 1 atm and T, = 300 K.
392
B. Y A N G ET AL.
1-
0.8-
0.6
0.4 L 0.2
J
I
0 0.5
.
.
.
.
.
016
.
.
.
i
017
.
.
.
.
0.8
i
0.9
Fig. 3. Variation of the parameters L and ~o with the equivalence ratio cb for p= latmand
introduce some inaccuracies in the calculated values of the e~. However, a closed form solution for Eq. 16 can be obtained in the limit o) ---, ~ only if Eq. 18 is used to estimate the value of I. Since the value of co is small for all values of ~b, the major contribution to L calculated from Eq. 21 is from L 0. Consequently inaccuracies arising from calculating the value of v, using the value of L . calculated from
T, = 3 0 0 K .
Eq. 20 can be expected to be small. In the Appendix, an expression for L 0 which is valid for all values of K0, 00, and ~0 is derived. The values of v, calculated using this expression to evaluate the value of L 0, and Eq. 20 for evaluating the value of L~, were found to agree closely with the values of v, calculated using Eq. 14 and Eq. 20 to estimate the values of L 0 and L~, respectively. Therefore, inaccuracies
/7
~
J
E
e° 0.1
0.01
.
0.5
.
.
.
,
0.6
.
.
.
.
,
.
.
.
0.7
.
.
.
.
018
qb
019
Fig. 4. Variation of the parameters K°, (4~, p?, E~,, tO, and X with the equivalence ratio q5 for p = 1 atm and Tu = 300 K.
STRUCTURE OF PREMIXED METHANOL-AIR FLAMES
393
DII
0.8
0.~
021 t 0
8 .
.
.
.
i
0.5
.
.
.
.
i
0.6
.
.
.
.
i
0.7
.
.
.
.
i
0.8
0.9
resulting from the use of Eqs. 14 and 20 in calculating the values of the burning velocities are expected to be small. Figure 5 shows the values of D n, S °, 8, and e and confirms the validity of the presumed ordering 8 -~ • -~ 1. The parameter D . roughly characterizes the relative influences of the recombination reaction III and the water-gas shift reaction II, on the structure of the oxidation layer. Since the value of D u is nearly close to unity, the structure of the oxidation layer is influenced everywhere by finite rates of
Fig. 5. Variation of D n, S °, 6, and • with the equivalence ratio ~ for p = 1 atm and 7~, = 300 K.
reactions III and IV. The parameter S Oroughly characterizes the relative contributions of the elementary reaction 5f, and the elementary reactions 10 and 11 to the overall recombination reaction III. Figure 5 shows the values of S O at ~b = 1.0 to be approximately 0.34; thus the elementary reactions 10 and 11 are responsible for roughly 34% of the recombination processes occurring in the oxidation layer. Figure 6 shows the values of the adiabatic flame temperature, T b and the characteristic temperature in the inner layer T O as a func-
2400-
2200-
2000-
a~ 0
1800-
% 1600 -
1400-
f
J
p = 1 (atm) I200 0.5
,
0.6
.
.
.
.
,
0.7
.
.
.
.
,
0.8
.
.
.
.
i
0.9
Fig. 6. Variation of the temperature at the inner layer T O with ~b for various values of p calculated for T, = 300 K.
394
B. Y A N G E T AL.
tion of 4), calculated for various values of p. At a fixed value of 4), the value of T O increases with increasing values of p, and at a fixed value of p the value of T O increases slightly with increasing values of 4). This functional behavior of T O with respect to ~b, and p are similar to those observed previously for premixed m e t h a n e - a i r flames [3]. Figure 7 shows results for the burning velocity v, as a function of ~b, calculated for various values of p. The results for p = 1 atm, calculated using the asymptotic model developed here, are compared with the values of v~ calculated by Miiller and Peters [6] using a detailed chemical-kinetic mechanism. At 4 ) = 1.0, the value of v, = 34.4 c m / s calculated using the present model is less than the value of v~ = 40 c m / s obtained by Miiller and Peters [6]. However, these discrepancies are well within the inaccuracies associated with calculating the value of )t/Cp appearing in Eq. 36. For example, if this ratio is presumed to be proportional to (T°) °8'~ rather than (T°) °7 as assumed here, then for 4) = 1.0, and p = 1 atm the present model will predict a value of v u, which is nearly identical to the value of v, obtained by Miiller and Peters [6] at these conditions. In Fig. 8 the values of v, calculated using the present model for various values of p with 4) = 1.0, are compared with those calculated by MOiler and Peters [6] using a detailed chemical-kinetic mechanism, and they agree reasonably well. Figure 9 shows the values of L and
to as a function of p for 4, = 1.0 (solid lines) and 4' = 0.73 (dotted lines). For lean flames at high pressure the value of to is no longer small; therefore the contribution of L~ to the value of L calculated from Eq. 21 is no longer negligible. Figures 10 and 11 show the values of the parameters K°, 0 °, /z°, E~, ~, and X as a function of p for 4 ) = 1.0 and 4 ) = 0 . 7 3 , respectively. Although the values of the parameters K° and 0 ° increase with increasing values of p, their magnitude is still less than unity; therefore the current expansions treating K° and 0 ° as small quantities continue to be reasonably accurate. The values of the parameters qJ and X decrease with increasing values of pressure; therefore the previously mentioned inaccuracies in computing the value of L~ using Eq. 20 at p = 1 atm are considerably decreased at higher pressures. Figures 10 and 11 show the value of ~0 to increase with increasing values of p. In the regions where the value of p0 is greater than unity the values of L.~ calculated from Eq. 20 may be inaccurate. These inaccuracies can be expected to have only a minor influence on the calculated values of v~ in the regions where to is small. For stoichiometrically lean flames at high pressures where the value of to is not small, inaccuracies in calculating the value of L~ using Eq. 20 could introduce inaccuracies in the calculated values of v,. In Figure 12 the values of D u, S °, 8, and E are plotted as a function p for 4) = 1.0 (solid lines) and th = 0.73 (dotted
45. M u l l e r a n d P e t e r s [6] p = 1 atm
40 35 q)
J
30 p = l a
t
m
~
[
J j.J"
25S ~
202 15 10 5 0.5
0.6
0.7
0.~
0.9
Fig. 7. T h e b u r n i n g velocity v. as a function of 05 for v a r i o u s v a l u e s of the p r e s s u r e p c a l c u l a t e d for 7", = 300 K. T h e d o t t e d line shows results of n u m e r ical c a l c u l a t i o n s of MiJller and P e t e r s [6] for p - 1 a t m a n d T, = 300 K.
STRUCTURE OF PREMIXED METHANOL-AIR FLAMES
395
45
40
~
ller and Peters 161
35
30
>=
25
20
4
2
Fig. 8. Comparison between the burning velocity v, as a function of p , calculated using the present model and the numerically calculated results of Miiller and Peters [6] for 4~ = l atm and T, = 300 K.
~
p (atm)
niques for analyzing the structure of premixed methane-air flames can be extended to premixed methanol-air flames. The model predicts values of the burning velocity, r', that agree well with results of numerical calculations obtained, using a detailed chemicalkinetic mechanism for values of the equivalence ratio q5 between 0.52 and 1.0, and for values of pressure p between 1 and 10 atm. The results of the asymptotic analysis were not used to predict the burning velocities and
lines) and again confirms the validity of the presumed ordering 6 < e < 1. Also the value of 6 and e decrease with increasing values of p. CONCLUDING REMARKS The asymptotic structure of premixed methanol-air flames are analyzed using a reduced four-step mechanism. The analysis presented here shows that previously developed tech-
1.4
S J
1.2
J
J J
0.8
0.6-
0.4-
0.2
0
-
0
2
4
6
p (atm)
8
10
Fig. 9. Variation of the parameters L and ~o with p, for T, = 3 0 0 K and 4~ = 1.0 (solid lines) and ~b = 0.73 (dotted lines).
396
B. YANG ET AL.
2
Z
1E
O°
0.1
po Eo
0.01 0
2
4
6
8
10
p (atm) structure of stoichiometrically fuel rich flames, because the elementary chemical-kinetic mechanism that was employed to deduce the reduced chemical-kinetic mechanism is not valid for values of 4, greater than unity. An ad hoc approximation is employed in the asymptotic analysis of the inner layer to evaluate a quantity L that contains the burning velocity.
Fig. 10. Variation of parameters K°, 0", p0, E~,, +, and X with p, for T, = 3 0 0 K a n d ~b= 1.0.
Future research should attempt to improve the analysis of the inner layer so that the value of L can be calculated more accurately. Also, it would be challenging to perform asymptotic analysis using a reduced five-step mechanism, although this would make the analysis of the inner layer more complicated than that developed here.
10
j
J
_ J
o 0. I
0.01 0
(90
2
4
6
p (atm)
8
+ 10
Fig. l l . Variation of the parameters K~, 00, /~0, E~,, ¢ and X with p, for Tu
=300 K and ~b = 0.73.
STRUCTURE
OF PREMIXED
METHANOL-AIR
FLAMES
397
s o
\
\ E 0.1-
8
0.01 2
4
6
8
10
p (atm) The authors are indebted to J. G6ttgens f o r providing the c o m p u t e r program that was used to perform all the numerical calculations reported in this paper. The authors also t h a n k Professor F. A. Williams f o r m a n y valuable suggestions. This work was supported by the U.S. A r m y Research Office under grant A R O D A A L 03-90-9-0084. The international collaboration was supported by the U.S. National Science Foundation, Division o f International Programs, under grant N S F I N T 9114461.
Fig. 12. Variation of the parameters Dli, S O, 6, and e with p, for Tu = 300 K and & = 1.0 (solid lines) and 4~= 0.73 (dotted lines).
tems (N. Peters and B. Rogg Eds.), Springer-Verlag, 1993, in press. 9. Peters, N., in Numerical Solution of Combustion Phenomena, Lecture Notes in Physics, Springer-Verlag, 1985, Vol. 241, p. 90. 10. Smooke, M. D., and Giovangigli, V., in Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames (M. D. Smooke Ed.), Lecture Notes in Physics, Springer-Verlag, 1991, Vol. 384, pp. 1-47. 11. Williams, F. A., Combustion Theory, 2nd ed., Addison-Wesley, Redwood City, CA, 1985. Received 18 February 1992; revised 24 July 1992
REFERENCES 1. Peters, N., and Williams, F. A., Combust. Flame 68:185-207 (1987). 2. Seshadri, K., and Peters, N., Combust. Flame 81:96-118 (1990). 3. Seshadri, K., and G6ttgens, J., in Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames (M. D. Smooke Ed.), Lecture Notes in Physics, Springer-Verlag, 1991, Vol. 384, pp. 111-136. 4. Bui-Pham, M., Seshadri, K., and Williams, F. A., Combust. Flame 89:343-362 (1992). 5. Paczko, G., Lefdal, P. M., and Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, pp. 739-748. 6. Miiller, C., and Peters, N., in Reduced Kinetic Mechanisms for Application in Combustion Systems (N. Peters and B. Rogg Eds.), Springer-Verlag, 1993, in press. 7. Chen, J. Y., Combust. Sci. Technol. 78:127-145 (1991). 8. Miiller, C., Seshadri, K., and Chen, J. Y., in Reduced Kinetic Mechanisms for Application in Combustion Sys-
APPENDIX In t h e limit ~o ~ 0, f r o m Eq. 8 it follows t h a t 6 -~ R °. H e n c e t h e fuel c o n s u m p t i o n r e a c t i o n I o c c u r s in a t h i n layer [2, 3]. F o r c o n v e n i e n c e , this t h i n layer will b e p r e s u m e d to b e l o c a t e d at x = 0. T h e s t r u c t u r e o f this layer r e s e m b l e s t h e s t r u c t u r e o f a d i f f u s i o n f l a m e in t h e B u r k e - S c h u m a n n limit [11]. O u t s i d e this diff u s i o n f l a m e layer, in t h e r e g i o n of p o s i t i v e x, t h e r e exists a r a d i c a l n o n e q u i l i b r i u m layer of t h i c k n e s s o f O ( 6 w - ~), w h e r e t h e c o n c e n t r a t i o n o f f u e l is zero. I n t r o d u c i n g t h e c o n t r a c t e d coo r d i n a t e ~ = ro~', E q . 10 w i t h YF = 0, c a n b e w r i t t e n as
d2yu d~ :2
A
1 -- •o _ OOyn _ y H 2 Yn, 1 + ~YH
(A1)
398
B. Y A N G ET AL.
where the quantity A - - L / w , is presumed to be of order unity. In the thin diffusion flame layer, if the influence of the overall reactions II, IlI, and IV is considered to be small, then from Eq. 5 the coupling relation dZ(2XF/LF -- X H / L H ) / d x 2 = 0 is obtained if the convective terms are neglected. Integrating once and matching with the preheat zone solution, the result d(2XF/L F - X H / L H ) / d x = --2 is obtained. Thus on the downstream side of the diffusion flame layer where dXF/NX= 0, it follows that d ( X n / L n ) / d x = - 2 . Thus introducing the expansion for X n shown in Eq. 7, it follows that
ayH - 1
at sc = 0 .
(A2)
Evaluating the integrals shown in Eq. A3, the result L,,=
216(1-
K°) - 3 0 ° 6tp
2
2(1 - K°)ln(1 + qs) - 20 o - 1 202
O°ln(1 + O) + 1
+
0 3
ln(1 + qs) 04
-1
(A4) is obtained. For small values of K°, 0 °, and ~h, Eq. A3 can be written as 1
1
-~ = A fo [ yH -- yn3 -- KOyH The boundary condition at sc ---) ~ is given by Eq. 13. With Yn~ = 1.0 a first integral of Eq. A1 can be written as 1
d y n ] 21
A'0fl t( 1 - K°l -+ O°yHoyn- yn2 Yn ]] '/YH. (A3)
-O°yn2 - tp(yn2 - yn4)] dyn.
(A5)
Integrating Eq. A5 and rearranging, the result shown in Eq. 14 is obtained. The burning velocities were recalculated with the values of L 0 calculated from Eq. A4. For p = 1 atm and for values of 4, between 0.52 and 1.0, the difference between the values of v u calculated using Eq. A4 and those calculated using Eq. 14 were less than 0.6%.
COMBUSTION AND FLAME
91:382-398 (1992)
The Asymptotic Structure of Premixed Methanol-Air Flames B. YANG and K. SESHADRI* Center for Energy and Combustion Research, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, CA 92093-0310, USA
N, PETERS Institut fiir Technische Mechanik, RWTH Aachen, W-5100 Aachen, Germany The asymptotic structure of methanol-air flames is analyzed using a reduced four-step mechanism, for values of equivalence ratio Ob from 0,52 to l.(I and values of pressure p from 1 to 10 arm. The reduced mechanism was deduced from a starting mechanism, containing 22 elementary chemical reactions. In the analysis, the overall flame structure is subdivided into four z o n e s - - a preheat zone with thickness of order unity, an inner layer with thickness of order 6, an oxidation layer with thickness of order e, and a postflame zone with thickness of order unity. The analysis is performed for 6 .~ e < 1.0. The inner layer is located between the preheat zone and the oxidation layer, and in this layer, finite-rate reactions related to the consumption of the fuel, to form primarily H 2 and CO and some H 2° and CO2, are considered. In the oxidation layer finite-rate reactions related to the oxidation of H 2 and CO to 1t20 and CO 2 are considered. Numerical integration of two coupled, nonlinear, second-order, ordinary differential equations is performed to resolve the structure of the oxidation layer. Analytical expressions are obtained for predicting the burning velocity, c,. At the stoichiometric conditions with p = 1 atm the model predicts values of t~u which are somewhat lower than those calculated from full numerical integration of the conservation equations. However, the model predicts the observed decrease in the value of c, with decrcasing values of ~b and increasing values of p.
INTRODUCTION
Asymptotic analysis, using reduced chemicalkinetic mechanisms, has been successfully employed previously to describe the structure and predict the burning velocities of laminar, premixed methane-air flames [1-4]. The analysis reported here attempts to provide a similar asymptotic description of the structure of laminar, premixed methanol-air flames, using reduced chemical-kinetic mechanisms. Numerical calculations using reduced chemical-kinetic mechanisms have been performed previously to describe the structure and to predict the burning velocities of laminar methanol-air flames [5-8]. These reduced mechanisms were deduced systematically from a detailed chemical-kinetic mechanism using procedures similar to those suggested by Peters [9]. The calculations show that the principal path of oxidation for methanol is CH3OH-CH2OH-CH20-HCO-CO,
*
Corresponding author.
0010-2180/92/$5.00
H 2-
C02, H 2 0 [5-8]. For premixed flames, a reduced four-step [5] and a reduced five-step [6] mechanism have been proposed. In the fivestep mechanism steady-state approximation is not introduced for the intermediate species C H 2 0 , whereas in deducing the four-step mechanism, steady-state approximation is introduced for CH20. It was found that the burning velocities, v, calculated at a value of pressure, p = 1 atm, using a detailed chemical-kinetic mechanism and a reduced five-step mechanism deduced from this detailed mechanism were in good agreement [6] over a wide range of values of the equivalence ratio, 4~. Similar comparisons were also made for values of p between 1 and 10 atm [6] and some discrepancies were found between the results of numerical calculations performed using the detailed chemical-kinetic mechanism and the five-step mechanism at high pressures. These discrepancies were attributed to inaccuracies associated with some of the approximations introduced in calculations using the reduced chemical-kinetic mechanism [6]. Also, the values of ~', calculated using a reduced four-step Copyright © 1992 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc.
STRUCTURE
OF
PREMIXED
METHANOL-AIR
mechanism were found to agree reasonably w e l l w i t h t h e v a l u e s o f vu c a l c u l a t e d u s i n g a d e t a i l e d c h e m i c a l - k i n e t i c m e c h a n i s m [5]. Reduced five-step mechanisms and reduced four-step mechanisms have also been proposed for characterizing the structure of counterflow m e t h a n o l - a i r d i f f u s i o n f l a m e s [7, 8]. T h e c r i t i cal c o n d i t i o n s o f e x t i n c t i o n o f t h e c o u n t e r f l o w diffusion flame predicted by these reduced m e c h a n i s m s w e r e g e n e r a l l y in g o o d a g r e e m e n t with those calculated using a detailed chemical-kinetic mechanism. With the exception of the profile of CHzO calculated using the fourstep mechanism, the profiles of a number of major species, intermediate species and radicals c a l c u l a t e d u s i n g t h e r e d u c e d m e c h a n i s m s were found to agree well with those calculated using the detailed mechanism. In general the results calculated using the five-step mechanism agreed better with the results calculated
FLAMES
383
using the detailed mechanism than those calculated using the four-step mechanism. For simplicity, in the asymptotic analysis des c r i b e d h e r e , t h e f o u r - s t e p m e c h a n i s m is e m ployed. The analysis closely follows previous asymptotic analysis of premixed methane-air f l a m e s [2, 3].
REDUCED CHEMICAL-KINETIC MECHANISM A chemical-kinetic mechanism describing the o x i d a t i o n o f m e t h a n o l is s h o w n in T a b l e 1. This particular set of elementary reactions was c h o s e n as t h e s t a r t i n g p o i n t o f t h e a n a l y s i s , because the calculated values of vu for prem i x e d f l a m e s [6] a n d c r i t i c a l c o n d i t i o n s o f ext i n c t i o n o f d i f f u s i o n f l a m e s [8] u s i n g t h i s m e c h anism and a reduced mechanism deduced from this set of elementary reactions were found to
TABLE 1 Chemical Kinetic Mechanism a No. If lb 2f 2b 3f 3b 4f 4b 5f 5b 6 7 8 9 10 11 12f 12b 13 14 15 16 17 18 19 20 21 22
Reaction H ~ OH + O OH + O ~ O z + H H 2 + O ~ OH + H OH + H ~ H z + O H 2 + OH ---, H 2 0 + H H 2 0 + H ~ H 2 + OH OH + OH ~ H 2 0 + O H 2 0 + O ---, OH + OH 0 2 + H + M* --* HO z + M* HO z + M* ~ O z + H + M* H O 2 + H ~ OH + OH HO 2 + H ~ H 2 + 0 2 HO 2 + OH ~ H 2 0 + 0 2 HO 2 + H --, H 2 0 + O H + H + M* ~ H 2 + M* OH + H + M* ~ H 2 0 + M* CO + OH ~ CO 2 + H CO 2 + H + CO + OH CHO + H ~ CO + H 2 CHO + OH --, CO + H 2 0 CHO + M* ~ CO + H + M* CHeO + H ---, C H O + H 2 C H 2 0 + OH --9 CHO + H 2 0 CH2OH + H --, C H 2 0 + H 2 CH2OH + 0 2 ~ C H 2 0 + HO 2 CH2OH + M* "--, C H 2 0 + H + M* CH3OH + H ---, CH2OH + H 2 CH3OH + OH ~ CH2OH + H 2 0 0 2 +
A~ 2.000E 1.568E 5.060E 2.222E 1.000E 4.312E 1.500E 1.473E 2.300E 3.190E 1.500E 2.500E 6.000E 3.000E 1.800E 2.200E 4.400E 4.956E 2.000E 1.000E 7.100E 2.500E 3.000E 2.000E 1.000E 1.000E 4.000E 1.000E
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
14 13 04 04 08 08 09 10 18 18 14 13 13 13 18 22 06 08 14 14 14 13 13 13 13 14 13 13
n
En
0.00 0.00 2.67 2.67 1.60 1.60 1.14 1.14 -0.80 -0.80 0.00 0.00 0.00 0.00 -1.00 -2.00 1.50 1.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
70.30 3.52 26.30 18.29 13.80 76.46 0.42 71.09 0.00 195.39 4.20 2.90 0.00 7.20 0.00 0.00 -3.10 89.76 0.00 0.00 70.30 16.70 5.00 0.00 30.10 105.10 25.50 7.10
Rate constants are in the form k, = AnTnexp[-E,,/(l~T)]. Units are moles, cubic centimeters, seconds, Kelvins and kJ/mol. Third-body efficiencies are H20: 6.5, CO2: 1.5, CO: 0.75, 02: 0.4, N2: 0.4, all other species 1.0.
384
B. Y A N G E T AL.
agree reasonably well with measurements. A four-step mechanism can be deduced from the detailed mechanism shown in Table 1 by introducing steady-state approximation for C H 2 O H , C H 2 0 , H C O , OH, O, and H O 2 and eliminating these species from the detailed mechanism with the elementary chemical-kinetic steps 18, 16, 15, 3, 2, and 8, respectively [6, 8, 9]. The resulting mechanism can be written as I.
C H 3 O H + 2H ~ 3H 2 + CO,
//.
CO + H 2 0 ~ CO 2 + H2,
III.
H + H + M ~ H 2 + M,
IV.
3H 2 + O 2 ~ 2H + 2 H 2 0 .
kl21CFCH CHc o =
k'13C n + klsC M ' k5fCMCo:
CHO2
,
,
k6
[klfCo2 d- "~kl2bfH CO =
+ ( k k,iC./k'6 )Co ]C. yklbCr t + k'2fCH2
,
(3)
where k, is the rate constant of the elementary reaction n. The subscripts f and b refer to the forward and the backward reactions, respectively, and subscript F refers to the fuel. Other quantities appearing in Eqs. 3 are defined as
The net reaction rates wk, k = I, II, Ill, IV for the overall reactions I - I V can be related to the elementary reactions w~, n = 1, 2 . . . . 22 shown in Table 1 and are W I = W21 q- W 2 2 ,
k'2f = k2f + "yK3k4b ,
k'2b = k2b + "Yk4f,
k'6 = k 6 + k 7 + y k 8 + k 9, k'13 = k13
+ Tkl4,
k~l = k21 + yk22.
k]6
= k16 +
Ykl7, (4)
WII = W l 2 , Wil I :
W 5 q- W10 -k- WI1 q- W13 q'- W14 - - W20 ,
WIV = W 1 q'- W 6 + W 9 .
(1)
T o minimize algebraic complexities, as in previous analysis [2-4], the elementary reaction 3 is assumed to be in partial equilibrium, yielding the algebraic relation Con = yCH, where C~ is the molar concentration of species i and CH2O "y =
K3CH-----~,
(2)
in which K 3 = 0.23 e x p ( 7 5 3 7 / T ) is the equilibrium constant of elementary reaction 3. The steady-state concentration of CH2OH, C H 2 0 , HCO, H O 2, and O determined from setting the source terms in the species balance equation for these quantities to zero can be written as
k'21C~C, CCHzO H =
kl8C H + kl9Coz + k20C M ' k'21C F
CcHzo
r k16
,
The concentration of the third body CM can be written in terms of the Chaperon efficiency 'lc/i of species i(i = 1,2 . . . . . N ) as C M = N
[ p W / ( t ~ T ) ] ~_~ (rliYJWi), where Y/ and W, are i=l
the mass fraction and the molecular weight of species i, respectively, T is the temperature, W is the average molecular weight of the mixture, the gas constant R = 82.05 atm cm3/(molK), and the pressure p is in atmospheres. THE CONSERVATION EQUATIONS FOR A STEADY PREMIXED FLAME For a steady, planar, adiabatic deflagration at low Mach numbers, the equation of motion implies that the pressure is constant. The equation for mass conservation can be integrated to yield the result p c = puvu, where p is the gas density and v is the gas velocity. The index u denotes conditions in the unburned gas. Lewis numbers for species i are defined as L i = h / ( p c p D i ) , where h is the thermal conductivity and Cp is the mean specific heat; the diffusion coefficient O i is taken to be that of species i with respect to nitrogen, and the binary-diffusion approximation is employed. The values of the I_~wis number for all species
STRUCTURE OF PREMIXED M E T H A N O L - A I R FLAMES are assumed to be constant. The nondimensionalized species and energy balance equations can be written as [2-4] =
.~H(XH) = --2W I
-
2o)i11
+
2Wlv,
preheat zone O( 1 )
L ~ '~-]
CHaOH~'~
" ~ H 2 ( X H 2 ) = 3 W l + °911 + OAIII -- 3OAIV, -~°H20(XH20)
.~
385
-
oxidation layer O(Q
To ~
[/
inner layer 0(5)
~--" --OAII q'- 2OAIV'
= -oA,v, 0 ~co(Xco)
x
lb
= OAI -- OAII' Fig. 1. A schematic illustration of the overall flame structure.
co2(Xco) = OA., ,~'~('7") = QIOA l + QIIOAtl -t- QIIIOAIII
+ Qlv OAw.
(5)
The operators are defined as S ~ / - d / d x and . 9 ' = d / d x - d2/dx 2. The nondimensional independent variable x is related to the spatial coordinate x' as x = f f ( p V C p / A ) dx'. The quantities X/ (i = 1 . . . . . N), r, wk, and Qk, with k = I - I V , are defined as
(1/Li)d2/dx 2,
r/wp
T-
rFuW, '
Tb - L '
L
I~WFW k OAk
with thickness of order e, where H 2 and CO are oxidized to H 2 0 and C 0 2 , and a postflame zone with thickness of order unity. It is presumed that 6 ~ e-~ 1. The concentration of fuel is zero in the oxidation layer, and the H radicals are in steady state. In the postflame zone the products are presumed to be in chemical equilibrium and the values of T b and Y/b in this zone are calculated by assuming that only the species 0 2, H2, CO, CO 2, and H z O are present and that the enthalpy and the element mass fraction are equal to those in the unburned gas.
cpVFu(p.v.) 2'
YF,(-AHk)
Qk = cp(Tt,
-
Tu)W
F '
(6)
where Yvu is the mass fraction of fuel in the initial reactant stream, Tb is the adiabatic flame temperature and ( - A H k) is the heat release in that step. The subscript b denotes conditions in the postflame zone. A schematic illustration of the presumed structure of the premixed flame, consistent with the results of numerical calculations [6], is shown in Fig. 1 and is similar to the asymptotic structure of the premixed m e t h a n e - a i r flames analyzed previously [2, 3]. As shown in Fig. 1 the structure consists of four z o n e s - - a preheat zone with thickness of order unity, an inner layer with thickness of order 6 where fuel is consumed by radicals to form H e and CO as well as some H 2 0 and CO 2, an oxidation layer
ASYMPTOTIC ANALYSIS OF THE INNER LAYER Since the inner layer is asymptotically thin, the convective terms in the operators ~ / and S z can be neglected when compared with the diffusive terms. The influence of reaction 5b will be neglected in this layer. Following the development in Ref. 2, the inner layer is presumed to be located at the origin x = 0 and at this point T = T °, X i = Xi O, i = 0 2 , H2, CO, H 2 0 , CO2, and N2, in which the superscript 0 implies that the quantities are evaluated at x = 0. The quantities T O and Xi ° represent the characteristic temperature and scaled concentrations of species i in the inner layer and are the leading terms in an asymptotic expansion. The quantities r °, Xco2 °, XH20 °, a n d XN20 are of order unity, whereas XH: °, Xco °, and Xo2 ° are of order e. To analyze the structure of this
386
B. YANG ET AL.
layer, the following expansions are introduced: X = 61~,
O=
'yklbksfCM(k 7 + k 9 + 3'ks) kl fk'2[k'6.,¥H2
X F = 6LFYF,
X n = (2 6 L n / t o ) y H,
(klo + ykll)CM
z = r ° + 6t, +
klfXo2
R °,
XM
WF --. YFuW
X i = X i ° + 3Liy i,
(11) (7)
i = H 2 , C O , O 2 , H 2 0 , CO 2 , a n d N 2,
where the quantities YF, YH, Y~, and t are presumed to be of order unity. The small parameter 8, which characterizes the thickness of the inner layer, and the parameter to are defined as
6
kt?Xo2 0
2 6L n
to
k 'zqL F
R°
and
,
klyk,2yXoXn2
R =
]
1/2
yk~b[yk':b + (kl0 + Tkl,)CM]
(8) Introducing the expansions of Eq. 7 and Eq. 8 into the first and the second expression of Eq. 5 and using Eqs. 1-4 and 6, the following equations are obtained to the leading order in 6: d2yF
(9)
d~2 = LyFYH,
With the exception of /x, the temperature dependence of all other parameters shown in Eqs. 11 is neglected, because the relevant activation energies are not large. Hence, these parameters are treated as constants and their values are evaluated at the inner layer. It will be shown later that it is necessary to include the temperature dependence of /~ to suppress a logarithmic singularity that would otherwise appear for large negative values of (. Boundary conditions for Eqs. 9 and 10 obtained from matching with the preheat zone solution are
dy~ d,~
- -1,
YH = 0 ,
XYH
+
tr + YH
]
i x + ~r+yH
I
1 - K °_ - O°y___p_H--yH 2 ] (10)
1 + 0YH
J
where
L = Ak'2] 62LF R°, k tl°3R O
X
o , k15XM
6
k 20 X M
I~
k~R o ,
T°klb °R° k '2f° X H2o ,
- oX 0
O"
(k 7 + Tk8)k5fCM
K=
k~ik ~
k19
O,
klsORO ,
(12)
For large positive values of ~', solution to Eqs. 9 and 10 must match with the solution in the oxidation layer, where H radicals are in steady state and YF = 0. Hence,
dyF d~
-- 0,
-0°+
d2y' =-----~H ddd2 LtoYH[YF(-1 +Xyn
as ~'-~ --~.
YH =YH~ = as ( -~ ~.
i f 0 ° = + 4(1 - K °) 2 (13)
It follows from Eq. 13 that dyH/d ~ = 0 as ,~ --, ~. The parameter to can be roughly viewed as representing the ratio of thickness of the fuel-consumption layer, which is of order 6, to the thickness of the radical nonequilibrium layer, which is of order R {}. The structure of the fuel-consumption layer is governed by the overall reaction I, whereas the structure of the radical nonequilibrium layer is governed by the overall reactions III and IV. Following previous analysis [2], two limiting solutions to Eq. 9 and Eq. 10 can be obtained depending on the value of to. In the limit to -~ 0, fuel consumption reaction occurs in a thin layer of thickness of O(8to-2/3). Following previous analysis [2] of the structure of the radical nonequilibrium
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES layer, where YF = 0, which is described briefly in the appendix, it can be shown that for small values of the parameters K°, 0 °, and 0, the quantity L is given by the expression L 0 =2o9 1 +
+ 30 0 +-~qJ
.
387
parameters K°, 0 °, X, 0, o-, and /x are sought by writing I as I=
(14)
[I]00 + K° ~
+x
0o +
+q, oo
In the limit ro ~ ~, the fuel consumption reaction occurs in a relatively broad layer, and embedded in this layer is a thin layer of thickness of 0 ( 6 w - 1 / 3 ) , where reaction IV is not in equilibrium. In this limit, it follows from Eq. 10 that H radicals are in steady state in the fuelconsumption layer. Thus, setting the source term on the right-hand side of Eq. 10 to zero the following result is obtained: (1
Y F ="
- K ° O°ytt --yH 2) (1 + XYH)( /x + o" + y H )
(15)
(1 + ~0yH)[~r + yH(1 + X/X" + 2 o ' x ) + 2XYH2]
where (1 -- K° -- O°yn --yH2) 2 (1 + XYH)2( IZ + O" + y H ) 2 y H
(l q- ~yH)2[O" 4- (1 q- X/z + 2 o ' x ) y H + 2XYH2] 2 0 + 2y H
X
1 - K ° -- OOyH - - y H 2
1 + XYH
1
tx + o- + yH
+
+
+ too"®+~ -~ oo'
(18)
where the subscripts 00 imply that these functions of YH must be evaluated setting K° = 0 ° = X = 0 = ~ r = / x = 0 . The expression for [I]oo and the partial derivatives of I can be written as [1]00 = 2YH2 -- 2yH 4,
¢,
+ 5yH 5,
[ 0 I / 0 o " ]oo [aI/ot~]oo
]
= O,
= 2yn - 3yH3 + (YH)-'
(19)
Integration of the expression for [OI/OlZ]oo shown in Eq. 19 with respect to YH leads to a singularity at YH = 0. To avoid this singular behavior, as in previous analysis [2], the temperature variation of /z must be considered near the region where Yrl is small, so that the exponential decrease in the value o f / x is more rapid than the decrease in the value of YH as YH---' 0. Introducing the approximate expansion for I shown in Eq. 18 into Eq. 16 followed by the use of Eq. 19 and including the temperature variation o f / z near the region where YH is small [2], after some rearrangement the expression L_]
1 + ~by H
1 + Xtx + 2 o x + 4XYH
[01/00°]oo =YH -- 3yH 3, [c71/0X]oo = [0I/0~O]oo = YH -- 6yH 3
(16)
×
oo
[ OI/OK 0 ]00 = -- 2yH 2,
An expression for L can be obtained by introducing Eq. 15 into Eq. 9 and integrating once, using the boundary conditions Eqs. 12 and 13, which yields the expression
I =
ooI[ 00°]0o ,1
= - - 8 I t 1 + - - 5K o + - - 15 0 ° 15 [~ 2 16 5
5
+ --~-/x
~
+gx+ ~0
)-1
~r + (1 + X/x + 2crX)y H + 2XyH 2 ]" (17) To simplify the integration, only the leading terms in an expansion for small values of the
(~ - In/x ° - In E . (20)
388
B. Y A N G E T AL.
is obtained [2], where
E~, =
E20(T~ - Tu)a }}TO2 ,
and C = 0.5772 is the Euler's constant and E20 is the activation energy of the elementary reaction 20. The parameter o- does not appear in the expression for L~ * shown in Eq. 20 because [ M ~ ao" ]00 = 0. An ad hoc approximation to determine L for all values of ~o was proposed and tested previously for m e t h a n e - a i r flames [2]. Since the structure of the inner layer and the governing equations in this layer for m e t h a n o l - a i r premixed flames are similar to those for m e t h a n e - a i r premixed flames, the same ad hoe approximation is used here, which can be written as
L =L~ 1-
1 + 0.18L--
"
(21)
The definition of the parameter ~0 shown in Eq. 11 is identical to the definition of the parameter A appearing in the analysis of Seshadri and Peters [2] of the structure of the inner layer of m e t h a n e - a i r flames. If all the parameters except A appearing in the analysis of Seshadri and Peters [2] are set equal to zero, then the value of L calculated as a function of to from the ad hoc approximation represented by Eq. 21 for various values of A was found to agree reasonably well with the values of L calculated from direct numerical integration of the equations governing the structure of the inner layer. For simplicity, Seshadri and Peters [2] assumed that the influence of other parameters on the value of L is similar to that of A. This reasoning is extended here to m e t h a n o l air flames. If all the parameters except ~0 are set equal to zero, then Eqs. 9 and 10 are identical to the equations governing the structure of the inner layer of m e t h a n e - a i r flames [2], if all the parameters appearing in these equations except A are set equal to zero. Therefore values of L calculated as a function of oJ from Eq. 21 can be expected to agree well with the values of L calculated from direct numerical integration of Eqs. 9 and 10 if all the parameters appearing in this equation
except w and ~O are set equal to zero. It is presumed that the influence of all parameters appearing in Eq. 9, on the value of L is similar to that of ~O. Therefore Eq. 21 together with the complete expressions of L 0 and L~ shown in Eqs. 14 and 20 will be used to calculate the value of L as a function of to. The burning velocity v,, can be calculated by using the results obtained in this section and the rate data shown in Table 1, together with knowledge of YFu, T~,, Tb, T °, and Xi °. The following analyses are needed in determining T O and X/°. ANALYSIS OF T H E OXIDATION LAYER In the oxidation layer X F = O, and H radicals are in steady state. Hence, from the second expression in Eq. 5 it follows that to m = wlV, which, neglecting reaction 5b and for small values of 0 and K, yields the result
XH=R
1
2
2 "
The parameters K and 0 depend on T, Xo2, and XH2; hence variation in the value of these parameters must be considered in the oxidation layer. Since this leads to considerable algebraic complexities, these parameters are treated as constants and their values are evaluated at T = T O and X i = X i °. With these simplifications, Eq. 5 reduces to d2xco t~2
-- (.OII
d2 dx2 [XH2 +
XCO]=
2tOin ,
d2 dx 2 [XH2 nt- XCO -- 2Xo2 ] = 0,
d2 d,x2 [Xco q- Xco2] : 0, d2
~[x.:
+ X.2o] = 0,
d2 dx2 [(Olll + O i v ) ( X H 2 + X c o ) / 2 + Q l l x C O + -r] = 0,
(23)
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES where for convenience the definition x i = X J L i is introduced. For stoichiometric and near stoichiometric flames the values of X , 2,b, Xo2, b and Xco, b are presumed to be of the order of e. Hence, the following expansions are introduced 2qx = erl,
qXH2 = ELH~( b + 0.5ZH~),
qXco = e L c o ( b a + 0.5Zco), 2qXo2 = eLoz( a + z ), 2qXH20 = 2qXu,.O,b -- eLHzOZHz ,
389
2.28 e x p ( - 9 6 3 / T ) are the equilibrium constants of elementary reactions 1 and 2, respectively. In order to express the temperature dependence of the source terms in Eq. 23 in an exponential form, the temperature dependence of the preexponential factor of the rate constants k, were expanded around a reference temperature Tref as T" = (Trer)nexp(n)exp(-nTreJT). The value of Tret was chosen to be 1600 K. The quantity O)ul appearing in Eq. 23 can be written in terms of the expansions shown in Eq. 24 using the third expression of Eq. 1, and Eqs. 6, 25, 26, as
2qXco2 = 2qXco>b - eLco2Zco, r= 1-
et,
(24)
O.)ii I =
X ((Z + a)3/Z(z + b -
where Z = 0.5(ZH2
KI2
XH2oLco '
qXH2,b b - - - ,
+ S°G'nlexp( - m 2 t )
+ OllZco ,
K3Xco2LH2
0 . 5 Z c o ) 3/2
- a l / 2 b 3 / 2 ( z + a)
"{-ZCO),
2qt = ( O [ l l + Qlv) z a=
2q•3DlliGxll
× [ ( z + a ) ( z + b - 0.5Zco) 2 - ab2]}
a - 2qXo:,b eLo~
×exp(mlt),
(27)
where
and
eLH z
A (ksfCM Kll/2K21/2K3 )°
(Qll + Qlll + QIv) q=
2
(25)
(LH2Lo2)3/2(1
' Dii I :
where KI2 -= 0.0089 exp(11169/T) is the equilibrium constant of the elementary reaction 12. The quantity E is the small expansion parameter and is identified later in the analyses. With zn~ --, 0, Zco ---, 0, as r/---, ~, these forms automatically satisfy the boundary conditions in the postflame zone as well as the coupling relations in Eq. 23. Introducing the expansions Eqs. 24 into the expression for R shown in Eq. 8, and neglecting ( k l o + " Y k l l ) C M / ( ] t k ' 2 b ) in comparison to unity, the quantity X H can be written as
E2Kll/2K21/2KsLH23/2Lo21/2 21/2XH20q 2 ×
1
K0 2
0° ] ~ ,
(26)
where K 1 = 12.76 e x p ( - 8 0 3 2 / T ) and K 2 =
K ° / 2 - 00/2)
25/2XH200q 4 (2K1K2LH2)I/2(klo/Y + kll ) (1- K°/2- 0°/2)
S =
Gill
,
(28)
k5fLo21/2 X 2-----~° ° exp(-mlt°), XH2O SOOO~(Tb - L )
m 1 =
m2=
( z + b -- 0.5Zco)3/2(z q- a) l/2
XH=
--
Tb 2 2578e(T b Tb2
-
GPIII =
exp(m2t°),
and
7".)
Here the variation of the values of the quantities 3', and k l o / k u have been neglected, and ml, m2, which are presumed to be of order unity, were determined from the rate data shown in Table 1 and assuming as in methane-air flames that ( a / c p ) o t T °'7 [10].
390
B. YANG E T AL.
The t e r m s al/2b3/2(z + a) and ab 2 arising from reaction 5b, and the backward rates of reactions 10 and 11, although small, are included to ensure that to m = 0 as r l ~ oc. Hence, the parameters multiplying these terms were evaluated at T = T b. For large values of Dnl, the small parameter • is chosen to be E = Dul 1/4
(29)
Introducing the expansions of Eq. 24 into the second expression of Eq. 23, followed by the use of Eqs. 27-29 with X H 2 0 ° / X H 2 0 = 1, in the leading order the following differential equation is obtained:
d2z dr/2
where the parameter m3, which is presumed to be of order unity, was determined from the data shown in Table 1. For simplicity, the quantity a is treated as a constant and is calculated using Eq. 25 with K 3, and Kl2 evaluated at T = T o, and Xco ' and XH,o set equal to their respective values in the postflame zone. Introducing the expansions of Eq. 24 into the first expression of Eq. 23, followed by the use of Eq. 31 and 32, in the leading order the following differential equation is obtained: d2zco d~ 2
-- {(Z q- a)3/2(z q- b -
- D n ( z + a)X/2( z + b - 0.5Zco) 1/2
0 . 5 Z c o ) 3/2
-al/2b3/2( z + a) +S°exp(m2(t
×
×exp[rn,(t1
d~
2
dzco
t°)],
1+ a - -
Z
ate=O,
d~
-Oas~-~
~.
The boundary conditions at rl = 0 and 7/--' oo were obtained from matching with the inner layer and the postflame zone, respectively. The source term %1 appearing in Eq. 23 can be written in terms of the expansions shown in Eq. 24 and using the second expression of Eq. 1, and Eqs. 6, 25, 26, 28, and 29 as
"b a )
1/2
E?
- to)],
1 a t rl = 0,
-
d~
dz
2qDiiGll(Z
2°)
--
×exp[-m3(t
dzco
(30)
¢.011 =
Zc°
° - t))
× [(z + a ) ( z + b - 0.5Zco) 2 - ab2]}
dz
(
d~
-
0
as
7---'
~-
(33)
The boundary conditions at r / = 0 and rt were obtained from matching with the inner layer and the postflame zone respectively. Equations 30 and 33 can be integrated numeri0 cally to determine the quantities z ° and Zco as a function of a, b, m 1, m 2, m 3, S °, and D n. These quantities with the exception of S O depend on e, which may be expressed as a function of T o according to
X ( z + b - 0.5Zco) 1/2
(
-1 -+ za
× Zc°
e x p ( - m 3 t ),
(31)
•
Tb - T o t°(T0 - T,)
(34)
where The quantity L defined in Eq. 11 can be expressed in terms of the expansions shown in Eq. 24 as
q(1 + a ° ) k l z f X n 2 o ° L c o D n =
Gu
EksfCMOK3OLH2Lo2
(1 + ~) (1 + a ) °
exp(m3t°),
4 4 4 5 • ( T b - T,) m 3 =
'
Tb 2
L=
• x3/2T kl/'2(z° +a)5/2(z°-O.Szco° +O) Co2
ks~'CM°k'2°LF(1 - K°/2 - 0o/2)
(32)
'
(35)
S T R U C T U R E OF P R E M I X E D M E T H A N O L - A I R FLAMES where use has been made of Eqs. 8, 28, and 29. Equation 35 provides an expression for L in terms of T °, z °, and Zco°. Since Eq. 21 provides an independent expression for L in terms of these quantities, Eqs. 21, 30, 33, and 35 provide three equations that can be used to determine T °, z °, and Zco °. In view of Eq. 29 and 34, the burning velocity can be calculated by rewriting the first expression of Eq. 28 and using the second of Eq. 11. Thus
""=wFT
1.2, Lo2 = 1.1, L H 2 0 = 0.83, Lco 2 = 1.39, Ln2 = 0.3, L n = 0.18, and Lco = 1.11. Calculations were performed for values of 4) between 0.52 and 1.0, for values of p between 1 atm and 10 atm, and for T, = 300 K. Calculations were not performed for values of ~b greater than 1.0 because the simplified chemical-kinetic mechanism employed here is not valid for stoichiometrically rich flames [6]. Calculations for values of p greater than 10 atm were not performed because the approximations introduced in the asymptotic analysis become inaccurate at higher values of p. Figures 2-5 show results obtained for various values of 4) with p = 1 atm. Figure 2 shows results for Zco °, z °, a, and b obtained from simultaneous integration of Eqs. 30 and 33, demonstrating that the values of Zco° and z °, are nearly constant. Figure 3 shows that the value of w decreases with increasing values of ~b; however, L is a weak function of 4). Figure 4 shows the values of the parameters K°, 0 °, /z°, E,~, 0, and g and confirms that the current expansions treating K°, 0 °, and /x° as small quantities to be reasonably accurate. However, all values of ~0, and the values of X for values of 4~ approximately greater than 0.85, are slightly greater than unity. Hence, the quantity I cannot be accurately estimated using the expansions shown in Eq. 18. This may lead to inaccuracies in the values of L 0 and L , calculated from Eqs. 14 and 20, which in turn may L F =
V
(I%CMK~I/2K[/:K3)°(1-,~'}'2- 0y2) ( LH2Lo2)3/2( Tb - To) 4 2S/eq4XH2o°(T b - T~)4(t°) 4 (36) RESULTS AND D I S C U S S I O N
Equations 21, 30, 33, 35, and 36 were used to calculate the characteristic temperature at the inner layer T °, and the burning velocity v, as a function of the equivalence ratio ~b, and the pressure p. In these calculations the value of (A/Cp) appearing in Eq. 36 was estimated from the relation (A/cp) = 2.58 × 10-4(T/298) °7 g / ( c m s). The Lewis numbers for the various species were presumed to be constant, with
2
0.3 0.2
10
-0.1
,,(--
Zco
.(-
zo
391
°
-0.01 ~
~
-0.001
O. I -
0.01 0.5
0.6
.
.
.
.
i
0.7
.
.
.
.
i
0.8
.
.
.
.
i
0.9
0.0001
Fig. 2. Results of numerical integration of Eqs. 30 and 33, showing z °, Zco °, a, and b as functions of 4~ for p = 1 atm and T, = 300 K.
392
B. Y A N G ET AL.
1-
0.8-
0.6
0.4 L 0.2
J
I
0 0.5
.
.
.
.
.
016
.
.
.
i
017
.
.
.
.
0.8
i
0.9
Fig. 3. Variation of the parameters L and ~o with the equivalence ratio cb for p= latmand
introduce some inaccuracies in the calculated values of the e~. However, a closed form solution for Eq. 16 can be obtained in the limit o) ---, ~ only if Eq. 18 is used to estimate the value of I. Since the value of co is small for all values of ~b, the major contribution to L calculated from Eq. 21 is from L 0. Consequently inaccuracies arising from calculating the value of v, using the value of L . calculated from
T, = 3 0 0 K .
Eq. 20 can be expected to be small. In the Appendix, an expression for L 0 which is valid for all values of K0, 00, and ~0 is derived. The values of v, calculated using this expression to evaluate the value of L 0, and Eq. 20 for evaluating the value of L~, were found to agree closely with the values of v, calculated using Eq. 14 and Eq. 20 to estimate the values of L 0 and L~, respectively. Therefore, inaccuracies
/7
~
J
E
e° 0.1
0.01
.
0.5
.
.
.
,
0.6
.
.
.
.
,
.
.
.
0.7
.
.
.
.
018
qb
019
Fig. 4. Variation of the parameters K°, (4~, p?, E~,, tO, and X with the equivalence ratio q5 for p = 1 atm and Tu = 300 K.
STRUCTURE OF PREMIXED METHANOL-AIR FLAMES
393
DII
0.8
0.~
021 t 0
8 .
.
.
.
i
0.5
.
.
.
.
i
0.6
.
.
.
.
i
0.7
.
.
.
.
i
0.8
0.9
resulting from the use of Eqs. 14 and 20 in calculating the values of the burning velocities are expected to be small. Figure 5 shows the values of D n, S °, 8, and e and confirms the validity of the presumed ordering 8 -~ • -~ 1. The parameter D . roughly characterizes the relative influences of the recombination reaction III and the water-gas shift reaction II, on the structure of the oxidation layer. Since the value of D u is nearly close to unity, the structure of the oxidation layer is influenced everywhere by finite rates of
Fig. 5. Variation of D n, S °, 6, and • with the equivalence ratio ~ for p = 1 atm and 7~, = 300 K.
reactions III and IV. The parameter S Oroughly characterizes the relative contributions of the elementary reaction 5f, and the elementary reactions 10 and 11 to the overall recombination reaction III. Figure 5 shows the values of S O at ~b = 1.0 to be approximately 0.34; thus the elementary reactions 10 and 11 are responsible for roughly 34% of the recombination processes occurring in the oxidation layer. Figure 6 shows the values of the adiabatic flame temperature, T b and the characteristic temperature in the inner layer T O as a func-
2400-
2200-
2000-
a~ 0
1800-
% 1600 -
1400-
f
J
p = 1 (atm) I200 0.5
,
0.6
.
.
.
.
,
0.7
.
.
.
.
,
0.8
.
.
.
.
i
0.9
Fig. 6. Variation of the temperature at the inner layer T O with ~b for various values of p calculated for T, = 300 K.
394
B. Y A N G E T AL.
tion of 4), calculated for various values of p. At a fixed value of 4), the value of T O increases with increasing values of p, and at a fixed value of p the value of T O increases slightly with increasing values of 4). This functional behavior of T O with respect to ~b, and p are similar to those observed previously for premixed m e t h a n e - a i r flames [3]. Figure 7 shows results for the burning velocity v, as a function of ~b, calculated for various values of p. The results for p = 1 atm, calculated using the asymptotic model developed here, are compared with the values of v~ calculated by Miiller and Peters [6] using a detailed chemical-kinetic mechanism. At 4 ) = 1.0, the value of v, = 34.4 c m / s calculated using the present model is less than the value of v~ = 40 c m / s obtained by Miiller and Peters [6]. However, these discrepancies are well within the inaccuracies associated with calculating the value of )t/Cp appearing in Eq. 36. For example, if this ratio is presumed to be proportional to (T°) °8'~ rather than (T°) °7 as assumed here, then for 4) = 1.0, and p = 1 atm the present model will predict a value of v u, which is nearly identical to the value of v, obtained by Miiller and Peters [6] at these conditions. In Fig. 8 the values of v, calculated using the present model for various values of p with 4) = 1.0, are compared with those calculated by MOiler and Peters [6] using a detailed chemical-kinetic mechanism, and they agree reasonably well. Figure 9 shows the values of L and
to as a function of p for 4, = 1.0 (solid lines) and 4' = 0.73 (dotted lines). For lean flames at high pressure the value of to is no longer small; therefore the contribution of L~ to the value of L calculated from Eq. 21 is no longer negligible. Figures 10 and 11 show the values of the parameters K°, 0 °, /z°, E~, ~, and X as a function of p for 4 ) = 1.0 and 4 ) = 0 . 7 3 , respectively. Although the values of the parameters K° and 0 ° increase with increasing values of p, their magnitude is still less than unity; therefore the current expansions treating K° and 0 ° as small quantities continue to be reasonably accurate. The values of the parameters qJ and X decrease with increasing values of pressure; therefore the previously mentioned inaccuracies in computing the value of L~ using Eq. 20 at p = 1 atm are considerably decreased at higher pressures. Figures 10 and 11 show the value of ~0 to increase with increasing values of p. In the regions where the value of p0 is greater than unity the values of L.~ calculated from Eq. 20 may be inaccurate. These inaccuracies can be expected to have only a minor influence on the calculated values of v~ in the regions where to is small. For stoichiometrically lean flames at high pressures where the value of to is not small, inaccuracies in calculating the value of L~ using Eq. 20 could introduce inaccuracies in the calculated values of v,. In Figure 12 the values of D u, S °, 8, and E are plotted as a function p for 4) = 1.0 (solid lines) and th = 0.73 (dotted
45. M u l l e r a n d P e t e r s [6] p = 1 atm
40 35 q)
J
30 p = l a
t
m
~
[
J j.J"
25S ~
202 15 10 5 0.5
0.6
0.7
0.~
0.9
Fig. 7. T h e b u r n i n g velocity v. as a function of 05 for v a r i o u s v a l u e s of the p r e s s u r e p c a l c u l a t e d for 7", = 300 K. T h e d o t t e d line shows results of n u m e r ical c a l c u l a t i o n s of MiJller and P e t e r s [6] for p - 1 a t m a n d T, = 300 K.
STRUCTURE OF PREMIXED METHANOL-AIR FLAMES
395
45
40
~
ller and Peters 161
35
30
>=
25
20
4
2
Fig. 8. Comparison between the burning velocity v, as a function of p , calculated using the present model and the numerically calculated results of Miiller and Peters [6] for 4~ = l atm and T, = 300 K.
~
p (atm)
niques for analyzing the structure of premixed methane-air flames can be extended to premixed methanol-air flames. The model predicts values of the burning velocity, r', that agree well with results of numerical calculations obtained, using a detailed chemicalkinetic mechanism for values of the equivalence ratio q5 between 0.52 and 1.0, and for values of pressure p between 1 and 10 atm. The results of the asymptotic analysis were not used to predict the burning velocities and
lines) and again confirms the validity of the presumed ordering 6 < e < 1. Also the value of 6 and e decrease with increasing values of p. CONCLUDING REMARKS The asymptotic structure of premixed methanol-air flames are analyzed using a reduced four-step mechanism. The analysis presented here shows that previously developed tech-
1.4
S J
1.2
J
J J
0.8
0.6-
0.4-
0.2
0
-
0
2
4
6
p (atm)
8
10
Fig. 9. Variation of the parameters L and ~o with p, for T, = 3 0 0 K and 4~ = 1.0 (solid lines) and ~b = 0.73 (dotted lines).
396
B. YANG ET AL.
2
Z
1E
O°
0.1
po Eo
0.01 0
2
4
6
8
10
p (atm) structure of stoichiometrically fuel rich flames, because the elementary chemical-kinetic mechanism that was employed to deduce the reduced chemical-kinetic mechanism is not valid for values of 4, greater than unity. An ad hoc approximation is employed in the asymptotic analysis of the inner layer to evaluate a quantity L that contains the burning velocity.
Fig. 10. Variation of parameters K°, 0", p0, E~,, +, and X with p, for T, = 3 0 0 K a n d ~b= 1.0.
Future research should attempt to improve the analysis of the inner layer so that the value of L can be calculated more accurately. Also, it would be challenging to perform asymptotic analysis using a reduced five-step mechanism, although this would make the analysis of the inner layer more complicated than that developed here.
10
j
J
_ J
o 0. I
0.01 0
(90
2
4
6
p (atm)
8
+ 10
Fig. l l . Variation of the parameters K~, 00, /~0, E~,, ¢ and X with p, for Tu
=300 K and ~b = 0.73.
STRUCTURE
OF PREMIXED
METHANOL-AIR
FLAMES
397
s o
\
\ E 0.1-
8
0.01 2
4
6
8
10
p (atm) The authors are indebted to J. G6ttgens f o r providing the c o m p u t e r program that was used to perform all the numerical calculations reported in this paper. The authors also t h a n k Professor F. A. Williams f o r m a n y valuable suggestions. This work was supported by the U.S. A r m y Research Office under grant A R O D A A L 03-90-9-0084. The international collaboration was supported by the U.S. National Science Foundation, Division o f International Programs, under grant N S F I N T 9114461.
Fig. 12. Variation of the parameters Dli, S O, 6, and e with p, for Tu = 300 K and & = 1.0 (solid lines) and 4~= 0.73 (dotted lines).
tems (N. Peters and B. Rogg Eds.), Springer-Verlag, 1993, in press. 9. Peters, N., in Numerical Solution of Combustion Phenomena, Lecture Notes in Physics, Springer-Verlag, 1985, Vol. 241, p. 90. 10. Smooke, M. D., and Giovangigli, V., in Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames (M. D. Smooke Ed.), Lecture Notes in Physics, Springer-Verlag, 1991, Vol. 384, pp. 1-47. 11. Williams, F. A., Combustion Theory, 2nd ed., Addison-Wesley, Redwood City, CA, 1985. Received 18 February 1992; revised 24 July 1992
REFERENCES 1. Peters, N., and Williams, F. A., Combust. Flame 68:185-207 (1987). 2. Seshadri, K., and Peters, N., Combust. Flame 81:96-118 (1990). 3. Seshadri, K., and G6ttgens, J., in Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames (M. D. Smooke Ed.), Lecture Notes in Physics, Springer-Verlag, 1991, Vol. 384, pp. 111-136. 4. Bui-Pham, M., Seshadri, K., and Williams, F. A., Combust. Flame 89:343-362 (1992). 5. Paczko, G., Lefdal, P. M., and Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, pp. 739-748. 6. Miiller, C., and Peters, N., in Reduced Kinetic Mechanisms for Application in Combustion Systems (N. Peters and B. Rogg Eds.), Springer-Verlag, 1993, in press. 7. Chen, J. Y., Combust. Sci. Technol. 78:127-145 (1991). 8. Miiller, C., Seshadri, K., and Chen, J. Y., in Reduced Kinetic Mechanisms for Application in Combustion Sys-
APPENDIX In t h e limit ~o ~ 0, f r o m Eq. 8 it follows t h a t 6 -~ R °. H e n c e t h e fuel c o n s u m p t i o n r e a c t i o n I o c c u r s in a t h i n layer [2, 3]. F o r c o n v e n i e n c e , this t h i n layer will b e p r e s u m e d to b e l o c a t e d at x = 0. T h e s t r u c t u r e o f this layer r e s e m b l e s t h e s t r u c t u r e o f a d i f f u s i o n f l a m e in t h e B u r k e - S c h u m a n n limit [11]. O u t s i d e this diff u s i o n f l a m e layer, in t h e r e g i o n of p o s i t i v e x, t h e r e exists a r a d i c a l n o n e q u i l i b r i u m layer of t h i c k n e s s o f O ( 6 w - ~), w h e r e t h e c o n c e n t r a t i o n o f f u e l is zero. I n t r o d u c i n g t h e c o n t r a c t e d coo r d i n a t e ~ = ro~', E q . 10 w i t h YF = 0, c a n b e w r i t t e n as
d2yu d~ :2
A
1 -- •o _ OOyn _ y H 2 Yn, 1 + ~YH
(A1)
398
B. Y A N G ET AL.
where the quantity A - - L / w , is presumed to be of order unity. In the thin diffusion flame layer, if the influence of the overall reactions II, IlI, and IV is considered to be small, then from Eq. 5 the coupling relation dZ(2XF/LF -- X H / L H ) / d x 2 = 0 is obtained if the convective terms are neglected. Integrating once and matching with the preheat zone solution, the result d(2XF/L F - X H / L H ) / d x = --2 is obtained. Thus on the downstream side of the diffusion flame layer where dXF/NX= 0, it follows that d ( X n / L n ) / d x = - 2 . Thus introducing the expansion for X n shown in Eq. 7, it follows that
ayH - 1
at sc = 0 .
(A2)
Evaluating the integrals shown in Eq. A3, the result L,,=
216(1-
K°) - 3 0 ° 6tp
2
2(1 - K°)ln(1 + qs) - 20 o - 1 202
O°ln(1 + O) + 1
+
0 3
ln(1 + qs) 04
-1
(A4) is obtained. For small values of K°, 0 °, and ~h, Eq. A3 can be written as 1
1
-~ = A fo [ yH -- yn3 -- KOyH The boundary condition at sc ---) ~ is given by Eq. 13. With Yn~ = 1.0 a first integral of Eq. A1 can be written as 1
d y n ] 21
A'0fl t( 1 - K°l -+ O°yHoyn- yn2 Yn ]] '/YH. (A3)
-O°yn2 - tp(yn2 - yn4)] dyn.
(A5)
Integrating Eq. A5 and rearranging, the result shown in Eq. 14 is obtained. The burning velocities were recalculated with the values of L 0 calculated from Eq. A4. For p = 1 atm and for values of 4, between 0.52 and 1.0, the difference between the values of v u calculated using Eq. A4 and those calculated using Eq. 14 were less than 0.6%.