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Flame fronts with complex chemical networks
Physica 12D (1984) 182-197 North-Holland, Amsterdam
FLAME FRONTS WITH COMPLEX CHEMICAL NETWORKS Paul C. FIFE Department of Mathematics, University of Arizona, Tucson, AZ 98721, USA
and Basil NICOLAENKO Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
For laminar flame fronts with general complex chemistry networks, we present a systematic method to reduce the detailed chemistry in the limit of high activation energy asymptotics. We detail the method on a global multistep hydrocarbon model. Generic flame profile structures are discussed, and extensions to networks with zero activation energies are outlined.
1. Introduction
Complex chemistry in flames involves a mix of reactions with both high and small activation energies. Since the pioneering works of FrankKamenetskii [11] and Zeldovich [16, 17], it has been recognized that large activation energies Ei determine a hierarchy of scales
RT+ T+ el= Ei T + - T _ '
ei~ 1 ,
where T is the cold mixture temperature, and T+ a typical adiabatic flame final temperature. The 8i are the inverse of the (large) Zeldovich numbers. In this paper we develop the asymptotic flame theory with complex chemistry first outlined in [7-8]. We specifically consider the case of finite ratios Ei/Ej # 1 for the subclass of High Activation Energy (H.A.E.) reactions. Cases where E~~ Ej, but the pre-Arrhenius factors are not the same order of magnitude, are also covered by our theory.
We first explain concepts which are common to
all reactions networks: admissible reactant allocations, which yield the possible final burned states
of the flame; and admissible reactant suballocations to the network, which determine feasible chemical network pathways as one passes from the unburned state to the burned state through the front (section 1.2). The latter correspond to subgroups of reactions which go to completion at temperatures lower than the final one. The notion of set of admissible allocations (section 2.2) turns out to be a convenient tool. Next, to determine the actual pathways within the network and along the flame profile, we construct normalized power gauge functions (section 1.3) for both H.A.E. and Z.A.E. (Zero Activation Energy) reactions. Unnormalized power rate functions have already been used implicitly by FrankKamenetskii [11] and Lifian [12]. We derive systematic comparison principles based on these gauge functions. In section 2, we detail our method on a global hydrocarbon two-step model. We evidence some generic flame structures present in complex networks: merged flames, competition flames and tandem flames. In section 3, we outline the methodology for general networks
0167-2789/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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1.1. The flame front equations for a chemical network
Example 1.1. Consider the network of sequential reactions
Let n chemical species Al, A2. . . . . An be involved in r reactions Rl, R2. . . . . R,; symbolically
Rl : Ai ~ A 2 R2 : A z ~ P
(P inert product)
n
v(j, i)Aj~ ~ ~.(j, i)Aj, j=l
i = 1. . . . . r.
(1.1)
j=l
where v(j, i) (resp. 2(j, i)) is the stoichiometric coefficient of species j on the left side (resp. right side) of reaction i. Both v and 2 are nonnegative integers; vij, i) is zero if species j is not a reactant in reaction i, while 2(j, i) is zero if it is not a product. Each reaction yields heat releases Q1, Q2. . . . . Q,. The flow field is described by the temperature T and the vector of rescaled mass concentrations (fractions): (1.2)
Y = ( Y~, Y~. . . . , Yn) ,
where the rescaled Yj, j = 1. . . . . n, are the actual mass fractions (concentrations) of species j divided by the corresponding molecular mass mj (this rescaling somewhat simplifies the flame front eqs. (1.6). The full state vector U is U = (T, Y) = (T, YI . . . .
, L).
(1.3)
Each reaction can be characterized by an ndimensional reaction vector ~, i = 1 , . . . , r, which represents the net change in the number of molecule grams (moles) of various species in reaction
then Y = (Y~, Y2), U = ( T , Y], Y2), Y] = ( - l , + l ) , F2 = (0, - 1), K~ = (Q,, - l, + 1),/(2 = (Q2, 0, - 1). Example 1.2. Consider the network of competing reactions RI : A l ~ P t (P~, P2 are both inert products) R2" A l ~ P2 then U = (T, Y), /(1 = (QI, - 1),/(2 = (Q2, - 1). With the above vector notations, the equations for a one-dimensional traveling flame front, in a system of coordinates moving with the front, and in the constant pressure approximation, reduce to the system of (n + 1) equations in T, Y I , . . . , Yn:
- (D(U)Ux)x + MUx = ~ og,(U)K,.;
(1.6)
i=1
where D ( U ) is a (n + 1) x (n + 1) diffusion matrix (positive definite;) M is the unknown mass flux, which equals wave velocity if the density is unity. The scalar functions ~o~ are the rates of the reactions R,.. The system 1.6 is closed by the perfect gas law,
K: V/= (2(I, i) - v(1, i), 2(2, i) - v(2, i ) , . . . , 2(n, i) - v(n, i)) = ( Z ~ I), VI 2). . . .
, Fit)) •
(1.4)
The extended (n + 1)-dimensional reaction vector K~, i = 1 , . . . , r, is defined as = (Qi, V3;
(1.5)
where p (x) is the mixture global density and/~c the constant pressure in the isobaric, small Mach number limit. Finally, the cold state u- = u(-oo)
= (r_, Y? .....
Y;)
(1.8)
is prescribed at x = - 0 o . The reaction rates o9i are given by the usual law
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
184
of mass action and Arrhenius factors,
t°i=[]YY'°B'exp(-~)
(1.9)
where E~ is the activation energy of reaction i and B,. the corresponding pre-Arrhenius factor; strictly speaking, B~ is a slowly varying function of T and p, hence of T and Yj through (1.7); we will in the sequel neglect its dependence w.r.t. Yj, taking for instance the context of a reacting mixture diluted in some inert species. In this presentation, we omit reversible reactions. The extension to reversible reactions has been completed and will be presented in [9]. We are concerned with exploiting the high activation energies E~ to obtain approximate wave front solutions. The Ei determine scales T+
-
T+
-
ei= Ei T + - T_
~ 1,
(1.10)
where T÷ is a typical final (burned) temperature and the ~ are inverse Zeldovich numbers (T has been renormalized so that the gas constant is unity). The sharp sensitivity of Arrhenius terms exp( - EJT) w.r.t, variations of T enable a systematic High Activation Energy (H.A.E.) asymptotic analysis. The procedures are linked to properties of the underlying reaction network, which supplies the vectors Kj. These procedures are well known in the case when the network is modeled by a single reaction A-~P. (See [2] and the review article [3] in these proceedings.) An approach similar to the one in this paper was used in treating two-reaction networks of competing, sequential, and competing-fuel types in [7] and [8], and it was clear that the extension of this approach to more complex schemes could uncover a baffling array of possibilities. A systematic formalism was called for, and hopefully was provided in part by the authors in [8]. The methods are those of asymptotic analysis. Roughly speaking, to the lowest order this means developing rational procedures for replacing the
original problem with a limit problem in which the e~ do not appear explicitly. The limit problem is expected to yield an approximate solution, and is easy to solve. In the limit, flame sheets appear as free boundaries with Ucontinuous across, but with jumps in the gradients. A rigorous justification of this expectation has been given recently for the case of a single reaction [1]. Reactions are concentrated on flame sheets (internal layers) with pure convective-diffusive flow in between, if there are no reactions with E~ = 0:
(D(V)Ux)x - MUx = 0 between the flame sheets. The flame sheets correspond to a Dirac-function in temperature limiting behavior of the o~ as ei~0. If there is a mix of reactions with H.A.E. Ei >> 1, i = 1 , . . . , p and with zero activation energies (Z.A.E.) E~=0, i = p + 1. . . . , r, there are still flame sheets (free boundaries) associated to the R~, i = 1 , . . . , p ; but the flow in between is convective-diffusive-reactive due to the Z.A.E. reactions. The latter "thicken" the wave front profiles. Many other parameters besides e~enter the problem, for example, the elements of the matrix D and of the unburned state vector of the gas, U-. Throughout the paper, we assume that their magnitudes do not interfere with the asymptotics. Particularly, we assume that any constants not depending explicitly on e;, B~, or E~ (as defined in (1.9) and (1.10)) are t~(1) quantities. This especially refers to elements of D and D-~, reference dimensionless Tj, their differences T~- Tj and the heat releases Q~. These restrictions are relaxed, and a more meticulous account of the other parameters is taken in [10]. For background and previous work on asymptotic methods in flame theory, the reader is referred to [2] and [3] in these proceedings. A number of papers have been written on the multiple reaction case; see the references in [2], [4], [12], and [17]. We remark that the consideration of realistic chemical networks avoids the so-called "cold boundary difficulty" [2], where U- is not an equi-
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
librium state of the chemical network. In realistic chemical networks, the initial zero concentrations of intermediate radicals species insures equilibrium at the cold side; a good example is the autocatalytic chain network investigated by Lifian [12]: R] : A1 + A2-*2A2
(1.11)
vectors lies in the linear span of the extended reaction vectors K~. Moreover, the ct~, i = 1. . . . . r, must satisfy constraints such that the final concentration vector Y+ ensures that the network is in equilibrium: Y+ = 0 =
Y ; + ~, o~iV~k)
where P and M are inert; the initial concentration of the intermediate radical A2 is null. Here, K1 = (01, - 1, + 1) and/(2 = (02, 0, - 2 ) .
1.2. The reactant allocations and suballocations to a chemical network The final state on the burned side
for some integer subset {k} such that o~,(U+)= 0, Vi. Because of (1.14) and (1.15), M must be considered as unknown nonlinear eigenvalue for the general system (1.6). We will call the sequence of positive or null numbers ~ , ~ . . . . . ~, an allocation of the initial concentration vector Y- to the network. To make this point of view more systematic, we first need a basic assumption about the reaction vectors:
(1.12)
is actually coupled with the initial state U through conservation laws which are the equivalent of the Rankine-Hugoniot conditions for nonreactive flows. To see this, integrate (1.6) from x=-oo tox=+~,use Ux=Oatx=+~ to obtain
M(U + - U - ) = ~ 12iK~,
(1.15)
i=l
R 2 : 2A 2 + M ~ 2 P + M ,
U + = U ( + ~ ) = U(T+, Y + , . . . , Y+)
185
(1.13a)
i=1
where 12,.is the integrated rate function for R~ along the flame profile (proportional to the thermal power due to R;):
Assumption 1. If ~Vi=O
and ~>~0, then a ~ - 0.
(1.16)
i=1
This assumption assures, as we shall see, that equilibrium can only be reached when each reaction rate is zero. It can easily be removed to cover reversible reactions [9].
Definition 1. Y E 8 is an equilibrium state if, for all
T, (o,(Y, r)v~ = 0.
q-oo
(1.17)
i=l
f2i = f og(U(x))dx ;
(1.13b)
now setting ~; = 12~/M, U + - U - = ~ ct,K~,
(1.14)
i=l
hence the difference between the asymptotic state
It follows from (1.16) that this can only happen if each (]-)i---0. For each i = 1. . . . . r, we denote by J(i) the set of all indices j of species Aj entering reaction Ri (appearing on the left side of the arrow). At this point we assume toi = 0 if and only if Yj = 0 for some j ~ J(i). We symbolize the equilibrium cone for each reaction, and the total equilibrium cone,
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
186
as follows: g, = { Y >I 0: Yj = 0 for some j ~ Y(i)},
i=l
(This is, of course, a cone imbedded in a set of coordinate hyperplanes.) For simplicity only we require
Assumption 2. Vi¢ gi, for all i. Definition 2. Given an input state Y->~ 0, an allocation of Y- to the network is a set of numbers 0t~t> 0, i = 1. . . . , r, such that
Y-+ ~ O~iE~¢.
(1.18)
i=l
Example. Consider the network of competing reactions described in Example 1.1. Clearly 8 = {0}. The T and Y~ components at eq. (1.14) are T+ = T_ + 0qQl + 0t2Q2; 0 = Y+ = Yi- - 0q - 0t2; but each ~ti can take values only between zero and Yf- and their sum must be Yi-- Setting oq = OY{, 0<0<1, we have ot2 = ( 1 - 0 ) Y ; - and T + = T_ + YI-(OQ1 + (1 - O)Q2). The final temperature can be any temperature between 7"1 = T_ + Y I Q I , adiabatic temperature of a pure reaction R1, and T2 = T_ + YIQ2, adiabatic temperature of a pure reaction R2. Indeed this network of competing reactions is the canonical example of nonuniqueness of allocations. In more complex networks, there is usually a whole array of possible allocation vectors which are restricted to the boundary of a specific convex set (simplex) defined by
(see section 2 for examples of such convex sets). In general, we must consider the final states U + ( a ~ , . . . , ~,) and T+(~q. . . . , ~t,) as functions of the a~. Nonunique allocations reflect the possibility of different chemical pathways in the network. Another consideration is that experimental and numerical simulations show that within a flame profile, there is a sequence of temperatures where some subgroups of reactions are completed, while some other reactions are still frozen [13]. For instance, in heavy hydrocarbon flames [13] peak concentrations of a typical radical occur at a distinctly different temperature than the temperature band where most of the fuel is consumed [14]. This leads to the notion of suballocations (to be developed in sections 2 and 3) such that a partial subgroup of reactions go to completion at temperatures lower than T+, within a partial subnetwork.
1.3. Normalized power functions The actual chemical pathways and chemical subnetworks in a flame are determined by the competing rates to~; the latter can vary sharply not only in absolute but also in relative magnitude as temperature increases across the flame front. Hence the need for some normalized gauge functions to systematically exploit these relative variations. Consider now the integrated rate functions fJi in eq. (1.13):
f2i= f co,(U(x)) dx ;
writing the first component in 1.13a yields the exact relation
M(T÷ - T_)= ~ Q,t2,,
(1.21)
i=I
oci>~O, i = l . . . . , r (1.19) j~-~- l ~ . . . ~ r l . i=l
(1.20)
i.e., the relative contribution of each reaction to the temperature profile is its heat release Qi weighted
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
by its integrated rate function. But if the reaction Ri with high activation energy E; is isolated then classical H.A.E. analysis for a single step reaction [2] going to completion at temperature T, gives
t2,(T) = C,e,(B~exp - EJ T) '/2 ,
(1.22)
where (7,- is some proportionality constant. Also, the rate function o9i is concentrated in the flame sheet. For networks with partial H.A.E., the mi for the H.A.E. reactions are still concentrated in a sequence of flame sheets (possibly merged) at various temperatures and it can be shown with a layer analysis that the g2~ are still of the order of magnitude given by (1.22). Rather than (1.22), we shall use the following choice for normalized power gauge functions H,(T) to compare the relative powers of reactions R~:
Definition 3. Power gauge functions: H,( T) = ~[ln B, - ~ I , = min ~ = min T+
(1.23a) T+
Ei T + - T _ '
i=l,...,r. (1.23b)
In (1.23a) we allow T t o vary, since the flame sheet temperatures are a priori unknown (nonunique allocations). We also allow the same definitions (1.23a-b) whether reaction Ri is H.A.E. or Z.A.E.
Indeed 1) if Ri is H.A.E., then
Hi(T) = g(21n Oi + tP (In ~)), which ensures that the variation of H,(T) over the whole flame temperature profile is 0(1); 2) if Ri is Z.A.E., then Hi = g In B; ~ constant, as B,(T) varies weakly with T; nevertheless we should keep in mind that physical Bi may be very
187
large indeed, anywhere from 10ll to 1019; SO the normalized Hi are still O(1) or smaller.
Remark 1.1. The power gauge functions Hi(T) are monotone increasing in T. Any linear combination of H,(T) is a monotone (decreasing or increasing) function of T. Our basic comparison principle is based on the following inequalities: if rip(T) >
HdT)
at a given T, and the difference is d~(1), then Op(T) >>Qq(T)
and Rq has negligible effect compared to Rp, at temperature T. An equilibrium configuration might still be obtained for some distinguished temperatures Tp and Tq such that Hp(Tp)= Hq(Tq) and t2p(Tp)/Oq(Tq)= t9(1). We will detail the systematic use of the comparison principle to construct flame fronts in sections 2 and 3. As a simple example, we show how the power gauge functions resolve the nonuniqueness of allocations in the competing reactions network (example 1.1). Recall that T÷ can vary anywhere between TI and T2 (single reaction adiabatic temperatures). Let E1 >> 1, E2 >> 1 and El~E2 =O(1); three cases are possible: 1) HI(T)> H2(T) for all T E[T l, T2]; then R 1 globally quenches R~, and there is a single flame sheet at T~; the allocation is (Yi-, 0); 2) H2(T) > HI(T) for all T ~ [Tt, T2]; then there is a single flame sheet at/'2, with allocation (0, Yi-); 3) HI(T.) = H2(T,) for some T., TI < T . < 7"2; T. is unique because of remark 1.1, and the two reactions are at equilibrium at the flame sheet temperature T÷ = T,. With T , = OT1 + (1 - O)T:, 0 < 0 < 1, the allocations are a~ = 0 Yi-, a 2 = ( 1 - O ) Y £ . The full layer analysis of this competition flame has been done by the authors in [7]. In section 2, we show that competition type flame regimes can in fact be generic for more complex networks, due to competing subnetworks.
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P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
Finally extending formula (1.23) to Z.A.E. reactions ensures that our comparison principle overlaps classical pseudo-steady state analysis: if Hp(T) > Hq(T) for Rp and Rq both Z.A.E.. then for all temperature ranges top >>toq (as Hp and Hq are nearly constant) and the usual pseudo-steady state reduction of Rp can be done.
RI:C, Hm+ ~ + ~
O2~nCO+
R 2:CO +~O 2~-CO 2,
H20,
(2.2)
with empirically derived reaction rates for both reactions; for R2: to2 ~--~-10146exp(-R4--~0T)[CO]l[H20]°5[O2]0'25 (2.3a)
2. A global hydrocarbon two-step model and for Rl:
2.1. The two-step Dryer-Westbrook model Hydrocarbon flames are usually modelled by global one-step models: Fuel + nl O:~n2 CO2 + n3 H20,
(2.1a)
where nl, n2, n3 are positive integers, and the global reaction rate to is t o = B ( T ) e x p ( - ~ T ) [Fuel]~[O2]#;
(2.1b)
the pre-Arrhenius factor B and the exponents and fl are usually adjusted to fit calculated flame parameters to some experimental data; ~t and fl turn out to be positive fractional numbers. As pointed out in [15], the single-step model predicts flame speeds reliably over a considerable range of reactions; but it systematically overpredicts the total heat of reaction and the final adiabatic temperature. Moreover, at temperatures characteristic of hydrocarbon fuels ( ~ 2000 K), substantial amounts of CO and H2 exist in equilibrium in the combustion products with CO2 and H20. It is also well recognized that typical hydrocarbons burn in a sequential manner; i.e., the fuel is partially oxidized to CO and H2, which are not appreciably consumed until all of the hydrocarbon species have disappeared [6]. From these observations, and following a similar model for methane by Dryer and Glassmann [5], Westbrook and Dryer [15] have proposed the following global two-step reaction mechanism:
to~ = Bl exp( -- R ~ ) [Fuel]~[O2]#.
(2.3b)
For Octane CsHls, B l = 5 . 7 x l0 II, E~=30.0 kcal/mol, 0t = 0.25 and fl = 1.5. In general, for all hydrocarbons except methane CH4, E 1 ~ 30 kcal, and B l ,~ B2 = 1014.6. So
Remark 2.1. For the global two-step model (2.2), the power gauge functions HI(T ) and H2(T) always intersect at some T , such that
,ln
)'
(2.4)
since BE >>B1 and E2 - E1 ~ 10 kcal. Competing flames and subnetworks are possible. In the sequel, we systematically investigate the simplified network R1 : Fuel + Oxygen~R + Pl, R2 : R + Oxygen-*P2,
(2.5)
where P1 and P~ are inert products (e.g., H20 and CO2) and R is an intermediate "radical" species (e.g., CO). We take Yf, Yox and Yr to be the respective (rescaled) concentrations of fuel, oxygen, and radical. We take the simplified extended reaction vectors (stoichiometric coefficients set equal to one): g~ ---(Q,, -I, -I, +I), (2.6) K~ = (Q2, o,
-
l, - I),
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
with both Qi > 0. Notice the alternation + 1 for the last component, as the Radical is produced by R,, but consumed by R2. The flame front equations for the state vector U---(T, Yf, Yox, Yr) are (DfU)Ux) x + MUx = o)I(U)K1 -Jr o)2(V)K2 , (2.7)
-
with the reaction rates (-/)l = BI YfYox exp(
-
092 = B2 Yox Y, exp( -
EUT), E21T),
E2>>l,
allocations in the positive quadrant ~t, ~2; but we must require that equilibrium be attained. Since the reactions are not reversible, chemical equilibrium corresponds to some points (~, ~2) on the boundary o f this simplex. However, equilibrium for the chemical network can be reached in two different ways: first both Yi ~ = 0 and Yr+ = 0 , which corresponds to an oxygen rich initial mixture. Or Yo+x= 0, which corresponds to a fuel rich mixture. More precisely:
(2.8) 1) In the oxygen rich case (fig. 1), a < b12 and Y~- = a - ~1 = 0, Y : = ~l - ~2 = 0, so
and E~>>I,
189
(2.12)
oq = ~ 2 = a = Y f ;
E2 d~(1). E =
the final state is The initial state is U - = (T_, a, b, 0),
(2.9)
with Y~-= a, Y~x = b and Yr = 0 (which is the physical situation). Notice that we do not assume reversibility of R2; the reversible case is fully treated in [9].
Y~=0,
(fuel depletion),
Yo+x=b-2a >O,
(excess oxygen),
Yr+ = O, (the radical concentration pulses to a maximum and then decreases to zero); T + = T- + a(Ql +
2.2. The convex set o f admissible allocations Let ~ and ~2 be the allocations to RI and R2 with at = (~l, ~2); possible final states are
(2.13)
Q2).
(2.14)
For the oxygen rich case, notice that the admissible allocation (a, a) is unique (edge o f the convex set). The case a = b/2 corresponds to an initially stoichiometric concentration Y-.
V + ( a t ) = U - + c t l K l -'1-o~2K2,
i.e. T+(1) = T_ + ~IQ~ + ~2Q2,
(2.10)
Y~" = a - - tXl~> 0 ,
Y+ox= b - - ~ 1 - ix2 ~> 0 ,
(2.11)
Y~+ = ~1 -- at2/> 0; inequalities (2.11) together with the conditions • l ~> 0, ~t2 I> 0 define the convex set o f eventual
"t U
I I
a
b/2
",,u
Fig. 1. The oxygen rich ease, a < b/2.
It--
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
190
2) In the fuel rich case (fig. 2), a > b / 2 , Yo+,-- b - 0q - ~t2= 0; since a2/>0, we have b /2 <<.a~ <~b, in fact:
b /2 <~ctI <~rain(a, b) ;
a2
(2.15)
the final allocations are b/2
Y/~ = a - ctl,
(excess fuel),
Y~ = 0,
(oxygen depletion),
i
t
d,
]
,
(2.16) /
Y+ = 2al - b.
I
I
b/2
There usually is some residual radical concentration on the burned side of the flame. The key feature of the fuel rich case is that allocations are not unique: (~t, ~2) covers a whole closed interval on the line ~ + ~t2= b, with (2.15) and b - min(a, b) ~
(2.17)
to fix ideas, take Q 2 2 Q 1 and a<~b, so min(a, b) = a; then T+(a0 is a decreasing function of ~ , which reaches its minimum T~, at ~q = a and its maximum T ~ at 0~ = b/2,
T~. = T_ + aQl + (b - a)Q2, (2.18)
Tma~ = T_ + b /2(Q~ + Q2).
x
!
N,
a
b
cL1
Fig. 2. The fuel rich case, a > b/2.
and H2(T) at T = T+. We consider the case H~(T+) = H~(T+) + ¢(g) as nongeneric; and in fact, the inequalities introducing cases 1-6 are meant to imply that the two sides differ by a quantity > ~(~).
Case 1. Hi(T+)> H2(T+): Sequential flames R1 would tend to dominate R2 at T÷; what we mean is that if R1 went to completion at T÷, then R2 would have negligible rate in comparison, and we would appear to have a one-step reaction. This would contradict the fact that both reactions must operate in order to achieve T+. A stable flame configuration exists only if R1 is completed in some flame sheet of temperature TI < T÷, such that the
2.3.Flame configurations in the oxygen rich case (a <<.b/2) This is characterized by a unique final temperature (2.14),
T+ = T_ + a(Ql + Q2) and a unique final allocation at = (a,a). Flame configurations are determined by the relative ordering of the power gauge functions (1.23) H~(T)
.....
~2
I TI
....
I T+
Fig. 3. Case 1.
~-
T
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P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
power H~ is lowered to a power equilibrium configuration, H,(T1) = H2(T+) ; such a temperature TI exists and is unique because of remark 1.1. However, to be physically attainable, Tt must be larger than the adiabatic temperature T_ +aQl = / ~ associated with RI in isolation [7, 8]. For simplicity, we assume that is so; if not, total equilibrium is only attained in stages (2 successive flames travelling at different velocities), rather than in a single steady flame. The flame configuration is sequential with R1 completed in a flame sheet (1) at Tx, where all the fuel is burned out; this is followed by a flame sheet at T÷, where R2 consumed all the radical created by the sheet (1). The temperature sequence (T l, T÷) equates the power rates and induces a stable chemical pathway along the flame profile: first R1 with R2 frozen, then R2 with Ri completed. We remark that there is a suballocation t~0) = (a, 0) to the flame sheet (1), and a final allocation ~(2)= ~ : (a, a) to the sheet (2). Remark 2.1. In practice the flame profile will be thickened because of Z.A.E. involving other radicals. Remark 2.2. The principle of determining a sequence of temperatures (possibly equal) to equate all the power gauge functions Hi, systematically induces stable chemical pathways and subnetworks in general complex networks. Case 2. H2(T+)> HI(T+): Pseodoreaction ~t RE is now much faster than RI at T+. We might think of R2 first taking place at some 7"2< T+, such that H2(T2) = H~(T+), followed by R~ at T+. However, the corresponding suballocation would be (0, 0); that is, such a separated flame sheet cannot exist, as there is no initial radical concentration (Y;- = 0) to trigger it. The only possible structure is a single flame layer at T+, with a pseudo-reaction
1~1 characterized by the same rate function o91, but with an instant heat release QI + Q2. The radical produced by i1[1 is instantly burned by. R2 within the same flame layer. Strictly speaking, this is a situation with merged layers, but to lowest order, R2 does not affect the layer analysis at T÷. 2.4. Flame configurations in the fuel rich case (a > b/2) Since allocations are not unique (fig. 2), we must compare 1t1(7') and H2(T) over the whole range [T~i~, T ~ ] as defined in (2.17)--(2.18). Hence, a larger variety of possible configurations than in the oxygen rich situation. Case 3. HI(T) > H2(T) over [T~an, T ~ j : sequential flames R1 is much faster than R2 over the whole permissible range of final temperatures. To determine allocations uniquely, we introduce the following physical principle: Maximum Allocation Principle. I f H j ( T ) > Hi(T) over some permissible range of final temperatures, then from all possible allocations, a stable chemical pathway maximizes aj for reaction Rj at the expense o f t~i. I n this case, maximizing ~l for Rl uniquely yields 0q=a,
ot2 = b - a ,
H1
.
) J T1
I
I
!
f
Tmi n
Fig. 4. Case 3.
Tma x
P.C. Fife and B. Nicolaenko / Flame fronts with complex chemical networks
192
i.e., the rightmost point on the interval of possible allocations (fig. 2). Moreover, T÷ - Tmi~--- T + aQ~ + (b - a)Q2, and Y+ = (0, 0, 2a - b). Since at T~., R~ is still much faster than R2, the situation is similar to case 1. R1 is completed in a f a m e sheet (1) at Tt such that HI(T1)
H2( Tmin) ,
=
while R 2 is completed in a'flame sheet (2) at T~n. (Again, we assume that 7", > T1 = T_ + aQv) Case 4. H i ( T ) > HI(T) over [T~,, Tm~]: pseudoreaction ~l R2 is much faster than RI over the whole permissible range of final temperatures. Applying the Maximum Allocation Principle to R2 maximizes ~2: ~z = b /2,
hence
~,
~ is uniquely determined by (2.17), T , - T+ = T_ + ~1Q1 + (b - ~,)Q2 • In this fuel rich context, there is genuine competition between the subnetworks R1 and R2 for the initial oxygen concentration. The layer analysis at T . is entirely similar to the canonical example 1.2 of competing reactions [7], hence the genericity of the latter. To insure stability of this competition fame, we need H2(T,~i.) > H,(T~.)
and Hl(Tmax) > H2(Tmax). The reasons will be made clear by the next case.
= b /2 ,
i.e., the leftmost point on the interval of possible allocations (fig. 2) and T+ =_ Tm~x= T_ + b /2 x (QI + Q2), Y+ = (a - b/2, 0, 0). Since at T÷ = Tm~x, R2 is still much faster than RI, the flame configuration is similar to case 2" pseudoreaction P-I at Tmx, with instant heat release Q1 + Q2-
Case 6. A non-uniqueness case This is again the case of intersecting power curves at T,, but now
Case 5. Genuine (stable) competition flame Here, there exists a unique T., Trmn< T . < /'max, where both power gauge functions are equal,
H~(T~) < ~(T~).
H1(T,)
=
and
We have three flame configurations possible: a competition f a m e at T , (as in case 5); sequential
H2(T,) ;
~
H1¸
H2
I
"2f 11 Tml n
J / ; I a2 I
T, Fig. 5. Case 5.
J
Tma x
,
"1"
Tmin
I
I
I
!
T.
Tmax
Fig. 6. Case 6.
T
P.C. Fife and B. Nicolaenko~Flamefronts with complex chemical networks
flames with R~ completed at T~n (as in case 3); or a pseudo-reaction ~, with a merged single flame layer at Tm~ (as in case 4). We conjecture that the competition flame at T . is unstable for reasons given in [7]; and that both flames at T~, and Tm~xare possible, depending on initial transient regimes. The previous analysis has been done with the assumption b /2 < a < b, to simplify the algebra. The same cases occur for a > b.
3. Extending the method to general networks
The detailed study of the hydrocarbon model evidences several features generic to complex networks: • depending on whether the initial mixture is fuel rich or fuel lean, the same network can exhibit tandem (sequential) flame sheets, or genuine competition flame sheets; or merged flame sheets dominated by a single reaction (the other ones being quasi-instantaneous); • for competition flames to occur, we need not only power gauge functions to intersect at some common temperature, but also a whole range of possible allocations; this is possible only if some deficient reactant is common to the two competing subnetworks; • for some power gauge curve configurations, three flame regimes are a priori possible; such non-uniqueness can only be resolved by a dynamical stability analysis, and is reminiscent of nonuniqueness and hysteresis in isothermal chemical reactors. With these points in mind, we will survey the extension of the method to general networks (without reversible reactions). Remark 3.1. We consider the case of three power gauge functions intersecting at the same T . as non -generic. Remark 3.2. For networks with reversible reactions, another type of flame sheet can appear,
193
where the sheet temperature locks onto the equilibrium temperature of some reversible reaction [9]. 3.1. The hierarchy o f power rate functions Let an allocation be given: at = (at,, at2,.-., at,)
(3.1)
and T+(o0 = T_ + ~ at~Q~
(3.2)
i=1
Our basic principle is to construct a temperature sequence T1, T2. . . . .
T,
such that power gauge functions are equal along this temperature sequence (fig. 7). Specifically, 1) set Tr = T÷(a); 2) pick the reaction Rm which has the minimal power at T÷: Hm(T+)=minHi(T+), i = 1 . . . . . r, ati ~ 0 ; (3.3) 3) define the T~ by H,~T~)= Hm(T+), l <<.i < r, i v~ m .
There exists a, unique temperature sequence {T~} satisfying (3.3) for a given at; this is ensured by remark 1.1. We now reorder the reactions in order of increasing T~, so that
~...~E=~,
(3.4)
where the new T, corresponds to the former reaction Rm. The new sequence {T~} defined in (3.4) equates the power rates of the reactions. It induces a hierarchy within the reactions and generates a stable chemical pathway in the flame profile. It is entirely possible that some of the T~ coincide, corresponding to competition flame sheets (cf.
194
P,C. Fife and B. Nicolaenko / Flame fronts with complex chemical networks
Hi(T) I
Hr-1 (T) Hr (Z)
I
I Hr-1 (T)
l
mr(T) / '
I .....
~
l
I
I,,,
Ti
Tr_ I
I1
I T+(~) = Tr
Fig. 7. Temperature sequence such that power gauge functions are equal along this sequence.
eases 5 and 6, section 2.4). Fig. 8 shows such a configuration. Here the possible allocations are a priori nonunique, with an allowable range for • on the boundary of the convex set of allocations. T÷ and • are actually determined by setting T+ = T,, intersection of the power gauge curves of R, and R,_t, the two reactions with the smallest power gauge functions over the interval [T,~,, Tm~] of possible final temperatures.
To cover such a situation, let {7~i}, i = 1 , . . . ,p <~ r be the possibly shorter sequence of distinct temperatures among the T~ in (3.4):
(3.5)
f,
we denote their number by p. It also may happen that two of the Tj almost coincide, that is, 0 < IT~ - Tjl = d9(8i). This case is /
I 1 1
Hr-2
i I t
I
~-z
_.
n
I
I
Tr_ 2
,
Tmin
Fig. 8. Competition flame sheets.
I
T+=T,
j
........ I
Tma x
~_
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
considered non generic, and is hereby excluded from consideration. Thus, we assume that two temperatures T,, T~ either coincide (this may be required by the feasibility criterion given later) or differ by a quantity of larger order than ~; + ej. We also assume that at each ~, all but perhaps two of the quantities H~(~) are distinct. Our attempt will be to construct a combustion front in which, roughly speaking, p sets of reactions go to completion at the temperatures ~ in that same order as one proceeds across the front's profile in the (upstream) direction of increasing temperatures. For that purpose, we use the notion of suballocation.
to be maximal. With it known, ~ 0 is then maximal, etc. In this case, ~(0 is uniquely determined. Next, suppose two of the H~(~) coincide, say H~,(Tt) = Hk2(~)- Then instead of maximizing ~ , we maximize ~0 + ~0. In this respect, we generalize the principle introduced in section 2, cases 3 and 4. For such suballocations, kt sets of reactions go to completion at ~t. We can now determine the temperature at which the suballocation ~(0 would bring the corresponding subnetwork to equilibrium, if no reaction outside that subnetwork were present. We call it
t= T Definition 4. For each 1 ~ kl. We say ~o ~< ~r) if ct~o ~< ~t~r~ for all i. The situation we envisage is when there exist suballocations ~(0 for all 1 <~p satisfying l=1 ..... p--l;
~,6o)=~.
kl
+ E ~OQ,.
(3.7)
i=1
3.2. Suballocations to subnetworks
~,(,)<~o+,),
195
(3.6)
There is a further condition which we shall require the ~(0 to satisfy which is a generalized maximum allocation principle: Let us fix 1 ~
FLAME FRONTS WITH COMPLEX CHEMICAL NETWORKS Paul C. FIFE Department of Mathematics, University of Arizona, Tucson, AZ 98721, USA
and Basil NICOLAENKO Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
For laminar flame fronts with general complex chemistry networks, we present a systematic method to reduce the detailed chemistry in the limit of high activation energy asymptotics. We detail the method on a global multistep hydrocarbon model. Generic flame profile structures are discussed, and extensions to networks with zero activation energies are outlined.
1. Introduction
Complex chemistry in flames involves a mix of reactions with both high and small activation energies. Since the pioneering works of FrankKamenetskii [11] and Zeldovich [16, 17], it has been recognized that large activation energies Ei determine a hierarchy of scales
RT+ T+ el= Ei T + - T _ '
ei~ 1 ,
where T is the cold mixture temperature, and T+ a typical adiabatic flame final temperature. The 8i are the inverse of the (large) Zeldovich numbers. In this paper we develop the asymptotic flame theory with complex chemistry first outlined in [7-8]. We specifically consider the case of finite ratios Ei/Ej # 1 for the subclass of High Activation Energy (H.A.E.) reactions. Cases where E~~ Ej, but the pre-Arrhenius factors are not the same order of magnitude, are also covered by our theory.
We first explain concepts which are common to
all reactions networks: admissible reactant allocations, which yield the possible final burned states
of the flame; and admissible reactant suballocations to the network, which determine feasible chemical network pathways as one passes from the unburned state to the burned state through the front (section 1.2). The latter correspond to subgroups of reactions which go to completion at temperatures lower than the final one. The notion of set of admissible allocations (section 2.2) turns out to be a convenient tool. Next, to determine the actual pathways within the network and along the flame profile, we construct normalized power gauge functions (section 1.3) for both H.A.E. and Z.A.E. (Zero Activation Energy) reactions. Unnormalized power rate functions have already been used implicitly by FrankKamenetskii [11] and Lifian [12]. We derive systematic comparison principles based on these gauge functions. In section 2, we detail our method on a global hydrocarbon two-step model. We evidence some generic flame structures present in complex networks: merged flames, competition flames and tandem flames. In section 3, we outline the methodology for general networks
0167-2789/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
183
1.1. The flame front equations for a chemical network
Example 1.1. Consider the network of sequential reactions
Let n chemical species Al, A2. . . . . An be involved in r reactions Rl, R2. . . . . R,; symbolically
Rl : Ai ~ A 2 R2 : A z ~ P
(P inert product)
n
v(j, i)Aj~ ~ ~.(j, i)Aj, j=l
i = 1. . . . . r.
(1.1)
j=l
where v(j, i) (resp. 2(j, i)) is the stoichiometric coefficient of species j on the left side (resp. right side) of reaction i. Both v and 2 are nonnegative integers; vij, i) is zero if species j is not a reactant in reaction i, while 2(j, i) is zero if it is not a product. Each reaction yields heat releases Q1, Q2. . . . . Q,. The flow field is described by the temperature T and the vector of rescaled mass concentrations (fractions): (1.2)
Y = ( Y~, Y~. . . . , Yn) ,
where the rescaled Yj, j = 1. . . . . n, are the actual mass fractions (concentrations) of species j divided by the corresponding molecular mass mj (this rescaling somewhat simplifies the flame front eqs. (1.6). The full state vector U is U = (T, Y) = (T, YI . . . .
, L).
(1.3)
Each reaction can be characterized by an ndimensional reaction vector ~, i = 1 , . . . , r, which represents the net change in the number of molecule grams (moles) of various species in reaction
then Y = (Y~, Y2), U = ( T , Y], Y2), Y] = ( - l , + l ) , F2 = (0, - 1), K~ = (Q,, - l, + 1),/(2 = (Q2, 0, - 1). Example 1.2. Consider the network of competing reactions RI : A l ~ P t (P~, P2 are both inert products) R2" A l ~ P2 then U = (T, Y), /(1 = (QI, - 1),/(2 = (Q2, - 1). With the above vector notations, the equations for a one-dimensional traveling flame front, in a system of coordinates moving with the front, and in the constant pressure approximation, reduce to the system of (n + 1) equations in T, Y I , . . . , Yn:
- (D(U)Ux)x + MUx = ~ og,(U)K,.;
(1.6)
i=1
where D ( U ) is a (n + 1) x (n + 1) diffusion matrix (positive definite;) M is the unknown mass flux, which equals wave velocity if the density is unity. The scalar functions ~o~ are the rates of the reactions R,.. The system 1.6 is closed by the perfect gas law,
K: V/= (2(I, i) - v(1, i), 2(2, i) - v(2, i ) , . . . , 2(n, i) - v(n, i)) = ( Z ~ I), VI 2). . . .
, Fit)) •
(1.4)
The extended (n + 1)-dimensional reaction vector K~, i = 1 , . . . , r, is defined as = (Qi, V3;
(1.5)
where p (x) is the mixture global density and/~c the constant pressure in the isobaric, small Mach number limit. Finally, the cold state u- = u(-oo)
= (r_, Y? .....
Y;)
(1.8)
is prescribed at x = - 0 o . The reaction rates o9i are given by the usual law
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
184
of mass action and Arrhenius factors,
t°i=[]YY'°B'exp(-~)
(1.9)
where E~ is the activation energy of reaction i and B,. the corresponding pre-Arrhenius factor; strictly speaking, B~ is a slowly varying function of T and p, hence of T and Yj through (1.7); we will in the sequel neglect its dependence w.r.t. Yj, taking for instance the context of a reacting mixture diluted in some inert species. In this presentation, we omit reversible reactions. The extension to reversible reactions has been completed and will be presented in [9]. We are concerned with exploiting the high activation energies E~ to obtain approximate wave front solutions. The Ei determine scales T+
-
T+
-
ei= Ei T + - T_
~ 1,
(1.10)
where T÷ is a typical final (burned) temperature and the ~ are inverse Zeldovich numbers (T has been renormalized so that the gas constant is unity). The sharp sensitivity of Arrhenius terms exp( - EJT) w.r.t, variations of T enable a systematic High Activation Energy (H.A.E.) asymptotic analysis. The procedures are linked to properties of the underlying reaction network, which supplies the vectors Kj. These procedures are well known in the case when the network is modeled by a single reaction A-~P. (See [2] and the review article [3] in these proceedings.) An approach similar to the one in this paper was used in treating two-reaction networks of competing, sequential, and competing-fuel types in [7] and [8], and it was clear that the extension of this approach to more complex schemes could uncover a baffling array of possibilities. A systematic formalism was called for, and hopefully was provided in part by the authors in [8]. The methods are those of asymptotic analysis. Roughly speaking, to the lowest order this means developing rational procedures for replacing the
original problem with a limit problem in which the e~ do not appear explicitly. The limit problem is expected to yield an approximate solution, and is easy to solve. In the limit, flame sheets appear as free boundaries with Ucontinuous across, but with jumps in the gradients. A rigorous justification of this expectation has been given recently for the case of a single reaction [1]. Reactions are concentrated on flame sheets (internal layers) with pure convective-diffusive flow in between, if there are no reactions with E~ = 0:
(D(V)Ux)x - MUx = 0 between the flame sheets. The flame sheets correspond to a Dirac-function in temperature limiting behavior of the o~ as ei~0. If there is a mix of reactions with H.A.E. Ei >> 1, i = 1 , . . . , p and with zero activation energies (Z.A.E.) E~=0, i = p + 1. . . . , r, there are still flame sheets (free boundaries) associated to the R~, i = 1 , . . . , p ; but the flow in between is convective-diffusive-reactive due to the Z.A.E. reactions. The latter "thicken" the wave front profiles. Many other parameters besides e~enter the problem, for example, the elements of the matrix D and of the unburned state vector of the gas, U-. Throughout the paper, we assume that their magnitudes do not interfere with the asymptotics. Particularly, we assume that any constants not depending explicitly on e;, B~, or E~ (as defined in (1.9) and (1.10)) are t~(1) quantities. This especially refers to elements of D and D-~, reference dimensionless Tj, their differences T~- Tj and the heat releases Q~. These restrictions are relaxed, and a more meticulous account of the other parameters is taken in [10]. For background and previous work on asymptotic methods in flame theory, the reader is referred to [2] and [3] in these proceedings. A number of papers have been written on the multiple reaction case; see the references in [2], [4], [12], and [17]. We remark that the consideration of realistic chemical networks avoids the so-called "cold boundary difficulty" [2], where U- is not an equi-
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
librium state of the chemical network. In realistic chemical networks, the initial zero concentrations of intermediate radicals species insures equilibrium at the cold side; a good example is the autocatalytic chain network investigated by Lifian [12]: R] : A1 + A2-*2A2
(1.11)
vectors lies in the linear span of the extended reaction vectors K~. Moreover, the ct~, i = 1. . . . . r, must satisfy constraints such that the final concentration vector Y+ ensures that the network is in equilibrium: Y+ = 0 =
Y ; + ~, o~iV~k)
where P and M are inert; the initial concentration of the intermediate radical A2 is null. Here, K1 = (01, - 1, + 1) and/(2 = (02, 0, - 2 ) .
1.2. The reactant allocations and suballocations to a chemical network The final state on the burned side
for some integer subset {k} such that o~,(U+)= 0, Vi. Because of (1.14) and (1.15), M must be considered as unknown nonlinear eigenvalue for the general system (1.6). We will call the sequence of positive or null numbers ~ , ~ . . . . . ~, an allocation of the initial concentration vector Y- to the network. To make this point of view more systematic, we first need a basic assumption about the reaction vectors:
(1.12)
is actually coupled with the initial state U through conservation laws which are the equivalent of the Rankine-Hugoniot conditions for nonreactive flows. To see this, integrate (1.6) from x=-oo tox=+~,use Ux=Oatx=+~ to obtain
M(U + - U - ) = ~ 12iK~,
(1.15)
i=l
R 2 : 2A 2 + M ~ 2 P + M ,
U + = U ( + ~ ) = U(T+, Y + , . . . , Y+)
185
(1.13a)
i=1
where 12,.is the integrated rate function for R~ along the flame profile (proportional to the thermal power due to R;):
Assumption 1. If ~Vi=O
and ~>~0, then a ~ - 0.
(1.16)
i=1
This assumption assures, as we shall see, that equilibrium can only be reached when each reaction rate is zero. It can easily be removed to cover reversible reactions [9].
Definition 1. Y E 8 is an equilibrium state if, for all
T, (o,(Y, r)v~ = 0.
q-oo
(1.17)
i=l
f2i = f og(U(x))dx ;
(1.13b)
now setting ~; = 12~/M, U + - U - = ~ ct,K~,
(1.14)
i=l
hence the difference between the asymptotic state
It follows from (1.16) that this can only happen if each (]-)i---0. For each i = 1. . . . . r, we denote by J(i) the set of all indices j of species Aj entering reaction Ri (appearing on the left side of the arrow). At this point we assume toi = 0 if and only if Yj = 0 for some j ~ J(i). We symbolize the equilibrium cone for each reaction, and the total equilibrium cone,
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
186
as follows: g, = { Y >I 0: Yj = 0 for some j ~ Y(i)},
i=l
(This is, of course, a cone imbedded in a set of coordinate hyperplanes.) For simplicity only we require
Assumption 2. Vi¢ gi, for all i. Definition 2. Given an input state Y->~ 0, an allocation of Y- to the network is a set of numbers 0t~t> 0, i = 1. . . . , r, such that
Y-+ ~ O~iE~¢.
(1.18)
i=l
Example. Consider the network of competing reactions described in Example 1.1. Clearly 8 = {0}. The T and Y~ components at eq. (1.14) are T+ = T_ + 0qQl + 0t2Q2; 0 = Y+ = Yi- - 0q - 0t2; but each ~ti can take values only between zero and Yf- and their sum must be Yi-- Setting oq = OY{, 0<0<1, we have ot2 = ( 1 - 0 ) Y ; - and T + = T_ + YI-(OQ1 + (1 - O)Q2). The final temperature can be any temperature between 7"1 = T_ + Y I Q I , adiabatic temperature of a pure reaction R1, and T2 = T_ + YIQ2, adiabatic temperature of a pure reaction R2. Indeed this network of competing reactions is the canonical example of nonuniqueness of allocations. In more complex networks, there is usually a whole array of possible allocation vectors which are restricted to the boundary of a specific convex set (simplex) defined by
(see section 2 for examples of such convex sets). In general, we must consider the final states U + ( a ~ , . . . , ~,) and T+(~q. . . . , ~t,) as functions of the a~. Nonunique allocations reflect the possibility of different chemical pathways in the network. Another consideration is that experimental and numerical simulations show that within a flame profile, there is a sequence of temperatures where some subgroups of reactions are completed, while some other reactions are still frozen [13]. For instance, in heavy hydrocarbon flames [13] peak concentrations of a typical radical occur at a distinctly different temperature than the temperature band where most of the fuel is consumed [14]. This leads to the notion of suballocations (to be developed in sections 2 and 3) such that a partial subgroup of reactions go to completion at temperatures lower than T+, within a partial subnetwork.
1.3. Normalized power functions The actual chemical pathways and chemical subnetworks in a flame are determined by the competing rates to~; the latter can vary sharply not only in absolute but also in relative magnitude as temperature increases across the flame front. Hence the need for some normalized gauge functions to systematically exploit these relative variations. Consider now the integrated rate functions fJi in eq. (1.13):
f2i= f co,(U(x)) dx ;
writing the first component in 1.13a yields the exact relation
M(T÷ - T_)= ~ Q,t2,,
(1.21)
i=I
oci>~O, i = l . . . . , r (1.19) j~-~- l ~ . . . ~ r l . i=l
(1.20)
i.e., the relative contribution of each reaction to the temperature profile is its heat release Qi weighted
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
by its integrated rate function. But if the reaction Ri with high activation energy E; is isolated then classical H.A.E. analysis for a single step reaction [2] going to completion at temperature T, gives
t2,(T) = C,e,(B~exp - EJ T) '/2 ,
(1.22)
where (7,- is some proportionality constant. Also, the rate function o9i is concentrated in the flame sheet. For networks with partial H.A.E., the mi for the H.A.E. reactions are still concentrated in a sequence of flame sheets (possibly merged) at various temperatures and it can be shown with a layer analysis that the g2~ are still of the order of magnitude given by (1.22). Rather than (1.22), we shall use the following choice for normalized power gauge functions H,(T) to compare the relative powers of reactions R~:
Definition 3. Power gauge functions: H,( T) = ~[ln B, - ~ I , = min ~ = min T+
(1.23a) T+
Ei T + - T _ '
i=l,...,r. (1.23b)
In (1.23a) we allow T t o vary, since the flame sheet temperatures are a priori unknown (nonunique allocations). We also allow the same definitions (1.23a-b) whether reaction Ri is H.A.E. or Z.A.E.
Indeed 1) if Ri is H.A.E., then
Hi(T) = g(21n Oi + tP (In ~)), which ensures that the variation of H,(T) over the whole flame temperature profile is 0(1); 2) if Ri is Z.A.E., then Hi = g In B; ~ constant, as B,(T) varies weakly with T; nevertheless we should keep in mind that physical Bi may be very
187
large indeed, anywhere from 10ll to 1019; SO the normalized Hi are still O(1) or smaller.
Remark 1.1. The power gauge functions Hi(T) are monotone increasing in T. Any linear combination of H,(T) is a monotone (decreasing or increasing) function of T. Our basic comparison principle is based on the following inequalities: if rip(T) >
HdT)
at a given T, and the difference is d~(1), then Op(T) >>Qq(T)
and Rq has negligible effect compared to Rp, at temperature T. An equilibrium configuration might still be obtained for some distinguished temperatures Tp and Tq such that Hp(Tp)= Hq(Tq) and t2p(Tp)/Oq(Tq)= t9(1). We will detail the systematic use of the comparison principle to construct flame fronts in sections 2 and 3. As a simple example, we show how the power gauge functions resolve the nonuniqueness of allocations in the competing reactions network (example 1.1). Recall that T÷ can vary anywhere between TI and T2 (single reaction adiabatic temperatures). Let E1 >> 1, E2 >> 1 and El~E2 =O(1); three cases are possible: 1) HI(T)> H2(T) for all T E[T l, T2]; then R 1 globally quenches R~, and there is a single flame sheet at T~; the allocation is (Yi-, 0); 2) H2(T) > HI(T) for all T ~ [Tt, T2]; then there is a single flame sheet at/'2, with allocation (0, Yi-); 3) HI(T.) = H2(T,) for some T., TI < T . < 7"2; T. is unique because of remark 1.1, and the two reactions are at equilibrium at the flame sheet temperature T÷ = T,. With T , = OT1 + (1 - O)T:, 0 < 0 < 1, the allocations are a~ = 0 Yi-, a 2 = ( 1 - O ) Y £ . The full layer analysis of this competition flame has been done by the authors in [7]. In section 2, we show that competition type flame regimes can in fact be generic for more complex networks, due to competing subnetworks.
188
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
Finally extending formula (1.23) to Z.A.E. reactions ensures that our comparison principle overlaps classical pseudo-steady state analysis: if Hp(T) > Hq(T) for Rp and Rq both Z.A.E.. then for all temperature ranges top >>toq (as Hp and Hq are nearly constant) and the usual pseudo-steady state reduction of Rp can be done.
RI:C, Hm+ ~ + ~
O2~nCO+
R 2:CO +~O 2~-CO 2,
H20,
(2.2)
with empirically derived reaction rates for both reactions; for R2: to2 ~--~-10146exp(-R4--~0T)[CO]l[H20]°5[O2]0'25 (2.3a)
2. A global hydrocarbon two-step model and for Rl:
2.1. The two-step Dryer-Westbrook model Hydrocarbon flames are usually modelled by global one-step models: Fuel + nl O:~n2 CO2 + n3 H20,
(2.1a)
where nl, n2, n3 are positive integers, and the global reaction rate to is t o = B ( T ) e x p ( - ~ T ) [Fuel]~[O2]#;
(2.1b)
the pre-Arrhenius factor B and the exponents and fl are usually adjusted to fit calculated flame parameters to some experimental data; ~t and fl turn out to be positive fractional numbers. As pointed out in [15], the single-step model predicts flame speeds reliably over a considerable range of reactions; but it systematically overpredicts the total heat of reaction and the final adiabatic temperature. Moreover, at temperatures characteristic of hydrocarbon fuels ( ~ 2000 K), substantial amounts of CO and H2 exist in equilibrium in the combustion products with CO2 and H20. It is also well recognized that typical hydrocarbons burn in a sequential manner; i.e., the fuel is partially oxidized to CO and H2, which are not appreciably consumed until all of the hydrocarbon species have disappeared [6]. From these observations, and following a similar model for methane by Dryer and Glassmann [5], Westbrook and Dryer [15] have proposed the following global two-step reaction mechanism:
to~ = Bl exp( -- R ~ ) [Fuel]~[O2]#.
(2.3b)
For Octane CsHls, B l = 5 . 7 x l0 II, E~=30.0 kcal/mol, 0t = 0.25 and fl = 1.5. In general, for all hydrocarbons except methane CH4, E 1 ~ 30 kcal, and B l ,~ B2 = 1014.6. So
Remark 2.1. For the global two-step model (2.2), the power gauge functions HI(T ) and H2(T) always intersect at some T , such that
,ln
)'
(2.4)
since BE >>B1 and E2 - E1 ~ 10 kcal. Competing flames and subnetworks are possible. In the sequel, we systematically investigate the simplified network R1 : Fuel + Oxygen~R + Pl, R2 : R + Oxygen-*P2,
(2.5)
where P1 and P~ are inert products (e.g., H20 and CO2) and R is an intermediate "radical" species (e.g., CO). We take Yf, Yox and Yr to be the respective (rescaled) concentrations of fuel, oxygen, and radical. We take the simplified extended reaction vectors (stoichiometric coefficients set equal to one): g~ ---(Q,, -I, -I, +I), (2.6) K~ = (Q2, o,
-
l, - I),
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
with both Qi > 0. Notice the alternation + 1 for the last component, as the Radical is produced by R,, but consumed by R2. The flame front equations for the state vector U---(T, Yf, Yox, Yr) are (DfU)Ux) x + MUx = o)I(U)K1 -Jr o)2(V)K2 , (2.7)
-
with the reaction rates (-/)l = BI YfYox exp(
-
092 = B2 Yox Y, exp( -
EUT), E21T),
E2>>l,
allocations in the positive quadrant ~t, ~2; but we must require that equilibrium be attained. Since the reactions are not reversible, chemical equilibrium corresponds to some points (~, ~2) on the boundary o f this simplex. However, equilibrium for the chemical network can be reached in two different ways: first both Yi ~ = 0 and Yr+ = 0 , which corresponds to an oxygen rich initial mixture. Or Yo+x= 0, which corresponds to a fuel rich mixture. More precisely:
(2.8) 1) In the oxygen rich case (fig. 1), a < b12 and Y~- = a - ~1 = 0, Y : = ~l - ~2 = 0, so
and E~>>I,
189
(2.12)
oq = ~ 2 = a = Y f ;
E2 d~(1). E =
the final state is The initial state is U - = (T_, a, b, 0),
(2.9)
with Y~-= a, Y~x = b and Yr = 0 (which is the physical situation). Notice that we do not assume reversibility of R2; the reversible case is fully treated in [9].
Y~=0,
(fuel depletion),
Yo+x=b-2a >O,
(excess oxygen),
Yr+ = O, (the radical concentration pulses to a maximum and then decreases to zero); T + = T- + a(Ql +
2.2. The convex set o f admissible allocations Let ~ and ~2 be the allocations to RI and R2 with at = (~l, ~2); possible final states are
(2.13)
Q2).
(2.14)
For the oxygen rich case, notice that the admissible allocation (a, a) is unique (edge o f the convex set). The case a = b/2 corresponds to an initially stoichiometric concentration Y-.
V + ( a t ) = U - + c t l K l -'1-o~2K2,
i.e. T+(1) = T_ + ~IQ~ + ~2Q2,
(2.10)
Y~" = a - - tXl~> 0 ,
Y+ox= b - - ~ 1 - ix2 ~> 0 ,
(2.11)
Y~+ = ~1 -- at2/> 0; inequalities (2.11) together with the conditions • l ~> 0, ~t2 I> 0 define the convex set o f eventual
"t U
I I
a
b/2
",,u
Fig. 1. The oxygen rich ease, a < b/2.
It--
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
190
2) In the fuel rich case (fig. 2), a > b / 2 , Yo+,-- b - 0q - ~t2= 0; since a2/>0, we have b /2 <<.a~ <~b, in fact:
b /2 <~ctI <~rain(a, b) ;
a2
(2.15)
the final allocations are b/2
Y/~ = a - ctl,
(excess fuel),
Y~ = 0,
(oxygen depletion),
i
t
d,
]
,
(2.16) /
Y+ = 2al - b.
I
I
b/2
There usually is some residual radical concentration on the burned side of the flame. The key feature of the fuel rich case is that allocations are not unique: (~t, ~2) covers a whole closed interval on the line ~ + ~t2= b, with (2.15) and b - min(a, b) ~
(2.17)
to fix ideas, take Q 2 2 Q 1 and a<~b, so min(a, b) = a; then T+(a0 is a decreasing function of ~ , which reaches its minimum T~, at ~q = a and its maximum T ~ at 0~ = b/2,
T~. = T_ + aQl + (b - a)Q2, (2.18)
Tma~ = T_ + b /2(Q~ + Q2).
x
!
N,
a
b
cL1
Fig. 2. The fuel rich case, a > b/2.
and H2(T) at T = T+. We consider the case H~(T+) = H~(T+) + ¢(g) as nongeneric; and in fact, the inequalities introducing cases 1-6 are meant to imply that the two sides differ by a quantity > ~(~).
Case 1. Hi(T+)> H2(T+): Sequential flames R1 would tend to dominate R2 at T÷; what we mean is that if R1 went to completion at T÷, then R2 would have negligible rate in comparison, and we would appear to have a one-step reaction. This would contradict the fact that both reactions must operate in order to achieve T+. A stable flame configuration exists only if R1 is completed in some flame sheet of temperature TI < T÷, such that the
2.3.Flame configurations in the oxygen rich case (a <<.b/2) This is characterized by a unique final temperature (2.14),
T+ = T_ + a(Ql + Q2) and a unique final allocation at = (a,a). Flame configurations are determined by the relative ordering of the power gauge functions (1.23) H~(T)
.....
~2
I TI
....
I T+
Fig. 3. Case 1.
~-
T
191
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
power H~ is lowered to a power equilibrium configuration, H,(T1) = H2(T+) ; such a temperature TI exists and is unique because of remark 1.1. However, to be physically attainable, Tt must be larger than the adiabatic temperature T_ +aQl = / ~ associated with RI in isolation [7, 8]. For simplicity, we assume that is so; if not, total equilibrium is only attained in stages (2 successive flames travelling at different velocities), rather than in a single steady flame. The flame configuration is sequential with R1 completed in a flame sheet (1) at Tx, where all the fuel is burned out; this is followed by a flame sheet at T÷, where R2 consumed all the radical created by the sheet (1). The temperature sequence (T l, T÷) equates the power rates and induces a stable chemical pathway along the flame profile: first R1 with R2 frozen, then R2 with Ri completed. We remark that there is a suballocation t~0) = (a, 0) to the flame sheet (1), and a final allocation ~(2)= ~ : (a, a) to the sheet (2). Remark 2.1. In practice the flame profile will be thickened because of Z.A.E. involving other radicals. Remark 2.2. The principle of determining a sequence of temperatures (possibly equal) to equate all the power gauge functions Hi, systematically induces stable chemical pathways and subnetworks in general complex networks. Case 2. H2(T+)> HI(T+): Pseodoreaction ~t RE is now much faster than RI at T+. We might think of R2 first taking place at some 7"2< T+, such that H2(T2) = H~(T+), followed by R~ at T+. However, the corresponding suballocation would be (0, 0); that is, such a separated flame sheet cannot exist, as there is no initial radical concentration (Y;- = 0) to trigger it. The only possible structure is a single flame layer at T+, with a pseudo-reaction
1~1 characterized by the same rate function o91, but with an instant heat release QI + Q2. The radical produced by i1[1 is instantly burned by. R2 within the same flame layer. Strictly speaking, this is a situation with merged layers, but to lowest order, R2 does not affect the layer analysis at T÷. 2.4. Flame configurations in the fuel rich case (a > b/2) Since allocations are not unique (fig. 2), we must compare 1t1(7') and H2(T) over the whole range [T~i~, T ~ ] as defined in (2.17)--(2.18). Hence, a larger variety of possible configurations than in the oxygen rich situation. Case 3. HI(T) > H2(T) over [T~an, T ~ j : sequential flames R1 is much faster than R2 over the whole permissible range of final temperatures. To determine allocations uniquely, we introduce the following physical principle: Maximum Allocation Principle. I f H j ( T ) > Hi(T) over some permissible range of final temperatures, then from all possible allocations, a stable chemical pathway maximizes aj for reaction Rj at the expense o f t~i. I n this case, maximizing ~l for Rl uniquely yields 0q=a,
ot2 = b - a ,
H1
.
) J T1
I
I
!
f
Tmi n
Fig. 4. Case 3.
Tma x
P.C. Fife and B. Nicolaenko / Flame fronts with complex chemical networks
192
i.e., the rightmost point on the interval of possible allocations (fig. 2). Moreover, T÷ - Tmi~--- T + aQ~ + (b - a)Q2, and Y+ = (0, 0, 2a - b). Since at T~., R~ is still much faster than R2, the situation is similar to case 1. R1 is completed in a f a m e sheet (1) at Tt such that HI(T1)
H2( Tmin) ,
=
while R 2 is completed in a'flame sheet (2) at T~n. (Again, we assume that 7", > T1 = T_ + aQv) Case 4. H i ( T ) > HI(T) over [T~,, Tm~]: pseudoreaction ~l R2 is much faster than RI over the whole permissible range of final temperatures. Applying the Maximum Allocation Principle to R2 maximizes ~2: ~z = b /2,
hence
~,
~ is uniquely determined by (2.17), T , - T+ = T_ + ~1Q1 + (b - ~,)Q2 • In this fuel rich context, there is genuine competition between the subnetworks R1 and R2 for the initial oxygen concentration. The layer analysis at T . is entirely similar to the canonical example 1.2 of competing reactions [7], hence the genericity of the latter. To insure stability of this competition fame, we need H2(T,~i.) > H,(T~.)
and Hl(Tmax) > H2(Tmax). The reasons will be made clear by the next case.
= b /2 ,
i.e., the leftmost point on the interval of possible allocations (fig. 2) and T+ =_ Tm~x= T_ + b /2 x (QI + Q2), Y+ = (a - b/2, 0, 0). Since at T÷ = Tm~x, R2 is still much faster than RI, the flame configuration is similar to case 2" pseudoreaction P-I at Tmx, with instant heat release Q1 + Q2-
Case 6. A non-uniqueness case This is again the case of intersecting power curves at T,, but now
Case 5. Genuine (stable) competition flame Here, there exists a unique T., Trmn< T . < /'max, where both power gauge functions are equal,
H~(T~) < ~(T~).
H1(T,)
=
and
We have three flame configurations possible: a competition f a m e at T , (as in case 5); sequential
H2(T,) ;
~
H1¸
H2
I
"2f 11 Tml n
J / ; I a2 I
T, Fig. 5. Case 5.
J
Tma x
,
"1"
Tmin
I
I
I
!
T.
Tmax
Fig. 6. Case 6.
T
P.C. Fife and B. Nicolaenko~Flamefronts with complex chemical networks
flames with R~ completed at T~n (as in case 3); or a pseudo-reaction ~, with a merged single flame layer at Tm~ (as in case 4). We conjecture that the competition flame at T . is unstable for reasons given in [7]; and that both flames at T~, and Tm~xare possible, depending on initial transient regimes. The previous analysis has been done with the assumption b /2 < a < b, to simplify the algebra. The same cases occur for a > b.
3. Extending the method to general networks
The detailed study of the hydrocarbon model evidences several features generic to complex networks: • depending on whether the initial mixture is fuel rich or fuel lean, the same network can exhibit tandem (sequential) flame sheets, or genuine competition flame sheets; or merged flame sheets dominated by a single reaction (the other ones being quasi-instantaneous); • for competition flames to occur, we need not only power gauge functions to intersect at some common temperature, but also a whole range of possible allocations; this is possible only if some deficient reactant is common to the two competing subnetworks; • for some power gauge curve configurations, three flame regimes are a priori possible; such non-uniqueness can only be resolved by a dynamical stability analysis, and is reminiscent of nonuniqueness and hysteresis in isothermal chemical reactors. With these points in mind, we will survey the extension of the method to general networks (without reversible reactions). Remark 3.1. We consider the case of three power gauge functions intersecting at the same T . as non -generic. Remark 3.2. For networks with reversible reactions, another type of flame sheet can appear,
193
where the sheet temperature locks onto the equilibrium temperature of some reversible reaction [9]. 3.1. The hierarchy o f power rate functions Let an allocation be given: at = (at,, at2,.-., at,)
(3.1)
and T+(o0 = T_ + ~ at~Q~
(3.2)
i=1
Our basic principle is to construct a temperature sequence T1, T2. . . . .
T,
such that power gauge functions are equal along this temperature sequence (fig. 7). Specifically, 1) set Tr = T÷(a); 2) pick the reaction Rm which has the minimal power at T÷: Hm(T+)=minHi(T+), i = 1 . . . . . r, ati ~ 0 ; (3.3) 3) define the T~ by H,~T~)= Hm(T+), l <<.i < r, i v~ m .
There exists a, unique temperature sequence {T~} satisfying (3.3) for a given at; this is ensured by remark 1.1. We now reorder the reactions in order of increasing T~, so that
~...~E=~,
(3.4)
where the new T, corresponds to the former reaction Rm. The new sequence {T~} defined in (3.4) equates the power rates of the reactions. It induces a hierarchy within the reactions and generates a stable chemical pathway in the flame profile. It is entirely possible that some of the T~ coincide, corresponding to competition flame sheets (cf.
194
P,C. Fife and B. Nicolaenko / Flame fronts with complex chemical networks
Hi(T) I
Hr-1 (T) Hr (Z)
I
I Hr-1 (T)
l
mr(T) / '
I .....
~
l
I
I,,,
Ti
Tr_ I
I1
I T+(~) = Tr
Fig. 7. Temperature sequence such that power gauge functions are equal along this sequence.
eases 5 and 6, section 2.4). Fig. 8 shows such a configuration. Here the possible allocations are a priori nonunique, with an allowable range for • on the boundary of the convex set of allocations. T÷ and • are actually determined by setting T+ = T,, intersection of the power gauge curves of R, and R,_t, the two reactions with the smallest power gauge functions over the interval [T,~,, Tm~] of possible final temperatures.
To cover such a situation, let {7~i}, i = 1 , . . . ,p <~ r be the possibly shorter sequence of distinct temperatures among the T~ in (3.4):
(3.5)
f,
we denote their number by p. It also may happen that two of the Tj almost coincide, that is, 0 < IT~ - Tjl = d9(8i). This case is /
I 1 1
Hr-2
i I t
I
~-z
_.
n
I
I
Tr_ 2
,
Tmin
Fig. 8. Competition flame sheets.
I
T+=T,
j
........ I
Tma x
~_
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
considered non generic, and is hereby excluded from consideration. Thus, we assume that two temperatures T,, T~ either coincide (this may be required by the feasibility criterion given later) or differ by a quantity of larger order than ~; + ej. We also assume that at each ~, all but perhaps two of the quantities H~(~) are distinct. Our attempt will be to construct a combustion front in which, roughly speaking, p sets of reactions go to completion at the temperatures ~ in that same order as one proceeds across the front's profile in the (upstream) direction of increasing temperatures. For that purpose, we use the notion of suballocation.
to be maximal. With it known, ~ 0 is then maximal, etc. In this case, ~(0 is uniquely determined. Next, suppose two of the H~(~) coincide, say H~,(Tt) = Hk2(~)- Then instead of maximizing ~ , we maximize ~0 + ~0. In this respect, we generalize the principle introduced in section 2, cases 3 and 4. For such suballocations, kt sets of reactions go to completion at ~t. We can now determine the temperature at which the suballocation ~(0 would bring the corresponding subnetwork to equilibrium, if no reaction outside that subnetwork were present. We call it
t= T Definition 4. For each 1 ~
~,6o)=~.
kl
+ E ~OQ,.
(3.7)
i=1
3.2. Suballocations to subnetworks
~,(,)<~o+,),
195
(3.6)
There is a further condition which we shall require the ~(0 to satisfy which is a generalized maximum allocation principle: Let us fix 1 ~
~0-~). We say that at(° is "maximal" if it allocates a maximal amount of reactants to those reactions with the largest values of H at the temperature 7~z. This, simple put, is the principle that among competing reactions, the one with the largest rate (as defined by the functions H,(T)) wins out. The precise definition is as follows. First, suppose all the quantities Hk(~) (~t fixed) are distinct. Let kl be the index that yields the largest of them; then k2 the next largest, etc. Out of all possible suballocations ac(° 1> ¢!(1-1), 0~ is required
We are now ready to give the exact mathematical criterion to select from all possible final allocations those which are feasible:
Definition 5. The allocation a is feasible if 1) there exists for each 1 ~
Example. Consider again case 3 in the fuel rich situation of section 2. There, ot = ( a , b - a) is determined so that ~tl = a maximizes allocations for the reaction R t with maximal power over ITch, T==]. This fixes T+---T~o. This suballocation at°)=(a,0) is such that /-/1(7'1)= H2(T~), and a¢O) = ( a , 0 ) < a~ = ( a , b - a ) ;
196
P.C. Fife and B. Nicolaenko /Flame fronts with complex chemical networks
the allocation is feasible ifi
Z.A.E. The initial concentration of the radical A2 is Y{ = 0. The unique final allocation is
TI = T_ + aQ, < TI , cq= Y?, in which case a steady tandem profile exists. Once feasible suballocations have been determined it remains to analyze the mathematical asymptotic limit problems, that is: 1) solve the various internal layer equations for each flame sheet; 2) determine in a self-consistent manner the transmission conditions between different sheets (in particular their geometric separation); 3) determine the mass flux eigenvalue M. For details, we refer to our papers [9] and [10]. We simply remark that if only H.A.E. reactions are involved, then the last layer equation on the burned side (T+) uniquely determines M. If this layer involves a single pseudoreaction, M is given by the usual explicit formulas [17, 2]. If it involves a competition flame layer, the analysis is more complex and only implicit formulas exist [7]. Nevertheless, successive flame sheet separations are calculated working backwards from the burned side. In any case, the qualitative structure of the profile can be immediately found without performing layer analysis; and if only H.A.E. reactions are present, then only one layer analysis is necessary to determine M.
3.3. Some remarks on the methodology for Z.A.E. reactions We briefly review how our methodology extends to a mix of H.A.E. and Z.A.E. reactions on the simple chain reaction model (1.11) investigated at length by Lifian [12]: Rl : AI + A2--)2A2, R2 : 2A2 q- M-+2P + M ,
ot2= Y~-/2,
with final temperature T+ = T_ + Y?(Q~ + Q:/2). Notice that the adiabatic temperature of a pure flame (1) involving only R~ is T~ = T + Y~-QI. By considering the normalized power gauge functions,
H~( T) = ~(In B~ - -~ ) , H2(T) = ~ In B2, ~----el=
r+
r+
E1 T + - T _ ' and comparing them over the temperature interval [7"1, T+], we immediately exhibit the three flame structures studied by Lifian: 1) if H~ > H1 over [T,, T÷], there is a merged flame structure at T÷; the radicals do not diffuse out of the flame sheet; 2) if H1 > HE over the interval, we have a pure flame (1) at T~, followed by very slow radical recombination; this is reflected by a suballocation:
~0) = (y~-, 0), 3) if H~(T,) = H2(T,) at some unique T , in the interval, we have a competition flame situation: a thin layer at T , for R1, imbedded in a thick radical flame. The general methodology for H.A.E. plus Z.A.E. reactions will be presented elsewhere. We simply remark that the generic flame cases evidenced in section 2 are still present, provided that one allows for thick radical flames.
(1.11) Acknowledgements
with K ~ = ( Q , , - 1 , +1), K2= ( Q 2 , 0 , - 2 ) , Q , > 0 , Q2 > 0. The branching reaction R~ has H.A.E. El >> 1; whereas the recombination reaction R E has
This research partially supported by the Center for Nonlinear Studies, Los Alamos National Lab-
P.C. Fife and B. Nicolaenko/Flame fronts with complex chemical networks
oratory; work also performed under the auspices of the U.S. Department of Energy under contract W-7405-ENG-36 and contract KC-04-02-01, Division of Basic and Engineering Sciences; and partial support was provided by NSF Grant number MCS-8202056.
References [1] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling Wave Solutions to Reaction Diffusion Systems Modeling Combustion, Contemporary Mathematics, vol. 17 (A.M.S., Providence, 1983), pp. 189-207. [2] J. Buckmaster and G.S.S. Ludford, Theory of Laminar Flames (Cambridge Univ. Press, London, 1982). [3] J. Buckmaster, Physica 12D (1984) 173 (these proceedings). [4] P. Clavin, Dynamical Behavior of Premixed Flame Fronts in Laminar and Turbulent Flow, Progress in Energy and Combustion Science, to appear, 1984. [5] F.L. Dryer and I. Glassman, High-t~-aperature oxidation of CO and CH4,Fourteenth Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1972), p. 987. [6] F.L. Dryer and C.K. Westbrook, "Chemical kinetic modelling for combustion applications," presented at the Propulsion and Energetics Panel 54th Specialists Meeting, Cologne, West Germany, Oct. 1979. NATO AGARD Conference Proceedings No. 275. University of California Lawrence Livermore National Laboratory report UCRL8177, September 1979. [7] P.C. Fife and B. Nicolaenko, "The singular perturbation
197
approach to flame theory with chain and competing reactions," Ordinary and Partial Differential Equations, W.N. Everitt and B.D. Slccman, eds., Lecture Notes in Mathematics, No. 964 (Springer, Berlin, 1982) pp. 232-250. [8] P.C. Fife and B. Nicolaenko, Asymptotic Flame Theory with Complex Chemistry, Contemporary Mathematics, vol. 17 (A.M.S., Providence, 1983), pp. 235-255. [9] P.C. Fife and B. Nicolaenko, "Asymptotic Flame Theory with Reversible Reaction Networks," Proc. I.N.R.I.A. Conf. on Combustion, Paris, 1983, to appear. [10] P.C. Fife and B. Nicolaenko, "Asymptotic Flame Theory with Complex Chemical Networks," in preparation. [11] D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics (Plenum, New York, 1965). [12] A. Lifian, "A theoretical analysis ofprcmixed flame propagation with an isothermal chain reaction," Instituto Nacional de Tecnica Aerospacial "Esteban Terradas" (Madrid), USAFOSR Contract No. E00AR 68-0031, Technical Report No. 1, 1971. Also P. Clavin and A. Lifian, to appear. [13] C.K. Westbrook and F.L. Dryer, "Chemical Kinetics Modelling of Hydrocarbon Combustion," University of California Lawrence Livermore National Laboratory report UCRL-88651, 1983. [14] C.K. Westbrook and W.J. Pitz, "Prediction of Laminar Flame Properties of Propane-Air Mixtures," Proc. 9th International Colloquium on Dynamics of Explosions and Reactive Systems, Poitiers, France, 1983, to appear. [15] C.K. Westbrook and F.L. Dryer, Simplified Reaction Mechanisms for the Oxydation of Hydrocarbon Fuels in Flames, Combustion Sc. and Tech. 27 (1981) 31-44. [16] Ia.B. Zeldovich, Theory of Flame Propagation, J. Phys. Chem. 22 (1948) 27-48. [17] Ia.B. Zcldovich, G.I. Barenblatt, V.B. Librovich and G.M. Mahviladze, Mathematical Theory of Combustion and Detonation (Nauka, Moscow, 1980).