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Effect of heat losses on the limits of stability of premixed flames propagating downwards
COMBUSTION
AND
FLAME
60: 1-14 (1985)
E f f e c t o f H e a t L o s s e s o n the Limits o f Stability o f P r e m i x e d F l a m e s Propagating Downwards P A U L CLAVIN and C O L E T T E N I C O L I Ddpartement de C O M B U S T I O N du L . A . 72 au CNRS, Universitd de Provence, centre de St Jdrdme, Rue H. Poincard--13397 M A R S E I L L E cedex 13, France
A previously developed theory of linear stability analysis of nonadiabatic premixed flames that was based on a diffusive thermal model is extended by eliminating the hypothesis of negligible expansion of the gas. The effects of heat losses on the limit of stability of slow burning flames propagating downward are investigated in detail. The influence of the proximity of the flammability limits upon the stability limits is studied with a view to describing the cellular threshold for flames propagating in tubes.
I. I N T R O D U C T I O N The quenching mechanism of premixed flames by heat losses has been completely described by using asymptotic analysis in large values of the reduced activation energy (/3 ~ 00) [1, 2]. The linear stability analysis of these nonadiabatic flames has been more recently investigated by Joulin and Clavin [3] (referred to as [I] in the following) in the diffusional-thermal model. This model [12] excluded density variations, thereby uncoupling the continuity and momentum conservation equations from the equations of species and energy conservation. This serves the useful purpose of describing the diffusive effects while eliminating hydrodynamics instability described in [7, 8]. Some mathematical justification for the adoption of this model can be given by the double limit
/5
~
-
E
(Tb-T.)
-
-
R T~2
-
Tb
and
3"= (p~- pb)/pu--,o. Copyright © 1985 by The Combustion Institute PubliShed by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017
u and b are subscripts referring to unburned and burned gases in adiabatic conditions, respectively. T is the temperature and p the density. But the troubling approximation 3' --* 0 is not realistic for the usual hydrocarbon flames and prevents a relevant description of the instability threshold for the appearance of cellular structures. In a recent work by Pelce and Clavin [5] (referred to as [II]) based on the analysis of the wrinkled flame structure of Clavin and Williams [4], the dynamical properties of adiabatic flame fronts have been investigated in the limit of large wrinkles A compared with the flame thickness d but for any gas expansion parameter 0 < 3" < 1, ~ = d / A ~ O. The coupling of the hydrodynamical effects with the diffusive phenomena has been thus described completely in the limit of small values of e and for large values of the reduced activation energy /5. A similar analysis has also been recently developed by Frankel and Sivashinsky [11]. It is worth noticing that, according to the results presented in [II], the acceleration of gravity associated with the diffusive effects taking place inside the flame is shown possibly to overcome the hydrodynamic instability mech-
O010-2180/85/$03.30
PAUL CLAVIN and COLETTE NICOLI anism produced by the gas expansion 3, :# 0. Under a critical value of the flame speed depending on the properties of the reactive mixture, planar fronts propagating downward are predicted to be stable. Thus, as recently verified in our laboratory [13], the cellular threshold is predicted to be experimentally observable on slow burning flat flames propagating downward. Furthermore, the size of cells at the threshold is found to verify effectively the basic assumption of the theory, e < 1. But the flame velocities corresponding to the threshold are found to belong to the range 5-17 cm/s, which is fairly close to the usual values of the flame velocity at the extinction limit in tubes ( - 7 cm/ s). Thus the stability limits are found to be close to the flammability limits. This explains the confusion that has been often caused in the past as explained in [10]. Nevertheless it can be expected that in the same experimental conditions as those used by Markstein [10] for studying the cellular threshold, the heat losses must play a nonnegligible influence on the stability limit. The present paper is devoted precisely to the study of such effects on planar fronts propagating downward in tubes, when the coupling between hydrodynamical and diffusive phenomena produced by the gas expansion is fully taken into account. This work is a natural extension of [I] and [II] and is developed in the same spirit as in [II]. The reader is referred to these references for a detailed technical presentation. For brevity, only the modifications of the calculations induced by the phenomena studied in this paper are explicitly written down. Nevertheless this paper is written in a self-contained form in the sense that a reader not necessarily concerned with the mathematical techniques can understand the general physical ideas and more especially the conclusions. Section II is devoted to recalling the basic concepts that are associated with the heat loss phenomena. Section III concerns the formulation of the problem as well as the presentation of the spirit of the method. The analysis is sketched in Section IV, and the results together with the
conclusions are presented in Section V, where an effort has been made to make easier the comparison of the predicted results with experiments. In particular, we have plotted the predicted stability and flammability limits on the same curves "flame velocity versus equivalence ratio" for different values of the dilution and also for different values of the radius of the tube controlling the intensity of the heat losses. II. T H E F L A M M A B I L I T Y L I M I T S The quenching mechanism of premixed flames by heat losses was studied in [I] with an asymptotic method valid in the limit of very large values of the reduced activation energy 3 (3 ~ oo). An Arrhenius law is adopted for the rate of the release of the chemical energy such that as 3 --' oo, the chemistry is confined to a surface [1, 6]. This reaction zone separates the preheated zone (where the species concentrations as well as the temperature evolve under only the transport processes) from the burned gases (where the concentration of the limiting component is zero). The foundation of the asymptotic method is based on matching the solutions that have been preliminarily obtained separately in the preheated zone and in the reaction zone by using the boundary conditions in the unburned and burned gases, respectively (see [I] and [liD. The quenching mechanism can be modeled by a volumetric heat loss that introduces a sink term, - K ( T - Tu), in the equation for the conservation of energy. K( T Tu) corresponds to the energy lost per unit time for a unit volume of reactive mixture at temperature T. For flames propagating in tubes, K is roughly inversely proportional to the radius R of the tube. One introduces the ratio of the actual transit time rr~ = P D t h / p u U t t 2 and of the characteristic time of cooling rc = puCp/K,
H=23rrt=23 ( K /PuUH2~ K(pDth) = 2 3 (ouUI02Cp ,
(1)
where urt is the laminar flame speed in the
STABILITY OF PREMIXED FLAMES presence of heat losses, Oth is the thermal diffusivity of the reactive mixture, and Cp is its specific heat per unit mass. The fact 2/3 is arbitrary in order that the extinction limit correspond exactly to H = 1. For heat loss intensities K such that H remains of order unity, in the limit/3 --. ~ [ H = O(1)], the modification of the flame temperature can be easily obtained by solving the conservation equations in the preheated zone to give [I]
diffusive flux (5) must equal, at the dominant order in the limit/3 ~ 0% the convective flux leaving the preheated zone and given by (PuUn)Cp(Tb -- Tu) in the plane case. This yields
~ ( 0 f - 1) = - H ,
\ $/Li
(2)
where 0 is the reduced temperature, g = ( T Tu)/(Tb - Tu), and where the subscript f refers to the flame temperature. But, as clearly shown by (1), H depends on the flame velocity un. Let us define a similar ratio of times as in (1) but using as a reference the transit time re (re = rn=o) of the flame sustained by the same reactive mixture but propagating in the adiabatic conditions with the laminar flame speed Ur (Ue = UH=O):
rc
\puCp" pDth ,/ "
(3)
This can be written as 2/3 rL
1 K
with
K* -
(#uUL)2Cp
e 2/3(ODth)
(3 ')
(4)
where K* is, as UL, a characteristic of the reactive mixture and K is a characteristic of the tube in which the flame propagates. The investigation of the reaction zone provides the expression (in terms of the flame temperature) of the diffusive flux of heat leaving the reaction zone toward the preheated zone: O u U L C p ( T b --
Tu)e a(af- u/2.
In
=
e K*
.
(5)
But the matching principle requires that this
(7)
This result has been obtained in [I]. Equation (7) describes an extinction process in the sense that, as soon as the heat losses are too strong, K > K*, there is no possible flame propagation. The extinction limit corresponds to K = K* and to a limiting flame velocity u . given, according to (7), by
(8)
u, = u d ~ e
in such a way that, according to (4), one has H = 1 at the extinction limit
,
(ua~21n [!(u.~27= u./ \u,/ 9
Thus (1) can be written as K 1 (UL") 2, H=K---~ e un/
(6)
to obtain, when (2) and (4) are used,
H=K,
rc e K* ' 1
UH) = e¢(6f- O/2,-
K K*
(8')
For heat loss intensity smaller than the critical value, K < K*, two possible steady solutions appear, one corresponding to u , < uH < UL ( H < 1) and the other corresponding to 0 < u , < u . (1 < H ) . This last regime has been proved in [I] to be unstable for one-dimensional (planar) perturbations (zero wave number) and thus cannot be relevant for the description of the cellular structures. Then, only the branch of solutions corresponding to 0 < H < 1 and reducing to the adiabatic flame velocity UL in the limit K --* 0 has to be considered. It is worth noticing that, according to ( 3 ' ) , the extinction limit K = K* corresponds to a characteristic time of cooling rc greatly larger than the transit time rL, re/7" L = 0 ( / 3 ) , and thus, for large values of/3, a small amount of heat
4
PAUL CLAVIN and COLETTE NICOLI
losses (compared with the heat released) is sufficient to quench the flame. It is interesting to consider the situation corresponding to the propagation in a given tube (K given) of a given reactive mixture (/3, pDth, Cp given) at different values of the composition. By varying the equivalence ratio ~ and/or the dilution, one varies the adiabatic flame speed UL as well as the flame propagation velocity in the tube uH given by (7). Usually the experiments are carried out by varying ~b with ~ kept constant. Beyond extinction values of ~b, there is no possible flame propagation. These extinction values of the composition depend in fact not only on the reactive mixture but also on the tube. This pointed out the difficulties involved in an intrinsic definition of the flammability limits. It turns out that the corresponding values of the adiabatic flame speed UL~Xas well as of the limiting speed of propagation in this tube, u . ~X ( = UL~X/ x/e), is a characteristic of the tube. These values UL~x and u , cx are directly obtained from (3 ') by replacing K* and UL by K and UL~x, respectively, to give (puu.eX) 2 = 2/3
pDth K
Cp
(9)
and (4) takes the form H=(
u*ex~2"uH /
(9')
This last expression ( 9 ' ) of the reduced heat losses is very useful when one works at a given value of K, as for the propagation in a given tube (K fixed, ~b and ~ variable). The other expression (8') is interesting when one works at a given composition (UL, U, and K* fixed) but with variable heat losses. As an example, when K is roughly evaluated by K = 23,/Rh (where X, R, and h are the thermal conductivity, the radius, and the thickness of the tube, respectively), the limiting speed of propagation in a Pyrex tube corresponding to R = 5 cm, and h = 0.5 cm is, according to (9), nearly equal to 6 cm/s: u . ~x 6 cm/s. And all the flames whose adiabatic flame speeds are smaller than 10 cm/s (ULex =
x/eu. ~x - 10 cm/s) are predicted not to propagate in this tube. Equation (9) means that when one considers the dependence of the velocities of the flame propagation (in a given tube) with the equivalence ratio ~ for different values of the dilution, all the corresponding isodilution curves are predicted to end at the same limiting speed value u . ~x (cf. Fig. 3). Such a result, which is obtained by assuming that the quantities K, ~, PDth, Cp are kept constant (independent of ~b and/~), has been roughly confirmed by the recent experimental work of J. Quinard [ 13]. Let us now go to the stability analysis of these steady state planar solutions whose range is limited by the above mentioned flammability limits. III. F O R M U L A T I O N The nonadiabatic flame model is identical to the one in [I] and the formulation is similar, but the thermal-diffusive approximation is removed herein. Attention will continue to be restricted to exothermic reactions amenable to a one step approximation, with the degree of progress describable in terms of the mass fraction Y of a single reactant, taken to be the limiting reactant in the sense that Y = 0 defines completion of the reaction. An Arrhenius rate is adopted so that, as the reduced activation energy goes to infinity (/3 ~ ~ ) , the chemistry is confined to a fluctuating surface, whose position is employed to define the origin of a moving coordinate system, pDth and the Lewis number Le, the ratio of the thermal diffusivity Dth of the mixture to the molecular diffusivity of the limiting component, are assumed to be constant. 1 The heat losses are modeled by a volumetric sink term as presented in the above section. The conservation equations concerning the unsteady wrinkled front are presented in detail for the adiabatic case in Ref. [4]. The nonadiabatic case is described by the same equations but with the above mentioned sink term -H0/2/3 added to J This restriction can be easily removed, and the temperature dependence of pD can be included easily as in Ref. [6].
STABILITY OF PREMIXED FLAMES the conservation equation of the energy. In the stability analysis, one is dealing with these equations linearized around the one-dimensional steady state solution (see [II] for the adiabatic case). At the dominant orders in the asymptotic expansion (/5 ~ oo), the study of the thin reactive zone is identical to that in [1] and [II]. Then, at these dominant orders, the matching principle reduces to jump conditions across the flame sheet that take the following form in the linearized approximation, a0 1 Off ~:o+ ~ + ~ ~=o-
00 ~=0-
= 1/2/50f,
=0,
(10.1)
(10.2)
where Of = 0(~ = 0) = O(1//5). Here 0 and are the perturbations of the nondimensional temperature ( T - Tu)/(Tb - Tu) and the nondimensional mass fraction Y ~ Y . of the limiting component, respectively (~ = 0, ff = 1, 0 = 0, ~b = 0 in the fresh mixture). 2 The location of the flame sheet is defined by x = or(y, z, t) in the laboratory frame, where x, y, z, t are nondimensional space and time coordinates defined with the flame thickness d = pDth/puUH and the transit time ~'H = d/UH choosen as units oflength and time. ~ = x - a , ~/ = y, ~" = z, z = t are the nondimensional moving coordinates. Equation (10.1) is valid in the asymptotic limit as/3 ~ 0o up to the order O(1//5). Equation (10,2) is valid at the dominant order O(1) and is restricted to a perturbation of the flame temperature Oe (defined at ~ = O) that is of order O ( 1 / /5).3 For heat losses such that H = O(1), this is satisfied whenever the Lewis number Le is close enough to unity to make (Le - 1) of order O ( 1 / /5). Thus, one introduces / = /5(Le - 1). Then, when / = O(1), the condition Of = O(1//5) is satisfied for reduced wavelengths of the wrinkled front A / d of order O(1) in the limit/3 --* 00. 2 The overbar denotes the one-dimensional steady state solution. 3 The perturbation of the flame temperature 0 is in fact of order O(1) in the preheated zone (~ < 0) but 0(~ = O) = O(1//3) (see [I]).
Nevertheless, as in [4], attention will continue to be focused on realistic situations for which the wavelengths A are large compared with the flame thickness. Thus, d / A will be considered as proportional to a small parameter 4 e (e < 1) independent of/5. Thus, after having used the limit 13 ---, 0o that confines the chemistry in a reactive sheet through which the jump conditions (10) are verified, one looks to find the outer solutions as power expansions in e. This later restriction is not necessary in the thermal diffusive approximation where, as in [I], the outer solutions can be found whatever e's value. But when the effects of the gas expansion are taken into account, the ronservation equations of energy and species are coupled with the conservation of momentum and the outer problem has to be solved in perturbation expansions around the one-dimensional solution. This has been done in [4] for the adiabatic case and has been used in [II] to study the corresponding stability limits of planar fronts propagating downward. As in Refs. [4] and [II] (see also Ref. [6]), the outer equations for the perturbations of temperature and concentrations in the nonadiabatic case reduce, at the second order e 2 in the power expansion in e, to the following quasi-planar and quasi-steady equations: 5 O0
020
H
-----+--
0
= - ( s ( O + O20dOZ 2) dO + O(e3),
(1 la)
d~ OXb O~
1
02~
LeO~ z
1
= - (s(~) + - - 0 2 . / O z 2) d ~ + O(e3), Le
(1 lb)
where 0(~) and ~(~) are the one-dimensional 4 ~ (defined here as the order of magnitude of d/A) has nothing to do with the quantity ~ defined in [I] as the amplitude of the perturbation around the one-dimensional steady solution. 5 To simplify the notation, only one transverse condinate z is explicitly specified in the following.
PAUL CLAVIN and COLETTE NICOLI steady state solutions of the nonadiabatic case characterized by H . The expressions for 0 and can be found in Refs. [1] and [I] [see also Eq. (18c)]. s(~) identifies the perturbation of the nondimensional longitudinal (x component) mass flux in the moving coordinate system, s = e(u - Oct/Oz) + ra,
~= ?t~,
(12)
where f(~) and a(~) are the steady state profiles of the reduced density and of the x component of the reduced flow velocity, respectively. These quantities are defined by r = p/pu and u = u / u n in such a way that ~ --, - ~ , f = 1 , r = 0, a = 1, u = 0. In fact, s(~) couples Eqs. (11) to the hydrodynamics through the continuity equation Or~Or + OslO~ + O(fv)/OT1= O,
(13a)
g(O = ~(~)f(~) = 1.
(13b)
Here v denotes the z component of the flow velocity and, according to the usual "isobaric approximation" Of flame theory (negligibly small Mach number), r is simply related to 0 through the perfect gas law ,y r =
-
~
(1 4a)
r20
1-3,
?=
7 5 ) -1 1+1_,7
s = eS_**(e/j, Z, T) + esl(~, Z, T) + ees2(~, Z, T) + O(e3), 0 = ~202(~, Z, T) + O(c3), = E21~2(~, Z, T) + O(e3),
s-~**= cS±**(~, Z, T), u~** = eU±**(~, Z, T), v±o.=cV±~,(~,, Z, T),
and if±** = 0± 00= r± o, = 0.
The quantities involved in (16) should be understood as being O(1) in the limit as/3 ~ oo and where ~ = e~, the subscripts - a o and +oo denoting the upstream and downstream incompressible gas flow, respectively. According to the definition given by (12), one has in these regions and
s+ = = (1 - 3`)(u+ = + Oct~Or).
u=eU_®(e~, Z, T)+eul(~, Z, T)
(17)
All the quantities appearing in (15) and labeled with the subscripts 1, 2 . . . . will be found to go to zero like e ~ when ~ --* - c o , and thus the matching between the upstream incompressible zone and the preheated zone is systematically satisfied. The connection of the preheated zone with the downstream incompressible flow is made through the thin reactive zone by the jump conditions (10) and those relating the characteristics of the gas flow p, u, v that can be found in [II, Eq. (13)].
(15a) IV. ANALYSIS
v=eV_**(e~, Z, T)+evl(~, Z, T)
"t- e21)2(~, Z, T) + O(e3),
(16)
(14b)
where 3' = (Ou - Pb)/0u = (Tb -- T,)/Tb is the gas expansion parameter. The x and z components u, o of the reduced velocity of the gas flow appearing in (12) and (13) are given by the same equation for the momentum conservation as in Refs. [41 and [II]. Thus, it is not necessary to reproduce explicitly these equations that have to be solved, as in the previous works [II, 4, 6], by perturbative expansions in powers of e. The following expansions are introduced,
+ e2u2(/j, Z, T) + O(e3),
(15d)
where Z = ez, T = er, and the quantities U_~., V_o,, S_** have to be developed around x = 0. Equations (15) are valid inside the preheated zone. Outside the flame, the gas flow varies on the same spacial scale A as the wrinkles of the front, and the solutions take the follow form:
s_ ~, = Uo. - Oa/ Ot ,
(15c)
(15b)
The integration of (11) in the preheated zone (x
STABILITY OF PREMIXED FLAMES asymptotic limit/5 --* + co. By adding the two equations (18a) and (18b) written for ~ = 0 and by using (19), the jump condition (10.1) leads to the following expression for Of = 0(~ = 0) = 0(1//5),
< 0) yields
~<0:
2
**
® (s+a2o~/az 2) ~ d~ = ~ ,
+
(18a)
d~
~<0:
ld¢
x I~** (s + 82ctlOz2)e ~ d~ - l 020l/OZ2 (18b)
Led~'
+(l-H)
o,(s+020t/Oz2)~e~ d~
where ti(~) = ~(0)e x- ~
and
~ = 1 - e I~
(18c)
+-2
(s+o,+O20t/Oz2)e-~ d~
with ,, X- = 1 + - - + 0 2/5
and where the term t °_~, 0 dx appearing in (18a) has been obtained from the dominant order of the asymptotic expansion (/5 ~ co) of (1 la):
,, ~t(0) = 1 - - - + O
In the downstream incompressible zone (~ > 0) located just behind the thin reactive zone, one has, according to [1] and [I], dJ//d~ = 0 and (11) yields ~>0:
~b=0,
~>0:
-'~ ~=o+:Ofx+
_
(19a) Furthermore Eq. (18a) leads to the dominant order of (dO/dOi ~o- :
dO
(S+O20I/OZ2)__dOe_x_ ~ d~, d~
_
(19b)
L
(s+02ot/Oz2)e ~ d ~ + O ( ~ ) . (22)
where X+ . . . .
I~. o d~
4- 0
2/5
0(0 = g(O)ex-L
and (19c)
It is clear that the dominant order of (19b) is O(1//5). Moreover, according to the above discussion, the expansion Le = 1 + l//5 is introduced in (18b) with / = O(1) in the
It is worth noticing that only the dominant order [O(1)] of the asymptotic expansion (/5 co) o f s and a is involved in (20) and (22). Thus, in the following s and ct should be understood as being O(1) in the limit/3 --* co. Then, after (20) and (22) have been introduced in the jump condition (10.2), one obtains an integral relation for s. As soon as the modification through the flame of the longitudi-
PAUL CLAVIN and COLETTE NICOLI nal mass flux [characterized by the terms sl(~) and s2(O of the expansion (15c) of s] is known, this integral relation provides the expression for s_**(E = O) = eS_~.(E = 0)and also, according to (17), the equation for the evolution of the flame front. These terms sl(~) and s2(O are given by the continuity equation (13a). At the order e 2, (13a) reduces to a quasi-static and planar equation which can easily be integrated when the transverse gas velocity v is known. This part of the analysis is quite similar to those in the previous works [4] and [II]. Equation (13a) yields sl =0,
Ii
/I-~ In(1 + 0
D=
1
~'
a(O)
f((gvl/(gZ) d~,
(23)
where v~ is simply given by the deflection of the streamlines through the tilted front (see [II, Eq. (23)1): 111
=
- -
((ga/(gz).
3'/(1 - 3")e~,
ul = 0.
(24)
dO
S
a(O) In 0
o 1 + [ 7 / ( 1 - 3,)]0 dO,
respectively, with a(O) denoting the ratio of the thermal diffusivity at the temperature 0 to its value at the unburned temperature. The calculation of (26) and (27) is carried out by introducing in (20) and (22) the expansions of s and s+ 0, in powers of ~ obtained from (15c) by using
S_ ~ ( ~ ) = S_ ®(0) + ~
This leads to (see [II, Eq. (24b)])
and
1 - 3' Jo 1 + [3"/(1 - ~,)]0
1 - 3"
+
#~.
W h e n the temperature dependence of the diffusivity is taken account (as in the adiabatic case-see Ref. [6]), ~3/d takes a more complex form than (27) but where in particular (I/3')In [I/(l -3")] and D have to be replaced by
3"
( e - 1) d~
s2(~)=(9O_o./(gZ[~=o
with
as_
j a ~ I~=0 + o(~2), (28a)
s2(O=((92od(gZ2+(gv+~/(gZ) In
1 + 1-3,
S_ ~.(~ = 0) = O(e).
' (25)
By following the procedure outlined just above, one obtains the following linear equation for the evolution of the front which is valid up to the order e 2, (ga
- - = U_ a.(~ = O) -i- ,~/d((920t/(gZ 2 q- (90_ ¢./(9Z1~ =0),
Or
(26) which is quite similar to Eq. (29) in [II] but where the length ~ is given now by
,[
~3/d= 1 - H
+
°
(I -3")
I-H+(1-3') 1
( / - H)
J
,
(28b)
where s+®(0) is given by the continuity of the mass flux through the reaction zone S+ ~(0) = ~S-oo(0) + e2S2(0)+ O(~ 3)
(29)
and where, according to the definitions (16) and (17), one has
as_,ja~ I~=o=
au_
J a ~ Iz=o,
as+Ja~,l~.=o=(l-3") aU+®la~l~:o.
(30)
The mass conservation upstream and downstream from the flame yields (gU+**/O~, = OV_**/OZ and OU+~/O~, = -OV+**/(gZ. The continuity of the transverse component of the flow velocity through the reactive zone leads to (9S+**1(9~ I ~ =o= - (1 - 3")(8 V_ ®/(gzl~ =o
~ In
1 - 3' D /
s+~=s+®(0)+e~ as+ja~[~°o+O(~Z),
(27)
+ (gul/(gZl~ =o),
which has to be introduced in (28b).
(31)
9
STABILITY OF PREMIXED FLAMES V. D I S C U S S I O N OF R E S U L T S The form of the local equation (26) for the evolution of the front is similar to that in the adiabatic case [4], but the heat losses affect the expression (27) of the Markstein length £ . In the adiabatic case, H = 0, (27) reduces effectively to the result in [4]. Furthermore, when the effects associated with the gas expansion are neglected (3~ = 0) u_ ~ = v_ ~. = 0, (26) with (27) reduces to the long wave limit of the result obtained previously for the thermal-diffusive model [I]:
0o1[0
(1 - H ) + ~
'] l
V2~.
(32)
As clearly exhibited by the presence of the factor (1 - /-/) in the denominator, the results (26) and (27) are no longer valid close to the extinction limit defined by H = 1 where (27) diverges. As in [I], near the extinction, the equation for evolution must involve an acceleration effect O2ol/Or 2 which becomes of the same order of magnitude as (1 - H ) 0cdOr. But the dimension of corresponding domain around H = 1 is very small. For example, for the usual value of the gas expansion (3' = 0.8), (27) yields 3, = 0.8 = ~/d: 1
l _ H (2.011-2.1544H+0.29331),
(33)
and thus for the usual Lewis number I = O(1) ( - 3 < I < 6), g,/d is seen to take a large value ( > 1 0 ) only for conditions very close to the extinction limit (0.9 < H < 1). For example, for H = 0.9, (33) gives 0.72 + 2.93 l, which is not really large, and the present analysis can be used without restriction up to more than H = 0.9. It is also worth noticing that according to (33) the value of l associated with g,/d = 0 is given by - 6 . 8 6 + 7.35 H . Contrary to the adiabatic case, when the heat losses are sufficiently important ( H > 0.5) this value of / corresponds to Lewis numbers which may be attained by the usual reactive mixtures of hydrocarbon/air flames (/ > - 3 ) . Thus the heat losses may produce an instability mechanism of
a purely diffusive nature which cannot occur in the adiabatic case [4, II]. But in any case, this purely diffusive instability cannot appear on planar fronts of flames propagating downward. The reason is that there is another mechanism of instability of a hydrodynamical nature which appears before, and corresponds to, a positive value of ~ / d . In fact, as explained in [II], the hydrodynamical instability mechanism, first pointed out by Darrieus [7] and Landau [8], cannot be overcome without sufficiently strong stabilizing effects of the acceleration of gravity and the diffusion processes. This requires sufficiently small burning velocities UL and sufficiently large positive values of the "Markstein coefficient" ~3/d. The corresponding detailed analysis requires investigation of the complete fluid mechanics problem associated with the wrinkling of the front. Such an analysis has recently been carried out for the adiabatic condition [II]. The intensity of the reduced heat losses involved in the nonadiabatic case is indeed very small in the limit of large values of the activation energy. In fact, at the extinction ( H = 1), this intensity is O(1/~) in the limit as ~ oo. Thus, outside the preheated zone of the flame, the nondimensional gradient of the temperature produced in the burned gases by the heat losses is sufficiently small [O(1//3) when/3 oo] to be neglected in the equations governing the gas flow outside the flame thickness. In the limit as/3 --* oo, only the dominant order 6 O(1) of the corresponding flow field is necessary to solve the problem. Thus, the fluid mechanical problem is identical with the adiabatic analysis [II] and will not be reproduced here. The result presented in [II] can be directly used by simply replacing the "Markstein length" by its modified value (27). The first important point is that the wavelength Ac of the cellular structure appearing at the instability threshold was found not to involve the "Markstein length" and to vary with only the laminar flame velocity urt,
A,:/d = 7ru~2/(1 - ~)gd,
(34)
6 Notice that this is not the case for the temperature field where the two orders O(1) and O(1//~) are required [see (18)].
PAUL CLAVIN and COLETTE NICOLI
10 Where g is the acceleration of gravity. Thus, according to the discussion given above, the relation (34) linking the dimension of the cells at the threshold to the flame speed uH is found to be unaffected by the heat losses. This is very encouraging for obtaining relevant experimental data for Ac by using laboratory burners in which the heat losses can never be completely suppressed. The critical value ~¢ of the "Markstein length" at the threshold obtained in [II] for the adiabatic condition is still valid [II, Eqs. (39)(41)1:
13e/d= 3" 2+3,
In - - 1-3,
1
-~ 4(1 - 3,)g But according to (27), the corresponding critical value Lec of the Lewis number is affected by the heat losses. For 3, = 0.8 and pDth/Ou = 0.35 cm2/s, one obtains from (27), (33), and (35) the following expression for the critical value /~ [--/~(Lee - 1)], l~=-3.85+4.34H+0.98(1-/-/)
\6.5/
' (36)
where UH has to be expressed in cm/s. Let us recall that the Lewis number Le is defined with the binary molecular diffusivity of the limiting component in the neutral component, which is assumed to be abundant. Thus, as shown in [9], this number varies with the equivalence ratio from its value associated with the oxidant for very rich mixtures to its value associated with the fuel for the lean limit. For example, for a propane-oxygen mixture diluted with nitrogen, 1 is expected to vary from - 2.5 for rich mixtures to + 6 for the lean limit. And it seems difficult in conventional reactive mixtures to encounter values of I outside the range [3, 10] [II]. Thus, in the adiabatic case, H = 0, (36) shows that, at any equivalence ratio, planar fronts with sufficiently large (uu -> 17 cm/s) and sufficiently small (uH -< 5 cm/s) flame speeds are unstable and stable, respectively [II]. Furthermore, in
mixtures where the fuel is heavier than oxygen, the instability threshold is predicted to be experimentally observable between 5 cm/s < uH < 17 cm/s at a fuel rich composition for the lower values of uH and at a lean composition for higher values of uH [II]. When the expression ( 9 ' ) for H in terms of ua and K is introduced, (36) provides us with the influence of the heat losses on the limits of stability of planar fronts propagating downward in a given tube corresponding to the K which is characterized by the limiting speed u , ex given by (9). These stability curves plotted in the plane (uH, 1 -= ~(Le - 1)) for different tubes (different K) are universal in the sense that they can be used whatever the chemical nature of the reactive mixture. These limits have been plotted in Fig. 1. The 1 axis can be considered as measuring the variation of the equivalence ratio ~b, with 4~ > 1 for the small values of l and ~b < I for larger values of l. The precise relation linking 1 and 4~ cannot be obtained without a detailed analysis of the flame structure including realistic multistep kinetics [14]; Ref. [9] provides this relation only for an overall one step model. In comparison with the adiabatic case ( H = 0), the heat losses are found to be stabilizing when the velocity of propagation is larger than 10 cm/s (uH > 10 cm/s) and destabilizing for ur~ < 10 cm/s. In this slow burning domain, the proximity of the flammability limits is found to modify drastically the shape of the stability limit curve that has to end at the limiting speed of propagation u . ex. It is worth noticing that the corresponding limiting value lc* obtained by introducing H = 1 in (36) is independent of the flame propagation. This means that when the intersection of the stability limits and the flammability limits can be reached experimentally, the corresponding equivalence ratio is predicted to be independent of the tube and can be considered as a characteristic of only the chemical nature of the burning mixture. Such a conclusion has to be used with caution because (26) and (27) are no longer valid at the extinction limit. But, as mentioned at the beginning of this section, the results are still valid if H < 0.9
STABILITY OF PREMIXED FLAMES
'
2.72 I
* {=/~(.1-1/L, )
11
0.34 I
0,1 I
io4
o.2s
o.s
1
ao2 I 1oli
ao~2
I 2.3
I
,*e" a./ o
3.1
Ac (cm) i R:w
I K=O
K/
PC x
z
4~ tU ..J
4
STABLE
. . . . . .
u~~-. 4~
-u%. - ~ . . . . ~ - .8. ~
• EXTINCTIO~I LIMIT . t # u / e • H= I Ulna'/~
0
I "
12 nc
I---.11-5cm
x =E =734cm
UNSTABLE
oc
-4
A D I A B A T I C ; t,,,Ab~ ~ " - ' "
Fig. 1. Limit of stability plotted in the plane (u., Le) for nonadiabatic planar flames propagating downward in different tubes (1 cm < R < oo). This representation does not depend on the reactive mixture. The size of the cells at the instability threshold is plotted at the top of the figure.
and the domain in question, 0.9 < H < 1, is found in Fig. 1 to cover a relatively small area. There is also another limitation of the theory in the very slow burning rate because the results have been obtained by assuming that d / A is small compared with unity. Thus, as shown in Fig. 1, the domain of validity is roughly given by u . eX > 4 cm/s. For larger tubes (R > 10 cm/ s for Pyrex tubes of 0.5 cm thickness), the predicted stability curves have to be considered with caution when the velocity of propagation is smaller than 4 cm/s ( u , < 4 cm/s). But under ordinary laboratory conditions (R < 10 cm/s, u . ex > 4 cm/s) the results presented in Fig. 1 are not limited by this last restriction because, even at the extinction limit (where the cell size at the instability threshold is minimum), d / A is small enough. Concerning the experimental data, it is usual for evident practical reasons to plot the flame propagation un as a function of the equivalence
ratio q~ for different values of the dilution (~ fixed, /~ = [02]/([02] + [N:])). Thus, it is worthwhile to present the shapes that the stability and flammability limit curves take in this representation. In Fig. 2, we have plotted the stability limit predicted by the theory in the adiabatic case. The reactive mixture is modeled by a variation of the effective Lewis number Le with the equivalence ratio so that i = - 2 in the rich limit (~ ~ oo) and l = 6 in the lean limit (~b ~ 0); the equivalence ratio dependence Le(~b) derived in [9] is used. These numbers correspond roughly to oxygen (l = - 2 ) and heavy hydrocarbons (such as propane, l = 6), respectively, taken as the limiting component of a reactive mixture characterized by a reduced activation energy of/$ = 10 and diluted in nitrogen [II]. The isolation curves UL(~b)have been computed from the one overall step approximate model solved in the asymptotic limit/3 ~ oo where the
12
P A U L C L A V I N and C O L E T T E N I C O L I
18.
UL(CM/S) \
l
•,Mi,
STABILITY
I:
--"""'O"S''BLE x~\
(CELLULAR
FLAME)
"\\\\
10, 3:1s.5%
/11///
~:14 %
Oo
'" I
I
l
1"
t
~
I
I
0.4 1o 2.20 Fig. 2. Limit of stability in adiabatic conditions plotted in the plane (4~, UL) for different values of the dilution 8 E (14%, 15.5 %). The conditions are chosen to model a propane flame (C3Hs-Oz-N2).
adiabatic t e m p e r a t u r e o f c o m b u s t i o n and the p r e f a c t o r o f the A r r h e n i u s law have been adj u s t e d to r e p r o d u c e r o u g h l y the e x p e r i m e n t a l results o f o r d i n a r y h y d r o c a r b o n flames (as C 3 H s - O 2 - N 2 flames).
18t,
The c o r r e s p o n d i n g curves, UH($), with the limits o f stability and o f f l a m m a b i l i t y , have been plotted in Fig. 3 for the same mixtures but for the flame p r o p a g a t i o n in the tube R = 5 cm r e p o r t e d in Fig. 1. A s e x p e c t e d from Fig. 1, the
-~ UH (CM/$)
|STABlY'T: C, ,T , ' - - - -
ll!J.! \\\\\
4~
L
i
I
I
1
I
r
I
I
0.400 1, 2.00 Fig. 3. Limit of stability of the same mixture and in the same representation as in Fig. 2, but for a nonadiatic propagation characterized by a given value of K associated roughly with a Pyrex tube of 0.5 cm thickness and 5 cm radius.
13
STABILITY OF PREMIXED FLAMES flammability limit is effectively represented by the line ul~ = 6 cm/s; furthermore, the isodilution curves u.(
10o
_~II.(cM/s) V~
bs l~s
f \g~ SrABiLITY--"~"X \\\X~ LIMIT IIr/~ .... "~ \ "l~\',k -~ 1o.
I/' '~-~1 \ \ \
I
I
:A\ ~ \ ~_STABILITy ~:)~'"lr,~l \ I'~. LIMIT
o.
# I
OOOOO
i
t
1.
I
I ~
2,-00
Fig. 4. Limit of stability of the same mixture as in Fig. 2, but for a given dilution 15.5% and for different values of the radius of the tube.
same mixture as in Fig. 2 but for a given dilution = 15.5 % and for different values R of the radius of the tube. As mentioned above, the second limit of stability ( 1) of the adiabatic case (R --, oo) disappears as soon as the radius R decreases to the usual values used in laboratory experiments (R < 1 m). It is also worth noticing that the relevant limit of stability (close to ~ = 1) is nearly vertical in such a way that the critical value of the equivalence ratio does not depend too much on the radius of the tube.
VI. CONCLUDING REMARKS The present study points out the effects of heat losses on the limit of stability of premixed flames propagating downward. These theoretical results are expected to be important in connection with the experimental studies of cellular instability because the corresponding threshold is found [10], as is clearly explained by recent studies [II, 6], to be observable for slow burning flames which are close to their extinction limit in tubes. This clarifies the
14 confusion reported, for example, in [10] concerning stability and flammability limits.
PAUL CLAVIN and COLETTE NICOLI 7. 8. 9.
REFERENCES 1. Joulin, G., and Clavin, P., Acta Astronaut. 3:223 (1976). 2. Buckmaster, J., Combust. Flame 26:151 (1976). 3. Joulin, G., and Clavin, P. Combust. Flame 35:139 (1979). 4. Clavin, P., and Williams, F. A., J. Fluid. Mech. 116:251 (1982). 5. Pelce, P., and Clavin, P., J. Fluid. Mech. 124:219 (1982). 6. Clavin, P., and Garcia, P., J. Mec. 2 (l January 1983).
10. 11. 12. 13.
14.
Darrieus, G., Paper presented at 6th International Congress of Applied Mathematics, Paris, 1946. Landau, L. D., Acta Physicochim. (USSR) 19:17 (1944). Joulin, G., and Mitani, T., Combust. Flame 49:235 (1981). Markstein, G. H., Nonsteady Flame Propagation, Pergamon Press, New York, 1964. Frankel, and Sivashinsky, G., J. Comb. Sciences and Tech. (1982). Barenblatt, G. I., Zeldovich, Y. B., and Istratov, A. G., Prikl. Mek. Tekh. Fiz. 2:21 (1962). Quinard, J., Searby, G., and Boyer, L., Paper presented at the 9th International Colloquium on Combustion in Reactive Systems, Poitiers, July 1983. Clavin, P., Progress in Energy and Combustion Sciences (1983).
Received 25 June 1982; revised 9 M a y 1983
AND
FLAME
60: 1-14 (1985)
E f f e c t o f H e a t L o s s e s o n the Limits o f Stability o f P r e m i x e d F l a m e s Propagating Downwards P A U L CLAVIN and C O L E T T E N I C O L I Ddpartement de C O M B U S T I O N du L . A . 72 au CNRS, Universitd de Provence, centre de St Jdrdme, Rue H. Poincard--13397 M A R S E I L L E cedex 13, France
A previously developed theory of linear stability analysis of nonadiabatic premixed flames that was based on a diffusive thermal model is extended by eliminating the hypothesis of negligible expansion of the gas. The effects of heat losses on the limit of stability of slow burning flames propagating downward are investigated in detail. The influence of the proximity of the flammability limits upon the stability limits is studied with a view to describing the cellular threshold for flames propagating in tubes.
I. I N T R O D U C T I O N The quenching mechanism of premixed flames by heat losses has been completely described by using asymptotic analysis in large values of the reduced activation energy (/3 ~ 00) [1, 2]. The linear stability analysis of these nonadiabatic flames has been more recently investigated by Joulin and Clavin [3] (referred to as [I] in the following) in the diffusional-thermal model. This model [12] excluded density variations, thereby uncoupling the continuity and momentum conservation equations from the equations of species and energy conservation. This serves the useful purpose of describing the diffusive effects while eliminating hydrodynamics instability described in [7, 8]. Some mathematical justification for the adoption of this model can be given by the double limit
/5
~
-
E
(Tb-T.)
-
-
R T~2
-
Tb
and
3"= (p~- pb)/pu--,o. Copyright © 1985 by The Combustion Institute PubliShed by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017
u and b are subscripts referring to unburned and burned gases in adiabatic conditions, respectively. T is the temperature and p the density. But the troubling approximation 3' --* 0 is not realistic for the usual hydrocarbon flames and prevents a relevant description of the instability threshold for the appearance of cellular structures. In a recent work by Pelce and Clavin [5] (referred to as [II]) based on the analysis of the wrinkled flame structure of Clavin and Williams [4], the dynamical properties of adiabatic flame fronts have been investigated in the limit of large wrinkles A compared with the flame thickness d but for any gas expansion parameter 0 < 3" < 1, ~ = d / A ~ O. The coupling of the hydrodynamical effects with the diffusive phenomena has been thus described completely in the limit of small values of e and for large values of the reduced activation energy /5. A similar analysis has also been recently developed by Frankel and Sivashinsky [11]. It is worth noticing that, according to the results presented in [II], the acceleration of gravity associated with the diffusive effects taking place inside the flame is shown possibly to overcome the hydrodynamic instability mech-
O010-2180/85/$03.30
PAUL CLAVIN and COLETTE NICOLI anism produced by the gas expansion 3, :# 0. Under a critical value of the flame speed depending on the properties of the reactive mixture, planar fronts propagating downward are predicted to be stable. Thus, as recently verified in our laboratory [13], the cellular threshold is predicted to be experimentally observable on slow burning flat flames propagating downward. Furthermore, the size of cells at the threshold is found to verify effectively the basic assumption of the theory, e < 1. But the flame velocities corresponding to the threshold are found to belong to the range 5-17 cm/s, which is fairly close to the usual values of the flame velocity at the extinction limit in tubes ( - 7 cm/ s). Thus the stability limits are found to be close to the flammability limits. This explains the confusion that has been often caused in the past as explained in [10]. Nevertheless it can be expected that in the same experimental conditions as those used by Markstein [10] for studying the cellular threshold, the heat losses must play a nonnegligible influence on the stability limit. The present paper is devoted precisely to the study of such effects on planar fronts propagating downward in tubes, when the coupling between hydrodynamical and diffusive phenomena produced by the gas expansion is fully taken into account. This work is a natural extension of [I] and [II] and is developed in the same spirit as in [II]. The reader is referred to these references for a detailed technical presentation. For brevity, only the modifications of the calculations induced by the phenomena studied in this paper are explicitly written down. Nevertheless this paper is written in a self-contained form in the sense that a reader not necessarily concerned with the mathematical techniques can understand the general physical ideas and more especially the conclusions. Section II is devoted to recalling the basic concepts that are associated with the heat loss phenomena. Section III concerns the formulation of the problem as well as the presentation of the spirit of the method. The analysis is sketched in Section IV, and the results together with the
conclusions are presented in Section V, where an effort has been made to make easier the comparison of the predicted results with experiments. In particular, we have plotted the predicted stability and flammability limits on the same curves "flame velocity versus equivalence ratio" for different values of the dilution and also for different values of the radius of the tube controlling the intensity of the heat losses. II. T H E F L A M M A B I L I T Y L I M I T S The quenching mechanism of premixed flames by heat losses was studied in [I] with an asymptotic method valid in the limit of very large values of the reduced activation energy 3 (3 ~ oo). An Arrhenius law is adopted for the rate of the release of the chemical energy such that as 3 --' oo, the chemistry is confined to a surface [1, 6]. This reaction zone separates the preheated zone (where the species concentrations as well as the temperature evolve under only the transport processes) from the burned gases (where the concentration of the limiting component is zero). The foundation of the asymptotic method is based on matching the solutions that have been preliminarily obtained separately in the preheated zone and in the reaction zone by using the boundary conditions in the unburned and burned gases, respectively (see [I] and [liD. The quenching mechanism can be modeled by a volumetric heat loss that introduces a sink term, - K ( T - Tu), in the equation for the conservation of energy. K( T Tu) corresponds to the energy lost per unit time for a unit volume of reactive mixture at temperature T. For flames propagating in tubes, K is roughly inversely proportional to the radius R of the tube. One introduces the ratio of the actual transit time rr~ = P D t h / p u U t t 2 and of the characteristic time of cooling rc = puCp/K,
H=23rrt=23 ( K /PuUH2~ K(pDth) = 2 3 (ouUI02Cp ,
(1)
where urt is the laminar flame speed in the
STABILITY OF PREMIXED FLAMES presence of heat losses, Oth is the thermal diffusivity of the reactive mixture, and Cp is its specific heat per unit mass. The fact 2/3 is arbitrary in order that the extinction limit correspond exactly to H = 1. For heat loss intensities K such that H remains of order unity, in the limit/3 --. ~ [ H = O(1)], the modification of the flame temperature can be easily obtained by solving the conservation equations in the preheated zone to give [I]
diffusive flux (5) must equal, at the dominant order in the limit/3 ~ 0% the convective flux leaving the preheated zone and given by (PuUn)Cp(Tb -- Tu) in the plane case. This yields
~ ( 0 f - 1) = - H ,
\ $/Li
(2)
where 0 is the reduced temperature, g = ( T Tu)/(Tb - Tu), and where the subscript f refers to the flame temperature. But, as clearly shown by (1), H depends on the flame velocity un. Let us define a similar ratio of times as in (1) but using as a reference the transit time re (re = rn=o) of the flame sustained by the same reactive mixture but propagating in the adiabatic conditions with the laminar flame speed Ur (Ue = UH=O):
rc
\puCp" pDth ,/ "
(3)
This can be written as 2/3 rL
1 K
with
K* -
(#uUL)2Cp
e 2/3(ODth)
(3 ')
(4)
where K* is, as UL, a characteristic of the reactive mixture and K is a characteristic of the tube in which the flame propagates. The investigation of the reaction zone provides the expression (in terms of the flame temperature) of the diffusive flux of heat leaving the reaction zone toward the preheated zone: O u U L C p ( T b --
Tu)e a(af- u/2.
In
=
e K*
.
(5)
But the matching principle requires that this
(7)
This result has been obtained in [I]. Equation (7) describes an extinction process in the sense that, as soon as the heat losses are too strong, K > K*, there is no possible flame propagation. The extinction limit corresponds to K = K* and to a limiting flame velocity u . given, according to (7), by
(8)
u, = u d ~ e
in such a way that, according to (4), one has H = 1 at the extinction limit
,
(ua~21n [!(u.~27= u./ \u,/ 9
Thus (1) can be written as K 1 (UL") 2, H=K---~ e un/
(6)
to obtain, when (2) and (4) are used,
H=K,
rc e K* ' 1
UH) = e¢(6f- O/2,-
K K*
(8')
For heat loss intensity smaller than the critical value, K < K*, two possible steady solutions appear, one corresponding to u , < uH < UL ( H < 1) and the other corresponding to 0 < u , < u . (1 < H ) . This last regime has been proved in [I] to be unstable for one-dimensional (planar) perturbations (zero wave number) and thus cannot be relevant for the description of the cellular structures. Then, only the branch of solutions corresponding to 0 < H < 1 and reducing to the adiabatic flame velocity UL in the limit K --* 0 has to be considered. It is worth noticing that, according to ( 3 ' ) , the extinction limit K = K* corresponds to a characteristic time of cooling rc greatly larger than the transit time rL, re/7" L = 0 ( / 3 ) , and thus, for large values of/3, a small amount of heat
4
PAUL CLAVIN and COLETTE NICOLI
losses (compared with the heat released) is sufficient to quench the flame. It is interesting to consider the situation corresponding to the propagation in a given tube (K given) of a given reactive mixture (/3, pDth, Cp given) at different values of the composition. By varying the equivalence ratio ~ and/or the dilution, one varies the adiabatic flame speed UL as well as the flame propagation velocity in the tube uH given by (7). Usually the experiments are carried out by varying ~b with ~ kept constant. Beyond extinction values of ~b, there is no possible flame propagation. These extinction values of the composition depend in fact not only on the reactive mixture but also on the tube. This pointed out the difficulties involved in an intrinsic definition of the flammability limits. It turns out that the corresponding values of the adiabatic flame speed UL~Xas well as of the limiting speed of propagation in this tube, u . ~X ( = UL~X/ x/e), is a characteristic of the tube. These values UL~x and u , cx are directly obtained from (3 ') by replacing K* and UL by K and UL~x, respectively, to give (puu.eX) 2 = 2/3
pDth K
Cp
(9)
and (4) takes the form H=(
u*ex~2"uH /
(9')
This last expression ( 9 ' ) of the reduced heat losses is very useful when one works at a given value of K, as for the propagation in a given tube (K fixed, ~b and ~ variable). The other expression (8') is interesting when one works at a given composition (UL, U, and K* fixed) but with variable heat losses. As an example, when K is roughly evaluated by K = 23,/Rh (where X, R, and h are the thermal conductivity, the radius, and the thickness of the tube, respectively), the limiting speed of propagation in a Pyrex tube corresponding to R = 5 cm, and h = 0.5 cm is, according to (9), nearly equal to 6 cm/s: u . ~x 6 cm/s. And all the flames whose adiabatic flame speeds are smaller than 10 cm/s (ULex =
x/eu. ~x - 10 cm/s) are predicted not to propagate in this tube. Equation (9) means that when one considers the dependence of the velocities of the flame propagation (in a given tube) with the equivalence ratio ~ for different values of the dilution, all the corresponding isodilution curves are predicted to end at the same limiting speed value u . ~x (cf. Fig. 3). Such a result, which is obtained by assuming that the quantities K, ~, PDth, Cp are kept constant (independent of ~b and/~), has been roughly confirmed by the recent experimental work of J. Quinard [ 13]. Let us now go to the stability analysis of these steady state planar solutions whose range is limited by the above mentioned flammability limits. III. F O R M U L A T I O N The nonadiabatic flame model is identical to the one in [I] and the formulation is similar, but the thermal-diffusive approximation is removed herein. Attention will continue to be restricted to exothermic reactions amenable to a one step approximation, with the degree of progress describable in terms of the mass fraction Y of a single reactant, taken to be the limiting reactant in the sense that Y = 0 defines completion of the reaction. An Arrhenius rate is adopted so that, as the reduced activation energy goes to infinity (/3 ~ ~ ) , the chemistry is confined to a fluctuating surface, whose position is employed to define the origin of a moving coordinate system, pDth and the Lewis number Le, the ratio of the thermal diffusivity Dth of the mixture to the molecular diffusivity of the limiting component, are assumed to be constant. 1 The heat losses are modeled by a volumetric sink term as presented in the above section. The conservation equations concerning the unsteady wrinkled front are presented in detail for the adiabatic case in Ref. [4]. The nonadiabatic case is described by the same equations but with the above mentioned sink term -H0/2/3 added to J This restriction can be easily removed, and the temperature dependence of pD can be included easily as in Ref. [6].
STABILITY OF PREMIXED FLAMES the conservation equation of the energy. In the stability analysis, one is dealing with these equations linearized around the one-dimensional steady state solution (see [II] for the adiabatic case). At the dominant orders in the asymptotic expansion (/5 ~ oo), the study of the thin reactive zone is identical to that in [1] and [II]. Then, at these dominant orders, the matching principle reduces to jump conditions across the flame sheet that take the following form in the linearized approximation, a0 1 Off ~:o+ ~ + ~ ~=o-
00 ~=0-
= 1/2/50f,
=0,
(10.1)
(10.2)
where Of = 0(~ = 0) = O(1//5). Here 0 and are the perturbations of the nondimensional temperature ( T - Tu)/(Tb - Tu) and the nondimensional mass fraction Y ~ Y . of the limiting component, respectively (~ = 0, ff = 1, 0 = 0, ~b = 0 in the fresh mixture). 2 The location of the flame sheet is defined by x = or(y, z, t) in the laboratory frame, where x, y, z, t are nondimensional space and time coordinates defined with the flame thickness d = pDth/puUH and the transit time ~'H = d/UH choosen as units oflength and time. ~ = x - a , ~/ = y, ~" = z, z = t are the nondimensional moving coordinates. Equation (10.1) is valid in the asymptotic limit as/3 ~ 0o up to the order O(1//5). Equation (10,2) is valid at the dominant order O(1) and is restricted to a perturbation of the flame temperature Oe (defined at ~ = O) that is of order O ( 1 / /5).3 For heat losses such that H = O(1), this is satisfied whenever the Lewis number Le is close enough to unity to make (Le - 1) of order O ( 1 / /5). Thus, one introduces / = /5(Le - 1). Then, when / = O(1), the condition Of = O(1//5) is satisfied for reduced wavelengths of the wrinkled front A / d of order O(1) in the limit/3 --* 00. 2 The overbar denotes the one-dimensional steady state solution. 3 The perturbation of the flame temperature 0 is in fact of order O(1) in the preheated zone (~ < 0) but 0(~ = O) = O(1//3) (see [I]).
Nevertheless, as in [4], attention will continue to be focused on realistic situations for which the wavelengths A are large compared with the flame thickness. Thus, d / A will be considered as proportional to a small parameter 4 e (e < 1) independent of/5. Thus, after having used the limit 13 ---, 0o that confines the chemistry in a reactive sheet through which the jump conditions (10) are verified, one looks to find the outer solutions as power expansions in e. This later restriction is not necessary in the thermal diffusive approximation where, as in [I], the outer solutions can be found whatever e's value. But when the effects of the gas expansion are taken into account, the ronservation equations of energy and species are coupled with the conservation of momentum and the outer problem has to be solved in perturbation expansions around the one-dimensional solution. This has been done in [4] for the adiabatic case and has been used in [II] to study the corresponding stability limits of planar fronts propagating downward. As in Refs. [4] and [II] (see also Ref. [6]), the outer equations for the perturbations of temperature and concentrations in the nonadiabatic case reduce, at the second order e 2 in the power expansion in e, to the following quasi-planar and quasi-steady equations: 5 O0
020
H
-----+--
0
= - ( s ( O + O20dOZ 2) dO + O(e3),
(1 la)
d~ OXb O~
1
02~
LeO~ z
1
= - (s(~) + - - 0 2 . / O z 2) d ~ + O(e3), Le
(1 lb)
where 0(~) and ~(~) are the one-dimensional 4 ~ (defined here as the order of magnitude of d/A) has nothing to do with the quantity ~ defined in [I] as the amplitude of the perturbation around the one-dimensional steady solution. 5 To simplify the notation, only one transverse condinate z is explicitly specified in the following.
PAUL CLAVIN and COLETTE NICOLI steady state solutions of the nonadiabatic case characterized by H . The expressions for 0 and can be found in Refs. [1] and [I] [see also Eq. (18c)]. s(~) identifies the perturbation of the nondimensional longitudinal (x component) mass flux in the moving coordinate system, s = e(u - Oct/Oz) + ra,
~= ?t~,
(12)
where f(~) and a(~) are the steady state profiles of the reduced density and of the x component of the reduced flow velocity, respectively. These quantities are defined by r = p/pu and u = u / u n in such a way that ~ --, - ~ , f = 1 , r = 0, a = 1, u = 0. In fact, s(~) couples Eqs. (11) to the hydrodynamics through the continuity equation Or~Or + OslO~ + O(fv)/OT1= O,
(13a)
g(O = ~(~)f(~) = 1.
(13b)
Here v denotes the z component of the flow velocity and, according to the usual "isobaric approximation" Of flame theory (negligibly small Mach number), r is simply related to 0 through the perfect gas law ,y r =
-
~
(1 4a)
r20
1-3,
?=
7 5 ) -1 1+1_,7
s = eS_**(e/j, Z, T) + esl(~, Z, T) + ees2(~, Z, T) + O(e3), 0 = ~202(~, Z, T) + O(c3), = E21~2(~, Z, T) + O(e3),
s-~**= cS±**(~, Z, T), u~** = eU±**(~, Z, T), v±o.=cV±~,(~,, Z, T),
and if±** = 0± 00= r± o, = 0.
The quantities involved in (16) should be understood as being O(1) in the limit as/3 ~ oo and where ~ = e~, the subscripts - a o and +oo denoting the upstream and downstream incompressible gas flow, respectively. According to the definition given by (12), one has in these regions and
s+ = = (1 - 3`)(u+ = + Oct~Or).
u=eU_®(e~, Z, T)+eul(~, Z, T)
(17)
All the quantities appearing in (15) and labeled with the subscripts 1, 2 . . . . will be found to go to zero like e ~ when ~ --* - c o , and thus the matching between the upstream incompressible zone and the preheated zone is systematically satisfied. The connection of the preheated zone with the downstream incompressible flow is made through the thin reactive zone by the jump conditions (10) and those relating the characteristics of the gas flow p, u, v that can be found in [II, Eq. (13)].
(15a) IV. ANALYSIS
v=eV_**(e~, Z, T)+evl(~, Z, T)
"t- e21)2(~, Z, T) + O(e3),
(16)
(14b)
where 3' = (Ou - Pb)/0u = (Tb -- T,)/Tb is the gas expansion parameter. The x and z components u, o of the reduced velocity of the gas flow appearing in (12) and (13) are given by the same equation for the momentum conservation as in Refs. [41 and [II]. Thus, it is not necessary to reproduce explicitly these equations that have to be solved, as in the previous works [II, 4, 6], by perturbative expansions in powers of e. The following expansions are introduced,
+ e2u2(/j, Z, T) + O(e3),
(15d)
where Z = ez, T = er, and the quantities U_~., V_o,, S_** have to be developed around x = 0. Equations (15) are valid inside the preheated zone. Outside the flame, the gas flow varies on the same spacial scale A as the wrinkles of the front, and the solutions take the follow form:
s_ ~, = Uo. - Oa/ Ot ,
(15c)
(15b)
The integration of (11) in the preheated zone (x
STABILITY OF PREMIXED FLAMES asymptotic limit/5 --* + co. By adding the two equations (18a) and (18b) written for ~ = 0 and by using (19), the jump condition (10.1) leads to the following expression for Of = 0(~ = 0) = 0(1//5),
< 0) yields
~<0:
2
**
® (s+a2o~/az 2) ~ d~ = ~ ,
+
(18a)
d~
~<0:
ld¢
x I~** (s + 82ctlOz2)e ~ d~ - l 020l/OZ2 (18b)
Led~'
+(l-H)
o,(s+020t/Oz2)~e~ d~
where ti(~) = ~(0)e x- ~
and
~ = 1 - e I~
(18c)
+-2
(s+o,+O20t/Oz2)e-~ d~
with ,, X- = 1 + - - + 0 2/5
and where the term t °_~, 0 dx appearing in (18a) has been obtained from the dominant order of the asymptotic expansion (/5 ~ co) of (1 la):
,, ~t(0) = 1 - - - + O
In the downstream incompressible zone (~ > 0) located just behind the thin reactive zone, one has, according to [1] and [I], dJ//d~ = 0 and (11) yields ~>0:
~b=0,
~>0:
-'~ ~=o+:Ofx+
_
(19a) Furthermore Eq. (18a) leads to the dominant order of (dO/dOi ~o- :
dO
(S+O20I/OZ2)__dOe_x_ ~ d~, d~
_
(19b)
L
(s+02ot/Oz2)e ~ d ~ + O ( ~ ) . (22)
where X+ . . . .
I~. o d~
4- 0
2/5
0(0 = g(O)ex-L
and (19c)
It is clear that the dominant order of (19b) is O(1//5). Moreover, according to the above discussion, the expansion Le = 1 + l//5 is introduced in (18b) with / = O(1) in the
It is worth noticing that only the dominant order [O(1)] of the asymptotic expansion (/5 co) o f s and a is involved in (20) and (22). Thus, in the following s and ct should be understood as being O(1) in the limit/3 --* co. Then, after (20) and (22) have been introduced in the jump condition (10.2), one obtains an integral relation for s. As soon as the modification through the flame of the longitudi-
PAUL CLAVIN and COLETTE NICOLI nal mass flux [characterized by the terms sl(~) and s2(O of the expansion (15c) of s] is known, this integral relation provides the expression for s_**(E = O) = eS_~.(E = 0)and also, according to (17), the equation for the evolution of the flame front. These terms sl(~) and s2(O are given by the continuity equation (13a). At the order e 2, (13a) reduces to a quasi-static and planar equation which can easily be integrated when the transverse gas velocity v is known. This part of the analysis is quite similar to those in the previous works [4] and [II]. Equation (13a) yields sl =0,
Ii
/I-~ In(1 + 0
D=
1
~'
a(O)
f((gvl/(gZ) d~,
(23)
where v~ is simply given by the deflection of the streamlines through the tilted front (see [II, Eq. (23)1): 111
=
- -
((ga/(gz).
3'/(1 - 3")e~,
ul = 0.
(24)
dO
S
a(O) In 0
o 1 + [ 7 / ( 1 - 3,)]0 dO,
respectively, with a(O) denoting the ratio of the thermal diffusivity at the temperature 0 to its value at the unburned temperature. The calculation of (26) and (27) is carried out by introducing in (20) and (22) the expansions of s and s+ 0, in powers of ~ obtained from (15c) by using
S_ ~ ( ~ ) = S_ ®(0) + ~
This leads to (see [II, Eq. (24b)])
and
1 - 3' Jo 1 + [3"/(1 - ~,)]0
1 - 3"
+
#~.
W h e n the temperature dependence of the diffusivity is taken account (as in the adiabatic case-see Ref. [6]), ~3/d takes a more complex form than (27) but where in particular (I/3')In [I/(l -3")] and D have to be replaced by
3"
( e - 1) d~
s2(~)=(9O_o./(gZ[~=o
with
as_
j a ~ I~=0 + o(~2), (28a)
s2(O=((92od(gZ2+(gv+~/(gZ) In
1 + 1-3,
S_ ~.(~ = 0) = O(e).
' (25)
By following the procedure outlined just above, one obtains the following linear equation for the evolution of the front which is valid up to the order e 2, (ga
- - = U_ a.(~ = O) -i- ,~/d((920t/(gZ 2 q- (90_ ¢./(9Z1~ =0),
Or
(26) which is quite similar to Eq. (29) in [II] but where the length ~ is given now by
,[
~3/d= 1 - H
+
°
(I -3")
I-H+(1-3') 1
( / - H)
J
,
(28b)
where s+®(0) is given by the continuity of the mass flux through the reaction zone S+ ~(0) = ~S-oo(0) + e2S2(0)+ O(~ 3)
(29)
and where, according to the definitions (16) and (17), one has
as_,ja~ I~=o=
au_
J a ~ Iz=o,
as+Ja~,l~.=o=(l-3") aU+®la~l~:o.
(30)
The mass conservation upstream and downstream from the flame yields (gU+**/O~, = OV_**/OZ and OU+~/O~, = -OV+**/(gZ. The continuity of the transverse component of the flow velocity through the reactive zone leads to (9S+**1(9~ I ~ =o= - (1 - 3")(8 V_ ®/(gzl~ =o
~ In
1 - 3' D /
s+~=s+®(0)+e~ as+ja~[~°o+O(~Z),
(27)
+ (gul/(gZl~ =o),
which has to be introduced in (28b).
(31)
9
STABILITY OF PREMIXED FLAMES V. D I S C U S S I O N OF R E S U L T S The form of the local equation (26) for the evolution of the front is similar to that in the adiabatic case [4], but the heat losses affect the expression (27) of the Markstein length £ . In the adiabatic case, H = 0, (27) reduces effectively to the result in [4]. Furthermore, when the effects associated with the gas expansion are neglected (3~ = 0) u_ ~ = v_ ~. = 0, (26) with (27) reduces to the long wave limit of the result obtained previously for the thermal-diffusive model [I]:
0o1[0
(1 - H ) + ~
'] l
V2~.
(32)
As clearly exhibited by the presence of the factor (1 - /-/) in the denominator, the results (26) and (27) are no longer valid close to the extinction limit defined by H = 1 where (27) diverges. As in [I], near the extinction, the equation for evolution must involve an acceleration effect O2ol/Or 2 which becomes of the same order of magnitude as (1 - H ) 0cdOr. But the dimension of corresponding domain around H = 1 is very small. For example, for the usual value of the gas expansion (3' = 0.8), (27) yields 3, = 0.8 = ~/d: 1
l _ H (2.011-2.1544H+0.29331),
(33)
and thus for the usual Lewis number I = O(1) ( - 3 < I < 6), g,/d is seen to take a large value ( > 1 0 ) only for conditions very close to the extinction limit (0.9 < H < 1). For example, for H = 0.9, (33) gives 0.72 + 2.93 l, which is not really large, and the present analysis can be used without restriction up to more than H = 0.9. It is also worth noticing that according to (33) the value of l associated with g,/d = 0 is given by - 6 . 8 6 + 7.35 H . Contrary to the adiabatic case, when the heat losses are sufficiently important ( H > 0.5) this value of / corresponds to Lewis numbers which may be attained by the usual reactive mixtures of hydrocarbon/air flames (/ > - 3 ) . Thus the heat losses may produce an instability mechanism of
a purely diffusive nature which cannot occur in the adiabatic case [4, II]. But in any case, this purely diffusive instability cannot appear on planar fronts of flames propagating downward. The reason is that there is another mechanism of instability of a hydrodynamical nature which appears before, and corresponds to, a positive value of ~ / d . In fact, as explained in [II], the hydrodynamical instability mechanism, first pointed out by Darrieus [7] and Landau [8], cannot be overcome without sufficiently strong stabilizing effects of the acceleration of gravity and the diffusion processes. This requires sufficiently small burning velocities UL and sufficiently large positive values of the "Markstein coefficient" ~3/d. The corresponding detailed analysis requires investigation of the complete fluid mechanics problem associated with the wrinkling of the front. Such an analysis has recently been carried out for the adiabatic condition [II]. The intensity of the reduced heat losses involved in the nonadiabatic case is indeed very small in the limit of large values of the activation energy. In fact, at the extinction ( H = 1), this intensity is O(1/~) in the limit as ~ oo. Thus, outside the preheated zone of the flame, the nondimensional gradient of the temperature produced in the burned gases by the heat losses is sufficiently small [O(1//3) when/3 oo] to be neglected in the equations governing the gas flow outside the flame thickness. In the limit as/3 --* oo, only the dominant order 6 O(1) of the corresponding flow field is necessary to solve the problem. Thus, the fluid mechanical problem is identical with the adiabatic analysis [II] and will not be reproduced here. The result presented in [II] can be directly used by simply replacing the "Markstein length" by its modified value (27). The first important point is that the wavelength Ac of the cellular structure appearing at the instability threshold was found not to involve the "Markstein length" and to vary with only the laminar flame velocity urt,
A,:/d = 7ru~2/(1 - ~)gd,
(34)
6 Notice that this is not the case for the temperature field where the two orders O(1) and O(1//~) are required [see (18)].
PAUL CLAVIN and COLETTE NICOLI
10 Where g is the acceleration of gravity. Thus, according to the discussion given above, the relation (34) linking the dimension of the cells at the threshold to the flame speed uH is found to be unaffected by the heat losses. This is very encouraging for obtaining relevant experimental data for Ac by using laboratory burners in which the heat losses can never be completely suppressed. The critical value ~¢ of the "Markstein length" at the threshold obtained in [II] for the adiabatic condition is still valid [II, Eqs. (39)(41)1:
13e/d= 3" 2+3,
In - - 1-3,
1
-~ 4(1 - 3,)g But according to (27), the corresponding critical value Lec of the Lewis number is affected by the heat losses. For 3, = 0.8 and pDth/Ou = 0.35 cm2/s, one obtains from (27), (33), and (35) the following expression for the critical value /~ [--/~(Lee - 1)], l~=-3.85+4.34H+0.98(1-/-/)
\6.5/
' (36)
where UH has to be expressed in cm/s. Let us recall that the Lewis number Le is defined with the binary molecular diffusivity of the limiting component in the neutral component, which is assumed to be abundant. Thus, as shown in [9], this number varies with the equivalence ratio from its value associated with the oxidant for very rich mixtures to its value associated with the fuel for the lean limit. For example, for a propane-oxygen mixture diluted with nitrogen, 1 is expected to vary from - 2.5 for rich mixtures to + 6 for the lean limit. And it seems difficult in conventional reactive mixtures to encounter values of I outside the range [3, 10] [II]. Thus, in the adiabatic case, H = 0, (36) shows that, at any equivalence ratio, planar fronts with sufficiently large (uu -> 17 cm/s) and sufficiently small (uH -< 5 cm/s) flame speeds are unstable and stable, respectively [II]. Furthermore, in
mixtures where the fuel is heavier than oxygen, the instability threshold is predicted to be experimentally observable between 5 cm/s < uH < 17 cm/s at a fuel rich composition for the lower values of uH and at a lean composition for higher values of uH [II]. When the expression ( 9 ' ) for H in terms of ua and K is introduced, (36) provides us with the influence of the heat losses on the limits of stability of planar fronts propagating downward in a given tube corresponding to the K which is characterized by the limiting speed u , ex given by (9). These stability curves plotted in the plane (uH, 1 -= ~(Le - 1)) for different tubes (different K) are universal in the sense that they can be used whatever the chemical nature of the reactive mixture. These limits have been plotted in Fig. 1. The 1 axis can be considered as measuring the variation of the equivalence ratio ~b, with 4~ > 1 for the small values of l and ~b < I for larger values of l. The precise relation linking 1 and 4~ cannot be obtained without a detailed analysis of the flame structure including realistic multistep kinetics [14]; Ref. [9] provides this relation only for an overall one step model. In comparison with the adiabatic case ( H = 0), the heat losses are found to be stabilizing when the velocity of propagation is larger than 10 cm/s (uH > 10 cm/s) and destabilizing for ur~ < 10 cm/s. In this slow burning domain, the proximity of the flammability limits is found to modify drastically the shape of the stability limit curve that has to end at the limiting speed of propagation u . ex. It is worth noticing that the corresponding limiting value lc* obtained by introducing H = 1 in (36) is independent of the flame propagation. This means that when the intersection of the stability limits and the flammability limits can be reached experimentally, the corresponding equivalence ratio is predicted to be independent of the tube and can be considered as a characteristic of only the chemical nature of the burning mixture. Such a conclusion has to be used with caution because (26) and (27) are no longer valid at the extinction limit. But, as mentioned at the beginning of this section, the results are still valid if H < 0.9
STABILITY OF PREMIXED FLAMES
'
2.72 I
* {=/~(.1-1/L, )
11
0.34 I
0,1 I
io4
o.2s
o.s
1
ao2 I 1oli
ao~2
I 2.3
I
,*e" a./ o
3.1
Ac (cm) i R:w
I K=O
K/
PC x
z
4~ tU ..J
4
STABLE
. . . . . .
u~~-. 4~
-u%. - ~ . . . . ~ - .8. ~
• EXTINCTIO~I LIMIT . t # u / e • H= I Ulna'/~
0
I "
12 nc
I---.11-5cm
x =E =734cm
UNSTABLE
oc
-4
A D I A B A T I C ; t,,,Ab~ ~ " - ' "
Fig. 1. Limit of stability plotted in the plane (u., Le) for nonadiabatic planar flames propagating downward in different tubes (1 cm < R < oo). This representation does not depend on the reactive mixture. The size of the cells at the instability threshold is plotted at the top of the figure.
and the domain in question, 0.9 < H < 1, is found in Fig. 1 to cover a relatively small area. There is also another limitation of the theory in the very slow burning rate because the results have been obtained by assuming that d / A is small compared with unity. Thus, as shown in Fig. 1, the domain of validity is roughly given by u . eX > 4 cm/s. For larger tubes (R > 10 cm/ s for Pyrex tubes of 0.5 cm thickness), the predicted stability curves have to be considered with caution when the velocity of propagation is smaller than 4 cm/s ( u , < 4 cm/s). But under ordinary laboratory conditions (R < 10 cm/s, u . ex > 4 cm/s) the results presented in Fig. 1 are not limited by this last restriction because, even at the extinction limit (where the cell size at the instability threshold is minimum), d / A is small enough. Concerning the experimental data, it is usual for evident practical reasons to plot the flame propagation un as a function of the equivalence
ratio q~ for different values of the dilution (~ fixed, /~ = [02]/([02] + [N:])). Thus, it is worthwhile to present the shapes that the stability and flammability limit curves take in this representation. In Fig. 2, we have plotted the stability limit predicted by the theory in the adiabatic case. The reactive mixture is modeled by a variation of the effective Lewis number Le with the equivalence ratio so that i = - 2 in the rich limit (~ ~ oo) and l = 6 in the lean limit (~b ~ 0); the equivalence ratio dependence Le(~b) derived in [9] is used. These numbers correspond roughly to oxygen (l = - 2 ) and heavy hydrocarbons (such as propane, l = 6), respectively, taken as the limiting component of a reactive mixture characterized by a reduced activation energy of/$ = 10 and diluted in nitrogen [II]. The isolation curves UL(~b)have been computed from the one overall step approximate model solved in the asymptotic limit/3 ~ oo where the
12
P A U L C L A V I N and C O L E T T E N I C O L I
18.
UL(CM/S) \
l
•,Mi,
STABILITY
I:
--"""'O"S''BLE x~\
(CELLULAR
FLAME)
"\\\\
10, 3:1s.5%
/11///
~:14 %
Oo
'" I
I
l
1"
t
~
I
I
0.4 1o 2.20 Fig. 2. Limit of stability in adiabatic conditions plotted in the plane (4~, UL) for different values of the dilution 8 E (14%, 15.5 %). The conditions are chosen to model a propane flame (C3Hs-Oz-N2).
adiabatic t e m p e r a t u r e o f c o m b u s t i o n and the p r e f a c t o r o f the A r r h e n i u s law have been adj u s t e d to r e p r o d u c e r o u g h l y the e x p e r i m e n t a l results o f o r d i n a r y h y d r o c a r b o n flames (as C 3 H s - O 2 - N 2 flames).
18t,
The c o r r e s p o n d i n g curves, UH($), with the limits o f stability and o f f l a m m a b i l i t y , have been plotted in Fig. 3 for the same mixtures but for the flame p r o p a g a t i o n in the tube R = 5 cm r e p o r t e d in Fig. 1. A s e x p e c t e d from Fig. 1, the
-~ UH (CM/$)
|STABlY'T: C, ,T , ' - - - -
ll!J.! \\\\\
4~
L
i
I
I
1
I
r
I
I
0.400 1, 2.00 Fig. 3. Limit of stability of the same mixture and in the same representation as in Fig. 2, but for a nonadiatic propagation characterized by a given value of K associated roughly with a Pyrex tube of 0.5 cm thickness and 5 cm radius.
13
STABILITY OF PREMIXED FLAMES flammability limit is effectively represented by the line ul~ = 6 cm/s; furthermore, the isodilution curves u.(
10o
_~II.(cM/s) V~
bs l~s
f \g~ SrABiLITY--"~"X \\\X~ LIMIT IIr/~ .... "~ \ "l~\',k -~ 1o.
I/' '~-~1 \ \ \
I
I
:A\ ~ \ ~_STABILITy ~:)~'"lr,~l \ I'~. LIMIT
o.
# I
OOOOO
i
t
1.
I
I ~
2,-00
Fig. 4. Limit of stability of the same mixture as in Fig. 2, but for a given dilution 15.5% and for different values of the radius of the tube.
same mixture as in Fig. 2 but for a given dilution = 15.5 % and for different values R of the radius of the tube. As mentioned above, the second limit of stability (
VI. CONCLUDING REMARKS The present study points out the effects of heat losses on the limit of stability of premixed flames propagating downward. These theoretical results are expected to be important in connection with the experimental studies of cellular instability because the corresponding threshold is found [10], as is clearly explained by recent studies [II, 6], to be observable for slow burning flames which are close to their extinction limit in tubes. This clarifies the
14 confusion reported, for example, in [10] concerning stability and flammability limits.
PAUL CLAVIN and COLETTE NICOLI 7. 8. 9.
REFERENCES 1. Joulin, G., and Clavin, P., Acta Astronaut. 3:223 (1976). 2. Buckmaster, J., Combust. Flame 26:151 (1976). 3. Joulin, G., and Clavin, P. Combust. Flame 35:139 (1979). 4. Clavin, P., and Williams, F. A., J. Fluid. Mech. 116:251 (1982). 5. Pelce, P., and Clavin, P., J. Fluid. Mech. 124:219 (1982). 6. Clavin, P., and Garcia, P., J. Mec. 2 (l January 1983).
10. 11. 12. 13.
14.
Darrieus, G., Paper presented at 6th International Congress of Applied Mathematics, Paris, 1946. Landau, L. D., Acta Physicochim. (USSR) 19:17 (1944). Joulin, G., and Mitani, T., Combust. Flame 49:235 (1981). Markstein, G. H., Nonsteady Flame Propagation, Pergamon Press, New York, 1964. Frankel, and Sivashinsky, G., J. Comb. Sciences and Tech. (1982). Barenblatt, G. I., Zeldovich, Y. B., and Istratov, A. G., Prikl. Mek. Tekh. Fiz. 2:21 (1962). Quinard, J., Searby, G., and Boyer, L., Paper presented at the 9th International Colloquium on Combustion in Reactive Systems, Poitiers, July 1983. Clavin, P., Progress in Energy and Combustion Sciences (1983).
Received 25 June 1982; revised 9 M a y 1983