Premixed-gas flames - Paul Ronney - University of Southern California

Transcription

Premixed-gas flames
Paul D. Ronney
Department of Aerospace and Mechanical Engineering
University of Southern California, Los Angeles, CA 90089-1453 USA
[email protected]
Keywords:
Microgravity; premixed-gas combustion; radiation; reabsorption;
flammability limits; ignition; instability; flame stretch; flame balls; cool flames;
turbulence.
Reference: Ronney, P. D., “Premixed-Gas Flames,” in: Microgravity Combustion: Fires in
Free Fall (H. Ross, Ed.), Academic Press, London, U.K., 2001, pp. 35-82.
1. INTRODUCTION
Premixed-gas flames occur in mixtures of fuel, oxidant and inert gases that are intimately
mixed on the molecular scale before combustion is initiated. Examples of premixed-gas flames
include Bunsen flames, gas appliance stoves and gasoline-fueled internal combustion engines.
Accidental explosions that occur in mine shafts and chemical refineries are also premixed-gas
flames that undergo a transition to a detonation (a combustion wave propagating at supersonic
speeds, driven by a leading gasdynamic shock), sometimes with disastrous consequences. Thus,
an understanding of premixed-gas flames is necessary for both energy conversion and fire safety
applications. This chapter discusses the studies performed to date and future challenges related
to premixed-gas flames at µg.
Perhaps the most important property of premixed-gas flames that distinguishes them
from non-premixed flames (e.g., gas-jet flames, liquid fuel droplet flames, fire spread over solid
fuel beds) is the fact that (with the exception of “flame balls” discussed in section 5.3) in
premixed flames the flame front propagates relative to the gas. This is because premixed flames
are not constrained to follow a contour of stoichiometric composition, whereas with nonpremixed flames, the fuel and oxidant must mix in stoichiometric proportions before chemical
reaction can occur. The propagation speed of the premixed flame with respect to the unburned
gases is called the burning velocity, SL. As indicated in recent reviews of µg combustion
(Sacksteder, 1990; Law and Faeth, 1994; Anon., 1995; Kono et al., 1996; Ronney, 1998) when SL
is low, i.e., comparable to or lower than convection velocities induced by buoyancy or forced
flow, gravity may have a significant effect on the burning characteristics of premixed-gas flames.
2. COMPARISON OF TIME SCALES
To estimate under what conditions gravity can affect premixed-gas flames, and thus µg
experiments might be enlightening, we compare the time scales for chemical reaction (tchem),
buoyant convection in inviscid flow (tinv), buoyant convection in viscous flow (tvis), heat loss to
tube walls via conduction (tcond) and radiant heat loss from the burned gases (trad).
Premixed flame structure is determined by a balance between chemical reaction and
diffusion of heat and reactants over a zone of thickness d, thus tchem ≈ d/SL. d in turn can be
estimated as a/SL, where a is the thermal diffusivity of the gas, thus t chem ≈ a/SL2. A buoyant
transport time scale can be estimated as d/U, where d is a characteristic length scale of the flow
and U is the velocity induced by buoyancy. U in turn can be estimated as (gd(Dr/r))1/2, where r
is the density and Dr is the density change across the flame front. Since Dr/r ≈ 1 for flames, U ≈
(gd)1/2 and thus the buoyant time scale for inviscid flow becomes d/(gd)1/2 = (d/g)1/2. For flames
propagating in tubes or stabilized on a burner, d would be the tube or burner rim diameter. For
viscous flow, d cannot be independently specified; instead d ≈ n/U, where n is the kinematic
viscosity. Combining the relations d ≈ n/U and U ≈ (gd)1/2 leads to U ≈ (gn)1/3 and thus t vis ≈ d/U
≈ (n/U)/U ≈ (n/g2)1/3. In general a Prandtl number (Pr ≡ n/a) should appear in this estimate, but
for gases Pr ≈ 1. tcond can be estimated as the ratio of the flame temperature to the rate of
temperature decay due to conductive heat loss, i.e., Tf/(dT/dt) ≈ Tf/(rCph(Tf-T∞)), where Cp is
the constant-pressure heat capacity, h the heat loss coefficient in the tube = 16l/d2, l the thermal
conductivity, Tf the flame temperature and T ∞ the ambient (wall) temperature, thus t cond ≈
d2/16a, where we have assumed (Tf-T∞)/Tf ≈ 1, which is reasonable for practical flames. t rad for
optically-thin gases can be estimated as Tf/(dT/dt) ≈ Tf(L/rCp), where L = 4sap(Tf4 - T ∞4) is the
radiative heat loss per unit volume of gas, s the Stefan-Boltzman constant and ap the Planck mean
absorption coefficient of the gas, thus t rad ≈ {g/(g-1)}{P/4sap(Tf4 - T ∞4)} where g is the specific
heat ratio and P the pressure.
For illustrative purposes two sets of time scales are generated, one for near-stoichiometric
hydrocarbon-air flames and the other for very weakly burning lean hydrocarbon-air flames near
the flammability limits, both at P = 1 atm. For the former case, SL = 40 cm/s, Tf = 2200K, a = n
= 1.5 cm2/s and ap = 0.56 m-1. For the latter case, SL = 2 cm/s, Tf = 1500K, a = n = 1.0 cm2/s and
ap = 0.83 m-1. For both cases g = 980 cm/sec2, g = 1.35, T∞ = 300K and d = 5 cm (a typical
diameter for burner or flame tube experiments.) The estimated time scales for these flames are
shown in Table 1.
< Table 1 near here >
2
Several observations can be made based on these simple estimates:
(1) Buoyant convection is unimportant for near-stoichiometric flames because both t vis
and tinv are much larger than tchem
(2) Buoyant convection strongly influences near-limit flames at earth gravity because in
this case both tvis and tinv are comparable to or smaller than tchem
(3) Radiation effects are unimportant at earth gravity because buoyant convection is a
much faster process (both tvis and tinv are much smaller than trad)
(4) Radiation effects will dominate the behavior of flames with very low SL since t rad and
t chem are comparable for the slower flame, but these effects can only be observed at
low gravity because of (3)
(5) For the representative conditions used here, the apparatus size, e.g., the tube
diameter, must be larger than about 2.6 cm if one is to observe radiation-induced
extinction, otherwise conduction losses to the tube wall will be comparable to
radiative losses.
(6) Many phenomena associated with radiative loss effects can be studied in drop tower
experiments, with test durations of 2 to 10 s, since these times are typically larger
than trad. As a result, combustion science has probably benefited more from the
utilization of short-duration drop tower experiments than any other microgravity
science discipline.
(7) Since tinv ~ g1/2 and tvis ~ g1/3, aircraft-based µg experiments at g ≈ 10-2 go, where go is
earth gravity, may not provide sufficient reduction in buoyancy effects to observe
radiative effects.
(8) Since tvis ~ n1/3 ~ P-1/3 and t rad ~ r/L ~ P1/P1 ~ P0, t vis/trad ~ P-1/3. Thus, the radiative
time scale is similar at all pressures, but at high pressures buoyancy effects interfere
more strongly with radiative effects.
(9) At Reynolds numbers Ud/n ~ (gd3/n2)1/2 ≡ Grd1/2, where Grd is a Grashof number, of
the order 103 or larger, thus Grd of the order 106 or larger, buoyant flow at 1g will
necessarily be turbulent, thus it is difficult to study steady laminar flames in large
systems at 1g (≈ 10 cm for the property values employed in the examples given here).
A key aspect of these predictions is that only for mixtures with large t chem and thus low
SL will buoyancy effects be significant. Low SL implies mixtures highly diluted with excess fuel,
oxidant, or inert gas. It is well known that combustible gases will not burn if sufficiently diluted.
The composition delineating flammable from nonflammable mixtures is called the flammability
3
limit. Much of the early µg research on premixed-gas flames was concerned with flammability
limits, thus our discussion will begin with this topic. The importance of flammability limits due
to radiative losses (tchem ≈ trad) in µg experiments cannot be overstressed because it leads to a new
type of limit at low flow velocities and long residence times in addition to the high-velocity, short
residence time limits that are well known from earth-based experiments. This dual-limit behavior
permeates many of the phenomena discussed below.
3. FLAMMABILITY LIMITS
3.1 Buoyancy effects
Despite many years of study, there is still controversy surrounding the mechanisms of
flammability limits, including the effects of hydrodynamic strain and flame front curvature
(collectively called “flame stretch”), buoyancy, heat losses and flame chemistry. A standardized
measurement of flammability limits using a cylindrical tube of 5 cm diameter and 180 cm length,
called a Standard Flammability Limit Tube (SFLT), was proposed long ago (Coward and Jones,
1952). The tube is filled with combustible gas, ignited at an end open to the atmosphere and
propagates toward a closed tube end. Mixtures are defined as flammable if they sustain flame
propagation throughout the tube. The data compiled in Coward and Jones (1952) show that
flammability limits are different for flame propagation in the upward, downward and horizontal
directions of propagation. This indicates that buoyancy effects play a role in these limits.
Apparently all flammability limit studies show that burning velocity at the flammability
limit (SL,lim) is nonzero. Computations (Lakshmisha et al., 1988; Giovangigli and Smooke, 1992)
have shown that there is no purely chemical flammability limit criterion for planar unstretched
flames; without losses SL decreases asymptotically to zero as dilution increases. In Lakshmisha
et al. (1988), it was shown that the solutions of the unsteady planar one-dimensional adiabatic
premixed flame equations do not predict any flammability limit for lean CH4-O2-N2 mixtures. In
Giovangigli and Smooke (1992), the steady version of these equations was solved for CH4-air and
H2-air mixtures and a similar conclusion was reached. Both are important findings, since it is
possible in principle that unsteady effects could suppress limits of steady flames or cause limits
to occur for mixtures that are flammable as steady flames. Together these works show that one
must identify loss mechanisms of the appropriate magnitude to explain flammability limits.
These computations also show that since d increases as SL decreases, an ever-larger
computational domain is needed to model progressively weaker mixtures. If the domain is too
small, an apparent extinction limit is observed that is purely a computational artifact.
Consequently, loss mechanisms such as those discussed below are needed to explain limit
mechanisms. The resulting predictions of SL,lim indicate that usually SL,lim depends only weakly
4
on chemical reaction rate parameters. Thus, limit mechanisms may be inferred by comparing
predicted and measured SL,lim without detailed chemical knowledge. The mixture composition at
the limit affects SL,lim only weakly through Tf, thus, comparing predicted and measured limit
compositions is not especially enlightening; comparisons of SL,lim values is much more useful.
Consequently, this discussion will emphasize comparisons of predicted and measured values of
SL,lim.
For upward propagation, experiments (Levy, 1965) show that at the flammability limit
the rise speed of the flame, Sb,lim, is the same as the rate of rise of a hot gas bubble up the tube:
(1).
Sb, lim = 0.33 gd
Equation (1) is consistent with the inviscid rise speed discussed in Section 1. This rise speed is
dictated by hydrodynamics alone. It represents a minimum speed for flame propagation, though
it does not in itself indicate an extinction mechanism. Subsequently, it has been shown
theoretically (Buckmaster and Mikolaitis, 1982a) how the hydrodynamic strain at the tip of a
flame (Fig. 1a) rising at this rate could cause extinguishment (see Section 5.2 for further
discussion of strain effects). These predictions can be expressed in the form (using temperature
averaging of the transport properties)
< Figure 1 near here >
SL ,lim
Ê ga 2 ˆ
= 2.8 f Á
˜
Ë d ¯
1/ 4
Èb Ê
1 ˆ
˘
T
; f ≡ exp Í Á1 - ˜ (1- e )˙ ; e ≡ •
Î4Ë
˚
Le ¯
Tf
(2),
where b = E/RgT f is the non-dimensional activation energy, E is the overall activation energy of
the heat-release reactions, Rg is the gas constant and Le the Lewis number defined below. Note
that f = 1 when Le = 1. As might be expected, the functional form of Eq. (2), except for the
Lewis number effect, can be obtained by setting t inv = t chem. Note that Sb,lim is different from
SL,lim; Sb,lim is the observed flame front propagation rate in the laboratory frame whereas SL,lim is
the value of SL for a planar steady flame in the mixture at the flammability limit and thus is a
property of the mixture (Williams, 1985). Because the rising flame is curved, its area is greater
than the cross-sectional area of the tube. Mass conservation dictates that the ratio of the flame
area to tube cross-sectional area be equal to the ratio of Sb to the mean SL averaged over the flame
surface. Consequently, in general Sb,lim > SL,lim, and there is no contradiction between Eqs. (1) and
(2).
5
In Eq. (2), SL,lim is seen to be strongly dependent on the Lewis number, defined as
Le ≡
Thermal diffusivity of the bulk mixture
Mass diffusivity of the scarce reactant into the bulk mixture
(3)
This parameter can be interpreted as the ratio of the rate of diffusion of thermal enthalpy from
the flame front to the unburned gas to the rate of diffusion of chemical enthalpy (in the form of
scarce reactant) from the unburned gas to the flame front. When Le differs from unity, flame
stretch (Section 5.2) causes changes in the rates of transport of chemical and thermal enthalpy
that in turn affects the temperature at the flame front. For flame in mixtures with low Le that are
concave towards the burned products (as in a rising flame cap in a tube or an expanding spherical
flame) the increase in the rate of chemical enthalpy to the flame front is greater than the increase
in the rate of thermal energy loss, and thus the curved flame will burn more intensely than a
planar flame in the same mixture. Since heat release reactions in most combustible mixtures have
high activation energies, these changes in flame front temperature lead to large changes in reaction
rate at the flame front and thus large changes in the local, instantaneous propagation rate.
For downward propagation, high-g centrifuge experiments (Krivulin et al., 1981) show
that SL,lim ~ g1/3. SL,lim is apparently independent of Le, which is reasonable since downwardpropagating near-limit flames in tubes are nearly flat and unstrained, hence SL ≈ Sb. An extinction
mechanism is suggested by observations (Jarosinsky et al., 1982) of a sinking of a layer of cooling
burned gas near the walls overtakes the flame front and “suffocates” the flame (blocking the flame
front from the fresh unburned gases) is an important factor in the extinction processes (Fig. 1b).
Detailed numerical computations (Patnaik and Kailasanath, 1992) support this mechanism also,
though the experimentally-observed g1/3 scaling was not tested. The g1/3 scaling can be obtained
by setting tchem = tvis, leading to
SL,lim ≈ (ga)1/3
(4).
The a 1/3 scaling has been confirmed by experiments on flames in tubes using a wide range of
diluent gases and pressures (Wang and Ronney, 1993), and together with the g1/3 scaling found in
centrifuge experiments (Krivulin et al., 1981) support the mechanisms proposed here.
Both upward and downward limit mechanisms indicate that as g Æ 0, SL,lim Æ 0 also,
implying that arbitrarily weak mixtures could burn at g = 0, albeit very slowly. As discussed in
the next section, an additional factor, namely heat losses, prevents arbitrarily weak mixtures from
burning even in the absence of gravity.
6
3.2 Radiative heat loss
Numerous authors (Williams, 1985; Spalding, 1957; Buckmaster, 1976; Joulin and Clavin,
1976; Jarosinsky, 1983; Aly and Hermance, 1981) have considered how dilution may lead to
extinction via heat loss due to radiation or conduction. Increasing dilution decreases the flame
temperature, which in turn increases tchem more than the characteristic loss time scale since the
former is generally a stronger function of temperature than the latter, i.e., exponential vs.
algebraic. Consequently, increasing dilution increases the impact of heat losses, which eventually
leads to a flammability limit. An estimate of SL,lim for radiative losses has been given in Williams
(1985), that, after temperature-averaging of transport properties, becomes approximately
SL ,lim =
1.2bLl f
1
r • Cp
Tf
(5),
where lf is the thermal conductivity evaluated at T = T f and L is the rate of radiative heat loss
per unit volume of gas discussed in Section 2. The functional form of Eq. (5) can be obtained by
equating tchem and trad. Using the same set of property values as employed in section 2, Eq. (5)
yields SL,lim ≈ 2.3 cm/s for lean-limit hydrocarbon-air mixtures at 1 atm, which is practically
identical to that predicted using the same gas radiation data (Hubbard and Tien, 1978) but a
detailed numerical model of chemistry and transport (Lakshmisha et al, 1990). Such small values
of SL,lim are not usually observed experimentally; note that Eq. (2) yields SL,lim ≈ 3.3 cm/s for
upward propagation of CH4-air flames and Eq. (3) yields SL,lim ≈ 7.8 cm/s for downward
propagation. Thus, radiation effects do not dominate at earth gravity because of buoyant
convection (tinv < trad and tvis < trad), however, at reduced gravity in sufficiently large diameter
tubes, radiation effects can be anticipated to be significant.
It is important to note that Eqs. (2), (4) and (5) apply only if the apparatus size is large
enough that conductive heat losses to the walls of the apparatus are not significant. Flame
quenching via conduction losses occurs when tchem ≈ tcond, leading to SL ~ a/d or (Spalding, 1957;
Joulin and Clavin, 1976).
Pelim ª 40; Pelim ≡
SL ,lim d
a•
(6),
where Pelim is the Peclet number at the flammability limit and a ∞ is the thermal conductivity
evaluated at T = T∞. Thus, for flames in small-diameter tubes or in gases with large a (e.g., low
pressures and light inert gases such as helium), even at µg it may be difficult to observe a limit
due to radiative loss rather than one due to conductive loss.
7
3.3 Microgravity experiments
The earliest study of premixed-gas flames at µg is probably that dating back to 1980
(Krivulin et al., 1980), in which an aircraft flying low-gravity trajectories was employed to study
the effects of buoyancy on flammability limits. The limits for lean H2-air and rich C3H8-air
mixtures in a 20 liter cylinder of equal diameter and length, ignited by a 17J spark at the center of
the chamber (that produced spherically expanding flames), were found to be 7.0% for H2 / 8.6%
C3H8 at µg (for propagation throughout the chamber), compared to about 4.0% H2 / 9.9% C3H8
for upward propagation at 1g and 8.5% H2 and 8.0% C3H8 for downward propagation. Thus,
with the definitions employed the limits at µg were found to lie between those for upward and
downward propagation at 1g. Surprisingly, these authors did not report SEFs (Section 5.1) for
the rich C3H8-air mixtures (Le ≈ 0.87) nor flame balls (Section 5.3) for the lean H2-air mixtures
(Le ≈ 0.3), both of which were readily observed in later µg experiments employing a similar
apparatus.
At about the same time, lean CH4-air mixtures were studied (Strehlow and Reuss, 1981)
in a SFLT using a 2.2 second drop tower facility. A 5.22% CH4 mixtures was determined to be
probably flammable whereas the flammability of a 5.10% CH4 was uncertain because it had not
extinguished nor reached a steady SL within the µg time available. The corresponding values of SL
for the experiments reported in Stehlow and Reuss (1981) have been inferred (Ronney and
Wachman, 1985) and are given in Table 2. Using a 12 liter cylindrical chamber with ignition by a
5 J spark at the center of the chamber, limit mixture compositions and near-limit values of SL
have been measured (Ronney and Wachman, 1985) (Table 2) that are very close to the SFLT
results (Strehlow and Reuss, 1981), suggesting that at least for lean CH4-air flames, µg may
provide a means to obtain a fundamental flammability limit, i.e., one that is independent of the
apparatus. Moreover, the value of SL,lim in both cases is close to the theoretical and
computational prediction of about 2 cm/s as discussed in the previous section. However, it
should be noted that the SFLT results may be somewhat fortuitous, for in 5 cm tubes, the
estimated SL,lim due to conduction losses (Eq. 6) is 1.6 cm/s, which is comparable to the estimated
SL,lim due to radiative losses (Eq. 5) of 2.3 cm/s. Thus, conductive losses to the tube wall and
radiative losses were probably of nearly equal importance in the SFLT experiment, whereas in
the expanding spherical flame experiment (Ronney and Wachman, 1985), conductive losses were
negligible.
< Table 2 near here >
8
The aforementioned SFLT experiments (Strehlow and Reuss, 1981) were later extended
Strehlow et al., 1986) using an aircraft to obtain longer µg durations. The flammability limits for
1g upward, µg and 1g downward propagation were 5.25%, 5.25% and 5.85%, respectively, for
CH4-air mixtures and 2.15%, 2.06% and 2.20% for C3H8-air mixtures. The corresponding values
for spherically expanding C3H8-air flames at µg are 2.02%, 2.09% and 2.07% (Ronney, 1988a).
Thus, again the µg limits are very similar in different apparatuses whereas the 1g limits are
different, indicating again that µg provides the closest approximation to an apparatusindependent limit. Strehlow et al. (1986) reported that the 1g upward limit was due to stretch at
the flame tip as discussed in Section 3.1, whereas both the µg and 1g downward limits were said
to be due to heat losses to the tube wall. These claims are difficult to evaluate because values of
SL,lim were not reported, but clearly since the downward limit occurs at a higher fuel concentration
than the µg limit, SL,lim must be higher than the value corresponding to Pe = 40, and thus it is not
a true quenching limit. Instead, the heat loss to the tube wall causes the sinking layer of
combustion products to form as discussed in Section 3.1 which leads to extinction, but in itself
this heat loss does not cause extinction in the manner described by the analysis leading to Eq. (6).
Finally, the predictions of Eq. (5) have been compared to experiments at µg (Ronney,
1988b; Abbud-Madrid and Ronney, 1990) and good agreement is found for a wide range of
pressures, fuels, and inert gases when Le not so high nor pressure so low that an ignition limit
rather than a flammability limit is reached (because minimum ignition energies increase as Le
increases or pressure decreases.) Thus radiative heat loss appears to be the cause of flammability
limits when extrinsic heat losses, e.g., due to conduction, buoyant convection and hydrodynamic
strain (Section 5.2) are eliminated. This is one instance where µg experiments have enabled
observation of a phenomenon that can probably never be observed at 1g.
3.4 Radiation reabsorption effects
The radiation effects described in the previous section are only valid if the gases are
“optically thin,” that is, there is no reabsorption of emitted radiation. This probably cannot be
true in systems of very large size, at very high pressures, or in mixtures with a high concentration
of sufficiently strong absorbers. With this motivation, lean CH4-air mixtures seeded with inert
solid particles have been studied (Abbud-Madrid and Ronney, 1993) to increase ap to values
sufficiently high that reabsorption effects were observable in a laboratory-scale combustion
apparatus. Since solids emit and absorb as black- or gray-bodies, whereas gases radiate in narrow
spectral bands, a particle-seeded gas can emit and absorb much more radiation than a particle-free
gas. Data reported in Abbud-Madrid and Ronney (1993) on flame shapes, propagation rates,
peak pressure, maximum rate of pressure rise, and thermal decay in the burned gases indicated
that at low particle loadings, the particles act to increase the radiative loss from the gases,
9
whereas at higher loadings, reabsorption of emitted radiation becomes significant, which in turn
acts to decrease the net radiative loss and augment conductive heat transport. For example, for
5.25% CH4-air mixtures, as the particle loading was increased, SL decreased at first then increased
to a value above that of particle-free mixtures (Fig. 2). In a leaner mixture (5.15% CH4) at particle
loadings of 0.00 g and 0.75g, the burning velocities were 1.70 and 1.30 cm/s, respectively. The
latter value is noteworthy because it is lower (by about 15%) than any value attainable in
particle-free lean CH4-air mixtures at 1 atm, which, according to Eq. (5), indicates that the net
radiative loss is lower than in any particle-free mixture. In principle, SL,lim may be reduced to
zero in optically-thick gases, though only minor decreases in SL,lim have been seen in experiments
performed to date. Based on these trends, one could speculate that for apparatuses in which the
absorption length (i.e., the inverse of the absorption coefficient ap) is much smaller than the
system size, flammability limits might not exist at µg conditions because emitted radiation would
not constitute a loss mechanism.
The maximum increase in SL attainable through radiation effects is related to the ratio of
the blackbody emissive power of the particle-laded gas per unit area, s(Tf4 - T ∞4), to the total
enthalpy flux per unit area of the particle-free flame, r ∞CpSLT f. The ratio of SL with radiative
reabsorption to that in the particle-free mixture (m) is given by (Joulin and Deshaies, 1986)
4
4
1 - w s (Tf - T• )
B
m = exp m ; B ≡ b
g
r •C p SLTf
( )
(7)
where B is the Boltzman number and the symbols w and g indicate, respectively, albedo
(scattering to attenuation ratio), and a constant ( 3 < g < 2, depending on w). Equation (7)
shows that away from flammability limits, where SL is large, B will be small and thus m will be
close to unity, indicating that radiative effects do not affect SL substantially. The predictions of
Eq. (7) have not been tested experimentally to date, though Eq. (7) is consistent with detailed
numerical computations (Ju et al., 1998a). When particles are used to decrease ap, the heat
capacity of the particles must be also be considered since this reduces Tf. Adding particles to the
gas might enable one to suppress flammability limits, but only if the heat capacity of the
particles were low enough to avoid a significant decrease in Tf.
Even for spectrally-radiating gases, computations (Ju et al., 1998a) using a detailed
statistical narrow-band radiation model show that flammability limits may be extended
remarkably by considering reabsorption, for example in CH4-O2-N2 mixtures seeded with CO2
10
(Fig. 3). Note that the equivalence ratio f (defined as the fuel to oxygen ratio of the mixture
divided by the stoichiometric ratio) at the flammability limit is 0.682 for optically-thin conditions
vs. 0.442 with reabsorption. The latter value is even lower than the computed value for CH4-air
despite the fact that CP is much higher for CO2 than N2, and thus the adiabatic T f is lower with
CO2. Still, in Ju et al. (1998a) it was found that there are two radiative loss mechanisms that lead
to flammability limits even with reabsorption. One is due to the difference in composition
between reactants and products; if a product of combustion that radiates significantly is not
present in the reactants (e.g., H2O), radiation from this species that is emitted upstream cannot
be reabsorbed by the unburned mixture unless, by coincidence, its spectrum overlaps completely
with the constituents of the unburned mixture. The second is that the emission spectra of most
molecules are broader at flame temperatures than ambient temperature, thus some radiation
emitted near the flame front cannot be absorbed by the reactants even if they are seeded with that
molecule. Via both mechanisms some net upstream heat loss due to radiation will always occur,
leading to extinction of sufficiently weak mixtures. These results suggest that fundamental
(domain-independent) flammability limits due to radiative heat loss may exist at µg, but these
limits are strongly dependent on the emission-absorption spectra of the reactant and product
gases and their temperature dependence, and cannot be predicted using gray-gas or optically-thin
model parameters. In fact because of the spectral nature of gas radiation, very significant
reabsorption effects were found in domains as small as 1 cm even for mixtures with aP-1 ≈ 24 cm.
This is because ap is a mean absorption coefficient weighted by the Planck function and averaged
over all wavelengths, whereas for some wavelengths the spectral absorption coefficient is much
higher (by orders of magnitude) than the mean.
4. FLAME INSTABILITIES
4.1 Low Lewis number (cellular) instability
It is well known that gravity affects the stability of plane flames through the RayleighTaylor effect, which states that when a dense fluid lies on top of a lighter fluid, gravity is a
destabilizing influence on the interface, whereas the opposite configuration is stabilizing. Darrieus
(1938) and Landau (1944) showed that, independent of gravity effects, the density decrease
across the flame front by itself causes wrinkling to be encouraged because the pressure and
density gradients become misaligned if a small perturbation to the flat front occurs, leading to a
torque on the fluid that acts to encourage the wrinkling. The combined effect of Rayleigh-Taylor
wrinkling (or stabilization) and Darrieus-Landau wrinkling can be written in the form of a
11
dispersion relation for the growth rate of infinitesimal disturbances (s) as a function of the
wavenumber of the wrinkle (k) (Pelcé and Clavin, 1982):
s
e 2 ga f ˆ ˘
1 È
1 - e 2 ÊÁ
˜
kSL = 1 + e -1 Í 1+ e Ë 1 - kS 3 ¯ - 1˙
L
Î
˚
(8).
Note that for g = 0 all wavenumbers are unstable (s > 0) and that only for downward propagating
flames (g < 0) can some wavenumbers, namely small ones (long wavelength disturbances), be
stabilized by gravity.
Besides thermal expansion driven and buoyancy driven instabilities, premixed-gas flames
are subject to diffusive-thermal instabilities when Le differs from unity (Williams, 1985; Clavin,
1985) due to the imbalance of the diffusion rates of thermal energy and reactants. For Le less
than a critical value (Lec ) slightly less than unity, cellular flames occur, which are characterized
mathematically by a growth rate that is real (no imaginary component) and maximum at a finite
wavenumber. Physically this instability occurs because wrinkled regions that are concave toward
the burned products (i.e., pointing toward the unburned gas) have a higher local burning velocity
than the flat flame (see the discussion of curvature effects in Sections 3.1 and 5.1) and wrinkled
regions pointing toward the burned gas have a lower local burning velocity than the flat flame,
thus the wrinkling is encouraged. Conversely, for Le > Lec , wrinkling is discouraged. Since this
instability is dependent on diffusional effects that occur on the scale of d, it has no effect on very
long wavelengths and at short wavelengths it is so dominant that diffusion damps out all
wrinkling. In the absence of buoyancy and thermal expansion, the dispersion relation for
adiabatic flames is given by solutions of the relation (Joulin and Clavin, 1979)
[
]
A(1 + A) = b ( Le - 1) s (1 + 1 + A ) - A / 2 ; A ≡ 4(s + k 2 )
(9),
Inspection of Eq. (9) shows that Lec = 1 - 2/b since only for Le > 1 - 2/b is s > 0 for any value of
k.
The effects of thermal expansion, buoyancy and diffusive-thermal instabilities are shown
schematically in Fig. 4. Diffusive damping suppresses instability at large k (small wavelengths)
while buoyancy suppresses instability at small wavenumbers (for downward propagating
flames). Thus, the growth rate is maximum at an intermediate wavelength and negative at short
and long wavelengths, leading to well-defined cellular structures. Even without buoyancy, s
exhibits a maximum, though in this case there is no mechanism to suppress growth entirely at
small k.
12
Cellular structures observed at µg can be expected to be different from those observed at
1g for at several reasons. First, mixtures that can be studied at µg may not be flammable at 1g for
the reasons discussed in section 3.1. Also, for upward propagation in a confined tube the
resulting hydrodynamic strain (Section 5.2) suppresses much of the fine-scale wrinkling (large k)
caused by the diffusive-thermal instability (Rakiv and Sivashinsky, 1987) (though the flame is
highly wrinkled on the scale of the tube diameter due to buoyancy effects). These factors may
affect the extinction mechanisms significantly because diffusive-thermal instabilities are an
important factor in flammability limits. In particular, it has been shown (Joulin and Sivashinsky,
1983) that in low-Le mixtures the wrinkling leads to an extension of flammability over that of
planar flames. This extension of flammability reaches its ultimate limit in flame balls, to be
discussed in Section 5.3.
Despite the rich behavior associated with premixed flame instabilities and the likely
influences of buoyancy, very few experimental studies of cellular flames at µg have been
conducted. Experiments (Dunsky and Fernandez-Pello, 1990) on the effects of buoyancy on rich
C3H8-air flames (Le ≈ 0.8), stabilized above a porous plug through which the combustible gases
were emitted, have shown that the flow induced by the plume at 1g had a substantial influence on
the overall flame shape and thus the size and shape of the cells, but otherwise the cells were not
significantly influenced by buoyancy. At first this seems surprising considering that t chem/tinv =
(a/SL2)/(d/g)1/2 ≈ 0.3 when d is based on the observed cell size (typically 1 cm), however, as
Dunsky noted, their experiments were performed on flames stabilized above a porous plug that
are not free to respond to buoyancy-induced flow to nearly the extent that freely propagating
flames can because of heat loss to the plug and the flow-straightening effect of the plug. Very
lean H2-air mixtures have been studied (Ronney, 1990) in a 12 liter cylindrical chamber with
spark ignition at the center of the chamber and observed cellular patterns in the resulting
expanding quasi-spherical flames (Fig. 5). While no quantitative information on cellular flame
structure was reported in (Ronney, 1990), it is evident that freely-propagating flames at µg
provide a means to study diffusive-thermal instabilities without buoyancy effects and minimal
influence of strain, heat loss to burners, etc.
4.2 High Lewis number (pulsating) instabilities
At sufficiently high Le, the diffusive-thermal theory (Eq. (9)) predicts flame fronts with
traveling-wave or pulsating characteristics (Re(s) > 0, Im(s) ≠ 0) (Joulin and Clavin, 1979). An
13
important application of this high Le instability is to the combustion of lean mixtures of heavy
hydrocarbons such as octane in air (Le ≈ 3), which is relevant to lean-burning automotive engines
because diffusive-thermal instabilities affect the wrinkling and thus propagation rates of turbulent
premixed flames (Williams, 1985) such as those occurring in automotive engines.
The high Le instability has been studied experimentally (Pearlman and Ronney, 1994;
Pearlman, 1997) using flames in lean C4H10-O2-He mixtures (Le ≈ 3.0) propagating downward in
a tube open at the ignition end and closed at the other end, i.e., a SFLT though with larger
diameters. Two types of instabilities were observed at 1g: (1) a rotating spiral wave (Fig. 6a) and
(2) a pure radial pulsation (Fig. 6b). The spiral waves occurred only very near the flammability
limit, whereas the radial pulsations could occur in these mixtures and also mixtures farther from
the limits. For mixtures sufficiently far from the limits, only stable flames were observed. For
µg drop tower experiments (Pearlman and Ronney, 1994) the pulsating mode was observed but
no spiral flames were seen; instead a flame consisting of six rotating azimuthally-distributed
bright and dark striped zones was found (Fig. 6c).
The existence of nonuniform modes of propagation near the limits but not farther from
the limits is consistent with theory (Joulin and Clavin, 1979) that indicates a wider range of
unstable k and Le exists for mixtures with greater impact of heat loss, i.e., closer to the
flammability limits. While no definitive explanation of the observed differences between the 1g
and µg flame instabilities has been advanced, the impact of gravity is probably to cause a
reduction in the effective heat loss, since theory (Kaper et al., 1987) shows that for the high-Le
instabilities, the influence of buoyancy and heat loss on instability can be combined into a single
parameter whose value decreases with increasing gravity for downward propagating flames.
Thus at 1g, the effective heat loss (from the standpoint of instability behavior) is lower. The
stability of various spinning modes of flame propagation having different numbers of stripes.
Predictions (Booty et al, 1987) for the parameters corresponding to the experimental conditions
at µg is that the first modes to bifurcate from the steadily-propagating solution as the combined
buoyancy/heat loss parameter (or Le) is increased correspond to a "four-headed" spinning wave
and a "one-headed" spinning wave, which are stable modes. Further increases in the
buoyancy/heat loss parameter indicate that a six-headed mode may appear. Since the
buoyancy/heat loss parameter is higher at µg than 1g for downward propagating flames, the µg
flame is farther into the unstable regime and thus might exhibit the higher-order instability mode,
i.e., the six-headed mode.
14
5. STRETCHED AND CURVED FLAMES
5.1 Self-Extinguishing Flames
Premixed flames are generally not flat and steady nor do they commonly propagate into a
quiescent flow. Consequently, flames are usually subject to “flame stretch” (S) defined by
(Williams, 1985)
S≡
1 dA
A dt
(10)
where A is the instantaneous flame surface area. The effects of flame stretch on propagation
rates and extinction conditions are discussed in various reviews (Williams, 1985; Clavin, 1985).
For flames at 1g, buoyancy imposes a flame stretch comparable to tinv-1 or tvis-1. In the
absence of gravity, weak flame stretch effects that are insignificant at earth gravity may become
dominant. One such example is expanding spherical flames for which, according to Eq. (10),
S=
1 d
2 dr*
4pr*2 ) =
(
2
4pr* dt
r* dt
(11),
where r* is the flame front radius. Using Eq. (11), an evolution equation describing the
propagation rate of an expanding spherical flame in the presence of heat loss effects was obtained
(Ronney and Sivashinsky, 1989):
dS
2S
+ S 2 ln S 2 =
-Q
dR
R
(12),
where S ≡ dR/dt is the propagation speed divided by SL(r∞/rf), R is the flame radius scaled by
bdI(Le, e), I(Le, e) is a scaling function that is positive for Le < 1 and negative for Le > 1 (but of
course the physical radius r* is always positive), t is the time scaled by b(d/SL(r∞/rf))I(Le, e)
and Q ≡ {bL( Tf)d2}/{l(Tf-T∞)} is the scaled heat loss. The terms in Eq. (12) represent
unsteadiness, heat release, curvature-induced stretch and heat loss, respectively. For steady
planar flames, Eq. (12) reduces to S2lnS2 = -Q, which has a turning point and thus a flammability
limit at a maximum value of Q = 1/e = 0.3679... where S = e1/2. This turning point corresponds to
the value of SL,lim given by Eq. (5).
Numerical integrations of Eq. (12) for Le < 1 transformed into the time domain are shown
in Fig. 7. Figure 7 shows that for Le < 1 the effect of curvature (2S/R) works opposite that of
heat loss (Q), allowing mixtures that are non-flammable as plane flames (Q > 1/e) to sustain
15
expanding spherical flames until R (or r*) grows so large that the enhancement of combustion due
to curvature is too small relative to the heat loss. Note that when Q is only slightly greater than
1/e, i.e., for mixtures just outside the flammability limit, the extinction radius may be very large.
Such behavior of spherically expanding flames, termed “self-extinguishing flames” (SEFs), has
been observed experimentally (Ronney, 1985; Ronney, 1988a, b) for a variety of fuels, diluents
and pressures in mixtures near the µg flammability limit when Le < 1 but not so low that
diffusive-thermal instabilities fragment the flame into a cellular structure. Figure 8 shows a
comparison of the temporal behavior of experimentally observed SEFs and normal expanding
spherical flames. Experiments (Ronney, 1988a, b; Abbud-Madrid and Ronney, 1990) also show
that SEFs do not occur for Le > 1 because in this case both curvature and heat loss weaken the
flame. This is consistent with the predictions of Eq. (12) for Le > 1 (R < 0).
While the analysis leading to Eq. (12) is instructive, there are several experimental
observations that it does not predict. The most significant of these are that (1) a narrow range of
mixtures can exhibit both SEFs and normal flames depending on the spark ignition energy (Eign)
and (2) Eign can affect the extinction radius substantially. In regards to (2), according to Eq. (12)
the initial conditions have almost no effect on the extinction radius whereas experimentally,
values of the ratio of the chemical enthalpy release before extinguishment (Echem) to Eign up to
70,000 have been observed. Thus, SEFs possess a remarkably strong “memory effect” for their
initial condition. An example of the effects of Eign and mixture strength on extinction radius is
shown in Fig. 8. SEFs occur for conditions just outside those resulting in normal flames (weak
mixtures and/or lower Eign), but sufficiently weak mixtures or small Eign result in the conventional
non-ignition behavior (Lewis and von Elbe, 1987) where Echem/Eign ≈ 10 (Ronney, 1985). This
causes the contours of constant extinction radius seen in Fig. 8 to bend over to horizontal for
sufficiently weak mixtures. A third difference between the predictions of Eq. (12) and
experimental observations is that experimentally-observed SEFs exhibit a radius increasing
roughly with the square root of time, i.e., r* ~ t 1/2 (Fig. 9) whereas Eq. (12) predicts a radius
increasing only slightly more slowly than linearly with t. Thus, while Eq. (12) provides insight
into SEFs, it does not provide a complete description.
< Figures 8, 9 near here >
Equation (12) is based on the Slowly Varying Flame assumption where the ratio of flame
radius to d is of order b, and the Activation Energy Asymptotics (AEA) analysis is performed in
16
the limit b Æ ∞. Thus, the analysis is not valid on the length scale corresponding to flame kernel
radii characteristic of ignition kernels in mixtures with Le < 1, where r* < d (Joulin, 1985).
Analysis using AEA but a different scaling approach (Buckmaster and Joulin, 1989) shows that
an expanding spherical flame in a mixture with Le < 1, starting from a spherical flame whose
structure was that of a flame ball (Section 5.3) would exhibit a radius increasing as t 1/2ln(t), which
might be indistinguishable from the apparent t1/2 behavior seen experimentally. Thus, this
analysis describes one additional aspect of SEF behavior, however, the memory effects were still
not predicted.
These analyses, based on AEA, are necessarily subject to certain scaling assumptions and
thus are not uniformly valid for the entire range of r* from very small values characteristic of
ignition kernels to much larger values characteristic of SEFs near their extinction radius. A
numerical model of nonadiabatic spherically expanding flames in mixtures with Le < 1 (Farmer
and Ronney, 1990), free of the scaling assumptions inherent in the analytical models, does
predict memory effects in good qualitative and fair quantitative agreement with experiments. As
discussed in Farmer and Ronney (1990), these memory effects seem to be a result of the
character of the temperature and concentration profiles that are established early in the life of the
flame but persist to much later times. In particular, at early times the ignition process in mixtures
with Le < 1 establishes profiles characteristic of flame balls (Section 5.3) with temperature
decaying in proportion to r-1, where r is the radial coordinate, and reactant concentration
increasing in proportion to 1 - r-1. These profiles persist on the longer time scale leading to the
development of a propagating flame when the profiles are much steeper, i.e., with temperature
decaying in proportion to exp(-(r-r*)/d), but the small additional temperature established early on
affects the propagation rate until this later stage
The calculations by Farmer and Ronney (1990) also show that for small initial radii, all
mixtures exhibit extinguishment, which corresponds to non-ignition behavior (Lewis and von
Elbe, 1987). This indicates that in mixtures capable of exhibiting SEFs, flames extinguish at
sufficiently small curvature due to high stretch rates and at large curvature due to radiative losses.
This dual-limit behavior is also observed in many of the other flame phenomena described below.
A one-step chemical reaction model was employed in Farmer and Ronney (1990),
indicating that, in accord with experimental observations (Ronney, 1988a, b; Abbud-Madrid and
Ronney, 1990), the details of the flame chemistry do not affect the qualitative behavior of SEFs.
Still, quantitative comparisons between experiment and computation using detailed chemical,
transport and radiation sub-models and comparisons with experiment would be instructive, and
in fact comparison to experimental observations of SEFs could provide a useful means of testing
models of near-limit premixed-gas flames.
17
5.2 Strained flames
The effects of hydrodynamic strain on flames has been studied for many years because of
the need to understand how turbulence-induced strain affects flame fronts in practical combustion
devices such as automotive engines. The most common apparatus for studying strained premixed
flames is the counterflow round-jet configuration. Since fresh reactants are emitted from both jets,
twin flames (one on either side of the stagnation plane) are produced with a burned gas region
between the twin flames. The axial flow velocity, U, is given by -2cz, where c is a constant and z
is the axial distance from the stagnation plane, which is independent of the radial coordinate r.
The radial velocity, V, is given by cr and is independent of z. The stretch acting on the flame is
sum of the extensional strains in the two directions orthogonal to the z-axis, i.e., ∂V/∂x + ∂V/∂y =
2c = -∂U/∂z. Thus, the flame stretch S is simply the magnitude of the axial velocity gradient.
The counterflowing jet configuration is popular because ideally S is constant within the entire
region between the two jets, so that a single parameter describes the flow, and it is very simple to
produce experimentally. The equilibrium location of the flame front is at the axial location where
the local axial velocity U is equal to SL for the given S = ∂U/∂z. Thus, as S is increased or SL is
decreased (e.g., by reducing the fuel concentration), the flame moves closer to the stagnation
plane (smaller U and z). This reduces the volume of burned gas, which affects the impact of
radiative loss as discussed in the following paragraph.
As with curvature-induced strain, for Le less than/greater than unity, in the counterflow
configuration SL is increased/decreased by moderate strain, and for all Le, sufficiently large S
extinguishes the flame (Buckmaster and Mikolaitis, 1982b). The combination of the nonmonotonic flame response to S at low Le, plus the reduced volume of burned gas (thus reduced
radiative heat loss) at larger S may lead to a variety of extinction behaviors for counterflow
flames depending on the relative magnitudes of trad and S-1. Recent µg experiments (Maruta et al,
1996; Guo et al., 1997) on counterflow flames in low-Le mixtures (Fig. 10) have revealed
extinction behavior somewhat reminiscent of SEFs and non-ignitions in spherical flames. For
large S, the short residence time of reactants within the flame front (~S-1) causes extinguishment
(S-1 ≈ tchem) along the “normal flame” branch, which is analogous to non-ignition behavior of
flames with small r*. For small S the residence time is large, the impact of heat losses is
significant (trad ≈ tchem) and the increase in SL due to Le effects is weak, so radiant heat losses
extinguish the flame along the “weak flame” branch, which is analogous to SEFs. Figure 10 shows
that extinguishment along the weak flame branch can occur even in mixtures far richer than the
lean planar flammability limit. The optimal S that produces the maximum increase in the
flammable range (S = 13 s-1) corresponds to a time scale of 0.08 s, which is less than t vis or t inv.
Thus, the C-shaped response and the entire weak-flame branch cannot be observed at 1g,
however, behavior on this time scale is readily observed in drop-tower experiments. The optimal
18
S is found to be nearly the same for model and experiment (Maruta et al., 1996; Guo et al., 1997),
suggesting that the loss rates are modeled well. In contrast, the computed limit composition is
leaner that the experimental one, suggesting that the chemical mechanism used is not accurate for
weak mixtures.
Interestingly, due to the decrease of radiant loss as S increases, the extension of the
flammability limit can also occur for mixtures with Le greater than unity (Ju et al., 1998b, 1999),
though for sufficiently high Le, e.g., lean C3H8-air (Le ≈ 1.7) (Fig. 11) it does not occur. No
analogous effects occur for SEFs because for the spherically expanding flame there is no
mechanism by which flame stretch can affect the total radiative loss.
Theory (Buckmaster, 1997) and numerical simulations (Ju et al., 1998b, 1999) predict
that strained premixed flames with radiative loss and Lewis number effects exhibit an even more
complex set of behaviors that those described here (Fig. 12). For example there is another branch
of solutions, called the Far-Standing Weakly Stretched Flame (FSWSF) in which the flame front
is far from the stagnation plane and thus has a very large burned-gas region. Only the FSWSF
behavior can be extrapolated to S = 0 to obtain the flammability limit for planar unstretched
flames. Additionally, there are jump limits between different modes that are not readily
explained based on the simple physical principles outlined here. It is uncertain whether any of
these solutions are physically observable, since they have not been identified experimentally and
stability analyses have not yet been performed.
5.3 Flame balls
Sections 5.1 and 5.2 discuss cases of stretched flames, where heat and mass transport are
influenced by the convective environment, and behavior resulting from differences in the
convective environment between 1g and µg conditions. This section discusses a phenomenon for
that can occur only in the absence of convection and the curvature of the flame front is the
dominant influence.
Over 50 years ago, Zeldovich (1944) showed that the steady heat and mass conservation
equations admit a solution corresponding to a stationary spherical flame or “flame ball” (Fig. 13),
19
just as the same governing equations in planar geometry admit a steadily propagating flame as a
solution for every mixture. In the former case the solutions are characterized by a radius (r*) and
in the latter case by SL. The mass conservation equation in a steady spherically symmetric
system with no sources or sinks, —⋅(rv) = 0, where r is the density and v the fluid velocity
vector, requires that v be identically zero everywhere. The solution to steady diffusion equations
— 2T = 0 and —2Yi for the temperature T and species mass fractions Yi in spherical geometry are
of the form c1 + c2/r, where c1 and c2 are constants. This form satisfies the requirement that T
and Y be bounded as r Æ ∞. For cylindrical and planar geometry the corresponding forms are c1
+ c2ln(r) and c1 + c2r, respectively, that are obviously unbounded as r Æ ∞. For this reason
theory admits steady flame ball solutions, but not “flame cylinder” or “flame slab” solutions.
Zeldovich (1944) showed that for an adiabatic flame ball, the energy and species conservation
equations could be combined to infer the temperature at the surface of the flame ball (T*):
T* = T• +
Tf - T•
(13),
Le
thus c1 = T∞ and c2 = (T* - T∞)r*. Zeldovich also predicted, as was supported much later by
more rigorous AEA analyses (Deshaies and Joulin, 1984; Buckmaster and Weeratunga, 1984),
that flame ball solutions are unstable and thus probably would not be physically observable, just
as planar flames are frequently subject to instabilities that prevent them from remaining planar
(Section 4). The unstable flame ball solutions, however, are related to the minimum flame kernel
size required for ignition (Joulin, 1985).
Forty years later after Zeldovich, seemingly stable flame balls were discovered
accidentally in drop-tower experiments in lean hydrogen-air mixtures (Le ≈ 0.3) (Ronney, 1990)
and later in drop tower and aircraft µg experiments using H2-air, H2-O2-CO2 (Le ≈ 0.2), H2-O2SF6 (Le ≈ 0.06) and CH4-O2-SF6 mixtures (Le ≈ 0.3) (Ronney et al., 1994). The µg environment
was needed to obtain spherical symmetry and to avoid buoyancy-induced extinction of the flame
balls. The following sequence of phenomena was observed as the mixtures were progressively
diluted. For mixtures sufficiently far from flammability limits, an expanding spherical front
composed of many individual cells (resulting from the diffusive-thermal instability discussed in
Section 4.1) was observed that regularly subdivided to maintain a nearly constant cell spacing (cf.
Fig. 5). For more dilute mixtures closer to the flammability limits, the cells formed initially did
not split but instead closed up upon themselves to form stable spherical flame structures (the
20
flame balls). For still more dilute mixtures all flame balls eventually extinguished. It was inferred
that stable, stationary flame balls would probably occur in all combustible mixtures with low Le
for mixtures close to the extinction limits, however, the short duration of drop tower experiments
and the substantial fluctuations in the acceleration level in aircraft-based µg experiments
precluded definite conclusions. Recent experiments performed on the STS-83 and STS-94 Space
Shuttle missions (Ronney et al., 1998) confirmed that flame balls can be stationary and stable for
at least 500 seconds, which was the entire test duration in these cases. Sample images of flame
balls from the space flight experiments are shown in Fig. 14.
These results were found to be qualitatively the same over the range 0.06 < Le < 0.3, with
H2 and CH4 fuels, with or without added CF3Br (a chemical inhibitor) and at pressures from 0.5
to 3 atm, indicating that variations in Lewis number over this range, chemical mechanisms, and
radiation spectra do not qualitatively influence these phenomena.
As predicted by Eq. (13), because T* - T ∞ ~ 1/Le, in mixtures with Le < 1 the flame ball
temperature T* can be much larger than the adiabatic homogeneous flame temperature T f. In the
case of H2-O2-SF6 mixtures (that have Le ≈ 0.06, the lowest of the mixtures tested to date),
values of Tf as low as 465K have been found to exhibit flame balls. This temperature is far below
the H2-O2 explosion limit temperature of about 850K at 1 atm (Lewis and von Elbe, 1987), thus
such mixtures could not possibly exhibit plane flames.
Zeldovich (1944) noted the possibility of heat losses stabilizing flame balls. The apparent
experimental discovery of stable flame balls in near-limit mixtures 40 years later motivated
additional theoretical studies. Volumetric radiative losses (e.g., due to gas radiation) are predicted
(Buckmaster et al., 1990, 1991) to lead to two stationary flame ball radii (Fig. 15), a “large” flame
ball that is strongly affected by heat loss and a “small” flame ball that is nearly adiabatic. When
the losses are sufficiently strong no solutions exist, indicating a flammability limit.* As the limit
is approached, the difference between the radii of the "large" and "small" balls decreases to zero.
Stability analyses predict that all small flame balls are unstable to radial disturbances, i.e., the
flame will either grow outward from the equilibrium radius (and possibly develop into a
propagating flame) or collapse and extinguish. The basic reason is that as the flame ball radius
increases, the radius to volume ratio decreases, thus the ratio of total heat release (which is
proportional to the flame ball radius) to total radiative heat loss (which is proportional to the
*As
a result, there are at least four steady solutions to the low Mach number conservation equations for nonadiabatic flames, namely the two solutions to the equation S 2ln(S2)=-Q for planar flames and the two flame ball
solutions.
21
flame ball volume) increases, thus the flame ball becomes weaker and shrinks. Conversely, if the
radius decreases, the flame ball grows stronger and expands. Thus, flame balls with sufficient
volumetric losses can be stable to radial disturbances, but only for the large flame balls because
the smaller balls have too little volume and thus too little volumetric loss for this mechanism of
stabilization to be effective. Large flame balls with weak heat loss effects, i.e., far from the
flammability limits, are predicted to be unstable to three-dimensional disturbances, which is
consistent with the observation of splitting cellular flames in these mixtures. A portion of the
large flame branch close to the extinction limits is stable to both types of disturbances, which is
consistent with the experimental observations.
It has also been predicted (Lee and Buckmaster, 1991) that stable flame balls can only
exist for mixtures with mixtures having Le less than a critical value that is less than unity, which
explains why flame balls are not observed for mixtures with Le less than but close to unity (e.g.
CH4-air) or larger than unity (e.g. C3H8-air), even for near-limit mixtures at µg. Instead,
conventional propagating flames are observed under these conditions, and SEFs in sub-limit
mixtures when Le is less than (but still close to) unity. The reason is that, according to Eq. (13),
for Le > 1, T* < Tf, thus flame balls are weaker than plane flames and cannot benefit from
curvature in the manner discussed above for Le < 1 mixtures.
All the aforementioned theories assume single-step Arrhenius kinetics with large
activation energy, constant thermodynamic and transport properties, and simple radiation
properties. Numerical simulations (Buckmaster et al., 1993) of the steady properties of nonadiabatic flame balls in H2-air mixtures employing detailed chemistry, diffusion and radiation
models were qualitatively consistent with these theories in that two solutions were predicted for
mixtures having fuel concentrations higher than the limiting value. It was shown that for mixtures
away from the flammability limit, the large flame ball is highly influenced by radiative loss, thus
its temperature is much lower than T* given by Eq. (11) (the cold giant (CG) flame) and the small
flame ball is too small for volumetric losses to be significant, thus its temperature is much closer
to T* from Eq. (11) (the hot dwarf (HD) flame). These correspond to the upper and lower
solution branches shown in Fig. 16; only the CG branch is stable, but the numerical methods used
in Buckmaster et al. (1993) allowed the unstable HD flames to be computed as well. These two
solutions are somewhat reminiscent of the “normal flame” and “weak flame” branches of the
strained flame extinction curves (Section 5.2). Thus, radiative heat losses can quench flames even
in mixtures far from the turning-point extinction limits on the solution branches corresponding to
large residence times (for strained flames) or large volumes (for flame balls).
22
The simulation of the steady properties of flame balls (Buckmaster et al., 1993) has been
extended (Wu et al., 1998, 1999) to consider dynamical properties. As Fig. 17 shows, when the
initial flame radius (ro) is close to the steady flame ball radius (r*), the flame eventually evolves to
r*, indicating stable solutions as predicted by the theories (Buckmaster et al, 1990, 1991; Lee and
Buckmaster, 1991), whereas for significantly larger or smaller ro the flames eventually quench.
As the lean and rich stability limits are approached, the range of ro leading to steady flames
narrows to zero. None of these CG-like initial conditions led to steadily propagating flames, but
obviously, sufficiently rich mixtures do exhibit propagating flames. Calculations showed that f
at the planar lean limit (due to radiative losses) is 0.298, which is higher than the rich stability
limit (f = 0.285) of the flame ball. Interestingly, then, for 0.285 < f < 0.298 there are no stable
flames of any kind. Additionally, the effects of various chemical and radiation models were
tested. It was found that three different widely-used models of H2-O2 chemistry gave widely
varying predictions of flame ball radius and radiant emission (Fig. 18), even though all predict the
burning velocities of propagating flames in H2-air mixtures very well (Fig. 19). The main
chemical step responsible for these discrepancies is the inhibiting step
H + O2 + H2O Æ HO2 + H2O, which, over the relevant range of temperatures, vary by a factor of
more than two between the H2-O2 reaction mechanisms shown in Fig. 18. Similar discrepancies
have been noted for near-limit propagating H2-air flames (Egolfopoulos and Law, 1990a).
< Figure 17, 18, 19 near here >
Another factor in flame ball properties is reabsorption of emitted radiation, which is an
important effect in mixtures diluted with radiatively-active gases such as CO2 and SF6. The
Planck mean absorption lengths are of the order 100 cm for H2O at the relevant conditions but 4
cm for CO2 and 0.3 cm for SF6. Since the chamber radius is 12 - 16 cm in the experiments
performed to date, H2O is optically thin (negligible reabsorption) in all cases, but CO2 and SF6
are optically thick, and thus at least a portion of their emitted radiation is reabsorbed within the
gas, and therefore is not a loss process. An upper bound on diluent reabsorption effects
(aP,diluentÆ ∞) can be obtained by neglecting diluent radiation entirely because as aP,diluentÆ ∞ there
is no radiative loss from the diluent and furthermore the “radiative conductivity” ≡ 16sT3/3aP
approaches zero, thus there is no additional heat transport due to radiative transfer. An example
of the differences in predictions obtained using the optically-thin and optically-thick
approximations is shown in Fig. 20 for H2-O2-CO2 mixtures. The agreement between predicted
23
and measured flame ball radii is much better when diluent radiation is neglected and the
experimental flammability limit composition is bracketed by numerical results with and without
diluent radiation. Recent simulations employing a detailed Statistical Narrow Band model (Wu et
al., 2000) yield results surprisingly similar to that of the upper bound estimate. These
observations strongly suggest that radiative reabsorption effects are needed for accurate numerical
simulation in these cases.
6. "COOL FLAMES"
It has been known for over 100 years than some combustible mixtures, particularly rich
hydrocarbon-O2-inert mixtures, can exhibit unusual behavior when preheated to moderate
temperatures (typically 300˚C - 600˚C) (Mallard and LeChatelier, 1880). The classical
apparatus used to study such behavior is a preheated, initially evacuated vessel into which
reactants are rapidly introduced. After some induction period, the vessel pressure begins to rise
and, depending on the conditions, either a single-stage autoignition that consumes most of the
reactants occurs, or one or more propagating fronts or "cool flames" occur that consume only a
portion of the reactants propagates through the vessel. Such cool flames occur because for these
mixtures, over a range of temperatures the net heat release rate decreases as temperature
increases, consequently the more typical self-accelerating thermal runaway is not observed. Such
cool flames result from complex thermokinetic interactions that occur within this temperature
range. Under favorable conditions, specifically in the presence of heat and radical losses, this
leads to oscillatory reaction or multiple cool flames. The study of autoignition and cool flames
has many practical applications, particularly to knock in premixed-charge (i.e. spark-ignition,
gasoline-fueled) internal combustion engines, which is the limiting factor in compression ratio and
thus thermal efficiency. Moreover, proposed homogeneous charge - compression ignition engines
(Stanglmaier, 1999) promise to provide high thermal efficiency and low emissions, but rely on
repeatable, controlled autoignition which is currently very difficult to obtain. Recently new
attention has been focussed on autoignition and cool flame phenomena because of their possible
role in the TWA Flight 800 Center Wing Tank explosion (DOT, 1998). Because of its practical
importance, numerous reviews of cool flame behavior appear in the literature, e.g., Griffiths and
Scott (1987), Scott (1997).
More than 30 years ago (Tyler, 1966; Griffiths et al., 1970) buoyant convection was
shown to be an important factor in the induction period and cool flame generation, since cool
flames are generally only observed at sufficiently high pressures and in sufficiently large vessels
24
that the Rayleigh number Rad ≡ GrdPr is larger than the critical value (≈ 600) for the onset of
buoyant convection. For this reason the continuously-stirred tank reactor, in which forced
turbulence is used to cause a homogeneous mixture, was developed to suppress temperature and
composition gradients. In this system oscillatory behavior can be observed, but spatiallyinhomogeneous cool flames cannot.
Experiments by Pearlman (2000a, b) in unstirred vessels showed that at 1g, multiple cool
flames can be observed in rich n-C4H10/O2 mixtures whereas at µg, usually only a single flame is
observed (Figure 21.) At 1g, the cool flames always originate at the top of the vessel whereas at
µg, the flame originates from the center of the vessel. In both cases this is thought to be due to
the temperature becoming highest at the location where the cool flames originate. (While the
initial temperature in both 1g and µg cases is uniform, at 1g exothermic chemical reaction
generates locally higher temperatures in some region and the hotter, less dense gas rises to the top
of the vessel, whereas at µg the walls act as a sink for the thermal energy generated by chemical
reaction and thus in quasi-steady state there is a net conductive heat flux from the center of the
vessel toward the walls.) Pearlman (2000a, b) also found that the induction period (time to the
first cool flame or autoignition) is always shorter at µg due to the lower heat losses associated
with conductive as compared to convective transport. Furthermore, at µg, when chemically-inert
helium is added, which increases both the thermal diffusivity and Lewis number of the mixture,
multiple cool flames were observed. When the same mole fraction of argon (which has the same
heat capacity as helium but leads to lower thermal diffusivity and Lewis number than helium) is
substituted for helium, only a single cool flame followed by homogeneous ignition is observed.
These results clearly indicate the need for sufficiently high heat and/or mass diffusion
rates in order to observe multiple cool flames at µg, and that transport due to buoyant convection
at 1g can act in a manner qualitatively similar to diffusive transport. Very recent modeling
studies (Fairlie et al., 2000) using a one-dimensional, time-dependent code in spherical geometry
show qualitative agreement with the experimental observations and provides some additional
insight. The following explanation for the effects of transport on the existence of multiple cool
flames at µg was proposed. After the passage of the first cool flame, the lowest temperatures are
present in the outer part of the reacting mass (nearer the wall in µg, or nearer the bottom of the
vessel at 1g), which enables the accumulation of reactive intermediates close to the edge of this
mass (due to the negative effect of temperature on reaction rate under these conditions).
Chemical reaction and heat release are thus sustained in this region and there is an accompanying
transport of the reactive intermediates towards the center, which then augment reaction of the
25
residual fuel sufficiently to lead to development of subsequent cool flame(s). Fairlie et al. also
found that when the mass diffusion of the intermediates was suppressed but the thermal
diffusion was maintained at a sufficiently high value, oscillations did not occur, indicating there is
a key role of the re-supply of the intermediate to the center to help re-establish reaction there
(since the cool flames always originate in the center.) Without this diffusion to the center, the
system was dormant because intermediates did not survive after the first cool flame passage. In
the model, it was found that a low Lewis number of the free radical specie responsible for chain
propagation encouraged multiple cool flames but a high Lewis number discouraged cool flames,
indicating that transport of radicals is a more significant factor that transport of temperature in
enabling cool flames. While the experiments (Pearlman, 2000b) showed that a higher Lewis
number encouraged rather than discouraged cool flames, this was in conjunction with the
substitution of helium for argon as an inert, and thus both diffusivities increased in this case.
Consequently, the overall propensity for cool flame behavior seems to depend more on increasing
diffusivities than increasing the ratio of mass to thermal diffusivity.
When spatially uniform conditions and a finite volumetric heat loss term are assumed (i.e.,
infinite diffusivity within the gas, but finite transfer from the gas to the wall) in the model, the
entire volume can cool down, leading oscillatory chemical activity (this is expected since the
chemical model used by Fairlie et al. was chosen to ensure oscillatory chemical activity under
spatially uniform conditions.) Of course, if the heat loss term were infinite, no cool flame or
homogenous oscillatory behavior could be observed because the gas temperature would be equal
everywhere to the wall temperature. To date, computations and experiments have not yet
established the upper and lower limits on Lewis and Peclet number (a measure of the heat loss
term that could be defined in this case as tdiff/tchem) for which cool flames, oscillatory homogenous
reactions, quenching, etc. are observed.
7. TURBULENT FLAMES
All of the phenomena described in the previous sections apply to flames in quiescent or
laminar flows. Of course, most practical flames occur in turbulent flows. In general, one would
expect that with the large increases in convective transport in turbulent flows as compared to
laminar flows, most turbulent flames would not be affected by buoyancy, but exceptions do
exist. The impact of buoyancy on turbulent flame speed (ST) has been predicted (Libby, 1989) to
be given by the relation
26
ST
u' 1 - e gLI
= 2.15 +
SL
SL 0.867 S2L
(14),
where LI is the integral scale of turbulence, u’ the turbulence intensity and g is positive/negative
for upward/downward propagation. Equation (14) indicates that the strongest effects occur for
large scale turbulence and small SL. This might be expected since a buoyancy-induced convection
velocity on the turbulent wrinkling scale would be proportional to (gLI)1/2, and thus the
buoyancy effect term in Eq. (14), gLI/SL2, is simply the square of the ratio of this buoyant
convection velocity to SL. It may also be considered the inverse of a Froude number. The effect
of the buoyancy parameter gLI/SL2, thermal expansion and diffusive-thermal effects on ST has
also been studied (Aldredge and Williams, 1991) for downward-propagating turbulent flames in
the limit u’/SL << 1. Still, there have been no experimental tests of these predictions. Flames in a
rapidly rotating tube (50 go < g < 850 go) have been examined (Lewis, 1970) and it is found that
ST ~ g0.387, though in this case there was no forced turbulence; instead the buoyant flow induced
turbulence (with the aid of a perforated plate near the ignition site to initiate turbulence).
Experiments at earth gravity in tubes of varying diameter with gases of varying viscosity (Wang
and Ronney, 1993) have shown that laminar-flow relation for rising flames in tubes (Levy, 1965)
Sb ~ (gd)1/2 applies even in large tubes and in gases with low viscosity where the burned gases are
turbulent, and thus the observations (Lewis, 1970) might be simply a manifestation of the
buoyant rise speed at very high g. Other investigators (Hamins et al., 1988) compared upward
and downward-propagating flames in which a rotating fan was used to generate turbulence. No
significant effect of gravity was found, but near-limit mixtures having small SL (which are the
mixtures most likely to exhibit buoyancy effects) were not tested and furthermore the turbulence
intensity (u’) was not measured.
Recently the effects of gravity on flame structure and stabilization of turbulent premixed
round-jet flames and flames stabilized downstream of a cylindrical rod (“V-flames”) has been
investigated (Kostiuk and Cheng, 1995; Bedat and Cheng, 1996). For these flames, as the flow
velocity (U) increases, both the turbulence levels and the volume of buoyant fluid increase, thus
buoyancy effects do not necessarily decrease. For example, in lean methane-air mixtures the
flame lengths at 1g and µg do not converge as U increases, in fact the difference in flame lengths
remains approximately constant (Kostiuk and Cheng, 1985). This was attributed to the
difference between the divergent (thus decelerating) flow at µg as opposed to the practically non-
27
divergent flow associated with the accelerating buoyant plume at 1g. In contrast, little effect of
buoyancy was found on flame stabilization limits (minimum and maximum U for which the jet
flame could be stabilized) for turbulent jets, whereas for laminar jets there was much more effect
(Bedat and Cheng, 1996), perhaps because the stabilization mechanism is dominated mainly by
the behavior near the flame base and thus is not affected by buoyancy when the jet exit velocity
is large (and thus the flow is turbulent).
8. RECOMMENDATIONS FOR FUTURE STUDIES
8.1 Flame propagation in optically-thick mixtures
This review of prior literature on premixed-gas flame studies at µg indicate the need for
additional studies in some areas. Perhaps the most pressing of these is optically-thick flame
propagation. All of the radiative effects discussed above are critically dependent on the degree of
reabsorption. To study reabsorption effects requires large systems, high pressures and/or
radiatively-active diluents such as CO2 and SF6 that have small n. All of these conditions lead to
high Grashof numbers at 1g and thus turbulent flow. Hence, µg experiments provide an excellent
opportunity to study reabsorption effects on combustion processes without the additional
complication of turbulent flow.
Reabsorption effects is a subject of importance not only to µg studies, but also
combustion at high pressures and in large industrial furnaces. For example, at 40 atm, a typical
pressure for premixed-charge internal combustion engines, ap ≈ 18 m-1, thus aP-1 = 4.5 cm, for the
products of stoichiometric combustion. This length scale is comparable to the cylinder radius,
thus reabsorption effects within the gas cannot readily be neglected in models of engine
combustion and heat transfer. The ratio of radiative conductivity (lR) to molecular conductivity
= (16sT f3/3ap)/lf for this example is about 1700. This indicates that even when turbulent
transport is considered (which increases lf significantly but has no effect on l R), much of the
thermal energy transport will be due to radiation rather than conduction or convection.
Understanding of these radiative effects is a relevant issue because simple estimates (Ronney,
1995, 1999) indicate radiative loss may influence flame quenching by turbulence, which limits the
use of clean, fuel-efficient lean mixtures in practical engines. For similar reasons, reabsorption
cannot be neglected in atmospheric-pressure furnaces larger than aP-1 ≈ 2.2 m. Moreover, many
combustion devices employ exhaust-gas or flue-gas recirculation; for such devices significant
amounts of absorbing CO2 and H2O will be present in the unburned mixtures, which will further
increase the importance of radiative transport.
28
While modeling of radiation effects in flames under optically thin conditions (e.g.
Lakshimisha et al., 1990; Ju et al., 1998b, 1999; Wu et al., 1998) is reasonably straightforward,
modeling of spectrally dependent emission and absorption is a challenging task because effects of
local fluxes depend on the entire radiation field, not just local scalar properties and gradients.
Some studies using gray-gas models have been reported (Marchese and Dryer, 1996) but recent
studies (Bedir et al., 1997; Ju et al., 1998a) have shown that the accuracy of these methods for
assessing reabsorption effects at high pressures or in large systems is questionable, because of the
wide variation in spectral absorption coefficient with temperature and species. A useful
comparison of various approximate radiative treatments for a nonpremixed flame of small
dimension (≈ 1 cm) has been given recently (Bedir et al., 1997). Comparisons for premixed
flames and larger, multi-dimensional systems would be useful.
Despite the value of investigations of radiative reabsorption effects, only very
preliminary experimental studies have been conducted to date (Abbud-Madrid and Ronney,
1990, 1993; Wu et al., 1998). Experimentally, there are at least two ways to control the optical
thickness, either by using diluent gases with small absorption lengths (e.g. CO2 or SF6,
particularly at high pressure), or by the addition of inert radiating particles. Both approaches
have limitations. In the former case the disadvantage is on the modeling side - there is a very
complicated spectral dependence of the absorption coefficient on wavelength, which in turn
depends on temperature and to a lesser extent on pressure, and there are always spectral regions
in which no emission or absorption occurs. In the latter case the problem is mostly on the
experimental side, namely in obtaining uniform particle dispersion and possibly complications
due to heterogeneous chemical effects on the surface of the particles.
8.2 Flame propagation in tubes
In sections 3.3 the relative importance of conductive vs. radiative heat loss on extinction
of flames propagating through cylindrical tubes was discussed. A more thorough assessment of
the mechanisms of flammability limits in tubes would be valuable, in particular the assessment of
the transition from limits induced by conductive loss to limits induced by radiative loss as the
tube diameter is increased. It would be expected that a transition from diameter-dependent SL,lim
due to conductive losses to diameter-independent SL,lim due to radiative losses would occur when
the values of SL,lim predicted by Eqs. (5) and (6) are equal, i.e.
d ª 16
l •T f
(14),
bL
29
where the same temperature-averaging of transport properties used in section 3.2 and 3.3 has
been applied here. For the example of lean hydrocarbon-air flames employed throughout this
paper, Eq. (14) predicts d ≈ 2.7 cm at the transition. The accuracy of this prediction should be
tested for a range of pressures and diluent gases to test the effect of l ∞ and L on this transition
diameter.
8.3 Three-dimensional effects
Linear stability analyses (Buckmaster et al., 1990, 1991) show that three-dimensional
instabilities are important in the development of flame balls from an ignition kernel. To date, it is
known that large flame ball branch is linearly unstable to three-dimensional disturbances and
exhibits splitting flame balls if the scaled heat loss is sufficiently small (Fig. 15), which will occur
mixtures sufficiently far from the extinction limit or if the local enthalpy is increased sufficiently
by an ignition source. Still, the transition from splitting flame balls to stable flames has not been
analyzed to date nor is there any method to predict the number of flame balls produced from a
given ignition source. Modeling using a three-dimensional code (Patnaik et al., 1996) could shed
some light on this subject.
8.4 Catalytic combustion
The potential benefits of catalytic combustion for reduced emissions and improved fuel
efficiency in many combustion systems is well known (Pfefferle and Pfefferle, 1986; Warnatz,
1992). Since catalysis occurs at surfaces, catalysis is inherently a multi-dimensional and/or
unsteady process requiring transport of reactants to the surface and heat and products away from
the surface. While boundary layer approximations can be incorporated into models of reaction at
catalytic processes, the only truly one-dimensional steady catalytic configuration would be a
spherical surface immersed in a nonbuoyant quiescent gas - i.e., a “catalytic flame ball.” In this
case the radius (r*) is fixed but the surface temperature T* and fuel concentration Y* are unknown.
T * and Y* can be related through energy conservation (including surface radiation) and the
diffusion equations, leading to the following expression for the surface reaction rate in moles per
second (w):
w (Y* ,T* ) = r* D* r* Y• (1 - Y* / Y• ) / M;
Ê T - T ˆ Ê ser* (T*4 - T•4 )ˆ
Y*
•
˜ Á1 +
= 1 - LeÁ *
˜
Y•
l * (T* - T• ) ¯
Ë Tf - T• ¯ Ë
(15)
where Y∞ is the ambient fuel mass fraction, M the fuel molecular weight, T* the measured surface
temperature, e the surface emissivity (not to be confused with the density ratio used earlier in
this paper) and the subscript * refers to properties evaluated at the temperature T *. Through
30
varying r*, Y∞, pressure, and diluent gas, w can be determined for a range of surface fuel
concentrations and temperatures and compared to models. Of course, the conditions must be
unfavorable for the initiation of a propagating flame or a flame ball that stands off from the
surface for the catalysis process to be examined.
8.5 Turbulent flame quenching
Turbulence may increase ST to values well above SL, however, increasing turbulence levels
beyond a certain value increases ST very little if any and may lead to complete quenching of the
flame (Abdel-Gayed and Bradley, 1985, Bradley, 1992). This effect is particularly pronounced
when SL is small compared to u', e.g., for lean fuel-air mixtures. This indicates that the
propagation rates of very lean mixtures cannot be increased ad infinitum merely by increasing u'.
Thus lean mixtures, which thermodynamically promise higher thermal efficiencies and lower
pollutant emissions, will exhibit unsatisfactory combustion rates in many practical systems. In
addition to its long-standing relevance to automotive applications (Heywood, 1988), lean
premixed turbulent combustion is now employed in stationary gas turbine applications because
NOx emissions can be reduced considerably compared to stoichiometric mixtures (Correa, 1992).
The mechanism(s) of flame quenching by turbulence are still not well understood.
Recently it has been suggested that radiative heat losses are a likely mechanism, leading to, for
hydrocarbons in air at 1 atm, the following predicted quenching criterion (Ronney 1995, 1999):
Ka ≈ 0.38 ReL0.76
(16),
where Ka ≡ 0.157 ReL-1/2(u'/SL)2Sc-1 is the turbulent Karlovitz number, ReL = u'LI/n is the
turbulent Reynolds number and Sc the Schmidt number (≈ 0.7 for most gases). One difficulty in
assessing this prediction and especially probing the structure of flames near this quenching
threshold is that relatively high u'/SL is required to obtain quenching, thus very high u' for nearstoichiometric mixtures with high SL. Moreover, the power required to generate the turbulence is
proportional to (u')3 and the size of the smallest scales of turbulence is proportional to 1/(u')3/4.
Consequently, it would be desirable to employ mixtures with low SL and thus low u' to obtain a
given u'/SL, but buoyancy effects preclude the use of such mixtures at 1g. A study of turbulent
flame quenching at µg using diluents of varying radiative properties, and comparison of these
results to Eq. (16) or other models of quenching (Abdel-Gayed and Bradley, 1985, Bradley,
1992), could provide considerable insight into this subject. For example, it may be useful to
study the effects of pressure (P) on Ka at quenching, since very few of these data are reported on
the literature. The analysis leading up to Eq. (16) predicts that the factor 0.38 is proportional to
31
P-1 because of the effect of pressure on radiative loss per unit volume. This leads to Ka ~ P0.24
for fixed u' and LI whereas published correlations (Bradley, 1992) predict Ka ~ P1 for such
conditions. Microgravity experiments could be quite useful for testing and comparing such
models.
8.6 Chemical models
One of the most important contributions of µg combustion experiments has been an
improved understanding of extinction processes. Of course, extinction processes are inherently
related to finite-rate chemistry effects. Thus, to obtain closure between µg experiments and
model predictions, accurate chemical models are needed. The near-limit flame studies at µg have
clearly indicated the inadequacy of our knowledge of the reaction rates of even the relatively
simple H2-O2 chemistry (a necessary subset of hydrocarbon-O2 chemistry) under weakly burning
conditions (cf. Figs. 18 and 19). In fact, µg experiments, for example SEFs (section 5.1) or flame
balls (section 5.3) may prove to be one of the most useful means of determining these rates
because the simplicity of the flow environment allows more computational resources to be
brought to bear on the chemical part.
To date, in practically all comparisons for lean premixed hydrocarbon-air flames the
models (Lakshmisha et al., 1990; Guo et al., 1997; Ju et al., 1998a) predict higher SL and leaner
flammability limits than the experimental observations (Strehlow and Reuss, 1981; Ronney,
1985; Abbud-Madrid and Ronney, 1990; Maruta et al., 1996). The discrepancy seems to be
more than experimental uncertainty or unaccounted heat losses could explain. In contrast, for
flame balls (Wu et al., 1998, 1999) and strained premixed H2-air flames at 1g (Egolfopoulos and
Law, 1990a) the same chemical reaction mechanisms predict smaller flame balls, lower SL and
richer flammability limits than the experimental observations. All of these chemical models
faithfully predict the burning velocities of flames in mixtures away from extinction limits. A
substantial part of the discrepancy seems to be due to differences in the 3-body recombination
rates for the H + O2 + M reactions, and in particular the third-body efficiency of various M
species (Wu et al., 1998). These reactions are extremely important in near-limit flames, but of
much lesser importance in mixtures away from limits, because of the competition between chainbranching and chain-inhibiting steps near limits (Egolfopoulos and Law, 1990b). Further
consideration of the proper rates of these reactions in the intermediate temperature range (11001400K in most cases) would be most welcomed.
All practical combustion engines operate at pressures much higher than atmospheric. The
relative importance of various elementary reaction steps changes as pressure increases
(Egolfopoulos and Law, 1990b). The impact of buoyancy scales as tchem/tvis ~ (ga/SL3)2/3 ~ Pn-4/3,
where n is the overall order of reaction (SL ~ Pn/2-1). Since typically n < 4/3 for weak mixtures
32
(Egolfopoulos and Law, 1990b) where buoyancy effects are likely to be important, the impact of
buoyancy generally increases with pressure. Also, as discussed in Section 1, the effects of
radiative transport are more difficult to assess at higher pressure due to increased interference
from buoyant transport. Consequently, further study of µg flammability limits and near-limit
burning velocities at high pressure would provide a useful assessment of flame chemistry models
at high pressure.
9. CONCLUSIONS
A wide variety of premixed-gas flame phenomena in mixtures with low burning velocities
are either seen only at µg conditions, or are much more clearly elucidated at µg. At 1g, heat and
mass transport affecting these weakly-burning flames is dominated by buoyant convection.
When buoyancy is eliminated, transport of thermal energy by diffusive and radiative mechanisms
becomes much more important. Correspondingly, transport of chemical species by diffusion also
becomes much more important at µg (though there is no analog to thermal radiation for transport
of chemical species). Most of the observed changes in flame behavior at µg can be attributed to
the increased importance of diffusive and radiative effects.
One of the most important results of µg combustion experiments is that they have helped
to integrate radiation into premixed flame theory. Although flame radiation has long been
recognized as an important heat transfer mechanism in large flames, its treatment has largely been
ad hoc because of the difficulty of predicting soot formation. Also, large-scale flames at 1g are
inevitably turbulent, leading to complicated flame-flow interactions. µg flames are laminar, often
soot-free and have significant influences of radiation. As a result, premixed flames have exhibited
dual-limit extinction behavior, with residence time limited extinction at high strain or curvature
and radiative loss induced extinction at low strain or curvature. The high-strain limit is readily
observed at 1g, and in fact in some cases buoyant flow causes the strain. For weak mixtures
these limits converge, but the convergence and the entire low-strain extinction branch can only be
seen at µg (Figs. 10 and 11). Similar behavior is also commonly found for non-premixed flames
(Maruta et al., 1998). Considering the rapid progress made recently in studies of premixed
flames at µg, further advances are certain to occur in the near future. Hopefully this report on the
current state of understanding can help motivate and inspire such advances.
ACKNOWLEDGMENTS
33
The author’s work on premixed-gas combustion at µg has been supported by the NASA
Glenn Research Center under grants NAG3-965, NAG3-1242, NAG3-1523 and NAG3-2124.
Discussions with John Buckmaster, John Griffiths, Guy Joulin, Yiguang Ju, Kaoru Maruta,
Takashi Niioka, Howard Pearlman, Howard Ross, Gregory Sivashinsky, Karen Weiland, Forman
Williams and others have been invaluable in extending the author’s knowledge of this subject and
in the preparation of this manuscript.
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39
Time scale
Chemistry (tchem)
Buoyant, inviscid (tinv)
Buoyant, viscous (tvis)
Conduction (tcond)
Radiation (trad)
Stoichiometric flame
0.00094 s
0.071 s
0.012 s
1.04 s
0.13 s
Near-limit flame
0.25 s
0.071 s
0.010 s
1.56 s
0.41 s
Table 1. Estimates of time scales for stoichiometric and near-limit hydrocarbon-air flames at 1
atm pressure.
40
Mole %
CH4
SL, µg, spherical expanding
flame (Ronney and
Wachman, 1985)
SL, µg, Standard Flammability
Limit Tube (Strehlow and
Reuss, 1981)
5.87
6.8
6.4
5.63
5.0
4.9
5.45
3.8
4.0
5.33
3.0
3.7
5.22
2.4
2.2
5.10
1.7
1.7
5.07
1.5
---
Table 2. Comparison of burning velocities of near-limit CH4-air mixtures at atmospheric pressure
measured at µg.
41
Figure captions
Figure 1. Schematic of buoyancy-induced flame extinction in tubes. Left: upward propagation;
right: downward propagation.
Figure 2. Effect of 0.6 µm SiC particle loading on burning velocity and peak pressure in 5.25%
CH4-air mixtures burning at µg (Abbud-Madrid and Ronney, 1993). Chamber volume is 12 liters,
initial pressure 1 atm.
Figure 3. Predicted values of burning velocity and peak flame temperature in CH4 - (0.21 O2 +
0.49 N2 + 0.30 CO2) mixtures under adiabatic conditions, with optically-thin radiative losses, and
with radiation including reabsorption effects (Ju et al., 1998a).
Figure 4. Diagram of non-dimensional growth rate for diffusive-thermal, Darrieus-Landau
(thermal expansion) and Rayleigh-Taylor (buoyancy-driven) instability mechanisms as a function
of wavenumber for an idealized downward propagating flame with b = 10, Le = 0.5, e = 0.8 and
ga/SL3 = 0.5.
Figure 5. Images of expanding cellular flame fronts in H2-air mixtures at µg (Ronney, 1990).
Left: 7.0% H2 in air, 0.4% CF3Br added for improved visibility, 1.18 s after ignition. Right: 7.0%
H2 in air, 2.25% CF3Br, 1.67 s after ignition. Field of view is 18 cm x 18 cm in both images.
Figure 6. Images of flames propagating in tubes in high Lewis number mixtures. All flames were
ignited at the open end of the tube and propagating towards the closed end. Images courtesy of
Howard Pearlman. (a) and (b) are axial views, (c) is a radial view.
(a) Spiral wave pattern, 0.80% octane - 21.00% oxygen - 78.20% helium, 28.5 cm
diameter tube, downward propagation at 1g
(b) Target pattern, 1.46% octane - 21.00% oxygen - 77.54% helium, 28.5 cm diameter
tube, downward propagation at 1g
(c) Spinning 6-arm flame, 1.21% butane - 21.00% oxygen - 77.79% helium, 14.3 cm
diameter tube, microgravity (Pearlman and Ronney, 1994).
Figure 7. Predicted evolution of non-dimensional flame radius (R) as a function of nondimensional time (t) for expanding spherical flames in a mixture with Le < 1 according to Eq.
(12), for various values of the non-dimensional heat loss (Q).
42
Figure 8. Measured minimum ignition energies at 1g and µg as a function of fuel concentration
and extinction radii of spherically expanding flames (dashed curves) at µg in mixtures outside the
µg flammability limit. Numbers in parenthesis refer to energy release before extinction.
Mixtures: NH3-air at 1 atm initial pressure (Ronney, 1988a).
Figure 9. Measured flame radius as a function of time for expanding spherical flames at µg in
5.07% CH4-air mixtures at 1 atm for varying values of spark ignition energy (Ronney, 1985).
Figure 10. Measured and predicted extinction strain rates for strained premixed CH4-air flames at
1g and µg (Guo et al., 1997).
Figure 11. Measured and predicted extinction strain rates for strained premixed C3H8-air flames
at 1g and µg (Ju et al., 1998b).
Figure 12. Predicted extinction branches for strained CH4-O2-N2-He mixtures (Le ≈ 1.2) with
radiative heat loss (Ju et al., 1999).
Figure 13. Schematic diagram of a flame ball, illustrated for the case of fuel-limited combustion at
the reaction zone. The oxygen profile is similar to the fuel profile except its concentration is nonzero in the interior of the ball. The combustion product profile is identical to the temperature
profile except for a scale factor.
Figure 14. Enhanced-contrast images of flame balls obtained during the STS-94 space flight
experiments (flame balls may appear to have widely varying sizes because of varying distances
from the camera).
Figure 15. Effect of heat loss (Q) on flame ball radius (R) and stability properties (Buckmaster et
al., 1990).
Figure 16. Computed effect of mixture strength on flame ball radii in lean H2-air mixtures
(Buckmaster et al., 1993).
Figure 17. Computed dynamical properties of flame balls in lean H2-air mixtures - eventual fate
as a function of initial radius (ro) normalized by the steady radius (r*) (Wu et al., 1999).
43
Figure 18. Predicted flame ball radii and radiant power for H2-air mixtures using three popular
chemical models (Wu et al., 1998).
Figure 19. Comparison of computed properties of propagating flames for 3 different H2-O2
chemical mechanisms - burning velocity (SL) as a function of equivalence ratio in H2-air mixtures
(Wu et al., 1998). A compilation of experimental results from several sources is also shown.
Figure 20. Computed flame ball radius at the location of maximum volumetric heat release as a
function of H2 mole fraction for steady flame balls for H2-O2-CO2 mixtures with H2:O2 = 1:2
(Wu et al., 1998). Preliminary experimental results from MSL-1 are also shown (filled circles).
Figure 21. Autoignition and cool flame behavior of a 67% n-C4H10- 33% O2 mixture at an initial
temperature 310oC and an initial pressure of 4.2 psia at 1g (left) and µg (right) (Pearlman, 2000b).
44
Buoyancy-induced
flame stretch
Cooling
combustion products near
wall cause sinking boundary
layer
Direction of flame
propagation
Direction of flame
propagation
Flame
front
Tube walls
Tube walls
Figure 1. Schematic of buoyancy-induced flame extinction in tubes. Left: upward propagation;
right: downward propagation.
45
4.2
2.50
Burning velocity (cm/s)
4
2.25
3.8
Burning velocity
3.6
2.00
3.4
1.75
Maximum pressure (atm)
Pressure
3.2
1.50
3
0
2
4
6
Mass of particles in chamber (g)
8
Figure 2. Effect of 0.6 µm SiC particle loading on burning velocity and peak pressure in 5.25%
CH4-air mixtures burning at µg (Abbud-Madrid and Ronney, 1993). Chamber volume is 12 liters,
initial pressure 1 atm.
46
Figure 3. Predicted values of burning velocity and peak flame temperature in CH4 - (0.21 O2 +
0.49 N2 + 0.30 CO2) mixtures under adiabatic conditions, with optically-thin radiative losses, and
with radiation including reabsorption effects (Ju et al., 1998a).
47
Diffusive-thermal
Thermal expansion
Buoyancy
Total
0.075
Growth rate ( s)
0.050
0.025
0.000
-0.025
-0.050
0
0.1
0.2
0.3
0.4
0.5
Wave number (k)
0.6
0.7
0.8
Figure 4. Diagram of non-dimensional growth rate for diffusive-thermal, Darrieus-Landau
(thermal expansion) and Rayleigh-Taylor (buoyancy-driven) instability mechanisms as a function
of wavenumber for an idealized downward propagating flame with b = 10, Le = 0.5, e = 0.8 and
ga/SL3 = 0.5.
48
Figure 5. Images of expanding cellular flame fronts in H2-air mixtures at µg (Ronney, 1990).
Left: 7.0% H2 in air, 0.4% CF3Br added for improved visibility, 1.18 s after ignition. Right: 7.0%
H2 in air, 2.25% CF3Br, 1.67 s after ignition. Field of view is 18 cm x 18 cm in both images.
49
(a)
(b)
(c)
Figure 6. Images of flames propagating in tubes in high Lewis number mixtures. All flames were ignited at the open end of the tube
and propagating towards the closed end. Images courtesy of Howard Pearlman. (a) and (b) are axial views, (c) is a radial view.
(a) Spiral wave pattern, 0.80% octane- 21.00% oxygen - 78.20% helium, 28.5 cm diameter tube, downward propagation at 1g
(b) Target pattern, 1.46% octane- 21.00% oxygen - 77.54% helium, 28.5 cm diameter tube, downward propagation at 1g
(c) Spinning 6-arm flame, 1.21% butane - 21.00% oxygen - 77.79% helium, 14.3 cm diameter tube, microgravity (Pearlman and
Ronney, 1994).
50
80
Q = 0.36
Dimensionless radius (R)
70
60
Q = 0.4
50
40
Q = 0.42
30
Q = 0.45
20
Q = 0.5
10
0
0
20
40
60
Dimensionless time ( t)
80
100
Figure 7. Predicted evolution of non-dimensional flame radius (R) as a function of nondimensional time (t) for expanding spherical flames in a mixture with Le < 1 according to Eq.
(12), for various values of the non-dimensional heat loss (Q).
51
Figure 8. Measured minimum ignition energies at 1g and µg as a function of fuel concentration
and extinction radii of spherically expanding flames (dashed curves) at µg in mixtures outside the
µg flammability limit. Numbers in parenthesis refer to energy release before extinction.
Mixtures: NH3-air at 1 atm initial pressure (Ronney, 1988a).
52
Figure 9. Measured flame radius as a function of time for expanding spherical flames at µg in
5.07% CH4-air mixtures at 1 atm for varying values of spark ignition energy (Ronney, 1985).
53
Figure 10. Measured and predicted extinction strain rates for strained premixed CH4-air flames at
1g and µg (Guo et al., 1997).
54
Figure 11. Measured and predicted extinction strain rates for strained premixed C3H8-air flames
at 1g and µg (Ju et al., 1998b).
55
Figure 12. Predicted extinction branches for strained CH4-O2-N2-He (Le ≈ 1.2) mixtures with
radiative heat loss (Ju et al., 1999).
56
T*
C ~ 1-1/r
Temperature
Fuel concentration
T ~ 1/r
To
Interior filled
with combustion
products
Reaction zone
Heat & products
diffuse outward
Fuel & oxygen
diffuse inward
Figure 13. Schematic diagram of a flame ball, illustrated for the case of fuel-limited combustion at
the reaction zone. The oxygen profile is similar to the fuel profile except its concentration is nonzero in the interior of the ball. The combustion product profile is identical to the temperature
profile except for a scale factor.
57
3.70% H2 - 96.30% air,
500 s after ignition, diameter 1.0 cm.
3.57% H2 - 96.43% air,
300 s after ignition, diameters 1.0 cm.
5.20% H2 - 10.40% O2 - 84.40% CO2,
50 s after ignition, diameters 0.7 cm.
4.00% H2 - 96.00% air,
15 s after ignition, diameters 1.0 -1.8 cm.
Figure 14. Enhanced-contrast images of flame balls obtained during the STS-94 space flight
experiments (flame balls may appear to have widely varying sizes because of varying distances
from the camera).
58
Dimensionless flame ball radius (R)
15
Unstable to 3-d disturbances
10
Stable
Equation of curve:
5
0
0
-2
R ln(R) = Q
Unstable to 1-d
disturbances
0.05
0.1
0.15
Dimensionless heat loss (Q)
0.2
Figure 15. Effect of heat loss (Q) on flame ball radius (R) and stability properties (Buckmaster et
al., 1990).
59
1
Radius (cm)
Cold
Giant
0.1
Hot
Dwarf
0.01
0.001
0.08
0.09
0.1
0.11
Equivalence ratio
0.12
0.13
Figure 16. Computed effect of mixture strength on flame ball radii in lean H2-air mixtures
(Buckmaster et al., 1993).
60
Quench
Steady flame ball
No stable flames
Planar flames possible
Quench
1
0.1
No flames possible
Stretching parameter (c) = or/r*
10
Steady
flame
balls
Steady
flame
balls
Quench
Quench
0.08
0.1
0.2
0.3
Equivalence ratio (f )
Figure 17. Computed dynamical properties of flame balls in lean H2-air mixtures - eventual fate
as a function of initial radius (ro) normalized by the steady radius (r*) (Wu et al., 1999).
61
10
Upper curves: radius
Lower curves: radiant power
8
0.6
6
0.4
4
Yetter
0.2
2
Peters
GRI
0
Radiant power (W)
Flame ball radius at max OH concentration
0.8
3
3.5
4
4.5
Mole percent H in air
5
0
2
Figure 18. Predicted flame ball radii and radiant power for H2-air mixtures using three popular
chemical models (Wu et al., 1998).
62
400
8
GRI chemical mechanism
350
250
4
w/o H O
2
radiation
2
w/ H O
2
radiation
0
200
0.26
0.28
0.3
0.32
0.34
L
S (cm/s)
300
6
150
Yetter
GRI
Peters
Expt.
100
50
0
0.1
1
Equivalence ratio
10
Figure 19. Comparison of computed properties of propagating flames for 3 different H2-O2
chemical mechanisms - burning velocity (SL) as a function of equivalence ratio in H2-air mixtures
(Wu et al., 1998). A compilation of experimental results from several sources is also shown.
63
Flame ball radius (cm)
1.6
1.2
predictions
(w/o CO radiation)
2
0.8
predictions
(w/ CO radiation)
2
0.4
0
2
experiments
4
6
8
10
Mole % H
12
14
16
2
Figure 20. Computed flame ball radius at the location of maximum volumetric heat release as a
function of H2 mole fraction for steady flame balls for H2-O2-CO2 mixtures with H2:O2 = 1:2
(Wu et al., 1998). Preliminary experimental results from MSL-1 are also shown (filled circles).
64
Cool Flames
5
Pressure
320
T1
315
T2
3
310
T3
2
305
1
300
0
295
-1
290
0
50
100
Time (s)
150
-2
200
325
5
Cool Flame
320
Pressure
315
T (o C)
P (psia)
4
4
T1
3
T2
310
2
T3
305
300
1
0
Acceleration
295
-1
290
P (psia), Acceleration (g/gearth)
T (o C)
325
-2
10
15
20
30
25
Time (s)
35
40
45
Figure 21. Autoignition and cool flame behavior of a 67% n-C4H10- 33% O2 mixture at an initial
temperature 310oC and an initial pressure of 4.2 psia at 1g (upper) and µg (lower) (Pearlman,
2000b).
65
NOMENCLATURE
aP
A
CP
d
D
E
g
go
Grd
h
LI
Le
P
r
rf
R
Rg
SL
SL,lim
Sc
t chem
t inv
t rad
t vis
T
T ad
u’
U
Uy
Y
Planck mean absorption coefficient
flame surface area
constant-pressure heat capacity
characteristic flow length scale or tube diameter
mass diffusivity
overall activation energy of the heat-release reactions
acceleration of gravity
earth gravity
Grashof number based on characteristic length scale (d) = gd3/n2
heat transfer coefficient in a cylindrical tube = 16l/d2
turbulence integral scale
Lewis number (a/D = thermal diffusivity / reactant mass diffusivity)
pressure
radial coordinate
flame radius
scaled flame radius (Eq. (12))
gas constant
premixed laminar burning velocity
burning velocity at the flammability limit
Schmidt number = n/D
chemical time scale
inviscid buoyant transport time scale
radiative loss time scale
viscous buoyant transport time scale
temperature
adiabatic flame temperature
turbulence intensity
convection velocity
local axial velocity in counterflow configuration
fuel mass faction
a
b
d
g
l
L
n
r
s
S
thermal diffusivity
non-dimensional activation energy = E/RgT f
flame thickness
gas specific heat ratio
thermal conductivity
radiative heat loss per unit volume = 4saP(Tf4 - T∞4)
kinematic viscosity
density
Stefan-Boltzman constant
flame stretch rate
66
Subscripts
f
∞
flame front condition
ambient conditions
67

Premixed-gas flames - Paul Ronney - University of Southern California

Transcription

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