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Nonlinear effects in the extraction of laminar flame speeds from expanding spherical flames
Combustion and Flame 156 (2009) 1844–1851
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Nonlinear effects in the extraction of laminar flame speeds from expanding spherical flames A.P. Kelley, C.K. Law * Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
a r t i c l e
i n f o
Article history: Received 14 January 2009 Received in revised form 9 April 2009 Accepted 10 April 2009 Available online 18 May 2009 Keywords: Flame speed extrapolation Butane Propagating spherical flame Laminar flame speed Markstein length
a b s t r a c t Various factors affecting the determination of laminar flames speeds from outwardly propagating spherical flames in a constant-pressure combustion chamber were considered, with emphasis on the nonlinear variation of the stretched flame speed to the flame stretch rate, and the associated need to nonlinearly extrapolate the stretched flame speed to yield an accurate determination of the laminar flame speed and Markstein length. Experiments were conducted for lean and rich n-butane/air flames at 1 atm initial pressure, demonstrating the complex and nonlinear nature of the dynamics of flame evolution, and the strong influences of the ignition transient and chamber confinement during the initial and final periods of the flame propagation, respectively. These experimental data were analyzed using the nonlinear relation between the stretched flame speed and stretch rate, yielding laminar flame speeds that agree well with data determined from alternate flame configurations. It is further suggested that the fidelity in the extraction of the laminar flame speed from expanding spherical flames can be facilitated by using small ignition energy and a large combustion chamber. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Early efforts at determining laminar flame speeds, even conducted with exceptional care, exhibited large scatter for data obtained using different experimental methods [1,2]. The fundamental reason that these measurements differed is due to the unquantified effects of stretch [3,4], which is induced by local flow straining, flame curvature, and flame unsteadiness [5,6]. As such, these effects must be subtracted from the experimental data in order to unambiguously determine the laminar flame speed [3]. Based on asymptotic analysis [7], Wu and Law [3] subtracted the stretch effect by linearly extrapolating the experimental stretched flame speed, determined as a function of the stretch rate, to zero stretch rate according to,
~s ¼ 1 r
ð1Þ
where ~s ¼ s=so , r ¼ Lj=so is a non-dimensional stretch parameter, s the flame speed, j the stretch rate, L a constant that measures the mixture’s sensitivity to stretch and is commonly referred to as the Markstein length, and the superscript o refers to the unstretched state. This linear extrapolation has been extensively adopted for flame speed determinations using the counterflow/stagnation flame [8] and the outwardly propagating flame [9,10]; the references cited are just representative of the many works that have since appeared.
* Corresponding author. E-mail address: [email protected] (C.K. Law).
In fact, works of this nature are appearing at an increasing rate because of the correspondingly increasing interest in accurate values of the laminar flame speeds for studies of fuel chemistry. While the linear extrapolation is convenient to apply, it has also been recognized that care needs to be exercised in order to ensure that the extrapolation not only is accurate but is also meaningful. For example, the range of data based on which the linear extrapolation is conducted should be sufficiently extensive while the magnitude of their stretch rates, when properly scaled through, say, the Karlovitz number, should also be sufficiently small so that the deviation of the measured flame speed from the unstretched value is correspondingly small. There are, however, situations in which the deviation can be more substantial, due to high stretch rates and/or strong mixture non-equidiffusion such that higher-order effects could be important. For these situations a linear approximation not only is inaccurate, but the act of performing a linear extrapolation from a dataset exhibiting curvature could also impart substantial uncertainty in the extrapolated value. Furthermore, additional processes and factors could also be present in the measurement technique that could lead to systematic deviations from linearity. For example, for the counterflow technique, Tien and Matalon [11] identified a regime of nonlinearity as j ! 0, showing that linear extrapolation would result in larger values. This has led to the use of larger separation distances between the opposing nozzles [12,13] in order to minimize this effect. The original objective of the present investigation was to determine the laminar flame speeds of large hydrocarbon/air mixtures,
0010-2180/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2009.04.004
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2. Experimental specifications The experiments were conducted using a dual-chamber, constant-pressure apparatus described in Ref. [22]. Briefly, the inner chamber (82:55 mm i.d., 127 mm length) is filled with a mixture of fuel and oxidizer based on partial pressures, while the outer chamber (273:05 mm i.d., 304:8 mm length) is filled with an inert mixture having the same density as that of the test gas in the inner chamber. The two chambers are initially partitioned from one another by two sleeves with holes that are off-set from each other. The mixture is then spark-ignited by discharging a capacitor with variable voltage through an ignition coil, resulting in a spark across vertically oriented electrodes in the inner chamber. The spark discharge occurs in approximately 0:5 ls. Simultaneous to the spark ignition, the holes on the two sleeves are aligned. The resulting outwardly propagating spherical flame is subsequently quenched upon reaching the partitioning wall and contacting the inert gas through the aligned holes. Since the volume of the inner chamber is 25 times smaller than that of the outer chamber, the pressure rise during flame propagation is small, rendering its propagation to be at essentially constant pressure. The history of the radius of the flame, r f ðtÞ, is imaged using Schlieren photography and recorded with a high-speed digital motion camera at up to 25,000 frames per second. The spatial resolution of the camera corresponds to roughly 0:1 mm. Since the burned gas is motionless for the outwardly propagating flame, the measured flame radius corresponds to that of the downstream boundary of the flame. Therefore, the instantaneous stretched flame speed, sb , and stretch rate, j, are respectively given by sb ¼ dr f =dt and j ¼ ð2=rf Þdr f =dt where the subscript b refers to the downstream, burned, state. Assuming that the stretched flame speed data can be rationally extrapolated to zero stretch, the downstream laminar flame speed, sob , can be determined, from which the upstream laminar flame speed, sou , can be correspondingly determined through, sou ¼ ðqob =qu Þsob , where the subscript u refers to the upstream condition, and q is the density.
Extensive experiments were conducted with lean and rich butane/air mixtures for their strong and opposite non-equidiffusive trends, recognizing that their controlling Lewis numbers, Le, are respectively greater and smaller than unity. The n-butane used in the study was of 99.98% purity. Initial temperature of the mixture was 25 °C. The experimental results and analysis of the factors affecting flame propagation are presented next. 3. Considerations of nonlinearity 3.1. Lean n-butane/air flames Figs. 1–3 show the evolution of the flame dynamics, subsequent to ignition, for a typical lean n-butane/air flame of equivalence ratio / ¼ 0:80. Fig. 1 shows that, apart from an initial period during which the flame motion is expected to be affected by the ignition energy, the trajectory appears to be fairly linear, suggesting that perhaps an approximately constant flame speed can be defined. This, however, is a rather inaccurate perception. As demonstrated in Fig. 2, the flame speed, obtained by differentiating a locally fitted second-order polynomial for each data point in Fig. 1, varies nonmonotonically with the instantaneous flame radius. Specifically, as the flame expands, the flame speed first decreases, then increases, and decreases again. Fig. 3 shows the relation between the response of the flame speeds and the stretch rate. This is the plot frequently used in discussing the response of stretched flames. It is also the one based on which linear extrapolation for the laminar flame speed would be conducted. Based on the qualitative trend shown in Figs. 1–3 as well as our subsequent quantitative analysis of the data, the flame propagation consists of three distinctive periods, namely an initial period dominated and then influenced by the ignition energy input, followed by a quasi-steady period of stretched flame propagation suitable for the extraction of the laminar flame speed, and a final period influenced by chamber confinement effects. These three periods, indicated in Fig. 3, will be discussed next. 3.1.1. Ignition-affected, early-stage flame propagation Upon spark ignition, the flame is initially driven by thermal conduction from the ignition kernel to the reaction front, resulting in an elevated flame speed. The flame speed subsequently decreases rapidly as the influence of the ignition kernel is dissipated [15,16]. Following this ignition-dominated phase, the flame speed increases rapidly and this causes the stretch to increase as well, in spite of the simultaneous increase in the flame radius.
30
time (ms)
such as those of n-butane and n-heptane, by using the spark-ignited outwardly propagating spherical flame. The lean mixtures of these hydrocarbons have large values of the Lewis number (Le) and as such are subjected to strong non-equidiffusive and thereby stretch effects. We have subsequently come to realize [14] that the extrapolation of the laminar flame speed from data obtained from the expanding spherical flame could be nonlinear in nature, and in addition could be affected by the ignition transient [14–17] during the early stage of flame propagation. Earlier [10] and further studies [18–21] have also suggested the importance of chamber asymmetry and confinement effects towards the later stage of flame propagation. This paper therefore serves as an update to our conference publication [14], reporting on our experimental investigations on the nonlinear propagation of the expanding spherical flame, with systematic quantifications of the extent of the earlyand late-stage system interference effects with the successful implementation of nonlinear extrapolation to yield meaningful values of the laminar flame speed. In the following we shall first briefly describe our experimental setup and methodology. We shall then present the experimental constraints emphasizing the effects of the ignition energy and chamber confinement. We will then present the nonlinear extrapolation leading to the accurate determination of the laminar flame speed and the Markstein length. The measured laminar flame speeds for the n-butane/air mixtures will be compared with those obtained by the flat flame burner and the counterflow. Additional experimental data, particularly those on hydrogen flames, can be found in [14–16].
n−butane/air φ=0.80 1 atm
20
10
0
0
1
2 rf (cm)
3
4
Fig. 1. Flame radius as a function of time for a typical lean n-butane/air outwardly propagating flame experiment.
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160
150
sb (cm/s)
sb (cm/s)
140 120
80
0
0
1
2 rf (cm)
3
Fig. 2. Flame speed as a function of flame radius for a typical lean n-butane/air outwardly propagating flame experiment.
quasi−steady
sb (cm/s)
140
100
80
chamber affected
ignition affected
n−butane/air φ=0.80 1 atm 100
200 κ (1/s)
300
100
200 κ (1/s)
300
400
Fig. 4. Two experiments with differing ignition energies. The region where the two experiments disagree is affected by the ignition energy.
3.1.2. Quasi-steady propagation During the second period of the flame evolution, the flame speed increases while the stretch rate decreases, indicating that the continuously increasing flame size finally dominates the stretch rate experienced by the flame, exhibiting the conventionally expected behavior that the stretch rate of the flame deceases as it expands. Data from this phase of propagation vary relatively slowly and hence can be considered to be quasi-steady. The influences of the ignition energy have been dissipated as discussed. Furthermore, the flow field of the unburned gas is unaffected by the confining walls of the chamber, to be substantiated next.
160
120
n−butane/air φ=0.80 1 atm
50
n−butane/air φ=0.80 1 atm
100
100
400
Fig. 3. Flame speed as a function of stretch for a typical lean n-butane/air outwardly propagating flame experiment. The experiment is influenced by ignition and wall confinement and care must be taken to determine data that are uninfluenced by these effects.
To assess the influence of the ignition kernel on the subsequent flame evolution, experiments were conducted with varying spark energy. Fig. 4 shows two typical experiments ignited with different ignition energies. It is seen that, after an initial, transient period during which results of the two experiments differ substantially due to different ignition energies, with the lower curve having a smaller ignition energy, the two flame trajectories eventually align as the influence of the ignition kernel is dissipated. Prior to the merging of the two experimental datasets shown in Fig. 4, the flame speed rapidly increases and a turning point results for the case with the lower ignition energy. It was also found that the flame may be driven beyond the turning point for excessively large ignition energy. Therefore, the region of disagreement between the two experimental results depends on the ignition kernel and as such cannot be readily used in the determination of the laminar flame speed. Consequently, we will discard the data up to and slightly after the turning point in order to eliminate any influence from the initial conditions and remove any transient effects, as described in Ref. [17]. For typical n-butane/air mixtures used in this study, data below 1.0 cm in radius was discarded for extrapolation purposes.
3.1.3. Chamber-affected, late-stage propagation During the third period of the flame evolution, the flame speed decreases which, together with the increasing flame radius, cause the stretch rate to decrease. The flame is now fairly large and its propagation can be affected by the confining nature of the inner chamber walls, even though the inner chamber gas is connected to the outer chamber inert gas [18–21]. Since at this stage the flame is still sufficiently far away from the wall in terms of its thermal thickness, influences that are fluid mechanical in nature due to the restriction of the flow of the un-reacted gas, instead of through conductive heat loss, are expected to be the mechanism in effecting the observed reduction. In particular, it was suggested [18– 21] that because of the asymmetrical dimension of the cylindrical chamber, with the radius of the inner chamber being smaller than its length for the present design, the flame propagation velocity will increase in the axial direction and decrease in the radial direction. This could explain the slowing down of the measured flame speed because the flame images recorded were the radial ones, taken along the axial direction. To assess this influence, a second inner chamber with a larger inner diameter of 114:3 mm and the same length of 127 mm was constructed. Flames in this larger chamber should experience reduced/delayed confinement effects. Fig. 5 shows two experimental results obtained from these two geometrically distinct chambers, with the open and closed symbols, respectively, designating the smaller and larger inner diameter chambers. We see that the flame responses initially differ due to the slight difference in the ignition energy, but they quickly merge. Subsequent to the turning point, the flames quasi-steadily propagate free of the ignition energy influence and their responses agree well. However, upon reaching a large radius and correspondingly small stretch rate, the two flames evolve differently, with the flame in the smaller inner diameter chamber slowing down as observed previously, while the
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time (ms)
sb (cm/s)
100
50
n−butane/air φ=1.80 1 atm
60
150
n−butane/air φ=0.80 1 atm
40
20
0 100
200 κ (1/s)
300
Fig. 5. Two experiments with different confinement. Disagreement at low stretch rates and high flame speed occurs as the flow of the unburned gas in the chamber interacts with the confining walls. Disagreement at high stretch rates and low flame speed occurs due to the ignition energy.
0
1
2 rf (cm)
3
4
Fig. 6. Flame radius as a function of time for a typical rich n-butane/air outwardly propagating flame experiment.
80
3.2. Rich n-butane/air flames Figs. 6 and 7 show the evolution of the flame dynamics, subsequent to ignition, for a typical rich butane flame of equivalence ratio / ¼ 1:8. Since the Lewis number is now less than unity, the flame dynamics are qualitatively changed. In particular, Fig. 7 shows that the flame speed now varies monotonically with stretch rate. The key factor that renders the difference in the response between the lean and rich flames is that while the flame speed increases with radius during the second period for the former, it decreases for the latter. Fig. 7 is marked to show regions which correspond to the three phases of evolution: ignition, quasi-steady propagation, and confinement affected. These regions have been experimentally studied similar to the lean case and as such will not be discussed in detail. 3.3. Range of useful experimental data Based on the above discussions and as seen in Figs. 3 and 7, there is a limited portion of the experimental data which is qua-
60
sb (cm/s)
flame in the larger diameter inner chamber continues with the apparently quasi-steady propagation. The point at which the two experiments begin to diverge identifies the stage at which the chamber geometry starts to influence the flow field of the unburned gas such that data beyond this stage for the smaller chamber should not be used for extrapolation purposes. Additional experiments were performed which varied the locations of the holes that connected the inner and outer chambers and similar results were obtained as those shown in Fig. 5, with experiments agreeing until reaching a large radius at which point the hole locations, and thus the geometry of confinement, begin influencing the flame propagation and motion of the unburned gases. Based on the above results, we have determined that, for typical n-butane/air experiments using our experimental setup, the radius at which the chamber walls and hole locations began influencing the propagation was approximately 1:7 cm. Therefore, all data with flame radius larger than 1.7 cm were not used for the extrapolation. The value of 1:7 cm corresponds to a flame radius that is roughly 40% of the radius of the inner chamber. This agrees well with the calculation of Burke et al. [19,21] which showed that the flame speed will be affected by less than 1% for flames less than 40% of the way to the chamber wall for cylindrical chambers whose length/diameter = 1.5.
quasi− steady
40
chamber affected
20 0
0
50
ignition affected
n−butane/air φ=1.80 1 atm 100 κ (1/s)
150
200
Fig. 7. Flame speed as a function of stretch for a typical rich n-butane/air outwardly propagating flame experiment. The experiment is influenced by ignition and wall confinement and care must be taken to determine data that are uninfluenced by these effects.
si-steady, independent of the ignition energy, and unaffected by the chamber confinement. Therefore, only data within this range should be used for extrapolation purposes. For our particular experimental setup and choice of fuel studied, a conservative assessment of this range is between radii of 1.0 and 1.7 cm. Consequently, all data reported in the following were obtained within this range. 4. Extrapolation methodology Now that we have identified the useful range of the experimental data that are quasi-steady and unaffected by ignition and confinement, a close inspection of these data, say those of Figs. 3 and 7, readily shows that they are not linear. Consequently, a nonlinear extrapolation procedure needs to be used in place of the linear expression of Eq. (1). The expression of Eq. (1) is derived assuming that the deviation of the flame speed from the planar, adiabatic value is Oð1=ZeÞ, where Ze is the Zeldovich number. For larger deviations, Ronney and Sivashinsky [23] derived an evolution equation for an outwardly propagating flame which is not subject to the small stretch assumption and also accounts for density variation. If we additionally restrict their analysis to flames that are adiabatic and propa-
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gate in a quasi-steady manner, the evolution equation for the propagation speed is given by,
2 2 sb sb Lb j ln o ¼ 2 o sob sb sb
ð2Þ
where we have defined a Markstein length as,
for experimental data taken around a normalized stretch rate, r , the error associated with linear extrapolation can be estimated by expanding ~sðrÞ around this location, r , as
~sðrÞ ~s þ
d~s ð r r Þ dr r
where ~s ¼ ~sðr Þ. Thus, the extrapolated value to zero stretch is
1 Ea o Lb ¼ ‘ I Le; T u =T ob 2 Ro T ob T
ð3Þ
~sð0Þ ~sðr Þ
and where,
IðLe; Þ ¼
Z
1
1 x
"
x 1
Le1
ð7Þ
# 1 dx
ð4Þ
‘oT is the unstretched flame thickness, Ea the activation energy of the one-step global chemical reaction, Ro the universal gas constant, T u the unburned gas temperature, and T ob the adiabatic flame temperature. Similar expressions have been derived for other situations. For example, the upstream flame speed for the counterflow flame is described in Ref. [6],
2 2 su su Lu j ln o ¼ 2 o sou su su
ð5Þ
while an expression similar to Eq. (5) has been derived for the planar flame subjected to volumetric heat loss, with the RHS replaced by a loss parameter [6,24]. The nonlinear relationship for loss-affected flames can therefore be generalized to [25],
~s2 ln ~s2 ¼ 2r
ð6Þ
where r is a generalized loss parameter. Eq. (6) reduces to the familiar Eq. (1) in the limit of small r. It is also noted that while the above expressions were derived on the basis of a one-step overall reaction with large activation energy, recent understanding [6] in the quantitative analysis of flame dynamics have shown that the flame responses, allowing for complex chemistry, can indeed be interpreted on the basis of a global one-step overall activation energy which is extracted from those of the adiabatic planar flame. Eq. (6) is plotted in Fig. 8, for both positive and negative values of r. It is clear that, due to the curvature of Eq. (6), significant uncertainty and hence error can occur if experimental data are linearly extrapolated from their stretched value. Furthermore, the extrapolated flame speed at r ¼ 0 is larger than the real value no matter where the linear extrapolation is conducted. Specifically,
d~s r dr r
ð8Þ
Evaluating Eq. (8) by using Eq. (6), the fractional amount of the overestimate through linear extrapolation is,
~sð0Þ 1 ð~s 1Þ þ
r~s 2 ð~s Þ
ð9Þ
2r
Fig. 9 shows the percent error in the flame speed measurement caused by the linear extrapolation as a function of the stretch parameter r as determined by Eq. (9). The error approaches infinity at r ¼ ð2eÞ1 0:184, the value at the turning point of Fig. 8. It is seen that the error associated with linear extrapolation grows very rapidly with positive r , corresponding to lean n-butane flames with Le > 1. For mixtures with Le < 1, r is negative and the error associated with linear extrapolation is seen to be less severe. Similarly, we can determine the fractional overestimate in the Markstein length from linearly extrapolating data at r to zero stretch rate,
~s ðd~s=drÞr 1¼ 1 2 ðd~s=drÞ0 ð~s Þ 2r
ð10Þ
Fig. 10 shows the percent error in the Markstein length measurement associated with linear extrapolation as a function of the stretch parameter, r , as determined by Eq. (10). It can be seen that even for very small values of r , both positive and negative, large errors occur in the determination of the Markstein length. While Eq. (6) is the fundamental relation used for the nonlinear extrapolation, fitting of the experimental data and hence the determination of the flame parameters can be conducted more directly by using the raw data for the instantaneous flame radius as a function of time, r f ðtÞ, instead of performing numerical differentiation to yield sb and j first. Thus, by integrating Eq. (2) we have,
t ¼ A E1 ln n2
1 n2 ln n
þC
ð11Þ
5
% error in flame speed
1.25
s / so
1.00 0.75 0.50 0.25 0.00 −0.25
0
0.25
σ = Lκ / so Fig. 8. Eq. (6) plotted as a function of both positive and negative values of the stretch parameter, r.
4 3 2 1 0
−0.3
−0.2
−0.1
0
0.1
σ* Fig. 9. Percent error in measuring the flame speed by linearly extrapolating experimental data from r to zero stretch rate.
1849
60
80
40
70
sb (cm/s)
% error in Markstein length
A.P. Kelley, C.K. Law / Combustion and Flame 156 (2009) 1844–1851
20
50
0
−0.05
0
0.05
0
0.1
σ* Fig. 10. Percent error in measuring the Markstein length by linearly extrapolating experimental data from r to zero stretch rate.
where
2Lb ; sob
rf ¼
2Lb ; n ln n
E 1 ð xÞ ¼
Z
1 x
ez dz z
ð12Þ
n 2 ½1=e; 1Þ for Lb > 0 and n 2 ½1; 1Þ for Lb < 0. In this manner, Eq. (11) can be used for constrained nonlinear least-square regression to determine the three constants of the fitting, namely A, Lb and C, from which the unstretched flame speed can be determined through sb ¼ 2Lb =A. A comparison of the results obtained through linear extrapolation and nonlinear extrapolation of typical n-butane/air experiments are shown in Figs. 11 and 12 for lean and rich mixtures, respectively. The linearly extrapolated flame speed and Markstein length associated with Fig. 11 are 28:3 cm=s and 0:28 cm, while the nonlinearly extrapolated values are 24:9 cm=s and 0:12 cm, respectively. The linearly extrapolated flame speed and Markstein length associated with Fig. 12 are 6:4 cm=s and 0:17 cm, while the nonlinearly extrapolated values are 5:0 cm=s and 0:60 cm, respectively. Thus, the linearly extrapolated flame speed results in an overestimate of the unstretched flame speed in both cases, as anticipated. The Markstein lengths are overestimated and underestimated by linear extrapolation in the lean and rich cases, respectively. We note in passing that while nonlinear extrapolation is clearly necessary for n-butane/air flames, we have found that linear
n−butane/air φ=0.80 1 atm
100 κ (1/s)
extrapolations are reasonably adequate for mixtures that are not as strongly affected by stretch, such as hydrogen/air and methane/air mixtures, because of the smaller Markstein lengths and the correspondingly smaller values of r involved. For example, similar hydrogen/air flame speed measurements had an error of less than 1% in flame speeds associated with linear extrapolation for all equivalence ratios. 5. Experimental results Figs. 13 and 14 show the measured flame speed as a function of stretch rate. These results show that mixtures that are farther from stoichiometric exhibit stronger nonlinear trends in the relationship between stretch and flame speed. These mixtures are therefore strongly influenced by stretch and necessitate the nonlinear extrapolation. For data that are near / ¼ 1:3, a linear relation between stretch and flame speed may be expected due to the small deviation of the flame speed from the unstretched flame speed. This is indeed the case. For these flames, both the linear and nonlinear extrapolations yield similar unstretched flame speeds. We note in passing that the above results show that mixtures with equivalence ratios less than 1:3 correspond to r > 0 while the richer mixtures correspond to r < 0, hence suggesting that a diffusionally neutral n-butane/air mixture correspond to an equivalence ratio of 1.3.
120
50 0
κ (1/s)
200
300
Fig. 11. Comparison of linear (dashed line) and nonlinear (solid line) extrapolations of lean n-butane/air experimental data.
0.90 0.85 0.80
150 100
100
1.00
200
140
0
φ=1.10
250
160
150
Fig. 12. Comparison of linear (dashed line) and nonlinear (solid line) extrapolations of rich n-butane/air experimental data.
sb (cm/s)
180
100
50
300
200
sb (cm/s)
n−butane/air φ=1.80 1 atm
40
−20 −0.1
A¼
60
φ=0.75 n−butane/air 1 atm 0
200
400 κ (1/s)
600
Fig. 13. Experimentally measured outwardly propagating stretched flame speed as a function of stretch rate for equivalence ratios 0.75–1.10. Nonlinear extrapolation is shown as solid black line.
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300
1.30
250
0
1.40
200
1.50
150
Lb (cm)
sb (cm/s)
0.2
φ=1.20
1.60 100 1.70 φ=1.80
50 0
0
200
n−butane/air 1 atm κ (1/s)
400
600
Fig. 14. Experimentally measured outwardly propagating stretched flame speed as a function of stretch rate for equivalence ratios 1.20–1.80. Nonlinear extrapolation is shown as solid black line.
Figs. 13 and 14 further show the nonlinear extrapolations plotted as solid lines. These extrapolations are fitted to the experimental data using Eq. (11) and differentiated for presentation purposes in the standard stretch versus flame speed plot. It is noted that while the datasets are necessarily limited as discussed above, the useful data provides a sufficient number of experimental measurements, usually on the order of 25 measurements, to obtain a statistically significant regression through the dataset. The regression parameters, flame speed and Markstein length, have typical 95% confidence intervals of less than 1—2% in flame speed and less than 0:04 cm in Markstein length for the cases considered. The nonlinearly extrapolated unstretched flame speeds are shown in Fig. 15. For comparison, the experimental measurements of Bosschaart et al. [26], obtained by using a flat flame burner in which the flame is not subjected to stretch, are also plotted. Additionally, the stretched flame speed data of Davis [27], obtained in the counterflow, have also been nonlinearly extrapolated using Eq. (5) and the resulting unstretched flame speeds are plotted in Fig. 15. As seen in Fig. 15, the outwardly propagating flame speed measurements, when nonlinearly extrapolated to zero stretch rate, agree quite well with the measurements of Bosschaart et al. The nonlinearly extrapolated counterflow flame speeds agree well for lean and rich mixtures, with a slightly higher peak value. As previ-
sou (cm/s)
40 30 20 10
n−butane/air 1 atm 0 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Equivalence Ratio, φ Fig. 15. Unstretched laminar flame speed for n-butane/air mixtures at 1 atm of pressure: nonlinear present work – circles, Davis [27] nonlinearly extrapolated – squares, Bosschaart [26] – triangles, USC Mech II [28] – solid line.
−0.2 −0.4 −0.6
n−butane/air 1 atm
−0.8 0.6 0.8
1 1.2 1.4 1.6 1.8 Equivalence Ratio, φ
2
Fig. 16. Experimentally measured downstream Markstein lengths obtained from nonlinear extrapolation.
ously stated, near-stoichiometric mixtures show very little sensitivity to linear-versus-nonlinear extrapolation. The discrepancy between the datasets is therefore not due to the extrapolation procedure. Nevertheless, the three datasets agree quite well and were obtained using three different measurement techniques. Also plotted in Fig. 15 are the predictions of USC Mech II [28]. It is seen that the mechanism tends to over-predict the flame speed on the lean side, but agrees well with the measurements on the rich side. Additionally, the peak flame speed measured tends to be slightly richer than the peak predicted by the mechanism. Fig. 16 plots the nonlinearly extrapolated downstream Markstein lengths, Lb , as a function of equivalence ratio. It is seen that the Markstein length changes sign around / ¼ 1:3, which corresponds to the change in r as shown in Figs. 13 and 14. 6. Concluding remarks While our original motivation for the present investigation was to determine the laminar flame speeds of higher hydrocarbon fuels, we were quickly confronted with the strong nonlinearity in the flame response as a consequence of the small diffusivity of these heavier fuels, casting considerable uncertainty on the feasibility and accuracy of the conventional method of linearly extrapolating the stretched flame speeds to zero stretch rate to yield the laminar flame speed for these fuels. In order to arrive at a rational methodology to extract the laminar flame speed from the experimental data, we were led to appreciate the importance of the nonlinear nature of the stretch effect on the present expanding spherical flame. Consequently, by using the well-established nonlinear expression governing stretch-affected flame propagation, the laminar flame speeds of n-butane/air mixtures were successfully determined and found to agree well with literature data determined using unstretched flames. The study also identified the roles of the initial ignition transient and the finite chamber confinement effects on the flame-front dynamics, and the feasibility and accuracy of the extrapolation can be enhanced by using larger chambers and smaller ignition energies. Acknowledgments This work was supported by the Air Force Office of Scientific Research under the technical monitoring of Dr. Julian M. Tishkoff. It is a pleasure to acknowledge helpful discussions with Professor John K. Bechtold of the New Jersey Institute of Technology.
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References [1] G.E. Andrews, D. Bradley, Determination of burning velocities: a critical review, Combust. Flame 18 (1972) 133–153. [2] C.K. Law, C.J. Sung, H. Wang, T.F. Lu, Development of comprehensive detailed and reduced reaction mechanisms for combustion modeling, in: Fortieth Aerospace Sciences Meeting and Exhibit AIAA 2002-0331. [3] C.K. Wu, C.K. Law, On the determination of laminar flame speeds from stretched flames, in: Twentieth Symposium (International) on Combustion, 1984, pp. 1941–1949. [4] C.K. Law, C.J. Sung, Structure, aerodynamics, and geometry of premixed flamelets, Prog. Energy Combust. Sci. 26 (2000) 459–505. [5] M. Matalon, On flame stretch, Combust. Sci. Technol. 31 (1983) 169–181. [6] C.K. Law, Combustion Physics, Cambridge University Press, 2006. [7] M. Matalon, B.J. Matkowsky, Flames as gasdynamic discontinuities, Journal of Fluid Mechanics 124 (1982) 239–259. [8] D.L. Zhu, F.N. Egolfopoulos, C.K. Law, Experimental and numerical determination of laminar flame speed of methane/(Ar;N2 ; CO2 )-air mixtures as function of stoichiometry, pressure, and flame temperature, in: Twenty-second Symposium (International) on Combustion, vol. 22, 1988, pp. 1537–1545. [9] D.R. Dowdy, D.B. Smith, S.C. Taylor, A. Williams, The use of expanding spherical flames to determine burning velocities and stretch effects in hydrogen/air mixtures, in: Twenty-third Symposium (International) on Combustion, 1990, pp. 325–332. [10] G. Rozenchan, D.L. Zhu, C.K. Law, S.D. Tse, Outward propagation, burning velocities, and chemical effects of methane flames up to 60 atm, Proc. Combust. Inst. 29 (2002) 1461–1469. [11] J.H. Tien, M. Matalon, On the burning velocity of stretched flames, Combust. Flame 84 (1991) 238–248. [12] B.H. Chao, F.N. Egolfopoulos, C.K. Law, Structure and propagation of premixed flame in nozzle-generated counterflow, Combust. Flame 109 (1997) 620–638. [13] C.M. Vagelopoulos, F.N. Egolfopoulos, C.K. Law, Further considerations on the determination of laminar flame speeds with the counterflow twin-flame technique, in: Twenty-fifth Symposium (International) on Combustion, 1994, pp. 1341–1347. [14] A.P. Kelley, C.K. Law, Nonlinear effects in the experimental determination of laminar flame properties from stretched flames, in: Meeting of the Eastern States Section of the Combustion Institute.
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[15] A.P. Kelley, G. Jomaas, C.K. Law, On the critical radius for sustained propagation of spark-ignited spherical flames, in: Forty-sixth AIAA Aerospace Sciences Meeting and Exhibit AIAA 2008-1054. [16] A.P. Kelley, G. Jomaas, C.K. Law, Critical radius for sustained propagation of spark-ignited spherical flames, Combust. Flame 156 (2009) 1006–1013. [17] Z. Chen, M.P. Burke, Y. Ju, Effects of Lewis number and ignition energy on the determination of laminar flame speed using propagating spherical flames, Proc. Combust. Inst. 32 (2009) 1253–1260. [18] M.P. Burke, X. Qin, Y. Ju, F.L. Dryer, Measurements of hydrogen syngas flame speeds at elevated pressures, in: Fifth U.S. Combustion Meeting. [19] M.P. Burke, Y. Ju, F.L. Dryer, Effect of cylindrical confinement on the evolution of outwardly propagating flames, in: Meeting of the Eastern States Section of the Combustion Institute. [20] M.P. Burke, Y. Ju, F.L. Dryer, Effect of flow field perturbations on laminar flame speed determination using spherical flames, in: Forty-sixth AIAA Aerospace Sciences Meeting and Exhibit. [21] M.P. Burke, Z. Chen, Y. Ju, F.L. Dryer, Effect of cylindrical confinement on the determination of laminar flame speeds using outwardly propagating flames, Combust. Flame 156 (2009) 771–779. [22] S.D. Tse, D. Zhu, C.K. Law, Optically accessible high-pressure combustion apparatus, Rev. Sci. Instrum. 75 (2004) 233–239. [23] P.D. Ronney, G.I. Sivashinsky, A theoretical study of propagation and extinction of nonsteady spherical flame fronts, SIAM J. Appl. Math. 49 (4) (1989) 1029– 1046. [24] Y.B. Zel’dovich, G.I. Barenblatt, V.B. Librovich, G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, 1985. [25] C.K. Law, Combustion at a crossroads: status and prospects, Proc. Combust. Inst. 31 (2007) 1–29. [26] K.J. Bosschaart, L.P.H. de Goey, The laminar burning velocity of flames propagating in mixtures of hydrocarbons and air measured with the heat flux method, Combust. Flame 136 (2004) 261–269. [27] S.G. Davis, An experimental and kinetic modeling study of the pyrolysis and oxidation of selected C3 —C8 hydrocarbons, Ph.D. thesis, Princeton University, Department of Mechanical and Aerospace Engineering, 1998. [28] H. Wang, X. You, A.V. Joshi, S.G. Davis, A. Laskin, F. Egolfopoulos, C.K. Law, USC Mech version II. High-temperature combustion reaction model of H2 =CO=C 1 —C 4 compounds. Available from:, May 2007.
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Nonlinear effects in the extraction of laminar flame speeds from expanding spherical flames A.P. Kelley, C.K. Law * Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
a r t i c l e
i n f o
Article history: Received 14 January 2009 Received in revised form 9 April 2009 Accepted 10 April 2009 Available online 18 May 2009 Keywords: Flame speed extrapolation Butane Propagating spherical flame Laminar flame speed Markstein length
a b s t r a c t Various factors affecting the determination of laminar flames speeds from outwardly propagating spherical flames in a constant-pressure combustion chamber were considered, with emphasis on the nonlinear variation of the stretched flame speed to the flame stretch rate, and the associated need to nonlinearly extrapolate the stretched flame speed to yield an accurate determination of the laminar flame speed and Markstein length. Experiments were conducted for lean and rich n-butane/air flames at 1 atm initial pressure, demonstrating the complex and nonlinear nature of the dynamics of flame evolution, and the strong influences of the ignition transient and chamber confinement during the initial and final periods of the flame propagation, respectively. These experimental data were analyzed using the nonlinear relation between the stretched flame speed and stretch rate, yielding laminar flame speeds that agree well with data determined from alternate flame configurations. It is further suggested that the fidelity in the extraction of the laminar flame speed from expanding spherical flames can be facilitated by using small ignition energy and a large combustion chamber. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Early efforts at determining laminar flame speeds, even conducted with exceptional care, exhibited large scatter for data obtained using different experimental methods [1,2]. The fundamental reason that these measurements differed is due to the unquantified effects of stretch [3,4], which is induced by local flow straining, flame curvature, and flame unsteadiness [5,6]. As such, these effects must be subtracted from the experimental data in order to unambiguously determine the laminar flame speed [3]. Based on asymptotic analysis [7], Wu and Law [3] subtracted the stretch effect by linearly extrapolating the experimental stretched flame speed, determined as a function of the stretch rate, to zero stretch rate according to,
~s ¼ 1 r
ð1Þ
where ~s ¼ s=so , r ¼ Lj=so is a non-dimensional stretch parameter, s the flame speed, j the stretch rate, L a constant that measures the mixture’s sensitivity to stretch and is commonly referred to as the Markstein length, and the superscript o refers to the unstretched state. This linear extrapolation has been extensively adopted for flame speed determinations using the counterflow/stagnation flame [8] and the outwardly propagating flame [9,10]; the references cited are just representative of the many works that have since appeared.
* Corresponding author. E-mail address: [email protected] (C.K. Law).
In fact, works of this nature are appearing at an increasing rate because of the correspondingly increasing interest in accurate values of the laminar flame speeds for studies of fuel chemistry. While the linear extrapolation is convenient to apply, it has also been recognized that care needs to be exercised in order to ensure that the extrapolation not only is accurate but is also meaningful. For example, the range of data based on which the linear extrapolation is conducted should be sufficiently extensive while the magnitude of their stretch rates, when properly scaled through, say, the Karlovitz number, should also be sufficiently small so that the deviation of the measured flame speed from the unstretched value is correspondingly small. There are, however, situations in which the deviation can be more substantial, due to high stretch rates and/or strong mixture non-equidiffusion such that higher-order effects could be important. For these situations a linear approximation not only is inaccurate, but the act of performing a linear extrapolation from a dataset exhibiting curvature could also impart substantial uncertainty in the extrapolated value. Furthermore, additional processes and factors could also be present in the measurement technique that could lead to systematic deviations from linearity. For example, for the counterflow technique, Tien and Matalon [11] identified a regime of nonlinearity as j ! 0, showing that linear extrapolation would result in larger values. This has led to the use of larger separation distances between the opposing nozzles [12,13] in order to minimize this effect. The original objective of the present investigation was to determine the laminar flame speeds of large hydrocarbon/air mixtures,
0010-2180/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2009.04.004
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2. Experimental specifications The experiments were conducted using a dual-chamber, constant-pressure apparatus described in Ref. [22]. Briefly, the inner chamber (82:55 mm i.d., 127 mm length) is filled with a mixture of fuel and oxidizer based on partial pressures, while the outer chamber (273:05 mm i.d., 304:8 mm length) is filled with an inert mixture having the same density as that of the test gas in the inner chamber. The two chambers are initially partitioned from one another by two sleeves with holes that are off-set from each other. The mixture is then spark-ignited by discharging a capacitor with variable voltage through an ignition coil, resulting in a spark across vertically oriented electrodes in the inner chamber. The spark discharge occurs in approximately 0:5 ls. Simultaneous to the spark ignition, the holes on the two sleeves are aligned. The resulting outwardly propagating spherical flame is subsequently quenched upon reaching the partitioning wall and contacting the inert gas through the aligned holes. Since the volume of the inner chamber is 25 times smaller than that of the outer chamber, the pressure rise during flame propagation is small, rendering its propagation to be at essentially constant pressure. The history of the radius of the flame, r f ðtÞ, is imaged using Schlieren photography and recorded with a high-speed digital motion camera at up to 25,000 frames per second. The spatial resolution of the camera corresponds to roughly 0:1 mm. Since the burned gas is motionless for the outwardly propagating flame, the measured flame radius corresponds to that of the downstream boundary of the flame. Therefore, the instantaneous stretched flame speed, sb , and stretch rate, j, are respectively given by sb ¼ dr f =dt and j ¼ ð2=rf Þdr f =dt where the subscript b refers to the downstream, burned, state. Assuming that the stretched flame speed data can be rationally extrapolated to zero stretch, the downstream laminar flame speed, sob , can be determined, from which the upstream laminar flame speed, sou , can be correspondingly determined through, sou ¼ ðqob =qu Þsob , where the subscript u refers to the upstream condition, and q is the density.
Extensive experiments were conducted with lean and rich butane/air mixtures for their strong and opposite non-equidiffusive trends, recognizing that their controlling Lewis numbers, Le, are respectively greater and smaller than unity. The n-butane used in the study was of 99.98% purity. Initial temperature of the mixture was 25 °C. The experimental results and analysis of the factors affecting flame propagation are presented next. 3. Considerations of nonlinearity 3.1. Lean n-butane/air flames Figs. 1–3 show the evolution of the flame dynamics, subsequent to ignition, for a typical lean n-butane/air flame of equivalence ratio / ¼ 0:80. Fig. 1 shows that, apart from an initial period during which the flame motion is expected to be affected by the ignition energy, the trajectory appears to be fairly linear, suggesting that perhaps an approximately constant flame speed can be defined. This, however, is a rather inaccurate perception. As demonstrated in Fig. 2, the flame speed, obtained by differentiating a locally fitted second-order polynomial for each data point in Fig. 1, varies nonmonotonically with the instantaneous flame radius. Specifically, as the flame expands, the flame speed first decreases, then increases, and decreases again. Fig. 3 shows the relation between the response of the flame speeds and the stretch rate. This is the plot frequently used in discussing the response of stretched flames. It is also the one based on which linear extrapolation for the laminar flame speed would be conducted. Based on the qualitative trend shown in Figs. 1–3 as well as our subsequent quantitative analysis of the data, the flame propagation consists of three distinctive periods, namely an initial period dominated and then influenced by the ignition energy input, followed by a quasi-steady period of stretched flame propagation suitable for the extraction of the laminar flame speed, and a final period influenced by chamber confinement effects. These three periods, indicated in Fig. 3, will be discussed next. 3.1.1. Ignition-affected, early-stage flame propagation Upon spark ignition, the flame is initially driven by thermal conduction from the ignition kernel to the reaction front, resulting in an elevated flame speed. The flame speed subsequently decreases rapidly as the influence of the ignition kernel is dissipated [15,16]. Following this ignition-dominated phase, the flame speed increases rapidly and this causes the stretch to increase as well, in spite of the simultaneous increase in the flame radius.
30
time (ms)
such as those of n-butane and n-heptane, by using the spark-ignited outwardly propagating spherical flame. The lean mixtures of these hydrocarbons have large values of the Lewis number (Le) and as such are subjected to strong non-equidiffusive and thereby stretch effects. We have subsequently come to realize [14] that the extrapolation of the laminar flame speed from data obtained from the expanding spherical flame could be nonlinear in nature, and in addition could be affected by the ignition transient [14–17] during the early stage of flame propagation. Earlier [10] and further studies [18–21] have also suggested the importance of chamber asymmetry and confinement effects towards the later stage of flame propagation. This paper therefore serves as an update to our conference publication [14], reporting on our experimental investigations on the nonlinear propagation of the expanding spherical flame, with systematic quantifications of the extent of the earlyand late-stage system interference effects with the successful implementation of nonlinear extrapolation to yield meaningful values of the laminar flame speed. In the following we shall first briefly describe our experimental setup and methodology. We shall then present the experimental constraints emphasizing the effects of the ignition energy and chamber confinement. We will then present the nonlinear extrapolation leading to the accurate determination of the laminar flame speed and the Markstein length. The measured laminar flame speeds for the n-butane/air mixtures will be compared with those obtained by the flat flame burner and the counterflow. Additional experimental data, particularly those on hydrogen flames, can be found in [14–16].
n−butane/air φ=0.80 1 atm
20
10
0
0
1
2 rf (cm)
3
4
Fig. 1. Flame radius as a function of time for a typical lean n-butane/air outwardly propagating flame experiment.
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160
150
sb (cm/s)
sb (cm/s)
140 120
80
0
0
1
2 rf (cm)
3
Fig. 2. Flame speed as a function of flame radius for a typical lean n-butane/air outwardly propagating flame experiment.
quasi−steady
sb (cm/s)
140
100
80
chamber affected
ignition affected
n−butane/air φ=0.80 1 atm 100
200 κ (1/s)
300
100
200 κ (1/s)
300
400
Fig. 4. Two experiments with differing ignition energies. The region where the two experiments disagree is affected by the ignition energy.
3.1.2. Quasi-steady propagation During the second period of the flame evolution, the flame speed increases while the stretch rate decreases, indicating that the continuously increasing flame size finally dominates the stretch rate experienced by the flame, exhibiting the conventionally expected behavior that the stretch rate of the flame deceases as it expands. Data from this phase of propagation vary relatively slowly and hence can be considered to be quasi-steady. The influences of the ignition energy have been dissipated as discussed. Furthermore, the flow field of the unburned gas is unaffected by the confining walls of the chamber, to be substantiated next.
160
120
n−butane/air φ=0.80 1 atm
50
n−butane/air φ=0.80 1 atm
100
100
400
Fig. 3. Flame speed as a function of stretch for a typical lean n-butane/air outwardly propagating flame experiment. The experiment is influenced by ignition and wall confinement and care must be taken to determine data that are uninfluenced by these effects.
To assess the influence of the ignition kernel on the subsequent flame evolution, experiments were conducted with varying spark energy. Fig. 4 shows two typical experiments ignited with different ignition energies. It is seen that, after an initial, transient period during which results of the two experiments differ substantially due to different ignition energies, with the lower curve having a smaller ignition energy, the two flame trajectories eventually align as the influence of the ignition kernel is dissipated. Prior to the merging of the two experimental datasets shown in Fig. 4, the flame speed rapidly increases and a turning point results for the case with the lower ignition energy. It was also found that the flame may be driven beyond the turning point for excessively large ignition energy. Therefore, the region of disagreement between the two experimental results depends on the ignition kernel and as such cannot be readily used in the determination of the laminar flame speed. Consequently, we will discard the data up to and slightly after the turning point in order to eliminate any influence from the initial conditions and remove any transient effects, as described in Ref. [17]. For typical n-butane/air mixtures used in this study, data below 1.0 cm in radius was discarded for extrapolation purposes.
3.1.3. Chamber-affected, late-stage propagation During the third period of the flame evolution, the flame speed decreases which, together with the increasing flame radius, cause the stretch rate to decrease. The flame is now fairly large and its propagation can be affected by the confining nature of the inner chamber walls, even though the inner chamber gas is connected to the outer chamber inert gas [18–21]. Since at this stage the flame is still sufficiently far away from the wall in terms of its thermal thickness, influences that are fluid mechanical in nature due to the restriction of the flow of the un-reacted gas, instead of through conductive heat loss, are expected to be the mechanism in effecting the observed reduction. In particular, it was suggested [18– 21] that because of the asymmetrical dimension of the cylindrical chamber, with the radius of the inner chamber being smaller than its length for the present design, the flame propagation velocity will increase in the axial direction and decrease in the radial direction. This could explain the slowing down of the measured flame speed because the flame images recorded were the radial ones, taken along the axial direction. To assess this influence, a second inner chamber with a larger inner diameter of 114:3 mm and the same length of 127 mm was constructed. Flames in this larger chamber should experience reduced/delayed confinement effects. Fig. 5 shows two experimental results obtained from these two geometrically distinct chambers, with the open and closed symbols, respectively, designating the smaller and larger inner diameter chambers. We see that the flame responses initially differ due to the slight difference in the ignition energy, but they quickly merge. Subsequent to the turning point, the flames quasi-steadily propagate free of the ignition energy influence and their responses agree well. However, upon reaching a large radius and correspondingly small stretch rate, the two flames evolve differently, with the flame in the smaller inner diameter chamber slowing down as observed previously, while the
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time (ms)
sb (cm/s)
100
50
n−butane/air φ=1.80 1 atm
60
150
n−butane/air φ=0.80 1 atm
40
20
0 100
200 κ (1/s)
300
Fig. 5. Two experiments with different confinement. Disagreement at low stretch rates and high flame speed occurs as the flow of the unburned gas in the chamber interacts with the confining walls. Disagreement at high stretch rates and low flame speed occurs due to the ignition energy.
0
1
2 rf (cm)
3
4
Fig. 6. Flame radius as a function of time for a typical rich n-butane/air outwardly propagating flame experiment.
80
3.2. Rich n-butane/air flames Figs. 6 and 7 show the evolution of the flame dynamics, subsequent to ignition, for a typical rich butane flame of equivalence ratio / ¼ 1:8. Since the Lewis number is now less than unity, the flame dynamics are qualitatively changed. In particular, Fig. 7 shows that the flame speed now varies monotonically with stretch rate. The key factor that renders the difference in the response between the lean and rich flames is that while the flame speed increases with radius during the second period for the former, it decreases for the latter. Fig. 7 is marked to show regions which correspond to the three phases of evolution: ignition, quasi-steady propagation, and confinement affected. These regions have been experimentally studied similar to the lean case and as such will not be discussed in detail. 3.3. Range of useful experimental data Based on the above discussions and as seen in Figs. 3 and 7, there is a limited portion of the experimental data which is qua-
60
sb (cm/s)
flame in the larger diameter inner chamber continues with the apparently quasi-steady propagation. The point at which the two experiments begin to diverge identifies the stage at which the chamber geometry starts to influence the flow field of the unburned gas such that data beyond this stage for the smaller chamber should not be used for extrapolation purposes. Additional experiments were performed which varied the locations of the holes that connected the inner and outer chambers and similar results were obtained as those shown in Fig. 5, with experiments agreeing until reaching a large radius at which point the hole locations, and thus the geometry of confinement, begin influencing the flame propagation and motion of the unburned gases. Based on the above results, we have determined that, for typical n-butane/air experiments using our experimental setup, the radius at which the chamber walls and hole locations began influencing the propagation was approximately 1:7 cm. Therefore, all data with flame radius larger than 1.7 cm were not used for the extrapolation. The value of 1:7 cm corresponds to a flame radius that is roughly 40% of the radius of the inner chamber. This agrees well with the calculation of Burke et al. [19,21] which showed that the flame speed will be affected by less than 1% for flames less than 40% of the way to the chamber wall for cylindrical chambers whose length/diameter = 1.5.
quasi− steady
40
chamber affected
20 0
0
50
ignition affected
n−butane/air φ=1.80 1 atm 100 κ (1/s)
150
200
Fig. 7. Flame speed as a function of stretch for a typical rich n-butane/air outwardly propagating flame experiment. The experiment is influenced by ignition and wall confinement and care must be taken to determine data that are uninfluenced by these effects.
si-steady, independent of the ignition energy, and unaffected by the chamber confinement. Therefore, only data within this range should be used for extrapolation purposes. For our particular experimental setup and choice of fuel studied, a conservative assessment of this range is between radii of 1.0 and 1.7 cm. Consequently, all data reported in the following were obtained within this range. 4. Extrapolation methodology Now that we have identified the useful range of the experimental data that are quasi-steady and unaffected by ignition and confinement, a close inspection of these data, say those of Figs. 3 and 7, readily shows that they are not linear. Consequently, a nonlinear extrapolation procedure needs to be used in place of the linear expression of Eq. (1). The expression of Eq. (1) is derived assuming that the deviation of the flame speed from the planar, adiabatic value is Oð1=ZeÞ, where Ze is the Zeldovich number. For larger deviations, Ronney and Sivashinsky [23] derived an evolution equation for an outwardly propagating flame which is not subject to the small stretch assumption and also accounts for density variation. If we additionally restrict their analysis to flames that are adiabatic and propa-
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gate in a quasi-steady manner, the evolution equation for the propagation speed is given by,
2 2 sb sb Lb j ln o ¼ 2 o sob sb sb
ð2Þ
where we have defined a Markstein length as,
for experimental data taken around a normalized stretch rate, r , the error associated with linear extrapolation can be estimated by expanding ~sðrÞ around this location, r , as
~sðrÞ ~s þ
d~s ð r r Þ dr r
where ~s ¼ ~sðr Þ. Thus, the extrapolated value to zero stretch is
1 Ea o Lb ¼ ‘ I Le; T u =T ob 2 Ro T ob T
ð3Þ
~sð0Þ ~sðr Þ
and where,
IðLe; Þ ¼
Z
1
1 x
"
x 1
Le1
ð7Þ
# 1 dx
ð4Þ
‘oT is the unstretched flame thickness, Ea the activation energy of the one-step global chemical reaction, Ro the universal gas constant, T u the unburned gas temperature, and T ob the adiabatic flame temperature. Similar expressions have been derived for other situations. For example, the upstream flame speed for the counterflow flame is described in Ref. [6],
2 2 su su Lu j ln o ¼ 2 o sou su su
ð5Þ
while an expression similar to Eq. (5) has been derived for the planar flame subjected to volumetric heat loss, with the RHS replaced by a loss parameter [6,24]. The nonlinear relationship for loss-affected flames can therefore be generalized to [25],
~s2 ln ~s2 ¼ 2r
ð6Þ
where r is a generalized loss parameter. Eq. (6) reduces to the familiar Eq. (1) in the limit of small r. It is also noted that while the above expressions were derived on the basis of a one-step overall reaction with large activation energy, recent understanding [6] in the quantitative analysis of flame dynamics have shown that the flame responses, allowing for complex chemistry, can indeed be interpreted on the basis of a global one-step overall activation energy which is extracted from those of the adiabatic planar flame. Eq. (6) is plotted in Fig. 8, for both positive and negative values of r. It is clear that, due to the curvature of Eq. (6), significant uncertainty and hence error can occur if experimental data are linearly extrapolated from their stretched value. Furthermore, the extrapolated flame speed at r ¼ 0 is larger than the real value no matter where the linear extrapolation is conducted. Specifically,
d~s r dr r
ð8Þ
Evaluating Eq. (8) by using Eq. (6), the fractional amount of the overestimate through linear extrapolation is,
~sð0Þ 1 ð~s 1Þ þ
r~s 2 ð~s Þ
ð9Þ
2r
Fig. 9 shows the percent error in the flame speed measurement caused by the linear extrapolation as a function of the stretch parameter r as determined by Eq. (9). The error approaches infinity at r ¼ ð2eÞ1 0:184, the value at the turning point of Fig. 8. It is seen that the error associated with linear extrapolation grows very rapidly with positive r , corresponding to lean n-butane flames with Le > 1. For mixtures with Le < 1, r is negative and the error associated with linear extrapolation is seen to be less severe. Similarly, we can determine the fractional overestimate in the Markstein length from linearly extrapolating data at r to zero stretch rate,
~s ðd~s=drÞr 1¼ 1 2 ðd~s=drÞ0 ð~s Þ 2r
ð10Þ
Fig. 10 shows the percent error in the Markstein length measurement associated with linear extrapolation as a function of the stretch parameter, r , as determined by Eq. (10). It can be seen that even for very small values of r , both positive and negative, large errors occur in the determination of the Markstein length. While Eq. (6) is the fundamental relation used for the nonlinear extrapolation, fitting of the experimental data and hence the determination of the flame parameters can be conducted more directly by using the raw data for the instantaneous flame radius as a function of time, r f ðtÞ, instead of performing numerical differentiation to yield sb and j first. Thus, by integrating Eq. (2) we have,
t ¼ A E1 ln n2
1 n2 ln n
þC
ð11Þ
5
% error in flame speed
1.25
s / so
1.00 0.75 0.50 0.25 0.00 −0.25
0
0.25
σ = Lκ / so Fig. 8. Eq. (6) plotted as a function of both positive and negative values of the stretch parameter, r.
4 3 2 1 0
−0.3
−0.2
−0.1
0
0.1
σ* Fig. 9. Percent error in measuring the flame speed by linearly extrapolating experimental data from r to zero stretch rate.
1849
60
80
40
70
sb (cm/s)
% error in Markstein length
A.P. Kelley, C.K. Law / Combustion and Flame 156 (2009) 1844–1851
20
50
0
−0.05
0
0.05
0
0.1
σ* Fig. 10. Percent error in measuring the Markstein length by linearly extrapolating experimental data from r to zero stretch rate.
where
2Lb ; sob
rf ¼
2Lb ; n ln n
E 1 ð xÞ ¼
Z
1 x
ez dz z
ð12Þ
n 2 ½1=e; 1Þ for Lb > 0 and n 2 ½1; 1Þ for Lb < 0. In this manner, Eq. (11) can be used for constrained nonlinear least-square regression to determine the three constants of the fitting, namely A, Lb and C, from which the unstretched flame speed can be determined through sb ¼ 2Lb =A. A comparison of the results obtained through linear extrapolation and nonlinear extrapolation of typical n-butane/air experiments are shown in Figs. 11 and 12 for lean and rich mixtures, respectively. The linearly extrapolated flame speed and Markstein length associated with Fig. 11 are 28:3 cm=s and 0:28 cm, while the nonlinearly extrapolated values are 24:9 cm=s and 0:12 cm, respectively. The linearly extrapolated flame speed and Markstein length associated with Fig. 12 are 6:4 cm=s and 0:17 cm, while the nonlinearly extrapolated values are 5:0 cm=s and 0:60 cm, respectively. Thus, the linearly extrapolated flame speed results in an overestimate of the unstretched flame speed in both cases, as anticipated. The Markstein lengths are overestimated and underestimated by linear extrapolation in the lean and rich cases, respectively. We note in passing that while nonlinear extrapolation is clearly necessary for n-butane/air flames, we have found that linear
n−butane/air φ=0.80 1 atm
100 κ (1/s)
extrapolations are reasonably adequate for mixtures that are not as strongly affected by stretch, such as hydrogen/air and methane/air mixtures, because of the smaller Markstein lengths and the correspondingly smaller values of r involved. For example, similar hydrogen/air flame speed measurements had an error of less than 1% in flame speeds associated with linear extrapolation for all equivalence ratios. 5. Experimental results Figs. 13 and 14 show the measured flame speed as a function of stretch rate. These results show that mixtures that are farther from stoichiometric exhibit stronger nonlinear trends in the relationship between stretch and flame speed. These mixtures are therefore strongly influenced by stretch and necessitate the nonlinear extrapolation. For data that are near / ¼ 1:3, a linear relation between stretch and flame speed may be expected due to the small deviation of the flame speed from the unstretched flame speed. This is indeed the case. For these flames, both the linear and nonlinear extrapolations yield similar unstretched flame speeds. We note in passing that the above results show that mixtures with equivalence ratios less than 1:3 correspond to r > 0 while the richer mixtures correspond to r < 0, hence suggesting that a diffusionally neutral n-butane/air mixture correspond to an equivalence ratio of 1.3.
120
50 0
κ (1/s)
200
300
Fig. 11. Comparison of linear (dashed line) and nonlinear (solid line) extrapolations of lean n-butane/air experimental data.
0.90 0.85 0.80
150 100
100
1.00
200
140
0
φ=1.10
250
160
150
Fig. 12. Comparison of linear (dashed line) and nonlinear (solid line) extrapolations of rich n-butane/air experimental data.
sb (cm/s)
180
100
50
300
200
sb (cm/s)
n−butane/air φ=1.80 1 atm
40
−20 −0.1
A¼
60
φ=0.75 n−butane/air 1 atm 0
200
400 κ (1/s)
600
Fig. 13. Experimentally measured outwardly propagating stretched flame speed as a function of stretch rate for equivalence ratios 0.75–1.10. Nonlinear extrapolation is shown as solid black line.
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300
1.30
250
0
1.40
200
1.50
150
Lb (cm)
sb (cm/s)
0.2
φ=1.20
1.60 100 1.70 φ=1.80
50 0
0
200
n−butane/air 1 atm κ (1/s)
400
600
Fig. 14. Experimentally measured outwardly propagating stretched flame speed as a function of stretch rate for equivalence ratios 1.20–1.80. Nonlinear extrapolation is shown as solid black line.
Figs. 13 and 14 further show the nonlinear extrapolations plotted as solid lines. These extrapolations are fitted to the experimental data using Eq. (11) and differentiated for presentation purposes in the standard stretch versus flame speed plot. It is noted that while the datasets are necessarily limited as discussed above, the useful data provides a sufficient number of experimental measurements, usually on the order of 25 measurements, to obtain a statistically significant regression through the dataset. The regression parameters, flame speed and Markstein length, have typical 95% confidence intervals of less than 1—2% in flame speed and less than 0:04 cm in Markstein length for the cases considered. The nonlinearly extrapolated unstretched flame speeds are shown in Fig. 15. For comparison, the experimental measurements of Bosschaart et al. [26], obtained by using a flat flame burner in which the flame is not subjected to stretch, are also plotted. Additionally, the stretched flame speed data of Davis [27], obtained in the counterflow, have also been nonlinearly extrapolated using Eq. (5) and the resulting unstretched flame speeds are plotted in Fig. 15. As seen in Fig. 15, the outwardly propagating flame speed measurements, when nonlinearly extrapolated to zero stretch rate, agree quite well with the measurements of Bosschaart et al. The nonlinearly extrapolated counterflow flame speeds agree well for lean and rich mixtures, with a slightly higher peak value. As previ-
sou (cm/s)
40 30 20 10
n−butane/air 1 atm 0 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Equivalence Ratio, φ Fig. 15. Unstretched laminar flame speed for n-butane/air mixtures at 1 atm of pressure: nonlinear present work – circles, Davis [27] nonlinearly extrapolated – squares, Bosschaart [26] – triangles, USC Mech II [28] – solid line.
−0.2 −0.4 −0.6
n−butane/air 1 atm
−0.8 0.6 0.8
1 1.2 1.4 1.6 1.8 Equivalence Ratio, φ
2
Fig. 16. Experimentally measured downstream Markstein lengths obtained from nonlinear extrapolation.
ously stated, near-stoichiometric mixtures show very little sensitivity to linear-versus-nonlinear extrapolation. The discrepancy between the datasets is therefore not due to the extrapolation procedure. Nevertheless, the three datasets agree quite well and were obtained using three different measurement techniques. Also plotted in Fig. 15 are the predictions of USC Mech II [28]. It is seen that the mechanism tends to over-predict the flame speed on the lean side, but agrees well with the measurements on the rich side. Additionally, the peak flame speed measured tends to be slightly richer than the peak predicted by the mechanism. Fig. 16 plots the nonlinearly extrapolated downstream Markstein lengths, Lb , as a function of equivalence ratio. It is seen that the Markstein length changes sign around / ¼ 1:3, which corresponds to the change in r as shown in Figs. 13 and 14. 6. Concluding remarks While our original motivation for the present investigation was to determine the laminar flame speeds of higher hydrocarbon fuels, we were quickly confronted with the strong nonlinearity in the flame response as a consequence of the small diffusivity of these heavier fuels, casting considerable uncertainty on the feasibility and accuracy of the conventional method of linearly extrapolating the stretched flame speeds to zero stretch rate to yield the laminar flame speed for these fuels. In order to arrive at a rational methodology to extract the laminar flame speed from the experimental data, we were led to appreciate the importance of the nonlinear nature of the stretch effect on the present expanding spherical flame. Consequently, by using the well-established nonlinear expression governing stretch-affected flame propagation, the laminar flame speeds of n-butane/air mixtures were successfully determined and found to agree well with literature data determined using unstretched flames. The study also identified the roles of the initial ignition transient and the finite chamber confinement effects on the flame-front dynamics, and the feasibility and accuracy of the extrapolation can be enhanced by using larger chambers and smaller ignition energies. Acknowledgments This work was supported by the Air Force Office of Scientific Research under the technical monitoring of Dr. Julian M. Tishkoff. It is a pleasure to acknowledge helpful discussions with Professor John K. Bechtold of the New Jersey Institute of Technology.
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