Measurements of Markstein Numbers and Laminar ... - ACS Publications

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Energy & Fuels 2004, 18, 316-326

Measurements of Markstein Numbers and Laminar Burning Velocities for Natural Gas-Air Mixtures S. Y. Liao,* D. M. Jiang, J. Gao, and Z. H. Huang School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China Received July 25, 2003. Revised Manuscript Received October 15, 2003

Spherically expanding flames of Chinese natural gas-air mixtures have been used to measure the laminar flame speeds, at equivalence ratios of φ ) 0.6-1.4, initial pressures of p ) 0.05, 0.1, and 0.15 MPa, and preheat temperatures of T ) 300-400 K. Following Markstein theory, one then obtains the corresponding unstretched laminar burning velocity after omitting the effect of stretch imposed on these flames. To study the effects of stretch on burning velocity, various Markstein numbers for both strain and curvature have been derived and the effects of initial temperature and pressure on these parameters have been discussed. Over the ranges studied, the laminar burning velocities are comparable with those previously reported for pure methaneair mixtures and fit by a functional form ul ) ul0(Tu/Tu0)RT(pu/pu0)βp, where the dependencies of RT and βp on the φ value of the mixture are also deduced. Furthermore, it is presented that the extrapolation results are still in good agreement with the previous work at relatively high pressure. The effects of the dilution gases on the burning velocity have been studied at φ ) 0.71.2, and the variations in burning velocities are plotted as functions of the dilution ratio and the equivalence ratio of the mixture.

Introduction Laminar burning velocities are fundamentally important in regard to developing and justifying the chemical kinetics mechanism, as well as in regard to predicting the performance and emissions of the internal and external combustion system.1 There are several techniques for measuring the laminar burning velocity of a combustible mixture, such as counterflow double flames (Yamaoka and Tsuji2), flat flame burner (Van Maaren et al.,3 Haniff et al.4), and spherically expanding flames.1,5-14 For spherically expanding flames, the stretch imposed on the premixed flames is well-defined * Author to whom correspondence should be addressed. E-mail: [email protected]. (1) Bradley, D.; Hicks, R. A.; Lawes, M.; Sheppard, C. G. W.; Woolley, R. Combust. Flame 1998, 115, 126-144. (2) Yamaoka, I.; Tsuji, H. In 22nd Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1988; p 1883. (3) van Maaren, A.; Thung, D. S.; Goey, L. P. H. Combust. Sci. Technol. 1994, 96, 327. (4) Haniff, M. S.; Melvin, A.; Smith, D. B.; Williams, A. J. Inst. Energy 1989, 62, 229. (5) Gu, X. J.; Haq, M. Z.; Lawes, M.; Woolley, R. Combust. Flame 2000, 121, 41-58. (6) Bechtold, J. K.; Matalon, M. Combust. Flame 2001, 127, 19061913. (7) Stone, R.; Clarke, A.; Beckwith, P. Combust. Flame 1998, 114, 546-555. (8) Metghalchi, M.; Keck, J. C. Combust. Flame 1982, 48, 191-210. (9) Metghalchi, M.; Keck, J. C. Combust. Flame 1980, 38, 143-154. (10) Hassan, M.; Aung, K. T.; Faeth, G. M. Combust. Flame 1998, 115, 539-550. (11) Kwon, S.; Tseng, L.-K.; Faeth, G. M. Combust. Flame 1992, 90, 230-246. (12) Lamoureux, N.; Djebaı¨li-Chaumeix, N.; Paillard, C.-E. Exp. Therm. Fluid Sci. 2003, 27, 385-393. (13) Karpov, V. P.; Lipatnikov, A. N.; Wolanski, P. Combust. Flame 1997, 109, 436-448. (14) Agrawal, D. D. Combust. Flame 1981, 42, 243-252.

and the asymptotic theories and experimental measurements have suggested that a linear relationship between flame speeds and flame stretches exists, which results in extensive use of spherically expanding flames to determine unstretched laminar burning velocities. In addition, following the previous idea of Markstein, the other laminar combustion properties, such as various Markstein lengths (Markstein numbers), can be derived from the experimental measurements simultaneously. The Markstein numbers characterize the variation in the local flame speed that is due to the influence of external stretching, which is important in expressing the onset of flame instabilities and the stretch rates at flame quenching.15 With the growing crisis of energy resources and the strengthening of pollutant legislations, the use of natural gas (NG) as an alternative fuel has been promoted. Although there is much literature on the laminar burning velocities of pure methane-air mixtures, and the main chemical component of natural gas is believed to be methane, the belief that the methane burning velocity data can be obtained for natural gas is not justified. The origin of the Chinese natural gas selected for the present study is north of the Shannxi province in the People’s Republic of China. The detailed chemical composition is listed in Table 1. The experimental measurements for spherical laminar premixed flames that freely propagate from spark ignition sources in an initially quiescent Chinese natural gas-air mixture are made at equivalence ratios of φ ) 0.6-1.4, initial (15) Bradley, D.; Gaskell, P. H.; Gu, X. J. Combust. Flame 1996, 104, 176-198.

10.1021/ef034036z CCC: $27.50 © 2004 American Chemical Society Published on Web 12/04/2003

Burning Velocities for Natural Gas-Air Mixtures

Energy & Fuels, Vol. 18, No. 2, 2004 317

Figure 1. Evolutions of the flame radius as a function of elapsed time. Table 1. Contents of CNG Used component

volume fraction (%)

CH4 C2H6 C3H8 N2 CO2 othera

96.160 1.096 0.136 0.001 2.540 0.067

a The remains will be butane, pentane, sulfureted hydrogen, and water.

pressures of p ) 0.05, 0.1, and 0.15 MPa, and preheat temperatures of T ) 300-400 K. The properties of premixed laminar NG-air flames are derived from measurements of the flame radius, as a function of elapsed time for outwardly expanding spherical flames. In addition, the results considered here are limited to conditions where the effects of ignition, chamber wall disturbance, buoyancy, and chamber pressure rise are small. The present study shows the importance, as primary quantities, of the Markstein numbers Mac, Mas, Macr, and Masr, and the dependencies of various Markstein numbers upon initial temperature and pressure are presented. Finally, the measurements provide an explicit expression of the laminar burning velocities for Chinese natural gas, and their dependencies on the equivalence ratio, unburned gas temperature, initial pressure, etc. The effects of dilution gases on burning velocity are also investigated and the results are plotted. Experimental Technique Quiescent natural gas-air mixtures and natural gas-air dilution mixtures are ignited in a constant combustion bomb. A detailed description of the experimental setup has been given by Ma.16 The combustion bomb has an inside size of 108 mm × 108 mm × 135 mm. Two sides of this bomb are transparent, to make the inside observable; these sides provide the optical (16) Ma, F. H. A Fundamental Study on Turbulent Premixed Combustion (in Chin.). Doctoral Dissertation, Xi’an Jiaotong University, Xi’an, PRC, 1996.

access, and the other four sides are enclosed with resistance coils, to heat the bomb to the desired preheat temperature. The combustible mixture is prepared within the chamber by adding the gases that are needed to appropriate partial pressures. The gases are then mixed through the motion of a perforated plate across the cubic chamber. The normal function of the perforated plate is to provide a turbulent combustion environment if needed, where it is only used to ensure that the reactants are well-mixed. A waiting period of at least 10 min is then implemented from the stop of the perforated plate to ignition, to allow the mixture to become quiescent. Two extended stainless-steel electrodes are used to form the spark gap at the center of this bomb. A standard capacitive discharge ignition system is used to produce the spark. The history of the shape and size of the developing flame kernel is recorded by a high-speed camera (NEC, model E-10) that is operating at 5000 pictures per second with the schlieren method. The dynamic pressure after spark ignition is measured with a piezoelectric absolute pressure transducer (Kistler, model 4075A) that had a calibrating element (Kistler, model 4618A).

Laminar Burning Velocity and Markstein Numbers The laminar burning velocity and Markstein lengths can be deduced from schlieren photographs as described by Bradley et al.1 From the definition, the stretched flame velocity, Sn, is derived from the data of flame radius versus time, as

Sn )

dru dt

(1)

where the flame size (ru) is defined as the isotherm that is located 5 K above the temperature of the reactants. It has been suggested that this radius is related to the flame front size (rsch), as defined in schlieren photography, and it is modified by

ru ) rsch + 1.95δl

() Fu Fb

0.5

(2)

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Figure 2. Variations of Sn with flame radius ru and equivalence ratio φ for NG-air mixtures at 300 K and 0.1 MPa.

Figure 3. Flame speeds plotted against flame radius for different ignition energies.

Here, Fu is the density of the unburned gas, Fb the density of the burned gas, and δl the laminar flame thickness (given by δl ) ν/ul, in which ν is the kinematic viscosity of the unburned mixture and ul is the unstretched laminar burning velocity). The evaluation of ru requires that the value of ul be known. Hence, ul is first estimated using rsch and then eq 2 is adopted to give ru. From the definition of the flame stretch, the R value of a flame front in a quiescent mixture is given by

1 dA R) A dt

(3)

However, in regard to a spherically outwardly expanding flame, the total stretch is well defined as

R)

()

1 dA 2 dru 2 ) ) S A dt ru dt ru n

(4)

Following Markstein theory, a linear relationship between the flame speed and the total stretch rate exists, as given in eq 5:

Sl - Sn ) LbR

(5)

As a consequence, the unstretched flame speed (Sl) can be obtained as the intercept value at R ) 0, in the plot of Sn against R, and the burned gas Markstein



in the present study is given as follows:1

S ) 1 + 1.2

[()] δl Fu ru Fb

2.2

[()]

- 0.15

δl Fu ru Fb

2.2 2

(8)

Another burning velocitysthe stretched mass burning velocity (unr)shas been proposed by Bradley et al.,1 and it is given as

unr )

Fb (u - Sn) Fb - Fu n

(9)

To obtain the separated Markstein values of Lc, Ls, Lcr, and Lsr (where Lc and Lcr are the Markstein lengths due to flame curvature for un and unr, respectively, and Ls and Lsr are those due to flow strain for un and unr, respectively), the parameters Rc and Rs must be evaluated. A spherical flame is subjected to stretch due to both curvature (Rc ) 2un/ru) and the flow field aerodynamic strain (Rs ) 2ug/ru), where ug is the gas velocity at ru due to flame expansion and is given by ug ) Sn un. After determinations of the various burning velocities and stretch rates, the Markstein lengths will then derived from the following equations:

ul - un ) LcRc + LsRs

(10)

ul - unr ) LcrRc + LsrRs

(11)

The value of the Markstein length of the burned gas (Lb) is readily obtained from the plot of experimental values of Sn against R, as mentioned in eq 5, using the linear regression shown in Appendix A1. In addition, the Markstein lengths of Lc, Ls, Lcr, and Lsr can be deduced using the multiple regression shown in Appendix A2. Usually, the Markstein lengths derived here can be expressed in their dimensionless form, as the Markstein numbers Mas, Mac, Masr, and Macr, through normalization by the flame thickness, δl. Experimental Observations and Measurements Figure 4. Variations of flame speeds with different stretch rates and (A) equivalence ratio φ, (B) initial temperature, and (C) initial pressure.

length Lb yields as well. When the observation is limited to the initial portion of the flame expansion, where the pressure does not vary significantly yet, then a simple relationship links the spatial flame velocity Sl to the fundamental one, i.e., the unstretched laminar burning velocity, ul,

ul )

FbSl Fu

(6)

as Equation 7 is used to determine the stretched laminar burning velocity (un):

[ ( )]

un ) S Sn

Fb Fu

(7)

Here, S is a generalized function that is dependent on the flame radius and density ratio, and it accounts for the effect of the flame thickness on the mean density of the burned gas. The generalized expression of S used

Influence of Stretch on Flame Speed and Burning Velocity. After the spark is obtained, a local ignition occurs, which propagates throughout the entire homogeneous mixture as a spherically growing flame. The flame radius varies quasi-linearly with elapsed time, and the typical cases are shown in Figure 1. To characterize the flame propagation regarding the stretch that is subjected to the flame, one can plot the evolutions of the flame speeds as a function of the flame radius, as shown in Figure 2. The characteristics of the igniter can affect the measured value of the burning velocity. The ignition energy, which is well above the minimum ignition energy, can lead to very high apparent flame propagation, because of the expansion of the spark plasma and the conductive energy transfer from it. As we know, the role of the spark is to initialize a flame kernel, which is to overcome the tendency for the flame to quench because of the high curvature stretch rate, during the early stages of flame propagation. In the present work, the measurements of initial flame development are made at various ignition energies to ascertain their subsequent effects on the flame development. The flame

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Figure 5. Measured burned gas Markstein length Lb of NG-air flames as a function of the equivalence ratio φ.

speeds Sn measured at the flame radius ru are shown in Figure 3, for the natural gas-air mixtures with equivalence ratios of φ ) 1.0 and 0.8. The flame speeds obviously show a decrease initially and then increases gradually as the radius increases and the flame stretch rate decreases. For higher spark energy, the initial flame speed is greater and the slope of the curve of flame speeds becomes flatter, which is indicative that the stable flame kernel is produced more easily with higher ignition energy. Over the ranges studied here, the propagation speeds become independent of the ignition energy from flame radii greater than ∼6 mm, and this speed becomes the practical flame speed. Thereby, the measurements are made at a spark energy of 45 mJ and the flame radii are analyzed only beyond 6 mm, at which point spark effects could be discounted. It is also of interest that oscillations occur in Sn and that their amplitude enlarges as the mixture becomes leaner, as also shown in Figure 2. These phenomena have also been observed for methane-air and isooctane-air mixtures by Gu et al., and Bradley et al.1,5 Figure 4 have presented a selection of experimental data showing the variations of Sn with a total stretch rate R for different equivalence ratios, initial temperatures, and pressures. As mentioned previously, extrapolations of these data to a zero stretch rate can yield Sl, and the gradient indicates the burned gas Markstein length, Lb. Obviously, the effect of stretch on the flame speed is dependent on the equivalence ratio of the combustible mixture. The flame accelerates with the decrease of stretch rate as the flame propagates, where the positive Lb values are located. For the leaner mixture, the flame speed is slower and the flame is less sensitive to flame stretch, although the Lb also remains positive, which can be an indication that the curve of the mixture with φ ) 0.8 is relatively flatter in Figure 4A. An increase in the unburned mixture temperature results in an acceleration of flame speed; however, the slopes of the fitted lines have no obvious change (i.e., the values of Lb remain unaltered). Generally speaking, the temperature has no significant influence on the role

of the stretch to flame, as shown in Figure 4B, within the range of 300-400 K. It is indicative of the effect of flame stretch on pressure, is slightly complex, as shown in Figure 4C. At relatively high pressure (p ) 0.15 MPa), the Lb value becomes slightly small. Because the measurements are only made at near-atmosphericpressure conditions, the cellular structures of the flame have not been observed. For all measured conditions, the value of Lb increases as φ increases, as shown in Figure 5. Generally speaking, increases in the unburned gas temperature and pressure result in a slight decrease of the Lb value. In addition, the results are comparable with those that have been measured for a pure methane-air mixture by Gu et al.,3 within the φ ranges that are considered here. The effects of stretch on un and unr for these mixtures are given in Figure 6. The values of un and unr are obtained using eqs 7 and 9. The results are plotted for different equivalence ratios, initial temperatures, and initial pressures. The difference between un and unr can be clearly observed in this figure. For these NG-air flames, the stretched burning velocity (un), which denotes the rate of mixture entrainment, always increases as the stretch increases. In contrast, unr, which is the burning velocity related to the production of burned gas, is usually reduced by the stretch. The value of (un unr) is largest at small radii (large stretch), where the flame thickness is of a similar order of the flame radius. The rich condition (φ ) 1.2) gives the largest differences between un and unr, as shown in Figure 6A. In Figure 6B, it is clear that the initial temperature has little effect on the quantity (un - unr). As the pressure increases, the flame thickness decreases, as does the difference between un and unr, as shown in Figure 6C. In practice, if the curves of un and unr against the total stretch rate are extrapolated to a zero stretch rate, they can yield almost the same value of ul. This provides an alternative method to determine the unstretched laminar burning velocity, in contrast with that obtained using eq 6.



flame quenching. Figure 8 shows the effects of initial temperature and pressure on the Markstein number Masr. We observe that the initial temperature and pressure have no significant influence on Masr. The value of Masr is shown to decrease slightly as the preheat temperature and pressure each increase. The measured values of Masr agree with those of previous work, which also are shown in Figure 8. The purity of methane in the NG is >96%; therefore, no large scatter is observed among these measured results. Laminar Burning Velocities. Figure 9 shows the experimental laminar burning velocities for a Chinese natural gas-air mixture with the preheat temperature of 300 K, at pressures of 0.05, 0.1, and 0.15 MPa. The results for methane-air mixtures also are shown. Generally speaking, the present experimental results of methane are in good agreement with the previous experimental measurement,17 which indicates that the experimental uncertainty is acceptable. Furthermore, comparable results are also obtained for NG-air mixtures, which shows that this NG can be assumed to be simply pure methane. To study the effects of the preheat temperature and the unburned gas pressure on laminar burning velocity, the systematic measurements of laminar burning velocities for a Chinese natural gas-air mixture are made. Figure 10 shows the selected experimental results for equivalence ratios of φ ) 0.8, 1.0, and 1.2. At each φ value, initial pressures of 0.05, 0.1, and 0.15 MPa are chosen, as well as a temperature of 300 K. The measurements then are extended to the higher temperatures, while maintaining the same pressure conditions. To obtain an explicit expression about laminar burning velocities that is dependent on pressure and temperature, the measured laminar burning velocities have been fit to a simple power-law relationship at the datum temperature (Tu ) 300 K) and the datum pressure (pu ) 0.1 MPa) as

( )( )

ul Tu ) ul0 Tu0 Figure 6. Stretch burning velocities at different flame stretch rates and (A) equivalence ratio, (B) initial temperature, and (C) initial pressure. Open points represent un calculated from eq 7, and the solid points are unr calculated from eq 9.

The four Markstein numberssMas, Mac, Masr, and Macrsare also measured in this work. Figure 7 shows their values under the datum conditions. Similar trends are observed for Mas, Mac, Masr, and Macr as that for Lb, within the φ range. The Markstein numbers reach their high values under the rich condition and decrease as the mixture becomes lean. Especially, Mac, Masr, and Macr always remain positive values; however, Mas is negative. For comparison, the Markstein number of Masr for isooctane is also plotted in Figure 7. Isooctane, which is a heavier hydrocarbon, shows a contrasting trend in Masr with natural gas that is considered here. It has been suggested that Masr is the most relevant Markstein number for many aspects of combustion,3 which expresses the effect of aerodynamic strain upon the mass burning velocity and, ultimately, upon the

RT

pu pu0

βp

(12)

where the coefficient RT is 1.94, 1.58, and 1.68 and the coefficient βp is -0.465, -0.398, and -0.405 for φ ) 0.8, 1.0, and 1.2, respectively. Over the ranges studied, they can be fit as functions of φ, given by the following relationship:

RT ) 5.75φ2 - 12.15φ + 7.98

(13)

βp ) -0.925φ2 + 2φ - 1.473

(14)

and

The power-law fit curves are also shown graphically by the dashed curves in Figure 10; an experimental standard deviation of (5% is also given in this figure. Over the ranges studied, the fit curves agree with the experimental results very well. Particularly for highpressure flames (pu ) 0.1 or 0.15 MPa), there are fewer points outside of the (5% standard deviation of the (17) Law, C. K. In Reduced Kinetic Mechanisms for Applications in Combustion Systems; Peters, N., Rogg, B., Eds.; Springer: 1992; pp 15-26.

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Figure 7. Markstein numbers plotted against the equivalence ratio φ under datum conditions, where the Masr of isooctane is obtained at 0.1 MPa at 358 K.

Figure 8. Measured Masr for NG-air flames as a function of the equivalence ratio φ.

measurements; however, some small systematic deviations seem to be present for pu ) 0.05 MPa. The datum laminar burning velocity (ul0) is fit best with the third-order polynomial described in eq 15, which is shown in Figure 9 with a solid curve.

ul0 ) -177.43φ3 + 340.77φ2 - 123.66φ - 0.2297 (15) The maximum burning velocity in eq 15 is 40.1 cm/s at φ ≈ 1.05, which is consistent with the 39.1 cm/s value determined from the experiment. The fitted burning velocities show good agreement with the results of the experiment; especially, the best and most-consistent estimations are observed for the lean flames. As mentioned previously, over wide ranges, the burning velocities of this NG are consistent with those of pure methane. For more-extensive use under relatively higherpressure conditions, the fitted formula (eq 12) is extrapolated and the results are plotted in Figure 11; data from previous work also are presented, where

Gu et al. reported a power-law formula for methaneair within T ) 300-400 K and p ) 0.05-1.0 MPa, and that reported by Stone et al. is within T ) 298-400 K and p ) 0.05-0.5 MPa. Interestingly, the present results still agree with those previously reported. Influence of Dilution Gas on Laminar Burning Velocities. Based on the concept of exhaust gas recirculation, which is regarded as an effective way to decrease NOx emissions in combustion engine, measurements of the burning velocity are then made, for diluting Chinese natural gas-air mixtures. Three different diluent mixtures are considered:, i.e., simulated combustion products, nitrogen (N2), and carbon dioxide (CO2), with diluting volume fractions up to 0.3. Because the volume fraction of methane in Chinese natural gas is >96%, the dilutions used to simulate combustion products consist of 88% N2 and 12% CO2 (by volume) in this work, which is in reasonable agreement with the



Figure 9. Experimental laminar burning velocities for Chinese natural gas-air mixtures.

the secondary CO2 content is only ∼2.5%, which corresponds to a dilution ratio of 0.2%. Experimental study on diluting natural gas with practical exhaust gas has been made, and the results are shown in Figure 13. In this figure, the ratio of the burning velocities with and without dilution are plotted as functions of φ for diluting volume fractions of 0-0.30. It can be seen that the decrease in burning velocity is dependent on the φ value of the combustible mixture. Obviously, the effect of diluting a nonstoichiometric mixture is much greater. The present measurements in regard to the effects of exhaust gases on the burning velocity have been done for NG with φ ) 0.71.2 under the datum conditions, and the results can be expressed by a polynomial, as

ul(φr) ul(0)

) Aφ2r + Bφr + C

(16)

where A and B are functions of the equivalence ratio φ, given by the relationships

A ) 85.662φ3 - 219.92φ2 + 178.61φ - 40.287 (17) B ) 39.081φ3 - 97.549φ2 + 76.763φ - 14.681 (18) and C is a constant (C ) 1.0). Conclusion Figure 10. Dependence of the laminar burning velocity on temperature at different pressures.

practical products for NG stoichiometric combustion at ambient temperature. Figure 12 shows the additive effects of these inert gas mixtures on the maximum burning velocity. As can be expected, in regard to the maximum burning velocities, CO2 addition is much more sensitive than N2 addition. And obviously, there is much difference between additive N2 and practical exhaust gas. From this figure, it is clear that the effect of CO2 on laminar burning velocities is small, because

The experimental study on the burning velocity of Chinese natural gas has been conducted using spherically expanding flames. Two burning velocities are derived under different stretch conditions, using linear or multiple regression. The difference between these two burning velocities are discussed and their responses to stretch due to flow strain and curvature are quantified by the appropriate Markstein numbers. The Markstein numbers increase as the equivalence ratio of the combustible mixture increase and slightly decrease as the initial temperature and pressure increase. The correlation formula of the fundamental ones, unstretched

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Liao et al.

Figure 11. Extrapolation of the burning velocities of NG-air mixtures; the results of Gu and Stone are for methane-air mixtures.

Figure 12. Comparison of maximal laminar burning velocities for the three dilute gases.

laminar burning velocities, which are dependent on the pressure, temperature, and equivalence ratio, is obtained. The volume fraction of methane in this studied natural gas is >96% and the dilution ratio of CO2 within NG is only ∼0.2%; therefore, the experimental measurements are in good agreement with those reported for pure methane over a wide range of temperatures, pressures, and equivalence ratios. Furthermore, although the present measurement is within a pressure

range of 0.05-0.15 MPa, the fitted formula is still valid almost up to 1.0 MPa, which is justified by the previous reports. The simulated combustion products (88% N2 + 12% CO2) are used as a diluent gas to study the effects on burning velocity also, and nondimensional burning velocities of diluting natural gas are fit well by a polynomial that involves the dilution ratio and equivalence ratio of the combustible gas.



Figure 13. Dependence of the nondimensional laminar burning velocity on the dilution ratio at different equivalence ratios with dilution gases of 88% N2 and 12% CO2 (by volume).

Acknowledgment. This work is supported by the State Key Project of Fundamental Research Plan entitled “New Generation of Engine of Alternative Fuels” (through Grant No. 2001CB209208) and is supported in part by the Doctorate Foundation of Xi’an Jiaotong University. Appendix A1. Linear Regression. Consider N pairs of computed or measured values of (R, Sn)i. They satisfy eq 5:

Sl - Sn ) LbR

N

(Ri - R j )(Sni - Sn) ∑ i)1 N

(Ri - R j) ∑ i)1

ul ) un + LsRs + LcRc

The values of Ls and Lc are determined from the equations

A10A22 - A20A12 Ls ) A11A22 - A122

(A6a)

A20A11 - A10A12 Lc ) A11A22 - A122

(A6b)

where the terms used in eq A6 have the following definitions:

(A2)

N

A11 )

2

in which

(Rsi - Rs)2 ∑ i)1

(A7)

N

Sn ) R)

1

∑Sni

Ni)1 1

∑Ri

(A3b)

x [∑ N-2

i)1

N

(Sn - Sn)2 + Lb

(Rsi - Rs)(Rci - Rc) ∑ i)1

(A9)

N

A12 )

(A3c)

The standard derivation is given by N

(A8)

(A3a)

N

j Sl ) Sn + LbR

1

(Rci - Rc)2 ∑ i)1

A22 )

N

Ni)1

e)

(A5)

(A1)

The value of Lb is calculated from

Lb ) -

A2. Multiple Regression. Consider N pairs of computed or measured values of (Rs,Rc,un)i. They satisfy the relationship

(Ri - R j )(Sni - Sn) ∑ i)1

]

(A4)

N

A10 )

(Rsi - Rs)(uni - Rn) ∑ i)1

A20 )

(Rci - Rc)(uni - Rn) ∑ i)1

(A10)

N

(A11)

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Liao et al.

N

A00 )

(uni - un)2 ∑ i)1

Rs )

Rc )

un )

1

N

∑Rsi

Ni)1 1

(A13)

N

∑Rci

Ni)1 1

(A12)

(A14)

N

∑uni

Ni)1

ul ) un + LsRs + LcRc

(A15) (A16)

The standard derivation is given by

e)

xN 1- 3(A

00

+ LsA10 + LcA20)

(A17)

Nomenclature A ) flame area Lb ) burned gas Markstein length Ls, Lc, Lsr, Lcr ) Markstein lengths, as defined Mas, Mac, Masr, Macr ) Markstein numbers, as defined p ) pressure

pu0 ) datum pressure pu ) pressure of reactant rsch ) schlieren front radius ru ) flame radius S ) flame speed factor Sl ) unstretched laminar flame speed Sn ) stretched laminar flame speed T0 ) datum temperature (300 K) Tu ) reactant temperature ul ) unstretched laminar burning velocity ul0 ) datum unstretched laminar burning velocity (at 0.1 MPa and 300 K) ul(φr) ) unstretched laminar burning velocity with dilution ratio of φr un ) stretched laminar burning velocity unr ) stretch mass burning velocity ug ) gas velocity ahead of flame front Greek Symbols R ) flame total stretch rate RT ) temperature coefficient in eq 12 Rc ) stretch rate due to flame curvature Rs ) stretch rate due to flow strain βp ) pressure coefficient in eq 12 δL ) laminar flame thickness, given by δl ) ν/ul Fb ) burned gas density Fu ) unburned gas density φ ) equivalence ratio φr ) dilution ratio ν ) unburned gas kinematical viscosity EF034036Z

Measurements of Markstein Numbers and Laminar ... - ACS Publications

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