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Laminar burning velocity of stoichiometric methane-air: pressure and temperature dependence
COMBUSTION A N D F L A M E 31, 5 3 - 6 8 (1978)
53
Laminar Burning Velocity of Stoichiometric Methane-Air" Pressure and Temperature Dependence A. M. GARFORTH and C. J. RALLIS School of Mechanical Engineering, University o f the Witwatersrand, Johannesburg, South Africa, 2001
Refinements to the spherical constant volume bomb method are reported which enable more accurate measurements of laminar burning velocity to be made than previously possible. Data is presented for stoichiometric methane-air mxitures in the ranges of unburnt gas pressures and temperatures of 0.06 ~< p < 0.228 MPa and 290 <~ T u <<.525 K respectively. These include an assessment of flame front thickness effects and a calculation of instantaneous spatially averaged burnt gas density. Over most of the range of values reported the error in burning velocity is less than 5%. A suggested simplification removes the need for much of the sophistication which has previously deterred many from using this method of measurement.
NOTATION A F K m M n p r S t T a t3 7 e p r q~
area defined as (1 - 7 ) correction factor mass molecular weight mass fraction absolute pressure radius velocity time absolute temperature defined (Pb/Po) defined as (Pu/Po) ratio of heat capacities defined as(pf/po) density flame front thickness equivalence ratio
Subscripts b burnt gas c center, correction e at the end of combustion
f o s t u ug w
flame front initial spatial transformation (or burning) u n b u r n t gas u n b u r n t gas at the bomb wall A bar over any symbol indicates an instantaneous spatially averaged value. The symbol ~ denotes nondimensional quantities. 1. INTRODUCTION At this stage in the history of combustion research, it is surprising that reliable data on the fundamental laminar burning velocity of many gaseous combustible mixtures is so scarce. Where data is available, usually at pressures and temperatures close to atmospheric, there still exists a considerable variation between published values [1]. There are, however, several reasons for the existence of this situation. Most of the methods for the measurement of burning velocity, although simpler in use, are Copyright © 1978 by The Combustion Institute Published by Elsevier North-Holland, Inc.
A. M. GARFORTH and C. J. RALLIS
54 severely range restricted and susceptible to experimental error. Of those that are not, the potentially most powerful, namely, the spherical constant volume bomb technique, requires considerable analytical and experimental expertise and has, therefore, not attracted as much effort as the simpler and quicker methods. Only Rallis [2, 3] and Babkin [4, 5, 6] and coworkers have recently devoted any significant attention to burning velocity determination using this method. A further reason for this situation is that the swing in research effort towards the chemical kinetic and theoretical studies in the last decade has left few workers to continue with the more experimental measurement approach. However, the need for this data still exists and the present study is an attempt t o provide a wide range of reliable data via a refinement of the constant volume bomb experimental method [7, 8, 9, 10,11].
2. REQUIREMENTS FOR SPHERICALBOMB INVESTIGATIONS The requirements for bomb studies, especially in the case of low burning velocity fuels with characteristically thick flame fronts, are both analytical and experimental, and include refined numerical processing techniques for handling the large volume of data generated.
Assumptions In the present study, the following assumptions used by previous bomb workers in the analysis of the process have been eliminated in that it is no longer necessary to consider that: (i)
(ii)
(iii) (iv)
the flame front has negligible thickness nor that the influence of its curvature is unimportant [9], the flame front remains spherical and centered at the point of ignition during propagation [2], the bomb and its contents constitute an externally adiabatic system [8], or that the burnt gas density immediately behand the flame front, Pr, is an accurate approximation to the instantaneous spa-
tially averaged gaspb [10].
density of the burnt
The following assumptions still apply: (0
(ii)
(iii)
(iv)
(v)
the mass and total volume of gas in the system remains constant (a statement of fact rather than an assumption), the flame front itself is an externally adiabatic system, in that it neither radiates heat nor allows the transfer bf heat from the burnt gas to the unburnt gas region (amenable to checking [8]), the instantaneous pressure variation through the flame front and the vessel is negligilbe (accurate for all but the fastest burning mixtures), the temperature immediately behind the flame front corresponds to the theoretical equilibrium adiabatic value, and no dissociation occurs in the unburnt gas region (a highly accurate assumption).
Of the remaining assumptions only item (iv) might introduce any significant error, which will remain until a method of measurement is devel.oped to check its magnitude. Analytical Requirements An analysis which requires only the assumptions mentioned above should thus be capable of producing equations for burning velocity calculation from experimental data which are applicable to the propagation of a thick, curved flame front throughout the bomb, in the presence of a density gradient behind the front in the moving burnt gas.
Combustion Data Requirements In order to apply the burning velocity equations derived under the above assumptions, it is necessary to calculate Po and Mu (= 31o) and, at each instant of recorded time during flame propagation, 7u, Tu, M r, 7r and T r. A routine to determine the adiabatic flame front values has been structured and is detailed in Refs [7, 10, 13]. The source of thermodynamic properties used in this case is that of McBride et al. [12].
BURNING VELOCITY OF METHANE-AIR
55
Experimental Necessities Using a fuel and oxidant of the highest possible purity, the burning velocity equations require the following experimental data: Po, To, fb, Ou, r t and t. The first three constitute the initial mixture specification while the remainder require transient recording. The flame front radius, rt, should preferably be obtained as a continuous record with time, while Pu may either be deduced from a transient pressure record, and the assumption of adiabatic unburnt gas behavior, or obtained by direct measurement. In Reference [8] it is shown, after direct measurement, that the adiabatic assumption, at least in the case of methaneair mixtures, provides adequately accurate data. It is also necessary to continuously measure flame front thickness, but, as pointed out in Reference [9] it is easier at present to correct the St values obtained using the thin flame assumption. The equipment used to carry out the above measurements is detailed in References [ 11, 13]. Data Reduction Scheme The equations for burning velocity [9] summarized in the following section are seen to require smoothed values of /3 = (Pu/Po), ~ = (-Pb/Po), (drf/dt) and (d/dt)(~/~). The routine which provides this facility is described in References [11,13].
3. EQUATIONS FOR THICK FLAME PROPAGATION As opposed to the equations previously published by the authors (3) the present set do not neglect the mass of gas contained within the flame front at any instant of time. Furthermore, neither the physical width of the front nor the mass of gas inside it are considered to be constant with time. The mass conservation equation is, thus: mo = m~ + m b + m r = const.
(1)
When this equation is rearranged and differentiated
with respect to time the following mass derivatives are obtained: (dmu/dt) = - ( d m b / d t ) -
(dmffdt).
(2)
This allows three forms of the burning velocity equation to be derived using the usual definition S t = - ( 1 / A p u ) ( d m u / d t ) = (1/moApu)(dn/dt),
(3)
where in this case a mass fraction 'burnt and burning' is defined as n = (mb + m~)/mo = 1 - (mu/mo).
(4)
Each of these masses may be expressed in terms of the instantaneous volume of the gas in the respective region of the vessel and the instantaneous spatially averaged density there. In the case of the unburnt gas there exists no significant density distribution so that Pu = Pu. In the burnt gas, however, a density distribution does exist, due both to the process of combustion there and the movement of gas as the flame propagates outward from the point of ignition. An analysis of this [10, 13] has enabled the instantaneous average burnt gas density to be calculated from the experimental observations of r r and p. This is further discussed in section 5 of this article. Using the above expressions for mass, three forms of the burning velocity equations can be derived, namely the unburnt gas equation, the burnt gas equation and the so-called combined equation [9, 13]. The unburnt gas equation has precisely the same form as that previously published [3]. Considering, then, that its form is independent of the flame thickness parameter, one might be led to the conclusion that it should be universally used. However, it is so very sensitive to experimental error as to be of little value. The combined equation, in this case, appears to offer little advantage over the burnt gas equation. It is also of a more complex form. The burnt gas equation appears to be the most suitable for the present work and, in its simplest form can be
56
A.M. GARFORTH and C. J. RALLIS averaged value of Pb is required at every time instant for which equations (8) and (8a) are used to calculate St. The determination of these spatial averages is detailed in Section 5.
written:
St =
( ~ / 3 ) [ F 2 + (~/a--')(1
-F2)] S~
+ (113){(rrF313)(daldt) + [rt(1 - F a ) / 3 ] ( d e l d t )
- (~ - ~)F2(d.r/dt)}
4. CORRECTIONS FOR FLAME THICKNESS
(5) where F = (1 - r) and 7 = (r/rr). For the purpose of comparison it is worthwhile to write down the 'thin flame' form of the burnt gas equation
St = (a/3)S, + (rr/3~)(d~/dt).
(6)
A consistency check shows that equation (5) reduces to equation (6) for infinitely thin flames (z = 0). Furthermore, an examination of the magnitude of each of the terms in equation (5) shows that if St is determined using equation (6), then accurate corrections, K c, to be applied to these values can be simply calculated from Kc
=
(1/3)~ - a-)(1 - F2)Ss.
(7)
Finally, it has been previously found [3] that the 'thin flame' combined equation yields the same results as equation [6] but has advantages from a data handling point of view. In the present study, therefore, this equation has been used to calculate St in the form
S t = UxSs + [rfl(1 - ~)/303 - ~)] (d/dtXa/3),
(8)
and these values were then corrected using equation (7). It is useful here to also write equation (8) in the less condensed form:
St = (Ob/po)Ss + [rrpu(1 - -fio)/3(p,, - Pb)] X (d/dt)(fib/pu).
(8a)
It is important to note that, whereas Pu = -Ou to a high degree of accuracy, the use of Ob = Of leads to errors of the order of 12% in burning velocity. Thus, even for 'thin flames' a spatially
Babkin et al (14) examined the effects of flame thickness and curvature on inwardly propagating spherical propane-air gas pockets. They concluded that these effects are only significant for flame front radii smaller than 3 mm. Andrews and Bradley [15] have analyzed the constant pressure flame with outward propagation and derive a correction to burning velocity which can be expressed in the form
Ke
=
(1/3)(e - o0(1 - Fa)Ss.
(9)
A comparison between equation (7) and (9) indicates the similarity of form. However, the fact that F a appears in their expression as opposed to F 2 in equation (7) results in too large a correction for burning velocity. For example their equation for a stoichiometric methane-air test with initial pressure and temperature of 0.101 MPa and 290.5 K gives a correction of + 145 mm/s at a flame radius of 13.7 mm, whereas at the same position for that test, equation (7) yields a value of + 97 mm/s. Furthermore, their correction drops below 10 mm/s only at a radius of 62.2 mm whereas the present correction is already less than that value at 52 mm. It is also worth noting that their equation is not written with ~, but simply a as no burnt gas density distribution exists with constant pressure flame propagation. The main difference between equation (7) and (9) arises because of a neglect of the term (d/dO (z/rt) in the analysis for equation (9). The present study has indicated that, even if r is constant, the variation of the ratio of flame thickness to outer flame front radius is an important factor. It is thus concluded that the expression of Andrews and Bradley results in corrected burning velocity values from bomb tests which are too large. Furthermore, a consideration of the variation of flame thickness with pressure and temperature leads, as is shown later, to the conclusion
BURNING VELOCITY OF METHANE-AIR
57 TABLE l
The Extent of Flame Thickness Influence a Radius (rf-mm) at which the correction to S t is: Initial test Pressure Initial test temperature (Po-MPa) (To-K)
+10%
+5%
+1%
0.0608
290.3
47
56
71
0.101
290.5
36
47
63
0.228
290.5
16
28
56
a Vessel diameter-160.2 mm.
that even the correction calculated from equation (7) may be too large at values o f flame radius less than about 50 mm in our vessel for a 1 atmosphere initial pressure test. The numerical corrections applied to the present results must be considered as tentative (see Section 6). The best documented values of r [16, 17] for a stoichiometric methane-air flame at 1 atmosphere are respectively 0.75 and 1.1 mm. Andrews and Bradley (15) consider Janisch's value of 1.1 mm to be the more reliable. However, in order to calculate values of ~, the form of the temperature profile through the flame front is also needed, and in this respect Dixon-Lewis and Wilson's result was the more useful. Furthermore, no flame thicknesses or temperature profiles have been reported at higher pressures. The process whereby the present values of r and ~ are determined for bomb tests (9), described in Section 6 of this article, is therefore open to criticism. Although the corrected St values presented here are thus possibly in error, one can nevertheless gauge the region of bomb tests over which thickness effects are important. This is sulnmarized in Table I above for stoichiometric methane-air tests. The table clearly shows that the region rf < 30 mm is, in general, one in which it is not possible to place great reliance on the values of burning velocity. It is noteworthy that the constant pressure region only extends approximately as far as 10 mm from the ignition p o i n t - a n d this is the region from which many bomb results have previously been determined. The corrections applied to the present data are illustrated in Figs. 1 and 2. Figure 1 shows the
average 1 atmosphere initial pressure test curve derived from three test runs in the bomb. Also shown are some of the corrected test values obtained using equation (7). It is clear from this diagram that the corrected points below a pressure of 0.15 MPa must be viewed with suspicion. Neither the pressure nor the temperature of the unburnt gas have changed by large amounts between ignition conditions and the 0.15 MPa 0.5
I REF. 21
I
I
®
value)
!
0.45 REF 21
I
value)
8
_
REf'. 2 3 ~ RE~ 24 ~"I,
OA
--
f
0.3~ REF. 5 (Corrected J accordlng~ / to this ~+/ study) /
02
Q2E
--
---
THIS STUDY-UNCORRECTEO
~o
THIS STUDY-CORRECTED USING EQN (7)
REF. 5 ._.__.~ + (Uncorrected)
1 0.1
THIS SrUDY- MOST LIKELY CORRECT RESULTS
p (MPa)
I 02
l 0.3
Fig. 1. Correction of bomb test curves in the region of
the initial pressure and temperature.
58
A.M. GARFORTH and C. J. RALLIS
0.60 , ~
UNCORRECTED TEST CURVE
0.55
CORRECTED
0.5t3
UNCORRECTED CORRECTED
,
TEST CURVE
CORRECT TEST CURVE
%
tg
0.45
,
i./
}
ISOTHERMS
f
0.40 0.35
j "
!
0.30
0.25
0.5
10 p (MPa)
1.5
20
Fig. 2. Illustration of flame front thickness corrections for stoichiometric methaneair tests.
point on the curve, so that the burning velocity must correspondingly have changed very little during the same period. It is therefore suggested that the corrections for flame front thickness over this range are indicative more of uncertainty bounds than of actual alterations which should be made to burning velocity. Certainly, at pressures above 0.15 MPa in this case, the corrected curve is a smooth one and follows the general trend of the uncorrected curve. In order to obtain corrected burning velocity values below 0.15 MPa therefore, the corrected curve has been extrapolated back to Po while maintaining the trends evident from the uncorrected curve in this region. This results in a value of St at 1 atmosphere and 290.5 K of 0.340 m/s as compared with the uncorrected value of 0.316 m/s. Examing the above extrapolation process further, it is seen that a value of burning velocity at Po and To higher than 0.35 m/s cannot be justified. The fact that this is low in comparison with currently accepted values for stoichiometric methane-air, some of which are included on Fig. 1, is discussed at a later stage. However, when deciding upon the form of the extrapolated curve used above, not only the test curve trends were taken into account, but also the shape of the isotherms as seen on Fig. 2. Here, only the two lowest isotherms are shown, together with test
curves at initial pressures of 0.0608, 0.101 and 0.228 MPa. 5. AVERAGE BURNT GAS DENSITY The details of the process whereby the instantaneous spatially averaged values of burnt gas density were obtained are presented in References [10, 11, 13]. In essence, the procedure is as follows: (/)
(ii)
(iii)
(iv)
The flame front radius and bomb pressure at approximately 100 evenly spaced time intervals between ignition and the end of flame travel are obtainted from the photographic test records. Using the thermodynamic relationships for the adiabatic compression of the unburnt gas (checked for every test with the interferometer method for measuring Pu), the unburnt gas temperature, Tu, is calculated for every recorded time mentioned above. The computer program detailed in Reference [7] is then used to calculate equilibrium flame front, burnt gas temperatures~ Tf, at the same instants of time. The technique presented in Reference (10) allows the position of each burnt gas
BURNING VELOCITY OF METHANE-AIR
(v)
(vi)
59
element to be radially located in the vessel for every flame front position. This, together with the assumptions that the burnt gas undergoes adiabatic compression and that 7~, remains constant with time for each burnt gas element, allows the instantaneous temperature distributions in the burnt gas, shown in Fig. 3, to be obtained. These assumptions are discussed in References [11, 13] and are judged to introduce almost negligible error. The above temperature distributions coupled with mixed mean burnt gas properties results in the required burnt gas density distributions. Finally, the density distributions are averaged on a mass basis to produce the instantaneous values of Pb-
Lewis and yon Elbe [18] calculated the burnt gas temperature distributions for hydrogen-oxygen and ozone-oxygen flames. In both cases the average temperature when the flame had reached the vessel wall, The, could be related to the center temperature, To, and the gas temperature adjacent to the wall, Tw, by (Tb e - Tw)/(Tc - Tw) ~ 0.36.
(10)
Recognising that they ignored gas movement and used an approximate expression for mass fraction burnt, n, their result is in surprisingly good agreement with the corresponding value of 0.29 for the present 0.1 MPa stoichiometric methane-air result. It seems that their two assumptions tended to cancel one another out, because the value without gas movement included for the methane-air case should be 0.54. In a very recent attempt to predict pressure rise ratio in closed vessels, Bradley and Mitcheson report calculated gas movement results for stoichiometric methane-air [19]. A comparison between their predictions and the combined analytical and experimental results of the present study is presented in Fig. 4. It is interesting that their predictions are for a vessel of diameter 160.2 m m - t h e precise size used in the present study.
3500
~'~'~
~.~
~[.~
250C Flame front ~ positions
~J
i
'lKTf
I
I I
i
I
I I I I
I I I
I I I
150C - - - -
-;II I I
50C
I
I I I I
I I
~T u
20
40
0
80
Radius-mm
Fig. 3. Typical instantaneous temperature profiles during flame propagation in the bomb. Some experimental particle tracks were accidentally obtained with a few of our tests [10, 13] and one of the clearest of these is shown and is seen to largely confirm the analysis of our study. A disturbing feature of Bradley and Mitcheson's computer prediction is the delay period of approximately 7 ms at the start of their flame front vs time curve. This materially affects the position of both their unburnt gas and burnt gas particle tracks (only one set of which is shown in the figure), and it makes comparison with the present ones difficult. This delay is certainly not realBledjian [20] shows that the delay is in the region of 30 tzs. One can only conclude that their delay is introduced by the computer programme. To effect a better comparison with the present results, their prediction in the region 7 to 50 ms has been expanded time-wise to correspond to the total flame propagation time of 50 ms found in this study. Although this produces better correspondence between the flame front positions, the unburnt gas track is definitely now in error. The general prediction of their burnt gas movement is in fair agreement with our experimental one, but the discrepancy becomes larger with increasing flame radius. No further comparison is worthwhile at present.
60
A.M. GARFORTH and C. J. RALLIS
7o
_'-_
.....
60
. . . . . . . . . . . . . , . . . . . . . . . c,, ,,,c,
/~""/ f~"j./
10' Z
/ .~
__.~
0
, . . . . . o.,
5
t
,
10
15
t
i
I
20 25 30 Time (ms)
i
i
35
i
40
45
50
Fig. 4. Comparisonof gaseous particle track predictions. The effect of the inclusion of both the temperature distribution and burnt gas movement in the S t equation on the values obtained is clearly shown in Fig. 5 for a single bomb test run. The neglect of either of these effects results in an error as large as 12% in burning velocity.
6. EXPERIMENTAL RESULTS In Table 2 the initial data for all the stoichiometric methane-air tests reported here are presented, together with some of the main test results. It should firstly be noted that although no direct control was exerted over the initial mixture temperature, T o never varied by more than -+1.6 K from the mean value of 290.5 K. The accuracy with which the initial pressure was set was better than -+0.1% while the mixture strength, ¢, was capable of control within -+0.1%. Some problems were encountered with initial mixture stratification but a controlled mixture preparation procedure and the provision of a stirrer reduced this to very acceptable levels. From the point of view of test reproducibility it is seen that the maximum pressures for individual tests never varied by more than +-1.3% from the mean for that series. Similarly, the deviations in t e and ae were never more than -+2.2% and +1.3%, respectively. The value of ae was usually
obtained by extrapolation [11] because it was not possible to transcribe the photographic flame trace fight up to the stage where the flame reached the wall; internal reflections in the bomb tended to obscure the last millisecond or so of the record of flame travel. The value of ae is considered to be a very good indication of test reliability during the later stages of combustion in the bomb. Since, in the constant mass and constant volume bomb system, ~e must equal 1.0 exactly, its departure from this value provides assessment of overall test inaccuracy. In
o55
r
F
0.50
~
~
i
r ..~ "~
I
"~ 0 . 4 5
.~'
%
"~ 0.40--
i~,"
"
"'-
,
~
~!
.
~
I
~ ~ '
/
035
/~'[
~
i
0.30
----
........
!
j NOI% AVERAGE OR BURNT GAS MOVEMENT
w,,.MOV,M~N,~,u, NO
~
WITH pbAND
°26o
i
o2
i
,
L
.ov,.,NT
0;6
'
o'a
p (MPa)
Fig. 5. Effects o f burnt gas movement and density averaging o n S t f o r a 0.101 MPa test.
BURNING VELOCITY OF METHANE-AIR
61 TABLE 2
Summarized Test Data
To
te
Pe
MPa
Po Test
(K)
ms
(MPa)
ae
~e a
(mm)
ro
0.0608
A B
290.3 289.6
47.6 47.1
0.540 0.546
1.073 1.082
0.953 0.961
2.073 2.068
0.0810
A B C
290.4 290.8 289.7
49.3 48.5 50.6
0.712 0.721 0.727
1.058 1.074 1.078
0.941 0.958 0.962
1.426 1.428 1.423
0.1013
A B C
290.4 290.5 292.1
52.2 53.3 53.5
0.882 0.905 0.897
1.049 1.074 1.071
0.933 0.956 0.953
1.065 1.065 1.071
0.1265
A B C
290.6 290.7 289.2
56.0 55.8 55.7
1.119 1.093 1.098
1.062 1.040 1.038
0.949 0.925 0.926
0.798 0.798 0.794
0.1519
A B
290.0 290.6
58.2 59.1
1.352 1.352
1.066 1.068
0.948 0.951
0.627 0.628
0.2025
A B
290.6 291.6
62.9 63.2
1.801 1.791
1.064 1.062
0.949 0.948
0.432 0.433
0.2279
A B C
291.2 290.2 290.5
67.6 65.9 65.1
2.043 2.024 2.042
1.074 1.061 1.071
0.957 0.950 0.956
0.371 0.369 0.370
a ave ae = 0.950.
Table 2 it is seen that, with the exception of four tests, ~e is usually close to 0.950 (its average value for this test series). It is thus possible, when reviewing the test results, to give greater credence to those tests with values of ~e closest to 1.0; tests with particularly low values are suspect. This selfcorroboration characteristic is an important advantage of the bomb method which has not always been appreciated by other users. It is used to good effect below when comparing St values from this study with those of other workers. The values of flame front thickness, T, and average flame front density, ~, for the present tests were determined as follows: (i)
using the data mentioned in Section 4, it was assumed that the temperature distributions through the curved flame fronts would be of the same form as that
(ii)
(t~i)
through Dixon-Lewis and Wilson"s plane flame [16], provided that at p = 1 atm and T = 300 K the value of T corresponded to the 1.1 mrn value of Janisch [ 17], it was assumed that r ~ 1/PuSt was an accurate enough representation of the variation of thickness for methane-air at other conditions, an approximate relationship for St in terms of p and Tu from the present 'thin flame' results was determined and, on substitution gave
~"= 1.848 X lO-7(To/Po°'265)p - 1 " ° 4 .
(1 1)
Values of ro at To and Po are reported in Table 2 for comparison purposes. The above procedure highlights the fact that,
62
A.M. GARFORTH and C. J. RALLIS 0.6 056
I_@°I
0.,6
046 " t
0.4 O.,3f.
02
N
L
4 t
0~2
2.0 (MPa) Fig. 6. Burning velocity pressure dependence for stoichiometric methane-air. strictly speaking, the process of St correction for flame front thickness effects should be an iterative one. However, with the present uncertainty in r values, this does not provide any reliable improvement over the currently reported data. For all tests reported here the spark gap was kept to a minimum, as was the ignition energy, consistent with reliable and reproducible ignition. Typical of the settings was that for the 0.101 MPa series for which the gap was 2.05 mm and the energy 6 mJ. In Figs. 6, 7 and 8, the present stoichiometric methane-air burning velocity results are presented
in the ranges 0.06 ~< p ~< 2.04 MPa and 290 ~< Tu <~ 525 K. In Fig. 6 the dependence of St on pressure can be seen to be initially very large at low unburnt gas temperatures while this dependence decreases considerably at higher pressures and temperatures. A comparison with our previously reported St data for stoichiometric acetylene-air [3] shows the methane pressure dependence to have a far less complex nature. However, in both cases the pressure dependence decreases to almost zero at higher pressures and temperatures. The nature of the methane temperature dependence is also simpler than that of acetylene and Fig. 7
06
P
0.5
mO~
'= ~' / , / o ? ' r"
03
021 3C
T.(K)
Fig. 7. Burning velocity temperature dependence for stoichiometric methane-air.
BURNING VELOCITY OF METHANE-AIR
63
!
,
1
_,= / o~
Fig. 8. Three-dimensional representation of burning velocity dependence on pressure and temperature for stoiehiometric methane-air. shows that this dependence varies little over the full range of pressures reported here. The threedimensional representation of Fig. 8 illustrates the combination of both pressure and unburnt gas temperature effects on the burning velocity of stoichiometric methane-air mixtures. An important observation is made on the basis of the present test results from the information in Fig. 9. Here the variation of 6 with pressure ratio, (P/Po), calculated for a stoichiometric methane-air test with Po = 0.101 MPa, including both density distribution and gas movement effects is plotted. It is seen to be slightly curved when compared with the straight line fitted between so at (P/Po) = 1 and ~ = 1 at (P/Po) = (Pe/Po). If the curvature of the actual curve is retained while ~e is adjusted from 0.956 to 1.0 it is seen that this modified curve never differs from the straight line by more than 2.8%. The importance of this conclusion lies in the fact that, provided 0~o is known, subsequent values of ~ can be obtained from the equation
nor of burnt gas movement calculations. Equation (12) will yield ~ values well within the experimental accuracy of the method. 7. COMPARISON WITH LITERATURE VALUES A feature of concern in the present burning velocity study for stoichiometric methane-air is that the value at p = 1 atmosphere and Tu = 290.5 K is 1.0
k
0.9 O,8 0.7
0~6 0,5 0.4 i
= % + (1 - ~ o ) [ ( P / P o )
-
11/[6oe/po)
-
02
l].
(12) It is therefore not necessary for prospective bomb users to enter into the complexity o f calculating the flame front adiabatic temperatures, nor of density distribution determination in the burnt gas
0.1
o
1
(~0o) Fig. 9. Comparison between accurate and simplified methods for ~ determination.
64
A.M. GARFORTH and C. J. RALLIS
p (MPa)
Fig. 10. Comparison with literature data. significantly lower than the commonly accepted value. As can be seen on Fig. 1, the data of Bradley and Hundy using hot wire anemometry in closed vessel explosions [21 ] ; that of Andrews and Bradley from'their double kernel, closed vessel experiments [22] ; the measurements of Gunther and Janisch on button flames above a Mache-Hebra burner [23]; and the laser-doppler anemometer data from the nozzle burner studies of France and Pritchard [24], all point to a value of St at 1 atmosphere and 293 K in excess of 0.4 m/s. The value from the present study at Tu = 293 K is in the region of 0.34 m/s. As mentioned in Section 4, the highest value of St under these conditions which could be justified on the basis of the test curves of Fig. 1 is 0.35 m/s. However, one major assumption remaining with the bomb method is that the burnt gas immediately behind the flame front is at the adiabatic equilibrium flame temperature. There is enough evidence in the literature to indicate that the actual temperature is lower than this, a reasonable estimate of the difference being about 5%. If this were taken into account, values of ~ = (-Pb/Po) would correspondingly be 5% higher throughout the flame propagation process in the bomb. This appears to agree with the current values of ~e which are generally 5% lower than the required value of 1.0. The value of St at the initial pressure and temperature could therefore be as high as 0.37 m/s. Whereas our present experience confirms that the constant pressure region in the bomb is an
unreliable one from the point of view of burning velocity determination, the experimental error is not greater than -+5% in the range 1.1 ~< (P/Po) <~ 6.0 for bomb tests. On Fig. 1 this corresponds to pressures above 0.11 MPa, while on Fig. 6 this is the range of values between the isotherms 300 K and 475 K. Certainly, at a value of pressure of 0.2 MPa on Fig. 1, the value of St of 0.37 m/s could be low only due to the 5% error mentioned above, since the flame front thickness correction is small by this stage, yielding a maximum value of 0.39 m/s here. Yet, although the burning velocity definitely increases during flame propagation in the bomb, this value is still below that of 0.4 m/s indicated by those authors using alternative, but range limited, methods at the initial conditions. It would thus require a considerable, and unjustifiable, distortion of the present test curves to bring about closer agreement with these other results. When considering the corresponding values for St obtained by Babkin et al. [4, 5, 25] on Figs. 1 and 10, they are seen to be even lower than those of the present study. Whereas the corrections applied to such results for flame thickness by Andrews and Bradley [15] would increase these values to yield better correspondence with those of the previously mentioned accepted studies, the present article shows that these corrections would be far too high. For example, the value at 1 atmosphere and 293 K applying Andrews and Bradley's correction factor of 1.22 would be 0.366 m/s, whereas the corrected value applying the techniques of this study is only 0.323 m/s. Recognising
BURNING VELOCITY OF METHANE-AIR that Babkin's results were originally calculated using simplified bomb equations, the discrepancy between his results and ours can be explained at the early stages of combustion. It thus seems that bomb studies yield values which are low in comparison with those obtained by other methods, even when they are corrected as judiciously as is presently possible. A reconsideration of some of the errors involved in the values from other methods might thus prove useful in accounting for this discrepancy. The recently published values of France and Pritchard [24] result from a sound experimental method and appear to be reliable. However, it seems that these authors did not measure the unburnt gas temperature immediately ahead of their conical flame. Some heating of the gas as it passed through the burner body near the port cannot therefore be ruled out. The corresponding rise in temperature is likely to have been small but would nevertheless have produced a value of St which was too high. A discrepancy of +13.5 °C would lower their value of 0.403 m/s to the present maximum value of 0.37 m/s. Such an increase of temperature through the burner does not seem unreasonable. Gunther and Janisch's value [23] of 0.41 m/s is close to that of France and Pritchard and was also measured under reliable conditions with considerable attention to experimental technique. These authors did measure the temperature of the unburnt gas in the burner port using a thermocouple. However, it appears that the thermocouple sensing element was unshielded and may therefore have radiated to the burner rim which was kept at 20 °C by circulating cooling water around it. In this case, however, an error of 13.5 °C in the temperature of the unburnt gas seems large. Unfortunately, these authors do not quote the size of the burner port and their previous studies on a 4 mm diameter burner [26] led Andrews and Bradley to conclude that the resultant burning velocity values would be too high [22]. The same might apply to their present studies if the burner diameter is not significantly larger than the flame thickness. An interesting facet of both france and Pritchard's and Gunther and Janisch's studies is that their burning velocity values at 1 atmosphere and 293 K are only just above 0.4 m/s, and are lower
65 than those of Bradley and Hundy [21] and Andrews and Bradley [22]. In the hot-wire studies, the scatter between burning velocity values is large as indicated on both Figs. 1 and 10. Furthermore, the sensitivity of the method to errors in the measurement of Sug produces a four times magnification of experimental error in the case of methaneair mixtures. Although the stoichiometric methane flame is non-sooting, some small radiation must have occurred to the hot-wire anemometer. In particular, the method requires the hot-wire to sense gas velocity immediately ahead of the high temperature flame, so that it is conceivable that an error of the order of 1% or even 2% in Sug measurement could have been caused by radiation effects, This could have produced St values which were 4% to 8% high, resulting in burning velocity in the range 0.39 to 0.44 m/s rather than the reported range of 0.42 to 0.48 m/s. In the case of the double kernel technique, we feel that until the kernels are simultaneously viewed from two mutually perpendicular directions, it will be impossible to rule out oblique flame propagation in the region of interest. This fact, if not established in the currently reported tests [22], would also yield high values of S t . In addition, the typical photographic records of Andrews and Bradley are evidently not of the same calibre as those of the pulsed scl'dieren method of Raezer and Olsen [27]. The accuracy of the burning velocity results obtained in the more recent study is therefore suspect. One distinct advantage of both of the above hot-wire and double kernel experiments is that they were performed in closed vessels, and could thus yield S t data over a greater pressure range than with burner studies. Unfortunately, the results from the double kernel tests were only reported for 10% CH4 mixtures. They are not, therefore, directly comparable with those of the present stoichiometric studies. Nevertheless the trend of the data is interesting and has been included in Fig. 10. Babkin's results as shown in Fig. 10 have not been corrected for flame front thickness and, are thus likely to be low at all stages. In addition, all his data points were obtained using the constant pressure combustion region of the vessel only, in which the bomb method is definitely inaccurate.
66 There also exists some doubt as to the precise method whereby the results were calculated from the bomb measurements. It is possible that the St values are for this reason a further 8% low [11, 13, 28]. Finally, it is not possible to use the selfchecking ability of the bomb method mentioned in Section 6 above when only recording in the constant-pressure region. Nevertheless, there is some quantitative agreement between their results and ours as is seen from a comparison of their 423 K curve and our 425 K line. At the higher pressures, however, the two sets of results diverge. At lower pressures their results are understandably low because of the importance of flame front thickness there. Both the hot-wire and the double kernel results of Bradley and his co-workers are in reasonable agreement with those of the present study in the region of 0.15 to 0.3 MPa. In fact the trend of the double kernel results is in closer agreement than the hot-wire one. It is unlikely that the small difference in mixture composition between 10% CH4-air mixtures and stoichiometric ones could account for the significant difference in trend between these two sets of results by the same team of workers. If it is also realized that the richer mixture values of S t are expected to be between 5% and 10% higher than the stoichiometric ones, the correspondence with our results is even closer. At this stage it is not possible to quantitatively account for the differences between the values of burning velocity from this study and those of other workers. Whereas it appears that results from bomb studies are generally lower than those from other methods, the discrepancies are only significant at the lower pressure and temperature states. The fact that the range of states at which St values can be obtained in closed vessels is much larger than in the case of most other methods, allows an examination of trends which is not otherwise possible. These values at the lower pressures and temperatures can therefore be more critically assessed than when only single data points are available. Furthermore, it appears that the burning velocity values recently reported for stoichiometric methane-air at 1 atmosphere and ambient temperatures may have been too high for a variety of reasons. It
A.M. GARFORTH and C. J. RALLIS would thus seem that an acceptable value for St at this condition lies in the range 0.37 to 0.4 m/s. 8. CONCLUSIONS The spherical constant volume bomb method has been used to provide laminar burning velocity data for stoichiometric methane-air mixtures over a wide range of pressures and temperatures [28]. In this study, the previously neglected effects of flame front thickness, burnt gas density distri. bution and burnt gas movement have all been in. cluded in an attempt to provide data which is as accurate as possible at the present time. It is felt that future studies will only improve upon the burning velocity results at the extremes of the pressure range reported here, where experimental inaccuracies may have been significant in the present series of tests. A simplification, applicable at least to methaneair tests in the bomb and probably to most other mixtures, has been suggested. It was only possible to make this suggestion after the effects of density distribution and gas movement had been considered. The simplification eliminates the need for such sophistication and does not introduce an error greater than 3% in burning velocity. The pressure and temperature dependence of burning velocity reported here indicates a complexity not evident in the literature values. Simple empirical fits to the data are thus not possible and, for this reason, none have been suggested. However, the trends are not as complex as those previously reported for stoichiometric acetylene-air mixtures. An error analysis of the present results indicates [13, 28] that the experimental error is not likely to be more than 5% in the range 1.1 ~< (P/Po) <~ 6.0 for each test while outside this region, the error does not exceed 9%. The values of burning velocity reported here are likely to be about 5% low due to the assumption that the flame temperature is equal to the instantaneous adiabatic equilibrium temperature.
This work was partially sponsored by the South African Council for Scientific and Industrial Research and the authors wish to express their appreciation to this body for their long-term support.
BURNING VELOCITY OF METHANE-AIR
67
REFERENCES 1. Andrews, G. E. and Bradley, D., Determination of Burning Velocities: A Critical Review, Combust. Flame 18, 133-153 (1972). 2. RaUis, C. J., A Critical Evaluation of the Spherical Constant Volume Method for Determining Laminar Burning Velocity, Ph.D. Thesis, Univ. of the Witwatersrand, Johannesburg (August 1963). 3. Rallis, C. J., Garforth, A. M. and Steinz, J. A., Laminar Burning Velocity of Acetylene-air Mixtures by the Constant Volume Method: Dependence on Mixture Composition, Pressure and Temperature, Combust. Flame 9,345-356 (1965), 4. Babkin, V. S., Kozaehenko, L. S. and Kuznetsov, I. L., The Influence of Pressure on the Burning Velocity of Methane-air Mixtures, Zh. Prikl. Mekhan. Tekn. Fiz. 145-149 (1964). 5. Babkin, V. S. and Kozachenko, L. S., Study of Normal Burning Velocity in Methane-air Mixtures at High Pressures, Fiz. Goren. Vzryva. 2(3), 77-86 (1966). (English Translation: Combust., Explosion and Shock Waves 2(2), 46-52 (1966).) 6. Babkin, V. S., V'yun, A. V. and Kozachenko, L. S., The Effect of Pressure on Normal Flame Velocity Investigated by the Initial-section Method in a Constant-Volume Vessel, Fiz. Goren. Vzryva. 2(2), 5260 (1966). (English translation: Combust. Explosion and Shock Waves 2(2), 32-37 (1966).) 7. Garforth, A. M., A Computer Programme for the Calculation of Equilibrium Composition and Adiabatic Flame Temperatures of Gaseous Mixtures Undergoing Constant Pressure Combustion, School of Mech. Eng., Univ. of the Witwatersrand, Johannesburg, Research Report 56 (1973). 8. Garforth, A. M., Unburnt Gas Density Measurement in a Spherical Combustion Bomb by Infinite-fringe Laser Interferometry, Combust. Flame 26, 343-352 (1976). 9. Garforth, A. M. and Rallis, C. J., The Sphericial Bomb Method for Laminar Bruning Velocity Determination: Equations for Thick Flame Propagation, School of Mech. Eng., Univ. of the Witwatersrand, Research Report. 59 (1975). (Also presented at The Second European Symposium on Combustion, Orleans, France, September 1975. ) 10. Garforth, A. M. and Rallis, C. J. The Spherical Bomb Method for Laminar Burning Velocity Determination: Analysis of Gas Movement during Flame Propagation, School of Mech. Eng., Univ. of the Witwatersrand, Research Report 64 (1975). (Also presented at The Fifth International Colloquim on Gas Dynamics o f Explosions and Reactive Systems. Bourges, France, September 1975. ) I1. Garforth, A. M. and Rallis, C. J., The Spherical Bomb Method for Laminar Burning Velocity De-
12.
13.
14.
15.
16.
17. 18.
19.
20.
21.
22.
23.
24,
25.
26.
termination: Equipment, Experimental Procedure and Data Handling, School of Mech. Eng., Univ. of the Witwatersrand, Research Report 65 (1976). McBride, B. J., Heimei, S., Ehlers, J. G. and Gordon, S., Thermodynamic Properties to 6000 K for 210 Substancies Involving the First 18 Elements, N A S A , SP 3001 (1963). Garforth, A. M. Theoretical and experimental developement of the constant volume method for determining laminar burning velocity, Ph.D. Thesis, Univ. of the Witwatersrand, Johannesburg (February 1975). Babkin, V. S., Kuznetsov, I. L. and Kozachenko, L. S., Influence of Flame Curvature on the Rate of Propagation of a Laminar Flame in a Lean Propaneair Mixture, Dokl. Akad. Nauk. SSSR 146(3), 625627 (1962). (English Translation: Prec. Acad. ScL USSR., Phys. Chem. Sect. 146. 677-679 (1962).) Andrews, G. E. and Bradley, D. The Burning Velocity of Methane-air Mixtures, Combust. Flame 19, 275-288 (1972). Dixon-Lewis, G. and Wilson, M. J. A Method for the Measurement of the Temperature Distribution in the Inner Cone of a Bunsen Flame, Trans. Far. Soc., 47, 1106-1114 (1951). Janisch, G. Chemi. Ing. Techn. 43,561 (1971). Lewis, B. and yon Elbe, G. Combustion, flames and explosions o f gases, 2nd Ed., Academic Press, New York, 1961. Bradley, D. and Mitcheson, A. Mathematical Solution for Explosions in Spherical Vessels, Combust. Flame 26, 201-217 (1976). Bledjian, L. Computation of Time-dependent Laminar Flame Structure, Combust. Flame 20. 5-17 (1973). Bradley, D. and Hundy, G. F., Burning Velocities of Methane-air Mixtures Using Hot-wire Anemometers in Closed Vessel Explosions, Thirteenth Symposium {International) on Combustion, Combustion Institute, Pitt sburgh, Pa., 575-581 (1971). Andrews, G. E. and Bradley, D. Determination of Burning Velocity by Double Ignition in a Closed Vessel, Combust. Flame 20, 77-89 (1973). Gunther, R. and Janisch, G. Measurements of Burning Velocity in a Flat Flame Front, Combust. Flame 19, 49-53 (1972). France, D. H. and Pritchard, R. Laminar Burning Velocity Measurements Using a Laser-doppler Anemometer, J. lnst. Fuel 79, 78-82 (1976). Babkin, V. S. and Kononenko, U. G. Equations for the Determination of Burning velocity in a Spherical Constant Volume Bomb, Fiz. Goren. Vzryva 3(2), 268-275 (1967). Gunther, R. and Janisch, G. Chemic. Ing. Tech. 43, 975 (1971).
68 27. Raezer, S. D. and Olsen, H. L., Measurement of Laminar Flame Speeds of Ethylene-air and Propaneair Mixtures by the Double Kernel Method, AppL Phys. Lab., Johns Hopkins Univ., CM-951 (June 1959). 28. Garforth, A. M., The Spherical Bomb Method for Laminar Burning Velocity Determination: Experi-
A.M. G A R F O R T H and C. ]. RALLIS mental Results and Critical Appraisal, School of Mech. Eng., Univ. of the Witwatersrand, Research Report 75, (January, 1977).
Received 10 January 1977; revised 22 June 1977
53
Laminar Burning Velocity of Stoichiometric Methane-Air" Pressure and Temperature Dependence A. M. GARFORTH and C. J. RALLIS School of Mechanical Engineering, University o f the Witwatersrand, Johannesburg, South Africa, 2001
Refinements to the spherical constant volume bomb method are reported which enable more accurate measurements of laminar burning velocity to be made than previously possible. Data is presented for stoichiometric methane-air mxitures in the ranges of unburnt gas pressures and temperatures of 0.06 ~< p < 0.228 MPa and 290 <~ T u <<.525 K respectively. These include an assessment of flame front thickness effects and a calculation of instantaneous spatially averaged burnt gas density. Over most of the range of values reported the error in burning velocity is less than 5%. A suggested simplification removes the need for much of the sophistication which has previously deterred many from using this method of measurement.
NOTATION A F K m M n p r S t T a t3 7 e p r q~
area defined as (1 - 7 ) correction factor mass molecular weight mass fraction absolute pressure radius velocity time absolute temperature defined (Pb/Po) defined as (Pu/Po) ratio of heat capacities defined as(pf/po) density flame front thickness equivalence ratio
Subscripts b burnt gas c center, correction e at the end of combustion
f o s t u ug w
flame front initial spatial transformation (or burning) u n b u r n t gas u n b u r n t gas at the bomb wall A bar over any symbol indicates an instantaneous spatially averaged value. The symbol ~ denotes nondimensional quantities. 1. INTRODUCTION At this stage in the history of combustion research, it is surprising that reliable data on the fundamental laminar burning velocity of many gaseous combustible mixtures is so scarce. Where data is available, usually at pressures and temperatures close to atmospheric, there still exists a considerable variation between published values [1]. There are, however, several reasons for the existence of this situation. Most of the methods for the measurement of burning velocity, although simpler in use, are Copyright © 1978 by The Combustion Institute Published by Elsevier North-Holland, Inc.
A. M. GARFORTH and C. J. RALLIS
54 severely range restricted and susceptible to experimental error. Of those that are not, the potentially most powerful, namely, the spherical constant volume bomb technique, requires considerable analytical and experimental expertise and has, therefore, not attracted as much effort as the simpler and quicker methods. Only Rallis [2, 3] and Babkin [4, 5, 6] and coworkers have recently devoted any significant attention to burning velocity determination using this method. A further reason for this situation is that the swing in research effort towards the chemical kinetic and theoretical studies in the last decade has left few workers to continue with the more experimental measurement approach. However, the need for this data still exists and the present study is an attempt t o provide a wide range of reliable data via a refinement of the constant volume bomb experimental method [7, 8, 9, 10,11].
2. REQUIREMENTS FOR SPHERICALBOMB INVESTIGATIONS The requirements for bomb studies, especially in the case of low burning velocity fuels with characteristically thick flame fronts, are both analytical and experimental, and include refined numerical processing techniques for handling the large volume of data generated.
Assumptions In the present study, the following assumptions used by previous bomb workers in the analysis of the process have been eliminated in that it is no longer necessary to consider that: (i)
(ii)
(iii) (iv)
the flame front has negligible thickness nor that the influence of its curvature is unimportant [9], the flame front remains spherical and centered at the point of ignition during propagation [2], the bomb and its contents constitute an externally adiabatic system [8], or that the burnt gas density immediately behand the flame front, Pr, is an accurate approximation to the instantaneous spa-
tially averaged gaspb [10].
density of the burnt
The following assumptions still apply: (0
(ii)
(iii)
(iv)
(v)
the mass and total volume of gas in the system remains constant (a statement of fact rather than an assumption), the flame front itself is an externally adiabatic system, in that it neither radiates heat nor allows the transfer bf heat from the burnt gas to the unburnt gas region (amenable to checking [8]), the instantaneous pressure variation through the flame front and the vessel is negligilbe (accurate for all but the fastest burning mixtures), the temperature immediately behind the flame front corresponds to the theoretical equilibrium adiabatic value, and no dissociation occurs in the unburnt gas region (a highly accurate assumption).
Of the remaining assumptions only item (iv) might introduce any significant error, which will remain until a method of measurement is devel.oped to check its magnitude. Analytical Requirements An analysis which requires only the assumptions mentioned above should thus be capable of producing equations for burning velocity calculation from experimental data which are applicable to the propagation of a thick, curved flame front throughout the bomb, in the presence of a density gradient behind the front in the moving burnt gas.
Combustion Data Requirements In order to apply the burning velocity equations derived under the above assumptions, it is necessary to calculate Po and Mu (= 31o) and, at each instant of recorded time during flame propagation, 7u, Tu, M r, 7r and T r. A routine to determine the adiabatic flame front values has been structured and is detailed in Refs [7, 10, 13]. The source of thermodynamic properties used in this case is that of McBride et al. [12].
BURNING VELOCITY OF METHANE-AIR
55
Experimental Necessities Using a fuel and oxidant of the highest possible purity, the burning velocity equations require the following experimental data: Po, To, fb, Ou, r t and t. The first three constitute the initial mixture specification while the remainder require transient recording. The flame front radius, rt, should preferably be obtained as a continuous record with time, while Pu may either be deduced from a transient pressure record, and the assumption of adiabatic unburnt gas behavior, or obtained by direct measurement. In Reference [8] it is shown, after direct measurement, that the adiabatic assumption, at least in the case of methaneair mixtures, provides adequately accurate data. It is also necessary to continuously measure flame front thickness, but, as pointed out in Reference [9] it is easier at present to correct the St values obtained using the thin flame assumption. The equipment used to carry out the above measurements is detailed in References [ 11, 13]. Data Reduction Scheme The equations for burning velocity [9] summarized in the following section are seen to require smoothed values of /3 = (Pu/Po), ~ = (-Pb/Po), (drf/dt) and (d/dt)(~/~). The routine which provides this facility is described in References [11,13].
3. EQUATIONS FOR THICK FLAME PROPAGATION As opposed to the equations previously published by the authors (3) the present set do not neglect the mass of gas contained within the flame front at any instant of time. Furthermore, neither the physical width of the front nor the mass of gas inside it are considered to be constant with time. The mass conservation equation is, thus: mo = m~ + m b + m r = const.
(1)
When this equation is rearranged and differentiated
with respect to time the following mass derivatives are obtained: (dmu/dt) = - ( d m b / d t ) -
(dmffdt).
(2)
This allows three forms of the burning velocity equation to be derived using the usual definition S t = - ( 1 / A p u ) ( d m u / d t ) = (1/moApu)(dn/dt),
(3)
where in this case a mass fraction 'burnt and burning' is defined as n = (mb + m~)/mo = 1 - (mu/mo).
(4)
Each of these masses may be expressed in terms of the instantaneous volume of the gas in the respective region of the vessel and the instantaneous spatially averaged density there. In the case of the unburnt gas there exists no significant density distribution so that Pu = Pu. In the burnt gas, however, a density distribution does exist, due both to the process of combustion there and the movement of gas as the flame propagates outward from the point of ignition. An analysis of this [10, 13] has enabled the instantaneous average burnt gas density to be calculated from the experimental observations of r r and p. This is further discussed in section 5 of this article. Using the above expressions for mass, three forms of the burning velocity equations can be derived, namely the unburnt gas equation, the burnt gas equation and the so-called combined equation [9, 13]. The unburnt gas equation has precisely the same form as that previously published [3]. Considering, then, that its form is independent of the flame thickness parameter, one might be led to the conclusion that it should be universally used. However, it is so very sensitive to experimental error as to be of little value. The combined equation, in this case, appears to offer little advantage over the burnt gas equation. It is also of a more complex form. The burnt gas equation appears to be the most suitable for the present work and, in its simplest form can be
56
A.M. GARFORTH and C. J. RALLIS averaged value of Pb is required at every time instant for which equations (8) and (8a) are used to calculate St. The determination of these spatial averages is detailed in Section 5.
written:
St =
( ~ / 3 ) [ F 2 + (~/a--')(1
-F2)] S~
+ (113){(rrF313)(daldt) + [rt(1 - F a ) / 3 ] ( d e l d t )
- (~ - ~)F2(d.r/dt)}
4. CORRECTIONS FOR FLAME THICKNESS
(5) where F = (1 - r) and 7 = (r/rr). For the purpose of comparison it is worthwhile to write down the 'thin flame' form of the burnt gas equation
St = (a/3)S, + (rr/3~)(d~/dt).
(6)
A consistency check shows that equation (5) reduces to equation (6) for infinitely thin flames (z = 0). Furthermore, an examination of the magnitude of each of the terms in equation (5) shows that if St is determined using equation (6), then accurate corrections, K c, to be applied to these values can be simply calculated from Kc
=
(1/3)~ - a-)(1 - F2)Ss.
(7)
Finally, it has been previously found [3] that the 'thin flame' combined equation yields the same results as equation [6] but has advantages from a data handling point of view. In the present study, therefore, this equation has been used to calculate St in the form
S t = UxSs + [rfl(1 - ~)/303 - ~)] (d/dtXa/3),
(8)
and these values were then corrected using equation (7). It is useful here to also write equation (8) in the less condensed form:
St = (Ob/po)Ss + [rrpu(1 - -fio)/3(p,, - Pb)] X (d/dt)(fib/pu).
(8a)
It is important to note that, whereas Pu = -Ou to a high degree of accuracy, the use of Ob = Of leads to errors of the order of 12% in burning velocity. Thus, even for 'thin flames' a spatially
Babkin et al (14) examined the effects of flame thickness and curvature on inwardly propagating spherical propane-air gas pockets. They concluded that these effects are only significant for flame front radii smaller than 3 mm. Andrews and Bradley [15] have analyzed the constant pressure flame with outward propagation and derive a correction to burning velocity which can be expressed in the form
Ke
=
(1/3)(e - o0(1 - Fa)Ss.
(9)
A comparison between equation (7) and (9) indicates the similarity of form. However, the fact that F a appears in their expression as opposed to F 2 in equation (7) results in too large a correction for burning velocity. For example their equation for a stoichiometric methane-air test with initial pressure and temperature of 0.101 MPa and 290.5 K gives a correction of + 145 mm/s at a flame radius of 13.7 mm, whereas at the same position for that test, equation (7) yields a value of + 97 mm/s. Furthermore, their correction drops below 10 mm/s only at a radius of 62.2 mm whereas the present correction is already less than that value at 52 mm. It is also worth noting that their equation is not written with ~, but simply a as no burnt gas density distribution exists with constant pressure flame propagation. The main difference between equation (7) and (9) arises because of a neglect of the term (d/dO (z/rt) in the analysis for equation (9). The present study has indicated that, even if r is constant, the variation of the ratio of flame thickness to outer flame front radius is an important factor. It is thus concluded that the expression of Andrews and Bradley results in corrected burning velocity values from bomb tests which are too large. Furthermore, a consideration of the variation of flame thickness with pressure and temperature leads, as is shown later, to the conclusion
BURNING VELOCITY OF METHANE-AIR
57 TABLE l
The Extent of Flame Thickness Influence a Radius (rf-mm) at which the correction to S t is: Initial test Pressure Initial test temperature (Po-MPa) (To-K)
+10%
+5%
+1%
0.0608
290.3
47
56
71
0.101
290.5
36
47
63
0.228
290.5
16
28
56
a Vessel diameter-160.2 mm.
that even the correction calculated from equation (7) may be too large at values o f flame radius less than about 50 mm in our vessel for a 1 atmosphere initial pressure test. The numerical corrections applied to the present results must be considered as tentative (see Section 6). The best documented values of r [16, 17] for a stoichiometric methane-air flame at 1 atmosphere are respectively 0.75 and 1.1 mm. Andrews and Bradley (15) consider Janisch's value of 1.1 mm to be the more reliable. However, in order to calculate values of ~, the form of the temperature profile through the flame front is also needed, and in this respect Dixon-Lewis and Wilson's result was the more useful. Furthermore, no flame thicknesses or temperature profiles have been reported at higher pressures. The process whereby the present values of r and ~ are determined for bomb tests (9), described in Section 6 of this article, is therefore open to criticism. Although the corrected St values presented here are thus possibly in error, one can nevertheless gauge the region of bomb tests over which thickness effects are important. This is sulnmarized in Table I above for stoichiometric methane-air tests. The table clearly shows that the region rf < 30 mm is, in general, one in which it is not possible to place great reliance on the values of burning velocity. It is noteworthy that the constant pressure region only extends approximately as far as 10 mm from the ignition p o i n t - a n d this is the region from which many bomb results have previously been determined. The corrections applied to the present data are illustrated in Figs. 1 and 2. Figure 1 shows the
average 1 atmosphere initial pressure test curve derived from three test runs in the bomb. Also shown are some of the corrected test values obtained using equation (7). It is clear from this diagram that the corrected points below a pressure of 0.15 MPa must be viewed with suspicion. Neither the pressure nor the temperature of the unburnt gas have changed by large amounts between ignition conditions and the 0.15 MPa 0.5
I REF. 21
I
I
®
value)
!
0.45 REF 21
I
value)
8
_
REf'. 2 3 ~ RE~ 24 ~"I,
OA
--
f
0.3~ REF. 5 (Corrected J accordlng~ / to this ~+/ study) /
02
Q2E
--
---
THIS STUDY-UNCORRECTEO
~o
THIS STUDY-CORRECTED USING EQN (7)
REF. 5 ._.__.~ + (Uncorrected)
1 0.1
THIS SrUDY- MOST LIKELY CORRECT RESULTS
p (MPa)
I 02
l 0.3
Fig. 1. Correction of bomb test curves in the region of
the initial pressure and temperature.
58
A.M. GARFORTH and C. J. RALLIS
0.60 , ~
UNCORRECTED TEST CURVE
0.55
CORRECTED
0.5t3
UNCORRECTED CORRECTED
,
TEST CURVE
CORRECT TEST CURVE
%
tg
0.45
,
i./
}
ISOTHERMS
f
0.40 0.35
j "
!
0.30
0.25
0.5
10 p (MPa)
1.5
20
Fig. 2. Illustration of flame front thickness corrections for stoichiometric methaneair tests.
point on the curve, so that the burning velocity must correspondingly have changed very little during the same period. It is therefore suggested that the corrections for flame front thickness over this range are indicative more of uncertainty bounds than of actual alterations which should be made to burning velocity. Certainly, at pressures above 0.15 MPa in this case, the corrected curve is a smooth one and follows the general trend of the uncorrected curve. In order to obtain corrected burning velocity values below 0.15 MPa therefore, the corrected curve has been extrapolated back to Po while maintaining the trends evident from the uncorrected curve in this region. This results in a value of St at 1 atmosphere and 290.5 K of 0.340 m/s as compared with the uncorrected value of 0.316 m/s. Examing the above extrapolation process further, it is seen that a value of burning velocity at Po and To higher than 0.35 m/s cannot be justified. The fact that this is low in comparison with currently accepted values for stoichiometric methane-air, some of which are included on Fig. 1, is discussed at a later stage. However, when deciding upon the form of the extrapolated curve used above, not only the test curve trends were taken into account, but also the shape of the isotherms as seen on Fig. 2. Here, only the two lowest isotherms are shown, together with test
curves at initial pressures of 0.0608, 0.101 and 0.228 MPa. 5. AVERAGE BURNT GAS DENSITY The details of the process whereby the instantaneous spatially averaged values of burnt gas density were obtained are presented in References [10, 11, 13]. In essence, the procedure is as follows: (/)
(ii)
(iii)
(iv)
The flame front radius and bomb pressure at approximately 100 evenly spaced time intervals between ignition and the end of flame travel are obtainted from the photographic test records. Using the thermodynamic relationships for the adiabatic compression of the unburnt gas (checked for every test with the interferometer method for measuring Pu), the unburnt gas temperature, Tu, is calculated for every recorded time mentioned above. The computer program detailed in Reference [7] is then used to calculate equilibrium flame front, burnt gas temperatures~ Tf, at the same instants of time. The technique presented in Reference (10) allows the position of each burnt gas
BURNING VELOCITY OF METHANE-AIR
(v)
(vi)
59
element to be radially located in the vessel for every flame front position. This, together with the assumptions that the burnt gas undergoes adiabatic compression and that 7~, remains constant with time for each burnt gas element, allows the instantaneous temperature distributions in the burnt gas, shown in Fig. 3, to be obtained. These assumptions are discussed in References [11, 13] and are judged to introduce almost negligible error. The above temperature distributions coupled with mixed mean burnt gas properties results in the required burnt gas density distributions. Finally, the density distributions are averaged on a mass basis to produce the instantaneous values of Pb-
Lewis and yon Elbe [18] calculated the burnt gas temperature distributions for hydrogen-oxygen and ozone-oxygen flames. In both cases the average temperature when the flame had reached the vessel wall, The, could be related to the center temperature, To, and the gas temperature adjacent to the wall, Tw, by (Tb e - Tw)/(Tc - Tw) ~ 0.36.
(10)
Recognising that they ignored gas movement and used an approximate expression for mass fraction burnt, n, their result is in surprisingly good agreement with the corresponding value of 0.29 for the present 0.1 MPa stoichiometric methane-air result. It seems that their two assumptions tended to cancel one another out, because the value without gas movement included for the methane-air case should be 0.54. In a very recent attempt to predict pressure rise ratio in closed vessels, Bradley and Mitcheson report calculated gas movement results for stoichiometric methane-air [19]. A comparison between their predictions and the combined analytical and experimental results of the present study is presented in Fig. 4. It is interesting that their predictions are for a vessel of diameter 160.2 m m - t h e precise size used in the present study.
3500
~'~'~
~.~
~[.~
250C Flame front ~ positions
~J
i
'lKTf
I
I I
i
I
I I I I
I I I
I I I
150C - - - -
-;II I I
50C
I
I I I I
I I
~T u
20
40
0
80
Radius-mm
Fig. 3. Typical instantaneous temperature profiles during flame propagation in the bomb. Some experimental particle tracks were accidentally obtained with a few of our tests [10, 13] and one of the clearest of these is shown and is seen to largely confirm the analysis of our study. A disturbing feature of Bradley and Mitcheson's computer prediction is the delay period of approximately 7 ms at the start of their flame front vs time curve. This materially affects the position of both their unburnt gas and burnt gas particle tracks (only one set of which is shown in the figure), and it makes comparison with the present ones difficult. This delay is certainly not realBledjian [20] shows that the delay is in the region of 30 tzs. One can only conclude that their delay is introduced by the computer programme. To effect a better comparison with the present results, their prediction in the region 7 to 50 ms has been expanded time-wise to correspond to the total flame propagation time of 50 ms found in this study. Although this produces better correspondence between the flame front positions, the unburnt gas track is definitely now in error. The general prediction of their burnt gas movement is in fair agreement with our experimental one, but the discrepancy becomes larger with increasing flame radius. No further comparison is worthwhile at present.
60
A.M. GARFORTH and C. J. RALLIS
7o
_'-_
.....
60
. . . . . . . . . . . . . , . . . . . . . . . c,, ,,,c,
/~""/ f~"j./
10' Z
/ .~
__.~
0
, . . . . . o.,
5
t
,
10
15
t
i
I
20 25 30 Time (ms)
i
i
35
i
40
45
50
Fig. 4. Comparisonof gaseous particle track predictions. The effect of the inclusion of both the temperature distribution and burnt gas movement in the S t equation on the values obtained is clearly shown in Fig. 5 for a single bomb test run. The neglect of either of these effects results in an error as large as 12% in burning velocity.
6. EXPERIMENTAL RESULTS In Table 2 the initial data for all the stoichiometric methane-air tests reported here are presented, together with some of the main test results. It should firstly be noted that although no direct control was exerted over the initial mixture temperature, T o never varied by more than -+1.6 K from the mean value of 290.5 K. The accuracy with which the initial pressure was set was better than -+0.1% while the mixture strength, ¢, was capable of control within -+0.1%. Some problems were encountered with initial mixture stratification but a controlled mixture preparation procedure and the provision of a stirrer reduced this to very acceptable levels. From the point of view of test reproducibility it is seen that the maximum pressures for individual tests never varied by more than +-1.3% from the mean for that series. Similarly, the deviations in t e and ae were never more than -+2.2% and +1.3%, respectively. The value of ae was usually
obtained by extrapolation [11] because it was not possible to transcribe the photographic flame trace fight up to the stage where the flame reached the wall; internal reflections in the bomb tended to obscure the last millisecond or so of the record of flame travel. The value of ae is considered to be a very good indication of test reliability during the later stages of combustion in the bomb. Since, in the constant mass and constant volume bomb system, ~e must equal 1.0 exactly, its departure from this value provides assessment of overall test inaccuracy. In
o55
r
F
0.50
~
~
i
r ..~ "~
I
"~ 0 . 4 5
.~'
%
"~ 0.40--
i~,"
"
"'-
,
~
~!
.
~
I
~ ~ '
/
035
/~'[
~
i
0.30
----
........
!
j NOI% AVERAGE OR BURNT GAS MOVEMENT
w,,.MOV,M~N,~,u, NO
~
WITH pbAND
°26o
i
o2
i
,
L
.ov,.,NT
0;6
'
o'a
p (MPa)
Fig. 5. Effects o f burnt gas movement and density averaging o n S t f o r a 0.101 MPa test.
BURNING VELOCITY OF METHANE-AIR
61 TABLE 2
Summarized Test Data
To
te
Pe
MPa
Po Test
(K)
ms
(MPa)
ae
~e a
(mm)
ro
0.0608
A B
290.3 289.6
47.6 47.1
0.540 0.546
1.073 1.082
0.953 0.961
2.073 2.068
0.0810
A B C
290.4 290.8 289.7
49.3 48.5 50.6
0.712 0.721 0.727
1.058 1.074 1.078
0.941 0.958 0.962
1.426 1.428 1.423
0.1013
A B C
290.4 290.5 292.1
52.2 53.3 53.5
0.882 0.905 0.897
1.049 1.074 1.071
0.933 0.956 0.953
1.065 1.065 1.071
0.1265
A B C
290.6 290.7 289.2
56.0 55.8 55.7
1.119 1.093 1.098
1.062 1.040 1.038
0.949 0.925 0.926
0.798 0.798 0.794
0.1519
A B
290.0 290.6
58.2 59.1
1.352 1.352
1.066 1.068
0.948 0.951
0.627 0.628
0.2025
A B
290.6 291.6
62.9 63.2
1.801 1.791
1.064 1.062
0.949 0.948
0.432 0.433
0.2279
A B C
291.2 290.2 290.5
67.6 65.9 65.1
2.043 2.024 2.042
1.074 1.061 1.071
0.957 0.950 0.956
0.371 0.369 0.370
a ave ae = 0.950.
Table 2 it is seen that, with the exception of four tests, ~e is usually close to 0.950 (its average value for this test series). It is thus possible, when reviewing the test results, to give greater credence to those tests with values of ~e closest to 1.0; tests with particularly low values are suspect. This selfcorroboration characteristic is an important advantage of the bomb method which has not always been appreciated by other users. It is used to good effect below when comparing St values from this study with those of other workers. The values of flame front thickness, T, and average flame front density, ~, for the present tests were determined as follows: (i)
using the data mentioned in Section 4, it was assumed that the temperature distributions through the curved flame fronts would be of the same form as that
(ii)
(t~i)
through Dixon-Lewis and Wilson"s plane flame [16], provided that at p = 1 atm and T = 300 K the value of T corresponded to the 1.1 mrn value of Janisch [ 17], it was assumed that r ~ 1/PuSt was an accurate enough representation of the variation of thickness for methane-air at other conditions, an approximate relationship for St in terms of p and Tu from the present 'thin flame' results was determined and, on substitution gave
~"= 1.848 X lO-7(To/Po°'265)p - 1 " ° 4 .
(1 1)
Values of ro at To and Po are reported in Table 2 for comparison purposes. The above procedure highlights the fact that,
62
A.M. GARFORTH and C. J. RALLIS 0.6 056
I_@°I
0.,6
046 " t
0.4 O.,3f.
02
N
L
4 t
0~2
2.0 (MPa) Fig. 6. Burning velocity pressure dependence for stoichiometric methane-air. strictly speaking, the process of St correction for flame front thickness effects should be an iterative one. However, with the present uncertainty in r values, this does not provide any reliable improvement over the currently reported data. For all tests reported here the spark gap was kept to a minimum, as was the ignition energy, consistent with reliable and reproducible ignition. Typical of the settings was that for the 0.101 MPa series for which the gap was 2.05 mm and the energy 6 mJ. In Figs. 6, 7 and 8, the present stoichiometric methane-air burning velocity results are presented
in the ranges 0.06 ~< p ~< 2.04 MPa and 290 ~< Tu <~ 525 K. In Fig. 6 the dependence of St on pressure can be seen to be initially very large at low unburnt gas temperatures while this dependence decreases considerably at higher pressures and temperatures. A comparison with our previously reported St data for stoichiometric acetylene-air [3] shows the methane pressure dependence to have a far less complex nature. However, in both cases the pressure dependence decreases to almost zero at higher pressures and temperatures. The nature of the methane temperature dependence is also simpler than that of acetylene and Fig. 7
06
P
0.5
mO~
'= ~' / , / o ? ' r"
03
021 3C
T.(K)
Fig. 7. Burning velocity temperature dependence for stoichiometric methane-air.
BURNING VELOCITY OF METHANE-AIR
63
!
,
1
_,= / o~
Fig. 8. Three-dimensional representation of burning velocity dependence on pressure and temperature for stoiehiometric methane-air. shows that this dependence varies little over the full range of pressures reported here. The threedimensional representation of Fig. 8 illustrates the combination of both pressure and unburnt gas temperature effects on the burning velocity of stoichiometric methane-air mixtures. An important observation is made on the basis of the present test results from the information in Fig. 9. Here the variation of 6 with pressure ratio, (P/Po), calculated for a stoichiometric methane-air test with Po = 0.101 MPa, including both density distribution and gas movement effects is plotted. It is seen to be slightly curved when compared with the straight line fitted between so at (P/Po) = 1 and ~ = 1 at (P/Po) = (Pe/Po). If the curvature of the actual curve is retained while ~e is adjusted from 0.956 to 1.0 it is seen that this modified curve never differs from the straight line by more than 2.8%. The importance of this conclusion lies in the fact that, provided 0~o is known, subsequent values of ~ can be obtained from the equation
nor of burnt gas movement calculations. Equation (12) will yield ~ values well within the experimental accuracy of the method. 7. COMPARISON WITH LITERATURE VALUES A feature of concern in the present burning velocity study for stoichiometric methane-air is that the value at p = 1 atmosphere and Tu = 290.5 K is 1.0
k
0.9 O,8 0.7
0~6 0,5 0.4 i
= % + (1 - ~ o ) [ ( P / P o )
-
11/[6oe/po)
-
02
l].
(12) It is therefore not necessary for prospective bomb users to enter into the complexity o f calculating the flame front adiabatic temperatures, nor of density distribution determination in the burnt gas
0.1
o
1
(~0o) Fig. 9. Comparison between accurate and simplified methods for ~ determination.
64
A.M. GARFORTH and C. J. RALLIS
p (MPa)
Fig. 10. Comparison with literature data. significantly lower than the commonly accepted value. As can be seen on Fig. 1, the data of Bradley and Hundy using hot wire anemometry in closed vessel explosions [21 ] ; that of Andrews and Bradley from'their double kernel, closed vessel experiments [22] ; the measurements of Gunther and Janisch on button flames above a Mache-Hebra burner [23]; and the laser-doppler anemometer data from the nozzle burner studies of France and Pritchard [24], all point to a value of St at 1 atmosphere and 293 K in excess of 0.4 m/s. The value from the present study at Tu = 293 K is in the region of 0.34 m/s. As mentioned in Section 4, the highest value of St under these conditions which could be justified on the basis of the test curves of Fig. 1 is 0.35 m/s. However, one major assumption remaining with the bomb method is that the burnt gas immediately behind the flame front is at the adiabatic equilibrium flame temperature. There is enough evidence in the literature to indicate that the actual temperature is lower than this, a reasonable estimate of the difference being about 5%. If this were taken into account, values of ~ = (-Pb/Po) would correspondingly be 5% higher throughout the flame propagation process in the bomb. This appears to agree with the current values of ~e which are generally 5% lower than the required value of 1.0. The value of St at the initial pressure and temperature could therefore be as high as 0.37 m/s. Whereas our present experience confirms that the constant pressure region in the bomb is an
unreliable one from the point of view of burning velocity determination, the experimental error is not greater than -+5% in the range 1.1 ~< (P/Po) <~ 6.0 for bomb tests. On Fig. 1 this corresponds to pressures above 0.11 MPa, while on Fig. 6 this is the range of values between the isotherms 300 K and 475 K. Certainly, at a value of pressure of 0.2 MPa on Fig. 1, the value of St of 0.37 m/s could be low only due to the 5% error mentioned above, since the flame front thickness correction is small by this stage, yielding a maximum value of 0.39 m/s here. Yet, although the burning velocity definitely increases during flame propagation in the bomb, this value is still below that of 0.4 m/s indicated by those authors using alternative, but range limited, methods at the initial conditions. It would thus require a considerable, and unjustifiable, distortion of the present test curves to bring about closer agreement with these other results. When considering the corresponding values for St obtained by Babkin et al. [4, 5, 25] on Figs. 1 and 10, they are seen to be even lower than those of the present study. Whereas the corrections applied to such results for flame thickness by Andrews and Bradley [15] would increase these values to yield better correspondence with those of the previously mentioned accepted studies, the present article shows that these corrections would be far too high. For example, the value at 1 atmosphere and 293 K applying Andrews and Bradley's correction factor of 1.22 would be 0.366 m/s, whereas the corrected value applying the techniques of this study is only 0.323 m/s. Recognising
BURNING VELOCITY OF METHANE-AIR that Babkin's results were originally calculated using simplified bomb equations, the discrepancy between his results and ours can be explained at the early stages of combustion. It thus seems that bomb studies yield values which are low in comparison with those obtained by other methods, even when they are corrected as judiciously as is presently possible. A reconsideration of some of the errors involved in the values from other methods might thus prove useful in accounting for this discrepancy. The recently published values of France and Pritchard [24] result from a sound experimental method and appear to be reliable. However, it seems that these authors did not measure the unburnt gas temperature immediately ahead of their conical flame. Some heating of the gas as it passed through the burner body near the port cannot therefore be ruled out. The corresponding rise in temperature is likely to have been small but would nevertheless have produced a value of St which was too high. A discrepancy of +13.5 °C would lower their value of 0.403 m/s to the present maximum value of 0.37 m/s. Such an increase of temperature through the burner does not seem unreasonable. Gunther and Janisch's value [23] of 0.41 m/s is close to that of France and Pritchard and was also measured under reliable conditions with considerable attention to experimental technique. These authors did measure the temperature of the unburnt gas in the burner port using a thermocouple. However, it appears that the thermocouple sensing element was unshielded and may therefore have radiated to the burner rim which was kept at 20 °C by circulating cooling water around it. In this case, however, an error of 13.5 °C in the temperature of the unburnt gas seems large. Unfortunately, these authors do not quote the size of the burner port and their previous studies on a 4 mm diameter burner [26] led Andrews and Bradley to conclude that the resultant burning velocity values would be too high [22]. The same might apply to their present studies if the burner diameter is not significantly larger than the flame thickness. An interesting facet of both france and Pritchard's and Gunther and Janisch's studies is that their burning velocity values at 1 atmosphere and 293 K are only just above 0.4 m/s, and are lower
65 than those of Bradley and Hundy [21] and Andrews and Bradley [22]. In the hot-wire studies, the scatter between burning velocity values is large as indicated on both Figs. 1 and 10. Furthermore, the sensitivity of the method to errors in the measurement of Sug produces a four times magnification of experimental error in the case of methaneair mixtures. Although the stoichiometric methane flame is non-sooting, some small radiation must have occurred to the hot-wire anemometer. In particular, the method requires the hot-wire to sense gas velocity immediately ahead of the high temperature flame, so that it is conceivable that an error of the order of 1% or even 2% in Sug measurement could have been caused by radiation effects, This could have produced St values which were 4% to 8% high, resulting in burning velocity in the range 0.39 to 0.44 m/s rather than the reported range of 0.42 to 0.48 m/s. In the case of the double kernel technique, we feel that until the kernels are simultaneously viewed from two mutually perpendicular directions, it will be impossible to rule out oblique flame propagation in the region of interest. This fact, if not established in the currently reported tests [22], would also yield high values of S t . In addition, the typical photographic records of Andrews and Bradley are evidently not of the same calibre as those of the pulsed scl'dieren method of Raezer and Olsen [27]. The accuracy of the burning velocity results obtained in the more recent study is therefore suspect. One distinct advantage of both of the above hot-wire and double kernel experiments is that they were performed in closed vessels, and could thus yield S t data over a greater pressure range than with burner studies. Unfortunately, the results from the double kernel tests were only reported for 10% CH4 mixtures. They are not, therefore, directly comparable with those of the present stoichiometric studies. Nevertheless the trend of the data is interesting and has been included in Fig. 10. Babkin's results as shown in Fig. 10 have not been corrected for flame front thickness and, are thus likely to be low at all stages. In addition, all his data points were obtained using the constant pressure combustion region of the vessel only, in which the bomb method is definitely inaccurate.
66 There also exists some doubt as to the precise method whereby the results were calculated from the bomb measurements. It is possible that the St values are for this reason a further 8% low [11, 13, 28]. Finally, it is not possible to use the selfchecking ability of the bomb method mentioned in Section 6 above when only recording in the constant-pressure region. Nevertheless, there is some quantitative agreement between their results and ours as is seen from a comparison of their 423 K curve and our 425 K line. At the higher pressures, however, the two sets of results diverge. At lower pressures their results are understandably low because of the importance of flame front thickness there. Both the hot-wire and the double kernel results of Bradley and his co-workers are in reasonable agreement with those of the present study in the region of 0.15 to 0.3 MPa. In fact the trend of the double kernel results is in closer agreement than the hot-wire one. It is unlikely that the small difference in mixture composition between 10% CH4-air mixtures and stoichiometric ones could account for the significant difference in trend between these two sets of results by the same team of workers. If it is also realized that the richer mixture values of S t are expected to be between 5% and 10% higher than the stoichiometric ones, the correspondence with our results is even closer. At this stage it is not possible to quantitatively account for the differences between the values of burning velocity from this study and those of other workers. Whereas it appears that results from bomb studies are generally lower than those from other methods, the discrepancies are only significant at the lower pressure and temperature states. The fact that the range of states at which St values can be obtained in closed vessels is much larger than in the case of most other methods, allows an examination of trends which is not otherwise possible. These values at the lower pressures and temperatures can therefore be more critically assessed than when only single data points are available. Furthermore, it appears that the burning velocity values recently reported for stoichiometric methane-air at 1 atmosphere and ambient temperatures may have been too high for a variety of reasons. It
A.M. GARFORTH and C. J. RALLIS would thus seem that an acceptable value for St at this condition lies in the range 0.37 to 0.4 m/s. 8. CONCLUSIONS The spherical constant volume bomb method has been used to provide laminar burning velocity data for stoichiometric methane-air mixtures over a wide range of pressures and temperatures [28]. In this study, the previously neglected effects of flame front thickness, burnt gas density distri. bution and burnt gas movement have all been in. cluded in an attempt to provide data which is as accurate as possible at the present time. It is felt that future studies will only improve upon the burning velocity results at the extremes of the pressure range reported here, where experimental inaccuracies may have been significant in the present series of tests. A simplification, applicable at least to methaneair tests in the bomb and probably to most other mixtures, has been suggested. It was only possible to make this suggestion after the effects of density distribution and gas movement had been considered. The simplification eliminates the need for such sophistication and does not introduce an error greater than 3% in burning velocity. The pressure and temperature dependence of burning velocity reported here indicates a complexity not evident in the literature values. Simple empirical fits to the data are thus not possible and, for this reason, none have been suggested. However, the trends are not as complex as those previously reported for stoichiometric acetylene-air mixtures. An error analysis of the present results indicates [13, 28] that the experimental error is not likely to be more than 5% in the range 1.1 ~< (P/Po) <~ 6.0 for each test while outside this region, the error does not exceed 9%. The values of burning velocity reported here are likely to be about 5% low due to the assumption that the flame temperature is equal to the instantaneous adiabatic equilibrium temperature.
This work was partially sponsored by the South African Council for Scientific and Industrial Research and the authors wish to express their appreciation to this body for their long-term support.
BURNING VELOCITY OF METHANE-AIR
67
REFERENCES 1. Andrews, G. E. and Bradley, D., Determination of Burning Velocities: A Critical Review, Combust. Flame 18, 133-153 (1972). 2. RaUis, C. J., A Critical Evaluation of the Spherical Constant Volume Method for Determining Laminar Burning Velocity, Ph.D. Thesis, Univ. of the Witwatersrand, Johannesburg (August 1963). 3. Rallis, C. J., Garforth, A. M. and Steinz, J. A., Laminar Burning Velocity of Acetylene-air Mixtures by the Constant Volume Method: Dependence on Mixture Composition, Pressure and Temperature, Combust. Flame 9,345-356 (1965), 4. Babkin, V. S., Kozaehenko, L. S. and Kuznetsov, I. L., The Influence of Pressure on the Burning Velocity of Methane-air Mixtures, Zh. Prikl. Mekhan. Tekn. Fiz. 145-149 (1964). 5. Babkin, V. S. and Kozachenko, L. S., Study of Normal Burning Velocity in Methane-air Mixtures at High Pressures, Fiz. Goren. Vzryva. 2(3), 77-86 (1966). (English Translation: Combust., Explosion and Shock Waves 2(2), 46-52 (1966).) 6. Babkin, V. S., V'yun, A. V. and Kozachenko, L. S., The Effect of Pressure on Normal Flame Velocity Investigated by the Initial-section Method in a Constant-Volume Vessel, Fiz. Goren. Vzryva. 2(2), 5260 (1966). (English translation: Combust. Explosion and Shock Waves 2(2), 32-37 (1966).) 7. Garforth, A. M., A Computer Programme for the Calculation of Equilibrium Composition and Adiabatic Flame Temperatures of Gaseous Mixtures Undergoing Constant Pressure Combustion, School of Mech. Eng., Univ. of the Witwatersrand, Johannesburg, Research Report 56 (1973). 8. Garforth, A. M., Unburnt Gas Density Measurement in a Spherical Combustion Bomb by Infinite-fringe Laser Interferometry, Combust. Flame 26, 343-352 (1976). 9. Garforth, A. M. and Rallis, C. J., The Sphericial Bomb Method for Laminar Bruning Velocity Determination: Equations for Thick Flame Propagation, School of Mech. Eng., Univ. of the Witwatersrand, Research Report. 59 (1975). (Also presented at The Second European Symposium on Combustion, Orleans, France, September 1975. ) 10. Garforth, A. M. and Rallis, C. J. The Spherical Bomb Method for Laminar Burning Velocity Determination: Analysis of Gas Movement during Flame Propagation, School of Mech. Eng., Univ. of the Witwatersrand, Research Report 64 (1975). (Also presented at The Fifth International Colloquim on Gas Dynamics o f Explosions and Reactive Systems. Bourges, France, September 1975. ) I1. Garforth, A. M. and Rallis, C. J., The Spherical Bomb Method for Laminar Burning Velocity De-
12.
13.
14.
15.
16.
17. 18.
19.
20.
21.
22.
23.
24,
25.
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Received 10 January 1977; revised 22 June 1977