A precise test of the flame-stretch theory of blow-off

A PRECISE TEST OF THE FLAME-STRETCH THEORY OF BLOW-OFF H. EDMONDSON AND M. P. HEAP

University of Salford, Lancashire, England The postulate that blow-off is caused by excessive flame stretch in the stabilization zone has been tested for methane-air flames inhibited with methyl bromide. This involved the determination of a series of burning velocity vs composition curves corresponding to constant methyl bromide/methane volume ratios of 0, 0.01, 0.02, and 0.03. The maximum burning velocity for methane-air flames was found to be 38.3 cm/sec at 10.25~ methane in air. The addition of methyl bromide reduced the maximum burning velocity to 35.1, 32.7, and 30.7 cm/sec for ratios of 0.01, 0.02, and 0.03, respectively, with a shift in the mixture composition at the maximum towards stoichiometric. Blow-off flow rates of inhibited methane-air flames were determined under laminar flow conditions on long cylindrical burners ranging from 0.62 to 1.30 cm diam. For any particular methane-air mixture, blow-off was fom~d to occur at a fixed value of the Karlovitz flame-stretch factor, irrespective of the inhibitor content of the mixture. It is concluded that blow-off is almost certainly caused by excessive flame stretch in the stabilization zone. The critical value of the flame-stretch factor at which blow-off occurs, appears to increase rapidly as the methane content of rich primary mixtures is increased. It is suggested that this apparent increase is due to the error involved in the assumption that the burning velocity in the flame-stabilization zone is the normal burning velocity of the primary mixture. The burning velocity in the stabilization zone at the base of the flame will be significantly affected by (a) intermixing of the primary mixture with the surrounding atmosphere prior to combustion, and (b) heat transfer to the primary flame front from the outer diffusion flame.

Introduction This paper is concerned with the elucidation of the mechanism of flame blow-off. Reed has recently proposed 1,2 that the blow-off of aerated burner flames is the result of excessive flame stretch in the stabilization region. Reed's theory was developed from experimental data of limited accuracy and a precise test of its validity is, therefore, necessary. Such a precise test has been carried out using accurate burning velocity and stability data obtained for this purpose. The results of this investigation provide accurate experimental support for Reed's theory of blow-off. The fundamental principles of flame stabilization may be stated concisely as follows: If, at a point in a combustible gas mixture, the flow velocity is equal to the burning velocity, a stationary element of flame can exist that will act as a source of ignition for the rest of the stream. Provided the flow velocity exceeds the burning velocity at all other points, a stationary flame will exist with the flame front sloping downstream from the point of ignition. If, at any point within the gas stream, the burning velocity

exceeds the flow velocity, the flame will move upstream (el. lightback). If, at all points in tile stream, the flow veloeity exceeds the burning velocity, the flame will move downstream (cf. blow-off and lift). The essential requirements for flame stabilization given above are, in practice, achieved by interaction between the flame and a solid surface, commonly the burner rim. According to the theory developed by Lewis and yon Elbe, a'4 blow-off results from an imbalance between two factors affecting the burning velocity of the primary mixture. As the mixture flow rate is increased, the flame base is considered to move away from the burner rim so that the local burning velocity tends to increase because of the reduced quenching effect. However, interdiffusion with the surrounding atmosphere also increases, and the burning velocity tends to be reduced by dilution. Lewis and yon Elbe postulate a critical flow rate at which these two effects are balanced, and above which the flow velocity will, everywhere, exceed the burning velocity, thus resalting in blow-off. Experimental support for the theory derives mainly from the good correlation of blow-off

1007

1008

FLAME STUDIES

data that is obtained for a 1)articular gas on the basis of the boundary-velocity gradient at blow-off, gb. This satisfactory correlation seems to have discouraged detailed examination of the validity of Lewis and yon Elbe's theory of blow-off. However, Reed 2 has noted that the physical picture of blow-off presented in the theory is not supported by detailed observation of the behavior of flames at blow-off. A fln'ther important criticism is that the theory fails to predict a quantitative relationship between burning velocity and blow-off flow rate: such a relationship might reasonably be expected to be inherent in any satisfactory theory of blow-off, and the relationship would be of great practical value. The mechanism of flame stabilization proposed by Lewis and von Elbe takes no account of the effects of shear flow in the stabilization region. Flow-velocity gradients will cause flame propagation to be divergent, and under such conditions the flame front is said to be stretched by the shear flow along it. The degree of flame stretch can be shown to be closely related to the dimensionless Karlovitz flame-stretch factor K = (~o/U) (d~'/dv),

(1)

where U is the velocity of the unburned gas, dU/dy the velocity gradient, and ~0 a measure of flame-front thickness given by

, 0 = k/pc~&,

(2)

in which k is the coefficient of thermal conductivity, Cp the specific heat, p the density, and &, the burning velocity. Karlovitz ~ has shown that flame temperature and burning velocity are somewhat reduced in divergent flame propagation, and that there exists a critical divergence or flame stretch above which it becomes impossible for a flame to propagate. Reed contends that the effect of shear flow in the stabilizing region cannot be ignored. The combination of low velocities and relatively high velocity gradients pertaining at the blow-off conditions are precisely those shown by Karlovitz to result in the extinction of a combustion wave. Reed, therefore, considers that blow-off "results from a catastrophic reduction in the reaction rate engendered by the enthalpy loss from the stabilizing region arising from the shear flow." In order to test this hypothesis, Reed calculated values of K at blow-off for a variety of aerated flames--K being interpreted for this 1)urpose as K = g~,d&.

Using stability data originally t)resented as evidence in support of the Lewis and yon Elbe theory, Reed plotted graphs of K vs equivalence ratio and fomnt that most of the points were reasonably close to one line. Although this graph provides an indication that the flame-stretch factor will correlate stability data, definite conclusions are inadvisable because of the high degree of scatter about the line. The scatter that Reed obtained may be due either to the limited accuracy of the experimental data or to some real effect. A precise test of Reed's theory was, therefore, considered to be necessalT. By careful choice of the experimental system, it was possible not only to test Reed's theory but also to obtain direct experimental evidence of the relationship between blow-off flow rate and burning velocity. The ideal requirement for this investigation was considered to be a gaseous system whose burning velocity could be controlled independently of its physical properties. This ideal requirement could only be approached practically by the use of inhibitors; consequently, blow-off flow rates and burning velocities of methane-air flames inhibited by methy[ bromide have been nmasured.

(3)

Experimental Complete details of the experimental techniques are available elsewhere, 6 and only a concise statement of the essential features of the experimental work is given here.

Flow System Air and methane were supplied from highpressure cylinders and, after drying and removal of carbon dioxide impurity from the inethane, both gases were controlled to a constant temperature and pressure prior to flow metering. This was achieved by measurement of the pressure drop across calibrated orifices. Flow regulators in each stream then permitted independent control of gas and air flow rates before mixing. Methyl bromide was stored as a liquid under pressure and could be obtained in gaseous form by "flashing off" from the storage cylinder. The flow of methyl bromide was measured at constant temperature and pressure by a calibrated rotameter fitted with a stainless-steel float. A flow regulator situated downstream of the rotameter was used to control the flow rate. The metered flow of methyl bromide was then mixed with the metered flow of methane prior to mixing with the air flow. This flow system was used to supply the combustible mixture fox' both burning-velocity and blow-off measurements,

FLAME-STRETCtI THEORY Burning-Velocity Measurement

Few systematic burning-velocity measurements have previously been made on inhibited flames. A major difficulty has been that, with such flames, the measured burning velocity was strongly dependent on the mixture flow rate at which the measurement was madeJ 's This ditficulty has also been generally experienced on noninhibited flames, but with these flames the del)endence was much tess pronounced. Such dependence clearly implies some important error in burning-velocity measurement, which becomes particularly significant with inhibited flames. Thus, an improved method of burning-velocity measurement had to be developed. Schlieren cone-angle methods using nozzle burners were selected as most suitable for further development. In such inethods, burning velocity is usually determined from the equation S,, = V o sin O,

(4)

where Vo is the velocity of the unburned gas approaching the flame front, and 0 is half the schlieren cone angle. Vg has usually been obtained by measurement or calculation under cold flow conditions, no allowance being made for the effect of the flame on the flow of unburned gas. There is experimental evidence to show that the unburned-gas velocity is significantly affected by the combustion process,~ and it was considered that Vo nmst, therefore, be determined in the presence of the flame. The present technique thus differs essentially froln previous methods ill that Vo was determined under flame conditions by the use of micro-pitot tubes and a highly sensitive micro-lnanometer. A combined pitot-static tube could not be fabricated to withstand flame conditions and, therefore, two separate tubes, measuring static head P~ and total head Pt were used. The velocity required to measure burning velocity was that of the unburned gas approaching the straight section of the flame fi'ont over which 0 was measured; provided the pitot probes were positioned so as to measure this velocity, the precise position of the probe was not considered to be critical. All measurements were made on a 89in. diana water-cooled nozzle burner similar to that used by Scholte and Vaags? Blow-off Measurement

The blow-off flow rates of inhibited methaneair flames were determined on cylindrical burners. Six lengths of pipe, two each of copper, iron, and

1009

stainless steel, were used as burners--each burner being of sufficient length to ensure fully developed streamline flow at its orifice. The internal diameters of the burners were as follows: copper, 1.01 and 1.30 cm; iron, 0.88 and 1.11 cm; and stainless steel, 0.94 and 0.62 cm. Blow-off rates were determined by keel)in~ the mcthane and methyl bromide flmvs constant trod increasing the air flow in small increments until hlow-off occurred.

Results Burning Velocity

The independence of measured burning velocity and mixture flow rate is illustrated by the typical results presented in Table I. It can be seen that there are no gross effects of flow on measured burning velocity for either uninhibited or inhibited flames. A series of burning velocity vs composition curves corresponding to constant methyl bromide/methane volume ratios of 0, 0.01, 0.02, and 0.03, were obtained and are reproduced in Fig. 1. The inaxilnum burning velocity for methane-air fames was found to be 38.3 era/see at 10.25% methane in air. The addition of methyl bromide reduced the maxinmm burning velocity to 35.1, 32.7, and 30.7 em/sec for ratios of 0.01, 0.02, and 0.03, respectively, with a shift in the mixture composition at the maximum towards stoiehiometrie. B low-oy

Blow-off flow rates were measured on each burner using the inhibitor/fuel volume ratios for which burning velocity curves had been determined. The burning velocity appropriate to each blow-off measurement could thus be obtained from the appropriate graph (Fig. 1). The boundary-velocity gradient at blow-off, gb, was also calculated for each measurement from the equation gb = 4Q/~R a,

(5)

in which Q is the volumetric flow-rate at blow-off, and R tile burner radius? Values of */0 were calculated froln Eq. (2) and substituted in Eq. (3) to give values of the Karlovitz flame-stretch factor K at blow-off in each case. A total of 204 blow-off determinations were made and the results are presented graphically in Figs. 2 and 3, representative points only being plotted hecause of the limited scale of the graphs.

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FLAME STUDIES TABLE I The burning velocity of inhibited 11.25~)~ methalm/air flames

Methyl bronfide/methane volume ratio

Volumetric f l o w (cc/sec)

Measured unburned gas velocity (cm/sec)

Burning velocity (cm/sec)

0

181.4 201.5 221.8 241.9 262 282.1 302.2

134.5 157 173 190 208.5 225.5 245

35.75 36.55 35.8 35.45 35.2 35.7 36

0.01

181.4 201.6 221.8 241.9 262 282

144 161 181 197 216 234

31.8 31.9 32.45 31.9 32.1 32

0.02

161.2 181.4 201.6 221.8 241.9 262

131 149.5 166.5 188 202.5 220.5

30.2 29.65 29.9 29.65 29 28.7

0.03

141.2 161.2 181.4 201.6 221.8 241.9

111 135.5 154 172 191.5 206

25 25.6 26.6 26.4 26.4 25.1

Discussion Figure 2 indicates the ability of the critical boundary-velocity-gradient concept to correlate the blow-off data. Four good correlations are obtained, each corresponding to a particular methyl bromide/methane ratio. Thus, so far as consideration of critical boundary-velocity gradicnt is concerned, the addition of inhibitor completely alters the behavior of the system, and the Lewis and von Elbe theory of blow-off fails to provide any means of calculating the effect. Reed's theory, however, offers a quantitative means of correlating the behavior of flames at different levels of inhibition, because it is based on the postulate that blow-off will occur at a critical value of the Karlovitz flame-stretch factor. Figure 3 constitutes impressive evidence of the validity of this postulate, since, for any particular methane/air mixture, blow-off does occur at a fixed value of K irrespective of the inhibitor content of the mixture. (The standard deviation of the 204 experimental points about

the line shown is 0.023.) This critical value of K is, moreover, practically independent of mixture composition insofar as lean flames are concerned. For rich flames, K increases rapidly as the methane content of the primary mixture is increased, but this is probably due to the complicating effect of secondary combustion with atmospheric air. It is appropriate at this point to compare the present results with those given by Reed. 1'2 This is done graphically in Fig. 4, in which the present correlation is shown as a solid line and Reed's correlation is shown by a broken line. The data used by Reed pertaining to methane and natural gas are also plotted. Considering that Reed's values of K are subject to a possible maximum error of • there is reasonably good agreement between the two correlations. It may be noted that although Reed's correlation gives values of K consistently lower than the present correlation, Reed's data for methane and natural gas tend to fall between the two lines. The rapid increase in the critical value of K with increase in the methane content of rich

FLAME-STRETCH THEORY

1011

Su cm Sr

-1

K

3.0

4O

3b

2-5

32

2.0 28 24 2O

I.O.

16

0.50.03

12

8

9

I0

II

12

13

8

14

9

I0

II

12

13

% Methane

~o M e t h a n e

Fm. 1. Burning velocity of methane-air flames inhibited withmethyl bromide9

FIG. 3. Correlation of blow-off data by flame-stretch factor.

flames requires further discussion. Reed has already demonstrated that this effect only occurs when rich flames are burned in air; K values for blow-off of flames burning in inert atmospheres being independent of the primary mixture earnposition. Thus, it is clear that secondary con> bustion in some way enables a flame to withstand

a greater degree of stretch9 Reed considers that the outer diffusion flame that forms when a rich mixture burns in air "acts as an energy source, or pilot, which can transfer heat to the stabilizing region of the inner cone." Because of this Reed claims that "the base of the inner cone can withstand more severe quenching th~n would otherwise be the ease and so the K factor, when blow-off eventually takes place, is higher." This is a plausible explanation, but the precise

gb se(~I 6000

0

K~ 30'

5O00. 4000-

/ /

.

0.01 /

X Methane - A i r (Reed) o Natural Cos-Air (Reed)

2,5-

0.02 2.0

3000-

O-03

1.5 2000-

//

I.O

I000-

J

0.5

os M e t h a n e

FIG. 2. Correlation of blow-off data by boundaryvelocity gradient.

0.8

0.9 Fuel:

I.O

I. I

Fraction of

1.2

1.3

Stoichiomctric

Fro. 4. Comparison of present correlation with that due to Reed.

1012

FLAME STUDIES of increased secondary combustion. It therefore seems possible that the rapid increase in the value of K at blow-off, with increase in the methane content of rich flames shown in Figs. 3 and 4, is caused by this error. We suggest that if the correct local burning velocity were used in the calculation of K values, blow-off might well be seen to occur at a constant critical value of K independent of secondary combustion effects. An alternative approach to this tentative conclusion is possible by reference to Fig. 5. Combining Eqs. (2) and (3) we obtain

gb sec-I O bOO0

5000

O.OI

4000

3000

2000

K = I000

,~

~

~

ic~

12~

i,~ Su 2

FIG. 5. llelationship between gb and S,,= for inhibited methane-air flames. mechanism by which the heat from the diffusion flame renders the primary flame front more resistant to flame stretch is neglected. In our opinion, this mechanism must involve enhancement of the local burning velocity in the stabilization zone. We contend that the heat supplied by secondary combustion increases temperature and reaction rates in the primary fame, particularly in the stabilization zone where primary and secondary combustion oeeur close together. Unfortunately, no measurements are available of local burning velocities in the stabilization zone of rich flames. However, several workers, including Kimura and Ukawa, m and Well] 1have reported results that show clearly that secondary combustion does increase the temperature and the average burning velocity of the primary flame front. A particularly significant feature of these experimental results is the dependence of average burning velocity on the burner diameter; the smaller the burner diameter, the greater the increase in average burning velocity resulting from secondary eombustion. It is difficult to interpret these results on any basis other than that burning velocity is most affected at the flame base (the stabilization zone) where seeondary combustion occurs very close to the primary flame front. If the burning velocity in the stabilization zone of rich flames is significantly enhanced in this way, it immediately follows that K values for such flames will be seriously overestimated in calculations based on normal burning velocities. The extent of the overestimate will increase as the primary mixture becomes more gas-rich, because

(6)

(k/pCp)(g~/&~).

Thus, for any system in which the term k/pCp is constant, blow-off at a constant critical value of K implies that gb must be proportional to S , 2. Figure 5 shows graphically the relationship between gb and S~2 for the methane air-methyl bromide flames used in the present investigation, for which k/pCp is practically constant. As predieted theoretically, gb is directly proportional to S,,2 for all lean mixtures, but the relationship does not hold for rich mixtures. We believe the apparent deviations from the theoretical prediction to be due primarily to the underestimate of local burning velocities in the stabilization zone of rich flames. The burning velocity in the stabilization zone will be affected not only by heat transfer from the outer diffusion flame, as previously discussed, but also by direct interdiffusion between the primary mixture and the surrounding atmosphere. The existence of such interdiffusion has been demon4,/ , / /"

/

ff

/

05

4/ 0.4 +J-~ 0.3

~

_

~

+

0

a

~

9 Helium Carbon D i o x i d e Nitrogen Air

o

Oxygen

0-2

- - ~

"\

0.1

o'.7

o:s

0:9

t.b

lh

,.2

Fuel: Fraction of Stoichiometric

FIG. 6. Effect of ambient atmosphere on flamestretch faetor at blow-off for natural gas-air flames.

FLAME-STRETCII TIIEORY strated with natural-gas flanies by Lewis and win Elbe. '~,t2 Figure 6 demonstrates the relevance of this interdiffusion to the flame-stretch theory of blow-off. Values of K at, blow-off for natural-gas flames burning in ambient atmospheres of oxygen, air, nitrogen, carbon dioxide, and helium, calculated from Lewis and yon Elbe's data, are shown. Clearly, interdiffusion has a significant effect on stability that can be attributed to its influence on burning velocity in the flame stabilization zone. When the 1)rimary gas-air mixture is fuel-lean, interdiffusion with the surrounding atmosphere generally dilutes the gas mixture in the stabilization zone with a consequent small reduction in burning velocity. However, when the primary mixture is gas-rich and the surrounding atmosphere is air, interdiffusion will lIroduee a shift in the mixture eomI)osition towards stoiehiometric and a substantial increase in burning velocity may result, t3 Thus, while interdiffusion may have only a sniall effect on the stability of lean flames, a much greater effect is to be expected for rich flames burning in air.

Conclusions 1. Accurate blow-off data for methane-air flames inhibited with methyl bromide are correlated precisely by the Karlovitz flame-stretch factor. 2. The excellence of the correlation indicates that blow-off is very probably caused by excessive flame stretch in the stabilization zone of the flame. 3. The critical value of the Karlovitz flamestretch factor at which blow-off occurs appears to increase rapidly as the methane content of rieh priniary nfixtures is increased. It is suggested that this apparent increase is due to the error involved in the assumption that the burning velocity in the flame stabilization zone is the normal burning velocity of the primary niixture. If the correct local burning velocity in the stabilization zone were used in the calculations, the critical flame-stretch factor might well be seen to be completely independent of mixture composition. 4. The burning velocity in the stabilization zone at the base of the flame will be significantly affected by: (a) Intermixing of the primary niixture with

1013

the surroulldillg atmoslohere prior lo combustioE: and (b) Heat transfer to the primary tlame front froni the outer diffusion flame. When fuel-rich flames are burned in an atmosphere of air, very large increases of local burning velocity in the flame stabilization zone may result. ACKNOWLEDGMENT

We are grateful to the Gas Council for the award of a Research Scholarship to one of us (M. P. H.) for the continuation of this work. REFERENCES 1. REED, S. B.: Combust. Flame 11, 177 (1967). 2. REED, S. B.: J. Inst. Gas Engr. 8, 157 (1968). 3. LEWIS, B. AND YON ELBE, G.: J. Chenl. Phys. 11, 75 (1943). 4. LEWIS, B. AND YON ELITE, G.: Combustion, Flames and Explosions of Gases, 2nd ed., p. 222, Academic Press, 1961. 5. KARLOVITZ, B., DENTNISTON, J. R., I{NAPSCHAEFER, D. W., AND WELLS, F. E.: Fonrth Symposium (International) on Combustion, p. 613, Williams and Wilkins, 1953. 6. HEAP, M. P.: Burning Velocity and Blow-Off of Burner Flames, M.Sc. thesis, University of Salford, 1967. 7. tIALPERN, C.: J. Research Natl. Bur. Std. 67A, 71 (1963). 8. HALPERN, C.: J. Research Natl. Bur. Std. 70A, 122 (1966). 9. SCHOLTE, T. G. AND VAAGS, P. B.: Combust. Flame 3, 495 (1959). 10. KIMURA, I. AND UKAWA, H.: Eighth Symposium (International) on Combustion, p. 521, Williams and Wilkins, 1962. 11. WEIL, S. A.: Burning Velocities of Itydrocarbon Flames, Chicago Institute of Gas Technology Research Bulletin No. 30, American Gas Association, 1961. 12. LEWIS, B. AND VON ELBE, G.: Combustion, Flames and Explosion of Gases, 2nd ed., p. 237, Academic Press, 1961. 13. WOHL, K., KAPP, N. s AND GAZLEY, C.: Third Symposium on Combustion, Flame and Explosion Phenomena, p. 3, Williams and Wilkins, 1949.

COMMENTS B. Lewis, Combustion and Explosives Research, Inc., 1007 Oliver Bldg., Pittsburgh, Penna. I think it proper to point out that the application of flame-stretch theory to blow-off

phenomena is discussed on page 241 et seq. in the 1961 edition of Lewis and Von Elbe's book "Combustion, Flames and Explosions of Gases." The authors' statement that no account is taken

1014

FLAME STUDIES

in this book of the effects of shear flow in the discussion of this topic is in error. The data presented in this paper provide a most interesting and valuable substantiation of the validity of the Karlovitz fame-stretch theory and is a welcome addition to the available information on this subject.

H. Edmondson. Lewis attributes to us a statement that we did not make. We are, of course, aware that Lewis and von Elbe applied the concept of critical flame stretch to several combustion phenomena including the blow-off of wirestabilized flames. Perhaps we should have referred to this in our paper. However, our test of the flame-stretch theory of blow-off relates entirely to cylindrical burners. Lewis and yon Elbe did not apply the critical flame-stretch concept to the blow-off of burner flames; indeed, they described the blow-off of such flames by an alternative mechanism. We are pleased that Lewis now seems to accept that blow-off of burner flames is caused by excessive flame stretch in the stabilization zone. B. Lewis. The test of the flame-stretch theory for the flame blow-off condition was carried out by Von Elbe and myself using data on the inverted flame (stabilized on an axially mounted wire), because the stabilization region of such a flame is unaffected by the ambient atmosphere. It thus affords a clean test of the theory. It should be evident that the flame-stretch theory applies to blow-off in any flame cmffiguration when allowance is made for secondary effects. It was so meant by us in discussing this subject in the 1961 edition of our book.

R. Lindow, Ruhrgas AG, Essen, Germany. The authors' calculations are based on flame-speed data, which cannot be regarded as precise enough

for this purpose. A maximum laminar flame speed of methane/air was used that lies more than 10% lower than the generally accepted value (e.g., data from the measurements done at the University of Karlsruhel). It would be interesting to know why the authors did not use known newer data or methods for their determination and to what extent these would affect their calculations. REFERENCE 1. LINDOW, R.: Eine verbesserte Breunermethode Zur Bestimmung der Laminaren Flammengeschwindigkeit, Breimstoff--Warme--Kraft "BWK," No. 1, 1968.

H. Edmondson. It is highly improbable that the excellent correlation represented by Fig. 3 would have been obtained with imprecise data. Lindow appears to confuse precision with absolute accuracy. The essence of his comment thus seems to be an unsupported assertion that his burning-velocity data are generally accepted as accurate, coupled with the inference that our data must be in error. We could just as easily assert that our data is accurate and that Lindow's is in error; but we prefer not to be dogmatic upon a matter that requires further investigation. Lindow determines the maximum burning velocity of methane/air mixtures to be 43.0, whereas our determination is 38.3 cm/sec. This significant difference indicates that one or both determinations are in error. Lindow asks why we did not use known burning-velocity data for our test: the answer is obvious--no suitable data was available. I t is made clear in the paper that we used a new, improved method of burning-velocity measurement. The effect of any error in burning velocity determination on the calculated values of K is obvious from consideration of Eqs. ( 2 ) a n d (3).