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A modified analysis for the determination of the burning velocity of a gas mixture in a spherical constant volume combustion vessel
A Modified Analysis for the Determination of the Burning Velocity of a Gas Mixture in a Spherical Constant Volume Combustion Vessel* K. H. O'DoNOVAN and C. J. RALLIS
An analysis, based on that of Lewis and von Elbe, is developed to provide a more accurate means of determining the burning velocity of a gas mixture in a spherical constant volume vessel with central ignition. Th approach is that of deriving an exact equation for the mass fraction burnt at any instant during combustion, valid throughout the combustion period. This involves a consideration of the temperature distribution in the vessel at any instant. The results given require that experimental records of pressure~time and flame radius~time be taken. The development avoids the possibility of magnification of errors of observation. The analysis proposed permits full use to be made of the chief advantage of the constant volume method, viz. that the burning velocity may be found over a large range oJ pressures in a single explosion. An approximate version of the analysis is also given to allow the burning velocity to be determined over the whole combustion range within reasonably close limits, using either a pressure~time or a radius~time record of combustion only.
THE burning velocity of a particular gas mixture is a constant under given conditions of temperature and pressure. Any experimental method used for its determination should give the same result, provided that the method is reliable. Due to the difficulty of obtaining accurate experimental conditions, and lack of agreement in the precise definition of the position of the flame front, not many results can be accepted without reservation. The data given by different investigators are often at variance and the reason for the discrepancies not explained. J . W . LINNETT and co-workers have made a valuable contribution to the clarification of the position by a critical examination of two methods used to determine burning velocity, viz. the soap bubble (constant pressure) method, and the burner (stationary flame) method, using both direct and schlieren photography TM. Such a systematic comparison does not appear to have been made between the constant volume method and other methods. The reason for this is perhaps due to the limited results available for this case, but it is more likely to be due to a tendency to suspect the method of being inaccurate. The analysis involved is inevitably more complicated than in other methods because of the variation in pressure during the explosion. This necessitates the introduction of simplifying assumptions, and the value of the method rests on the validity of these assumptions and the extent of their influence on the results. *University of the Witwatersrand Report No. 1/58.
201
Dewey decimal classification: 536.461.
K. H. O'DONOVAN AND C. J. RALLIS CONSTANT VOLUME METHOD
Before the constant volume method can be established as suitable for determining burning velocity, it is important first to examine how the analysis may best be effected to ensure sufficient accuracy. The ideal approach is to make such necessary assumptions as can be accepted as reasonable or verifiable by experiment, and to develop the analysis in such a way that the effect of errors, either in observation or interpretation, is reduced to an acceptable minimum. A method of determining burning velocity from constant volume explosions in a spherical vessel was developed by E. F. FLOCK et al. 5. The equation developed is unwieldy in application and magnifies observational errors. B. LEWIS and G. YON ELBE6 introduced additional assumptions to give a result more readily applicable and which was not subject to the disadvantage of magnification of errors. However, the range was then limited to the early stages of combustion where the pressure was low, besides being admittedly approximate even within the range utilized. The analysis developed by these investigators has been included here (in a concise but slightly different form), mainly for purposes of reference, and logical presentation. The limitations of this analysis are discussed, and a modified version is proposed which avoids many of the drawbacks inherent in the original. ANALYSIS
The combustion process in a spherical combustion vessel involves compression of the unburnt gas ahead of the flame front, expansion within the flame front, and compression of the burnt gas behind it, at any stage of the combustion. Notation Suffix b denotes state of burnt gas Suffix i denotes initial condition of gas before ignition Suffix u denotes state of unburnt gas mb -- mass of gas burnt at any instant m~= initial mass of unburnt gas n = mass fraction of gas burnt at any instant P = pressure at any instant during combustion P ' = pressure when flame is at any given radius P, = pressure at end of combustion P~ = calculated pressure at end of combustion P~= initial pressure of unburnt gas Rb = specific gas constant for burnt gas Ru = specific gas constant for unburnt gas r,-- radius of vessel rb = radius of flame at any instant during combustion r~= radius of burnt gas at any instant under initial conditions Su = burning velocity T~ = mean temperature of burnt gas at end of combustion Tb = mean temperature of burnt gas at any instant Tb = temperature of flame at initial pressure T{ = temperature of flame at given radius
202
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
Tbr = temperature at centre of vessel at any instant T~,,, = temperature at given radius at any instant T',, = temperature rise due to combustion at given radius T~ = initial temperature of unburnt gas T~ = temperature of unburnt gas at any instant T;, = temperature of unburnt gas at given radius t = interval of time after ignition for any instant during combustion ~, = volume of vessel ~,~= volume of gas burnt at any instant ~,~=volume occupied by burnt gas at any instant under initial conditions v,, = volume of unburnt gas at any instant ~, = ratio of heat capacities of unburnt gas -/,, = ratio of heat capacities of burnt gas Assumptions made Fiock et al.:' listed the assumptions which they made and justified them by the statement that they were consistent with physical observations of the behaviour of the gas mixture during combustion. Lewis and yon Elbe ~ made the same assumptions, being satisfied that the error involved in making them was negligible. These assumptions are: (I) The pressure at any instant during combustion is equalized throughout the vessel (2) No heat is lost to the walls of the container during combustion (3) The temperature and pressure of the unburnt gas rise during combustion in accordance with the law of adiabatic compression for perfect gases (4) The flame front remains spherical throughout combustion (5) The flame front is a surface of discontinuity across which the change from the unburnt to the burnt state takes place (6) The ratio of the gaseous specific heats for the unburnt gas remains constant (7) The total mass of gas and the change in the number of molecules remain constant during combustion. In addition to the above, B. LEWIS and G. VON ELBE7 made the following assumptions: (8) The temperature and pressure of the burnt gas rise during combustion according to the law of adiabatic compression for perfect gases (9) The ratio of the gaseous specific heats for the burnt gas is constant and equal to the mean value between beginning and end of combustion. Condition before ignition The instant of ignition is considered to be the starting point of combustion, i.e. the instant of zero time. The initial unburnt state of the gas mixture is the condition immediately prior to ignition as denoted by the sutT-LX i. The initial pressure, temperature and mass of unburnt gas are given by P , T~ and m~ respectively. Also initial volume of unburnt gas =volume of vessel v =4~r~/3 = m~R~Ti/Pi 203
K. H. O'DONOVAN AND C. J. RALLIS
Condition at time t after ignition Here the flame front is at radius rb. Pressure throughout vessel Mass of burnt gas Mass fraction of total gas burnt .'. Mass fraction of unburnt gas Mass of unburnt gas, m, Temperature of unburnt gas, T, Volume of unburnt gas, v~
=P mb
=m~/mi=n =1 - n =mi(1 - n ) = T, (p,/p)ca-7~)/r, --4~ (r~ - r g ) / 3 --m~(1 - n ) R , T , / P
(1 -n) (P,Ipp/,.
=~
Volume of burnt gas, vb
-- 4~r~/3 = v { 1 - (1 - n) (PdP)II,.}
Therefore flame front radius = rb = ra{ 1 - (1 - n) (PdP)l/r~} 1/3
. . . . [1]
Alternatively n = 1 - ( 1 - r ~ / r ] ) (P,/P)~/,.
.
.
.
.
[21
Volume occupied by burnt gas under conditions obtaining before ignition v, =4~r~/3 = mbR,Ti/ P~= nv Hence initial radius of burnt gas
r,=ran 11~ i.e.
. . . . [31
r,=ra{1 - ( 1 - r ] / r ] ) (p/p~)~/7,}~/~
. . . . [4]
Condition at time (t + dO triter ignition During the element of time dr, the flame front advances from radius rb to radius (rb + d r 0 with burning velocity S,. The flame advance may be considered to be made up of two parts, viz. that due to burning of the gas at constant pressure P and that due to mass movement of the flame caused by expansion of the gas at the flame front and adiabatic compression of the unburnt gas ahead of the front. Increase in volume enclosed by the flame front, dvb
= 4~r~drb
Also advance of front due to combustion during time dt
= S, dt
Therefore volume consumed by flame in time dt
= 4~rgSu dt
Volume increase due to compression of unburnt gas = dv, = d v~ - 4~r~ S, dt = 4~r~(drb - S~ dt) Now for adiabatic compression of unburnt gas
dP PFrom which
dv~ ~" z,~ -
7,
( -4~r~(drb -S~ dt) } 4~:(r~-r~)/3
S,= I1- ,,~:
dt
~_j dr~
.... [5]
This equation, due to Fiock et al?, gives the burning velocity of the gas mixture at any stage in the combustion. 204
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
Limitations of burning velocity equation In order to find the value of the burning velocity of a gas mixture using equation 5, the quantities drb/dt, dP/drb, rb, P, "/, and r, must be known. The value drb/dt is the rate of change of flame radius with time at any instant during combustion, i.e. the flame speed. It can be found directly from a photographic trace of the flame movement. The radius rb is derived from the width of the flame trace at the instant considered. Here an error of observation is possible due to the fact that the precise position of the flame front is not necessarily coincident with the edge of the luminous zone. The pressure P at any instant is observed from a pressure/time record. This pressure is most conveniently measured at the radius of the vessel and is considered uniform throughout the vessel at the instant. The value of dP/dr~ is derived from the radius/time and pressure/time records and involves the same error of observation as the term rb. The chief drawback in using the equation is the fact that the second term in the square brackets is very nearly equal to unity in the early stages of the process. Thus a small error in measuring rb or P leads to a greatly magnified error in the value deduced for S,. Alternative burning velocity equations To avoid the difficulty of magnification of errors, the burning velocity equation can be rendered in a different form by the following considerations. During the time interval dt at time t, an elementary mass of gas mixture dm is burnt. This mass of gas is at a temperature T, and a pressure P immediately prior to burning, and occupies a volume dr, given by dvu = dm R , T , / P In terms of the initial temperature T, and pressure P, the volume would be d-o~as given by d vi = dm RuTdPi Therefore
dr, -- (P~/ P) (T, / TO dv~ = (P,/P) (T,/T~)4~r~ dr,
Also
d~, = 4~r~ S, dt
Therefore
S, = (dr~/dt) (r~ / r~ ) (e,/p)l/~ . . . . .
[6]
This equation is not immediately applied, as the values drddt and r~ must first be determined. Equation 3 gives the values of r, in terms of the radius of the vessel and the mass fraction burnt at the time t after ignition. Using this relationship, equation 6 may then be written S,=½ (dn/dt) (r3,/r~)(Pde)~/r .
.
.
.
.
[7]
If the value of n as given in equation 2 is differentiated with respect to t and the result inserted in the above equation, the original burning velocity equation 5 is obtained. This process serves no purpose other than to verify equation 7.
205
K. H. O'DONOVAN AND C. J. RALLIS
The value rb can be expressed in terms of n, P; and P, using equation 1, giving
Su=
7] (dn/dt) ra (p,/p)x;~ { 1 - (1 - n) (p,/p)l;r,}.-,/:,
. . . . [8]
The pressures P~ and P are given by the pressure record. The mass fraction n remains to be evaluated in order to solve for burning velocity from equation 8. DISCUSSION ON EVALUATION OF MASS FRACTION
It is clear that n is a function of the radius rb and the pressure P at time t, as evidenced by equation 2. Lewis and von Elbe ~ give the following approximate relation between n and P n = (P - e o / (P~ - P,)
. . . . [9]
Using this relationship, the burning velocity can be computed for a gas mixture from a pressure record of the combustion only. The relationship is approximate because it assumes a linear proportionality between r~, and P. The error involved in this assumption is not large in the early stages of combustion, so that the application of equation 9 must be confined to small values of n. Since constant pressure is assumed to exist throughout the vessel at any instant, the radius of the flame front, as deduced from the pressure record, does not indicate a particular edge in the flame front, but it gives rather some kind of mean value over the whole reaction zone. The pressure P, in equation 9 is taken to be the final pressure at the end of combustion as computed from thermodynamic data, assuming that thermodynamic equilibrium is established in the flame. Consequently the evaluation of Pe is based on the assumption that the temperature distribution in the burnt region is uniform. It is claimed 8 that the values of n computed from equation 9 are precise for small values of n. This is justified by using the values of n so obtained to derive the values of rb at different instants during combustion, and comparing these with the corresponding experimental values of rb. The results given show that in every case the difference was small, being at worst about one per cent of the radius rb. However, it can be shown that a large variation in the value taken for P~ will not affect the resulting value of r,, to any great extent. For example, even for a five per cent variation of the value of Pc, the resulting calculated radius rb is, in most cases, still within one per cent of the observed radius in the cases tested. The actual final pressure obtained from the pressure/time trace is well within five per cent of the fictitious pressure Pe, so that it would appear far more convenient to employ the actual pressure, since the final result is within the same limits of accuracy. The suitability of equation 9 for use in finding burning velocity is somewhat doubtful because, although the error involved may be small in the early stages of combustion in the cases where it has been used, there is 206
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
no guarantee that this would be so for other gas mixtures, particularly when the rate of combustion is high. If attention is confined to the early stages of combustion, as it must be using this equation, the constant volume method loses its greatest advantage, viz. that the burning velocity of a gas mixture can be found over a large range of pressures in a single explosion. Other methods, particularly the soap bubble method, can then give as much information with far less experimental and analytical difficulty. It has been shown that equation 7 makes it possible to achieve greater accuracy in the determination of the burning velocity of a flame than can be obtained by using equation 5 in the early stages of combustion, even when employing what is in effect an assumed value of n. Thus, if the fraction n can be evaluated accurately over the whole combustion range, valuable results for burning velocity would be obtained. Experimental records of the variation of pressure, temperature or flame front radius with time are obtainable, so that the fraction n would need to be found in terms of one or more of these parameters. At the same time the original disadvantage of magnification of error must be avoided, which precludes the use of equation 2 for the purpose. DERIVATION OF MASS FRACTION
The following analysis has for its object a more rigorous derivation of the mass fraction n. The assumptions quoted above are retained, as all of these have been shown by earlier experimenters to be valid. One additional assumption is made here; namely, that the temperature of the burnt gas at any instant is given by the mean temperature throughout the burnt gas volume at that instant. The validity of this assumption is investigated below.
Temperature distribution An element of gas at the centre of the spherical combustion vessel will be at a pressure P~ and temperature Ti immediately before ignition. This element will be the first to burn after ignition. Therefore the combustion of the element will take place at constant pressure P~ and its temperature will rise to Tb, equal to the flame temperature, during combustion (if the external effect of ignition on the element is negligible). Subsequently, its temperature will be further increased due to adiabatic compression as the flame progresses beyond it, until at time t after ignition it reaches a value T~, corresponding to the pressure P at that instant. A second element of gas remote from the centre is first compressed adiabatically. This element then burns at a constant pressure, higher than P~, and is subsequently compressed adiabatically until at time t after ignition it reaches a temperature T~,p less than T~,,,. The process can be illustrated by means of Figure 1. The curves (~) and (~,,) indicate adiabatic compression of burnt and unburnt gas respectively. The rise in temperature with time during adiabatic compression as the flame progresses is greater for burnt than for unburnt gas. An element at radius r0 burns to temperature T~ and its temperature is then raised by compression to T~F after time t, giving point A. Elements at radii r, and r., are compressed as unburnt gas, then burned, and then compressed further as burnt gas during the time interval t, giving 207
K. H. O'DONOVAN AND C. J. RALLIS
the curves 0MNB and 0PQC respectively. An element at r8 burns at time t after being compressed as unburnt gas, giving the curve 0RD. The temperature rise during burning decreases slightly with increase of combustion pressure, so that R D < PQ < MN < L0 and the curve joining
I Ii
r,I
f
I
0
.
t Time
Figure 1
LNQD has a slightly smaller slope than curve 0MPR at corresponding points. A family of curves of temperature versus radius at time t can thus be plotted using such points as A, B, C and D, for radii r0, rl, !"2 and rs respectively. In this case t represents the interval of time that has elapsed from the instant of ignition to the instant when the flame front reaches the radius considered. For example, when the flame front reaches the radius r2, a time t2 has elapsed since ignition and the temperature gradient throughout the vessel is given by the curve EFGH in Figure 2. The temperature of the burnt gas at the instant t2 varies from radius r0 to radius r2 according to the curve EF. In particular, the curve ABCD gives the temperature gradient in the whole vessel at the instant the flame front reaches the wall (radius ra). A
B
to
~ u H lI _ _ ~ ~ ~L- ~ ~ ~2 --'-i.
~ro
.
.
.
.
q
r2 Radius
Figure 2
208
i
5
J
DETERMINATION OF THE BURNING V E L ( ~ I T Y OF A GAS MIXTURE
If, in general, curves such as E F are expressed as T = T (r0, then the mean temperature of the burnt gases at time t is given by rb
~ = (1/r~) J T (r~) drb
....
[Io]
0
At the end of combustion, the mean temperature is ra
T, = (1/r,) f T (rb) drb 0
A general expression for the temperature gradient in the burnt region at any instant is obtained as follows: In Figure 1 the points A, B, C, D, for time t are points on the required curve. Such points are in general arrived at in three stages, viz. (a) adiabatic compression of unburnt gas, (b) constant pressure temperature rise due to burning, and (c) adiabatic compression of burnt gases. Thus
temperature rise, stage (a) = T" - T~ temperature rise, stage (b) = T~ - T'~ temperature rise, stage (c) = T~p - T~
where Tg--temperature at beginning of adiabatic compression of burnt gases (pressure P'), and 7~p =temperature after adiabatic compression of burnt gases to pressure P from pressure P'. Total temperature rise
= ( ~Z t - T , ) + ( T o - TI ~ )
t
4 - ( TI~ - T ' b )
= Tip - T~
= [(P'/P)~I-~Plrb] T'b - T, Let T b - T u = T u = temperature rise due to combustion from temperature T u l'herefore t
•
•
•
temperature rise = (P' / P)~-~b)/~b [T', + T~ (P~/ P')~-~.'~] - T~ Therefore the temperature at any point on the curve EF (Figure 2) is
T = (P" l p)(~-rb)!rb IT', + T, (P, I P')<~-~,)I~,]
. . . . [11]
The pressure P" is a function of rb, since it is the pressure at which burning takes place at any particular radius rb. Hence equation 11 is an expression for the temperature gradient in the burnt region at time t in terms of the radius of the flame front at that instant. T~ is the temperature rise of the gas when burnt at pressure P'.
Mass equations The mass of the system as a whole is the same at the end as at the beginning of combustion. Hence m~----ma
i.e.
Piv / RuTi = P~v~ / RbTa 209
K. H. O'DONOVAN AND C. J. RALLIS
Also, the volume v at the beginning is the same as the volume v~ at the end of combustion. Therefore Ti / T~ = PiR~ I PaR. or
(TJTO (Tb/To)=P,R~IP~R .
.
.
.
[12]
.
The ratio n is given by m b / m . i.e. n = (Pvb/R~Tb)/(P~v/R~TO
Therefore
. . . . [13]
n =PTar~/P~T-br~
Now
mi = mb + m,,
i.e.
P~v / R,, T~ = Pv b / R bTb + Pv,, / R,,T,,
or
P e a3 / R . T ~ = P r 3b/RbTb + P (r~3 - r b3) / R u T ~
Hence
r~ /r~ = ( P U P T ~ - I / T u ) / ( R ~ / R b T b - 1/T=)
. . . . [14]
where Tu = (PIPi)%-')l~',,. T,
Substituting equation 14 in 13 gives
n : To .~
P-P,(PIP,) ~"-'/~
and when using equation 12 T. ~ n=
~--
~a'~/D (Tu l)/Tu P-l"itr!Pi)
~'nx(y
Tb t P . - ( T . / T O P , ( r / r o
u
"]
-
1)l~
....
[151
u
Comparing equations 15 and 9, it is noted that two additional terms, T. and Tb, must be evaluated in order to solve for n. Equation 15 has the considerable advantage of being valid for all values of n, as well as being subject to fewer assumptions, thus rendering it more exact. PROCEDURE FOR DETERMINING BURNING VELOCITY The solution of the burning velocity of a gas mixture using a constant volume explosion in a spherical vessel is obtained by the application of equations 8, 10, 11 and 15. For convenience, these equations are rewritten below, in the order in which they are to be applied. T=
(P"IP)<~-~'Pi~'~[T'n + T~ (PU P')<'-~>/q 210
....
[11]
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE rb ¢.
= 1/rb ~ T fib) dr, .
.
.
.
.
[10]
0
n:S, =
T~, t. p , _ (T,/T~,) P~ (P/P3
....[15]
½ ( d n / d t ) r~ (P,/P)~/r,
{ 1 - (l - n) (P,/P)~/v,} ''/~
. . . . [8]
Consider a series of radii r~ taken at equal intervals. The pressure obtaining in the vessel when the flame reaches radius rb is P' and is given by a correlation of the pressure/time and radius/time traces. At a particular time t after ignition, the pressure is P throughout the vessel. The temperature at the radii taken are given by equation 11. The curves of temperature versus radius are plotted for various time intervals between the beginning and end of combustion. The mean temperatures obtaining at these time intervals are most easily found by graphical integration of these curves, the result being a solution of equation 10. The mass fraction n is evaluated using equation 15 for various pressures corresponding to the time intervals chosen, and plotted against time. Finally, the burning velocity is computed from equation 8, inserting the corresponding values of d n / d t , n, P and t as found above. Observations required
To carry out the procedure outlined above it is necessary to have available sufficient experimental results. The following information is required: (a) Constituents and proportion of gas mixture (b) Initial temperature T~ and pressure P~ immediately before ignition (c) Record of pressure P versus time t throughout combustion (d) Record of flame radius rb versus time t throughout combustion (e) Radius of spherical vessel. T h e r m o d y n a m i c constants
In addition to the experimental values enumerated above, it is necessary that 7,, 7~, and T~ be evaluated for each mixture at the particular condition of temperature and pressure. For a mixture of perfect gases, the mean value of ,/is defined by: x~C~,,, ~£ m ~ A H . 7 = ~ x~,C~,,, -- ~ m,,AE~,
. . . . . [16]
where the suffix q refers to each constituent present, xq=mole fraction, C,0 and Cvq=mean molal heat capacities at constant pressure and constant volume respectively, AH,, and AEq = change in specific enthalpy and internal energy respectively, and m~ = mass of constituent q present in the mixture. The determination of 7, presents no difficulties, since the mole fractions 211
K. H, O'DONOVAN AND C. J. RALLIS
of the various constituents are not significantly affected by temperature. Also, the use of a mean value for ~/ is justifiable, since the temperature changes are relatively small. For the evaluation of 7b, it is first necessary to determine Tb""(= T"n+ T u). " This is done by the simultaneous solution of the two equations--the dissociation and energy equations--for the layer of mass dm, from the given initial conditions of temperature T~ and pressure P'. From this the composition of the equilibrium mixture at temperature Tg may be found. During subsequent adiabatic compression of these combustion products, the equilibrium shifts, and alters the mole fraction distribution in a determinable manner. Thus, since all the xqs are known, and the AH~s and AEqs over the temperature range are available from tables, the mean value of ~b can be calculated. SIMPLIFICATION
It is possible to reduce the computational and experimental labour involved in the above analysis by making certain simplifying approximations. These would inevitably lead to somewhat less accurate results. However, this may often be acceptable as a check or, if the differences are insignificant, such approximation may be utilized directly. The only really satisfactory way of validating such approximation is by a sufficient number of experimental comparisons.
General approximation It seems reasonable to assume that the temperature rise due to combustion (T~) is constant, since the actual variation of this quantity is relatively small. This renders equation 11 and hence 10 fairly easily soluble. Burning velocity using pressure~time records only The value of the burning velocity at any instant during combustion can be obtained using only equations 15 and 8, if a satisfactory average value for the mean temperature T~ is used. The advantage of doing this is that the burning velocity can then be obtained using only a pressure/time record Of the combustion. If the ratio of Ta to Tb in equation 15 is assumed equal to unity, then results of the same order of accuracy as those obtained by Lewis and yon Elbe ~ using equation 9 can be expected. This approximate form of equation 15 is equivalent to equation 9 except that it is valid throughout the full range of combustion. It is possible to improve on this by noting that the ratio Ta/T~ varies from T~/Tb at the beginning, to unity at the end, of combustion. The values of T~ and Tb can be determined by taking a temperature/time record at the centre of the spherical vessel during combustion. The manner in which the ratio varies between Ta/Tb and unity can be assumed or further points on the curve plotted by using temperature/time records at various radii in the vessel. Burning velocity using radius~time records only Consider the combustion process at any instant when the flame has propagated to a radius rb. Here m b + m~ =mt 212
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
i.e. Pr~ P (r~ - r~) P,r~ RbT"~-~b+ RuT. = R.TI Substituting for Tu=T~(P/P~)Cr-I)/ru and rearranging gives p,/~. + [- r~ R . T , . p,,_y,,)/r. 1 P ra" t_(r] - r~) " R j ' ~ (r~ - r~,) Pi'/*" i.e.
P~ /~ + CaP = C~
. . . . [17]
Here it is apparent that for any set of initial conditions and observed flame radius rb, C1 and C2 can be evaluated provided Tb can be determined or estimated. Values for P at different radii rb then follow. Use of these results in equation 2 leads to values for n and thus d n / d t . Substituting in equation 7 then yields the burning velocity. Alternatively, of course, if both rb and P are measured, equation 17 can be used to solve for T~. The burning velocity can then be determined by using equations 15 and 8. CONCLUSION
The general analyses outlined above are intended to provide a method of solving for the burning velocity of a gas mixture that is at once accurate and relatively simple. The accuracy is, of course, dependent on the validity of the assumptions made. These assumptions are, in general, the same as those made in the references quoted 5, ~. The accuracy obtainable by this method is not implicitly greater than that of the previous methods, but the possibility of obtaining satisfactory results is increased by avoiding any magnification of incidental small errors of observation and calibration. The usefulness of the constant volume method of obtaining burning velocities of gas mixtures is in this way extended. Department of Mechanical Engineering, University of the Witwatersrand, Johannesburg, S. Africa (Received September 1958) REFERENCES
'Burning velocity determinations, Part III. Burning velocities of ethylene-air-carbon dioxide mixtures.' Trans. Faraday Soc. 1951, 47, 179 2 Lir~,~yr, J. W., PmKERING,H. S. and Wm~^TLEY,P. J. 'Burning velocity determinations, Part 1V. The soap bubble method of determining burning velocities.' Trans. Faraday Soc. 1951, 47, 981 3 CON^~, H. R. and L~'~rETr, J. W. 'Burning velocity determinations, Part V. The use of schlieren photography in determining burning velocities by the burner method.' Trans. Faraday Soc. 1951, 47, 981 4 PmKER~NO,H. S. and LI~3"T, J. W. 'Burning velocity determinations, Part VI. The use of schlieren photography in determining burning velocities by the soap bubble method.' Trans. Faraday Soc. 1951, 47, 989 LINt,rETr, J. W. and Ho^RE, M. F.
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K. H. O'DONOVAN AND C. J. RALLIS FiocI~, E. F., MARVIN, C. F., CALDWELL,F. R. and ROEDER, C. ]-I. 'Flame speeds and energy considerations for explosions in a spherical bomb.' Nat. Adv. Comm. Aero., Wash., Rep. No. 682 (1940) LEWIS, B. and YON ELBE, G. Combustion, Flames and Explosions of Gases, p 448 ft. Academic Press: New York, 1951 z LEwis, B. and voN ELBE, G. Combustion, Flames and Explosions of Gases, p 651 ft. Academic Press: New York, 1951 s MANTON, J., VON ELBE, G. and LEWIS, B. 'Burning velocity measurements in a spherical vessel with central ignition.' Fourth Symposium (International) on Combustion, p 358. Williams and Wilkins : Baltimore, 1953
214
An analysis, based on that of Lewis and von Elbe, is developed to provide a more accurate means of determining the burning velocity of a gas mixture in a spherical constant volume vessel with central ignition. Th approach is that of deriving an exact equation for the mass fraction burnt at any instant during combustion, valid throughout the combustion period. This involves a consideration of the temperature distribution in the vessel at any instant. The results given require that experimental records of pressure~time and flame radius~time be taken. The development avoids the possibility of magnification of errors of observation. The analysis proposed permits full use to be made of the chief advantage of the constant volume method, viz. that the burning velocity may be found over a large range oJ pressures in a single explosion. An approximate version of the analysis is also given to allow the burning velocity to be determined over the whole combustion range within reasonably close limits, using either a pressure~time or a radius~time record of combustion only.
THE burning velocity of a particular gas mixture is a constant under given conditions of temperature and pressure. Any experimental method used for its determination should give the same result, provided that the method is reliable. Due to the difficulty of obtaining accurate experimental conditions, and lack of agreement in the precise definition of the position of the flame front, not many results can be accepted without reservation. The data given by different investigators are often at variance and the reason for the discrepancies not explained. J . W . LINNETT and co-workers have made a valuable contribution to the clarification of the position by a critical examination of two methods used to determine burning velocity, viz. the soap bubble (constant pressure) method, and the burner (stationary flame) method, using both direct and schlieren photography TM. Such a systematic comparison does not appear to have been made between the constant volume method and other methods. The reason for this is perhaps due to the limited results available for this case, but it is more likely to be due to a tendency to suspect the method of being inaccurate. The analysis involved is inevitably more complicated than in other methods because of the variation in pressure during the explosion. This necessitates the introduction of simplifying assumptions, and the value of the method rests on the validity of these assumptions and the extent of their influence on the results. *University of the Witwatersrand Report No. 1/58.
201
Dewey decimal classification: 536.461.
K. H. O'DONOVAN AND C. J. RALLIS CONSTANT VOLUME METHOD
Before the constant volume method can be established as suitable for determining burning velocity, it is important first to examine how the analysis may best be effected to ensure sufficient accuracy. The ideal approach is to make such necessary assumptions as can be accepted as reasonable or verifiable by experiment, and to develop the analysis in such a way that the effect of errors, either in observation or interpretation, is reduced to an acceptable minimum. A method of determining burning velocity from constant volume explosions in a spherical vessel was developed by E. F. FLOCK et al. 5. The equation developed is unwieldy in application and magnifies observational errors. B. LEWIS and G. YON ELBE6 introduced additional assumptions to give a result more readily applicable and which was not subject to the disadvantage of magnification of errors. However, the range was then limited to the early stages of combustion where the pressure was low, besides being admittedly approximate even within the range utilized. The analysis developed by these investigators has been included here (in a concise but slightly different form), mainly for purposes of reference, and logical presentation. The limitations of this analysis are discussed, and a modified version is proposed which avoids many of the drawbacks inherent in the original. ANALYSIS
The combustion process in a spherical combustion vessel involves compression of the unburnt gas ahead of the flame front, expansion within the flame front, and compression of the burnt gas behind it, at any stage of the combustion. Notation Suffix b denotes state of burnt gas Suffix i denotes initial condition of gas before ignition Suffix u denotes state of unburnt gas mb -- mass of gas burnt at any instant m~= initial mass of unburnt gas n = mass fraction of gas burnt at any instant P = pressure at any instant during combustion P ' = pressure when flame is at any given radius P, = pressure at end of combustion P~ = calculated pressure at end of combustion P~= initial pressure of unburnt gas Rb = specific gas constant for burnt gas Ru = specific gas constant for unburnt gas r,-- radius of vessel rb = radius of flame at any instant during combustion r~= radius of burnt gas at any instant under initial conditions Su = burning velocity T~ = mean temperature of burnt gas at end of combustion Tb = mean temperature of burnt gas at any instant Tb = temperature of flame at initial pressure T{ = temperature of flame at given radius
202
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
Tbr = temperature at centre of vessel at any instant T~,,, = temperature at given radius at any instant T',, = temperature rise due to combustion at given radius T~ = initial temperature of unburnt gas T~ = temperature of unburnt gas at any instant T;, = temperature of unburnt gas at given radius t = interval of time after ignition for any instant during combustion ~, = volume of vessel ~,~= volume of gas burnt at any instant ~,~=volume occupied by burnt gas at any instant under initial conditions v,, = volume of unburnt gas at any instant ~, = ratio of heat capacities of unburnt gas -/,, = ratio of heat capacities of burnt gas Assumptions made Fiock et al.:' listed the assumptions which they made and justified them by the statement that they were consistent with physical observations of the behaviour of the gas mixture during combustion. Lewis and yon Elbe ~ made the same assumptions, being satisfied that the error involved in making them was negligible. These assumptions are: (I) The pressure at any instant during combustion is equalized throughout the vessel (2) No heat is lost to the walls of the container during combustion (3) The temperature and pressure of the unburnt gas rise during combustion in accordance with the law of adiabatic compression for perfect gases (4) The flame front remains spherical throughout combustion (5) The flame front is a surface of discontinuity across which the change from the unburnt to the burnt state takes place (6) The ratio of the gaseous specific heats for the unburnt gas remains constant (7) The total mass of gas and the change in the number of molecules remain constant during combustion. In addition to the above, B. LEWIS and G. VON ELBE7 made the following assumptions: (8) The temperature and pressure of the burnt gas rise during combustion according to the law of adiabatic compression for perfect gases (9) The ratio of the gaseous specific heats for the burnt gas is constant and equal to the mean value between beginning and end of combustion. Condition before ignition The instant of ignition is considered to be the starting point of combustion, i.e. the instant of zero time. The initial unburnt state of the gas mixture is the condition immediately prior to ignition as denoted by the sutT-LX i. The initial pressure, temperature and mass of unburnt gas are given by P , T~ and m~ respectively. Also initial volume of unburnt gas =volume of vessel v =4~r~/3 = m~R~Ti/Pi 203
K. H. O'DONOVAN AND C. J. RALLIS
Condition at time t after ignition Here the flame front is at radius rb. Pressure throughout vessel Mass of burnt gas Mass fraction of total gas burnt .'. Mass fraction of unburnt gas Mass of unburnt gas, m, Temperature of unburnt gas, T, Volume of unburnt gas, v~
=P mb
=m~/mi=n =1 - n =mi(1 - n ) = T, (p,/p)ca-7~)/r, --4~ (r~ - r g ) / 3 --m~(1 - n ) R , T , / P
(1 -n) (P,Ipp/,.
=~
Volume of burnt gas, vb
-- 4~r~/3 = v { 1 - (1 - n) (PdP)II,.}
Therefore flame front radius = rb = ra{ 1 - (1 - n) (PdP)l/r~} 1/3
. . . . [1]
Alternatively n = 1 - ( 1 - r ~ / r ] ) (P,/P)~/,.
.
.
.
.
[21
Volume occupied by burnt gas under conditions obtaining before ignition v, =4~r~/3 = mbR,Ti/ P~= nv Hence initial radius of burnt gas
r,=ran 11~ i.e.
. . . . [31
r,=ra{1 - ( 1 - r ] / r ] ) (p/p~)~/7,}~/~
. . . . [4]
Condition at time (t + dO triter ignition During the element of time dr, the flame front advances from radius rb to radius (rb + d r 0 with burning velocity S,. The flame advance may be considered to be made up of two parts, viz. that due to burning of the gas at constant pressure P and that due to mass movement of the flame caused by expansion of the gas at the flame front and adiabatic compression of the unburnt gas ahead of the front. Increase in volume enclosed by the flame front, dvb
= 4~r~drb
Also advance of front due to combustion during time dt
= S, dt
Therefore volume consumed by flame in time dt
= 4~rgSu dt
Volume increase due to compression of unburnt gas = dv, = d v~ - 4~r~ S, dt = 4~r~(drb - S~ dt) Now for adiabatic compression of unburnt gas
dP PFrom which
dv~ ~" z,~ -
7,
( -4~r~(drb -S~ dt) } 4~:(r~-r~)/3
S,= I1- ,,~:
dt
~_j dr~
.... [5]
This equation, due to Fiock et al?, gives the burning velocity of the gas mixture at any stage in the combustion. 204
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
Limitations of burning velocity equation In order to find the value of the burning velocity of a gas mixture using equation 5, the quantities drb/dt, dP/drb, rb, P, "/, and r, must be known. The value drb/dt is the rate of change of flame radius with time at any instant during combustion, i.e. the flame speed. It can be found directly from a photographic trace of the flame movement. The radius rb is derived from the width of the flame trace at the instant considered. Here an error of observation is possible due to the fact that the precise position of the flame front is not necessarily coincident with the edge of the luminous zone. The pressure P at any instant is observed from a pressure/time record. This pressure is most conveniently measured at the radius of the vessel and is considered uniform throughout the vessel at the instant. The value of dP/dr~ is derived from the radius/time and pressure/time records and involves the same error of observation as the term rb. The chief drawback in using the equation is the fact that the second term in the square brackets is very nearly equal to unity in the early stages of the process. Thus a small error in measuring rb or P leads to a greatly magnified error in the value deduced for S,. Alternative burning velocity equations To avoid the difficulty of magnification of errors, the burning velocity equation can be rendered in a different form by the following considerations. During the time interval dt at time t, an elementary mass of gas mixture dm is burnt. This mass of gas is at a temperature T, and a pressure P immediately prior to burning, and occupies a volume dr, given by dvu = dm R , T , / P In terms of the initial temperature T, and pressure P, the volume would be d-o~as given by d vi = dm RuTdPi Therefore
dr, -- (P~/ P) (T, / TO dv~ = (P,/P) (T,/T~)4~r~ dr,
Also
d~, = 4~r~ S, dt
Therefore
S, = (dr~/dt) (r~ / r~ ) (e,/p)l/~ . . . . .
[6]
This equation is not immediately applied, as the values drddt and r~ must first be determined. Equation 3 gives the values of r, in terms of the radius of the vessel and the mass fraction burnt at the time t after ignition. Using this relationship, equation 6 may then be written S,=½ (dn/dt) (r3,/r~)(Pde)~/r .
.
.
.
.
[7]
If the value of n as given in equation 2 is differentiated with respect to t and the result inserted in the above equation, the original burning velocity equation 5 is obtained. This process serves no purpose other than to verify equation 7.
205
K. H. O'DONOVAN AND C. J. RALLIS
The value rb can be expressed in terms of n, P; and P, using equation 1, giving
Su=
7] (dn/dt) ra (p,/p)x;~ { 1 - (1 - n) (p,/p)l;r,}.-,/:,
. . . . [8]
The pressures P~ and P are given by the pressure record. The mass fraction n remains to be evaluated in order to solve for burning velocity from equation 8. DISCUSSION ON EVALUATION OF MASS FRACTION
It is clear that n is a function of the radius rb and the pressure P at time t, as evidenced by equation 2. Lewis and von Elbe ~ give the following approximate relation between n and P n = (P - e o / (P~ - P,)
. . . . [9]
Using this relationship, the burning velocity can be computed for a gas mixture from a pressure record of the combustion only. The relationship is approximate because it assumes a linear proportionality between r~, and P. The error involved in this assumption is not large in the early stages of combustion, so that the application of equation 9 must be confined to small values of n. Since constant pressure is assumed to exist throughout the vessel at any instant, the radius of the flame front, as deduced from the pressure record, does not indicate a particular edge in the flame front, but it gives rather some kind of mean value over the whole reaction zone. The pressure P, in equation 9 is taken to be the final pressure at the end of combustion as computed from thermodynamic data, assuming that thermodynamic equilibrium is established in the flame. Consequently the evaluation of Pe is based on the assumption that the temperature distribution in the burnt region is uniform. It is claimed 8 that the values of n computed from equation 9 are precise for small values of n. This is justified by using the values of n so obtained to derive the values of rb at different instants during combustion, and comparing these with the corresponding experimental values of rb. The results given show that in every case the difference was small, being at worst about one per cent of the radius rb. However, it can be shown that a large variation in the value taken for P~ will not affect the resulting value of r,, to any great extent. For example, even for a five per cent variation of the value of Pc, the resulting calculated radius rb is, in most cases, still within one per cent of the observed radius in the cases tested. The actual final pressure obtained from the pressure/time trace is well within five per cent of the fictitious pressure Pe, so that it would appear far more convenient to employ the actual pressure, since the final result is within the same limits of accuracy. The suitability of equation 9 for use in finding burning velocity is somewhat doubtful because, although the error involved may be small in the early stages of combustion in the cases where it has been used, there is 206
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
no guarantee that this would be so for other gas mixtures, particularly when the rate of combustion is high. If attention is confined to the early stages of combustion, as it must be using this equation, the constant volume method loses its greatest advantage, viz. that the burning velocity of a gas mixture can be found over a large range of pressures in a single explosion. Other methods, particularly the soap bubble method, can then give as much information with far less experimental and analytical difficulty. It has been shown that equation 7 makes it possible to achieve greater accuracy in the determination of the burning velocity of a flame than can be obtained by using equation 5 in the early stages of combustion, even when employing what is in effect an assumed value of n. Thus, if the fraction n can be evaluated accurately over the whole combustion range, valuable results for burning velocity would be obtained. Experimental records of the variation of pressure, temperature or flame front radius with time are obtainable, so that the fraction n would need to be found in terms of one or more of these parameters. At the same time the original disadvantage of magnification of error must be avoided, which precludes the use of equation 2 for the purpose. DERIVATION OF MASS FRACTION
The following analysis has for its object a more rigorous derivation of the mass fraction n. The assumptions quoted above are retained, as all of these have been shown by earlier experimenters to be valid. One additional assumption is made here; namely, that the temperature of the burnt gas at any instant is given by the mean temperature throughout the burnt gas volume at that instant. The validity of this assumption is investigated below.
Temperature distribution An element of gas at the centre of the spherical combustion vessel will be at a pressure P~ and temperature Ti immediately before ignition. This element will be the first to burn after ignition. Therefore the combustion of the element will take place at constant pressure P~ and its temperature will rise to Tb, equal to the flame temperature, during combustion (if the external effect of ignition on the element is negligible). Subsequently, its temperature will be further increased due to adiabatic compression as the flame progresses beyond it, until at time t after ignition it reaches a value T~, corresponding to the pressure P at that instant. A second element of gas remote from the centre is first compressed adiabatically. This element then burns at a constant pressure, higher than P~, and is subsequently compressed adiabatically until at time t after ignition it reaches a temperature T~,p less than T~,,,. The process can be illustrated by means of Figure 1. The curves (~) and (~,,) indicate adiabatic compression of burnt and unburnt gas respectively. The rise in temperature with time during adiabatic compression as the flame progresses is greater for burnt than for unburnt gas. An element at radius r0 burns to temperature T~ and its temperature is then raised by compression to T~F after time t, giving point A. Elements at radii r, and r., are compressed as unburnt gas, then burned, and then compressed further as burnt gas during the time interval t, giving 207
K. H. O'DONOVAN AND C. J. RALLIS
the curves 0MNB and 0PQC respectively. An element at r8 burns at time t after being compressed as unburnt gas, giving the curve 0RD. The temperature rise during burning decreases slightly with increase of combustion pressure, so that R D < PQ < MN < L0 and the curve joining
I Ii
r,I
f
I
0
.
t Time
Figure 1
LNQD has a slightly smaller slope than curve 0MPR at corresponding points. A family of curves of temperature versus radius at time t can thus be plotted using such points as A, B, C and D, for radii r0, rl, !"2 and rs respectively. In this case t represents the interval of time that has elapsed from the instant of ignition to the instant when the flame front reaches the radius considered. For example, when the flame front reaches the radius r2, a time t2 has elapsed since ignition and the temperature gradient throughout the vessel is given by the curve EFGH in Figure 2. The temperature of the burnt gas at the instant t2 varies from radius r0 to radius r2 according to the curve EF. In particular, the curve ABCD gives the temperature gradient in the whole vessel at the instant the flame front reaches the wall (radius ra). A
B
to
~ u H lI _ _ ~ ~ ~L- ~ ~ ~2 --'-i.
~ro
.
.
.
.
q
r2 Radius
Figure 2
208
i
5
J
DETERMINATION OF THE BURNING V E L ( ~ I T Y OF A GAS MIXTURE
If, in general, curves such as E F are expressed as T = T (r0, then the mean temperature of the burnt gases at time t is given by rb
~ = (1/r~) J T (r~) drb
....
[Io]
0
At the end of combustion, the mean temperature is ra
T, = (1/r,) f T (rb) drb 0
A general expression for the temperature gradient in the burnt region at any instant is obtained as follows: In Figure 1 the points A, B, C, D, for time t are points on the required curve. Such points are in general arrived at in three stages, viz. (a) adiabatic compression of unburnt gas, (b) constant pressure temperature rise due to burning, and (c) adiabatic compression of burnt gases. Thus
temperature rise, stage (a) = T" - T~ temperature rise, stage (b) = T~ - T'~ temperature rise, stage (c) = T~p - T~
where Tg--temperature at beginning of adiabatic compression of burnt gases (pressure P'), and 7~p =temperature after adiabatic compression of burnt gases to pressure P from pressure P'. Total temperature rise
= ( ~Z t - T , ) + ( T o - TI ~ )
t
4 - ( TI~ - T ' b )
= Tip - T~
= [(P'/P)~I-~Plrb] T'b - T, Let T b - T u = T u = temperature rise due to combustion from temperature T u l'herefore t
•
•
•
temperature rise = (P' / P)~-~b)/~b [T', + T~ (P~/ P')~-~.'~] - T~ Therefore the temperature at any point on the curve EF (Figure 2) is
T = (P" l p)(~-rb)!rb IT', + T, (P, I P')<~-~,)I~,]
. . . . [11]
The pressure P" is a function of rb, since it is the pressure at which burning takes place at any particular radius rb. Hence equation 11 is an expression for the temperature gradient in the burnt region at time t in terms of the radius of the flame front at that instant. T~ is the temperature rise of the gas when burnt at pressure P'.
Mass equations The mass of the system as a whole is the same at the end as at the beginning of combustion. Hence m~----ma
i.e.
Piv / RuTi = P~v~ / RbTa 209
K. H. O'DONOVAN AND C. J. RALLIS
Also, the volume v at the beginning is the same as the volume v~ at the end of combustion. Therefore Ti / T~ = PiR~ I PaR. or
(TJTO (Tb/To)=P,R~IP~R .
.
.
.
[12]
.
The ratio n is given by m b / m . i.e. n = (Pvb/R~Tb)/(P~v/R~TO
Therefore
. . . . [13]
n =PTar~/P~T-br~
Now
mi = mb + m,,
i.e.
P~v / R,, T~ = Pv b / R bTb + Pv,, / R,,T,,
or
P e a3 / R . T ~ = P r 3b/RbTb + P (r~3 - r b3) / R u T ~
Hence
r~ /r~ = ( P U P T ~ - I / T u ) / ( R ~ / R b T b - 1/T=)
. . . . [14]
where Tu = (PIPi)%-')l~',,. T,
Substituting equation 14 in 13 gives
n : To .~
P-P,(PIP,) ~"-'/~
and when using equation 12 T. ~ n=
~--
~a'~/D (Tu l)/Tu P-l"itr!Pi)
~'nx(y
Tb t P . - ( T . / T O P , ( r / r o
u
"]
-
1)l~
....
[151
u
Comparing equations 15 and 9, it is noted that two additional terms, T. and Tb, must be evaluated in order to solve for n. Equation 15 has the considerable advantage of being valid for all values of n, as well as being subject to fewer assumptions, thus rendering it more exact. PROCEDURE FOR DETERMINING BURNING VELOCITY The solution of the burning velocity of a gas mixture using a constant volume explosion in a spherical vessel is obtained by the application of equations 8, 10, 11 and 15. For convenience, these equations are rewritten below, in the order in which they are to be applied. T=
(P"IP)<~-~'Pi~'~[T'n + T~ (PU P')<'-~>/q 210
....
[11]
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE rb ¢.
= 1/rb ~ T fib) dr, .
.
.
.
.
[10]
0
n:S, =
T~, t. p , _ (T,/T~,) P~ (P/P3
....[15]
½ ( d n / d t ) r~ (P,/P)~/r,
{ 1 - (l - n) (P,/P)~/v,} ''/~
. . . . [8]
Consider a series of radii r~ taken at equal intervals. The pressure obtaining in the vessel when the flame reaches radius rb is P' and is given by a correlation of the pressure/time and radius/time traces. At a particular time t after ignition, the pressure is P throughout the vessel. The temperature at the radii taken are given by equation 11. The curves of temperature versus radius are plotted for various time intervals between the beginning and end of combustion. The mean temperatures obtaining at these time intervals are most easily found by graphical integration of these curves, the result being a solution of equation 10. The mass fraction n is evaluated using equation 15 for various pressures corresponding to the time intervals chosen, and plotted against time. Finally, the burning velocity is computed from equation 8, inserting the corresponding values of d n / d t , n, P and t as found above. Observations required
To carry out the procedure outlined above it is necessary to have available sufficient experimental results. The following information is required: (a) Constituents and proportion of gas mixture (b) Initial temperature T~ and pressure P~ immediately before ignition (c) Record of pressure P versus time t throughout combustion (d) Record of flame radius rb versus time t throughout combustion (e) Radius of spherical vessel. T h e r m o d y n a m i c constants
In addition to the experimental values enumerated above, it is necessary that 7,, 7~, and T~ be evaluated for each mixture at the particular condition of temperature and pressure. For a mixture of perfect gases, the mean value of ,/is defined by: x~C~,,, ~£ m ~ A H . 7 = ~ x~,C~,,, -- ~ m,,AE~,
. . . . . [16]
where the suffix q refers to each constituent present, xq=mole fraction, C,0 and Cvq=mean molal heat capacities at constant pressure and constant volume respectively, AH,, and AEq = change in specific enthalpy and internal energy respectively, and m~ = mass of constituent q present in the mixture. The determination of 7, presents no difficulties, since the mole fractions 211
K. H, O'DONOVAN AND C. J. RALLIS
of the various constituents are not significantly affected by temperature. Also, the use of a mean value for ~/ is justifiable, since the temperature changes are relatively small. For the evaluation of 7b, it is first necessary to determine Tb""(= T"n+ T u). " This is done by the simultaneous solution of the two equations--the dissociation and energy equations--for the layer of mass dm, from the given initial conditions of temperature T~ and pressure P'. From this the composition of the equilibrium mixture at temperature Tg may be found. During subsequent adiabatic compression of these combustion products, the equilibrium shifts, and alters the mole fraction distribution in a determinable manner. Thus, since all the xqs are known, and the AH~s and AEqs over the temperature range are available from tables, the mean value of ~b can be calculated. SIMPLIFICATION
It is possible to reduce the computational and experimental labour involved in the above analysis by making certain simplifying approximations. These would inevitably lead to somewhat less accurate results. However, this may often be acceptable as a check or, if the differences are insignificant, such approximation may be utilized directly. The only really satisfactory way of validating such approximation is by a sufficient number of experimental comparisons.
General approximation It seems reasonable to assume that the temperature rise due to combustion (T~) is constant, since the actual variation of this quantity is relatively small. This renders equation 11 and hence 10 fairly easily soluble. Burning velocity using pressure~time records only The value of the burning velocity at any instant during combustion can be obtained using only equations 15 and 8, if a satisfactory average value for the mean temperature T~ is used. The advantage of doing this is that the burning velocity can then be obtained using only a pressure/time record Of the combustion. If the ratio of Ta to Tb in equation 15 is assumed equal to unity, then results of the same order of accuracy as those obtained by Lewis and yon Elbe ~ using equation 9 can be expected. This approximate form of equation 15 is equivalent to equation 9 except that it is valid throughout the full range of combustion. It is possible to improve on this by noting that the ratio Ta/T~ varies from T~/Tb at the beginning, to unity at the end, of combustion. The values of T~ and Tb can be determined by taking a temperature/time record at the centre of the spherical vessel during combustion. The manner in which the ratio varies between Ta/Tb and unity can be assumed or further points on the curve plotted by using temperature/time records at various radii in the vessel. Burning velocity using radius~time records only Consider the combustion process at any instant when the flame has propagated to a radius rb. Here m b + m~ =mt 212
DETERMINATION OF THE BURNING VELOCITY OF A GAS MIXTURE
i.e. Pr~ P (r~ - r~) P,r~ RbT"~-~b+ RuT. = R.TI Substituting for Tu=T~(P/P~)Cr-I)/ru and rearranging gives p,/~. + [- r~ R . T , . p,,_y,,)/r. 1 P ra" t_(r] - r~) " R j ' ~ (r~ - r~,) Pi'/*" i.e.
P~ /~ + CaP = C~
. . . . [17]
Here it is apparent that for any set of initial conditions and observed flame radius rb, C1 and C2 can be evaluated provided Tb can be determined or estimated. Values for P at different radii rb then follow. Use of these results in equation 2 leads to values for n and thus d n / d t . Substituting in equation 7 then yields the burning velocity. Alternatively, of course, if both rb and P are measured, equation 17 can be used to solve for T~. The burning velocity can then be determined by using equations 15 and 8. CONCLUSION
The general analyses outlined above are intended to provide a method of solving for the burning velocity of a gas mixture that is at once accurate and relatively simple. The accuracy is, of course, dependent on the validity of the assumptions made. These assumptions are, in general, the same as those made in the references quoted 5, ~. The accuracy obtainable by this method is not implicitly greater than that of the previous methods, but the possibility of obtaining satisfactory results is increased by avoiding any magnification of incidental small errors of observation and calibration. The usefulness of the constant volume method of obtaining burning velocities of gas mixtures is in this way extended. Department of Mechanical Engineering, University of the Witwatersrand, Johannesburg, S. Africa (Received September 1958) REFERENCES
'Burning velocity determinations, Part III. Burning velocities of ethylene-air-carbon dioxide mixtures.' Trans. Faraday Soc. 1951, 47, 179 2 Lir~,~yr, J. W., PmKERING,H. S. and Wm~^TLEY,P. J. 'Burning velocity determinations, Part 1V. The soap bubble method of determining burning velocities.' Trans. Faraday Soc. 1951, 47, 981 3 CON^~, H. R. and L~'~rETr, J. W. 'Burning velocity determinations, Part V. The use of schlieren photography in determining burning velocities by the burner method.' Trans. Faraday Soc. 1951, 47, 981 4 PmKER~NO,H. S. and LI~3"T, J. W. 'Burning velocity determinations, Part VI. The use of schlieren photography in determining burning velocities by the soap bubble method.' Trans. Faraday Soc. 1951, 47, 989 LINt,rETr, J. W. and Ho^RE, M. F.
213
K. H. O'DONOVAN AND C. J. RALLIS FiocI~, E. F., MARVIN, C. F., CALDWELL,F. R. and ROEDER, C. ]-I. 'Flame speeds and energy considerations for explosions in a spherical bomb.' Nat. Adv. Comm. Aero., Wash., Rep. No. 682 (1940) LEWIS, B. and YON ELBE, G. Combustion, Flames and Explosions of Gases, p 448 ft. Academic Press: New York, 1951 z LEwis, B. and voN ELBE, G. Combustion, Flames and Explosions of Gases, p 651 ft. Academic Press: New York, 1951 s MANTON, J., VON ELBE, G. and LEWIS, B. 'Burning velocity measurements in a spherical vessel with central ignition.' Fourth Symposium (International) on Combustion, p 358. Williams and Wilkins : Baltimore, 1953
214