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Effect of ignition geometry on initial flame acceleration in a spark ignited explosive gas
Effect of Ignition Geometry on Inith l Flame Acceleration in a Spark Ignited Explosive Gas* A. J. LADERMAN, P. A. URTIEW and A. K. OPPENHEIM University o] California, Berkeley (Received April 1962) The influence of the geometry o] ignition on the initial flame acceleration in spark igmted, stoichiometric hydrogen-oxygen mixtures at n.t.p, was investigated experimentally. Observations were made using sehlieren photography with simultaneous pressure measurements in the vicinity of the igniter. It was found that flame acceleration ,ncreased when the distance between the ignition source and the backwaU became larger provided that the flame was in continuous contact with the backwall. When the ignition source was extended ]arther into the tube so that the flame contacted the sidewalls before reaching the back end, its acceleration became independent of the distance of the point of ignition from the backwall. Experimental results were interpreted quite satisfactorily by means of a one-dimensional gas wave dynamic analysis, demonstrating a significantly larger dependence of the flame acceleration process upon the normal burning velocity than on the net amount of heat released per unit mass.
of detonation, the present study was carried out in order to examine the importance of the geometry of ignition in this respect. In an effort to interpret some of the experimental results obtained previously TM, we analysed recently '5 the effect of pressure waves reflected from the closed end (backwall) of the detonation tube and demonstrated how the separation between the backwall and the igniter influenced the strength of these waves and their subsequent interaction with the flame. We concluded, however, that although some significant details of the process were indeed altered, the detonation induction distance was relatively insensitive to variations in the backwall distance. The analysis was performed under a tacit assumption that during the time when the flame front was expanding over the cross section of the tube, the heat transfer to the walls was the same for each igniter position. While this assumption was justified by the operating conditions of these experiments ~4 (pilot flame ignition in Plexiglas tubes), it became of interest to investigate the initiation of explosion under circumstances where variations in the backwall distance would be associated with significant changes in the heat transfer to the wall.
Introduction THE onset of detonation in gaseous mixtures has been studied experimentally by H. LE CHATELIER 1, S . B . DIXON 2, P . LAFFITTE 3, A . SOKOLIK a n d K . I. SHCHELKIN 4, A . C . EGERTON and S. F. GATES "~, W. PAYMANand H. T I T I A N ~,
W. A. BONE et al. 7, E. SCHMIDT et al. s, J. J. TURIN and J. HUEBLER% B. GREIFER el al. ~°, and, more recently, by G. D. SALAMANDRAet al. 1~, F. J. MARTINr', L. E. BOLLINGERet al. 1"~, W. BAUMANN et al. 1'1, and by A. J. LADERMAI~ and A. K. OPPENHEIM15'Is. While our understanding of the processes of transition from deflagration to detonation was definitely enhanced by these investigations, the development of a complete theory has been hampered by the lack of attention given to initial conditions, especiallT to the mode of ignition and, in particular, to the position of the igniter in the detonation tube. Since it was reasonable to expect that the initiation is of considerable importance to the subsequent development of the highly non-linear process of the development *This research was supported by the United States Air Force. through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract AF 49(638)166 and the National Aeronautics and Space Administration under Grant No. NSG-10-59. 325
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A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
For this purpose we performed a series of experiments in our 1 in. x 1{ in. detonation tube, whose 1½ in. walls and the closed end are steel and the 1 in. walls are glass, with spark ignited stoichiometric hydrogen-oxygen mixtures initially at n.t.p., varying the distance between the spark gap and the backwall. When the spark gap was very close to the backwall, the flame spread out as a hemisphere until it reached the sidewalls. When the spark gap was sufficiently far away from the backwall, the flame grew as a sphere until it became distorted by contacting sidewalls before it reached the backwall. From then on further shift of the gap away from the backwall did not affect the heat transfer to the walls. On the other hand, until this instant the flame was sufficiently close to the backwall so that the influence of waves reflected from the end was negligible. As pointed out before 15, the backwall plays then the same role, in this respect, as the plane of symmetry of the flame kernel, the effects of reflected waves becoming noticeable when the ignition centre is much farther away from the end than it was in these experiments. The sole reason for the observed changes in initial flame acceleration could be ascribed, therefore, to heat transfer effects. To ensure this, the influence of ignition energy over a limited range of values was also investigated, but within the range of operating conditions used in our tests it was found to be negligible.
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the undisturbed baseline) it was considered as due entirely to the noise inherent in the electronics rather than indicative of any real process. The amplitude of the noise was 0.50 lb/in -°, significantly smaller than the real signal. A complete description of the apparatus can be found in ref. 16. Ignition was performed by means of a spark discharge. The spark igniter, Figure 1, was installed at the geometric centre of the backwall, with the centre electrode concentric with
ExperimentalApparatus Observations of wave processes that accompany the initiation of explosion by spark discharge in stoichiometric hydrogen-oxygen mixtures at n.t.p, were made in a lin. x 1½in. cross section detonation tube 1~ by means of streak schlieren photographs with simultaneous measurement of pressure at several positions within the induction distance. For this purpose a Kistler Instrument Corporation PZ-6 SLM pressure transducer was used and the measurement was recorded by means of a Tektronix 551 oscilloscope. Although the gauge was shock mounted to eliminate ringing, a small amplitude, high frequency signal was observed on the oscilloscope trace. Since this signal occurred under varied operating conditions (and even on
Figure 1. Typical spark igniter the tube axis. The end of the spark plug, with the exception of the electrodes, was flush with the inner surface of the tube. The centre electrode (anode), 0-063 in. diameter, was insulated with a 0-006 in. layer of Teflon, leaving, at the free end, an exposed length 0-01 to 0-02 in. long. The ground electrode, a bare 0.09.0 in. diameter wire the same length as the anode, was silver
December 1962
Ignition geometry and flame acceleration in explosive gas
soldered to the outer housing of the spark plug. Spark plugs with electrode lengths of 0, ¼, ½, a, 4 1, and 1½in. were used. As a result of the spark plug construction and since the spark discharge path was restricted to the free end of the electrodes, the electrode length was identical with the backwall distance. Furthermore, because of its small surface area and mass, heat transfer to the electrodes was negligible. Hence, extending the electrodes into the test section was considered to serve only as a means of positioning the point of ignition with respect to the baekwall. The spark electrodes were positioned parallel to each other, separated by a 0'02 to 0.03in. gap. For streak photography the windows of the tube were masked off with the exception of a narrow slit parallel to the axis of the tube. In this instance it was necessary to have the slit width equal to, or greater than, the electrode gap*, otherwise, some error of interpretation could have been introduced into the photographic records. If the gap were partially obscured from the view, then the flame would reach the edge of the slit with already some finite size radius, but giving a false appearance of a point explosion. The flame would subseqently appear to move faster than it actually does. A displacement between the gap and the slit would, therefore, produce false information on the initiation of the process. In order to avoid this the slit was positioned with particular care. This was accomplished by shining a parallel beam of light through the test section forming thus a shadow of the edges of the slit on a translucent screen which was placed behind the slit. The slit was then adjusted until the inner edges of the electrodes appeared on the screen, superposed on the edges of the slit. The spark was produced by discharging a capacitor, charged to 200V, through a 2D21 thyratron into the primary of a Thordarson T22R44 step-up pulse transformer. The spark electrode was connected to the transformer secondary, Figure 2. Capacitors ranging from
0"1 to 80#F were used producing a variation in ignition energy of 0.1 to 6 mJ. To oscilloscope trigger
0"1-80 pF
Delay
unit
~0"-~ _ ~ ~Peacrtkr odes, Thyratron ' = ~ trigger _L -[_
._.~ "---
Thordarson T22R44 D,_
To light source trigger
Thyratron trigger
Figure 2.
Schematic diagram of ignition circuit
Results Position of ignition source Typical streak schlieren photographs of initial flame acceleration for backwall distances of 0, ¼ and ½ in. are shown in Figures 3 to 5, respectively. The bright curved band represents.
0 *Ideally. the slit should be as narrow as possible to avoid i m a g e smearing, the minimum slit width being determined by the m i n i m u m light intensity required for exposure. T h e use of a 0'04 in. slit width, however, did not affect the optical quality as evidenced by the experimental records.
327
0,10
0-20 X
Figure 3.
0.30
0"40 rn
Streak schlieren photograph--O backwall distance
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A . J . Laderman, P. A. Urtiew and A. K. Oppenheim
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tube have been transformed to profiles at a fixed distance from the point of ignition whose position in the tube was varied. This was accomplished by taking advantage of the fact that the pressure wave is initially a simple one, and that consequently the process across it is isentropic while state parameters along its world-lines are invariant. The results are shown in Figure 7 for x = 2-62 cm, and in Figure 8 for x = 12.78 cm. Figure 7 depicts the increase in the rate of pressure rise that results from the shift of the point of ignition from 0 to ½ in. away from the backwall. The records are terminated when the flame contacts the glass sidewalls of the tube across the 1½in. width; the discontinuity in the slope of the profile is due to the flame contacting the steel sidewalls across the 1 in. width. Figure 8 shows the full pressure profile for each backwall distance at x = 1 2 . 7 8 c m from the igniter. In this instance all the flames were in contact with the sidewalls of the tube within the period of the 0
0"10
020 X
Figure 4.
0'30
0"40 m
Streuk schlieren photograph--~ in. bac,% wall distance
the flame front, while the dark areas and lines ahead of the flame represent pressure waves generated during the process. The flame worldlines have been reproduced in Figure 6 where, for comparison, the points of ignition have been shifted so as to coincide at the origin. Also included there are flame world-lines for backwall distances of ~, 1 and l~r in. The influence of the backwall distance is quite pronounced, the flame acceleration increasing significantly when it is varied between 0 and ½in. With further displacement of the spark gap, however, the development of the process remains almost unchanged, only a slight difference in flame velocity being observed when it was shifted from ½ to 1½in. from the backwall. The effect of the backwall distance is reflected quite clearly in pressure records which have been obtained at positions 2.62 and 12-78cm from the end of the tube. In order to set a proper basis for interpretation the pressure/time profiles measured at a given cross section of the
E 125-
100-
0750.50-
025-
0
Figure 5.
0.10
0'20 X
0 30
0 40
rn
Streak schlieren photograph--~ in. buckwall distance
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Ignition geometry and flame acceleration in explosive gas
measurement for, at least, the last several hundred microseconds. Judging from Figure 7 it appears that the pressure has about the same value when the flame first contacts the sidewalls, irrespective of the distance between igniter and backwall.
329
negligible and the flame temperature and hence the average overall reaction rate are then greater than for x~:=O. This is then reflected in the subsequent progress of the process so that at the distance of 12"78 cm from the ignition source as ~26
1-O
S qo Sq m/s catlg mls A 45 2050 B5 B 55 2050 -C 45 2050 100 D 45 2050 -E 40 2050 -xE, Distance between backwa[[ and ~gnlter, in ~ Observed pressure - - - - Computed pressure
Curve
c
E 0.9
24
08
22
07
20
B
06
1)
~114
z,, 2 /
,5"
18
/
/
D
xE=O
Ao /
O5 16
0'4
14
03
I
100
i
I
200
(
300
la sec
t
i
02
XE, D i s t a n c e and
between i g n i t e r , in.
backwat[
Figure
7.
Pressl~re histories
at
location
2.62
cm
from ignition
0.1 i
0
Figure 6. h~storics
5 lrO 1~5 2'0 Distance from igniter, x
I
25
30 cm
Comparison of experimental space~time of accelerating flames for all backwalI dis tanees
Since from then on the contact area between the flame and the sidewalls is essentially independent of the preceding process, one could expect that the rest of the pressure profiles would eventually become identical, the effect of the position of the igniter being reflected only in the shift of the profile in space. Figure 8 demonstrates, however, that evidently this is not so. The maximum pressure and the rate of pressure rise for xT~>0.Sin, are both larger than for x~ = 0. This can be explained by noting that when the flames contact the sidewalls, they are physically dissimilar, although they appear geometrically identical. The reason for this is indeed the fact that heat losses to the backwall occur so late for x~ ~> 0-5 in. that their effect is
shown in Figure 8, the pressure waves are much steeper and attain higher values, if at initiation the flame did not touch the backwall. Ignition energy To determine the effect of ignition energy on the development of the process, extensive tests were performed with the various igniters for ignition energies over the range from 0-1 to 6 m J, several tests being run at each condition. In contrast to the influence of backwall distance, no change in the process was observed within the variability of its progress. In fact, for a given backwall distance, it would be expected that flame acceleration should remain independent of the ignition energy until the amount of energy deposited in the gas was either large enough to generate pressure waves of sufficient strength to disturb the unburnt gas from its initial state, or too small to maintain the reaction. Apparently neither limit was reached in the present case.
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A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
Analysis Effect of heat loss to wall on the growth pressure w a v e
of
the
The relatively slower flame acceleration associated with the shorter electrodes is attributed to heat losses to the backwall. Let us consider a simple model of a flame expanding at first spherically about the point of ignition. This
/
Recently we reported lz on measurements of the rate of heat transfer from the combustion zone to tube walls during the development of detonation. We found it to be a rapidly decaying function of distance behind the flame almost independent of its position and state. The rate of heat release per unit frontal area of the flame propagating at a steady state can then be expressed as follows: co = qlplS, = I~p2So - I"~p~S~
+½(s~-s~,)p,s,+g,.
"iS l III W"
:0'
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°f,o
I
20
x.c
,
and tgniter, Distance between in backwaI]
36o 4bo sbo Gbo 76o 8~o 9oo' t
~l.$ec
Figure 8. Pressure histories at location 12.78 cm ~rom ignition
model has been largely substantiated by flash schlieren photographs of the initial growth of the flame front. When the flame contacted the walls of the detonation tube, heat losses occurred which tended to retard the process of flame acceleration. For electrode lengths less than 0.Sin., the flame contacted the backwall of the tube before reaching the sidewalls, while for electrodes extended farther than 0"5 in. the converse was true, so that in this case two flames propagating away from each other were eventually produced. When this happened heat losses to the backwall were not sensed by the forward running flame which exhibited then the largest acceleration. This would explain why the development of the process remained essentially unchanged when the backwall distance was increased from 0-5 to 1.5 in. Such a conclusion is, of course, based on the premise that the effects of reflected pressure waves were then still negligible but this has been already demonstrated ~4 to be quite true under present circumstances.
....
[1]
where p denotes density, p is pressure, S is relative flame speed with respect to the gas, I~,i=7;/(7(- I) (7( being the specific heat ratio), q is the heat released per unit mass and QL is the rate of heat transfer to the walls per unit area of flame surface; the gas immediately ahead and behind the flame is denoted by subscripts 1 and 2 respectively, while its behaviour has been assumed to obey the perfect gas equation of state (with different specific heats and molecular mass, however, for reactants and products). A disturbance in the flame propagation process changes the heat release rate from to to to + ~to, and pressure waves are generated at the flame front, increasing pl to pl+~p~, S~ to S~+~SI etc. With restriction to the first order terms, equation 1 leads to the following expression for the pressure wave generated b y the flame
~p,_
~,- 1
(go -~QL)
. . . . [2]
where a 1 and a~. are local velocities of sound in front of and behind the flame respectively. The above formula was derived first by BOA-TEH CHO1~ and its validity verified by us IB with reference to the problem at hand, but without consideration given to heat losses. When the flame front surface is curved rather than plane, the total heat release rate is A1plS q where A t is the flame surface area and S the normal, relative flame speed. The rate of heat release per unit frontal area referred to the tube cross section A, is then = (AdA,)
p,Sq
....
[S]
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Ignition geometry and flame acceleration in explosive gas
and equation 2 becomes 7.~ - 1 l 3A I ) ('t._,/ 7~) a, + a~ [piSq ~ - ~Q,.. . . . [4] where the change in the product p~Sq is assumed negligibly small in comparison with the change in A 1. The effect of heat losses on the generation of pressure waves can be estimated as follows. For ignition at the backwall, the flame expands almost as a hemisphere during the few hundreds of microseconds that it requires to reach the sidewalls. According to our measurements ~7 the heat loss to the wall can be expressed approximately as Q::Qo e -~1' . . . . [5] where Q is the heat loss per unit time per unit surface area of the tube, Qo ~ 800 c a l / c m 2 sec, T ~ 10 -4 sec, and r is time measured from the arrival of the flame at a given position, x. The surface area of the backwall exposed to hot combustion gases is ~x ~. Assuming that the absolute flame velocity, S~=S~+u~ (% being the particle velocity immediately ahead of the flame) is constant and equal to 8 5 m / s , taking r = t - ( x / S ~ , ) and integrating equation 5 over the exposed surface area between the limits x = 0 and x = S . t , we obtain the expression for the total heat transfer rate to the wall at any time t. The change in the average heat transfer rate with reference to the frontal area of the flame, At, that occurs in the time interval 3t, can be then determined simply by differentiation with respect to time, yielding g~QL= (2~S]QoT/At)[1 - e -'/~] ~;t . . . [6]
Substituting numerical values into equation 6 yields 3QL ~ 60 eal/cm-" sec for i~t=20/~sec. At the same time, taking for example the interval between 80 and 100#see after ignition, the flame, moving at an average speed of S = 5 0 m / s , increases in surface area by 3Ai= 1.8em 2. With the effective heating value for stoichiometric hydrogen-oxygen at n.t.p., q ~ 2 kcal / g, equation 3 gives ~co=p~Sq~Ai/At~ 1100 cal/cm 2 sec. The heat loss to the backwall constitutes then 6 per cent of the rate of heat released b y combustion.
331
Influence of Normal Burning Speed In order to assess the influence of burning speed it is convenient to lump the rates of heat release and heat loss into a single term by introducing an effective heat of reaction q e = q - ( Q L / p ~ S ~ ) so that equation 4 becomes 3'2 - 1 p,SqfiA, 3P1= (7./'yl)a~+a~ - At
. . . . [7]
As the spark electrodes are moved away from the backwall, the amount of heat transferred to it decreases, the flame temperature and reaction rate approach their adiabatic values, and both q~ and S increase. Since the values of S and q, are difficult to predict a priori, and since the growth of the flame surface area is itself a function of S, in order to arrive at a correct interpretation of experimental data the relative influence of changes in S and q, on the growth of the pressure wave have first to be determined. This is accomplished by a linearized analysis of the growth of the pressure wave on the basis of equation 7. For this purpose we assume first that to the first approximation a_, = [ m - (v_,/'/,)] . ,
where m is a constant determined from an energy balance for an adiabatic combustion process. Equation 7 then becomes 3p p0
_. 72 - 1 p~Sq_e3A l m~
. . . . [8]
a,p,,
Assuming that the coefficient is essentially constant, equation 8 can be integrated simply to P / P o = 1 + KSq~A t
. . . . [9]
where K=T.- -1
Po
mA, aop~
(7~ - 1) 70 mata~ subscript 0 referring to the state of the undisturbed medium ahead of the pressure wave. Within the simple pressure wave which precedes the flame, the pressure and particle velocity are related by the isentropic equation P/Po = [1 + ½ (To- 1) U/ao]2,d(,o -1)
332
A.J. Laderman, P. A. Urtiew and A. K. Oppenheirn
which, upon expanding the RHS and retaining only the first order term, can be reduced to p/po=l+you/ao
....
[10]
Equations 9 and 10 yield then the following simple expression for the initial 'piston action' of the flame u= a°Ksqeat
....
[11]
7o For a hemisphericaUy expanding flame front A t = 2wx ~, so that equation 11 expresses in effect the following velocity/displacement law of the flame 'piston': u=KlSq~x 2 . . . . [12] where K1 = (2~a0 / Yo) K 27 (V: - 1) As a consequence of the relative burning speed, S, such a piston is, however, not impervious; its actual displacement is given b y dx/dt=S+u
....
[13]
which with equation 12 yields, upon integration,
1
x = (K~q~)~/~ tan t (K~S2qe) ~/~
...
[14]
The growth of the pressure wave is then described in terms of S and q, by means of equation 9 with A t = 2 w x ~ and x expressed by equation 14 giving + Y°Stan: y
. . . . [15]
a 0
where y --- t (KiS:qe) x/°"
Expanding tan2y as a power series in terms of the argument y, we obtain finally
P/Po=
1 + X C,K~S2"+~q,'t ~-~ n
{t ( K t S 2 7 ~ ) ' / 2 < ~ / 2 }
....
[16]
with C1 = "/o
a,,'
C2 _ 2y o 3a0 '
they have been evaluated for a 2H2+ 02 mixture, initially at n.t.p, in our l in. x 1½in. detonation tube. With S = 5 5 m / s , qe=2050 c a l / g and m = 4 , equation 16 yields P / P o = 1 + 3.84 × 10-6t 2
+ 7.28 x 10-11t a + 9.32 x 10-1~t 6 where t is in microseconds. From the form of equation 16 it becomes obvious that when changes in S and qe are small and of the same order of magnitude the influence of the former on the development of the pressure field ahead of the flame obscures completely the effects of the latter. This applied in the present investigation. Both S and qe changed approximately by 15 to 20 per cent over a range in x~: from 0.5 to 0 in. so that the influen.ce of S on the pressure w a v e w,as threeto /our-fold that o/ qe. This effect is reflected
m A ta2o
p/po=l
VoL 6
17y,,
C~ = £gh~
where, for the time interval considered, higher order terms can be neglected. To demonstrate the relative magnitude of each term in the series,
clearly in the following results obtained by a numerical analysis. A n a l y s i s of flame acceleration
The formation of the pressure wave ahead of the accelerating flame was calculated for the two limiting cases, that is for backwall distances of 0 and 0.5 in., by means of equations 4 and 6. In the former one, the flame grew in the form of a hemisphere, in the latter it expanded as a sphere, but, according to our theory the surface area only on the positive x side from the point of ignition contributed toward the generation of pressure waves moving in that direction. Since the cross section of the detonation tube is rectangular, the flame contacted first only two walls. Between this time and that of reaching the second pair of walls, the flame surface was still hemispherical in shape, but its area was decreased by the segments intercepted by the sidewafls. The flame surface area is given then b y A ~ = 2 ~ x 2 =6-28x 2 . . . . [17] when x < r~ and A t = 2 ~ r l x = 7"91 x when r ~ < x < r 2. In the above 2 r , = 2 . 5 4 c m and 2r2=3.81 cm are the width and depth of the tube, respectively. The change in the growth of the area at x = r 1 is reflected in the abrupt
December
1962
I g n i t i o n g e o m e t r y a n d f l a m e a c c e l e r a t i o n in e x p l o s i v e g a s
change in the slope of the experimental pressure profile, Figure 7. The change in A i during a time interval 3t was computed using the relation ;L~
=
(S
+
u)
~t
....
300
:1.
33.3
Curve A of Figure 7 S : 45 m/s, xE : 0
/ ~_67.Q /~-f~~609
......
[181
with
&,- (a, l./,p,) @, On the basis of schlieren records the following values were established for the normal burning speed : for x r - 0 S = 4 5 m / s + 10 per cent for xT,:=0"fin. S = 5 5 m / s + 1 0 p e r c e n t As a consequence of the assumption that the combustion mixture obeys the perfect gas equation of state, the heat of reaction q had to be evaluated so as to produce compatible results. Accordingly, q was taken as 2 0 5 0 c a l / g , the value that would produce the actual Mach number of detonation computed for the same perfect gas as that used in our theory. The flame world-lines and the pressure rise immediately ahead of the flame during the course of its acceleration were calculated using equations 4, 6, 17 and 18. For comparison with experimental observations the computed pressure profi]e was transformed to a pressure/time history at a section 2.62cm from the point of ignition. The flame world-line was determined from the relation between the flame front radius, x, and the time, t. Each point on the flame world-line represents in effect the origin, in space/time, of the pressure wave world-line associated with the pressure corresponding to that point. Consequently, the pressure wave world-lines could be positioned on the x / l plane, and the desired pressure/time profile determined from the intersection of the worldlines and the time coordinate at x - 2 . 6 2 cm.
Results and discussion Flame and pressure wave world-lines are shown in Figures 9 and I0 for backwall distances of 0 and 0'5 in., respectively, giving the comparison between the analytical and experimental results. The numbers adjacent to pressure waves represent their velocities in m / s . The computed pressure/time profiles for xT;=0 and 0-5 in. are shown in Figure 7 as curves A and B, respectively.
~'~~-I
Yz_"
200
100
~
/ 7 - ......
# - ~
--
~ ~
....
5~z' - s s I J s~9 ~ - - "
......
/,, ~// ¢
0
Calculated world- hnes Observed
-----
I
I
{
10
2'0
30 cm
Figuz; 9. Computed space~time history of accelerating flame--O bachwall distance (Numerals adjacent lo 'world-lines denote velocities in m / s )
In view of the simplifying assumptions used for the analysis, the agreement with experimental observations should be considered quite satisfactory. In particular, the close coincidence between the points of sudden change in the slope of the experimental and theoretical pressure profiles gives certainly a satisfactory justification for the analytical model.
;oo]
Curve B of Figure 7 S : 55 m/s, x E- 1/2m e s-
.,~.%:-
200[
_ _~'67.5 62~1
~
~71~
-
/ j -.,7._---
-
/ f
lOO
/
Ca
....
::~ -
:
;:-u.622gou 694i J
6o.
AI ssq ~Sg/ 5L~3 ---E
[cu[at ed world- hnes
Observed
1EO
2"I0 x
I
3"0 crn
Figure lO. Computed space~time history of accelerating flame--½ in. backwall distance (Numerals adjacent to world-lines denote velocities in rn /s)
334
A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
The sensitivity of the results to the heat loss term, ~QL, was tested in several ways. First, although the absolute flame speed, Sa, actually varied from 45 to 125 m / s during the time interval under consideration, ~QL was evaluated from equation 6 which was derived under the assumption of constant Sa with S ~ = 8 5 m / s . This then led to an over-estimate of ~QL at the initiation of the process and an under-estimate at later times. Some idea of the effect of simplifications underlying equation 6 was obtained by repeating the calculations with S~ = 100 m / s . The larger value of S~ exaggerated the influence of heat loss over a greater portion of the entire time interval of interest and, consequently, produced a retardation in the development of the process. However, the resulting pressure profile (curve C, Figure 7) was found to differ by an insignificant amount from the original pressure profile (curve A). Secondly, the calculations for x E = 0 were repeated using equation 7 in place of equations 4 and 6 so that the effects of ~QL were absorbed in the effective heat of reaction, qe- The pressure profile computed with S = 4 5 m / s and q~= 1 750 cal/g is for all intents and purposes identical to curve A (Figure 7) and for this reason it has not been plotted there. This procedure provided moreover direct means for estimating the total heat loss which is then expressed simply by the ratio ( q - q,)/q. In this instance 15 per cent of the energy released by chemical reaction has been absorbed by the backwall of the detonation tube. On the basis of these considerations then, it is concluded that, when the heat loss is small, the analysis is insensitive to the form of the term which expresses this quantity. In particular under these circumstances heat loss effects may be accounted for quite accurately by the use of a constant 'effective' heat of reaction. The relative influence of changes in S and q~ associated with changes in x~ was tested by repeating the calculations using equation 7 with the same value for the effective heat of reaction qe = 2 050 cal / g but with different normal burning speeds. The results, shown in Figure 7 as curves D for S--45 m / s and E for S = 4 0 m / s are in much closer agreement with
Vol. 6~
the experimental pressure profile for x~ = 0 than with that corresponding to xE=0"5 in spite of the fact that heat loss effects on the value of qe have been neglected. As demonstrated by equation 16 the influence of heat losses on the development of the process is reflected considerably more in the decrease of the flame speed, S, than in the associated decrease of q, (or increase in QL). This is, of course, a direct consequence of the hemispherical model assumed for the expanding flame and may not apply under different conditions. It should be concluded, however, that under the experimental conditions of this study, i.e. point source ignition in a combustion tube, the flame speed, S, plays a prominent role in the acceleration of the flame front.
Conclusions The influence of the position of the igniter on initial flame acceleration is manifested in two ways, by the action of pressure waves reflected from the backwall of the tube and by heat transfer to the backwall. Interestingly enough, the two effects are uncoupled, at least when the tube diameter (or width, in the case of a rectangular cross section) is not greater than several centimetres. Furthermore, the scale of the two effects is different, i.e. a much larger change in the separation between the igniter and the backwall is necessary for the production of noticeable change by reflected waves than by heat losses to the backwall. It has been shown T M 1 5 that in order to produce significant changes in the wave interaction processes a shift of several tube diameters in the backwall distance is required. On the other hand, the influence of heat losses is important only when the igniter is located in the close vicinity of the backwall. By varying the backwall distance from zero to about one tube radius, the growth of the flame front changes from that corresponding to the case when (I) the flame is in continuous contact with the backwall, to that when (2) the flame contacts the sidewalls of the tube before reaching the backwall. Consequently changes in the backwall distance of much less than one tube radius produce large effects on the heat loss during the
December 1962
Ignition geometry and ltame acceleration in explosive gas
initial stages of t h e process, as reflected b y significant v a r i a t i o n s in flame a c c e l e r a t i o n . More specifically, d u r i n g t h e initial p h a s e of t h e process, w h e n the flame e x p a n d s s p h e r i c a l l y a b o u t the p o i n t source of ignition, it h a s b e e n s h o w n t h a t the p r e s s u r e rise at t h e f r o n t of an e q u i v a l e n t o n e - d i m e n s i o n a l flame c a n be expressed w i t h a g o o d a p p r o x i m a t i o n in t e r m s of a p o w e r series w i t h respect to (S"-~+iq~t~'"), n = 1, 2, 3. H e n c e the initial t a m e a c c e l e r a t i o n is i n f l u e n c e d to a m u c h l a r g e r d e g r e e b y the m a g n i t u d e of t h e b u r n i n g v e l o c i t y t h a n b y the net h e a t of r e a c t i o n .
References I LE CI-IATELIER, H. C. R. Acad. Sci., Paris, 1900, 130, 1755-1758 _'2DIXON, H. B. Phil. Trans. A, 1893, 184, 97-188 3 LAFFITTE, P. C. R. ,4cad. Sci., Paris, 1923, 176, 1392-1395; 1923, 177, 178-180; 1924, 179, 13941396 4 SOKOLIK, A. and SHCHELKIN, K . I. f . phys. Chem., Moscow, 1934, 5, 1459-1463 5 EGERTON, A. C. and GATES, S. F. Proc. Roy. Soc. ,4, 1927, 114, 137-151 and 152-160; 1927, 116, 516-519 PAYMAN, ~V. and TITIAN, H. Proc. Roy. Soc. A, 1935, 152, 418-445
335
7 BONE, W. A., FRASER, R. P. and WHEELER, \V. H. Phil. Trans. ,4, 1938, 235, No. 747, 29-68 8 SCHMIDT, E., STEINICKE, H. and NEUBERT, U. Forschungsh. Ver. dtsch. Ing. No. 431, Ausgabe B, Band 17, 31 pp. Deutscher Ingenieur-Verlag: Dusseldorf, 1951 '~ TURIN, J. J. and I-IUEBLER, J. 'Advanced studies in the combustion of industrial gases', Report of Commission on Industrial and Commercial Gas Research, American Gas Association, Project IGR-59, Interim Report, August 1950, 17 pp; Final Report, April 1951, 14 pp. 10 GREIFER, B., COOPER, J. C., GIBSON, F. C. and MASON, C. M. J. appl. Phys. 1957, 28, 289-294 11 SALAMANDRA, G. D., BAZHENOVA, T. V. and NABOKO, I. M. Seventh Symposium (International) on Combustion, pp 851-855. Butterworths: London, 1959. See also Zh. tekn. Fiz. 1959, 29,
1354-1359 ,2 MARTIN, F. J. Physics of Fluids, 1958, 1, 399-407 13 BOLLINGER, L. E., FONG, M. G. and EDSE, R. J. Amer. Rocket Soc. 1961, 31, 588-594 1~ BAUMANN, W., URTIEW, P. A. and OPPENHEIM, A. K. Z. Elektrochem. 1961, 65, 898-902 1.~ LADERMAN, A. J. and OPPENHEIM, A. K. Physics of Fluids, 1961, 4, 778-782 16 LADERMAN,A. J. and OPPENHEIi, A. K. Proc Roy. Soc. ,d, 1962, 268, 153-180 ,7 LADERMAN, A. J., HECI~T, G. J. and OPPENHEIM, A. K. Temperature--Its Measurement and Control in Science and Industry, Vol. III, Part 2, 943-947. Reinhold: New York, 1962 Is BoA-TEH CHU. Tech. Note Nat. ,4dr. Comm. ,4ero., Wash., No. 3683 (1956)
of detonation, the present study was carried out in order to examine the importance of the geometry of ignition in this respect. In an effort to interpret some of the experimental results obtained previously TM, we analysed recently '5 the effect of pressure waves reflected from the closed end (backwall) of the detonation tube and demonstrated how the separation between the backwall and the igniter influenced the strength of these waves and their subsequent interaction with the flame. We concluded, however, that although some significant details of the process were indeed altered, the detonation induction distance was relatively insensitive to variations in the backwall distance. The analysis was performed under a tacit assumption that during the time when the flame front was expanding over the cross section of the tube, the heat transfer to the walls was the same for each igniter position. While this assumption was justified by the operating conditions of these experiments ~4 (pilot flame ignition in Plexiglas tubes), it became of interest to investigate the initiation of explosion under circumstances where variations in the backwall distance would be associated with significant changes in the heat transfer to the wall.
Introduction THE onset of detonation in gaseous mixtures has been studied experimentally by H. LE CHATELIER 1, S . B . DIXON 2, P . LAFFITTE 3, A . SOKOLIK a n d K . I. SHCHELKIN 4, A . C . EGERTON and S. F. GATES "~, W. PAYMANand H. T I T I A N ~,
W. A. BONE et al. 7, E. SCHMIDT et al. s, J. J. TURIN and J. HUEBLER% B. GREIFER el al. ~°, and, more recently, by G. D. SALAMANDRAet al. 1~, F. J. MARTINr', L. E. BOLLINGERet al. 1"~, W. BAUMANN et al. 1'1, and by A. J. LADERMAI~ and A. K. OPPENHEIM15'Is. While our understanding of the processes of transition from deflagration to detonation was definitely enhanced by these investigations, the development of a complete theory has been hampered by the lack of attention given to initial conditions, especiallT to the mode of ignition and, in particular, to the position of the igniter in the detonation tube. Since it was reasonable to expect that the initiation is of considerable importance to the subsequent development of the highly non-linear process of the development *This research was supported by the United States Air Force. through the Air Force Office of Scientific Research of the Air Research and Development Command under Contract AF 49(638)166 and the National Aeronautics and Space Administration under Grant No. NSG-10-59. 325
326
A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
For this purpose we performed a series of experiments in our 1 in. x 1{ in. detonation tube, whose 1½ in. walls and the closed end are steel and the 1 in. walls are glass, with spark ignited stoichiometric hydrogen-oxygen mixtures initially at n.t.p., varying the distance between the spark gap and the backwall. When the spark gap was very close to the backwall, the flame spread out as a hemisphere until it reached the sidewalls. When the spark gap was sufficiently far away from the backwall, the flame grew as a sphere until it became distorted by contacting sidewalls before it reached the backwall. From then on further shift of the gap away from the backwall did not affect the heat transfer to the walls. On the other hand, until this instant the flame was sufficiently close to the backwall so that the influence of waves reflected from the end was negligible. As pointed out before 15, the backwall plays then the same role, in this respect, as the plane of symmetry of the flame kernel, the effects of reflected waves becoming noticeable when the ignition centre is much farther away from the end than it was in these experiments. The sole reason for the observed changes in initial flame acceleration could be ascribed, therefore, to heat transfer effects. To ensure this, the influence of ignition energy over a limited range of values was also investigated, but within the range of operating conditions used in our tests it was found to be negligible.
Vol. 6
the undisturbed baseline) it was considered as due entirely to the noise inherent in the electronics rather than indicative of any real process. The amplitude of the noise was 0.50 lb/in -°, significantly smaller than the real signal. A complete description of the apparatus can be found in ref. 16. Ignition was performed by means of a spark discharge. The spark igniter, Figure 1, was installed at the geometric centre of the backwall, with the centre electrode concentric with
ExperimentalApparatus Observations of wave processes that accompany the initiation of explosion by spark discharge in stoichiometric hydrogen-oxygen mixtures at n.t.p, were made in a lin. x 1½in. cross section detonation tube 1~ by means of streak schlieren photographs with simultaneous measurement of pressure at several positions within the induction distance. For this purpose a Kistler Instrument Corporation PZ-6 SLM pressure transducer was used and the measurement was recorded by means of a Tektronix 551 oscilloscope. Although the gauge was shock mounted to eliminate ringing, a small amplitude, high frequency signal was observed on the oscilloscope trace. Since this signal occurred under varied operating conditions (and even on
Figure 1. Typical spark igniter the tube axis. The end of the spark plug, with the exception of the electrodes, was flush with the inner surface of the tube. The centre electrode (anode), 0-063 in. diameter, was insulated with a 0-006 in. layer of Teflon, leaving, at the free end, an exposed length 0-01 to 0-02 in. long. The ground electrode, a bare 0.09.0 in. diameter wire the same length as the anode, was silver
December 1962
Ignition geometry and flame acceleration in explosive gas
soldered to the outer housing of the spark plug. Spark plugs with electrode lengths of 0, ¼, ½, a, 4 1, and 1½in. were used. As a result of the spark plug construction and since the spark discharge path was restricted to the free end of the electrodes, the electrode length was identical with the backwall distance. Furthermore, because of its small surface area and mass, heat transfer to the electrodes was negligible. Hence, extending the electrodes into the test section was considered to serve only as a means of positioning the point of ignition with respect to the baekwall. The spark electrodes were positioned parallel to each other, separated by a 0'02 to 0.03in. gap. For streak photography the windows of the tube were masked off with the exception of a narrow slit parallel to the axis of the tube. In this instance it was necessary to have the slit width equal to, or greater than, the electrode gap*, otherwise, some error of interpretation could have been introduced into the photographic records. If the gap were partially obscured from the view, then the flame would reach the edge of the slit with already some finite size radius, but giving a false appearance of a point explosion. The flame would subseqently appear to move faster than it actually does. A displacement between the gap and the slit would, therefore, produce false information on the initiation of the process. In order to avoid this the slit was positioned with particular care. This was accomplished by shining a parallel beam of light through the test section forming thus a shadow of the edges of the slit on a translucent screen which was placed behind the slit. The slit was then adjusted until the inner edges of the electrodes appeared on the screen, superposed on the edges of the slit. The spark was produced by discharging a capacitor, charged to 200V, through a 2D21 thyratron into the primary of a Thordarson T22R44 step-up pulse transformer. The spark electrode was connected to the transformer secondary, Figure 2. Capacitors ranging from
0"1 to 80#F were used producing a variation in ignition energy of 0.1 to 6 mJ. To oscilloscope trigger
0"1-80 pF
Delay
unit
~0"-~ _ ~ ~Peacrtkr odes, Thyratron ' = ~ trigger _L -[_
._.~ "---
Thordarson T22R44 D,_
To light source trigger
Thyratron trigger
Figure 2.
Schematic diagram of ignition circuit
Results Position of ignition source Typical streak schlieren photographs of initial flame acceleration for backwall distances of 0, ¼ and ½ in. are shown in Figures 3 to 5, respectively. The bright curved band represents.
0 *Ideally. the slit should be as narrow as possible to avoid i m a g e smearing, the minimum slit width being determined by the m i n i m u m light intensity required for exposure. T h e use of a 0'04 in. slit width, however, did not affect the optical quality as evidenced by the experimental records.
327
0,10
0-20 X
Figure 3.
0.30
0"40 rn
Streak schlieren photograph--O backwall distance
328
A . J . Laderman, P. A. Urtiew and A. K. Oppenheim
Vol. 6
tube have been transformed to profiles at a fixed distance from the point of ignition whose position in the tube was varied. This was accomplished by taking advantage of the fact that the pressure wave is initially a simple one, and that consequently the process across it is isentropic while state parameters along its world-lines are invariant. The results are shown in Figure 7 for x = 2-62 cm, and in Figure 8 for x = 12.78 cm. Figure 7 depicts the increase in the rate of pressure rise that results from the shift of the point of ignition from 0 to ½ in. away from the backwall. The records are terminated when the flame contacts the glass sidewalls of the tube across the 1½in. width; the discontinuity in the slope of the profile is due to the flame contacting the steel sidewalls across the 1 in. width. Figure 8 shows the full pressure profile for each backwall distance at x = 1 2 . 7 8 c m from the igniter. In this instance all the flames were in contact with the sidewalls of the tube within the period of the 0
0"10
020 X
Figure 4.
0'30
0"40 m
Streuk schlieren photograph--~ in. bac,% wall distance
the flame front, while the dark areas and lines ahead of the flame represent pressure waves generated during the process. The flame worldlines have been reproduced in Figure 6 where, for comparison, the points of ignition have been shifted so as to coincide at the origin. Also included there are flame world-lines for backwall distances of ~, 1 and l~r in. The influence of the backwall distance is quite pronounced, the flame acceleration increasing significantly when it is varied between 0 and ½in. With further displacement of the spark gap, however, the development of the process remains almost unchanged, only a slight difference in flame velocity being observed when it was shifted from ½ to 1½in. from the backwall. The effect of the backwall distance is reflected quite clearly in pressure records which have been obtained at positions 2.62 and 12-78cm from the end of the tube. In order to set a proper basis for interpretation the pressure/time profiles measured at a given cross section of the
E 125-
100-
0750.50-
025-
0
Figure 5.
0.10
0'20 X
0 30
0 40
rn
Streak schlieren photograph--~ in. buckwall distance
December 1962
Ignition geometry and flame acceleration in explosive gas
measurement for, at least, the last several hundred microseconds. Judging from Figure 7 it appears that the pressure has about the same value when the flame first contacts the sidewalls, irrespective of the distance between igniter and backwall.
329
negligible and the flame temperature and hence the average overall reaction rate are then greater than for x~:=O. This is then reflected in the subsequent progress of the process so that at the distance of 12"78 cm from the ignition source as ~26
1-O
S qo Sq m/s catlg mls A 45 2050 B5 B 55 2050 -C 45 2050 100 D 45 2050 -E 40 2050 -xE, Distance between backwa[[ and ~gnlter, in ~ Observed pressure - - - - Computed pressure
Curve
c
E 0.9
24
08
22
07
20
B
06
1)
~114
z,, 2 /
,5"
18
/
/
D
xE=O
Ao /
O5 16
0'4
14
03
I
100
i
I
200
(
300
la sec
t
i
02
XE, D i s t a n c e and
between i g n i t e r , in.
backwat[
Figure
7.
Pressl~re histories
at
location
2.62
cm
from ignition
0.1 i
0
Figure 6. h~storics
5 lrO 1~5 2'0 Distance from igniter, x
I
25
30 cm
Comparison of experimental space~time of accelerating flames for all backwalI dis tanees
Since from then on the contact area between the flame and the sidewalls is essentially independent of the preceding process, one could expect that the rest of the pressure profiles would eventually become identical, the effect of the position of the igniter being reflected only in the shift of the profile in space. Figure 8 demonstrates, however, that evidently this is not so. The maximum pressure and the rate of pressure rise for xT~>0.Sin, are both larger than for x~ = 0. This can be explained by noting that when the flames contact the sidewalls, they are physically dissimilar, although they appear geometrically identical. The reason for this is indeed the fact that heat losses to the backwall occur so late for x~ ~> 0-5 in. that their effect is
shown in Figure 8, the pressure waves are much steeper and attain higher values, if at initiation the flame did not touch the backwall. Ignition energy To determine the effect of ignition energy on the development of the process, extensive tests were performed with the various igniters for ignition energies over the range from 0-1 to 6 m J, several tests being run at each condition. In contrast to the influence of backwall distance, no change in the process was observed within the variability of its progress. In fact, for a given backwall distance, it would be expected that flame acceleration should remain independent of the ignition energy until the amount of energy deposited in the gas was either large enough to generate pressure waves of sufficient strength to disturb the unburnt gas from its initial state, or too small to maintain the reaction. Apparently neither limit was reached in the present case.
330
A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
Analysis Effect of heat loss to wall on the growth pressure w a v e
of
the
The relatively slower flame acceleration associated with the shorter electrodes is attributed to heat losses to the backwall. Let us consider a simple model of a flame expanding at first spherically about the point of ignition. This
/
Recently we reported lz on measurements of the rate of heat transfer from the combustion zone to tube walls during the development of detonation. We found it to be a rapidly decaying function of distance behind the flame almost independent of its position and state. The rate of heat release per unit frontal area of the flame propagating at a steady state can then be expressed as follows: co = qlplS, = I~p2So - I"~p~S~
+½(s~-s~,)p,s,+g,.
"iS l III W"
:0'
Vol. 6
°f,o
I
20
x.c
,
and tgniter, Distance between in backwaI]
36o 4bo sbo Gbo 76o 8~o 9oo' t
~l.$ec
Figure 8. Pressure histories at location 12.78 cm ~rom ignition
model has been largely substantiated by flash schlieren photographs of the initial growth of the flame front. When the flame contacted the walls of the detonation tube, heat losses occurred which tended to retard the process of flame acceleration. For electrode lengths less than 0.Sin., the flame contacted the backwall of the tube before reaching the sidewalls, while for electrodes extended farther than 0"5 in. the converse was true, so that in this case two flames propagating away from each other were eventually produced. When this happened heat losses to the backwall were not sensed by the forward running flame which exhibited then the largest acceleration. This would explain why the development of the process remained essentially unchanged when the backwall distance was increased from 0-5 to 1.5 in. Such a conclusion is, of course, based on the premise that the effects of reflected pressure waves were then still negligible but this has been already demonstrated ~4 to be quite true under present circumstances.
....
[1]
where p denotes density, p is pressure, S is relative flame speed with respect to the gas, I~,i=7;/(7(- I) (7( being the specific heat ratio), q is the heat released per unit mass and QL is the rate of heat transfer to the walls per unit area of flame surface; the gas immediately ahead and behind the flame is denoted by subscripts 1 and 2 respectively, while its behaviour has been assumed to obey the perfect gas equation of state (with different specific heats and molecular mass, however, for reactants and products). A disturbance in the flame propagation process changes the heat release rate from to to to + ~to, and pressure waves are generated at the flame front, increasing pl to pl+~p~, S~ to S~+~SI etc. With restriction to the first order terms, equation 1 leads to the following expression for the pressure wave generated b y the flame
~p,_
~,- 1
(go -~QL)
. . . . [2]
where a 1 and a~. are local velocities of sound in front of and behind the flame respectively. The above formula was derived first by BOA-TEH CHO1~ and its validity verified by us IB with reference to the problem at hand, but without consideration given to heat losses. When the flame front surface is curved rather than plane, the total heat release rate is A1plS q where A t is the flame surface area and S the normal, relative flame speed. The rate of heat release per unit frontal area referred to the tube cross section A, is then = (AdA,)
p,Sq
....
[S]
December 1962
Ignition geometry and flame acceleration in explosive gas
and equation 2 becomes 7.~ - 1 l 3A I ) ('t._,/ 7~) a, + a~ [piSq ~ - ~Q,.. . . . [4] where the change in the product p~Sq is assumed negligibly small in comparison with the change in A 1. The effect of heat losses on the generation of pressure waves can be estimated as follows. For ignition at the backwall, the flame expands almost as a hemisphere during the few hundreds of microseconds that it requires to reach the sidewalls. According to our measurements ~7 the heat loss to the wall can be expressed approximately as Q::Qo e -~1' . . . . [5] where Q is the heat loss per unit time per unit surface area of the tube, Qo ~ 800 c a l / c m 2 sec, T ~ 10 -4 sec, and r is time measured from the arrival of the flame at a given position, x. The surface area of the backwall exposed to hot combustion gases is ~x ~. Assuming that the absolute flame velocity, S~=S~+u~ (% being the particle velocity immediately ahead of the flame) is constant and equal to 8 5 m / s , taking r = t - ( x / S ~ , ) and integrating equation 5 over the exposed surface area between the limits x = 0 and x = S . t , we obtain the expression for the total heat transfer rate to the wall at any time t. The change in the average heat transfer rate with reference to the frontal area of the flame, At, that occurs in the time interval 3t, can be then determined simply by differentiation with respect to time, yielding g~QL= (2~S]QoT/At)[1 - e -'/~] ~;t . . . [6]
Substituting numerical values into equation 6 yields 3QL ~ 60 eal/cm-" sec for i~t=20/~sec. At the same time, taking for example the interval between 80 and 100#see after ignition, the flame, moving at an average speed of S = 5 0 m / s , increases in surface area by 3Ai= 1.8em 2. With the effective heating value for stoichiometric hydrogen-oxygen at n.t.p., q ~ 2 kcal / g, equation 3 gives ~co=p~Sq~Ai/At~ 1100 cal/cm 2 sec. The heat loss to the backwall constitutes then 6 per cent of the rate of heat released b y combustion.
331
Influence of Normal Burning Speed In order to assess the influence of burning speed it is convenient to lump the rates of heat release and heat loss into a single term by introducing an effective heat of reaction q e = q - ( Q L / p ~ S ~ ) so that equation 4 becomes 3'2 - 1 p,SqfiA, 3P1= (7./'yl)a~+a~ - At
. . . . [7]
As the spark electrodes are moved away from the backwall, the amount of heat transferred to it decreases, the flame temperature and reaction rate approach their adiabatic values, and both q~ and S increase. Since the values of S and q, are difficult to predict a priori, and since the growth of the flame surface area is itself a function of S, in order to arrive at a correct interpretation of experimental data the relative influence of changes in S and q, on the growth of the pressure wave have first to be determined. This is accomplished by a linearized analysis of the growth of the pressure wave on the basis of equation 7. For this purpose we assume first that to the first approximation a_, = [ m - (v_,/'/,)] . ,
where m is a constant determined from an energy balance for an adiabatic combustion process. Equation 7 then becomes 3p p0
_. 72 - 1 p~Sq_e3A l m~
. . . . [8]
a,p,,
Assuming that the coefficient is essentially constant, equation 8 can be integrated simply to P / P o = 1 + KSq~A t
. . . . [9]
where K=T.- -1
Po
mA, aop~
(7~ - 1) 70 mata~ subscript 0 referring to the state of the undisturbed medium ahead of the pressure wave. Within the simple pressure wave which precedes the flame, the pressure and particle velocity are related by the isentropic equation P/Po = [1 + ½ (To- 1) U/ao]2,d(,o -1)
332
A.J. Laderman, P. A. Urtiew and A. K. Oppenheirn
which, upon expanding the RHS and retaining only the first order term, can be reduced to p/po=l+you/ao
....
[10]
Equations 9 and 10 yield then the following simple expression for the initial 'piston action' of the flame u= a°Ksqeat
....
[11]
7o For a hemisphericaUy expanding flame front A t = 2wx ~, so that equation 11 expresses in effect the following velocity/displacement law of the flame 'piston': u=KlSq~x 2 . . . . [12] where K1 = (2~a0 / Yo) K 27 (V: - 1) As a consequence of the relative burning speed, S, such a piston is, however, not impervious; its actual displacement is given b y dx/dt=S+u
....
[13]
which with equation 12 yields, upon integration,
1
x = (K~q~)~/~ tan t (K~S2qe) ~/~
...
[14]
The growth of the pressure wave is then described in terms of S and q, by means of equation 9 with A t = 2 w x ~ and x expressed by equation 14 giving + Y°Stan: y
. . . . [15]
a 0
where y --- t (KiS:qe) x/°"
Expanding tan2y as a power series in terms of the argument y, we obtain finally
P/Po=
1 + X C,K~S2"+~q,'t ~-~ n
{t ( K t S 2 7 ~ ) ' / 2 < ~ / 2 }
....
[16]
with C1 = "/o
a,,'
C2 _ 2y o 3a0 '
they have been evaluated for a 2H2+ 02 mixture, initially at n.t.p, in our l in. x 1½in. detonation tube. With S = 5 5 m / s , qe=2050 c a l / g and m = 4 , equation 16 yields P / P o = 1 + 3.84 × 10-6t 2
+ 7.28 x 10-11t a + 9.32 x 10-1~t 6 where t is in microseconds. From the form of equation 16 it becomes obvious that when changes in S and qe are small and of the same order of magnitude the influence of the former on the development of the pressure field ahead of the flame obscures completely the effects of the latter. This applied in the present investigation. Both S and qe changed approximately by 15 to 20 per cent over a range in x~: from 0.5 to 0 in. so that the influen.ce of S on the pressure w a v e w,as threeto /our-fold that o/ qe. This effect is reflected
m A ta2o
p/po=l
VoL 6
17y,,
C~ = £gh~
where, for the time interval considered, higher order terms can be neglected. To demonstrate the relative magnitude of each term in the series,
clearly in the following results obtained by a numerical analysis. A n a l y s i s of flame acceleration
The formation of the pressure wave ahead of the accelerating flame was calculated for the two limiting cases, that is for backwall distances of 0 and 0.5 in., by means of equations 4 and 6. In the former one, the flame grew in the form of a hemisphere, in the latter it expanded as a sphere, but, according to our theory the surface area only on the positive x side from the point of ignition contributed toward the generation of pressure waves moving in that direction. Since the cross section of the detonation tube is rectangular, the flame contacted first only two walls. Between this time and that of reaching the second pair of walls, the flame surface was still hemispherical in shape, but its area was decreased by the segments intercepted by the sidewafls. The flame surface area is given then b y A ~ = 2 ~ x 2 =6-28x 2 . . . . [17] when x < r~ and A t = 2 ~ r l x = 7"91 x when r ~ < x < r 2. In the above 2 r , = 2 . 5 4 c m and 2r2=3.81 cm are the width and depth of the tube, respectively. The change in the growth of the area at x = r 1 is reflected in the abrupt
December
1962
I g n i t i o n g e o m e t r y a n d f l a m e a c c e l e r a t i o n in e x p l o s i v e g a s
change in the slope of the experimental pressure profile, Figure 7. The change in A i during a time interval 3t was computed using the relation ;L~
=
(S
+
u)
~t
....
300
:1.
33.3
Curve A of Figure 7 S : 45 m/s, xE : 0
/ ~_67.Q /~-f~~609
......
[181
with
&,- (a, l./,p,) @, On the basis of schlieren records the following values were established for the normal burning speed : for x r - 0 S = 4 5 m / s + 10 per cent for xT,:=0"fin. S = 5 5 m / s + 1 0 p e r c e n t As a consequence of the assumption that the combustion mixture obeys the perfect gas equation of state, the heat of reaction q had to be evaluated so as to produce compatible results. Accordingly, q was taken as 2 0 5 0 c a l / g , the value that would produce the actual Mach number of detonation computed for the same perfect gas as that used in our theory. The flame world-lines and the pressure rise immediately ahead of the flame during the course of its acceleration were calculated using equations 4, 6, 17 and 18. For comparison with experimental observations the computed pressure profi]e was transformed to a pressure/time history at a section 2.62cm from the point of ignition. The flame world-line was determined from the relation between the flame front radius, x, and the time, t. Each point on the flame world-line represents in effect the origin, in space/time, of the pressure wave world-line associated with the pressure corresponding to that point. Consequently, the pressure wave world-lines could be positioned on the x / l plane, and the desired pressure/time profile determined from the intersection of the worldlines and the time coordinate at x - 2 . 6 2 cm.
Results and discussion Flame and pressure wave world-lines are shown in Figures 9 and I0 for backwall distances of 0 and 0'5 in., respectively, giving the comparison between the analytical and experimental results. The numbers adjacent to pressure waves represent their velocities in m / s . The computed pressure/time profiles for xT;=0 and 0-5 in. are shown in Figure 7 as curves A and B, respectively.
~'~~-I
Yz_"
200
100
~
/ 7 - ......
# - ~
--
~ ~
....
5~z' - s s I J s~9 ~ - - "
......
/,, ~// ¢
0
Calculated world- hnes Observed
-----
I
I
{
10
2'0
30 cm
Figuz; 9. Computed space~time history of accelerating flame--O bachwall distance (Numerals adjacent lo 'world-lines denote velocities in m / s )
In view of the simplifying assumptions used for the analysis, the agreement with experimental observations should be considered quite satisfactory. In particular, the close coincidence between the points of sudden change in the slope of the experimental and theoretical pressure profiles gives certainly a satisfactory justification for the analytical model.
;oo]
Curve B of Figure 7 S : 55 m/s, x E- 1/2m e s-
.,~.%:-
200[
_ _~'67.5 62~1
~
~71~
-
/ j -.,7._---
-
/ f
lOO
/
Ca
....
::~ -
:
;:-u.622gou 694i J
6o.
AI ssq ~Sg/ 5L~3 ---E
[cu[at ed world- hnes
Observed
1EO
2"I0 x
I
3"0 crn
Figure lO. Computed space~time history of accelerating flame--½ in. backwall distance (Numerals adjacent to world-lines denote velocities in rn /s)
334
A.J. Laderman, P. A. Urtiew and A. K. Oppenheim
The sensitivity of the results to the heat loss term, ~QL, was tested in several ways. First, although the absolute flame speed, Sa, actually varied from 45 to 125 m / s during the time interval under consideration, ~QL was evaluated from equation 6 which was derived under the assumption of constant Sa with S ~ = 8 5 m / s . This then led to an over-estimate of ~QL at the initiation of the process and an under-estimate at later times. Some idea of the effect of simplifications underlying equation 6 was obtained by repeating the calculations with S~ = 100 m / s . The larger value of S~ exaggerated the influence of heat loss over a greater portion of the entire time interval of interest and, consequently, produced a retardation in the development of the process. However, the resulting pressure profile (curve C, Figure 7) was found to differ by an insignificant amount from the original pressure profile (curve A). Secondly, the calculations for x E = 0 were repeated using equation 7 in place of equations 4 and 6 so that the effects of ~QL were absorbed in the effective heat of reaction, qe- The pressure profile computed with S = 4 5 m / s and q~= 1 750 cal/g is for all intents and purposes identical to curve A (Figure 7) and for this reason it has not been plotted there. This procedure provided moreover direct means for estimating the total heat loss which is then expressed simply by the ratio ( q - q,)/q. In this instance 15 per cent of the energy released by chemical reaction has been absorbed by the backwall of the detonation tube. On the basis of these considerations then, it is concluded that, when the heat loss is small, the analysis is insensitive to the form of the term which expresses this quantity. In particular under these circumstances heat loss effects may be accounted for quite accurately by the use of a constant 'effective' heat of reaction. The relative influence of changes in S and q~ associated with changes in x~ was tested by repeating the calculations using equation 7 with the same value for the effective heat of reaction qe = 2 050 cal / g but with different normal burning speeds. The results, shown in Figure 7 as curves D for S--45 m / s and E for S = 4 0 m / s are in much closer agreement with
Vol. 6~
the experimental pressure profile for x~ = 0 than with that corresponding to xE=0"5 in spite of the fact that heat loss effects on the value of qe have been neglected. As demonstrated by equation 16 the influence of heat losses on the development of the process is reflected considerably more in the decrease of the flame speed, S, than in the associated decrease of q, (or increase in QL). This is, of course, a direct consequence of the hemispherical model assumed for the expanding flame and may not apply under different conditions. It should be concluded, however, that under the experimental conditions of this study, i.e. point source ignition in a combustion tube, the flame speed, S, plays a prominent role in the acceleration of the flame front.
Conclusions The influence of the position of the igniter on initial flame acceleration is manifested in two ways, by the action of pressure waves reflected from the backwall of the tube and by heat transfer to the backwall. Interestingly enough, the two effects are uncoupled, at least when the tube diameter (or width, in the case of a rectangular cross section) is not greater than several centimetres. Furthermore, the scale of the two effects is different, i.e. a much larger change in the separation between the igniter and the backwall is necessary for the production of noticeable change by reflected waves than by heat losses to the backwall. It has been shown T M 1 5 that in order to produce significant changes in the wave interaction processes a shift of several tube diameters in the backwall distance is required. On the other hand, the influence of heat losses is important only when the igniter is located in the close vicinity of the backwall. By varying the backwall distance from zero to about one tube radius, the growth of the flame front changes from that corresponding to the case when (I) the flame is in continuous contact with the backwall, to that when (2) the flame contacts the sidewalls of the tube before reaching the backwall. Consequently changes in the backwall distance of much less than one tube radius produce large effects on the heat loss during the
December 1962
Ignition geometry and ltame acceleration in explosive gas
initial stages of t h e process, as reflected b y significant v a r i a t i o n s in flame a c c e l e r a t i o n . More specifically, d u r i n g t h e initial p h a s e of t h e process, w h e n the flame e x p a n d s s p h e r i c a l l y a b o u t the p o i n t source of ignition, it h a s b e e n s h o w n t h a t the p r e s s u r e rise at t h e f r o n t of an e q u i v a l e n t o n e - d i m e n s i o n a l flame c a n be expressed w i t h a g o o d a p p r o x i m a t i o n in t e r m s of a p o w e r series w i t h respect to (S"-~+iq~t~'"), n = 1, 2, 3. H e n c e the initial t a m e a c c e l e r a t i o n is i n f l u e n c e d to a m u c h l a r g e r d e g r e e b y the m a g n i t u d e of t h e b u r n i n g v e l o c i t y t h a n b y the net h e a t of r e a c t i o n .
References I LE CI-IATELIER, H. C. R. Acad. Sci., Paris, 1900, 130, 1755-1758 _'2DIXON, H. B. Phil. Trans. A, 1893, 184, 97-188 3 LAFFITTE, P. C. R. ,4cad. Sci., Paris, 1923, 176, 1392-1395; 1923, 177, 178-180; 1924, 179, 13941396 4 SOKOLIK, A. and SHCHELKIN, K . I. f . phys. Chem., Moscow, 1934, 5, 1459-1463 5 EGERTON, A. C. and GATES, S. F. Proc. Roy. Soc. ,4, 1927, 114, 137-151 and 152-160; 1927, 116, 516-519 PAYMAN, ~V. and TITIAN, H. Proc. Roy. Soc. A, 1935, 152, 418-445
335
7 BONE, W. A., FRASER, R. P. and WHEELER, \V. H. Phil. Trans. ,4, 1938, 235, No. 747, 29-68 8 SCHMIDT, E., STEINICKE, H. and NEUBERT, U. Forschungsh. Ver. dtsch. Ing. No. 431, Ausgabe B, Band 17, 31 pp. Deutscher Ingenieur-Verlag: Dusseldorf, 1951 '~ TURIN, J. J. and I-IUEBLER, J. 'Advanced studies in the combustion of industrial gases', Report of Commission on Industrial and Commercial Gas Research, American Gas Association, Project IGR-59, Interim Report, August 1950, 17 pp; Final Report, April 1951, 14 pp. 10 GREIFER, B., COOPER, J. C., GIBSON, F. C. and MASON, C. M. J. appl. Phys. 1957, 28, 289-294 11 SALAMANDRA, G. D., BAZHENOVA, T. V. and NABOKO, I. M. Seventh Symposium (International) on Combustion, pp 851-855. Butterworths: London, 1959. See also Zh. tekn. Fiz. 1959, 29,
1354-1359 ,2 MARTIN, F. J. Physics of Fluids, 1958, 1, 399-407 13 BOLLINGER, L. E., FONG, M. G. and EDSE, R. J. Amer. Rocket Soc. 1961, 31, 588-594 1~ BAUMANN, W., URTIEW, P. A. and OPPENHEIM, A. K. Z. Elektrochem. 1961, 65, 898-902 1.~ LADERMAN, A. J. and OPPENHEIM, A. K. Physics of Fluids, 1961, 4, 778-782 16 LADERMAN,A. J. and OPPENHEIi, A. K. Proc Roy. Soc. ,d, 1962, 268, 153-180 ,7 LADERMAN, A. J., HECI~T, G. J. and OPPENHEIM, A. K. Temperature--Its Measurement and Control in Science and Industry, Vol. III, Part 2, 943-947. Reinhold: New York, 1962 Is BoA-TEH CHU. Tech. Note Nat. ,4dr. Comm. ,4ero., Wash., No. 3683 (1956)