1. science
2. mathematics
3. geometry

SOME ASPECTS OF IGNITION BY LOCALIZED SOURCES,
AND OF CYLINDRICAL AND SPHERICAL FLAMES
G. DIXON-LEWIS AND I. G. SHEPHERD
Houldsworth School of Applied Science, The University, Letds~ England
The time dependent conservation equations governing flame propagation in cylindrical
and spherical systems have been set up and solved by finite difference methods for the case of
a 60% hydrogen-air flame. By this means it is possible (a) numerically to follow the sequence
of events Iollowing an "ignition" at the axis of a cylinder or the center of a sphere, or (h) to
investigate the effect of flame curvature on burning velocity and other flame properti~.
It was found that the minimum ignition energy depended on the form in which the energy
wa~ supplied. For a constant total energy, ignition was facilitated by increasing the proportion
supplied as H atoms rather than as thermal energy.
The velocities of movement of the freely propagating flames from the ignitions were found
to be slightly different from those of the inward propagating, cylindrical and spherical stationary flames. The velocities of the latter were independent of the flame diameter. The effect
of curvature on the flame properties is shown to be an effect on reaction rate distribution,
which also leads to differences in H atom concentration profiles. Unlike the situation in planar
flames, the detailed structure of freely propagating curved flames may not be the same ms
that of the corresponding stationary flames, a~d this may lead to the apparent differences in
burning velocity.
Introduction
A local source of ignition, such as an electric
spark, initiates chemical reaction by energizh~g
a small volume of gas around itself. Energy is
lost from this core by transport processes, aud
the conversion of the chemJcM ettergy of the
unreaeted mixture into thermal energy must
take place at a sufficient rate for the core to be
able to grow into a burnt gas kernel behind a
steadily propagating flame. For constant conditions of energy supply this occurs only when the
energy is above some critical value, the minimum
ignition energy for the defined conditions.
Amongst other factors, the minimum ignition
energy may depend upon the form in which the
energy is supplied--fur example, whether it is
supplied principally as thcrmal energy to raise
the temperature of the central core, or whcther
it is supplied as chemical energy in the form of
free radical dissociation products from the reactants, as from a photochemical source. The papcr
investigates this problem by following theoretically the growth of several eylh~drical and spherical flame keniels in a 60~0 hydrogen-air mixture,
to which energy has been added partly as lI
atoms and partly as thermal energy to raise the
temperature. For each fixed ratio of added chemical energy to added thermal energy, the minimum
total ignition energy was found by trial and
error, and these energies were then compared.
During the course of the igatition investigation
it soon became apparent that the outward radial
velocities of propagation of the developing
cylitldrical and spherical flames were not the
same. A second part of the investigation has
therefore been concerned with the properties
of the steady state cylindrical and spherical
flames, and their comparison with the planar
flame properties.
Continuity Equations and Method of
Computation
The development of the continuity equations
for cylindrical and spherically symmetric systems
is essentially similar to that recently described
by Bledjian, ~ and represents a more general
development of the equations governing the
p/anar flame?-'~ For any quantity denoted by
subscript i the generalized continuity equation
1483
1484
IGNITION
may be written as
Opl/Ot+(1/r~)(O/cgr)(r~F~)=qi
(1)
For a planar flame k = 0 ; for a cylindrical flame
k = l , and for spherical symmetry k = 2 . Expressing the composition of the gas mixture at
a n y point in the flame in terms of the weight
fractions w , the species continuity equations
become
(Olcgt)(pwi)A- (1/r ~)(O/Or){r~(Fzcw~-~-jl)} = q, (2)
Using the simple formulation (3) for the diffusional flux
ji= -- pDi(cgwi/Oy)
(3)
then since by conthmity we have
Opl'Ot+(1/r ~)(cg/'0r)(r~F~,) = o
(4)
equations (2) and (3) may be combined to give
P(CgWl/O0"~FM(OWi/Or)
= (1/r~)(O/Or)Er~pD~(OwjOr)]q-q~ (5)
The energy equation may be similarly derived
by considering the energy fluxes of the form
F~=~'[,~(F.~"f-.i~)Hi--X(dT/Oy). The conservation of energy is then
Developing for quiescent gas rrdxtures along
similar lines to Dixon-Lewis 5 for plane onedimensional flames, we make the transformations
du= prk dr
(8)
and then collsider equal intervals du in the integration instead of equal intervals dr. Since
elements of equal mass are then always considered, the corrvective fluxes across clement
boundaries always become equal to zero, regardless of effects of tcmperature rise and stoichiometry on the density. At the same time (a)
neglecting energy transport associated with the
diffusional fluxes in Eqs. (6) and (7), (b) assuming
all the components of the mixture to have the
same specific heat gp, which rermtios constant
at its mean value between 7",, and Tb, i.e.,
~iw,%~=cv, and (c) putting r=(T--T~)/
(T~--T~), the transformation (8) gives in the
general case
Species
OwJcgtTr~FM(cgw~/cgu)
= (O/Ou)[Dip2~rr~(cgw~/cgu)]+qjp (9)
Energy
&r/cgt'-}-~l,:,.z(cgr/Ou)
= (cgtcgu)[l~pl~},,,,z,(o~/ou)]
(O/O0{p~ (w,H0 }
+ O/rq (0/0~){r~EE ( ~ , + j , ) H ,
i
--X(OrlOr)~}=fl
The
(~)
cheruical rate of production of heat,
- - ~ q~Itl, is included in the derivatives on the
left of Eq. (6). The kinetic energ~ change involved
in accelerating the gases through the flame is
assumed small enough to be neglected, as also arc
radiation fluxes. Since OHi/Or=O(cv~T)/Cgr, Eq.
(6) may be re-arranged and expanded by use of
(3), (4) and (5) to give Eq. (7)
p E {w,(otcgO(c~,~)l+F~E {~,(OtOr)(e~,T)}
i
Further aesummg for convenience that (kp) is
independent of temperature, and taking Dp~v/~=
1, we have, with the additional transformation
~= (cJo•
Owi/cgt+[r~FM/ (Do2)lt~](Owjcg~)
= (D,/D){r~(c92wj&~)
-~-2Dll~kr~-~(Owjc9~.l')}-I-qJp (11)
Or~c9t'4-[r~F,~l (DI~)1/2"](cgr/0~b)
= r ~ (c9~r/0r~)+ 2D ~t2kr~-~(&/c~b)
i
=p ~.. Di(Owe/Or)(O/Or)(c~iT)
+(1/r~')(O/cgr)[-h(cgT/Or)]--~ q~H, (7)
i
(12)
For a fixed source strength M ( g e m -~ see-I
for a planar flame, g e m -1 see-1 at the axis of a
cylinder, or g see-x at the center of a sphere) the
SOME ASPECTS OF IGNITION
fluxes FM across element boundaries are given by
FM= M
(k= O; Cartesian co-ordinates)
l,~=M/r(2r) ~
(k= 1, 2; Polar co-ordinates)
148,5
flames the analogous summations provide similar
informatiou. In these cases the range of integration commences at r = 0, and we have
(q'/p)Au~-" ~ qlr~ dr= (~f /2k~')(Wlb--Wiu)
(13)
For a quiescent gas mixture the source strength
M is zero, leading to F M = 0 lit the equations.
The study of the development of a flame is
now essentially the solution of the appropriate
set of simultaneous differential eqtmtions formed
from (11), (12), and (13), subject to bomldary
conditions fixed by the specific problem. These
boundary conditions will be discussed later. The
integrations in t were performed numerically
using the explicit finite difference approach
employed earlier by Adams and Cook, ~ Zeldovieh
and Barrenblatt,a and Dixon-Lewis3 The space
derivatives were replaced by a central difference
approximation and the time derivatives b y a
forward difference approximation. Although this
approach restriet~ the size of the integration
time step compatible with stability, s,~ tlm a W
proaeh was considered to be adequate for the
present objectives.
In the treatment of planar flames, Dixon-Lewis~
followed the progress of the flame towards a
steady state burning velocity by means of the
space integral rates
f2
qi dr,
which for sufficiently small intervals At, may be
replaced by the summation
05)
where M is now the source strength (in g c m -~
see-~ or g see-t) to be associated with the stationary flame profiles which develop as the
integration proceeds. The information derived
directly from this summation is therefore information about the source strength.
Reaction Mechanism and Rale Constants
The reaction mechanism assumed for the
60v~vH~-air flame was the mechanism established
by Day, Dixon-Lewis, and Thompson ~ from
structure studies of lower temperature flames. It
eon~prises the following steps
OH+H2.~H20+H
(i)
H+(IN= Oli+O
(ii)
0+H~---0H+H
(iii)
H+O2+M= HO2+M
(iv)
H+H0.2= OH+0H
(vii)
H + H ( h = H2+02
(xii)
0H+HO2=
0 + I t 0 2 = O1 I + 0 2
n ~
The steady value of this summation which is
obtained as integration proceeds in time is related with the mmss burning velocity M of the
stationary flame by
f _ T q~dr= ~
(qd'p)Au
= M(G~b--G~,,)= M (we~--w.,)
H~O+02
(14)
Since the diffusioual fluxes are zero at both the
hot and cold boundaries, G,=w, at these boundaries. The sumnmtions of (qi/P)&u thus provide
a good method of extracting burning velocity
information from the calculation.
In the cases of the cylindrical and spherical
(xifi)
(xiv)
I : / + H + M = H2+M
(xv)
H'+0H'+M= H20+M
(xvi)
HWOWM= OH+M
(xvii)
Following the previous treatment of hydrogenrich flames, b~ which OH and O, once formed,
were assumed to react immediately by reactions
(i) and (iii), and HO~ was assumed to react
immediately by (vii), (xii), (xiii), or (•
the
mechanism may be reduced effectively to a
number of reaction cycles controlled by the rate
constants k2, k4, kl~, k16, and k17, and the ratios
kT/km, k~/km, and k14/k~. The appropriate cycles,
IGNITION
1486
which involve only H atoms and the stable
molecular species in their stoichiometry, become
(iva)
In computing the properties of the hydrogenrich flame, explicit conservation equations were
set up only for the energy, :[t atoms and molecular
oxygen. At the same time the weight fraction of
nitrogen in the mixture was assumed to remain
constant throughout, while the hydrogen and
steam weight fractions were calculated by
0vc)
wa,=w , , , ~ - (i/8) (wo~.~- wo~)
(20)
wn2o= (9/8)(wo,.=-wo2)
(21)
~2
H--}-O~(-~-3H2)----*2H~O-~-3H
(iia)
H + O~-4-M (-4-H-4-2H~)--~2tI~O-4-2H-4- M
HWCh+ M ( + H)--~H2-4-O~+M
I-IA-0,z-4-M (h- O H + HA- H20 )
--.*H20+ Or4- OH-4- He+ M
(ivd)
tI+O~+M(+O+ll+OH)
---~OH-4-02+ O-f- H2-~ M
(ive)
HWH+M---~H2WM
(xv)
tI+OH-4-M (+H+H~O)--~H=O-4-M-4- OH+Hz
(xvia)
H + O + M ( + H + 0 H )-"-'~OH+M --}-O--}-H2 (xviia)
The cycles (ivd), 0re), (xvia) and (xviia) all
assume that the occurrence of the primary steps
(xiii), (xiv), (xvi) a~ld (xvii), which remove OH
and O, is immediately followed by reaction (-i)
or (-hi) to restore the small quasi-steady state
concentration of the appropriate radical. On this
basis cycles (ive), (ivd), and (ive) are precisely
equivalent in their effect, as also are reaction
(xv) and the cycles (x-via) and (xviia). In the ease
of reaction (iv), the effective rate constants for
the chain propagating and breaking cycles become
These additional simplifications ignore the
effects of the rapid diffusional properties of
molecular hydrogen on the concentrations in the
hydrogen-rich system, but this is unlikely appreciably to affect the overall conclusions. However, in treating the one-dimensional lower
temperature flames of Day et al., e the optimmn
value of the ratio kT/k~. With these simplifications
was slightly different from that found using a
rigorous transport, property calculation. The rate
parameters used in the present investigation were
kt = 3.3X 10~3exp(-- 2700/T)
K3= 0.21 exp(+7640/T)
k~= 2.05X 1014exp(-- 8250/T)
k3= 1.8X lfl~aexp(-- 4700/T)
Ka= 2.27 exp(-- 938/T)
2k~/k4.n~= 0.091 exp(-- 90(0/T)
k4,o2 = 0.35k4.n2,
~mp
k4,~r =
0.44k4,u,
k4.mo= 6.5k4.n~
k~k4/k:
I.O+kT/k=+(k,3/k=)(EOH]/[H])
(16)
"4- (k14/km) ([O]/[H])
kbreak: k4-- kprop
kjz/kx~ = 0.3,
(17)
where
[oH]_ k_~[H:01+2~[O~]+2~.~[C~][M]
[H]
kT/k~ = 6.7
k u / k ~ = 2.5
k~.~=4.5X101~
(M=H~, 02, N~, H=O)
k~a,~= 2.0X 10x6
(M=H~, O~, N2)
km~r~o= 2.4X 10t7
k~[H2]
(18)
[o]= k__,EOH]+k~[O~]
[H]
ksEH~]
ktT.~=0.25k~6.g
(M=H~, O~, Ne, H20)
The adiabatic flame temperature Ta= 1630~
The group DOz
sontrolhng the transport properties was given
0 9) led to ~=0.564 cal enW3 ~
SOME ASPECTS OF IGNITION
the value 1.525X 10-~ g c m ~ sec-L Individual
D~ were assumed inversely proportional to square
root of molecular weight.
Results
and
Discussion
1487
i
i
1200
(a) Ignition Studies
For ignition studies, the sequence of events
following an instantaneous supply of energy to a
small central sphere or eyliader in a quiescent
gas mixture was followed numerically. In Eqs.
(11) and (12) the fluxes FM are zero, and the
boundary condition becomes
~-[-~,
I~--*0,
wc-+w+~
(22)
The conditions at the start of the integration
and the size of the central core to which the
energy is initially supplied may clearly be of
considerable importance in the study of ignition
energies. In order to provide computational
smoothness in the degradation of the step temperature and H atom profiles initially introduced,
several concentric shells should be included in
the volume energized. On the other hand, if the
shells are too small, stability limitations on the
values of ht/A~ used in the integration begin
to dictate prohibitively large computation times.
The effect of this is cumulative as the flame moves
outwards, since actual radial increments decrease
as (l/r) ~. In a typical spherical ignition with an
energy of 3.7 m J, the initial core of gas h a d a
diameter of 2.4 mm when hot, and was divided
into twenty concentric spheres representing equal
intervals 89
where d4b is the normal reduced
distance interval. Constant values of At~At,z were
nmintalned throughout. Extremely short time
intervals were used initially in the integration
a.~seclated with the central region, and these
were lengthened progressively by up to a factor
of 64 as the ignition developed. A single time step
involved starting at the eeoter of the sphere and
moving outwards by successive intervals d~b
(when outside the central core) until the reduced
temperature r fell to some preset small value
above the cold boundary condition. At the start
of each time cycle the conditions within the
innermost, spherical volume clement of radius
d~b/2 were assumed to be uniformly those at its
outer edge. Thus, a t this stage it was arranged
t h a t the values of the dependent variables at
~b= 0 were set equal to those calculated a t the
first mesh point in the previous time cycle.
400
0
I
5 -5
TIPIE ,t I 0 SEC
I
I0
Fro. l. Temperature histories at axis of cylinder
following axial injection of ignition energy. Letters
refer to conditions in Table I.
Effect of Mode of Energy Supply
was supplied to the small central core of unburnt
gas, with different proportions entering as thermal energy a n d as energy in the form of H atoms.
For a cylindrical system with a total energy of 7
mJ cm-1, flames A to E of Table I give a series
of starting conditions with a number of different
ratios of E~/Etot~l, and Fig. 1 shows the corresponding temperature histories at the cylinder
axis. In all the relevant cases studied, the total
time of transition from ignitiou conditions to
those of a freely propagating flame was of the
order of 0.2 in sec or less. It is also clear that
energy supplied in the form of H atoms is a more
efficient igniting agent than thermal energy.
Similar results were found for the spherical flame.
By considering different total energies in the
above manner, it is possible to determine a range
of minimum ignition energies, depending on the
form of energy supply. Tables I and II give a
selection of results obtained for the cylindrical
and spherical flames respectively. In the cylindrical case it was found that above 8 mJ cm-1
all the systems studied ignited, but that below
5 mJ cm-1 all were extinguished. In the spherical
case the upper and lower energy limits were 3.0
and 1.0 mJ respectively. Unfortunately ill both
cases it was not possible to study situations where
the whole of the ignition energy was supplied as
H atoms, because the calculation becomes unstable. Both the energy ranges found compare
favorably with the minimum ignition energy of
3 mJ measured by Lewis and von E l b d for this
mixture, using a "spherical" spark source of
diameter 2 mm. The indeterminancy found in
the minimum ignition energies also reflects a
situation observed experimentally.
In a single series of runs to investigate the
effect of mode of energy supply, a fixed energy
with time were examined in two separate ways:
(i) The velocity of advance of the position of
The velocities of pragrcssion of the fiar~ fronts
IGNITION
1488
TABLE I
Initial core conditions used in simulation of cylindrical ignition in 60% hydrogen-air flame
R,m
Etot~l
EH
mJ cm -1
Etot~i
Radius
:/'/~
10~Xn
nml
A
B
C
D
E
7.0
7. O
7.0
7.0
7.0
0.5
O. 4
0.3
0.2
O. 1
490
528
566
605
643
2.64
2.12
1.59
1,06
0.53
0.92
0.95
0.98
1.01
1.04
F
G
H
I*
J
K
L
5.25
6.72
8.14
8.0
24.5
4.0
2,0
0.26
9.42
0.11
9.1
O.O1
0.1
0.2
1363
831
698
692
1630
495
365
5.09
5.35
0.64
0.60
O. 12
0.30
0,45
0.71
0.77
1.09
1.08
1.6
0.93
O. 80
Max. axial
temp. during
simulation/~
Time to max,
axial temp./
10-~ sec
1725
1718
:>1708
> 1684
949
(No ignition)
1908
1845
1718
>147l
2518
No ignition
No ignition
10.4
10.4
>13.0
> 13.0
7.8
0.8
2.6
19.5
>7.8
3.9
---
* Ignition devdops only with difficulty.
TABLE II
Initial core conditions used in simulation of spherical ignition in 60% hydrogen-air flame
E~,
Eu
Radius
Run
mJ
Etotal
:/'/~
lO~Xa
mm
M
N
O
P
Q
R
3.67
6.83
3.0
2.0
2.0
2.0
0.71
0.22
0.1
0.7
0.5
0.1
564
1630
971
447
547
746
8.9
5.1
1.0
4.8
3.4
0.69
1.19
1 . 68
i. 4
1.09
1.17
1.29
T
U
V
W
1.5
1.5
1.0
1.0
0.3
0,1
0.4
0.1
560
634
447
522
1.5
0.52
1.4
O. 34
1.17
1.22
1.09
1.14
m a x i m u m heat release rate was measured directly from the gradient of the graph of radial
distance vs time. This gives a value of Sb, the
velocity of m o v e m e n t of the flame front relative
to the burnt gas. I t was found t h a t the velocity
of progression became uniform eve,, a t quite
small flame radii of about 2 ram. Values of Sb
equal to 1230 and 1120 cm sec -~ were fomld for
the cylindrical and spherical cases respectively
(Tb = 1630~
These are equivalent to S ~ = 246
a n d 205 e m sec- t a t 298~
Max. central
temp. during
sinmlation/~
Time to max.
central temp./
10 -~ sec
1962
2519
1802
1664
1650
1500
(Doubtful
ignition)
> 1213
866
993
697
2.5
1.8
4.0
4.0
4.4
11.4
Ignition
No ignition
No ignition
No ignition
(it) As the flames propagate outwards, the
effective negative source strengths increase, and
a t a n y t i m e these can be computed b y m e a n s of
the space integral rate. Hence, from Eq. (15), if
the graph of space integral rate vs (radius) ~ is a
straight llue passiug through the origill (or in
practice close to the origin because of the difficulty of defining tile flame radius), then the flame
is m o v i n g uniformly outwards. The slope of the
line is a measure of an effective mass burning
velocity (per unit area), which in t u r n can be
SOME ASPECTS OF IGNITION
converted into an effective velocity of movement
relative to either the unburat or the burnt gas.
Figure 2 shows the variation of space integral
rate with flame radius for several of the cylindrical
conditions of Table I. A similar curve is shown
in Fig. 3 for the spherical case, where space
integral rate is plotted against (radius) 2. The
results in Fig. 2 (inset) suggest that for a given
initial core size and total energy supplied there
is a common radius at which all successful ignitions converge to give a (negative) source strength
corresponding ~ith a more or less uniform propagation. At lower kernel sizes than this, the ignitions having higher ratios EK//Et~tal give higher
space integral rates of oxygen consumption, and
this presumably accounts for the greater ease
of ignition in such cases. The greater the energy
supplied, and the larger the initial core size, the
larger the radius at which the initial transient
conditions die out.
(b) Studies of A pproximately Statio~ary Flames
In these studies a gas flow into which the flame
propagates was simulated by supplying non-zero
values of the source strengths M in Eqs. (11),
(12), and (13). The integration was then allowed
to proceed into the steady state, entry into which
was shown by the development of a constant
value of the space integral rate. Because of
problems of flame stability towards small disturbances, it is only possible to study inward
propagating flames in this manner, i.e., only
positive source strengths M will lead to stable
stationary flames. The boundary conditions then
become
r--*~
r----~l
w,--~,b
(23)
1489
T
N
-~r..~
ee~176
~z 5
H x2.5
o r <be~
3
~
oO
_
_
-
oO
-
0
1.3
2,3
0.5
RADIUS / CPI
1.0
FIG. 2. Space integral rates for consumption of
oxygen in some cylindrical flames. Line S refem to
inward propagating stationary flame. Points series
F, H, and J refer to ignitions of Table I, and have
values on both axes multiplied by 2.5. Space integral
rates multiplied by v.
i
i
o~
,d
~~176
o
:-
o
%
~J
Q
o
O
~
o
9
O
I
I
I.O
RADIUS2/CH 2
The position of maximum heat release rate
was again taken as defining the flame radius. The
lines S in Figs. 2 and 3 shows the space integral
rates of the cylindrical a~ld spherical flames, the
actual calculations having been performed for
radii of approximately 1.5 ram, 5 mm, and 10
mm. At radii above one or two millimeters the
space integral rates are directly proportional to
rk, indicating little or no effect of curvature on
the flame velocity. In agreement with this, the
temperature and composition profiles of the 5
and 10 mm radius flames are virtually indistinguishable in each geometry. By measuring
the gradients of the lines S the cylindrical and
spherical flame velocities at the larger radii were
found to be 234 and 220 cm see-~ respectively.
These figut~es amy be compared with a planar
Fro. 3. Space integral rates for consumption of
oxygen in some spherical flames. Line $ refem to
inward propagating stationary flazne. Points series
M, Q, and R refer to ignitions in Table II, and have
values on both axes multiplied by 10. Space integral
rates multiplied by 2~.
flame burning velocity of 226 cm sec-~ computed
by Dixon-Lewis and Thompson8 using the same
kinetic and diffusion constants. At radii above
one or two millimeters there is clearly little effect
of curvature on the linear burning velocity of
this flame~ and even at smaller radii the effect
does not appear large in the stationary flames.
The first result is in agreement with observation.
Burning velocities measured by the soap bubble
14917
IGNITION
~"
"
TEMP
~
I
2001 .-,P__
I0
I
I
12
"I0
DISTANCE ] I~M
Fro. 4. Temperature and 1t atom profiles for
three inward propagating stationa~- flames. A,
cylindrical flame, 10.2 mm radius; B~ spherical flamo,
10.2 m m radius; C, cylindrical flame, 1.4 mm radiuu
(flame C has radial distances increased by 9.0 ram).
i
~
9
tY/
~
profilee very close to those of the larger spherical
flame, except for the tT a t o m maximum.
Figure 5 shows the heat release rates in all
the cyli,tdrieal flame~, and J.n the 10 rmn radius
spherical flame, plotted a~ainst msidenc~ titn~s
in the flames (starting at arbitrary zeros). F r o m
t h e ~ results, it, is clear thai the adjust,ment~
,~tl~in the tiame to take aet:nu~tt, of eha~tgia~g
curvature take place by way of modifications in
the distribution of reaction rate, aml it ks tails
which lea(l~ to the increasexl 1-1 atom concentratiuns izt the large cylindrical flazae.
\\~-
(c) Comparison of Sfalior~rg at~l Freshly
Propagating Flamc~
Unlike the situation in planar flame theory, it
has been found t h a t the detaih'd ~ r u e t u r e of a
cylimtrical or spherical tlame is not independent
(if the Ittmle positi~m. This impli,s that a freely
propagating flame, as from an ig.ition, may have
It co,ltinulmsly P.hall~ing Stl'l.lP'iure ',vheo the
rndith~ is ~mall, even though the flame is sufficiently renmte from it~ source for the igqtition
transients to have died out. The radius at which
the latter occurs is not eertaia, but it is likely to
coincide approxin~tely with the positions of
eonflueztee of the i~lition curves in Figs. 2 and
3. Event after this the liaear bm~it~g velocity
does not attain precisely tlm ~tatioztary flame
value. For flame F of Fig. 2, the temperature, II
a t o m and heat relr~so rate profiles at a radius
of 3.7 m m are compared ill Fig. 6 with the "in-
1800
%u
r
,
.S
O
/
IO
TIME / IO-5SEC
Fzc-. 5. Relative beat, release ra~
-- TEMP
A
HE TI
~.~ a fnn~tion of
r e , deuce Iinle ill three inward propagating stationary flames (arbitrary. time zeros). A, cylindrical
flame, 10.2 mm radius; B, spherical flame, 113.2mm
radius; C, cyliadrical flame, 1.4 m m radiu~.
method conform with those dotcrmim~d by other
nleaUS. 9
Curves A, B, at~(l C in Fig. 4 show the temperature and II agora profiles for the 10 m m
rs.dius cyliudrieM a~d sgheriea[ flames, and the
1.5 m m radius cylindrical flame respectively. The
major observable differcaee here is in the II atom
concentration profiles. "the 1 0 m m spherical
flame has X H . ~ = 0 . 0 1 7 6 , whereas the eorrespoudiag cylindrical flame has I~.,~.~=0.0199.
The small, 1.5 m m radius, eyliadrical flame has
IOOC
20
2.5
2~
3.5
O
DISTANCE /MH
Fro. 6. Comparison of " i n v e r ~ l " temperature
pmfiles, I! amm profiles and beat release rate profiles nf inward propagating cylindrical ~taiiou~y
/lame S of radius 52. mm with profiles of freely outward propagating flame F of radius .q.7 mm (arbitrary d~tance zen~, and temperature profiles moved
to left by 4.0 nun compared with o~,hers).
SOME ASPECTS OF IGNITION
verted" stationary flame profile for a radius of
5 nun. The differences are apparent. Parallel
behavior was observed in the spherical ease M.
Further investigation, including that of free
"inward" propagation, is in progress.
A final point of interest is that in bnrderline
ignitions such as F in Fig. 2 or Q and R in Fig.
3, the space integral rate during the ignition
transient may drop considerably below its
"steady" rate of increase. In flame F, the minimum velocity of flame movement, corresponding
with the tangent from the origin to curve F, is
about i75 cm sac-~. In this ignition also, the
temperature history at the axis is slightly more
complex than normal. After the maxhrmm at
1908~ given in Table I, the temperature at the
axis decreases to a minimum of 1562~ at a
time of 0.i7 msec, and then rises slowly again,
due to residual recombhmtion of H atoms still
present there.
Nomenclature
c~
tYv
D
specific heat of i at constant pressure
average specific heat at constant pressure
diffusion coefficient of major component of
mixture
D~ diffusion coefilcicnt of i
Fi
radial flux of i
FE radial energy flux
FM radial flux of overall mass
G, mass ffilX fraction of i
H~ enthalpy per gram of i
j~
diffusional flux of i
k
integer which takes values 0, 1, and 2 for
planar, cylindricM and spherical flames
respectively
M
ql
r
l
T
Tb
Tu
u
w,
X
p
1491
SOllrce strength
chemicM rate of fnrmatlon/g cm-a see-1
radius
time
temperature
burnt gag temperature
unburnt gas temperature
transformed radial co-ordinate
weight fraction of i
thermal conductivity
density
t~duced temperature
transformed radial co-ordinate
REFERENCES
1. BL:~DJIAN, L.: CorniEst. Flame 20~ 5 (1973).
2. SPALD1NO, D. B.: Philos. Trans. E. See. Lend.
A2~9, 1 (1956).
3. ZELDOVICH, Y. S. AND BARRENELA~I~, G. I.:
Combust. Flame 3, 61 (1959).
4. ADAMS,G. K. AND COON., a . B.: Combnst. Flame
4, 9 (1960).
5. DIXON-LEwis, G.: Prec. R. Soc. Lend. A~98,
495 (1967).
6. DAY, M. J., DtXoN-LEwIS~G.~ AND THOMI'SON~
K.: Prec. R. Soc. Lend. A330, 199 (1972).
7. LEwis, B. ANDYON ELBE, G.: Combustion, Flames
and Explosions of Gases, p. 335, Academic Press,
1961.
8. DIXON-LEwiS, G. AND T[~OMPSON, K.: TO be
published.
9. I,~NNET'r, J. W.: Fourth Symposium (International) on Combustion, p. 20. Williams and
Wilkins, 1953.