1. science
2. mathematics
3. geometry

Twenty-first Symposium (International) on Combustion/The Combustion Institute, 1986/pp, 1891-1897
A N A N A L Y S I S OF L E W I S N U M B E R A N D F L O W EFFECTS O N T H E I G N I T I O N
O F P R E M I X E D GASES
P.S. TROMANS AND R.M. FURZELAND
Shell Research Ltd., Thornton Research Centre,
P.O. Box 1,
Chester CHI 3SH, England
We have analysed numerically the ignition of a combustible gas to which heat is added over a
specified period at a spherical boundary of small diameter. We solve conservation equations for
reactant and sensible enthalpy and allow the gas to undergo a one step chemical reaction. The
numerical technique involves a finite difference scheme with an adaptive grid. We find that,
until the flame has grown beyond a critical radius, it relies upon the heat source to prevent its
extinction. A minimum energy is required for successful ignition. This minimum increases as
the Lewis number of the mixture increases. The minimum ignition energy also increases if the
flame front is stretched. This behaviour is in very close agreement with experimental observations of the spark ignition of turbulent mixtures.
1. Introduction
To explain their observations of spark ignition Lewis and von Elbe I m a d e intuitive arguments in terms of"excess enthalpy" and "flame
stretch". T h o u g h their description provides
useful scales for the m i n i m u m ignition energy
and predicts its relationship with pressure and
b u r n i n g velocity, it does not completely define
the process. To gain f u r t h e r insight several
numerical studies have been m a d e of ignition
initiated by small, hot pockets of gas. Generally,
this work has involved complex chemical kinetic
schemes. Dixon-Lewis and S h e p h e r d z'3 examined the ignition of hydrogen-air mixtures by
spherical and cylindrical hot spots. They investigated the effect of pocket size and radical
addition on m i n i m u m ignition energy. Overley,
Overholser and Reddien 4 studied the influence
on m i n i m u m ignition energy of hot pocket size
and shape; they looked at spherical and spheroidal pockets. Kailasanath, O r a n and Boris"
included the effect of a d d i n g heat to the hot
pocket over some prescribed p e r i o d rather than
instantaneously.
T h e present authors studied the ignition of
a gas with a Lewis n u m b e r of one, which
could u n d e r g o a one step chemical reaction. 6
We c o m p a r e d the effects o f a d d i n g heat at a
planar b o u n d a r y and at cylindrical and
spherical boundaries of very small diameter.
Only the spherical geometry provided results
which were similar to e x p e r i m e n t a l observa-
tions o f spark ignition: there was a minimum
radius below which deceleration and extinction o f a spherical flame could occur, a
m i n i m u m energy for ignition, an optimum
source duration and a sensitivity to imposed
strain fields. T h e importance o f spherical
geometry was illuminated by the asymptotic
analysis of Deshaies and Joulin.; T h e i r work
reveals a m i n i m u m radius for the unsupp o r t e d propagation of a spherical flame.
Kailasanath and O r a n s and Sloane 9 included
detailed chemical kinetic schemes in their
numerical studies of the ignition o f hydrogenoxygen-nitrogen mixtures and methane-oxygen-argon mixtures, but still f o u n d that flame
curvature had a major influence on their
results. Kailasanath and O r a n observed large
flame decelerations and a tendency to extinction in the spherical system, while Sloane
found that spherical geometry was essential
for his calculated m i n i m u m ignition energies
to show the correct d e p e n d e n c e on equivalence ratio.
In view o f the importance of geometrical and
physical factors it is worthwhile to investigate
the effect of variation o f the parameters in a
model involving only a one step chemical
reaction. This is the p u r p o s e o f the present
paper. In particular we investigate two areas:
the role o f an effective Lewis n u m b e r in
controlling the d e p e n d e n c e of ignition energy
on equivalence ratio and the possibility of
modelling the influence o f turbulent motion on
spark ignition in terms of flame stretch.
1891
1892
IGNITION AND EXTINCTION
2. The Conservation Equations and the
Method of Solution
We wish to analyse the initiation of a spherical flame in a p r e m i x e d gas by a small, spherical
source of heat. We consider a one step reaction
controlled by the concentration of one limiting
reactant. T h e conservation equations for the
mass fraction of this reactant and sensible
enthalpy are:
superscript asterisk indicates conditions in the
burnt gas behind a planar, adiabatic, laminar
flame. T h e non-dimensional conservation
equations are:
aY
or
1 0/
0yX
-/32MYexp j~0 - - l )
c~T
aT
i7 [" 4 0T" ~
+RB2MYexpl3(1-1)
and
o-7 +v Ox
x2
(2)
xty
(5)
(6)
We have eliminated the continuity equation by
introducing the mass weighted coordinate, t~,
such that:
where
a~
O-T = r 2 0 = r 2 ~
A nomenclature is p r o v i d e d later. T h e continuity equation is:
Ot~
_
1 0.2_
`
-~T -t- p e q- -~ff ~x ( X p v j = O
=
T*
-
30T
E Ok
(7)
(4)
T = ~1~*, To = ~1"o/7"*, Y = ?17o,
5 = ko/Oo ut Cp, r = x/8 , t = "rud8,
P = P/Po, f3 = E/RT"*, M = koB/e~Oo u2Cp~ ~,
R
02(r 3)
OC
The heat addition takes place at a spherical
surface with its centre at x = 0. T h e term e in
Eq. (4) corresponds to flows in directions
normal to the radial one. It leads to a velocity
gradient in the radial direction o f e/3 for the
constant density case and a flow g e n e r a t e d
flame stretch of the same value. Such a straining flow field could be generated by a distribution of sinks on an axis o f the sphere. T h o u g h
such a flow field would be difficult to p r o d u c e
practically, it allows us to investigate the influence o f flow g e n e r a t e d flame stretch in a one
dimensional problem. We shall assume ~-1 oc k
~/'. Equations (1) to (4) may be non-dimensionalised by substituting:
= e g/ul,
withr= 0atqJ=0.
To obtain the relationship between r and
we solved the additional differential equation:
To,
T h e b o u n d a r y conditions for the equations
are set by the heat addition at the spherical
surface and far field conditions. T h e heat
addition is at a surface defined by + = a. A
small value, a -- 0.05, was chosen so that the
heat addition reasonably models a point source.
Thus, we have:
3r
H(t)
T=To,
Y = I at large
(8)
The term (To/T) 4/3 is a d d e d so that a constant
value of H c o r r e s p o n d s to a constant rate of
heat addition. H(t) was given the form shown in
Fig. 1. T h e initial conditions are:
T = To, Y =
1 for a l l 0
(9)
It is straightforward to show that the total
energy a d d e d is:
E=4~(3a)413
where ul is the adiabatic burning velocity, g is
the adiabatic flame thickness and the tilde
indicates a dimensional variable, the subscript
zero indicates u n b u r n t gas conditions, and the
at~=a
flH(t)dt
(10)
We note that the dimensional energy is:
= OoCp~ 3 ? ' * E
(11)
ANALYSIS OF IGNITION OF PREMIXED GASES
/
"1Z
0
FZ
LL
er
I--
I
0
tI
t2
t3
TIME, t
FIG. 1. Form of heat source function H (t)
T h e governing Eqs. (5), (6), (7), are discretised in space using an adaptive mesh and
corresponding finite-difference approximations, to give a system of ordinary differential
equations in time for T and Y (of order 2N,
where N is the n u m b e r of space points). Within
certain limits the n o n - u n i f o r m space mesh is
generated by the co-ordinate transformation:
~ J( ~,t) d~b
XOk, r) = i ~ j ( 4 , , t) d4"
in the literature. Lavoie 11 gave activation energy as a function of equivalence ratio for
propane-air mixtures. T h o u g h it is often assumed that activation energy is i n d e p e n d e n t of
concentration, his results show a strong peak at
an equivalence ratio of one. The activation
energy falls off faster than adiabatic flame
temperature as the mixture is weakened or
richened. In view of the n u m e r o u s uncertainties it seems appropriate to use a constant
value of non-dimensional activation energy, [3,
in the present work: we have performed our
calculations for 13 = 8 and [3 -= 10. Our results
later will show that, though [3 is important, it is
a much less significant parameter than the
Lewis number. For extremely rich and extremely weak mixtures the Lewis n u m b e r
should be based on the diffusivity of the
deficient component. However, we require
some intermediate value for other mixtures.
We obtained an effective Lewis n u m b e r by
means of the formula derived by Joulin and
Mitani lz in their study of the instability of two
reactant flames. Their formula is:
L = L2 - G (L1 - L2)
(12)
where J(+,~), the monitor function, is some
measure of space derivatives of T and Y. For a
given u n i f o r m X mesh Xi = (i - 1) ~ , i = 1,
. . . . N with (N - 1) s
=- 1, constant for all
time, the above transformation adapts the
spacing of the corresponding I]/i points in
proportion to the gradients of T and Y. For the
present problem the monitor function we use
lS:
The mesh allows accuracy and fine resolution of
the flame to be combined with computational
efficiency. The limits on the mesh spacing are
provided by user chosen criteria on maximum
and m i n i m u m mesh sizes At~i. T h e finite-difference discretisations, space remeshing and time
integration are performed automatically by the
SPRINT package (Software for PRoblems IN
Time) developed jointly by Leeds University
and T h o r n t o n J ~
T h e calculations have been performed over
a range of values of Lewis n u m b e r , L, activation energy, [3, and initial temperature, To. The
initial temperature a n d flame temperature are
easily calculated to give To. However, a wide
range of values of activation energy are quoted
1893
(14)
where L2 and L1 are the Lewis numbers of the
deficient and a b u n d a n t reactants. G is a function of the equivalence ratio a n d the orders of
the reaction which we have assumed to be one
with respect to both reactants. The expression
is valid for L1 and L2 near unity. M, the
eigenvalue of the adiabatic flame problem, is
well approximated by 13
M = 1/2L + (0.656 + L)/[3L
In our previous work we found that for
Lewis number, L, equal to unity the m i n i m u m
ignition energy showed little sensitivity to the
duration, t3, of the heat source for t3 less than
one, For the present calculations we set t3 = 1
except for those at large Lewis n u m b e r where
large energies were required and we found
much longer source durations were acceptable.
3. Results
As indicated by Kapila 14 a n d discussed in our
previous paper the c o m m e n c e m e n t of the heat
source is followed by heating, induction, local
explosion and propagation stages. Ignition failure can occur as a result of flame extinction
early in the p r o p a g a t i o n stage as the flame
front moves away from the source. Sets of
temperature distributions illustrating a successful ignition and an extinction are shown in Figs.
2 and 3 respectively. Some position diagrams
1894
IGNITION AND EXTINCTION
2
C
10
b-
?A
~-
r
O
n
ku
0
5
10
--I
LL
RADIUS, r
FIG. 2. T e m p e r a t u r e distributions for successful
ignition
A, t = 0.4; B, t = 1.1; C, t = 2.6; D, t = 4.0;
To = 0.145, L = 1.00, 13 = 8, t3 = 1.0, E = 58
f o r f l a m e s i n i t i a t e d in d i f f e r e n t m i x t u r e s b y
d i f f e r e n t h e a t s o u r c e s a r e s h o w n in Fig. 4. W e
h a v e d e f i n e d f l a m e p o s i t i o n as t h e p o i n t w h e r e
t h e s t e e p e s t t a n g e n t to t h e c o n c e n t r a t i o n p r o file t h a t m a y b e d r a w n w i t h i n t h e f l a m e f r o n t
cuts t h e Y = 0 axis. Initially t h e f l a m e p r o p a g a t e s quickly, at a s p e e d close to t h e n o r m a l
b u r n i n g velocity, w h i l e it is d r i v e n by t h e h e a t
source. After cessation of the source the flame
s p e e d is m u c h r e d u c e d . F o r e a c h m i x t u r e t h e r e
is a critical r a d i u s w h i c h t h e f l a m e m u s t r e a c h ,
a i d e d by t h e h e a t s o u r c e , b e f o r e it m a y c o n t i n u e
A
i
I
i
2
I
I
I
4
|
I
6
I
8
TIME, t
FIG. 4. Position d i a g r a m for flames
A, To = 0.131, L = 1.19, 13 = 8, t3 = 1.1, E = 201;
B, To = 0.131, L = 1.19, 13 = 8, t~ = 1.1, E = 195;
C, To = 0.155, L = 1.00, 13 = 8, t3 = 1.0, E = 48.4
u n s u p p o r t e d . T h i s b e h a v i o u r is in q u a l i t a t i v e
agreement with the experimental observations
o f f l a m e g r o w t h b y L i n t i n a n d W o o d i n g , 15 d e
S o e t e 16 a n d A k i n d e l e , B r a d l e y , M a k a n d
McMahon) 7 The minimum energy for ignition
i n c r e a s e s w i t h t h e critical r a d i u s .
We have investigated the influence of the
p a r a m e t e r s To, ~ a n d L o n t h e m i n i m u m
ignition energy. Figure 5 shows the depend e n c e o n To: as w e m i g h t e x p e c t , t h e m i n i m u m
i g n i t i o n e n e r g y d e c r e a s e s as t h e initial t e m p e r a t u r e is b r o u g h t c l o s e r to t h e f l a m e t e m p e r a t u r e .
uJ
>-
F-
80
70
z
I-
60
I--
~
50
~
40
z
I
5
I
10
"~
3o
I
0.13
|
I
I
0.14
0.15
0.16
INITIAL
RADIUS,
TEMPERATURE, T O
r
FIG. 3. T e m p e r a t u r e distributions for extinction
A, t = 0.4; B, t = 1.1; C, t = 2.6; D, t = 4.0;
To = 0.145, L = 1.00, 13 = 8, t3 = 1.0, E = 53
FIG. 5. Variation o f m i n i m u m ignition energy with
initial t e m p e r a t u r e , To, for activation energy, 13 = 8
and Lewis n u m b e r L = 1; o Ignition, x Extinctiontinction
ANALYSIS OF IGNITION OF PREMIXED GASES
2O0
-
180
m
LU
160
Z
140
~_
120
Z
100
-
60
8
.~=10
/
so
z
40
~
0.95
/
"
1.00
i
i
i
i
1.05
1.10
1.15
1.20
LEWIS
NUMBER, L
FIG. 6. Effect of Lewis number, L, on minimum
ignition energy; o ignition 13= 8, [Nignition 13= 10, x
extinction 13 = 8, + extinction 13 = 10, To = 0.135
The influence is quite strong: in dimensional
terms a 10~ rise in ambient temperature
reduces the m i n i m u m energy for ignition by
about ten per cent. Figure 6 is of greater
interest: it shows how m i n i m u m ignition energy
increases monotonically with Lewis number.
The Lewis n u m b e r is large when the mixture is
very deficient in a heavy, slowly diffusing gas: it
will be small when the mixture is very deficient
in a light, rapidly diffusing gas. There is a
transition region between the two extremes,
which, for typical alkanes, occupies most of the
flammable range. Thus, we should expect the
equivalence ratio at which methane-air mixtures are most easily ignited to be the lowest of
all the alkanes. Propane-air mixtures will have
the next lowest equivalence ratio; the equivalence ratios of lowest spark ignition energy will
become richer as the molecular weight of the
fuel increases. This is exactly as found in
experiments.1
It can be seen from Fig. 6 that the effect of
increasing activation energy is to increase
slightly the sensitivity of m i n i m u m ignition
energy to Lewis n u m b e r . This arises from the
imbalance between chemical a n d sensible enthalpy fluxes that can occur in flames of
non-unity Lewis n u m b e r , is W h e n Lewis number is greater than one the imbalance leads to a
reduction in the flame temperature compared
with the unity Lewis n u m b e r case. Increasing
the activation energy increases the sensitivity of
the flame to these temperature changes.
We have used Eq. (11) in conjunction with
the data such as that listed in Table 1 to
calculate dimensional m i n i m u m ignition energies for methane-air and propane-air mixtures over a range of equivalence ratios. The
results are shown in Fig. 7. Although the
m i n i m u m ignition energies for the most ignitable mixtures are a factor of 5 smaller than the
values measured in spark ignition experiments,
1895
the curves are qualitatively correct. The difference is not unreasonable considering that our
model is a spherical one taking no account of
the real shape of the spark source or heat losses
to the electrodes. Moreover, quite large variations may arise from small errors in the flame
thickness, g.
Finally, we have explored the influence of
the strain rate term, e, in Eqs. (5) and (6). As
shown in Fig. 8 the addition of this term
increases the m i n i m u m ignition energy particularly of mixtures with large Lewis numbers.
This can be explained in terms of the known
sensitivity of flames, especially large Lewis
n u m b e r flames, to strain fields. It seems likely
that this sensitivity to strain rate reflects the
relationship between the m i n i m u m energy for
spark ignition and the root mean square,
turbulent strain rate found by T r o m a n s and
O ' C o n n o r 19 in their experiments on the spark
ignition of turbulent mixtures. To investigate
this we have replotted our theoretical curves in
Fig. 9 together with the results of T r o m a n s and
O ' C o n n o r and some taken from Ballal and
Lefebvre. 2~ All the experimental results are for
cases where the smallest eddies of the turbulent
motion are larger than or approximately equal
to the laminar flame thickness.
T h e i n d e p e n d e n t variable used in Fig. 9 is
the rate of flame stretch, that is the increase of
area of material surface in the flame front. This
equals ~/3 in the numerical analysis and approximates one half of the root mean square
turbulent strain rate in the experiments. The
m i n i m u m ignition energy was normalised on
the value at zero stretch rate. T h e agreement is
remarkably good.
4. D i s c u s s i o n
O u r model is restricted to one step chemistry; the combustible mixture is described only
by its Lewis number, activation energy and, via
the non-dimensionalization,its b u r n i n g velocity
and flame thickness. Omission of complex
chemistry in combination with advanced numerical techniques has allowed us to perform
over 300 simulations economically on a readily
available computer, an IBM 4341. It has emphasised the physical aspects of the problem.
Perhaps the best justification is found in the
results; at least qualitative agreement with
experiments is achieved without parameter
fitting. T h e implication is that m i n i m u m spark
ignition energy is mainly controlled by physical
mechanisms such as flame curvature and
stretch. These processes influence flame propagation through balances and gradients in the
1896
IGNITION AND EXTINCTION
TABLE I
Typical data used to generate Figure 7
Mixture
equivalence
ratio
Temperature
ratio,
T*/T0
Lewis
number,
L
Flame
thickness,
8 x 106m
Calculated
non-dimensional
minimum ignition energy,
E
Methane-air
0.8
0.9
1.0
1.1
6.70
7.16
7.47
7.42
0.967
0.977
0.991
1.00
103
78
63
53
45.5
54.4
63.0
65.1
Propane-air
0.9
1.0
1.28
1.40
7.31
7.61
7.18
6.88
1.38
1.24
1.04
1.00
59
54
59
69
492
286
77.5
56.1
L = 1.19
3.
uJ"
T O = 0.131
1000
300
w
o
z
O
z
(.9
o x
L = 1.00
250
T o = 0.141
500
o
LU
100
200
g
z__
I-
50
|
0.8
I
1.0
EQUIVALENCE
I
1.2
!
1.4
RATIO
90
z
5;
o
Fro. 7. Calculated minimum ignition energies for
methane-air and propane-air mixtures.
convection-diffusion zone of the flame front; ]8
they couple with the chemistry in a gross and
indirect way. Lewis n u m b e r will remain crucial
and our present theory will be applicable so
long as the sheet-like flame structure and
reaction-diffusion zone are maintained.
The practical limits and possibilities of our
theory remain to be tested by comparison of
our predictions of ignition energy with experimental results over a range of combustible
mixtures and conditions. A most exciting future application of the analysis is to the flame
development period following ignition, which is
vital to the performance of gasoline engines.
5, Conclusions
(i) O u r numerical calculations predict the
spherical flame growth and extinction that
may follow heat addition to a gas.
(ii) A m i n i m u m ignition energy and critical
o~
i
x
70
T O = 0.142
50
30
0
0.1
0.2
0.3
0.4
0.5
VELOCtTY GRADIENT, e
Fro. 8. Effect of velocity gradient, E, on minimum
ignition energy. [3 = 8, o Ignition x Extinction
flame radius are f o u n d in qualitative agreement
with experimental observations of spark ignition.
(iii) Calculations of the effect of flame stretch
on m i n i m u m ignition energy are in remarkable agreement with experimental
observations of influence of turbulence on
spark ignition.
(iv) The broad features of spark ignition are
generated by the physical mechanisms of
flame curvature and flame stretch; the
ANALYSIS OF IGNITION OF PREMIXED GASES
1897
REFERENCES
m
uJ
1. LEwis AND VON ELBE: Combustion, Flames and
E
g
2.
z
N
3.
1
I
0.1
FLAME
4.
0.2
STRETCH
RATE
5.
FIG. 9. Comparison of the predicted effect of flame
stretch with the measured effect of turbulent motion
on spark ignition, o methane-air, pressure = 1 bar,
ignition energy at zero stretch = 0.32 mJ, from ref,
18; 9 propane-air, pressure = 0.17 bar, ignition
energy at zero stretch = 3.4 mJ, from ref. 19
m a j o r controlling
number.
parameter
is the
Lewis
Nomenclature
a
B
Cp
e
E
H
J
k
L
m
N
Q
r
R
t
T
ul
v
W
z
Y
13
e
radius o f the cylindrical source
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d i m e n s i o n a l strain rate
ignition e n e r g y
f u n c t i o n giving t i m e d e p e n d e n c e o f the
heat source
monitor function
t h e r m a l conductivity
Lewis n u m b e r
n o n - d i m e n s i o n a l reaction rate constant
n u m b e r o f m e s h points in space
h e a t release by chemical reaction
n o n - d i m e n s i o n a l space c o o r d i n a t e
t e m p e r a t u r e rise across an adiabatic flame
n o n - d i m e n s i o n a l time
temperature
adiabatic b u r n i n g velocity
velocity
chemical reaction rate
d i m e n s i o n a l space c o o r d i n a t e
reactant c o n c e n t r a t i o n
n o n - d i m e n s i o n a l activation e n e r g y
adiabatic flame thickness
n o n - d i m e n s i o n a l strain rate
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
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