©
Combust. Sci. and Tech., 1990, Vol. 73. pp. 395-413
Photocopying permitted by license only
Gordon and Breach Science Publishers. Inc.
Printed in Great Britain
Dependence of Minimum Ignition Energy on Ignition
Parameters
ABOELKAOER FRENOI and MERWIN SIBULKIN
Division of Engineering,
Brown University, Providence, R./. 02972
Downloaded By: [University of Southern California] At: 22:09 18 January 2011
(Received January 30,1990; in final form May 14. 1990)
Abstract-Using two mathematical models (one allowing for pressure variations, and the other assuming
the pressure to be uniform), the minimum ignition energies of a stoichiometric methane-air mixture are
calculated. The effect of the initial pressure wave generated by the spark on minimum ignition energy is
studied by comparing the results of the two models. The pressure wave is found to affect the values of
minimum ignition energy for ignition times shorter than 2 us. The dependence of minimum ignition energy
on the kernel radius and ignition time are also investigated. The results show that as the kernel radius is
decreased to small values, the minimum ignition energy reaches a constant value. The ignition times that
give an optimum value of minimum ignition energy are in the range of 3 J.1.S to lOOps. Our results show that
minimum ignition energy is strongly dependent on the ignition parameters, and a minimum value of
0.005 mJ is found for an ignition kernel radius of 0.0635 mm and an ignition time of 27.5I's. This value is
70 times lower than the experimentally reported values. When a kernel radius of 0.5 rnm is used, fair
agreement with experiments is obtained.
NOMENCLATURE
A,
A
Cp
Da
o,
E
E,ig
s;
F
H( ...)
h,
k
K
/,
u,
M
o
P
Pr
p
Frequency factor
Frequency factor (see Eq. (17))
Average specific heat (see Eq. (18))
Non-dimensional number (see Eq. (26))
Binary diffusion coefficient for species k
Activation energy
Ignition energy
Ignition power
Fuel
Heaviside unit function
Heat of combustion per unit mass of fuel
Reaction rate constant
Geometry factor (I, 2n, 4n)
Reference length (/ = (K", r,)'/2)
Molecular weight of species k
Average molecular weight
Oxidiser
Products
Prandtl number
Pressure
Non-dimensional pressure
p
q
R
r
(fJ --
L)
p",
Integer controlling the spatial distribution of ignition energy
Rate of energy
Universal gas constant
Radius
395
A. FRENDI AND M. SIBULKIN
396
Non-dimensional time (I = ~)
r,
Temperature
T
T(O, ei,) Kernel center temperature at completion of energy input
Radial gas veloci ty
II
II,
Non-dimensional enthalpy (
pc;, (T - T",) )
P",cp (1J.uJ - T",)
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Rate of production (or consumption) of species k per unit volume
Non-dimensional radius (r/I,)
Mass fraction of species k
Greek Symbols
p
Geometry factor (0, I,2)
Gas viscosity
Thermal conductivity
Thermal diffusivity
Mixture equivalence ratio
Stoichiometric coefficient of species k
Gamma function
Gas mixture density
Non-dimensional density (p/p",)
pu
Non-dimensional momentum flux (PII/p",(!c..»
r
r,
Time
Reference time ([p~ y~", A exp ( -
IX
Ii
).
K
</>
Vk
f( ...)
P
r,
PW 1)
Subscripts
f,ad
C
comb
d
f
ig
k
00
Ad iaba tic flame
Critical values
Combustion
Diffusion
Flame
Ignition
Species k (F, 0, P)
Outerflow condition
Superscripts
per unit volume
INTRODUCTION
The knowledge of ignition limits of a combustible gas mixture is of importance in
many combustion systems. In order to successfully ignite a flammable gas mixture, a
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MINIMUM IGNITION ENERGY
397
sufficient amount of energy is needed. The smallest amount of energy that results in
a successful ignition is known as the minimum ignition energy. The experimental
determination of minimum ignition energy is done using different types of sparks. The
minimum ignition energies of various hydrocarbon fuels were determined by Blanc et
al. (1947, 1948) using a capacitance spark. They found that minimum ignition energy
varied with mixture composition, fuel type, pressure and electrode configuration. A
summary of these results and other experimental results is given in Lewis and von Elbe
(1987). For a stoichiometric methane-air mixture at Too = 298 K and p ; = latm, the
minimum ignition energy was found to be 0.35 mJ; the values of spark gap and spark
duration were not reported.
Ronney (1985) measured minimum ignition energies for methane-air mixtures both
at earth gravity and zero gravity for different pressures. He did not report any
significant difference between the minimum ignition energies except near the zero
gravity flammability limit, where the zero gravity minimum ignition energies were
higher than the earth gravity values. The minimum ignition energy for a stoichiometric methane-air mixture at the same conditions as above was found to be 0.5 mJ
for a spark gap of 3 mm and a spark duration of 6 us. Kono et al. (1976) studied the
effects of electrode configuration, electrode gap and spark duration on the minimum
ignition energy of lean propane-air mixtures. Their results showed the existence of an
optimum spark duration for which the energy is minimum. Decreasing the spark gap
increased the minimum ignition energy, and a similar result was obtained by increasing the electrode diameter. Kravchenko et al. (1973) determined the minimum ignition
energy for a near stoichiometric methane-air mixture as a function of circuit inductance. They found in the cases they studied that there exists a critical ignition time in
the range of 100-120 IlS, for which minimum ignition energy is minimal. This critical
time was found to be independent of the electrical and mechanical conditions of the
spark formation, and they concluded that it was dependent only on the nature and
state of the mixtures.
Even though the physical phenomena involved in ignition are complicated,
theoretical and numerical attempts at modeling the early stages of flame initiation
have been made, and a good understanding of the experimental results has been
reached. A detailed model for a hydrogen-air flame was presented by Dixon-Lewis
and Shepherd (1974). They studied the effects of adding hydrogen atoms to a cornbustible mixture, and found that the thermal energy needed to ignite the mixture was
reduced. This conclusion was also reached by Deshaies et al. (1984) when studying the
effect of molecular diffusion on flame initiation. Using matched asymptotic expansions for large activation energy they showed the existence of a critical radius for
successful flame propagation. When some of the fuel was replaced by hydrogen, the
critical radius became smaller, and if a constant heat source was placed at the center
of the kernel, no critical radius was obtained. Champion et al. (1986) calculated the
critical ignition parameters for premixed lean propane-air mixtures and used them as
the basis for their experiments. The effects of molecular diffusion, activation energy
and unburned gas temperature on minimum ignition energy for methane-air and
propane-air mixtures were examined by Tromans and Furzland (1986). Molecular
diffusion was found to have the greatest effect on minimum ignition energy. Their
results predicted the correct trend, but the values were five times lower than the
experimental ones. Sloane and Schoene (1988) used a detailed chemistry model for the
combustion of methane in air to study the relative importance of the different
reactions on minimum ignition energy. In their model, ignition energy was added
instantaneously into a spherical kernel.
In a recent paper by Maas and Warnatz (1988), an attempt at simulating a spark
A. FRENOI AND M. SIBULKIN
39H
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ignition experiment was made. They used in their calculations two models; one
allowing the pressure to vary in the gas mixture and the other assuming the pressure
to be uniform. They also studied the effects of spark gap and spark duration on the
minimum ignition energy of a hydrogen-oxygen flame. They concluded that both
models predict the same minimum ignition energy for long ignition times, and that for
small kernel radii the energy became constant.
In this paper, two theoretical models are presented; one allowing the pressure to
vary in the gas mixture and the other assuming the pressure to be uniform. These
models are then used to calculate the minimum ignition energies of a stoichiometric
methane-air mixture for different ignition times and ignition radii. The results of these
two models are compared to existing numerical and experimental results.
MATHEMATICAL FORMULATION
The mathematical models presented in this paper are obtained from the conservation
laws for a chemically reacting gas mixture. The unsteady, one-dimensional models are
derived using the following simplifying assumptions;
• No gravitational force.
• The chemistry is modeled by a one-step, irreversible reaction.
• The gas properties used assume a constant specific heat (chosen to give the correct
adiabatic flame temperature, '0.ad)' constant values for p). and PJ1, and constant
values of p2 D for methane and oxygen.
• No volume heat losses (such as radiation) are included.
Based on these assumptions, the unsteady onedimensional equations written in a
conservation form are:
.
Mass conservation
I a
r' ar
ap
- + - - (ra pu) = 0,
aT
(I)
where ex = 0, I, 2 for planar, cylindrical and spherical geometries, respectively.
Species conservation
a (pYd + .!.. aa (rapYku)
r':l r
ar
la(a
aYk)
a - r pD k r ar
ar
+
(2)
lI'k
Momentum conservation
-a (pu)
aT
+ -I -a
r' ar
+
'
(ra
pu')
=
- ap
- -
ar
2-Ia -a [;
r II -au] + -ex [
r ar
ar
r
-2 -I -a
3 r" ar
[ J1a (ra)]
u
ar
a (r, u)]
- -2J1U
+ -32 -r'J1 -ar
r
(3)
Energy conservation
a - T)
-a (p
T
Cp
I
a
r
r
-
+ -;: -a (rap cp Tu)
.!..r" ~
or [ra A aD
orJ +
q....
com
b
+
q':"
19
(4)
MINIMUM IGNITION ENERGY
399
State equation
For an ideal gas mixture, the equation of state is
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(5)
In the energy equation written above, the pressure work and the dissipation have
been neglected. The source terms q;;,m; and q;; represent the chemical power per unit
volume released and the ignition power per unit volume input respectively. For the
derivation of these equations, the reader is refered to Toong (1983). For many
combustion systems the pressure change across the combustion wave is neglegible and
therefore the pressure can be considered as uniform. This assumption results in
neglecting the momentum equation. In order to calculate the fluid velocity when the
constant pressure assumption is used, the continuity equation can be integrated to
give
=
pu
I
-r'
f,' r' -dr.
op
or
(6)
constant.
(7)
0
The equation of state (5) becomes
pT
=
Equations (6) and (7) along with Eqs. (3) and (4) define the constant (or uniform)
pressure model; whereas eqns. (I) through (5) define the variable pressure model .
• The energy source terms
The energy equation given by Eq. (4) is a general one; in order to apply it to a specific
case the source terms need to be defined. The following is the definition of the ignition
and combustion source terms.
The ignition source term
The study we are to perform is that of minimum ignition energy. Most of the recent
experimental work done on this subject used spark discharge systems which delivered
energy at a constant rate. Therefore we choose the ignition source term to be of the
form;
(8)
In this calculation, the spatial distribution used is close to a step function (p
derived in Sibulkin and Siskind (1987), the function g(P) is such that for
g(P)
=
[I ( I
+a -+-I)J/(' +
P
I)
6). As
(9)
the ignition energy is given by;
K
......,+1
- - I Lig'ig riga
+
(10)
A. FRENDI AND M. SIBULKIN
400
where K
=
1, 27': and 47':, for
Cl
=
0, 1 and 2 respectively.
The combustion source term
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The gas mixture considered in this paper is stoichiometric methane-air. The combustion of methane involves over 100 simultaneous exothermic and endothermic chemical reactions. Depending upon the physical problem being solved, the number of
reactions can be reduced significantly using a sensitivity analysis. However, what is
needed in solving the energy equation is only the net local energy release rate q;~mb'
This is the motivation for the development of "global reaction" models which use
empirical reaction rate constants to model the combustion chemistry.
A single-step irreversible reaction can be represented by
(II)
where V F, "o and Vp are the stoichiometric coefficient for fuel, oxygen and products
respectively. The source terms Wk of the species equations can be written as
WF
= _k
Wo
=
F
M (pYS (PYa)""
F
F
M\'PM\'o
'
F
a
v
-kv M (pYF)"F(pYa)'O
a
0
M'FM'o
F
0
(12)
(13)
The combustion source term of the energy equation is given by
(14)
where h, is the heat of combustion per unit mass of fuel. Taking the rate constant of
the reaction to be of the Arrhenius form
k
=
A, exp (- E/RT),
(15)
leads to an expression for the combustion source term of the form
q;':m'b =
h,(pYp)'F (PYa)"" A exp (- E/RT)
(16)
where
A =
( 17)
The kinetic parameters A and E used in this calculation were determined by Coffee
et al. (1983). In their calculation for a steady, one-dimensional, laminar premixed
flame they tried to match the heat release rate of the global reaction to that of the
detailed chemistry model. After an initial choice of V F and Va, "best fit" values for A
and E were chosen for each mixture to give the correct value of q~"mb' The adiabatic
flame temperature and the heat of combustion h, are also taken from Coffee et al's
data. The specific heat was calculated using the relation
(18)
MINIMUM IGNITION ENERGY
401
• Initial and boundary conditions
The system of partial differential equations defining both the variable pressure and the
constant pressure models are subject to the following initial and boundary conditions;
r
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r
=
=
0
T
T""
r
0
a;
r",
a;
p
op
Or
ar
OYk
op
Or
ar
Yk
P""
oYk
oT
oT
=
=
=
Yk ,,,, and u
o and u =
0
0,
(19)
(20)
0,
ou
Or
and
=
exu
r",
- -,
(21)
where roo is the outerflow boundary.
METHOD OF SOLUTION
The conservation Eqs. (I )-(7) along with the initial and boundary conditions (19)(21) are then put in a non-dimensional form using an approach developed in Sibulkin
and Siskind (1987) and Sibulkin and Frendi (1989). In the variable pressure model,
the momentum equation is non-dimensionalized as follows
2
op~, +--;,x-.I 0 [ , p~, ]
-a
t
x ox
P
= -
rj)/fJ ax
~3 Pr~x ~
[~~ (x'P~)]
Ox pax
p
(22)
0 [X' 0(PU)]
+
I
2Pr--- x' ax p Ox p
+
Pr -ex [ - ~
x
px
2pu + -32
I
0[x.,PU]]
7--;,
px uX
-e-
p
.
Similarly, the equation of state (5) becomes
...
P =
...
P
U
1
+ -.
y
(23)
The non-dimensional variables of Eqs. (22) and (23) are defined in the nomenclature.
The remaining PDEs were non-dimensionalized in Sibulkin and Frendi (1989). The
non-dimensional parameters rj)p, Pr and yare given by
7J.ad -
Too'
(24)
where Pr is the Prandtl number. The non-dimensional system of equations and
boundary conditions is integrated using a "method of lines" algorithm. The spatial
derivatives are discretized using fourth order finite differences, and the resulting
system of ordinary differential equations in time is integrated using the Gear-Hindmarsh programs for solution of ODEs. The specific package used was developed by
Hyman (1979). The problem is solved on a Cyber205 supercomputer. The average
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402
A. FRENO I AND M. SIBULKIN
computational cost of a run is 3600 SBUs for a spatial mesh of 150 points (I SBU "" 1
second in CPU time).
When the variable pressure model is used, the instantaneous ignition energy addition by the spark generates a pressure wave which travels at the speed of sound. As
the wave reaches the outer boundary it gets reflected and therefore causes oscillations
in the density, velocity and pressure profiles. In order to overcome this difficulty, a
method is devised to convect the pressure wave out of the domain of integration. It
is observed that the decay of the pressure wave is fast, and behind the pressure front
the ambient condition is recovered quickly; therefore a two-stage calculation is used.
In the first stage, the calculation is done on a domain three times larger than the
desired domain. The integration is then stopped at a known time which corresponds
to the time it takes the pressure wave to reach the outer boundary. The data corresponding to the desired domain is stored. By the end of this stage the pressure in the
desired domain has come back to the ambient value. In the second stage, the calculation is continued using the stored results as input. The combustion wave is now the
only propagating wave in the domain. The details of this method are given in Frendi
( 1990).
Besides the built-in accuracy checks on the time integration required by the GearHindmarsh programs; an additional check on the accuracy of the calculation was
monitored by applying conservation of energy to the entire computational domain.
The thermal energy stored in the computational domain at a given time is compared
to the algebraic total of the ignition and combustion inputs to the system up to that
time. As an initial check, a calculation of the burning velocity for a stoichiometric
methane-air mixture in a planar geometry is performed. A burning velocity of 40 cm/s
is obtained, which is in good agreement with the detailed chemistry value of 39.6 cm/s
(Coffee et al., 1983) and with experimental values.
RESULTS AND DISCUSSION
The results reported in this paper are for a stoichiometric methane-air mixture at
Poo = I atm and Too = 298 K and for a spherical geometry (0: = 2). The method of
finding the minimum ignition energl' is as follows. For a given set of ignition parameters (r,g, r,g), the power input E'K is increased (or decreased) until the relative
difference in energy between ignition and non-ignition with respect to non-ignition is
approximately 10%.
Effect oj spark-induced pressure wave on minimum ignition energy
We first compare the results obtained using the constant pressure assumption to that
obtained through the variable pressure model. Figure I shows that for a fixed kernel
radius of r'g = 0.167 mm the two models predict the same minimum ignition energies
for long ignition times (r,g > 2 liS). However; for short ignition times (r,g < 2ps), the
variable pressure model predicts higher minimum ignition energies. This result is
explained by the generation of a strong pressure wave (Kono et al. (1988); Bradley et
al. (1987) and Lim et al. (1987)) which leads to lower temperatures in the ignition
kernel and therefore non-ignition. The minimum ignition energy predicted by the
constant pressure model remains constant for short ignition times.
Effect oj ignition parameters on minimum ignition energy
The effect of changing the kernel radius on minimum ignition energy is shown on
Figure 2 for different ignition times. For kernel radii larger than 0.3 mm, the minimum
MINIMUM IGNITION ENERGY
403
10-'--,---------------,
Ignition
o Non-ignition
--,
E
riG; 0.167 mm
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_ _ _ Variable pressure
........... Constant pressure
10-7
10-6
T. (S)
'9
FIGURE I Comparison of minimum ignition energies obtained by the variable pressure and constant
pressure models. Stoichiometric methane-air mixture, spherical geometry.
ignition energy increases in proportion to the volume (r!g) of the ignition kernel. As
the kernel radius is decreased below r;g = 0.1 mm, the minimum ignition energy
reaches a constant value for !;g = 2.75 Ji,S and 27.5 us. For !;g = 100/ls and 500 Ji,S
asymptotic behavior is approached, but due to the computational cost for small
ld - = r - - - - - - - - - - - - - ,
_ _ _ 2.75
____________ .27.5
----100.0
_ _._500.0
FIGURE 2
Variation of minimum ignition energy with kernel radius for different ignition times.
Stoichiometric methane-air mixture, spherical geometry.
404
A. FRENDI AND M. SIBULKIN
108 =
------------,
:
-
-
\
2.75
\
27.5
\\
-.
\,
~
100.0
500.0
"-
<,
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". ' 't ; ~
0.1
"-----. - " " - ' "
,
,
0.2
0.3
0.4
r. (mm)
'9
FIGURE 3 Variation of minimum ignition energydensity with kernel radius for different ignition times.
Stoichiometric methane-air mixture, spherical geometry.
values of rig the calculation was stopped at rig = 0.04 mm. The variation of the
ignition energy density (E;;) with kernel radius for different ignition times is shown
on Figure 3. For rig > 0.2 rnm, the ignition energy density is a constant for values of
rig ,;:; 100ps, and tends to a constant value for larger ignition times. For kernel radii
smaller than 0.2 mm, the ignition energy density increases rapidly for all ignition
times.
The variation of minimum ignition energy with ignition time for different kernel
radii is shown on Figure 4. The vertical dotted line separates the regions where the
two models predict the same values (r i, > 2ps) and different values (r iK < 2ps) of
minimum ignition energy. For rig < 2 us, only the constant pressure results are
plotted, and, as the figure shows, the minimum ignition energy is constant (the
minimum ignition energies predicted by the variable pressure model would be higher
in this region). For 2 us < rig < 100 JiS the minimum ignition energy is nearly constant; however for rig > 100 JiS it increases in proportion with ignition time (E i• o: rig)'
The increase in minimum ignition energy at long ignition times is caused by the larger
diffusional heat losses. Figure 5 shows the variation of the ignition energy density with
ignition time for different kernel radii. The general trend of the curves is similar to that
of the minimum ignition energy, the only difference is that the ignition energy density
is higher for smaller kernel radii.
To summarize these results, Figure 6 shows a three dimensional plot of the minimum ignition energy surface, Eig = Eig(rig, rig)' It is important to observe that for
small kernel radii and short ignition times, the minimum ignition energy surface
shows a plateau; therefore supporting the conclusion of constant minimum ignition
energy for this range of parameters. This result is in agreement with the conclusions
of Maas and Warnatz (1988), and disagrees with that of Kailasanath et al. (1982) who
found an increase in minimum ignition energy as the kernel radius is decreased to
smaller values.
MINIMUM IGNITION ENERGY
405
10-'-::r----------,--------,
rig (mm)
_ _ 0.0635
........... 0.1100
___ .0.1253
_ _ .0.1670
/.
j(/ ~/
If"
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----------//:
=. -.=.:.:-::::::-.:.:t::--::".:.~;./
Constant pressure
iVoriable pressure
:
and
jConstont pressure
-3
10 , '"
10-9
FIGURE 4
Variation of minimum ignition energy with ignition time for different kernel radii. Stoichio-
metric methane-air mixture, spherical geometry.
108 - = r - - - - - - - - - - - - , - - - - - , - - - - ,
_ _ 0.0635
7
10
~
-€.
...,
........... 0.1100
___ .0.1253
_ . _ .. 0.1670
.~
W
6
10
'vcrlobte pressure
Constant pressure
~
and
iConstant pressure
I 1,111
10-4
I
10-3
FIGURE 5 Variation of minimum ignition energy density with ignition time for different kernel radii.
Stoichiometric methane-air mixture, spherical geometry.
ROLE OF HEAT DIFFUSION
The nondimensional enthalpy profiles (u,) at the end of ignition are shown on Figure
7. These profiles correspond to the successfully ignited cases, and to an ignition kernel
radius of rig = 0.0635 mm. As the ignition time is increased, more energy is needed
for successful ignition (except for rig = 27.5 J1S where less energy is used). This
406
A. FRENDI AND M. SIBULKIN
Minimum Ignition Energy Surface
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0.0_------1------,
FIGURE 6
Dependence of minimum ignition energy on ignition time and kernel radius. Stoichiometric
methane-air mixture. spherical geometry.
0.20 - , - - - - - - - - - - - - - - - - - - - - ,
T" (ILS ) E" ( ILJ )
_ _ _ 2.75
6.05
............ 27.5
5.5
0.16 ____ 100.0
7.0
_. _ _ 500.0
18.0
0.12-
~
"".
..... '.~
\'\\
'\\
: \
\ ' \
\ \\
\ ' \
0.08-
r .,
. = 0.0635 mm
\.
0.04
,
0.0
0.2
'.
\
<,
,
0.6
0.8
FIGURE 7 Non-dimensional enthalpy profiles (u,) at the end of ignition for different ignition times.
Stoichiometric methane-air mixture, spherical geometry. (Successful ignitions only)
407
MINIMUM IGNITION ENERGY
10.0,..------------------,
8.0
rill
= 0.0635 mm
EIIl
= 5.S pJ
T"
"
2.75 fl.S
6.0
~
27.5 fl.S
I
~
>-
4.0
T
=
T
ig
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2.0
0.0
--I------,---..::.:;===:::;:::==::,::==:::j
I
0.0
0.1
0.2
0.3
0.4
r (mm)
FIGURE 8a Temperature profiles at the end of ignition for two different ignition times. Stoichiometric
methane-air mixture, spherical geometry.
increase in energy is explained by the higher diffusional heat losses at longer ignition
times (the energy spreads over a larger volume). However, the value of UI at the center
becomes smaller as the ignition time is increased which corresponds to lower kernel
temperatures.
Figures (Sa) and (Sb) show an example of ignition and non-ignition cases for two
ignition times (r,g = 2.751ls and 27.5Ils), a kernel radius of r,g = 0.0635 mm and
using the same ignition energy E,g = 5.5IlJ. Figure Sa shows the temperature profiles
at the end of ignition. For rig = 2.751ls the temperatures near the kernel center are
higher than that for rig = 27.5Ils; however, more energy has diffused out of the
ignition kernel in the latter case. The kernel trajectories for the two cases are shown
on Figure Sb. Both kernels expand initially to nearly the same kernel size
(rj = 0.17 mm). For rig = 2.75 IlS the kernel decayed, and for rig = 27.5 Ils a combustion wave is propagated. Notice that in the latter case the kernel radius remains
constant for some time (0.03 ms < r < 0.12 ms) before it propagates. This example
shows that for small kernel radii (rig < 0.17 mm), in order to obtain a successful
ignition a critical size and temperature need to be reached during the decay phase. For
this particular case, the size and temperature are 0.17 mm and 1700 K respectively.
Variations of the kernel center temperature
Figure 9 shows the kernel center temperature at the completion of energy input
(r = rig) as a function of kernel radius for different ignition times. For r,g = 2.75,
27.5 and IOOIlS the curves show the existence of two regions; one in which the kernel
center temperature is constant and equal to approximately 1000 K (for r,g > 0.2 mm),
and the other where the temperature rapidly increases to high values (rig < 0.2 mm).
This region of rapid increase is followed by another constant temperature region
above the adiabatic value for r'g = 27.5 and IDOllS. For rig = 500 us, only two regions
are shown by the figure; the constant temperature region above the adiabatic value
which extends over a wide range of r'g values and a region of rapid change in kernel
408
A. FRENDI AND M. SIBULKIN
0.50
,---~-------------,
T
0.45
0.40
0.35
'.
2.75 1'5
_ _ _ 27.5 f1S
rig
= 0.0635 mm
E" = 5.5 j.LJ
0.30
E
E
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...-
0.25
0.20
0.15
0.10
0.05
0.00 +--,-----,,..-----,----,--------1
0.00 0.06
0.12
0.18
0.24
0.30
T (
ms )
FIGURE 8b Kernel trajectories for ignition and non-ignition cases. Stoichiometric methane-air mixture,
spherical geometry.
center temperature (r;K > 0.3 mm). This figure illustrates the role of heat diffusion and
chemical reaction on the kernel center temperature. For short ignition times there is
less diffusion and therefore the kernel center temperature is high for small kernel radii.
At longer ignition times both heat diffusion and chemical reaction contribute to the
final value of the kernel center temperature. Notice that the temperatures plotted on
Figure 9 are those that give successful ignition.
The change in kernel center temperature with ignition times for different kernel
radii is shown on Figure 10. As mentioned previously, the combination of heat
diffusion and chemical reaction contribute to the value of the kernel center temperature. The constant temperature region for short ignition times corresponds to
neglegible diffusional heat losses and chemical energy production. As the kernel
radius becomes smaller. this constant temperature region becomes smaller. For
r;K = 0.0635 mm no results are obtained at short ignition times due to numerical
instabilities.
In the absence of diffusional heat losses and chemical reaction, the energy equation
reduces to a simple form and the temperature distribution can be found analytically
when the ignition source given in the mathematical model is used (see Frendi (1990)
for details). In particular the kernel center temperature is given by
(25)
MINIMUM IGNITION ENERGY
409
10.0,-------------_
,-..
'::L
2.75
27.5
100.0
500.0
8.0
~
,
9
~
6.0
x
.'"
t-
o
00'9:::::
1.0
4.0 -
)=:"
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2.0
,
,
0.0 +----;r----,--,.------,---I
0.0
0.1
0.2
0.3
0.4
0.5
';9 ( mm )
FIGURE 9
Variation of kernel center temperature with kernel radius for different ignition times.
Stoichiometric methane-air mixture, spherical geometry. (Successful ignitions only)
Using the energy densities corresponding to the successfully ignited cases for kernel
radii of 0.167 mm, 0.1253 mm and 0.11 mm, the kernel center temperature is calculated and the results are shown on Figure I I. For short ignition times, Figures II
and 10 predict the same values of kernel center temperatures, implying the absence of
10.0.,-----------------,
Tr,ad
'" (mm)
_ _ 0.0635
8.0
........... 0.1100
•
_ _ .. 0.1253
6.0
= 1.0
Do
"
_ _ .0.1670
4.0
2.0
0.0
I
I I
10- 9
10- 7
10- 6
T,
'9
(s)
FIGURE to Variation of kernel center temperature with ignition time for different kernel radii. Stoichiometric methane-air mixture, spherical geometry. (Successful ignitions only)
410
A. FRENDI AND M. SIBULKIN
10.0
"Y:
.......,
",
~
x
~
fIg
1
(mm)
Ii
0.1100
___ .0.1253
•••• C••••••
8.0
f
_:'-.0.1670
6.0
-'I'
1-.
,..:'1
0
):::'
. . .:.. . . j
4.0
T,.,;·················
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2.0
0.0
•
::=::-:="'::.=-:---=.:::::::- _911111111
10
_611111111
10
_1711111111
10
.==-_.--./
_1611111111
10
T.
'0
..-I/-:
-5
10
11111
111111
4
10.
10"
(s)
FIGURE II Kernel center temperatures obtained using Eq. (25), for three different kernel radii.
Stoichiometric methane-air mixture, spherical geometry. (Successful ignitions only)
heat losses by diffusion or gains by chemical reaction. However, as the ignition time
is increased Eq. (25) predicts higher temperatures. When the chemical source term in
the energy equation is dropped we obtain the diffusion equation. A comparison of
kernel center temperatures given by the total energy equation, the diffusion equation
and Eq. (25) is shown on Figure 12. These results are obtained using the same ignition
parameters. At short ignition times rig < 10- 7 s, all three equations predict the same
temperatures (this means neglegible diffusion and chemical energy production). For
10- 7 s < rig < 2 x 10- 6 s, the diffusion equation and the thermodynamic equation
(Eq. (25» predict the same temperatures; however the energy equation gives higher
temperatures. This means that diffusion is still neglegible but there is some chemical
energy production. As the ignition time is increased above 2 x 10- 65 , the three
equations predict different values of kernel center temperature. This is because for
these values of ignition times both diffusion and chemical reaction are important. The
lowest temperatures are those predicted by the diffusion equation.
Comparison of time scales
Consider the nondimensional parameter defined by;
(26)
where Da ig is similar to the so-called 'Darnkohler number' and the subscript "ig"
means ignition. In Eq. (26), rdis the diffusional time scale (r~/Kig) and rig is the ignition
time, Equation (26) expresses the relative importance of diffusion time to ignition
time, therefore;
Da, <
I
--+
Diffusion is important
411
MINIMUM IGNITION ENERGY
10.0
........... Energy eq.
~
8.0
",
\
- - - - Diffusion eq.
I
- . -.. Thermodynamic eq.
I
9
x
.,
.
I-
r lt
6.0
• DOIIl = 1.0
a
)::'
4.0
0.0
/
.......
/
/
"./
/
TL:--"='~<:"'.
<,
2.0
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= 0.11 mm
---- -
10- '9 ' "';'O-'B'
"';'0-7' "';'0- 6 "';'0- ;'05
4
10-3
T '9 ( S )
FIGURE 12 Kernel center temperatures obtained using three different equations; energy equation,
diffusion equation and Eq. (25). Stoichiometric methane-air mixture, spherical geometry. (Successful
ignitions only)
Da, >
I
---+
Diffusion is not important.
The value Da.; = I can be regarded as the boundary between the region of high
diffusional losses and the region of low diffusional losses. This point is clearly marked
on Figures 9,10 and 12. In particular, Figure 12 shows that this point is indeed located
at the time when the diffusion equation starts to predict different values from those
of Eq. (25).
COMPARISONS AND CONCLUSIONS
We have shown in this paper that in order to make any meaningful comparison to
available experimental data, one needs to know the ignition parameters used. However, a large number of published works on minimum ignition energy fail to give those
parameters. The value of minimum ignition energy given in Lewis and von Elbe (1987)
for stoichiometric methane-air mixtures at p = I atm and T = 298 K is 0.35 mJ for
a spark gap of the order of the quenching distance (2.1 mm) and using a fast spark.
For the same conditions of temperature and pressure, and for a spark gap of 3 mm
and spark duration of6'/S, Ronney (1985) found a minimum ignition energy of 0.5 mJ
at zero gravity. The minimum ignition energy found in our calculation is 0.005 mJ for
a kernel radius of 0.0635 mm and an ignition time of27.5/1s. This value of minimum
ignition energy is 70 times lower than the experimental one. In making these comparisons it is important to recall that spark ignition is a complex sequence of chemical
and fluid dynamic processes. The presence of the relatively cold electrodes leads to
both heat extraction from the ignition kernel and the development of convective
motion (Kono et al. 1988), and the high temperatures during spark initiation result
in additional chemical reactions. However, the authors believe that a major cause of
the difference between experimental and calculated minimum ignition energies is the
Downloaded By: [University of Southern California] At: 22:09 18 January 2011
412
A. FRENDI AND M. SIBULKIN
use of electrode spacings based on quenching distance in these experiments. When a
kernel radius of rig = 0.5 mm and an ignition time 2/ls < rig < 500/ls are used, our
model predicts an ignition energy of 0.3 mJ as shown by Figure 2, which is in fair
agreement with the experimental values. (Complete agreement could be obtained by
adjusting rig')
Kono et al. (1976) studied the effects of electrode configuration, electrode gap and
spark duration on the minimum ignition energy of near stoichiometric propane-air
mixtures. As they varied the spark gap from 0.5 mm to 2.0 mm, the minimum ignition
energy decreased, which is in disagreement with our calculations (Figure 2). This may
be due to the heat losses to the electrodes; since by varying the electrode configuration
the minimum ignition energy varied in a similar way. Their results showed that there
exits an optimum ignition time for which the minimum ignition energy has the
smallest value. This result is reproduced by our calculation using the two models
(Figure I). The increase in minimum ignition energy obtained by Kono et al. for short
ignition times may be explained by the strong pressure wave generated by the spark.
The increase in minimum ignition energy at long ignition times is due to heat losses
by diffusion as found using both models (Figure 4). The critical ignition time obtained
in our calculation (Figure 4) is in good agreement with that measured by Kravchenko
et al. (1973).
Using an overall reaction model, Tromans and Furzland (1986) calculated minimum ignition energies for propane and methane. Their study was directed towards the
effects of initial temperature, pressure and activation energy on minimum ignition
energy. For stoichiometric methane-air, the minimum ignition energy they reported
was 0.1 mJ for an ignition time of 10- 3 s, but there is nothing directly comparable to
our ignition radius in their calculation. This result is 20 times higher than our
minimum ignition energy value. Sloane and Schoene (1988) using a detailed chemistry
model and an instantaneous heat addition, found a minimum ignition energy of
0.16 mJ for a stoichiometric methane-air mixture. The size of their ignition kernel was
not reported. In a recent paper, Maas and Warnatz used a variable pressure model
and a constant pressure model to calculate the minimum ignition energy of hydrogenoxygen mixtures. They concluded that as the ignition radius was decreased the
minimum ignition energy reached a constant value, which agrees with our results and
disagrees with Kailasanath et al. (1982). The same result was obtained by Sibulkin and
Siskind (1987) in their study of ignition of solid fuels. Maas and Warnatz found that
at long ignition times (10- 3 s), both the variable pressure and constant pressure
models predict the same results. However, no calculation was done for short ignition
times. In this paper, the same conclusion is reached for long ignition times; however,
the pressure wave effect is found to be important for short ignition times (Figure I).
It was shown in Figures 8a and 8b that there exists a set of critical parameters,
kernel size and temperature, for a successful ignition to occur. Such parameters were
also predicted by Champion et al. (1986), who used large activation energy asyrnptotics to study the existence of steady state flame propagation. The calculated critical
parameters (r,., r" E,) obtained from their expressions are in good agreement with our
numerical results. The role of heat diffusion in flame initiation is found both to
contribute to successful ignition for small kernel radii (rig < rJ, and to increase the
minimum ignition energy at long ignition times.
ACKNOWLEDGEMENTS
Computational support for this work was provided by the John von Neumann National Supercomputer
Center under NAC grant 13031.A scholarship was awarded to the first author by the Algerian government.
~IN!~V~ !CiNITION ENERGY
413
His current address is Mech. Eng. Dept., 334 Snell Engineering Center, Northeastern University. Boston,
MA 02115. and correspondence should be directed to this address.
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