the temperature and pressure dependences of the … · the temperature and pressure dependences of...
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* Corresponding Author: [email protected]
Proceedings of the European Combustion Meeting 2015
The Temperature and Pressure Dependences of the Laminar Burning Velocity:
Experiments and Modelling
Alexander A. Konnov *
Division of Combustion Physics, Lund University, Lund, Sweden
Abstract
The laminar burning velocity at standard conditions, i.e. atmospheric pressure and initial temperature of 298 K is
invaluable for characterization of combustion properties of the given fuel, for understanding of the underlying
chemistry, validation of kinetic models, calibration of turbulent flame models, etc. However, in general
applications, from domestic appliances to engines and gas turbines, pressure and initial temperature of the mixture
are often higher than standard ones. Elucidation of the temperature and pressure dependences is therefore important
both from practical and fundamental points of view: many engineering and CFD codes implement empirical power
law equations to evaluate burning velocity at elevated conditions often not covered in laboratory experiments;
comparison of the power exponents α (for temperature dependence) and β (for pressure dependence) obtained from
experiments and derived from different kinetic mechanisms provides an independent tool for model validation and
analysis; finally, interpreting measurements obtained at different temperatures and/or pressures using these relations
helps to check that the data are not biased by some systematic or random errors.
1. Introduction
The adiabatic laminar burning velocity is a
fundamental parameter of any combustible mixture,
which depends on the stoichiometric ratio, pressure and
temperature. It is remarkable that different expressions
for this concept are still used in textbooks on
combustion and in many research papers. For instance
Lewis and von Elbe [1, 2], and Tanford and Pease [3]
used the term “burning velocity”, while Coward and
Hartwell [4, 5] and others used the term “fundamental
speed of flame propagation” or “fundamental flame
speed”. Coward and Payman [6] discussed the
appropriateness of this term, which originates from the
expression “la vitesse normale” introduced by Mallard
and Le Chatelier [7], and suggested it as preferable.
Nowadays, however, the words “flame speed” are often
used instead of the term “fundamental flame speed”, the
replacement which can be misleading. Indeed, the
flame speed of the burner stabilized flames is zero,
while at other conditions it can be affected by stretch,
heat losses and bulk flow movement. Therefore, the use
of the words “flame speed” without adjective
“fundamental” should be discouraged.
By definition, the laminar burning velocity, SL, is the
velocity of a steady one-dimensional adiabatic free flame
propagating in the doubly infinite domain. Other
definitions specific for a particular method of
measurements were discussed as well, yet it was
concluded that “it seems impossible to give a precise
definition for burning velocity which will be equally
applicable to plane and spherical flames” [8]. It is,
however, unfeasible to perform experiments with plane
flames in the doubly infinite domain both from obvious
practical and from fundamental points of view; indeed,
planar flames are intrinsically unstable in open domains
due to thermal expansion across the flame front. This
Landau-Darrieus instability leads to wrinkling and
flame surface growth producing flame acceleration.
The laminar burning velocity is, therefore, not a
measurable quantity; it is derived from other observables
using different assumptions or theoretical models. The
ultimate uncertainty of the experimentally obtained
burning velocity is thus defined both by the uncertainty
(accuracy) of the measurements and by the accuracy
(validity) of the underlying models.
The laminar burning velocity at standard conditions,
that is, atmospheric pressure and initial temperature of
298 K is invaluable for characterization of combustion
properties of the given fuel, for understanding of the
primary chemistry and for validation of kinetic models.
Although comparison of the laminar burning velocities
with 1-D modelling is often used for validation of
detailed reaction mechanisms, one should remember
that the majority of the measurements performed prior
to 1980’s possess significant spread and could deviate
from the “real” values by as much as a factor of 2, e.g.
[9]. A greater part of the methods for measuring
laminar burning velocities involves curved (Bunsen
type), stretched (counter-flow) or spherical time-varying
flames. Wu and Law [10] demonstrated experimentally
that flame stretch due to flame front curvature and/or
flow divergence must be taken into account in the data
processing. Experimental difficulties generating
uncertainties in measuring burning velocities could be
common or different for different methods. They are
analyzed in the present overview to demonstrate a range
of remaining problems and challenges in the
measurements.
2. Temperature and pressure dependences
In general applications, from domestic appliances to
engines and gas turbines, pressure and initial
temperature of the mixture are often higher than
standard ones. Certain applications like industrial gas
turbines require pressures as high as 30 atm and operate
at temperatures close to minimum ignition temperature.
It is therefore important to quantify the effects of
pressure and initial temperature on the adiabatic laminar
burning velocity of many practical fuels. These effects
were often interpreted using empirical correlations as
2
outlined below. Moreover, in determining turbulent
burning velocities or other flame properties, an accurate
correlation of the laminar burning velocity is essential
[11, 12]. It is thus not surprising that investigations of
the laminar burning velocity at different initial
temperatures and pressures are numerous and have quite
long history.
2.1. Correlations for temperature dependence
Different empirical correlations describing variation
of the laminar burning velocity, SL, with initial
temperature have been developed. For example,
experiments performed in 1950’s and 1960’s [13, 14]
were often approximated by equations in the form
SL = A + BTα (1),
where T is initial temperature of the mixture. The most
popular correlation describing the effect of initial
temperature on the laminar burning velocity
SL = SL0 (T/T0)α (2),
where T0 is the reference temperature and SL0 is the
burning velocity at this temperature, was used in this or
equivalent form also since 1950’s, e.g. [15, 16].
Variations of the mixture composition and pressure in
other measurements indicated that power exponent
coefficient, α, varies with pressure and equivalence
ratio. Therefore complex equation in the form
SL = CTα P
-D+ET (3)
was proposed by Babkin and Kozachenko [17].
Coefficients C, α, D and E in Equation (3) were found
different for different equivalence ratios and pressure
ranges. Pressure dependence of the power exponent
coefficient, α, in Eq. (2) was sometimes neglected or
averaged, e.g. [18], yet its variation with equivalence
ratio was studied both in lean and rich flames, as it will
be discussed in the following.
2.2. Correlations for pressure dependence
The first experiments demonstrating the effect of
pressure on the burning velocity have been performed
by Ubbelohde and Koelliker [19], who covered the
range from 1 to 4 bars. Since then the measurements
were often interpreted using empirical power-law
pressure dependence
SL = SL0 (P/P0)β
1 , (4),
where SL0 is the burning velocity at reference conditions
(usually at 1 atm), and P0 is the reference pressure. This
power-law Eq. (4) has its rationale in early theories of
flame propagation. The very first (on the priority see
Manson [20]) thermal theory of Mallard and Le
Chatelier [7] considered only heat and mass balance and
indicated inverse dependence of SL with pressure.
Jouguet [21], Crussard [22], Nusselt [23], and Daniell
[24] developed this theory explicitly introducing an
overall reaction rate W. Zeldovich and Frank-
Kamenetsky [25] further advanced this theoretical
approach showing that the mass burning rate m = ρ SL
is proportional to the square root of the overall reaction
rate W. Since W is proportional to Pn exp(-Ea/RT),
where n is the overall reaction order, the mass burning
rate shows a power exponent dependence of n/2. The
power exponent β is therefore
β = n/2 -1 (5),
which is 0 for bimolecular reactions and -0.5 for the
first order reactions. This consideration is still in use for
explaining changes of the overall reaction order with
pressure, e.g. [26, 27].
Another type of the pressure dependence correlation,
SL = SL0 (1 + β2 log(P/P0)) , (6)
was first proposed by Agnew and Graiff [28]. Sharma
et al. [18] and Bose et al. [29] interpreted their
measurements using Eq. (6) and presented discrete
coefficients β2 for several equivalence ratios. Iijima and
Takeno [30] derived a similar correlation with the
parameter β2 linearly dependent on the stoichiometry.
Equations (4) and (6) are in fact related through series
expansion as was noticed by Dahoe and de Goey [31],
although the values of β1 and β2 are not equal.
An entirely different correlation
SL = SL0 exp (b(1 - (P/P0)x) , (7)
was proposed by Smith and Agnew [32]. Since the
parameters b and x were found to be dependent on the
composition of the oxidizer and the burning velocity at
reference conditions, respectively, it was not attempted
by others, except by Konnov et al. [33] for sub-
atmospheric methane + hydrogen + air flames.
Other empirical correlations describing pressure and
temperature dependence of the burning velocity were
suggested as well, since, as it is mentioned above, the
power exponent β is different at different pressures,
initial temperatures and varies with mixture
composition. Babkin and Kozachenko [17] performed
extensive studies of methane + air flames covering a
wide range of equivalence ratios, initial temperatures
and pressures. They demonstrated that the power
exponent β decreases with the increase of pressure
within the range from 1 to 8 atm, while in the range
from 8 to 70 atm the exponent is almost constant. It has
its maximum value in near-stoichiometric mixtures
decreasing towards lean and rich mixtures. Even after
this seminal study of Babkin and Kozachenko [17],
correlations in the form of equation (4) or (6) are often
used, knowing that the power exponent β1 or coefficient
β2 does not have a constant value.
2.3. Data consistency – role of α and β
Typical presentation of the laminar burning velocity
measurements is in coordinates SL vs. equivalence ratio,
ϕ. Figure 1 shows such an example for methane + air
flames at selected pressures up to 5 atm [34].
Experiments denoted as “present work” were performed
from P = 1.5 to 5 atm and for 0.8 < ϕ < 1.4 using the
heat flux method. For comparison, experimental data
from the literature [27, 35, 36, 37, 38, 39] have been
included together with predictions of GRImech 3.0 and
USC Mech II. For this type of graphs further discussion
can be focused on agreement/disagreement of different
sets of measurements and on agreement/disagreement
between experiments and model predictions. It is,
however, difficult to assess if the data obtained at
different pressures, even from the same set, are
3
consistent or not. It will be shown in the following that
analysis of the pressure (or temperature) dependence
provides a powerful tool for this assessment. Indeed,
since the laminar burning velocity of the given fuel
depends on the stoichiometric ratio, pressure and
temperature only, SL = f(ϕ, P, T), interpretation of
experimental and modeling results using correlations
outlined above provides additional dimensions which
allow for elucidation of possible outliers or
inconsistency.
Fig. 1. Comparison of the burning velocities of methane + air flames from experimental (symbols) and
simulation (lines) results for 2, 3, 4 and 5 atm as functions of equivalence ratio [34].
2.3.1. Pressure dependence of methane flames
The pressure dependence of methane flames has
been extensively studied. Andrews and Bradley [14]
reviewed available experimental data before 1972 and
proposed the correlation SL = 43 P-0.5
cm/s for
stoichiometric mixtures at pressures above 5 atm.
Dahoe and de Goey [31] reviewed and interpreted
available experimental data using Equation (4) or (6).
Goswami et al. [34] extended this analysis using their
own data shown in Fig. 1 and comparing with the
literature as summarized below.
In the log-log coordinates, the most frequently used
power-law pressure dependence (Eq. (4)) should
become a straight line. This was indeed observed for all
equivalence ratios with minor deviations at ɸ = 0.8 and
1.4 and pressure of 1.5 atm. Calculations performed
using GRImech 3.0 [39] and USC Mech II [40] have
also been processed to derive power exponents β1 (Eq.
(4)). Although the modeling results plotted in a log-log
scale look quite linear, careful analysis shows that they
do not exactly follow Eq. (4). This is in line with earlier
observations [17, 26, 27], which demonstrated that the
overall reaction order decreases with the pressure
increase from 1 to 5 atm.
The power exponents β1 (Eq. (4)) derived from the
experiments of Goswami et al. [34] are compared with
the selected literature data in Fig. 2. In this graph the
older measured values for which no stretch correction
was implemented are not shown. Uncertainties in the
power exponents were evaluated using the least-squares
method taking the typical uncertainty of the measured
burning velocity of 0.8 cm/s into account. Most
recently Dirrenberger et al. [41] obtained close values of
the power exponent β1 in lean flames of methane using
the same heat flux method. Also shown are values of
4
the power exponents derived from the modeling using
both GRImech 3.0 and USC Mech II.
Fig. 2. Variation of the power exponent β1 (Eq. (4))
with equivalence ratio at elevated pressures. Open
squares: [36], open triangles: [42], solid triangle: [43],
star: [44], solid line: [45], dash-dot line: [31], dashed
line: [46], circles: [34], green stars: [41]. Red line: GRI-
mech., blue line: USC Mech II.
What can be concluded and learned from the
comparison shown in Fig. 2?
Both the power exponents obtained
experimentally and derived using two kinetic models
show parabola-like variation with equivalence ratio
from lean to moderately rich mixtures in qualitative
agreement with the literature [17, 36, 42, 45, 46].
Linear variation of the power exponent β1 suggested by
Dahoe and de Goey [31] is therefore not supported by
these results.
Power exponents β1 obtained by Halter et al.
[42] are obvious outliers, or they possess rather high
uncertainty.
The difference between calculations using
GRImech 3.0 and USC Mech II is minor in rich flames,
increasing toward lean flames. This observation is
consistent with Fig. 1, where burning velocities
calculated using these two models remain close in rich
flames with increasing divergence in lean flames with
the pressure increase. GRImech 3.0 has been efficient
in capturing the pressure behavior of SL especially in
lean mixtures.
USC Mech II uses certain reaction rate
constants from GRImech 3.0 and still shows significant
divergence especially in lean mixtures. Simple
comparison of the model predictions with experimental
data shown in Fig. 1 cannot be conclusive for the model
screening since the difference between two mechanisms
is close to the scattering of the SL measurements.
Analysis of the pressure dependence as shown in Fig. 2
thus provides an additional tool for kinetic model
validation.
2.3.2. Pressure dependence for other hydrocarbons
Combustion of methane as the main constituent in
natural gas has been a key field of research for many
years. Methane (or natural gas) is burnt, premixed or
non-premixed, in a large variety of conditions: at room
and elevated initial temperatures, at atmospheric and
high pressures. Much less experiments have been
performed with other hydrocarbons also present in
natural gas.
Goswami [47] has recently measured burning
velocities of ethane + air and propane + air flames at
pressures up to 5 atm using the heat flux method.
Figures 3 and 4 taken from Goswami [47] compare
available stretch-corrected data for these mixtures at
atmospheric pressure. Significant spread of the data
even at 1 atm is obvious, yet it is well established that
burning velocities of hydrocarbon flames decrease with
the pressure rise very similar to methane flames, e.g.
[48, 49]. In other words, the pressure dependence of the
burning velocity in the form of Eq. (4) satisfactorily
describes existing measurements [50].
Detailed analysis of the burning velocities for many
hydrocarbons aimed at the evaluation of their
consistency is yet to be done, therefore Figure 5 [47]
presents only power exponents β1 (Eq. (4)) derived from
the experiments performed using the heat flux method.
In lean mixtures ethane and propane flames possess
close values of the power exponents, while in rich
mixtures those for ethane are higher than for other fuels
studied. One may note that Metghalchi and Keck [50]
found significantly different values of the power
exponent for propane in the range of equivalence ratios
0.8 – 1.2.
Fig. 3. Comparison of the burning velocities of
ethane + air flames at 1 atm as a function of equivalence
ratio. Available stretch-corrected data: [49, 51, 52, 53,
54], present work denotes Goswami [47].
0.6 0.8 1.0 1.2 1.4
Equivalence ratio
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
Po
wer
exp
one
nt
5
Fig. 4. Comparison of the burning velocities of
propane + air flames at 1 atm as a function of
equivalence ratio. Available stretch-corrected data: [49,
51, 52, 53] , present work denotes Goswami [47].
Fig. 5. Variation of the power exponent β1 (Eq. (4))
with equivalence ratio at elevated pressures for
methane, ethane, and propane flames, taken from
Goswami [47].
Interestingly, in the recent study of Dirrenberger et
al. [41] devoted to the pressure dependence of n-pentane
flames, the power exponents determined in the range 0.7
< ɸ < 0.9 are found increasing from -0.37 to -0.33 in
good agreement with those shown in Fig. 5 for ethane
and propane. It could be speculated that the pressure
dependence of the burning velocities of lean flames for
normal alkanes has close similarity, yet further analysis
and more experiments should be performed.
2.3.3. Different behavior for H2 pressure dependence
Hydrogen + air flames behave largely different as
compared to the flames of hydrocarbons. It was early
realized that burning velocity of hydrogen flames may
increase with the pressure rise, e.g. [30]. Moreover, the
overall reaction order, n, (see Equation (5)) varies non-
monotonically with pressure and is considerably
different for different equivalence ratios [55]. At
certain combinations of equivalence ratio and pressure,
the effective reaction order of hydrogen flames can
assume negative values.
Unstretched laminar burning velocities of hydrogen
flames at high pressures are available mostly from
constant volume bombs [56, 57, 58, 59, 60, 61]. The
systematic measurements in hydrogen + air mixtures
[56] are shown in Fig. 6 and compared with the Konnov
[62] hydrogen model predictions. It is interesting that
the calculated burning velocities converge with the
experiments as the pressure increases. Good agreement
of these experiments with the calculations could not be
expected in rich mixtures at atmospheric and moderate
pressures because the measurements of [56, 63] were
found consistently lower than other data from the
literature [62]. This was recently confirmed by Dayma
et al. [61] who covered pressure range from 0.2 till 3
atm. Non-monotonic behavior in this study was
substantiated and supported by the modeling using
kinetic schemes of Burke et al. [64] and of Keromnes et
al. [65].
Fig. 6. Unstretched laminar burning velocities for
hydrogen + air flames at standard temperature as a
function of initial pressure [62]. Measurements:
symbols; calculations: lines with corresponding small
symbols. Equivalence ratios: crosses: 0.75; diamonds:
1.05; squares: 1.8; [56]; solid circles: 3 [56]; open
circles: 3 [12].
Measurements of the laminar burning velocities
were also conducted using different technique based on
particle tracking velocimetry and image processing for
burner-stabilized flames in a high-pressure chamber
[12]. These results for equivalence ratio = 3 are shown
in Fig. 6 to illustrate significant disagreement of the
available experimental data at elevated pressures, which
is much higher than expected experimental uncertainty.
Negative pressure dependence of the mass burning
rates was experimentally proved and analyzed in the
0 1 2 3 4 5
Pressure, atm
100
150
200
250
300
350
Bu
rnin
g v
elo
city, cm
/s
6
mixtures H2 + O2 + diluent (He, Ar or CO2) also studied
in spherical flames by Burke et al. [66]. It was
demonstrated that many contemporary hydrogen kinetic
schemes fail to predict these observations and require
further improvement.
Fig. 7. Laminar burning velocity of 85:15 % H2 +
N2 with 1:7 O2:He oxidizer at ɸ = 0.6 and 298 K.
Symbols: Goswami et al. [67]; lines: predictions of
different models.
The only measurements of the burning velocities in
non-stretched flat flames of hydrogen at high pressure
have been recently performed by Goswami et al. [67].
They were compared with several kinetic models as
well, as shown in Fig. 7. Again, all tested mechanisms
except GRImech 3.0 overpredict these results with
increasing relative divergence at higher pressures
approaching 10 atm.
The composition of the mixtures studied in flat non-
stretched [67] and spherical [61, 66] flames was
different. However, analysis of the comparison between
different experimental data and essentially the same
collection of hydrogen kinetic schemes indicate that the
measurements, especially in lean flames, are probably
inconsistent and call for further investigation.
Experimental difficulties generating uncertainties in
measuring burning velocities from spherical flames
have recently been discussed in several publications,
e.g. by Wu et al. [68] and Varea et al., [69]. This
inconsistency is also manifested in the analysis of the
temperature dependence of hydrogen flames outlined in
the following Section 2.3.5.
2.3.4. Temperature dependence of methane flames
Temperature dependence of the laminar burning
velocity provides another dimension for the analysis of
the data consistency. Andrews and Bradley [14]
reviewed experiments with methane flames prior to
1972 and noted considerable scatter in both the
magnitude of the burning velocity and its rate of
increase with temperature. For stoichiometric flames
they derived correlation SL = 10 + 0.000371T2. Since
then it was generally accepted that the power exponent
α for methane flames is 2, while for other hydrocarbon
+ air mixtures it ranges between 1.5 and 2 [70].
Metghalchi and Keck [50, 71] investigated the effect of
temperature on the burning velocities of several fuels
using Equation (2) and proposed linear function of the
power exponent α with equivalence ratio independent of
the fuel type: α = 2.18 - 0.8(ɸ-1) in the range of
equivalence ratios 0.8 – 1.2. Dahoe and de Goey [31]
processed essentially the same experimental data as did
Andrews and Bradley [14] using both Equations (1) and
(2). They concluded that mean value of α in Equation
(2) for stoichiometric methane + air flames is 1.89 with
apparent scattering from 1.5 to 2.2.
Konnov [72] further analyzed the temperature
dependence of methane flames focusing on comparison
of the power exponents α obtained experimentally and
derived from predictions of two kinetic schemes: the
Konnov mechanism and the GRI-mech. 3.0 in the range
of equivalence ratios from 0.5 to 2. This comparison
together with recent updates is shown in Fig. 8. Both
models are in good agreement with the measurements in
methane + air flames in the range of ɸ from 0.8 to 1.2
[18, 44, 73, 74]. Furthermore, they qualitatively
reproduce rapid increase of α from stoichiometric to
very lean methane mixtures.
Fig. 8. Power exponent coefficient, α, in Eq. (2) for
methane + air flames. Crosses: [74], open diamonds:
[44], solid diamonds: [73], open circles: [29], solid
circle: [75], open squares: [36], solid triangles: [17],
open triangles: [18], long-dash line: [30], dash-dot line:
[76], dash- double dot line: [45], dash-triple dot line:
[77], red solid line: [78], blue stars: [41]. Solid line:
modeling using the Konnov mech., short-dash line:
modeling using the GRI-mech. 3.0.
Unexpected non-monotonic behavior of α was found
in rich methane + air flames [72]. Both calculated
dependences peak at ɸ = 1.4, at higher equivalence
ratios they level off approaching α = 2. In the dedicated
experiments of Yan et al. [79] notable decrease of the
0.4 0.8 1.2 1.6 2.0
Equivalence ratio
1
2
3
Tem
pe
ratu
re p
ow
er
expo
ne
nt,
7
power exponent was confirmed from ϕ = 1.3 towards ϕ
= 1.45. Most recently Dirrenberger et al. [41] extended
determination of the power exponent α up to ϕ = 1.7
essentially confirming predictions of the two models.
This variation of the power exponents α clearly
indicates significant changes in the combustion
chemistry that is modification of the relative role of the
elementary reactions, as discussed by Konnov [72] and
Yan et al. [79]. Moreover, Goswami et al. [34] pointed
out on the resemblance and relation of the anomalous
behavior of the power exponent β1 (Fig. 2) and power
exponent α suggesting a general coupling of the flame
response on the pressure and temperature variation.
What can be concluded and learned from the
comparison shown in Fig. 8?
The temperature power exponent α obtained
experimentally and derived using two kinetic models
shows parabola-like variation from lean to moderately
rich mixtures. Linear variation expressions of α with
equivalence ratio suggested by Metghalchi and Keck
[50, 71], Iijima and Takeno [30], Stone et al. [76], and
Takizawa et al. [77] should not be used.
The powers exponents derived from earlier
experiments of Babkin and Kozachenko [17] and Lauer
and Leukel [73] may possess rather high uncertainty due
to inaccuracies in the measurements of the laminar
burning velocity itself caused by the lack of stretch
correction.
The results of Bose et al. [29] are obvious
outliers.
Two essentially different kinetic schemes: the
Konnov mechanism and the GRI-mech. 3.0 predict very
close power exponents for methane flames. This
unexpected insensitivity was observed for other fuels as
well and will be discussed in the following.
2.3.5. Temperature dependence of hydrogen flames
Measurements of the laminar burning velocity of
hydrogen flames are as numerous as those of methane.
However, available data on the effect of temperature on
the burning velocities of H2 + air flames interpreted
using correlation SL = SL0 (T/T0)α are rather scarce.
Heimel [16] and Milton and Keck [80] determined
discrete values of the power exponent α, while Liu and
MacFarlane [81] proposed linear correlations below and
above the maximum of the burning velocity as a
function of equivalence ratio with the junction value of
α = 1.571. Iijima and Takeno [30] covered practically
the same range of equivalence ratios (0.5 < ϕ < 4) and
observed modest increase of α without a minimum.
These measurements were analyzed by Konnov [72],
who calculated laminar burning velocities of hydrogen
+ air flames in the range of equivalence ratios from 0.35
to 8.5. From the calculations the power exponents α
were derived as shown in Fig. 9. Modeling results
confirmed small variation of α in the range of ϕ from 1
to 3. In the very lean and very rich mixtures, however,
the temperature power exponent increases dramatically.
Sensitivity analysis revealed the reason of this behavior
[72].
Apart from the studies mentioned above, where the
power exponent α was derived directly, a number of
works [82, 83, 84, 85, 86, 87] presented burning
velocities of H2 + air flames at room and elevated
temperatures without evaluation of the power exponent
in the form of Eq. (2). For these studies, the power
exponent α was extracted with a least-square fit
procedure by Alekseev et al. [88]. All available and
extracted values of α from the experimental data are
shown in Figure 9.
Fig. 9. Power exponent α for H2 + air flames at
standard conditions. Solid symbols and thick lines:
experiments [16, 30, 80, 81, 89], open symbols: fit to
experimental data [82, 83, 84, 85, 86, 87], made by
Alekseev et al. [88]; thin lines: modeling.
A new mechanism was employed to investigate the
temperature dependence of the laminar burning velocity
of H2 flames by Alekseev et al. [88]. The conditions of
Figure 9, i.e. the standard conditions of p = 1 atm and T0
= 298K, where the majority of available literature data
were obtained, were further analyzed. For the modeling
with the new mechanism indicated as “current
mechanism” in Fig. 9, the power exponent α was
derived from a least-square fit to the simulated data in
the range of temperatures 250-500K. It was found that
the value of α does depend on the fitted temperature
range, which also means that the power law in the form
of Eq. (2) with constant α is not valid for a wide
temperature intervals. However, for 250-500 K such
correlation is still reasonably accurate, and it
corresponds to the range of the majority of the H2 + air
burning velocity measurements. Also shown for
8
comparison in Figure 9 is the performance of the most
recent H2 model of Keromnes et al. [65] at the same
conditions.
The two models, as well as the previous version of
the mechanism [62], were found to give almost identical
results. Comparing the calculations to the experimental
data, it can be observed that they are in a reasonable
agreement with the most recent measurements of Krejci
et al. [86]. The data of Krejci et al. [86] also possess
less scattering for the α(ϕ) dependence than the
measurements of Hu et al. [82]. However, in the very
lean mixtures (ϕ<0.5), a solid conclusion cannot be
drawn due to the large discrepancies in the experimental
data. Notably, at ϕ = 0.3 there are three measurements
available, from Verhelst et al. [89], Bradley et al. [84]
and Das et al. [85], which differ by about a factor of 5.
Also, all modeling studies shown in Figure 9 predict a
trend of rapid rise of α in lean mixtures, which is
supported by the data of Krejci et al. [86] and Das et al.
[85]. The latter is the only study, which indicates a very
high α in a very lean mixture. However, it should be
noted that in this case α is based on just two temperature
points in a narrow interval of 25K (298K and 323K),
though the rise of the burning velocity was
considerable. Nevertheless, the results extracted from
Das et al. [85] are valuable as a possible evidence for a
trend found in the modeling, yet should not be seen as
an attempt to determine exact value of α at ϕ = 0.3.
It should be also noted that Das et al. [85] employed
the counterflow technique. In the other studies of very
lean H2 + air flames, Bradley et al. [84] and Verhelst et
al. [89], a spherical bomb method was used. For these
last two cases, the flame configuration was unstable due
to a high Lewis number at ϕ = 0.3. This could be seen as
one of the possible reasons for the remaining
discrepancy.
What can be concluded and learned from the
comparison shown in Fig. 9?
The contemporary kinetic models for hydrogen
combustion are capable to reproduce the experimental
data in a wide range of conditions.
It was shown that the temperature exponent
cannot be treated as a constant for hydrogen flames,
even to a first approximation. For a given fuel-oxidizer
mixture, it strongly depends on equivalence ratio,
diluent content and correlation temperature interval.
A rapid increase of α in very lean mixtures was
observed.
New experimental measurements in the very
lean H2+air mixtures are required to confirm the quality
of the model prediction for these conditions.
2.3.6. Peculiarities of the temperature dependence
Nowadays it is a common practice to compare new
measurements of the burning velocity with the
modelling using several kinetic schemes, if available, to
validate them and to recommend the most suitable one
for the given fuel. When measurements are performed
at different initial gas temperatures, the temperature
dependence in the form of empirical Equation (2) can
also be analysed. As was exemplified by Konnov [72],
the power exponent, α, can be derived from the
modelling and compared with experimental data. Such
analyses are illustrated for methane and hydrogen
flames in the previous sections. They clearly indicate
outliers and help to reveal inconsistency in the
measurements.
On the other hand, close examination of the models’
behaviour demonstrates that the mechanisms showing
large difference in prediction of the burning velocities
can be very close with respect to the prediction of the
power exponents α. This has been earlier observed for
methane [72], ethanol [90], hydrogen [88] and other
fuels. To elucidate the underlying reasons of this
observation, a dedicated study for methyl formate, MF,
flames was performed by Christensen et al. [91] as
summarized in the following.
Laminar burning velocities of methyl formate and
air flames were determined at atmospheric pressure and
initial gas temperatures of 298, 318, 338 and 348 K.
Measurements were performed in non-stretched flames,
stabilized on a perforated plate burner at adiabatic
conditions, generated using the heat flux method. The
measurements at 338 K shown in Fig. 10 are compared
to modelling using the mechanisms of Glaude et al. [92]
and Dievart et al. [93]. The Dievart mechanism is in
better agreement with the present experiments in lean
and slightly rich flames, while the Glaude mechanism is
systematically over predicting the experimental results.
Included in the figure as well are the data of Wang et al.
[94] obtained at 333 K with good agreement.
Fig. 10. Laminar burning velocities of MF+air as a
function of ϕ at 338 K. Symbols: experimental data and
lines: modelling. Squares and circles: Christensen et al.
[91]; diamonds: Wang et al. [94] at 333 K. Solid line:
Dievart et al. [93] and dotted line: Glaude et al. [92].
The power exponents α derived from the new
experimental data are shown in Fig. 11. Presented in
the figure as well are the power exponents derived from
the modelling results using the mechanisms of Glaude et
9
al. [92] and Dievart et al. [93]. The calculated values are
in relatively good agreement although there is an
increasing discrepancy between the experimental and
calculated coefficients α towards richer flames, as the
experimental coefficients are larger. Both mechanisms
show a minimum located at equivalence ratio 1.2.
Fig. 11. Power exponent, α, as a function of ϕ.
Symbols: experiments [91] and lines: modelling.
Dashed line: Glaude et al. [92], solid line: Dievart et al.
[93].
Despite over predicting the experimental results of
SL, the Glaude et al. [92] mechanism is good at
predicting the temperature dependence of the laminar
burning velocity. To investigate the deviations seen at
rich conditions as well as the kinetics behind the
temperature dependence, a separate sensitivity analyses
were performed for the laminar burning velocity and for
the power exponents α. To identify the chemistry
responsible for this discrepancy, a sensitivity analysis
was performed with the Glaude et al. and the Dievart et
al. mechanisms at 298 K. The normalized sensitivity of
the laminar burning velocity with respect to the rate
constant, k, is commonly defined as
𝑆𝑒𝑛𝑠(𝑆𝐿, 𝑘) =𝜕𝑆𝐿
𝜕𝑘
𝑘
𝑆𝐿 (8)
The 20 most sensitive reactions for SL of
stoichiometric mixture are presented in Fig. 12. The
results indicate that the laminar burning velocity is
sensitive to reactions from the H2 - O2 subset as well as
to C1 chemistry. The sensitivity spectra for both
mechanisms are essentially very close.
The sensitivity analysis of the laminar burning
velocity as above is commonly used to reveal key
reactions responsible for the differences between model
and experiment or between models. Intriguing was,
however, that two mechanism largely different in
predicting the burning velocities (Fig. 10) show close
performance with respect to the temperature dependence
prediction, Fig. 11.
Fig. 12. Normalized sensitivity coefficients for the
laminar burning velocity of MF+air flames at ϕ = 1.0
To gain further insight into the relation of the
temperature dependence and the key reactions, a
sensitivity analysis of the power exponent α has been
performed for the first time [91]. As mentioned above,
the normalized sensitivity of the laminar burning
velocity can be described by Eq. (8). In a similar
manner the normalized sensitivity of the power
exponent with respect to the rate constant, k, can be
defined as
𝑆𝑒𝑛𝑠(𝑎, 𝑘) =𝜕𝑎
𝜕𝑘
𝑘
𝑎 (9)
By utilizing that the sensitivity of the power
exponent and the laminar burning velocity both are
functions of k, and that the temperature dependence is
described by Eq. (2), the partial derivatives with respect
to k can be calculated using the product rule. It can then
be shown that the normalized sensitivity of the power
exponent with respect to k can be obtained as
𝑆𝑒𝑛𝑠(𝑎, 𝑘) =𝑆𝑒𝑛𝑠(𝑆𝐿,𝑘)−𝑆𝑒𝑛𝑠(𝑆𝐿0,𝑘)
𝑙𝑛(𝑇
𝑇0) 𝛼
(10)
In this equation T0 refers to the reference
temperature of 298 K and T to the arbitrary elevated
temperature (398 K in the study of Christensen et al.
[91]).
Figure 13 shows normalized sensitivity coefficients
of the power exponent for stoichiometric mixture.
When comparing the sensitivity of SL, Fig. 12, and the
sensitivity of α, Fig. 13, one can see some important
differences. Most noticeable is that the sensitivity
coefficients of the power exponent are roughly 1 order
of magnitude smaller than the sensitivity coefficients of
SL. There are also several reactions in the sensitivity of
SL that are displayed with opposite sign in the
sensitivity spectra of α. This indicates that as the
temperature is increased from 298 to 398 K, several
chain promoting reactions become less important and
10
these reactions are now displayed with negative
sensitivity. The same is true for several chain inhibiting
reactions that display positive sensitivities in Fig. 13.
One last observation is that while the two mechanisms
displayed similar results for the sensitivities of the
laminar burning velocity in shape and reaction ranking,
the sensitivity coefficients of α are more scattered.
Fig.13. Normalized sensitivity coefficients for the
power exponent α of MF+air flames at ϕ = 1.0.
This analysis explains why the mechanisms showing
large difference in prediction of the burning velocities
can be very close with respect to the prediction of the
power exponents α. In fact, the normalized sensitivity
coefficients for the power exponent are roughly one
order of magnitude smaller than commonly used
normalized sensitivity coefficients for the burning
velocity. This means that if the differences in reaction
rate constants of two models are manifested in, say,
20% difference in the burning velocity, one may expect
only 2% difference in the power exponent α.
It also implies that comparison of the experimental
alpha coefficients and those derived from the modelling,
as shown in Fig. 11, cannot be used for kinetic model
validation, as has been suggested in other studies, unless
the model is significantly incapable in the burning
velocity prediction. On the other hand, this comparison
can serve for analysis of experimental data consistency.
Indeed, since the models are more “robust” in
calculation of the alpha coefficients, substantial
deviation between experiments and modelling most
probably indicates potential problems in the
measurements. This analysis of experimental data
consistency has recently been demonstrated for
hydrogen + air mixtures [88], where contradiction of
several datasets and modelling was evidenced in lean
flames.
3. Experimental uncertainties
Quite often measurements of the laminar burning
velocities are reported with a stated accuracy of 1-2 %.
These evaluations are usually based on the accuracy of
equipment used. In practice, however, scattering of
available data at the same conditions of temperature,
pressure and equivalence ratio often exceeds 5 and even
10% as illustrated in Figs. 1, 3, 4, and 6 above. There
are several reasons for this inconsistency.
Firstly, technical details important in data
processing are most often not reported in archival
publications. Thus each attempt of measurements in a
new laboratory and even re-visiting the same
experiment after some time has passed at the same place
is unique since operating experiment and data
processing involves the effect of operator.
Second, both equipment and theory behind
each of the methods mentioned above are in a constant
development.
Moreover, there are no protocols in place to
renounce older data even if the author himself or herself
realized that the published results were incorrect or
highly uncertain. This leads to an accumulation of
results hardly suitable for model validation due to large
degree of scatter.
Several reviews devoted to comparison of the
experimental approaches and of the results obtained
deserve to be mentioned. In 1953 Linnett [8] described
the following methods: (1) Egerton-Powling flat flame
method; (2) Soap bubble method; (3) Closed spherical
vessel method; (4) Cylindrical tube method; (5) Bunsen
burner method. In the seminal review of Andrews and
Bradley [95] in 1972 essentially the same methods were
further scrutinized and the maximum burning velocity
of methane + air flame at 1 atm and 298 K of 45 ±2
cm/s was recommended. Enlightening Fig. 14
composed by Law [96] is basically an extension of the
similar figure compiled by Andrews and Bradley [95].
Law [96] has emphasized that after the work of Wu and
Law [10], which demonstrated the importance of the
stretch correction in the data processing, the uncertainty
of the burning velocity measurements was reduced to
less than ±8 cm/s (the scattering in Andrews and
Bradley [95] plot was of the order of ±25 cm/s), while
contemporary measurements together with non-linear
stretch correction are indeed consistent within ±2 cm/s.
Although the stretch correction is extremely
important, other experimental difficulties generating
uncertainties in measuring burning velocities were
realized and reviewed. These difficulties could be
common or different for different methods; some
methods for measuring burning velocity are now
considered obsolete. Ranzi et al. [97] and Nilsson and
Konnov [98] limit their discussion to four methods:
Bunsen burner, spherical flames, counter-flow flames
and flat flames stabilized using the heat flux method. In
the recent review of Egolfopoulos et al. [99] only the
last three were carefully analyzed and compared. This
review summarizes the state of the art in the data quality
and demonstrates a range of remaining problems and
11
challenges in the measurements. Many exciting
developments in the field presented and analyzed by
Egolfopoulos et al. [99] open new prospects in the
improvement of the theoretical and experimental
approaches.
Fig. 14. Variation of measured maximum laminar
burning velocities of CH4 +air mixtures at standard
conditions as a function of year of publication, taken
from Law [96].
Concluding remarks
In most practical situations combustion is
accompanied by flames. With a few exceptions like
detonations or explosions, flames of different natures
and appearances propagate in engines, in domestic
appliances, in gas-turbine combustion chambers, etc. It
is therefore natural that laboratory studies of flames
provide valuable information on combustion
characteristics important both for solution of
engineering problems and for development and
validation of combustion models.
The laminar burning velocity and its temperature
and pressure dependences is important both from
practical and fundamental points of view: many
engineering and CFD codes implement empirical power
law equations to evaluate burning velocity at elevated
conditions often not covered in laboratory experiments.
Moreover, comparison of the power exponents α (for
temperature dependence) and β (for pressure
dependence) obtained from experiments and derived
from different kinetic mechanisms provides an
independent tool for model validation and analysis. The
power exponents α are not very sensitive to the rate
constants implemented in the modeling; therefore they
are more suitable for analysis of consistency of the
measurements and helps to check that the data are not
biased by some systematic or random errors.
Acknowledgements
The author is grateful to all his co-authors who
contributed to the present analyses of the pressure and
temperature dependences, especially to Mayuri
Goswami, Vladimir A. Alekseev, Moah Christensen,
Jenny D. Nauclér, and Elna J.K. Nilsson.
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