Combustion and Flame 232 (2021) 111525
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Combustion and Flame
journal homepage: www.elsevier.com/locate/combustflame
Exploring the fuel structure dependence of laminar burning velocity:
A machine learning based group contribution approach
Florian vom Lehn a,∗, Liming Cai b, Bruno Copa Cáceres a, Heinz Pitsch a
a
b
Institute for Combustion Technology, RWTH Aachen University, 52056 Aachen, Germany
School of Automotive Studies, Tongji University, 201804 Shanghai, China
a r t i c l e
i n f o
Article history:
Received 12 January 2021
Revised 22 May 2021
Accepted 22 May 2021
Keywords:
Laminar burning velocity
QSPR model
Fuel design
Functional group analysis
Machine learning
a b s t r a c t
The laminar burning velocity (LBV) is a fundamental property of a fuel/oxidizer mixture with high impact
on combustion processes in practical engines. Profound knowledge of its dependence on the underlying
molecular structures of hydrocarbon and oxygenated hydrocarbon fuels is of high interest. In the present
work, a quantitative structure-property relationship model is developed for the first time to predict the
LBVs of a wide range of fuels. For this purpose, an artificial neural network is trained based on a training
set consisting of both the experimental LBV values of 124 fuel compounds and additional data obtained
from numerical simulations with a detailed kinetic model. Twelve molecular groups as well as pressure,
temperature, and fuel/air equivalence ratio serve as input features to the model. Cross-validation reveals a
mean absolute error of 3.3 cm/s when applying the model to fuels, whose LBV datapoints were not used
for training. In order to gain insights into the underlying fuel structure dependence of LBV, the model is
then applied to analyze the functional group effects at unified conditions by means of sensitivity analysis
and detailed fuel comparisons. It is found that unsaturation increases the LBV, while methyl substitution
consistently has a negative effect for the wide range of fuel structures considered, which confirms similar findings in the literature. More interestingly, while carbonyl groups in ketones and aldehydes, ether
groups in ethers, acetals, furanics, and oxygenated benzenoids, as well as hydroxy groups in n-alcohols
tend to increase the LBV compared to corresponding non-oxygenated fuels of similar structures, ester and
carbonate functional groups have a clearly negative impact. Overall, the results demonstrate that a group
contribution approach in combination with a machine learning methodology is capable of predicting the
LBVs of a wide range of fuel structures with acceptable accuracy, which can be useful for future fuel
design.
© 2021 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction
The required reduction of green-house gas emissions and the
demand for cost-effective energy conversion systems motivate research into improved efficiencies, for instance, for internal combustion (IC) engines [1,2]. IC engine efficiencies are greatly dependent
on operating conditions and compressions ratios, which are for
spark-ignition engines typically limited by engine knock and thus
by the knock resistance of the fuel in use [3]. The knock resistance
of a fuel is on the one hand strongly dependent on its autoignition propensity, and recent studies have thus been dedicated
to the exploration of auto-ignition dependence on fuel structure
so as to design knock resistant fuels for future high-performance
engines [4]. On the other hand, knock resistance, efficiency, as well
∗
Corresponding author.
E-mail address: f.vom.lehn@itv.rwth-aachen.de (F. vom Lehn).
as undesired phenomena such as flame extinction are significantly
affected by the laminar burning velocity (LBV) of the fuel [5–8].
The LBV is a fundamental property of a fuel/oxidizer mixture,
which depends for a specific fuel heavily upon initial pressure, temperature, fuel/air equivalence ratio, and dilution. Its
unstretched value, which can be modeled by one-dimensional
premixed flame simulations, is driven by the fuel combustion
chemistry and thus also serves as kinetic model validation target.
For specific fuels, the LBV dependence on the physical conditions
of the unburnt mixture is often modeled by means of analytical
approximation formulae, such as asymptotics-based [9,10] and
fully empirical power law [11] expressions, whose parameters are
commonly determined through regression analysis based on experimental data or results from simulations using detailed kinetic
models. Nevertheless, while large sets of experimental LBV data
have become available over the years for well-known fuels such as
hydrogen or methane [7], the LBVs of many novel fuel candidates,
especially oxygenated ones, have often only been measured at few
https://doi.org/10.1016/j.combustflame.2021.111525
0010-2180/© 2021 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
F. vom Lehn, L. Cai, B. Copa Cáceres et al.
Combustion and Flame 232 (2021) 111525
conditions, if at all, and detailed kinetic models are not always
available either. Since the LBV at a specific condition can differ
significantly among certain types of fuels, detailed knowledge
about its dependence on fuel structure and functional groups is
thus of high interest, as it may support the design of novel fuels
for which no experimental data or well-validated detailed kinetic
models have been reported.
A few early studies have made first attempts to analyze this fuel
structure dependence. Gerstein et al. [12] experimentally compared
the LBVs of various saturated and unsaturated hydrocarbons, observing a marginal dependence of burning velocity on carbon chain
length of longer n-alkanes. Unsaturation was further reported
to increase the LBV, while a slightly negative impact was observed by methyl substitution [12]. This behavior was confirmed by
Gibbs and Calcote [13], who conducted an experimental study on
the LBVs of a wider range of hydrocarbon types. From theoretical
considerations, the adiabatic flame temperature has been shown
to be of leading order importance for the LBV magnitude of a fuel
[9,14]. Since the adiabatic flame temperature correlates for hydrocarbons with their H/C ratios [15] and thus their atomic compositions, this thermal effect of the adiabatic flame temperature on the
LBV has been considered as one contributing factor to explain the
LBV dependence on fuel structure. However, by correlating LBVs of
various fuels with their heats of combustion, it was also shown
that such correlation only exists approximately within subgroups
of fuels, such as linear alkanes or alkenes, while it cannot explain
many of the differences between different types of fuels [16].
Davis and Law [17] measured the LBVs of a range of saturated
and unsaturated hydrocarbons as well as alcohols and mostly confirmed the findings from earlier studies [12,13] regarding the fuel
structure effects. In particular, they attributed the differences of
LBVs among different fuels to the underlying oxidation kinetics
and especially the stability and reactivity of the initially generated radicals, in addition to the flame temperature effect [17]. Farrell et al. [18] also conducted a comparative study by measuring
the LBVs of more than 40 mostly non-oxygenated hydrocarbons
at unified pressure and temperature, highlighting the characteristic effects of fuel structure on LBV, where those fuels whose combustion easily forms methyl radicals were demonstrated to burn
slowly, while those producing preferably H radicals burn relatively
faster due to the chain branching effect of H radicals at high
temperatures. By comparing the LBVs and corresponding reaction
pathways through detailed kinetic modeling, Ranzi et al. [19] further promoted the understanding that the low burning velocities
of methyl-substituted fuels are the result of increased production
of methyl radicals and resonantly stabilized radicals. Various other
recent studies have compared the LBVs and the underlying reaction kinetics of selected components of certain fuel types, such
as alkanes [20–26], alkenes [27–31], aromatics [32–38], alcohols
[39–49], ketones/aldehydes [50,51], esters [52–56], ethers [57,58],
or relatively similar fuels (e.g. isomers or similar carbon number
fuels) with different functional groups [59–65]. These have thus
promoted the understanding of structural group impacts on detailed fuel decomposition pathways and resulting LBVs considerably. Nevertheless, only a small number of compounds was considered in each of these respective studies. Comparative analyses
considering wide ranges of fuels and fuel types remain scarce, not
least because of the fact that LBV data are usually reported in different studies at varying experimental conditions, making their direct comparison without the use of scaling laws [66] unfeasible.
Such wide-range comparisons would be very valuable from
a practical point of view, as they may support the selection of
optimal fuel candidates for highly efficient engines [67–69]. The
availability of a quantitative structure-property relationship (QSPR)
model for LBV would greatly facilitate such comparisons, and
would additionally allow to estimate the LBV magnitudes of novel
fuel candidates only based on a set of fuel structural descriptors
even before experimental data or kinetic models are available. A
number of QSPR models have recently been developed for different fuel combustion properties, including cetane number [70–75],
octane numbers [73–78], octane sensitivity [78], sooting tendency
[79–83], and ignition delay time [84,85]. Regarding the LBV, a correlation between maximum values of different fuels and fuel structure has been derived many decades ago for a limited set of hydrocarbon fuels [86], while more recently a “targeted QSPR” approach was demonstrated by predicting the LBV of ethanol based
on a model trained with LBV data of a small number of other fuels
[87]. These studies provided first evidence of the general feasibility of LBV prediction based on fuel structure. Nevertheless, they
were very limited in terms of considered fuels, and no attempt has
been made to the authors’ best knowledge to correlate the LBVs of
a wide range of fuels relevant for engine application to molecular
structure in form of a group contribution based QSPR approch.
Therefore, the present study has two main goals. First, it targets
the development of a QSPR model to estimate the LBVs of a wide
range of pure hydrocarbon and oxygenated hydrocarbon fuels in
dependence of their molecular groups. Artificial neural networks
(ANNs) have recently been demonstrated to be advantageous in
terms of predictive accuracy for QSPR modeling of complex combustion properties [73,77,78] and were also successfully applied
for modeling the LBV dependence on compositions of specific fuel
mixtures [88] or to increase the accuracy of LBV prediction by reduced kinetic models [89]. An ANN is thus employed in the present
study to model the LBV dependence on fuel structure as well as on
pressure, temperature, and equivalence ratio, where the fuel structure is represented by a set of molecular groups. The second goal
of this study lies in the application of the developed model for a
detailed analysis of the impacts of different functional groups on
the LBV, in order to provide guidelines for the future design of
high-performance fuels with high flame speeds.
The remainder of this article is structured as follows.
Section 2 outlines the used methodology. This includes summaries
of the used training dataset and the employed group contribution
approach, an overview of the ANN training methodology, and the
explanation of the model validation procedure. Section 3 first assesses the validation results, followed by a sensitivity analysis on
the model input features, a detailed analysis on functional group
dependence of the LBV based on predictions of the developed
model, exemplary fuel ranking at specific conditions, and the additional development of a simplified multivariate linear regression
model for LBV estimation at a specific condition. Section 4 concludes the paper.
2. Methodology
In the present work, an ANN is trained to predict the LBV in dependence of both the fuel structure and the physical conditions of
the unburnt mixture, i.e., pressure, temperature, and equivalence
ratio, where the fuel structure dependence is modeled through a
set of molecular groups. In this section, the employed group definitions are introduced first, followed by the description of the training dataset and details on the ANN training methodology. Thereafter, three cross-validation approaches, which are used to assess
the model prediction accuracy for different model application scenarios, are discussed.
2.1. Training dataset
The success of a data-driven model depends heavily upon the
availability of a sufficient amount of accurate training data. For
this reason, a literature review has been undertaken in order to
collect experimental LBV datasets reported for a wide range of
2
F. vom Lehn, L. Cai, B. Copa Cáceres et al.
Combustion and Flame 232 (2021) 111525
Table 1
Overview of experimental data from the literature that were employed as training
data in the present study.
Fuel type
# Datapoints
# Fuels
Linear alkanes
Branched alkanes
Cycloalkanes
Linear alkenes
Branched alkenes
Cycloalkenes
Dienes
Aromatics
Acyclic alcohols
Cyclic alcohols
Acyclic ketones
Cyclic ketones
Aldehydes
Esters
Acyclic ethers
Cyclic ethers
Furanics
Oxygenated benzenoids
Sum
208
82
153
326
126
86
92
366
781
22
232
18
34
411
168
133
83
42
3363
10
7
6
10
4
4
2
14
20
1
7
1
3
17
9
5
2
2
124
Fig. 1. Distribution of all 3363 experimental training data points over the pressuretemperature domain. The data points obtained by numerical simulation, which were
additionally considered in the training process, are not included in the diagram. For
most pressure-temperature tuples of specific fuels, data covering equivalence ratios
from 0.7 to 1.4 were considered.
pure fuel compounds. The emphasis of this study lies on practically relevant fuels, which could be potentially used in IC engines.
The compounds considered in our recently established property
database of spark-ignition engine fuels [69] were thus considered
first, identifying experimental studies with LBV data for more than
100 different fuel compounds. In addition, a number of other fuel
compounds, most of which are rather suitable for compressionignition engine applications, such as long-chain alkanes, ethers,
oxymethylene ethers, and acetals, were taken into account as
well in order to include all relevant fuel structures and molecular
groups. The resulting experimental dataset finally covers 124 fuel
compounds and overall 3363 single LBV datapoints for fuel/air
mixtures. The considered fuels comprise linear, branched, and
cyclic alkanes and alkenes, aromatics, acyclic and cyclic alcohols,
acyclic and cyclic ketones, aldehydes, esters, acyclic and cyclic
ethers (including acetals and oxymethylene ethers), furanics, and
oxygenated benzenoids. These are listed in Table 1 and in more
detail in Table S1 of the Supplementary Material (SM). With very
few exceptions, only data from original research studies published within the past ten years were taken into account, where
sufficiently high consistency among the experimental data and
limited associated uncertainties are expected. Note that the small
compounds hydrogen and methane were not considered, as their
structures do not contain any molecular groups present in the
larger fuel compounds of interest here.
The considered experimental LBV data were determined in their
respective original studies using a range of well-established measurement techniques (see e.g. Konnov et al. [7] for a detailed
overview of commonly employed experimental setups). A reasonable interpretation of the validation results discussed later requires
a general understanding of the uncertainty levels in these experimental data. The uncertainties in experimental LBV data depend
upon various factors and sources. For instance, LBVs measured by
the spherical flame method can exhibit significant uncertainties
due to the extrapolation procedure used to derive the unstretched
flame speed from the actually measured stretched one [7]. In an
uncertainty quantification study for constant pressure spherically
expanding flames, Xiouris et al. [90] determined 1σ uncertainties
of up to ±5% in the extracted LBV values, which would correspond to ±3 cm/s for a flame speed of 60 cm/s. The measurement
uncertainties tend to decrease with higher pressures [90], where
however less experimental data are available in the literature
and thus the present training set. Note that comparisons of LBV
data reported for specific fuels at similar conditions by different
Table 2
Overview of simulated training data employed in the
present study. These were obtained by numerical simulation with the FlameMaster software package [92] using the
kinetic model of Cai et al. [91].
Fuel
# Datapoints
Ethanol
iso-Octane
Sum
1722
1722
3444
studies have demonstrated considerable data scatter, which is often even higher than the respective uncertainties reported by the
original studies [7]. Obviously, the achievable prediction accuracies
of a QSPR model evaluated through validation against these experiments, as performed in the present work, are limited by these uncertainties in the experimental data.
Overall, the 3363 used experimental LBV data points (see
Table 1) cover pressures of 1–5 bar, temperatures of 298–500 K,
and equivalence ratios of 0.7–1.4. Their distribution in dependence
of pressure and temperature is illustrated in Fig. 1. It is seen that
the majority of experimental data are those at atmospheric pressures, since relatively less results are reported in the literature at
higher pressures.
In order to improve the recognition of the typical LBV dependence on pressure and temperature by the model, simulations
using a recently developed chemical kinetic model [91] and the
FlameMaster software package [92] were conducted so as to generate comprehensive LBV datasets for two representative compounds
with rather high and low LBVs, respectively, namely ethanol and
iso-octane (see Table 2). These data cover evenly the considered
p/T /φ domain (1–5 bar, 298–500 K, φ =0.7–1.4) and were used as
additional training data. A validation example of the mechanism
[91] for these two fuels is shown in Fig. S1 of the SM. While the
exact pressure and temperature dependencies may obviously be
different for other fuels compared to these two compounds, the
qualitative trends are similar as the flame propagation is strongly
governed by the chemistry of small species [19]. Moreover, it
should be noted that for many fuels, experimental data at multiple pressures and temperatures are in fact available (compare
Table S1 in the SM), which further promotes the recognition of the
pressure and temperature dependencies for different fuel types
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Combustion and Flame 232 (2021) 111525
Table 3
Molecular groups considered as input features.
Group
Type
-CH3
-CH2 >CH- / >C<
Primary carbon group (saturated)
Secondary carbon group (non-ring, saturated)
Tertiary or quaternary carbon group (non-ring,
saturated)
Primary carbon group (non-ring, unsaturated)
Secondary carbon group (non-ring, unsaturated)
Tertiary carbon group (non-ring, unsaturated)
Saturated ring carbon group (secondary, tertiary, or
quaternary)
Unsaturated ring carbon group (secondary or tertiary)
Alcohol group
Ether group
Carbonyl group (in ketones & aldehydes)
Ester group
=CH2
=CH=C<
Csat,ring
Cunsat,ring
-OH
-O>C= O / -CH=O
-COO- / CHOO-
ester functionalities are considered, where no distinction is made
in terms of cyclic/non-cyclic type or the positions of carbonyl and
ester groups in the molecule structure, i.e., inside the carbon chain
or at the carbon chain end. Finally, only two carbonate fuel compounds (dimethyl carbonate and diethyl carbonate) were available
in the training set. Hence, in order to reduce overfitting by minimizing the number of input features, the carbonate functionality,
which consists of a carbonyl moiety with two ether groups located
on both sides, is modeled here by an ester group (which represents
a carbonyl group neighbored by one ether group) and an additional
ether group, instead of using a separate carbonate group as in the
original scheme of Joback and Reid [96].
It should be noted that some fuels in the training set comprise
the same molecular groups according to the employed group
definitions, and thus cannot be distinguished by the model. This
pertains, for instance, to the dimethylbenzene isomers, which
only differ by the relative positions of methyl side chains. The
incorporation of additional input features or a more refined group
definition would in principle allow to discriminate between such
fuels [78]. Multiple input features were thus tested in preliminary
investigations of the present work in addition to the molecular
groups shown in Table 3 for such distinctions, yet this did not improve the prediction accuracies in the model validations. A reason
for this is that most of the fuels which cannot be distinguished
by the molecular groups in Table 3, such as the dimethylbenzene
isomers [97], in fact exhibit relatively similar LBVs, and the incorporation of additional input features thus possibly only leads to an
increased degree of overfitting, rather than providing useful additional information to the model. Generally, it should be noted that
the twelve-dimensional space of molecular groups used as input
features is not densely covered in all dimensions by the training
dataset, as firstly certain combinations of structural groups are
not possible in general (e.g. secondary/tertiary non-cyclic carbon
groups without primary carbon groups). Secondly, for many other
combinations, such as different oxygenated groups (e.g. an alcohol
and an ether group) in one molecule, experimental data are not
available in the training set for the corresponding fuel structures
either, as such fuels have not been investigated in the literature.
with specific molecular groups by the model. The consideration
of the additional simulation data in the training set was found
to have a clearly positive impact on prediction accuracies of the
model and to result in reasonable prediction accuracies for the
purpose of the present work, as discussed later. Nevertheless, if
more accurate predictions over wide pressure and temperature
ranges were desired for specific types of fuels, the incorporation of
similar simulated LBV data for other fuels could be advantageous,
provided that accurate well-validated kinetic models are used.
2.2. Group contribution approach
Group contribution methods rely on the prediction of molecular properties based on contributions from characteristic underlying groups [93,94]. The exact definition of these groups varies
depending on the group contribution approach used. Well-known
group contribution approaches include those of Benson [95] and
Joback and Reid [96]. While these were originally proposed as
group additivity methods with linear additive contributions of the
different groups of a molecule to its predicted property, schemes
with similar group definitions have later been applied in combination with more complex models, such as ANNs [73,78]. This has
the advantage of being able to incorporate possible interaction effects between different groups in a molecule, which may not be
captured by linear group additivity schemes but can be important for accurate prediction of complex molecule properties, such
as fuel combustion properties. Hence, a set of molecular groups is
used in the present work as inputs to the ANN-based model.
The group definition employed here is based on that proposed
by Joback and Reid [96], where groups are defined according to
the type of heavy atom as well as the types of bonds of that
atom to neighboring atoms, and additional distinction is made as
to whether the heavy atom is part of a cyclic or non-cyclic structure. However, several of the original groups [96] are summarized
here into fewer groups in order to improve the prediction accuracy for the specific case of LBV prediction based on the available training data and to reduce overfitting. These modifications
mainly involve original Joback groups [96] that only occur in few
of the fuels in the training set and for which no sufficient constraints are thus provided by the training data if used as separate model inputs. All resulting groups employed in the present
LBV model are listed in Table 3. In contrast to the original group
definition of Joback and Reid [96], tertiary and quaternary carbon
groups of non-cyclic species are considered here as one unified
group. Furthermore, no distinction is made as to whether ring carbon groups are of secondary, tertiary, or quaternary nature, but
only one unified group is employed for saturated and one for
unsaturated ring carbon groups. With respect to the oxygenated
groups, only four general groups for alcohol, ether, carbonyl, and
2.3. Artificial neural network training methodology
The ANN employed here for the prediction of LBV was trained
based on the set of experimental target data (see Tables 1
and 2) using the open-source library Keras [98] with the TensorFlow package [99]. The input data consisted of both the set of
molecular groups (see Table 3) as well as temperature, pressure,
and equivalence ratio as additional physical input parameters, as
outlined earlier. For good training convergence, the input feature
values xi j and target data y j were normalized as
xi j,norm =
xi j − x̄i
σx,i
and y j,norm =
y j − ȳ
σy
,
(1)
where x̄i , ȳ and σx,i , σy are the mean values and standard deviations, respectively, of the absolute feature and target values over
the entire fuel dataset, following our previous study [78]. The fuel
component and feature type indices are labeled here and later
as j and i, respectively. The input data were then passed to the
hidden layers, where the output of each neuron consists of the
sum of outputs of neurons in the previous layer and an additional
bias, weighted by the neuron’s pre-defined activation function.
The final output quantity of the output layer is similarly obtained
based on the outputs of neurons in the previous hidden layer and
corresponding bias. This general structure is depicted in Fig. 2.
The selection of the ANN’s hyperparameters is important for
good training convergence and prediction accuracy. The Rectified
Linear Unit (ReLU) activation function provided by TensorFlow
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Combustion and Flame 232 (2021) 111525
Fig. 2. Schematic structure of a feed-forward dense neural network (3-5-3-1), as
used in the present work. The numbers of neurons in input and hidden layers do
not represent the actual numbers of neurons employed here, compare Table 4. The
depicted biases represent vectors with the bias values of individual neurons.
Fig. 3. Schematic overview of the k-fold cross-validation employed in the present
study.
Table 4
Network structure (layers and neurons), dropout, and training hyperparameters of
the Adam optimizer [100] of the final LBV model.
Network structure
Dropout
Adam hyperparameters α , β1 , β2
15-512-256-1
0.2
0.0002, 0.99, 0.9995
evaluated, e.g. in terms of mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square percentage
error (RMSPE), while the final overall prediction accuracy is determined by the values of these respective metrics averaged over all
iterations. Compared with a fixed split into training and validation
set, k-fold cross-validation has the advantage that the resulting
prediction accuracy is not biased towards the specific choice of
the training and validation sets. In order to further reduce the
potential remaining bias towards a specific split of the data into
k folds, multiple seeds were always employed here, where the
data were in each seed again split into k folds, and for each seed
a separate cross-validation was performed. The overall prediction
accuracies presented later in Section 3.1 are thus based on averaging the results of these separate cross-validations. Note that the
final model, which was used for the LBV predictions discussed in
Sections 3.2–3.4, was ultimately trained based on the full available
training set, after the cross-validations have been performed.
In the standard form of k-fold cross-validation, all available
data are regarded as equally significant for model validation and
they are thus distributed randomly over the k folds. However, in
the present work, the set of training data consists on the one
hand of a large variety of experimental data for different fuels
(see Table 1), but on the other hand of a large number of simulated data for ethanol and iso-octane (see Table 2), as outlined
previously. The purpose of the model development lies in the LBV
prediction of a wide range of possible fuel components, hence
considering the very large number of simulated data for only two
fuels in the model validation would strongly bias the evaluated
prediction accuracies towards only these two compounds. Therefore, the simulated training data were exempted as validation
data here and were always retained in the training set. Only the
experimental LBV values, which also include data for ethanol and
isooctane, were consequently distributed over the k folds, which
were used as part of either the training or validation sets. This is
illustrated in Fig. 3.
If the experimental LBV data are fully randomly distributed
over the k folds, some data points (at certain p, T , φ ) for a specific
fuel can be in the training set and others (for other p, T , φ ) of the
same fuel can simultaneously be in the validation set. However,
the LBVs of a specific fuel at different physical conditions are
naturally correlated to a considerable extent. For instance, the LBV
of n-heptane with varying equivalence ratio is consistently higher
than that of iso-octane at the same respective equivalence ratio
(see Fig. S3 of the SM). Hence, having some n-heptane data in the
training set, the model will already recognize the approximate LBV
level of n-heptane, and higher prediction accuracies for n-heptane
data in the validation set would thus be expected compared to
a case where no n-heptane data would be in the training set at
[99] was used, owing to its good convergence performance. The
adaptive moment estimation algorithm (Adam) [100] without
mini-batching and weight decaying was employed as optimizer,
following previous work [78]. Different combinations were tested
in a preliminary grid search study for the remaining hyperparameters, including network structure, dropout [101], and training
hyperparameters of the optimizer. The use of dropout allows to
minimize overfitting despite relatively large network sizes [101].
In the grid search study, the prediction accuracies were assessed
with cross-validations based on single data points employing only
a subset of the overall training/validation database as experimental
target data (see next subsection for details on model validation).
Good prediction performance was achieved with the ANN structure and training parameters shown in Table 4, which were thus
considered for training of the final model. For illustration, the
prediction errors for different sizes of the ANN structure are compared in Table S2 of the SM. It is worth noting that the network
size chosen here is of similar magnitude as the network sizes in
other recent ANN-based QSPR models for combustion property
prediction [77]. The optimal training epoch count was determined
for each training case by automatically sampling over a range of
epochs and choosing the epoch count with minimum validation
error in cross-validation based on single data points (see next
subsection for details on model validation). Exemplary loss curves
with the evolutions of training and validation errors are shown in
Fig. S2 of the SM. While the validation error is slightly lower than
the training error, it does not increase with increasing number
of training epochs, which demonstrates that the model is not
overfitting the data strongly during the training process.
2.4. Model validation and testing
The model validation is of crucial importance for ANN-based
models, as it allows to assess the expected accuracy of model
predictions of new data, i.e., data which were not used for training. A common approach of model validation, namely k-fold
cross-validation [102], is applied in the present work. Here, the
available dataset is split into k so-called folds. Overall, k validation
iterations are then performed, where in each iteration step, one
fold serves as validation set which is exempted from training,
while the other folds serve as training set. In each iteration,
the prediction accuracy of the corresponding validation set is
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Combustion and Flame 232 (2021) 111525
sentative of the model application to new data and not significantly biased by the specific choice of the hyperparameters. To further verify this assumption, an additional test set is used, consisting of 607 experimental data points at pressure/temperature combinations for which no other data of the same fuels are considered in the training/validation set yet, compare Table S3 of the
SM. The achieved prediction accuracies of model testing against
this additional set will thus later be compared to those of crossvalidation based on p − T subsets, to assess the suitability of the
cross-validation for model evaluation and to further test the model
for application to completely new data. We chose to not fully separate another test dataset consisting of all experimental data of certain “test” fuels (analogously to cross-validation based on fuel subsets), as this would firstly lead to a deterioration of the predictive
capabilities of the final model for certain fuel classes with fuels excluded from the training/validation set, considering that for a number of fuel classes, only one or two fuels are available (see Table 1).
Secondly, testing results based on such a potential dataset composed of only relatively few test fuels would be strongly dependent
on the random choice of fuels used in this dataset, thus making
the results even less meaningful than those of the cross-validation
based on fuel subsets, where all fuels are used for validation. This
will later be demonstrated by performing an exemplary fixed-split
validation based on fuel subsets, where even significantly lower
prediction errors than in the corresponding cross-validation are
observed due to the specific random choice of validation fuels.
Note that the prediction accuracy of the present model at
pressures, temperatures, and equivalence ratios outside of the
domain of conditions covered by the training and validation data
(as depicted in Fig. 1) is expected to be limited. If more accurate
model predictions for such conditions were desired, the incorporation of additional training data at such conditions, obtained for
instance by numerical simulations using kinetic models, could be
necessary. Another possibility to improve the prediction accuracy
at high pressure or temperature could lie in the incorporation of
analytical approximation formulae, where the empirical parameters could be estimated based on group contributions using ANNs.
This is beyond the scope of the present work, as the model will
be applied later for fuel comparisons at selected pressure and
temperature conditions within the domain covered by the set of
training and validation data. The final trained model is available at
https://doi.org/10.17632/d3ghjdz4sb.2.
Fig. 4. Illustration of the three approaches of k-fold cross-validation employed in
the present study, which distinguish themselves by the distribution of experimental
data over the k folds. Only two exemplary folds are shown for brevity. Each pair
of curly braces represents one LBV data point or a subset of LBV data points, as
described in the text.
all. Such a validation would not be adequate for evaluation of the
model’s predictive capabilities when applied to completely new
fuels without available LBV data. In order to account for specific
application scenarios of the model (i.e., application to fuels from
the training set or application to completely new fuels), three
different cross-validation approaches are therefore used here,
which distinguish themselves in terms of how the experimental
LBV data are distributed over the k folds.
• Cross-validation based on single data points: The distribution of
experimental LBV data over the k folds is completely random.
The evaluated prediction accuracy can be regarded to be on
average representative for a case where the model is applied
to predict the LBV at a pressure and temperature, for which
training data of the corresponding fuel were considered in the
training set, but at a different equivalence ratio.1
• Cross-validation based on p − T subsets: All LBV data points
corresponding to a specific fuel at a specific pressure and
temperature form a “p − T subset”. The subsets are randomly
distributed over the k folds, but all data points corresponding
to one subset remain in the same fold. The evaluated prediction
accuracy can be regarded to be on average representative for a
case where the model is applied to predict the LBV of a fuel,
for which training data were considered in the training set, but
at a different pressure and/or temperature.
• Cross-validation based on fuel subsets: All LBV data points corresponding to a specific fuel form a “fuel subset”. The subsets
are randomly distributed over the k folds, but all data points
corresponding to one subset remain in the same fold. The
evaluated prediction accuracy can be regarded to be on average
representative for a case where the model is applied to predict
the LBV of a fuel, for which no training data were considered
in the training set.
3. Results and discussion
This section presents the results of the model development and
the subsequent analysis of LBV dependence on fuel structure. First,
the model prediction accuracies as results of the model validation
approaches introduced previously are discussed. Thereafter, the
overall impact of molecular groups on LBV is first assessed by
means of a local sensitivity analysis on the molecular groups,
followed by a systematic detailed investigation on the effects of
functional group additions and removals to given fuel structures.
Finally, the fuels from the training set with highest LBV values
are exemplarily ranked at a unified condition, accompanied by the
development of a simplified multivariate linear regression model
for straightforward estimation of LBV at this condition without the
necessity to use the ANN.
The distribution of data points among different folds in these
validation approaches is illustrated in Fig. 4. In the following, results from all three validations are presented, as they jointly allow
for an understanding of model prediction accuracy in different application scenarios. In all cases, the simulated data points remain
always in the training set, as outlined before and shown in Fig. 3.
As the ANN hyperparameters were selected by assessing the results of cross-validation based on single data points and only employing a subset of the overall training/validation database (compare Section 2.3), it is expected that the results of cross-validations
based on p − T subsets and fuel subsets, respectively, are repre-
3.1. Prediction accuracies
1
The LBV data of each fuel at specific pressure and temperature in the training
set cover always a number of different equivalence ratios. Hence, for each LBV data
point in the validation set, there will mostly be at least some data points in the
training set which correspond to the same fuel at same pressure and temperature,
if the data points are distributed randomly over the k folds.
Ten-fold cross-validations were performed according to the
three validation approaches described in Section 2.4. Correspondingly, comparisons of the predicted and measured LBV values for
all experimental training data points are shown in form of parity
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the LBV of the specific fuel to some extent based on training data
points within this domain, which may lead to higher errors than
if the LBV is predicted at equivalence ratios close to stoichiometric conditions. Note that the prediction errors of the final model
on the training set, where the predicted data were already used
for model training, are obviously lower than those in k-fold crossvalidation, as seen in Fig. S5 and Table S4 of the SM. This was similarly reported by other recent studies developing QSPR models for
fuel combustion properties based on neural networks [75,83].
When the 10-fold cross-validation is performed based on random distribution of the fixed p − T data subsets over the 10 folds,
higher prediction errors are observed as shown in Fig. 5(b), with
an MAE of 2.3 cm/s. In this case, the LBV data of a specific fuel
at a specific pressure and temperature are always kept together in
either the training or validation set, as outlined earlier. Hence, the
model only receives data of this fuel at other pressure and temperature as input. The resulting prediction accuracies give an estimate
of the model performance when applied to fuels which are considered in the training set, but at pressure or temperature at which
no training data were available. This is relevant for the analyses
performed later (see Sectons 3.3–3.4), where the LBVs of the fuels
from the training set are predicted by the model at unified pressure and temperature.
Figure 5(c) depicts the results of the 10-fold cross-validation
based on fuel subsets, where all LBV datapoints of a specific fuel
are used only in either the training or the validation set. The
resulting prediction accuracy is thus representative for the model
application to “new” fuel candidates, for which no data were
available yet. While the prediction errors are obviously higher
than in the other two validation cases, the MAE is with 3.3 cm/s
still acceptable, considering that the LBVs of the different fuels in
the training set vary over a range of more than 30 cm/s at the exemplary condition of 1 atm, 373 K, and φ = 1.1 (compare Fig. 13).
Hence, the present modeling approach based on group contributions and an ANN is capable of estimating the LBV of a wide variety of non-oxygenated and oxygenated hydrocarbons over a range
of pressures, temperatures, and equivalence ratios with reasonable
accuracy, even if no experimental data are available yet. The largest
deviations between predicted and measured LBV values in Fig. 5(c)
are observed for the data points corresponding to cyclopentanone
(cyan diamond symbols in the top right corner). Cyclopentanone
is the only cyclic ketone which was available in the training set. It
will be shown in the functional group analysis in Secton 3.3 that
the addition of the carbonyl group to cyclopentane, yielding cyclopentanone, has a significantly more pronounced increasing effect on the LBV than adding a carbonyl group to n-pentane, which
yields 2- or 3-pentanone. Hence, the model is presumably incapable of capturing the strong positive effect of a carbonyl group
in combination with a cyclic ring correctly, if no data for cyclic
ketones are provided in the training set.
The dependence of the prediction error on pressure and temperature of the validation data points is depicted for the case of
cross-validation based on fuel subsets in Fig. S6 of the SM. The
prediction errors are found to be relatively low at 1 atm between
290 and 380 K. This is the range in which the highest number of
experimental data were available (compare Fig. 1), which underlines the necessity of sufficient amounts of training data to achieve
satisfactory prediction accuracies. Additionally, as the error metrics
averaged over all fuels are not necessarily representative for each
fuel class, the prediction errors of the different fuel classes are
presented in detail in Table S5 of the SM. Besides, the errors
conditioned on the values of each of the different input features
(including the molecular groups) are shown in Fig. S7 of the SM.
The errors are found to be only rather weakly dependent on the
group quantity, where a tendency of error increase is observed
with higher numbers of unsaturated carbon groups, while the
Fig. 5. Comparison of measured and predicted values for the LBV model based on
10-fold cross-validations with five seeds, respectively, according to the three approaches of data distribution described in Section 2.4.
plots in Fig. 5, while results of exemplary validations with fixedsplit training/validation sets are provided in Fig. S4 of the SM.
Overall, the highest prediction accuracies are observed when performing the 10-fold cross-validation based on single data points,
i.e., with fully random distribution of the experimental data over
the 10 folds, see Fig. 5(a). Here, the RMSPE of 5.7% is slightly higher
than the expected experimental 1σ uncertainty, if the 1σ experimental uncertainty value of 5% reported by Xiouris et al. [90] is
considered as estimate. The data points with highest prediction errors are found to be those with equivalence ratios at the boundary of the considered equivalence ratio domain, i.e., 0.7 or 1.4. At
these equivalence ratios, the model essentially has to “extrapolate”
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Fig. 6. Comparison of measured LBV values with the predictions by the final
model for an additional test set of LBV data not used for training and validation.
Data of fuels considered in the training/validation database, but at different pressure/temperature combinations are considered here, with conditions and references
shown in Table S3 of the SM (analogously to the cross-validation based on p - T
subsets).
Fig. 7. Averaged semi-normalized local sensitivities Si of the LBV on the different
molecular groups i, determined by averaging over the local sensitivity values Si j of
all fuels j that were considered in the training set, at 1 atm, 373 K, and φ = 1.1. The
error bars denote the standard deviations of averaging over the local sensitivities of
the single fuel compounds j.
errors tend to decrease with higher numbers of saturated carbon
groups. To further analyze the model prediction performance
for specific exemplary fuels in detail, the LBV dependence on
equivalence ratio is demonstrated in Fig. S8 of the SM for multiple
fuel compounds at selected pressures and temperatures. While
the final model trained on all experimental data achieves better
predictions of the experiments at certain conditions where these
are systematically over- or underpredicted in the cross-validation,
it does not fit the random noise of single experimental data points
(compare Fig. S8). As mentioned earlier, the model performance
for conditions outside the range of pressures, temperatures, or
equivalence ratios considered in the training set is expected to
be limited. To illustrate this, the model predictions are compared
against experimental LBV data of n-heptane and iso-octane at
high pressures of 10 and 15 bar in Fig. S9 of the SM. While the
model only moderately underpredicts the experiments at 10 bar,
the deviations become larger at 15 bar and are expected to further
increase towards higher pressures.
The model predictions are finally compared in Fig. 6 against
the experimental data of the external test set. The resulting prediction accuracies are in the same range as those in the crossvalidation based on p − T subsets, thus confirming the representativeness of the cross-validation errors for model application to
completely new data.
with x j being the vector of nominal feature values of fuel j and
LBV0 being the unperturbed predicted burning velocity. The dependence of the predicted LBV on variation of the values xi j of different molecular groups, as performed in the sensitivity analysis,
is illustrated for the example of cyclopentanone in Fig. S10 of the
SM. To assess the average impact of each molecular group on the
model prediction, an average sensitivity Si was then determined for
each input feature by averaging the sensitivities Si j over all fuels j.
Note that the specific choice of the used perturbation affects the
resulting averaged sensitivity values only marginally for perturbations up to 50% of the feature standard deviations, as shown in
Fig. S11 of the SM.
The results are shown in Fig. 7, where the error bars denote
the standard deviations of averaging over the single sensitivity values for the considered fuels. A clearly negative sensitivity is observed with respect to the primary carbon group -CH3 . This is in
agreement with the conclusion of Ranzi et al. [19], who considered
methyl substitution as a major factor with negative impact on LBV.
The effect of the secondary carbon group -CH2 - is only very weak,
which corresponds well to the observation that the lengthening of
carbon chains has typically no strong effect on the resulting LBVs
of hydrocarbons, such as longer n-alkanes [17]. In contrast to the
-CH3 group, the unsaturated primary carbon group =CH2 has a
clearly positive impact, highlighting the effect of unsaturation on
LBV, especially for double bonds at chain ends.2 Interesting effects
are observed when comparing the four oxygenated groups. The alcohol group shows on average a weak sensitivity with respect to
the LBV. Strongly positive sensitivities are observed for the ether
and carbonyl groups. In contrast, the ester group, which essentially
combines a carbonyl group with an ether group, exhibits a very
strong negative sensitivity.
Finally, it is noted that the standard deviations of the averaged
sensitivity values (as denoted by the error bars) are high for all
3.2. Sensitivity analysis
In the following, the developed model is employed to analyze
the dependence of LBV on fuel structure in detail. To gain a first
overview of how the different molecular groups affect the model
predictions, a local sensitivity analysis on the input features has
been performed based on the perturbation method [103]. For this
purpose, semi-normalized local sensitivities were first determined
for all fuels from the training set at 1 atm, 373 K, and φ = 1.1 by
consecutively perturbing the feature values (i.e., the numbers of
the molecular groups) by 20% of their respective standard deviations σx,i over the entire fuel set. Hence, the local sensitivity of the
LBV on the input feature xi of a fuel j is determined as
1 ∂ LBV Si j =
LBV0 ∂ xi xj
LBV(xi j + 0.2σx,i ) − LBV(xi j − 0.2σx,i ) ,
=
LBV0 · 0.4σx,i
2
It should be noted here that the addition of some of these respective carbon
groups is dependent on the numbers of the other groups [78]. For instance, the
addition of a methyl side chain to a straight-chain fuel molecule will not only add
a primary carbon group in terms of the present group contribution method, but
also replaces a secondary by a tertiary carbon group.
xj
(2)
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Fig. 8. Predicted LBVs of alkanes and alkenes at 373 K, 1 atm, and φ = 1.1. An experimental value [104] is used for methane.
features, demonstrating that the effect of a particular group on
the LBV depends to a considerable extent on the presence of other
groups in the molecule structure and is not the same for the different fuels. In order to explore to which extent a linear group additivity approach would still be able to capture the LBV dependence
on fuel structure at a specific measurement condition in terms of
pressure, temperature, and equivalence ratio, a corresponding multivariate linear regression model will be trained in Section 3.5 by
employing the molecular groups as input parameters. Since the
averaged sensitivities discussed here only provide a first general
overview of the effects of specific functional groups on LBV, the
fuel structure effects will be investigated in more detail for particular fuel types in the following.
3.3.1. Alkanes and alkenes
For insight into the effects of methyl groups, double bonds,
and chain length, the LBVs of alkanes and alkenes are depicted
in Fig. 8 in dependence on the carbon number. First, it is demonstrated again that the LBV is relatively independent of the carbon
chain length of higher n-alkanes. This behavior is well-known and
can be explained by the formation of similar amounts of small
radicals, in particular H and CH3 , from the consecutive decomposition reaction sequences of these fuels [17,19]. Methane’s LBV
is relatively low due to the formation of methyl radicals by H
atom abstraction reactions, which can further react in recombination reactions to ethane, leading to chain termination [19]. In
contrast, ethane exhibits significantly higher LBV values compared
with methane and the longer n-alkanes, which can be largely attributed to the formation of reactive H radicals through a dehydrogenation reaction of the formed ethyl radicals and to the formation
of large amounts of reactive vinyl radicals [19].
Second, a negative effect of methyl group addition is consistently observed in Fig. 8 for both alkane and alkene fuels. For the
smallest branched alkane, iso-butane, the low LBV compared to its
linear counterpart n-butane has been explained by formation of
alkene intermediate species from the decomposition of iso-butyl
radicals, which subsequently react into resonantly stable allylic
radicals that proceed through chain terminating recombination reactions [19]. As larger branched alkanes such as iso-octane produce similar small branched alkenes during their combustion, similar underlying mechanisms are responsible for their lower flame
speeds as well. As seen from the present LBV predictions, the LBV
decreasing effect of methyl group addition becomes weaker for
longer alkanes such as n-heptane, compared to the short chain
compounds like propane.
Third, the addition of a double bond to a saturated fuel
structure increases the flame speed, and a higher degree of unsaturation consequently correlates with even further increased values,
as demonstrated here for n-butane, the butene isomers, and 1,3butadiene. The effect of double bond addition on LBV tends to
get weaker for larger fuel structures. This can be explained by
the fact that the increased LBVs of longer alkenes, compared to
their respective alkane counterparts, are mainly the result of the
higher adiabatic flame temperatures of these alkenes [19]. The
difference between the adiabatic flame temperatures of alkanes
and alkenes decreases with increased size of the respective fuel
molecules (see Table S6 of the SM), and so does thus the flame
speed.
3.3. Functional group analysis
The ANN-based group contribution model allows to predict
the LBVs of different fuel compounds over a range of conditions
with reasonable accuracy, as shown before. This makes direct
LBV comparisons for a wide range of components and fuel types
straightforward. In the following, we take advantage of this possibility by analyzing the impacts of the different functional groups
on the predicted LBV in detail for a unified condition of 1 atm and
373 K, as well as an equivalence ratio of 1.1, where the majority
of fuels exhibit their maximum LBV values. Such comparisons are
otherwise hampered by the necessity of scaling the data from
other conditions for each component for which experiments are
not available at the particular chosen condition. In a range of
±0.5 bar, ±15 K and an equivalence ratio of ±0.05 around this
condition, the model exhibits relatively low MAE and MAPE values
of 1.7 cm/s and 3.4%, respectively, in the cross-validation based on
p − T subsets. In the cross-validation based on fuel subsets, the
corresponding MAE and MAPE values around this condition are
2.5 cm/s and 4.9%, respectively. If not specified explicitly, the fuels
shown in the following were part of the training set (see Table S1
of the SM) with at least some data points. Therefore, the prediction
accuracies for these fuels can be expected to be in the range of the
cross-validation errors based on p − T subsets. It should be noted
that small differences between the predicted LBVs of single fuel
molecules in Figs. 8–12 can be affected by these prediction uncertainties and the main conclusions made later will thus be based
on systematic observations on functional group effects for various
fuel components rather than differences between single molecules.
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Fig. 9. Predicted LBVs of alcohols at 373 K, 1 atm, and φ = 1.1. An experimental value [104] is used for methane.
Fig. 10. Predicted LBVs of ketones, aldehydes, esters, and carbonates at 373 K, 1 atm, and φ = 1.1. An experimental value [104] is used for methane. The compounds in
rectangular boxes have the same molecular groups according to the applied group definitions, and the model thus predicts identical values for them.
Last, the position of the double bond relative to the carbon
chain end also slightly affects the LBV, with the values of the
1-alkenes being predicted to be consistently higher than those of
the 2-alkenes. A reason is expected to lie in the availability of
allylic carbon sites on both sides of the double bond for 2-alkenes
compared to only one available allylic site for 1-alkenes, which
promotes the formation of allylic radicals and subsequent chain
termination reactions. Note that the present group definition does
not allow to discriminate between 2- and 3-alkenes, and overall
only small differences were observed, for instance, between the
linear hexene isomers in experiments [30].
longer alcohols, since mainly formaldehyde rather than methyl radicals (as in the case of methane) are produced in its combustion [105]. Similar to the corresponding n-alkanes, the LBVs of the
longer 1-alcohols approach a relatively constant level. Methyl substitution has a comparably negative effect as for alkanes, with e.g.
the LBV of 2-methyl-2-propanol (tert-butanol) being significantly
decreased compared to that of 2-propanol. Interestingly, the effect
of adding a hydroxy moiety to a given n-alkane strongly depends
on its position in the carbon chain, where increased flame speeds
are only observed for 1-alcohols. In contrast, the 2-alcohols such
as 2-propanol and 2-butanol are even predicted to have slightly
lower flame speeds than the n-alkanes. From the standpoint of
the present group contribution approach, these lower flame speeds
of 2-alcohols compared with the 1-alcohol analogues can be interpreted as the result of their higher numbers of primary carbon groups, which on average have a strongly negative impact on
LBV (compare Fig. 7). Kinetically, this can be again traced back to
3.3.2. Alcohols
Next, functional group effects are assessed for alcohols in Fig. 9,
where the linear alkanes are shown as well as reference values.
The flame speed of methanol is increased considerably more relative to the corresponding alkane (methane) than those of the
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Fig. 11. Predicted LBVs of aromatics at 373 K, 1 atm, and φ = 1.1.
the resulting radical pools, where for instance 1-propanol produces
rather reactive radicals (e.g. formyl, ethyl, vinyl), while 2-propanol
reactions yield less reactive radicals (e.g. methyl, allyl) [46]. Double
bond addition to a given alcohol is also found here to be advantageous in terms of increased LBV, as seen for the example of 3methyl-1-butanol and its unsaturated analogues 3-methyl-2-buten1-ol (prenol) and 3-methyl-3-buten-1-ol (iso-prenol).
3.3.3. Compounds with carbonyl and ether functionalities
The flame speeds of fuel compounds comprising a carbonyl
group are depicted in Fig. 10, where those of the corresponding nalkanes are again shown for comparison. These include aldehydes
(carbonyl group attached to carbon chain end), ketones (carbonyl
group attached inside carbon chain), esters (carbonyl group adjacent to one ether oxygen), and carbonates (carbonyl group adjacent
to two ether oxygens on both sides). First, it is seen that for chain
lengths higher than C4 and C3 , ketones and aldehydes, respectively,
exhibit higher LBVs than the corresponding n-alkanes, where the
aldehyde burning velocities are higher than those of their respective ketone isomers. Hence, the difference between the effects of
adding a carbonyl group to the end of or inside a given n-alkane
carbon chain is qualitatively similar as for the alcohol group addition discussed before, with the addition at the chain end leading to
higher flame speeds. It is also observed that the addition of a second carbonyl group, as seen here for the case of 2-butanone and
2,3-butanedione, strongly increases the LBV.
Interestingly, esters exhibit a completely different behavior with
significantly decreased flame speeds compared to alkanes, despite
the fact that they only distinguish themselves from the aldehydes/ketones by the presence of one additional oxygen atom near
the carbonyl group. Again, those compounds where the ester group
is located at the carbon chain end (i.e., formates) have higher
flame speeds than their counterparts where it is located inside
the carbon chain. The two carbonate compounds dimethyl carbonate and diethyl carbonate are depicted as well, where the LBV of
dimethyl carbonate is even lower than that of the corresponding
ester (methyl acetate) and is in fact the lowest of all 124 fuels in
the training set at this condition.
For a better understanding of the underlying mechanisms
responsible for these different flame speeds, the adiabatic flame
temperatures of all fuel compounds discussed here have been
Fig. 12. Predicted LBVs of 5-membered ring compounds (top panel) and 6membered ring compounds (bottom panel) at 373 K, 1 atm, and φ = 1.1. Furan is
marked here, as no LBV data were available in the training set and its predicted
value may thus potentially be associated with higher uncertainties than those of
the other fuels.
determined and are compiled in Table S6 of the SM. For the
example of the C3 fuels in Fig. 10, it is found that the order of
their flame temperatures correlates mostly to the order observed
here for their LBV values, where dimethyl carbonate exhibits the
lowest flame temperature, propionaldehyde exhibits the highest,
and 2-propanone, ethyl formate, and methyl acetate lie in between. It is thus expected that these differences in the adiabatic
flame temperature are at least to a significant degree responsible
for the different LBVs. Still, kinetic effects are expected to contribute to the differences as well. For instance, for both formates
and aldehydes with the same chain lengths, the production of
α -radicals is expected to be favorable because of the lower C-H
bond dissociation energies [106,107] at this site. However, while
the following β -scission of α -radicals of aldehydes can produce
an alkyl radical and ethenone, the β -scission of α -radicals of
formates would give an alkyl radical but with a stable CO2 . This
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Combustion and Flame 232 (2021) 111525
may contribute to the lower LBVs of formates compared with
their aldehyde counterparts. Future comparative experimental and
kinetic modeling studies exploring in detail the contributions of
thermal and kinetic effects to the differences in LBVs between
aldehydes, ketones, esters, and carbonates are thus of interest.
The LBVs of ethers and polyethers with different chain lengths
are compared with the corresponding alkanes in Fig. S12 of the SM.
Linear ethers exhibit significantly higher values than their alkane
counterparts, as will be discussed in more detail in Section 3.4 in
the context of fuel design. On the other hand, highly branched
ethers, such as the octane boosters methyl and ethyl tert-butyl
ether, exhibit relatively low LBVs, which is expected from their
high numbers of methyl groups.
of the 1,3-dioxolanyl radicals, mostly reactive species such as
ethylene, ethyl radicals, and formaldehyde are formed [108], which
may contribute to the high LBV of 1,3-dioxolane.
High flame speeds are also observed for 2-methylfuran, where
the addition of a second methyl group, yielding 2,5-dimethylfuran,
has a clearly negative effect, as expected. Hence, for furan itself, one would expect an even higher flame speed, which is confirmed by the model prediction. It should be noted though, that
to our best knowledge, no flame speed data of furan in air are
available in the literature which could have been considered in
the training set. Hence, this prediction is possibly associated with
higher uncertainties, and the LBVs of furan thus deserve to be investigated experimentally for more profound understanding of its
burning characteristics. Generally, the furanic compounds have significantly higher adiabatic flame temperatures than the saturated
cyclic ethers (see Table S6 of the SM), which explains the consistently higher LBVs of furan and its methyl-substituted derivatives
compared with their respective tetrahydrofuran counterparts.
The LBVs of 6-membered ring compounds from the present
training set are finally compared in Fig. 12 (bottom panel). The effect of ether groups is seen to be similar as for the 5-membered
ring analogues, with 1,3-dioxane exhibiting an LBV almost as high
as 1,3-dioxolane. Methyl substitution has a slightly negative effect,
while lengthening of the alkyl side chain has no significant impact,
as discussed in previous work [22].
3.3.4. Aromatic compounds
The LBV dependence on side chain characteristics and functional groups of aromatic fuel compounds is depicted in Fig. 11.
Benzene exhibits a relatively high LBV, which is largely the result of its high adiabatic flame temperature. Consecutively adding
methyl side groups to benzene, toluene, and dimethylbenzene (the
isomers are not distinguished by the present group definition), respectively, consistently lowers the flame speed. The trimethylbenzene isomers, which are not distinguished by the group definition
either, are clearly predicted to have the lowest flame speeds among
the compounds shown. The reason lies, besides having slightly
lower flame temperatures with increased degree of saturation, in
the preferred H atom abstraction from their methyl groups, which
yields resonantly stabilized benzyl-like radicals [19]. In contrast to
the increase of the number of methyl groups, the increase of side
chain length affects the flame speed only slightly. The addition of a
double bond to the side chain, as seen when comparing ethylbenzene and its counterpart ethenylbenzene (styrene), increases the
LBV, as expected. Similarly, the addition of an ether group in the
side chain has a clearly positive impact, which is demonstrated
here for anisole and 4-methylanisole.
3.4. Fuel ranking and design
The previous analysis of functional group effects has demonstrated the strong dependence of flame speed on the underlying
structure of hydrocarbon and oxygenated hydrocarbon fuels. At the
same time, the LBV constitutes an important fuel property with
high practical relevance for engine performance [6]. This suggests
that the LBV may be considered as target for future fuel design
studies, similar to previous works where performance indicators
such as octane or cetane numbers have been commonly considered [67–69]. Here, we demonstrate the application of the present
model for such purposes, by ranking the fuels at selected conditions according to their LBVs. Note that besides the LBV, other fundamental combustion properties such as ignition delay time and
extinction strain rate affect the engine performance of a fuel as
well. Hence, in a full fuel design study, the relevant fuel properties
should be jointly taken into account [8,68,69]. This is beyond the
scope of the present work, and not all high-LBV fuels identified
here thus necessarily represent optimal fuels for practical engine
applications.
The highest-ranked fuel components of the present training
set at 373 K, 1 atm, and φ = 1.1 are shown in Fig. 13, where the
primary reference fuel (PRF) compounds n-heptane and iso-octane
as well as the compound from the training set with lowest LBV,
dimethyl carbonate, are additionally shown as references. The
comparison clearly demonstrates again that various oxygenated
fuel compounds, which can be produced from biomass or as
synthetic fuels from renewable electricity, are among the highestranked compounds, with significantly increased LBVs compared
with the PRF components n-heptane and iso-octane. With the
exception of ethylene, which is gaseous and not a typical fuel
candidate for practical applications, the highest LBVs are observed
for the cyclic acetals 1,3-dioxolane and 1,3-dioxane, as briefly discussed before. These have very recently been proposed as future
generation “bio-hybrid” fuels produced from bio-based feedstock
and CO2 as carbon source in combination with renewable electricity [108]. Here, we thus demonstrate the capability of these fuels
to significantly increase the laminar flame speeds in comparison
with gasoline (which lie in the range of the PRF compounds) and
even ethanol. Future engine studies on these fuels are thus of high
3.3.5. Cyclic non-aromatic compounds
The effect of functional groups on 5-membered ring compounds
is illustrated in Fig. 12 (top panel). First, it is noted that cyclopentane has approximately the same LBV as its non-cyclic analogue npentane. The same holds for cyclopentanol, whose LBV is predicted
to be in a similar range as that of cyclopentane, which is in agreement with recent experimental measurements [66]. In contrast, cyclopentanone’s LBV is significantly higher. Qualitatively, the effects
of hydroxy and carbonyl group additions to the cyclic ring are thus
similar to their additions to non-cyclic alkanes, where the LBV’s
of 2-ketones were predicted to be higher than those of the corresponding 2-alcohols (see Figs. 9–10). Still, quantitatively the LBV of
cyclopentanone is significantly higher than that of its non-cyclic
counterpart 2-pentanone, and the positive effect of a carbonyl
group on LBV is thus particularly pronounced for cyclic ketones.
The effect of unsaturation on LBV is predicted to be weak when
comparing cyclopentene with cyclopentane, while the addition of
a second double bond, yielding 1,3-cyclopentadiene, significantly
increases the LBV. Furthermore, the replacement of carbon atoms
by oxygen atoms in form of ether groups has a clearly positive impact. While tetrahydrofuran’s LBV is already significantly increased
compared to cyclopentane’s, 1,3-dioxolane exhibits (except for
ethylene) the highest LBV of all 124 fuel compounds considered
in the training set. Comparing the adiabatic flame temperatures
of the 5-membered ring compounds (see Table S6 of the SM), it
is seen that these exhibit only slight differences for the saturated
components, indicating that the significant flame speed increases
with the addition of carbonyl or ether groups seen here must
mainly originate from the underlying reaction kinetics. For instance, in the decomposition reactions following the ring-opening
12
F. vom Lehn, L. Cai, B. Copa Cáceres et al.
Combustion and Flame 232 (2021) 111525
Fig. 13. Ranking of fuels in the present training set with highest predicted LBVs at 373 K, 1 atm, and φ = 1.1. The values of n-heptane, isooctane, and dimethyl carbonate
(which has the lowest value of all considered fuels) are included for comparison.
interest to investigate whether this laminar flame speed advantage
can be utilized to achieve higher engine efficiencies.
Similar to these cyclic acetals, a number of non-cyclic
ethers and polyethers are ranked highly, including the series
of oxymethylene ethers (methoxymethane=OME0 /dimethyl ether,
dimethoxymethane=OME1 , OME2 , OME3 ) as well as the related
compounds diethoxymethane and ethoxyethane (diethyl ether).
Due to their high cetane numbers, these fuels have been primarily
considered as diesel fuel substitutes or additives [109–113], or as
fuel candidates for gasoline controlled auto-ignition concepts [114].
Comparing the LBVs of these highly ranked ether fuels, relatively
similar values are observed, with a slight tendency of increased
LBV with increased mass fraction of oxygen in the molecule structure [115].3
As indicated earlier, two very promising highly ranked fuel candidates, which can be potentially produced from lignocellulosic
biomass, are cyclopentanone and anisole. In addition, these exhibit research octane numbers above 100 [69], which in combination with the high LBV makes them primary candidates for future
highly efficient spark-ignition engines.
Finally, the impact of stoichiometry on the resulting fuel rankings is to be highlighted. While the previous comparisons have
focused on an equivalence ratio of 1.1, a similar comparison is
made for a lean condition of 373 K, 1 atm, and φ = 0.7 in Fig. 14.
In a range of ±15 K, ±0.5 bar and an equivalence ratio of ±0.05
around this condition, the model exhibits MAE and MAPE values
of 2.0 cm/s and 7.3%, respectively, in the cross-validation based on
fuel subsets, and of 1.5 cm/s and 5.5%, respectively, in the crossvalidation based on p − T subsets. The highest-ranked compounds
still include ethylene, 1,3-dioxolane, and 1,3-dioxane, though
noteworthily some fuels rank considerably higher than at the
slightly rich condition considered earlier. This includes especially
the prenol isomers (3-methyl-2-buten-1-ol and 3-methyl-3-buten1-ol), which have been proposed as promising biofuel candidates
for highly efficient spark-ignition engines and whose maximum
burning velocities were observed at a relatively low equivalence
Fig. 14. Ranking of fuels in the present training set with highest predicted LBVs
at 373 K, 1 atm, and φ = 0.7. The values of ethanol, n-heptane, and isooctane are
included for comparison.
ratio of 1.0 [48], as well as anisole (ranking significantly higher
than at φ = 1.1) and 2-methylfuran, which similarly constitute
octane boosters derived from lignocellulosic biomass [117,118].
These fuels may thus be particularly suitable for engines operating
with lean-burn concepts [119].
3.5. Multivariate linear regression model
While the group-based ANN model allows for LBV predictions
over a wide range of conditions, the availability of a group-based
multivariate linear regression (MLR) model may be advantageous
in certain cases due to its simplicity in application. To assess the
suitability of this approach, a simplified MLR model is finally developed here for the condition of 1 atm, 373 K, and φ = 1.1. The
3
Large amounts of reactive formaldehyde are produced due to successive β scissions of primary OME radicals in OME flames [116], which is expected to be
a main reason for their high LBVs. Their adiabatic flame temperatures are in the
same range as those of the corresponding n-alkanes.
13
F. vom Lehn, L. Cai, B. Copa Cáceres et al.
Combustion and Flame 232 (2021) 111525
Table 5
Regression coefficients of the MLR model at 373 K, 1 atm, and φ = 1.1, compare
Eq. (3).
Coefficient
Value [cm/s]
Coefficient
Value [cm/s]
β0
β−CH3
β−CH2−
β>CH−/>C<
β=CH2
β=CH−
β=C<
62.9
- 4.2
- 0.2
2.3
2.1
0.0
1.0
βCsat,ring
βCunsat,ring
β−OH
β−O−
β>C=O/−CH=O
β−COO−/CHOO−
- 1.5
- 1.0
- 3.2
3.7
1.0
- 7.4
equivalence ratio served as input parameters to an ANN, which has
been trained based on a large database of training data for 124 different compounds. Validation of the model based on three different cross-validation approaches demonstrated good prediction performance over a range of pressures, temperatures, and equivalence
ratios. The model was then applied to assess the impacts of functional groups on LBV in detail by conducting sensitivity and detailed functional group analyses at unified conditions. The LBV was
demonstrated to consistently increase with increased degree of unsaturation, while it was shown to decrease with addition of methyl
groups. Furthermore, different types of oxygenated groups were
found to exhibit very distinct effects on LBV. While the effect of
alcohol group addition to a given alkane with higher chain length
on LBV was found to be either slightly positive or slightly negative, depending on the position of its addition in the carbon chain,
moderately positive effects on LBV were observed for addition of
carbonyl groups to higher alkanes, yielding ketones or aldehydes.
Similarly, ethers and polyethers such as acetals consistently exhibited high LBV values. In contrast, esters and carbonates were found
to have significantly lower LBVs than the corresponding alkanes,
alcohols, or ketones. Besides, the LBV values were observed to be
consistently higher for additions of alcohol, carbonyl, or ester functional groups at the chain end compared to the addition of these
respective functionalities inside a given alkane carbon chain. The
fuels from the present training set were then exemplarily ranked
at unified conditions based on the model predictions in order to
demonstrate the capability of identifying high-LBV fuels, which can
be very valuable in the context of fuel design. Finally, a simplified
MLR model has been developed to predict the LBV in dependence
of the molecular groups at one specific condition, achieving a reasonable prediction accuracy, which was however lower than that
of the ANN-based model.
Future work should investigate whether high-LBV fuel compounds identified by relative comparisons at the conditions of
laboratory-scale experiments, as performed in the present study,
indeed allow for improved efficiencies when applied in real engines. Besides, comparative experimental and detailed kinetic modeling studies focusing on specific types of fuels are clearly desirable in order to further enhance the fundamental understanding on
the underlying thermal and chemical kinetic effects contributing to
the global impacts of different functional groups on LBV, that were
demonstrated here.
Fig. 15. Leave-one-out cross-validation of the MLR model predictions against the
predictions of the ANN model. The asterisks were added to the error metrics to indicate that these are not directly comparable with the error metrics shown in previous sections, as the cross-validation is performed here against the ANN predictions
rather than against experimental data.
LBV is thus expressed as
LBV j = β0 +
βi xi j ,
(3)
i
with xi j denoting the number of groups i in the fuel molecule j.
Since experimental data at a specific condition such as the one
chosen here are commonly only available for few fuels, the predictions of the ANN model for all 124 fuels listed in Table S1 of the
SM are used instead as training data to determine the regression
coefficients βi . The resulting coefficients are provided in Table 5.
Leave-one-out cross-validation against the ANN predictions exhibits a reasonable prediction accuracy as shown in Fig. 15, considering the simplicity of the model. Note that due to the validation
against the ANN predictions rather than experimental data, the error metrics depicted in Fig. 15 are not directly comparable to those
of the ANN model validation discussed in Section 3.1. For a small
set of 19 fuels, for which experimental data are available at the exact condition of 1 atm, 373 K, and φ = 1.1, a validation of the MLR
model against these experimental data was performed as well, observing an MAE of 3.5 cm/s and an MAPE of 6.2%. The ANN model
itself exhibits obviously lower MAE and MAPE values of 2.5 cm/s
and 4.9%, respectively, at this condition in the cross-validation
based on fuel subsets, as mentioned in Section 3.3. Still, the results show that a linear group contribution model can serve as a
simple alternative with moderate accuracy at a specific condition,
as long as sufficient data are available at this particular condition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
This work was performed as part of the Cluster of Excellence “The Fuel Science Center”, which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – Cluster of Excellence 2186
“The Fuel Science Center” ID: 390919832. Simulations were performed with computing resources granted by RWTH Aachen University under project rwth0626. The authors thank Mr. Rafal Broda
for technical assistance.
4. Concluding remarks
Supplementary material
In the present study, a QSPR model has been developed for
the estimation of laminar flame speeds of hydrocarbons and oxygenated hydrocarbons based on their underlying fuel structures.
A set of molecular groups as well as pressure, temperature, and
Supplementary material associated with this article can be
found, in the online version, at doi:10.1016/j.combustflame.2021.
111525.
14
F. vom Lehn, L. Cai, B. Copa Cáceres et al.
Combustion and Flame 232 (2021) 111525
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