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During their high-temperature oxidation, complex hydrocarbons and their early fragments are short-lived and figure prominently only during the pyrolysis stage. However, they are quickly replaced by smaller hydrocarbons at the onset of the oxidation stage, resulting in simpler chemistry requirements past pyrolysis. In this study, we develop a data-based hybrid chemistry approach to accelerate chemistry integration for complex fuels. The approach is based on tracking the evolution of chemistry through representative species for the pyrolysis and coupling their reactions with simpler foundational chemistry. The selection of these representative species is implemented using principal component analysis (PCA) based on simulation data. The description of chemistry for the representative species is implemented using an artificial neural network (ANN) model for their reaction rates followed by the description of their chemistry using a foundational chemistry model. The selection of the transition between these models is trained a priori using an ANN pattern recognition classifier. This data-based hybrid chemistry acceleration model is demonstrated for three fuels: n-dodecane, n-heptane and n-decane and investigated with two foundational chemistry, C0–C2 and C0–C4, models. The hybrid scheme results in computational saving, up to one order of magnitude for n-dodecane, two orders of magnitudes for n-heptane, and three orders of magnitudes for n-decane. The accuracy and saving in computational cost depend on the number of selected species and the size of the used foundational chemistry. The hybrid model coupled with the more detailed C0–C4 foundational performs, overall, better than the one coupled with the C0–C2 foundational chemistry.
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A Data-Based Hybrid Model for Complex Fuel Chemistry
Acceleration at High Temperatures
Sultan Alqahtania,b,
1
, Tarek Echekkia
a Department of Mechanical and Aerospace Engineering, North Carolina State University,
Raleigh, NC 27695
b Department of Mechanical Engineering, King Khalid University, Abha, Saudi Arabia
*Corresponding Author Email: ssalqaht@ncsu.edu
Abstract: At high-temperature oxidation, complex hydrocarbons and their early fragments are
short-lived and figure prominently only during the pyrolysis stage. However, they are quickly
replaced by smaller hydrocarbons at the onset of the oxidation stage, resulting in simpler
chemistry requirements past pyrolysis. In this study, we develop a data-based hybrid chemistry
approach to accelerate chemistry integration for complex fuels. The approach is based on
tracking the evolution of chemistry through representative species for the pyrolysis and coupling
their reactions with simpler foundational chemistry. The selection of these representative species
is implemented using principal component analysis (PCA) based on simulation data. The
description of chemistry for the representative species is implemented using an artificial neural
network (ANN) model for their reaction rates followed by the description of their chemistry
using a foundational chemistry model. The selection of the transition between these models is
trained a priori using an ANN pattern recognition classifier. This data-based hybrid chemistry
acceleration model is demonstrated for three fuels: n-dodecane, n-heptane and n-decane and
investigated with two foundational chemistry, C0-C2 and C0-C4, models. The hybrid scheme
results in computational saving, up to one order of magnitude for n-dodecane, two orders of
magnitudes for n-heptane, and three orders of magnitudes for n-decane. The accuracy and saving
in computational cost depend on the number of selected species and the size of the used
foundational chemistry. The hybrid model coupled with the more detailed C0-C4 foundational
performs, overall, better than the one coupled with the C0-C2 foundational chemistry.
Keywords: Data-based hybrid chemistry model, PCA, ANN
1
Corresponding Author. Address: Department of Mechanical and Aerospace Engineering, North
Carolina State University, 1840 Entrepreneur Drive, Campus Box 7910, Engineering Building III,
Room 3252, Raleigh, NC 27695-7910, USA. Fax: +1 919 515 7968, E-mail address:
ssalqaht@ncsu.edu (S. Alqahtani).
2
1. Introduction
The integration of the chemistry of complex fuels is computationally challenging for
simulations beyond 0D. Therefore, chemistry reduction or other strategies to accelerate chemistry
integration have been an active area of research over the past 4 decades [1]. Strategies to accelerate
chemistry include in situ adaptive tabulation (ISAT) [2] and regression (e.g. the piecewise reusable
implementation for solution mapping or PRISM [3]), manifold-based methods, such as intrinsic
low-dimensional manifolds (ILDM) [4] and computational singular perturbation (CSP) [5], and
adaptive chemistry methods, including dynamic approaches (see for example Refs. [6-7]).
Additional strategies to accelerate chemistry can be adopted under certain mixture or
combustion conditions. During high-temperature oxidation in complex hydrocarbon fuels, there
are distinct stages associated with a fast pyrolysis followed by “slower” oxidation [9]. This
behavior has enabled a few strategies to describe complex fuel chemistry, including the recent
hybrid chemistry (HyChem) approach by Wang and co-workers [10]-[16]. HyChem relies on
fuel/fragment measurements to develop a reduced chemistry description for fuels without the prior
requirement for an existing detailed or reduced chemistry description for these fuels. In this
approach, the global reactions for the fuel fragments are coupled with established foundational
chemistry that models the oxidation of C0-C4 or C0-C2 species.
Recently, we have proposed a similar approach to HyChem where the temporal profiles of
measured species can be used directly to determine their chemical reaction rates [17,18]. In Ref.
[18], we proposed that soon after the pyrolysis stage, the foundational chemistry can be further
simplified to account primarily for C0-C2 species. The hierarchical development during complex
fuel oxidation from complex to simpler hydrocarbons has been a key observation on which the
HyChem approach and our approach have been based. In the section below, we will illustrate this
observation for two fuels considered in this study.
The hybrid chemistry models proposed in [17,18] are inspired by the HyChem approach [10]-
[17], with some key distinctions. Both models combine a reaction rate models for the fuel
fragments with foundational chemistry for the remaining species. The HyChem approach [10]-
[17] uses global reaction steps for the fragments whose rate parameters are obtained through an
optimization process to capture key observables, such as ignition delay times. In contrast, Refs.
[17] and [18] derive these reaction rates directly from temporal profiles of multi-scalar
measurements using shallow artificial neural networks (ANNs). Moreover, the reaction rates for
the fragments are modeled using ANNs in terms of the modeled fragments. The chemistry for a
subset of these fragments may not be captured by a simpler foundational C0-C4 chemistry. The
same set of strategies, of course, can be implemented for chemistry reduction/acceleration when
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an adequate (e.g. detailed) chemistry description is available. The use of such mechanisms may
not be practical beyond 0D or 1D simulations of relevance to practical applications. Moreover,
important questions remain as related to whether how many fragments need to be modeled within
a hybrid chemistry framework and whether there are alternative species that can be used instead
within this framework.
An additional reduction of the complexity of fuel oxidation and a potential impetus for
chemistry acceleration is associated with the adoption of a reduced chemistry description under
certain combustion conditions. As in HyChem and our hybrid chemistry approach, this can be
achieved based on available data for these conditions. Indeed, the present study explores this and
attempts to develop strategies for chemistry acceleration for the high-temperature oxidation of
complex fuels within a data-based framework. Principal component analysis (PCA) a fundamental
tool for dimensionality reduction is designed to identify representative species that can be used to
track and integrate the evolution of the chemical system instead of the fuel fragments.
The objective of the present study is to develop and investigate such an approach. In the next
section, Sec. 2, we provide a brief motivation for the proposed approach and discuss the
implementation steps. In Sec. 3, results of the model’s predictions are presented for 3 different
fuels, n-dodecane, n-heptane and n-decane, using two foundational chemistry mechanisms and
compared against predictions from detailed chemical mechanisms. In Sec. 4, the results are
summarized, and future extensions of the present work are discussed.
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2. Motivation and model implementation
In this section, we provide a motivation for the present approach followed by the model
implementation.
2.1. Motivation for the approach
As discussed in our recent work [18], the hierarchical evolution in the complexity of the fuels
representing the pyrolysis and oxidation stages provides a mechanism for simplifying the
foundational chemistry within a hybrid chemistry framework. This evolution is illustrated in Fig.
1 for the high-temperature chemistry of stoichiometric mixtures in air of n-dodecane and n-heptane
at an initial temperature of 1500 K and 1 atm. Hydrocarbons are grouped based on their carbon
composition to distinguish groups, C0-C2 (orange lines), C3-C4 (green lines) and C5 and higher (>
C4) (blue). The fuel, indicated in red lines, decays much faster to produce fragments. The vertical
dashed lines delineate the transition in time to primarily C0-C4 chemistry, while the vertical solid
lines delineate the transition in time to primarily C0-C2 chemistry. The figure clearly shows that
the hierarchical evolution in the complexity of the hydrocarbons as the oxidation proceeds in time.
The strategy that we will adopt in this work exploits this hierarchical evolution.
2.2. Model formulation
In our recent study [18], we proposed a formulation for chemistry acceleration for complex
fuels at high-temperature based on a hybrid chemistry model that combines the chemistry of fuel
fragments with a simpler foundational chemistry based on a C0-C2 mechanism. The fuel fragments’
chemistry is modeled using a regression based on artificial neural networks (ANN) to relate the
fragments reaction rates to their concentrations starting with their measured temporal profiles.
The reliance on temporal profiles is motivated by their potential availability in experimental
measurements.
In the present approach, we propose a model to accelerate chemistry starting from detailed or
skeletal mechanisms for complex fuels. Moreover, in contrast with the model in Ref. [18] where
the modeled species are the fragments, we model, using ANN, the reaction rates of representative
species, which are used to track the evolution of the mixture from pyrolysis to the subsequent
oxidation stages. These species, compared to fragments, tend to be based on simpler hydrocarbons
and other oxidation species. Although they can be modeled using this foundational chemistry at
later stages, they are modeled differently during pyrolysis stages to account for their interactions
with larger fragments and the fuel.
Figure 2 illustrates the process of integration in time of all species in the mechanism for the
case of using a C0-C4 or a C0-C2 foundational chemistry. The representative species are integrated
during the initial stages of the chemistry integration using an ANN regression of the reaction rate
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built from numerical data of the evolution of the detailed mechanism. Subsequently, these species
are integrated with the foundational chemistry. The remaining species (other than the
representative species) are handled entirely by the foundational chemistry. The selection of the
representative species out of the foundational chemistry species is carried out using a priori
simulation data and principal component analysis [19] as discussed below. As a data-based
framework, the proposed hybrid model is designed to be adaptive to a set of conditions dictated
by the data on which the model is trained. This should not be perceived as a deficiency in the
model; rather, it is an attribute of the model to enable a more aggressive reduction in the description
of the chemistry if we know in advance the nature of the composition space to be accessed during
simulations. Generalizing to a broader set of conditions requires a diverse set of data that is
applicable to more scenarios for mixing and chemistry. From this perspective, the training for the
hybrid model is not much different from that based on conventional chemistry reduction schemes.
And, potentially, a hybrid chemistry model can be combined with a conventional chemistry
reduction approach as a prior step before the development of the hybrid model.
The training steps to construct the chemistry acceleration model are summarized in Fig. 3.
First, the model must be built on an existing solution that is representative of the problem under
consideration, such as homogeneous chemistry for systems characterized by faster chemistry
compared to transport, or canonical premixed/non-premixed flames or other stochastic 0D reactor
models. This solution is built using a reference detailed or skeletal mechanism that we wish to
reduce to accelerate chemistry integration. Of course, it can also be based on a HyChem hybrid
chemistry models that couples global reaction steps for the fuel fragments with foundational
chemistry. The second key step is the reduction step, which involves principal component analysis
(PCA) [19] on the simulation data. PCA is used to select the representative species for which the
reaction rates are modeled using an ANN regression.
The representative species reaction rate during the bulk of the pyrolysis stage and potentially
part of the oxidation stage is carried out using ANN regression based on an input defining the
mixture state primarily using the representative species and temperature or a subset of this vector.
At this step, criteria for determining the transition from the ANN regression model to the
foundational chemistry model for the representative species’ reaction rates are determined using a
pattern recognition network (PRN) classifier discussed below. Finally, the validation step is carried
out to compare the predictions of the hybrid model to results from detailed chemistry simulations.
Once the hybrid model is validated, it can now be used in simulations as a chemistry acceleration
scheme.
6
As indicated above, we use PCA for the representative species selection. PCA is a
feature/variable reduction approach that can reduce the representation of data from a set of
correlated variables to a linear transform of uncorrelated variables. By this transformation, most
of the data variance resides in the first principal components (PCs). PCA is becoming increasingly
popular as a method for composition space reduction [20-28]. Its implementation in mechanism
reduction is not new in the combustion literature [19,29-34]. However, it has been implemented
primarily on variables that represent the sensitivities of thermo-chemical scalars or global
measures of combustion (e.g. the flame speed) on reaction rate parameters to identify the key
reactions to retain. Another novel implementation of PCA by D’Alessio et al. [35] implemented
PCA as a method to partition the composition space into different clusters where different reduced
mechanisms can be implemented. In the present study, we implement PCA on thermo-chemical
scalars’ reaction rates to identify the key species to retain.
PCA operates on existing data, which correspond in the present model to detailed chemistry
simulations of a combustion problem. More specifically, in the present study, we consider a
homogeneous system of an initial fuel-air mixture at high temperature. PCA takes the data from
the temporal evolution of the temperature and species profiles until equilibrium is reached. It is
implemented first on the full set of species reactions     , excluding the fuel,
in the detailed mechanism over time increments. The first step selects the first set of species; and
a second PCA is implemented on the retained species from the first PCA. With a single-step PCA
as a potential alternative, the species’ selection may vary significantly by small changes in their
criteria. This prospect is eliminated by the 2-step PCA hierarchical procedure.
To carry out the PCA, each specie’s reaction rate is centered by subtracting its mean over all
variable as follows:
  . (3)
The two-step PCA procedure is implemented as follows. For the first PCA, the constant matrix
Q of eigenvectors of the symmetric covariance matrix C of size   is evaluated based on the
solution of over the different time increments and initial conditions. Mathematically,
components  correspond to the covariance of the reaction rates of the ith and jth species,
and
over all values of the solution and is expressed as follows:
  


. (4)
where M is the total number of discrete data points from the solution of The PCs’ vector is
related to the original thermo-chemical scalars’ reaction rate vector as follows:
7
  . (5)
At this stage, it is important to order the PCs based on the magnitude of the eigenvalues in a
descending order. Every PC, , also may be expressed as a linear combination of centered the
thermo-chemical scalars’ reaction rate vector:
 

  
(6)
where the  are the coefficients of matrix corresponding to thermo-chemical scalar
The
first few PCs represent most of the variance in the solution vector . Therefore, we retain only the
first NPC PCs that contain a cumulative variance of 99%. These retained PCs can be expressed in
terms of the normalized thermo-chemical scalars as follows:
   . (7)
The matrix A contains the leading NPC eigenvectors of Q. The same expression (7) applies to
relate the retained PCs to the species’ reaction rates vector. However, j ranges from 1 to NPC instead
of from 1 to N.
At this stage, we reorder the sum in Eq. (6) such that the terms with the highest magnitudes of
the coefficients  are listed first in a descending order again. This reordering also identifies the
importance of the associated thermo-chemical scalars. To identify the most important species, we
implement a cut-off criterion as follows:




        (8)
The goal is to determine the number b of the reordered thermo-chemical scalars that represents
a cumulative magnitude of the coefficients  that represent the dominant contributions to the
PCs. For example, in the present study of n-C12H26, n-C7H16 and n-C10H22, we get a value for b
ranging between 40 and 50 species that correspond to a cutoff percentage of 99%. The identified
species from this range include C1-C4 hydrocarbons along with a few more complex hydrocarbons,
including C5H10, C6H12 and C7H14.
The next PCA is carried out on the selected species from the first PCA. For this step, 3
conditions are imposed on the number of retained PCs: 1) these retained PCs must account for
more than 99% (or similar threshold) of the data variance and 2) the cutoff condition (based on
Eq. (8)) and 3) the selected species from the list must be a part of the foundational chemistry. In
this study, the cutoff condition percentage is investigated in the range of 80% to 99% and two
foundational chemistries are used based on two widely-used mechanisms, USC Mech-II [36], a
C0-C4 mechanism, and GRI-Mech 1.2 [37], a C0-C2 mechanism. Although this earlier version of
the GRI mechanism is not as comprehensive as the later versions, especially as far as including
more C2-related steps, it remains a valid foundational chemistry to capture the oxidation phase of
8
complex hydrocarbons. The number of selected species, from the second PCA ranges from 12
to 19 species for the 3 fuels considered.
The next step in the model development is an ANN-based regression based on a multi-layer
perceptron (MLP) ANN architecture for the reaction rate of the final selection of the species (the
output) in terms of a subset of the representative species and temperature (the input). We have
found that specifying the entire list of selected representative species for input in the ANN is not
necessary and that a subset of these species is sufficient. In fact, the ANN in this study is
established with only 3 species and the temperature as input. These species include the fuel, O2,
and CO. These scalars are adequate markers for the progress of reaction from pyrolysis to
oxidation. Also, they are the same variables used to select the reaction rate model using PRN. For
the ANN/PRN training and most importantly during a posteriori simulations, the selection of a
subset of the representative species as an input for the ANN/PRN reduces the computational cost
of training and selection of the chemistry model. Based on the present results, this subset recovers
all representative species’ reaction rates. Each representative species is modeled by a single ANN
to independently optimize the ANN architecture for accuracy and computational efficiency.
The criterion for transition from ANN regression to foundational chemistry for the
representative species is set during the training process based on the simulation data. For
homogeneous chemistry, we set a criterion when 99% of the fuel C and H are represented by the
species in the foundational chemistry. We use this criterion to train an ANN-based classifier, a
pattern recognition network (PRN), to identify which model for chemistry, ANN-based or
foundational chemistry, to use during a posteriori simulations. The PRN classifier may not be
needed within the context of the homogeneous system considered in the present study. However,
it will be an important tool in more complex configuration when both mixing and reaction are
present or when the model is trained on different initial mixtures of fuel and oxidizer.
Figure 4 shows the network architecture for a PRN. It includes an input layer, a single hidden
layer, and an output layer. In principle, the entire set of transported thermo-chemical scalars or the
entire list of representative species can be represented by neurons in the input layer. Again, we
have found that a subset of the thermo-chemical scalars’ vector is sufficient to decide the model
for chemistry to be used for the representative species based on the state of the mixture. In the
present study, this subset consists of temperature, the fuel, O2 and CO mass fractions. Also, for
effective training, at least one hidden layer in the PRN architecture is needed. In the present study,
we use one hidden layer with 15 neurons.
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The network calculates a probability based on the input representation of the thermo-chemical
state of whether the chemistry model falls into the ANN-based class or the foundational chemistry
class. Based on the value of this probability, the decision is made to use either model based on its
higher probability.
To illustrate the procedure for implementing the PRN as a classification algorithm, we briefly
discuss the essential steps of training (on the training data) and implementing the network in situ
during a simulation involving the hybrid model. In the beginning, the training data is labeled with
either a value of 0 or 1, depending on whether a given state needs to be modeled with ANN
chemistry or with the foundational chemistry models, respectively, based on an a priori criterion
for classification. First, we select the PRN architecture, which in this case corresponds to 4 neurons
at the input, one hidden layer with 6 neurons, and one output neuron, which will be labeled as
belonging to one of two classes.
In a second step, the PRN is trained by matching the input training data with their labeled
classes. This results in the determination of weights that measure the strengths of the connections
from inputs to neurons in the hidden layers and these neurons to the output. Mathematically, the
value of the output, which is interpreted as a probability, Prob, is expressed as:
6(1) (1) (1)
1
Prob ii
i
f w a b
=

=+


(9)
where f is the activation function, which is, in the present study, the sigmoid function.
(1)
i
w
is the
weight of the connection between the ith neuron in the hidden layer and the output, which measures
the strength of this connection.
is the bias neuron value in the hidden layer and
(1)
i
a
is the value
of the ith neuron in the hidden layer. This value, in turn, can be related to the values at the input
using a similar expression to Eq. (9):
4
(1) (0) (0) (0)
1
i ji j
j
a f w a b
=

=+


(10)
where
(0)
ji
w
is the weight of the connection between the jth neuron in the input layer and the ith
neuron in the hidden layer.
(0)
j
a
and
(0)
b
are the values of the jth neuron and the bias neuron in the
input layer. The bias neurons serve to shift the activation function f, hence enabling more flexibility
in training the network.
Once the training is completed and the connections’ weights are determined, the network is
ready to be implemented in situ during a simulation using the hybrid model. Given a state of the
solution, a subset of the solution vector represented by the 4 inputs is used to determine the
10
probability according to Eqs. (9) and (10). These probabilities, then, are converted to 0’s or 1’s,
depending to their relative proximity to these 2 limits, thus identifying the class of reaction models
to use for the state.
It is important to emphasize the importance of developing a priori criteria for the
“classification” of the reaction model for the representative species. With a single set of training
conditions, it is easier to set a threshold criterion for the transition from a reaction to another.
However, if the model data is to be derived from a broad range of simulations (e.g. a combination
of reactor models or under non-homogeneous conditions), the criteria for transition can be set
during the training using a PRN and the corresponding network can provide an unbiased
classification of the model during a posteriori simulations.
The feedforward network for ANN-based regression shares common elements of its
architecture with PRN architecture shown in Fig. 4. The input layer is identical to that of the PRN
architecture and uses only 4 inputs. Given that the data is relatively simple, also only one hidden
layer is used with the number of neurons varying from 5 to 15 depending on the complexity of the
reaction rate to be modeled. Training for a broader range of mixture conditions may require a
deeper network to accommodate for the range of mixing and reaction interactions. Moreover, the
training is done individually for each representative species reaction rate. Therefore, the output
has only one neuron and different networks, one for each representative species, are used. The
transfer function connecting a given neuron to neurons from a previous layer is the tangent sigmoid
transfer function. The ANN weights and biases are updated during the training process by
minimizing the loss function by a Levenberg-Marquardt backpropagation algorithm. The loss
function is chosen as the mean squared error (MSE). The training data used to build the hybrid
model is divided randomly into three sets, a learning set, a validation set, and a test set with
proportions of 60/20/20 %, respectively, and fed to the training algorithm. The training times with
3000 epochs/iterations for each reaction rate takes less than 10 minutes on a single CPU
workstation.
It is important to note that there is significant room for optimization of the network architecture,
including its depth and number of neurons in each hidden layer, as well as whether to train all
representative species’ reaction rates individually or in groups. This optimization will be attempted
in the future to accommodate larger datasets with more complex composition spaces.
Finally, in a posteriori simulations, the representative species are integrated with the remaining
species of the foundational chemistry. The solutions for the temperature and a subset of the
representative species is used to distinguish the regime for modeling the representative scalars
reaction rates using the PRN network. The same set of inputs is used to determine the
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representative species and temperature chemical source terms in the governing equations if the
ANN model is needed. In addition to the input variables for the networks, the weights associated
with each network and its architecture are used. As described above, these weights are determined
during the modeling step (see Fig. 3).
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3. Results and discussion
3.1. Run conditions
The present model is validated using a simple reactor model based on homogeneous chemistry
at constant volume for 3 fuels: n-dodecane, n-heptane and n-decane. The characteristics of the
detailed mechanisms for these fuels are summarized in Table 1. The JetSurF2.0 mechanism is used
as a detailed chemistry mechanism for the n-dodecane. It consists of 348 species and 2163
reactions [38]. For n-heptane, a detailed mechanism by Mehl et al. [39] is used. It consists of 654
species and 2827 reactions. A jet fuel surrogate mechanism by Dooley et al. [40] is used for n-
decane. It consists of 1599 species and 6633 reactions. Of course, it is possible to select a subset
of the reactions and species in each mechanism or based on the detailed chemistry calculations on
skeletal mechanism. However, one of our principal goals to investigate that the present model can
describe the chemistry of these fuels by using a smaller foundational chemistry mechanism, either
USC Mech-II [36] with 111 species and 784 reactions or GRI-Mech 1.2 with 32 species and 177
reactions [37]. Based on the information in Table 1, a large portion of the detailed mechanisms are
for describing the chemistry of species larger than C4 and an important portion is represented by
C3C4 species. Therefore, by eliminating the complex hydrocarbons and their associated reactions,
potential computational saving and acceleration can be achieved. Further chemistry acceleration
can be achieved by a reduction of the foundational chemistry. Standard approaches for chemistry
reduction can be coupled with our present hybrid chemistry model.
Table. 1 Number of species in detailed mechanism for n-dodecane, n-heptane, and n-decane.
Fuel
n-dodecane
n-heptane
n-decane
Detailed mechanism
JetSurF2.0 [38]
Mehl et al. [39]
Dooley et al. [40]
Total species
348
654
1599
Species with > C4
250
381
1207
Species with C3 C4
57
200
313
Species with C0 C2
41
73
79
The governing energy and species equations for the homogeneous reactions are:

 



 (11)

    (12)
where T is the temperature and , and
 are the lth specie’s internal energy, reaction rate
and molecular weight, respectively, is mixture density, is mixture average specific heat at
constant volume, and is the kth specie’s concentration. In the above equations, corresponds
to number of species in the foundational chemistry mechanism. The governing equations (11) and
(12) are integrated using Senkin [41].
In this study, we train the model for an initial pressure of one atmosphere, a stoichiometric
composition of fuel and air and vary the initial temperature from 1400 K to 1500 K at 10 degrees
13
increments. For validation, we also investigate the ability of the model to “interpolate” by
investigating initial temperatures at 1425 and 1475 K.
Our present results show that the combination of 4 PCs and a cutoff percentage of 90% yield
the best performance in term of solution accuracy and ANN training time for all studied fuels. The
representative species for all studied fuels are listed in Table 2. The data-based hybrid model
predicts a common set of representative species among all studied fuels with both foundational
chemistries. They include: O2, CO2, CO, H2O, H2, OH, C2H4, H, C2H2, O, CH3, and CH4.
Table 2. List of representative species for n-dodecane, n-heptane, and n-decane.
Fuel
n-C12H26
n-C10H22
n-C7H16
Hybrid model
Foundational chemistry used with the hybrid model
USC-
Mech
GRI-
Mech
USC- Mech
GRI- Mech
USC- Mech
GRI- Mech
common
species
O2
CO2
CO
H2O
H2
OH
H
O
CH3
CH4
C2H2
C2H4
C2H6
C2H6
C2H5
CH2O
C2H6
C2H6
C3H6
HCCO
C3H6
C3H6
C4H8-1
C4H8-1
C4H8-1
HCCO
CH2O
3.2. Model performance
In this section, we present comparisons of the predictions of data-based hybrid chemistry
model with the detailed chemistry results for the 3 fuels considered and the two foundational
chemistries adopted. Figures 5 and 6 show temporal profiles’ comparisons among the data-based
hybrid model coupled with USC Mech-II, data-based hybrid model coupled with GRI-Mech 1.2
and the detailed mechanism at 1425 K and 1475 K, respectively, initial mixture temperatures for
n-dodecane. The results are presented for temperature and select species mole fractions that are
either modeled with the hybrid chemistry approach or the foundational chemistry. Here, the
modeled species by ANN in the hybrid model for n-dodecane combined with USC Mech-II are
O2, CO2, CO, H2O, H2, OH, C2H4, H, C2H2, O, CH3, CH4, HCCO, C2H6, C4H8-1 and C3H6.
Therefore, in the figures, HO2, CH3OH, CH3O and H2O2 are modeled entirely by the foundational
chemistry. The results show that the data-based model can predict the evolution of a subset of these
14
species as well as the ones modeled initially by ANN and the temperature accurately. Nonetheless,
the present discrepancies still capture the trends of the temporal profiles. The resulting
computational saving is approximately 5 times; although, there was no attempt to optimize the
performance of the model.
Next, we substitute the foundational chemistry by using the simpler GRI-Mech 1.2 instead of
the USC Mech-II. The modeled species by ANN with the GRI-Mech 1.2 include: O2, CO2, CO,
H2O, H2, OH, C2H4, H, C2H2, O, CH3, CH4, HCCO, and C2H6. Therefore, 4 of the species shown
in Figs. 5 and 6 are modeled with the foundational chemistry. The use of the simpler GRI-Mech
1.2 pushes the transition point for the hybrid reaction rate model to a later time. This is expected
since a transition to a dominant C0-C2 chemistry occurs later than the transition to a dominant C0-
C4 chemistry. The figure shows an excellent agreement for the temperature and a number of major
species modeled by ANN, reasonable predictions for HO2 and CH3O, which are modeled by the
GRI-Mech 1.2 and some discrepancies for CH3OH and H2O2. Again, despite the presence of some
quantitative discrepancies, the trends of the temporal profiles are well captured by the model and
the final equilibrium conditions are reasonably matched with the GRI-Mech 1.2 as the foundational
chemistry model. The coupling of the model with the GRI-Mech 1.2 is achieved with a
computational saving of an order of magnitude.
Next, we carry out comparisons based on n-heptane simulations with the data-based hybrid
model. Figures 7 and 8 mirror the conditions in Figs. 5 and 6, except that the fuel now is n-heptane.
Moreover, the list of modeled species with ANN is different for this fuel. With the USC Mech-II
as the foundational chemistry model, the following species are selected using the PCA procedure
outlined above: O2, CO2, CO, H2O, H2, OH, C2H4, H, C2H2, O, CH3, CH4, C2H6, C4H8-1and C3H6.
Again, in Figs. 7 and 8, HO2, CH3OH, CH3O and H2O2 are modeled with the foundational
chemistry.
The following list is adopted for coupling with the GRI-Mech 1.2: O2, CO2, CO, H2O, H2, OH,
C2H4, H, C2H2, O, CH3, CH4, and C2H6. Figures 7 and 8 confirm essentially the same observations
made for n-dodecane. The use of the USC Mech-II as foundational chemistry yields better
predictions than the model simulations based on the GRI-Mech 1.2 for foundational chemistry.
The computational saving with the model for n-heptane is one order of magnitude with USC Mech-
II and two orders of magnitude with the GRI-Mech 1.2.
The same trends are established for the similar comparisons between model predictions and
detailed chemistry calculations for n-decane, which are shown in Figs. 9 and 10. For this fuel, the
following species are selected for ANN modeling with the USC Mech-II: O2, CO2, CO, H2O, H2,
OH, CH4, C2H4, H, C2H2, O, CH3, CH2O, C4H8-1 and C3H6; while a different set as listed in Table
15
2 is used with the GRI-Mech 1.2. The computational saving with the model for n-decane is even
higher compared to the other fuels, yielding 3 orders of magnitude in acceleration.
The above results demonstrate that a hybrid chemistry model that is based on an ANN-
based reaction model for a set of representative scalars (species and temperature) coupled with
simpler foundational chemistry can be a viable chemistry acceleration scheme. The model
performs slightly better with the C0-C4 foundational chemistry; but we can also see that the simpler
foundational chemistry also can yield comparable or better results for some species in the n-decane
simulations. As suggested in Fig. 1, the accuracy of the model may be attributed primarily to the
value of the assumption associated with the transition as well as the role modeled representative
scalars in the reaction of other species in the foundational chemistry.
At this stage, it is useful to reiterate that the hybrid approach presented in Ref. [17] and the
present approach are based on tracking the pyrolysis stage during high-temperature oxidation
(HTO) using a set of representative species. Both approaches yield very good comparisons with
detailed mechanism results. In Ref. [17], as proposed in the HyChem approach [10]-[16], a
common set (among a wide range of complex fuels) of pyrolysis fragments represent a natural
choice for representative species during HTO. The present approach develops a more systematic
strategy for the selection of these species using PCA. This strategy has been recently implemented
to derive similar representative species during low-temperature fuel oxidation (LTO) [42]. In
contrast with HTO with its common fragments, the LTO process is initiated through the formation
of fuel-specific intermediates [43]. The present strategy enabled the selection of representative
species that are simpler than these intermediates; and yet they can track even the early stages of
LTO.
16
4. Conclusions
A data-based model for chemistry acceleration to integrate the chemistry of complex fuels at
high temperature is developed. The model is based on a hybrid scheme that involves data-based
models for representative species combined with the solution of the remaining species using
foundational chemistry mechanism. The selection process for the representative species is carried
out using a 2-step PCA on the reaction rate data from detailed simulations.
Once the selection of the representative species is completed, their reactions rates are modeled
using ANN starting with the available computational data. These species’ modeled reactions
account for their interactions with the complex hydrocarbons. This ANN-based reaction rate model
is coupled in time with the solution of the remaining species using a smaller foundational
chemistry. The criteria for transition from the model to the foundational chemistry are
implemented using a classification approach using a pattern recognition network. This approach
is implemented during the training stage using the simulation data and a reduced input.
The validation of the hybrid model by comparison with detailed chemistry calculations for 3
fuels, n-dodecane, n-heptane and n-decane, show that the model can yield reasonable results by
adopting either C0-C2 or C0-C4 foundational chemistries and resulting in acceleration of the
chemistry integration process from 5 times to 3 orders of magnitude. The reduction process also
sifts through the complexity of the detailed mechanism to identify the key species to model and
transport.
The model is built on adapting the chemistry for a specific class of problems. It can be
generated based on data from surrogate problems that spans the same composition space of the
configuration of interest. Such problems include 0D reactor models, such as perfectly stirred
reactors, 1D flame models or low-dimensional stochastic models, such as the linear-eddy model
(LEM) or the one-dimensional turbulence (ODT) model. The generation of data from these models
is inherently cheaper than the detailed multi-dimensional reacting flow simulations of interest. In
these simulations, accounting for chemistry often makes up a good fraction of the computational
cost.
Finally, the choice of the surrogate model is critical to the success of the implementation of
the hybrid chemistry model proposed here. This choice can determine how accurate these
predictions are. However, an equally important consideration is that the accessed composition
space during multi-dimensional simulations must not exceed the bounds of the corresponding
composition space of the surrogate models. Criteria should be implemented to identify such events.
17
However, and most importantly, strategies must be implemented to predict conditions outside the
bounds of the hybrid chemistry model. Such events can be handled by accessing a different model,
such as a more detailed chemical description, or to retrain in situ the hybrid chemistry approach to
accommodate the new events.
Acknowledgements
The first author would like to acknowledge the support of King Khalid University in Abha, Saudi
Arabia.
18
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22
Figure 1. Species temporal profiles from detailed homogenous chemistry simulations of n-
dodecane and n-heptane/air stoichiometric mixtures at initial temperatures 1500 K and 1 atm
pressure.
Time (ms)
Mole Fraction
0 0.03 0.06 0.09
0
0.004
0.008
0.012
0
0.08
0.16
0.24
0.32
n-C12H26
1500 K
C3-C4C0-C2
> C4
Time (ms)
Mole Fraction
0 0.04 0.08 0.12
0
0.006
0.012
0.018
0
0.1
0.2
0.3
n-C7H16
1500 K
C3-C4
C0-C2
> C4
23
Data-based hybrid model chemistry
Time
Representative Scalars ( temperature and species)
Other Species
Foundational chemistry
ANN model for heat release
rate and selected species
reaction rates
Foundational chemistry
Transition condition (99% of fuel mass is represented in
foundational chemistry species)
Figure 2. Data-based hybrid model reaction rates calculation by ANN chemistry before
transition to the foundational chemistry.
24
ANN
Species selection by PCA
Data preprocessing
Solution
Validation Coupling
Modeling
Reduction
classifier
Figure 3. Data-based hybrid model scheme.
25
output layerInput layer
Fuel
Temperature
CO
Hidden
layer
O2
ANN class
Foundationa l
chemistry clas s
Figure 4. A pattern recognition network (PRN) architecture of 4 nodes in the input layer, one
hidden layer with 6 neurons, and two classes.
26
Figure 5. Stoichiometric n-dodecane/air at 1425 K. Temporal profiles’ comparisons of the
hybrid model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed
mechanism (solid green) predictions. The vertical dashed blue line and the vertical solid blue line
represent the transitions between the ANN regression model and the foundational chemistry
based on the USC Mech-II and the GRI-Mech 1.2, respectively.
Time (ms)
Temperature (K)
0 0.05 0.1 0.15 0.2
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
0.002
0.004
0.006
0.008
0.01 n-C12H26
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
0.02
0.04
0.06
0.08 H2O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
0.02
0.04
CO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
0.002
0.004 CH3OH
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
1E-05
2E-05
3E-05 CH3O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2
0
5E-05
0.0001 H2O2
27
Figure 6. Stoichiometric n-dodecane/air at 1475 K. Temporal profiles’ comparisons of the
hybrid model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed
mechanism (solid green) predictions. The vertical blue lines correspond to the same criteria as in
Fig. 5.
Time (ms)
Temperature (K)
0 0.05 0.1
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.05 0.1
0
0.002
0.004
0.006
0.008
0.01 n-C12H26
Time (ms)
Mole Fraction
0 0.05 0.1
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.05 0.1
0
0.02
0.04
0.06
0.08 H2O
Time (ms)
Mole Fraction
0 0.05 0.1
0
0.02
0.04 CO2
Time (ms)
Mole Fraction
0 0.05 0.1
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.05 0.1
0
0.002
0.004 CH3OH
Time (ms)
Mole Fraction
0 0.05 0.1
0
1E-05
2E-05
3E-05 CH3O
Time (ms)
Mole Fraction
0 0.05 0.1
0
2E-05
4E-05
6E-05
8E-05
0.0001 H2O2
28
Figure 7. Stoichiometric n-heptane/air at 1425 K. Temporal profiles’ comparisons of the hybrid
model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed mechanism (solid
green) predictions. The vertical blue lines correspond to the same criteria as in Fig. 5.
Time (ms)
Temperature (K)
0 0.05 0.1 0.15 0.2 0.25
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
0.005
0.01
0.015 n-C7H16
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
0.02
0.04
0.06
0.08
0.1 H2O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
0.02
0.04 CO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
5E-05
0.0001
0.00015 CH3OH
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
5E-06
1E-05
1.5E-05
2E-05
2.5E-05 CH3O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15 0.2 0.25
0
5E-05
0.0001 H2O2
29
Figure 8. Stoichiometric n-heptane/air at 1475 K. Temporal profiles’ comparisons of the
hybrid model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed
mechanism (solid green) predictions. The vertical blue lines correspond to the same criteria as in
Fig. 5.
Time (ms)
Temperature (K)
0 0.05 0.1 0.15
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
0.005
0.01
0.015 n-C7H16
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
0.02
0.04
0.06
0.08 H2O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
0.02
0.04 CO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
5E-05
0.0001
0.00015 CH3OH
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
5E-06
1E-05
1.5E-05
2E-05
2.5E-05 CH3O
Time (ms)
Mole Fraction
0 0.05 0.1 0.15
0
5E-05
0.0001 H2O2
30
Figure 9. Stoichiometric n-decane/air at 1425 K. Temporal profiles’ comparisons of the hybrid
model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed mechanism
(solid green) predictions. The vertical blue lines correspond to the same criteria as in Fig. 5.
Time (ms)
Temperature (K)
0 0.08 0.16 0.24
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
0.0025
0.005
0.0075
0.01
0.0125 n-C10H22
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
0.02
0.04
0.06
0.08 H2O
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
0.02
0.04
0.06 CO2
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
2E-05
4E-05 CH3OH
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
1E-05
2E-05
3E-05
4E-05 CH3O
Time (ms)
Mole Fraction
0 0.08 0.16 0.24
0
3E-05
6E-05
9E-05
0.00012
0.00015
0.00018
0.00021 H2O2
31
Figure 10. Stoichiometric n-decane/air at 1475 K. Temporal profiles’ comparisons of the
hybrid model (dashed red: USC Mech-II, dashed black: GRI-Mech 1.2) and the detailed
mechanism (solid green) predictions. The vertical blue lines correspond to the same criteria as in
Fig. 5.
Time (ms)
Temperature (K)
0 0.08 0.16
1500
1800
2100
2400
2700 T
Time (ms)
Mole Fraction
0 0.08 0.16
0
0.0025
0.005
0.0075
0.01
0.0125 n-C10H22
Time (ms)
Mole Fraction
0 0.08 0.16
0.05
0.1
0.15
0.2 O2
Time (ms)
Mole Fraction
0 0.08 0.16
0
0.02
0.04
0.06
0.08 H2O
Time (ms)
Mole Fraction
0 0.08 0.16
0
0.02
0.04
CO2
Time (ms)
Mole Fraction
0 0.08 0.16
0
0.0002
0.0004
0.0006
0.0008 HO2
Time (ms)
Mole Fraction
0 0.08 0.16
0
2E-05
4E-05
6E-05 CH3OH
Time (ms)
Mole Fraction
0 0.08 0.16
0
2E-05
4E-05
6E-05 CH3O
Time (ms)
Mole Fraction
0 0.08 0.16
0
3E-05
6E-05
9E-05
0.00012 H2O2
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We propose and test an alternative approach to modeling high-temperature combustion chemistry of multicomponent real fuels. The hybrid chemistry (HyChem) approach decouples fuel pyrolysis from the oxidation of fuel pyrolysis products. The pyrolysis (or oxidative pyrolysis) process is modeled by seven lumped reaction steps in which the stoichiometric and reaction rate coefficients are derived from experiments. The oxidation process is described by detailed chemistry of foundational hydrocarbon fuels. We present results obtained for three conventional jet fuels and two rocket fuels as examples. Modeling results demonstrate that HyChem models are capable of predicting a wide range of combustion properties, including ignition delay times, laminar flame speeds, and non-premixed flame extinction strain rates of all five fuels. Sensitivity analysis shows that for conventional, petroleum-derived real fuels, the uncertainties in the experimental measurements of C2H4 and CH4 impact model predictions to an extent, but the largest influence of the model predictability stems from the uncertainties of the foundational fuel chemistry model used (USC Mech II). In addition, we introduce an approach in the realm of the HyChem approach to address the need to predict the negative-temperature coefficient (NTC) behaviors of jet fuels, in which the CH2O speciation history is proposed to be a viable NTC-activity marker for model development. Finally, the paper shows that the HyChem model can be reduced to about 30 species in size to enable turbulent combustion modeling of real fuels with a testable chemistry model.
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A correlated dynamic adaptive chemistry and transport (CO-DACT) method is developed to accelerate numerical simulations with detailed chemistry and transport properties in a reactive flow. Different sets of phase parameters, which govern the transport properties and chemical reaction pathways, respectively, are proposed to identify the correlated groups for transport properties and reaction pathways in both temporal and spatial coordinates. The correlated transport properties and reduced chemical mechanisms in phase space are dynamically updated by different user-specified threshold values. For the calculation of detailed transport properties, the mixture-averaged diffusion model is employed. For the on-the-fly generation of reduced chemistry, the multi-generation path flux analysis (PFA) method is used. In the present method, the chemical reduction and transport properties calculation are only conducted once for all the computation cells in the same correlated group within the pre-specified thresholds. Therefore, without sacrificing accuracy within the range of uncertainty of mechanisms and transport properties, the CO-DACT method can eliminate all redundant chemistry reductions and transport properties calculations in temporal and spatial coordinates when the transport properties and chemical reaction pathways are correlated due to the similarities in phase space. The CO-DACT method is further integrated with the hybrid multi-timescale (HMTS) method to achieve efficient integration of chemistry. Simulations of outward propagating spherical premixed flames and one dimensional (1D) diffusion ignitions of a jet fuel surrogate mixture, as well as an unsteady spherical propagating diffusion flame of a DME/air mixture are conducted to validate the present algorithm. The impact of the selection of threshold values as well as the dependence of numerical errors on pressure and equivalent ratio are also examined. The results demonstrate that the CO-DACT method can increase the computation efficiency for transport properties by at least two-order of magnitudes. Moreover, it is robust, accurate, and improves the overall computation efficiency involving a large kinetic mechanism.
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The hybrid chemistry modeling approach, termed HyChem, was used to explore the combustion chemistry of blended petroleum and bio-derived jet fuels. The pyrolysis products of conventional petroleum derived-fuels, such as Jet A, are dominated by ethylene and propene, whereas in many bio-derived fuels, such as alcohol to jet (ATJ) fuels, the fuel comprises highly branched alkanes and produces isobutene as a main pyrolysis product. We report here an investigation of blends of Jet A (designated A2) and an ATJ fuel (designated C1) with the central question of whether the HyChem models for neat fuels can be combined to model the blend combustion behaviors. The pyrolysis and oxidation of several blends of A2 and C1 were investigated. Flow reactor experiments were carried out at 1 atm, 1030 and 1140K, with equivalence ratios of 1.0 and 2.0. Shock tube measurements of blended fuel pyrolysis were performed at 1 atm from 1025 to 1325 K. Good agreement between measurements and model predictions was found showing that formation of the products in the blended fuels were predicted by a simple combination of the HyChem models for the two individual fuels, thus demonstrating that the HyChem models for two jet fuels of very different compositions are “additive.”
Conference Paper
With increasing use of alternative fuels, approaches that can efficiently model their combustion chemistry are essential to facilitate their utilization. The hybrid chemistry (HyChem) method incorporates a basic understanding about the combustion chemistry of multicomponent liquid fuels that overcomes some of the limitations of the surrogate fuel approach. The present work focused on a comparative study of one conventional, petroleum-derived Jet A fuel (designated as A2), with an alternative, bio-derived fuel (designated as C1), using a variety of experimental techniques as well as HyChem modeling. While A2 is composed of a mixture of n-, iso-, and cyclo-alkanes, and aromatics, C1 is composed primarily of two highly branched C12 and C16 alkanes. Upon decomposition, A2 produces primarily ethylene and propene, while C1 produces mostly isobutene. HyChem models were developed for each fuel, using both shock tube and flow reactor speciation data. The developed models were tested against a wide range of experimental data, including shock tube ignition delay time and laminar flame speed. The stringent validations and agreement between the models and experimental measurements highlight the validity as well as potential wider applications of the HyChem concept in studying combustion chemistry of liquid hydrocarbon fuels.