Combustion and Flame 262 (2024) 113330
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Combustion and Flame
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A novel method for quantifying variations in NO formation pathways using a
sub-mechanism accounting and reaction tracing algorithm
Hannah Helbig∗, Myles D. Bohon
Chair of Pressure Gain Combustion, Technische Universität Berlin, 10623, Berlin, Germany
ARTICLE INFO
Keywords:
NOx
Reaction path analysis
Counterflow flames
Sub-mechanisms
ABSTRACT
This work presents a new implementation of reaction path analysis that traces molar fluxes attributable
to various sub-mechanisms from inception through to their final product. The post-processing procedure
involves creating a network of these fluxes and quantifying the contributions made by the specified formation
and destruction mechanisms through an iterative process starting from the individual initiation reactions.
A key distinction of this approach is that the reaction fluxes are directly correlated to their initiation
reaction, which is then followed throughout the entire reaction network. This allows for the distinction of
contribution to the formation of a target species from an arbitrary number of sources. The effectiveness of
this approach is demonstrated by examining the formation of NO in three different case studies: (1) the
impact of equivalence ratio on methane/air premixed counterflow flames, (2) the effect of hydrogen addition
on ultra-lean methane/air premixed counter flames, and (3) the influence of pressure elevation in partially
premixed methane/air flames. The results are presented in emission indexes, NO production rates throughout
the flame, and multidigraphs illustrating the allocated flows between the species. Compared to replicated
analysis methods previously used for these case studies, this new approach provides a more comprehensive
and detailed analysis for several reasons. Firstly, the method does not require duplicated simulations with
modified kinetic mechanisms, but is applied purely post-processing. This avoids the previous problems of
needing to re-simulate the flame multiple times and allows for the fully coupled chemistry set. Secondly, the
method allows for an arbitrarily defined set of initiation pathways which propagate through all possible routes
within the network. This allows for the identification of spatial/temporal variability in the sub-mechanisms
and thereby yields more information than the integral values of previous methods.
Novelty and significance statement
Understanding reaction pathways that lead to the formation of pollutants such as NO is an essential
aspect in the development of new sustainable and low-emission combustion concepts. However, evaluating
the different underlying sub-mechanisms is difficult due to the high degree of interconnectedness and
spatial/temporal overlap of the reaction pathways. This work, therefore, presents a new algorithm that can
be used to investigate the contributions of an arbitrary number of sub-mechanisms to the formation or
decomposition of a species. The results can be expressed not only as integral values, but also as spatially and/or
temporally resolved production rates and reaction path figures and graphs, leading to a more comprehensive
and detailed analysis. The approach is also applied as a purely post-processing technique, which does not
require multiple simulations with different chemical kinetic mechanisms as has previously be applied.
1. Introduction
The mitigation of emissions to avert the negative impacts of the
anthropogenic emissions of greenhouse gases is a pressing issue in the
combustion community; but even with the transition of the energy
sector to low-carbon or carbon-free fuels, the combustion of sustain-
able fuels, like hydrogen, ammonia, or biofuels in air will still emit
nitrogen oxides (NO𝑥), predominantly as nitric oxide (NO) and with
∗Corresponding author.
lower amounts nitrogen dioxide (NO2) or nitrous oxide (N2O). The
emission of NO𝑥is not only hazardous to the environment but also to
health. Comprehending the formation pathways of NO is, therefore, an
urgent matter to promote designs of efficient, sustainable combustion
processes that meet the necessary emission regulations.
NO is formed by either fixing molecular nitrogen contained in the
combustion air or by the oxidation of fuel-bound organic nitrogen. The
https://doi.org/10.1016/j.combustflame.2024.113330
Received 14 June 2023; Received in revised form 15 January 2024; Accepted 16 January 2024
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
underlying chemical kinetic processes controlling the formation and
depletion of NO have been a subject of research for half a century,
however, unresolved issues remain, and a comprehensive understand-
ing continues to be a challenge. Nevertheless, several sub-mechanisms
for the formation and destruction of NO have been identified, including,
but not limited to, the Thermal [1], the Prompt [2,3], N2O [4], the
NNH [5], the NO2[6], HNO [7], the Reburn [8,9], NH3[10] (with
Thermal DeNO𝑥[11]) and fuel-nitrogen [12]. The importance of each
mechanism depends on the fuel oxidation chemistry and the resulting
radical pool. A more detailed overview on the characteristics of the
mechanisms is beyond the scope of this work and a recent review by
Glarborg et al. [13] provides an extensive summary of the specifics of
these processes and their modelling.
Beyond accurately modelling NO chemistry, interpreting the results
in terms of the contributions of the various sub-mechanisms to NO
production in a flame is challenging because the mechanisms are often
concurrent and have shared intermediate species or reactions along
the pathways. In the past, efforts have been made to separate the
distinct contributions by grouping relevant reactions [14], by simu-
lating with only the chemical kinetics of the mechanism of interest
present [15], or by removing pathway initiation reactions from the
kinetic mechanism and extracting the contributions from the difference
to the full-chemistry solution [16–19]. These approaches have also been
applied in further works [20–23]. Another interesting approach to the
analysis of combustion kinetics was proposed by Laurent et al. [24].
The developed atom tracking algorithm has been used to predict the
fate of individual atoms of multi-component fuels during the oxidation.
However, each of the methods comes with merits and restrictions.
While simple grouping and summing reaction rates might be the sim-
plest and fastest method in a manageable kinetic mechanism, this
method is unable to separate contributions from sub-mechanisms that
are coupled by common reactions. For instance, when atomic nitrogen
reacts with molecular oxygen to form nitric oxide and atomic oxygen,
multiple sub-mechanisms (Thermal, Prompt, N2O) could be the source
of the reacting nitrogen. Attributing the reaction to only one sub-
mechanism would result in an incorrect allocation of contributions.
Whereas, the multiple simulation methods with the removal of sub-
sets from the kinetic mechanism removes the coupling problem, but
increases the simulation effort and might cause unforeseeable discrep-
ancies in the simulation due to the reduced mechanism. Additionally,
the interactions of these sub-mechanisms are then not captured.
In general, reaction path analyses are exemplified by reaction path
diagrams that are either schematic or quantitative, i.e., they show all
possible interspecies connections resulting from the reaction mecha-
nism, or they provide quantitative information as weighted interspecies
connections, most commonly as aggregated reaction fluxes over space
(integral reaction paths analysis). These diagrams are used not only to
understand the formation of pollutants such as NO𝑥but also to identify
relevant reaction pathways in the decomposition of fuels. Integral reac-
tion pathway diagrams are useful in observing changes in the relevant
reaction pathways for different boundary conditions, i.e., low/high
temperature decomposition pathways [25]. However, regional varia-
tions due to different flame/ fluid regimes are not captured because of
the aggregation. One solution is conditional reaction pathway analysis,
which integrates over defined sub-regimes, e.g. conditioned by stoi-
chiometry in diffusion flames [26]. But the success of this approach
depends on the selection and definition of the sub-regimes. A better
solution would be to resolve and monitor the contributions of the
reaction pathways over the entire regime.
The objective of this work is to introduce a novel algorithm for
reaction path analysis, that overcomes the limitations of previous ap-
proaches and allows for a reliable and comprehensible interpretation
of the contributions to NO production from different formation mech-
anisms in total and, more importantly, throughout the flame. The
Sub-Mechanism Accounting and Reaction Tracing (SMART) algorithm
is a post-processing tool for reacting path analysis. As opposed to other
methods, it allocates molar fluxes to sub-mechanisms without manip-
ulating the included reactions in the simulation or tying intermediate
reactions in a reaction path to a single sub-mechanism. The algorithm
is mathematically based on element flux graphs, which have already
proven successful in mechanism reduction [27]. A prototype version
of the algorithm was proposed by Bohon et al. and applied to account
for the contributions to NO production from several sub-mechanisms in
alcohol and alkane flames [28,29]. Since then the algorithm has been
further developed and tested on various flame applications.
The goal of this work is to demonstrate the capabilities of the
SMART algorithm as a new analysis method and use it to identify
new results from several case studies from the literature. For this, we
replicated selected flame simulations from three former studies of NO
formation and compare the recreated results to the outcomes of our
analysis. The investigated case studies involve NO formation analysis:
•in premixed methane-air counterflow flames with equivalence
ratio variation from 0.5 to 1.4 by Cho and Chung [14],
•in ultra-lean premixed methane-air counterflow flames with hy-
drogen addition up to a volume fraction of 0.8 by Guo et al. [16],
•and in partially premixed methane-air counterflow flames under
elevated pressure up to 15 atm by Naik and Laurendeau [19].
In each of these studies, the authors present a different approach for
analysing the contributions of the sub-mechanisms to NO formation
and demonstrate the viability of their method, including advantages
and limitations. The application of SMART analysis to the replicated
flames from these studies is not intended to contradict these works
but to extend the previous results, extract new understanding and
highlight the additional capabilities of this new approach. Moreover,
the diversity of the selected case studies, which investigate different
effects on NO formation in various flame configurations, demonstrates
the wide applicability of the presented algorithm.
2. Methodology
The SMART algorithm uses an element graph approach to trace mo-
lar fluxes. The graph proceeds from initiation reactions to the formation
of the target species and accounts for the contributions of the distinct
sub-mechanisms in this process. It is not the goal to trace the fate of
individual atoms, but to trace impact of fluxes specific to individual
sub-mechanisms in an element flux graph. The graph nodes are species
containing the tracing element, e.g., 𝑁for NO formation analysis. The
edges are element fluxes between species and represent the rate at
which the atoms of the tracing element are transferred between these
species due to all possible reactions. For the corresponding adjacency
matrix of the directed graph, it is defined that the fluxes flow from
columns to rows. The reaction path analysis (RPA) of the SMART
algorithm is an iterative procedure that allocates these global element
fluxes to sub-mechanisms at each flame position until convergence is
reached. Fig. 1 demonstrates the general idea of the SMART algorithm
for a single intermediate species 𝑆𝑖at a fixed position in the flame 𝑧.
The subscript 𝑧indicating flame position, however, is excluded from
all local variable descriptions to enhance clarity. On the left, the figure
shows a section of a global graph centred around the species 𝑆𝑖. The
global inflow fluxes to species 𝑆𝑖from the species 𝑆𝑗are notated 𝑔𝑖,𝑗
and the outflow fluxes from species 𝑆𝑖to species 𝑆𝑗𝑔𝑗,𝑖, respectively.
For the accounting of fluxes, the SMART algorithm compares the total
formation to the total depletion of species 𝑆𝑖due to all species 𝑆𝑗. The
difference (when ∑𝑗𝑔𝑖,𝑗 ≠∑𝑗𝑔𝑗,𝑖) is captured as the source flux 𝑟𝑖.
If inflow surpasses outflow, the species 𝑆𝑖is produced. Conversely, if
outflow exceeds inflow, the species is depleted. In order to keep track
of the record of production and consumption of a species throughout
the flame, the pool 𝑝𝑖is introduced. The pool accounts for all positive
and negative contributions from the source for each step of time 𝑡𝑧in
the flame simulation, according to:
𝑝𝑖,𝑧+1 =𝑝𝑖,𝑧 + (𝑡𝑧+1 −𝑡𝑧)𝑟𝑖,𝑧 (1)
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
Fig. 1. Representation of the division on the global molecular fluxes (left) into their corresponding sub-graphs (right), where each sub-graph represents a single sub-mechanism,
i.e. sub-mechanisms 𝑀𝑘=1 and 𝑀𝑘=2 in the top-right and bottom-right, respectively.
Fig. 2. Flux cases in the allocation process dependent on the total inflow ∑𝑗𝑔𝑖,𝑗 and total outflow ∑𝑗𝑔𝑗,𝑖, the resulting direction of the source 𝑟𝑖, and accumulation or overdrawing
of species A in the pool 𝑝𝑖. The value of the case factor 𝛼is conditioned on the ratio between total inflow and outflow and assumes either 0 or 1 for the determination of the
allocation factor.
Thus, at each position in the flame, the pool is an indicator of whether a
species was overall more produced or consumed. A positive pool means
that the species has been predominantly produced, and a negative
pool means that the species has been predominantly consumed. It is
important to recognize that the pool is not the local concentration, but
rather an accounting record of the differences between the inflows and
outflows. The difference between total inflow and total outflow and the
status of the pool give two accounting indicators from which four flux
cases can be derived. Fig. 2 shows the considered flux cases. The flux
cases determine the allocation process.
The primary objective of the SMART algorithm is to distribute
global fluxes among the sub-mechanisms. To accomplish this, the al-
gorithm creates a sub-graph for each sub-mechanism 𝑀𝑘. In each
sub-graph, the edges are the fraction of each global flux 𝑔𝑖,𝑗 (or 𝑔𝑗,𝑖)
that can be assigned to a sub-mechanism 𝑀𝑘. These allocated fluxes
are denoted by 𝑓𝑖,𝑗 (𝑓𝑗,𝑖). The source flux 𝑟𝑖and the pool 𝑝𝑖are also
fractionally attributed to the sub-mechanisms. The allocated sources
and the allocated pools are then denoted by 𝑢𝑖,𝑘 and 𝑞𝑖,𝑘, respectively.
Fig. 1 shows the breakdown of the global fluxes, the source, and pool
allocated to the sub-graphs of sub-mechanisms 𝑀1and 𝑀2.
In the allocation process, the outflow 𝑔𝑗,𝑖 is always allocated based
on the distribution of the inflow. Therefore, the allocation factor 𝜂𝑖,𝑘
is the ratio between the inflow from sub-mechanism 𝑀𝑘and the total
inflow.
𝑓𝑗,𝑖,𝑘 =𝜂𝑖,𝑘𝑔𝑗,𝑖 (2)
where
𝜂𝑖,𝑘 =∑𝑗𝑓𝑖,𝑗,𝑘 +𝛼𝑢𝑖,𝑘
∑𝑗𝑔𝑖,𝑗 +𝛼𝑟𝑖
(3)
The composition of the inflow into species 𝑆𝑖is dependent on the flux
case. It may either proceed exclusively from species 𝑆𝑗(flux case 1 and
2) or result from both species 𝑆𝑗and the source (flux case 3 and 4).
For flux cases 1 and 2, the total inflow from all species 𝑆𝑗exceeds the
total outflow to all species 𝑆𝑗. Accordingly, the source represents an
additional outflow towards the pool. Setting the case factor 𝛼to zero
eliminates the source component from the allocation factor equation
(Eq. (3)). In contrast, if the outflow from species 𝑆𝑗exceeds the inflow
(flux cases 3 and 4), the source’s direction changes and it becomes an
additional inflow to species 𝑆𝑖. The case factor 𝛼is equal to 1 for these
cases.
If 𝛼= 1, the allocated source 𝑢𝑖,𝑘 is required for the calculation of the
allocation factor. The allocation of the source is dependent on the pool’s
status. Thus, it is necessary to make a further distinction between flux
case 3 (𝑝𝑖>0) and flux case 4 (𝑝𝑖<0). For flux case 3, the pool of 𝑝𝑖is
positive, indicating that there has been an overproduction of species 𝑆𝑖
and the ledger is positive. For the distribution of the source that is now
drawing from the pool, it is assumed that the sub-mechanisms deplete
the pool in the same proportion as they previously added to the pool.
Consequently, only sub-mechanisms with a positive pool (𝑞𝑖,𝑘 >0) are
considered in the allocation.
𝑢𝑖,𝑘 =max(0, 𝑞𝑖,𝑘)
∑𝑘max(0, 𝑞𝑖,𝑘)𝑟𝑖(case 3) (4)
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
The allocated pool 𝑞𝑖,𝑘 is determined in the same manner as the global
pool 𝑝𝑖but with the allocated sources 𝑢𝑖,𝑘,𝑧 instead of the global sources
𝑟𝑖.
𝑞𝑖,𝑘,𝑧+1 =𝑞𝑖,𝑘,𝑧 + (𝑡𝑧+1 −𝑡𝑧)𝑢𝑖,𝑘,𝑧 (5)
The overall iterative process for the determination of the allocated
sources and pools is explained in more detail in the supplementary
material.
For flux case 4, when the overall consumption of species 𝑆𝑖has
surpassed its production (𝑝𝑖<0), indicating that the species is produced
further downstream but diffuses forward, the source 𝑟𝑖is allocated
using a flux correction factor.
𝑢𝑖,𝑘 =𝜈𝑖,𝑘𝑟𝑖(case 4) (6)
𝜈𝑖,𝑘 =∫𝑡𝑒𝑛𝑑
𝑡=0 ∑𝑗𝑓𝑖,𝑗,𝑘𝑑𝑡
∫𝑡𝑒𝑛𝑑
𝑡=0 ∑𝑗𝑔𝑖,𝑗 𝑑𝑡
.(7)
The flux correction factor 𝜈𝑖,𝑘 is the relative contribution of the sub-
mechanism 𝑘to the total formation of the species over the whole flame.
This yields the assumption that the species that is consumed at the
local flame position is produced by the sub-mechanisms at another
stage in the flame and with the distribution of their overall share in
the production of the species throughout the entire flame. This pool
term then serves to account for the transport between neighbouring
points in the flame. While it is a reasonable conjecture that the sub-
mechanisms producing the species along the span of the flame are
involved to that extent in the local consumption of the species, this
does neglects the proximity of the sub-mechanism’s production rate to
the local flame position and the concentration gradient of the species.
However, this assumption is only necessary for source fluxes in flux
case 4, which account for a very small fraction of the total flux for the
flames analysed in this work. Additionally, investigations with more
complicated methods to determine the flux correction factor resulted
in negligible differences to this method at much greater computational
cost.
For this description of the allocation process, it was assumed that
none of the global fluxes include initiation fluxes, which is true for the
majority of the fluxes. Initiation fluxes are by definition allocated to
their corresponding sub-mechanism. A more comprehensive explana-
tion of the allocation including the initiating fluxes is provided in the
supplementary material, supported by a pseudocode of the procedure.
The supplementary material contains the breakdown of flux cases
across all the examined flames and a model distribution for a single
flame.
The accounting is an iterative process, where the allocation of out-
flow is dependent on the allocation of the inflow. This way the allocated
fluxes are traced through the entire network, starting from the individ-
ual initiation reactions of the sub-mechanisms. Sub-mechanisms that
lead to the formation of a target species are initiated by reactions that
unfix the tracing element from the initiation species, whereas depletion
routes are initiated by reactions that decompose the target species. For
example, for the investigation of NO in a flame, the SMART algorithm
accounts for the contributions of the Thermal, Prompt and/or further
arbitrary involved sub-mechanism(s) by defining the corresponding
initiation reactions that break the triple bond of N2. Additionally, the
algorithm accounts for sub-mechanisms that are defined as NO deple-
tion routes, like the Reburn chemistry, where the initiation begins with
the decomposition of NO. All initiation reactions that are not allocated
to a sub-mechanism, are captured in undefined sub-mechanisms for
formation and depletion routes.
To visualize the characteristics of the SMART algorithm, Fig. 3
shows sample solutions of the algorithm for a simple example of a
constant pressure 0D reactor at atmospheric pressure and an initial
mixture of methane and air at 1500 K under stoichiometric conditions.
The analysis in Fig. 3(a) shows the prompt mechanism’s typical steep
progression in the reaction zone triggered by the rapid increase in CH-
radicals concentration. The thermal mechanism takes over shortly after
and becomes the primary NO producer in the post-flame zone due to
high temperature. The NNH mechanism, like the thermal mechanism,
increases with the rising amount of O and OH radicals. Nonetheless,
it produces NO at a relatively lower rate compared to the thermal
mechanism. Fig. 3(b) displays the outcome of the analysis after one
iteration. The solution has not yet reached convergence, and the flux
correction factor for the distribution of the source is currently based
on the initial guess, which consists of an equal distribution over all
sub-mechanisms. The findings indicate that the flux correction factor
has a negligible impact on this configuration. Additionally, the sum of
all sub-mechanisms (black, dashed line labelled Total) differs slightly
from the simulated values shown in the circle markers from the original
simulation. In Fig. 3(c), the SMART analysis focuses solely on the Ther-
mal mechanism. The findings highlight that the solution is independent
of the number of sub-mechanisms examined. The SMART algorithm
reaches the same conclusion for the distribution for the Thermal mech-
anisms. All other production rates that result in NO are stored in the N2
undefined mechanism, while those that deplete NO are attributed to the
NO undefined mechanism. Residuals for each iteration are exhibited in
Fig. 3, with a set tolerance of 10−5. For these analyses, convergence
is reached after 4 and 5 iterations, respectively. This indicates that
there is only a negligible change in all assigned quantities throughout
the entirety of the flame between two iterations. This convergence
behaviour is representative of the flames analysed in this work. For
the majority of the SMART analyses, convergence was reached after
the fifth iteration. The iterative process is comprehensively explained
in the supplementary material.
While this work focuses on demonstrating the algorithm for NO𝑥
formation through several case studies, it is intended to be applicable
in other areas of flame chemistry as well. For example the oxidation
of carbon originating through the various branching pathways of fuel
decomposition could be traced using the branching steps as initiating
reactions and observing when these sub-mechanisms oxidize to CO2.
3. Results
In the following section, we will demonstrate the results of the
SMART algorithm by applying it to several case studies extracted from
the existing literature. The objective here is to highlight the ability of
this approach to provide additional insight into the flame chemistry
that would not otherwise be accessible using traditional approaches.
In these case studies, the authors have generally acknowledged the
limitations or shortcomings of their methods for distinguishing the
contribution of the various NO𝑥sub-mechanisms and we endeavor
to show how the SMART algorithm can overcome these issues. The
case studies investigate NO formation in different counterflow flames,
which we then replicate and compare the reported results with those
of the SMART algorithm. Numerical computations for all replicated
counterflow flames were conducted using Cantera [30]. Any differences
to the original simulations concerning the solver are pointed out in
the respective sections. All authors used the GRI 3.0 [31] as a kinetic
mechanism for the simulation, therefore the same mechanism was
chosen for this study. In the following, the reactions are referred to by
their numbers, and most of the corresponding reaction equations can be
found in the tables in the supplementary material. The complete listing
is available under Ref. [31].
3.1. Case study 1: Equivalence ratio effects on NO formation in methane-air
counterflow premixed flames
In this work, the authors Cho and Chung [14] present a numerical
evaluation of NO emission characteristics for methane-air premixed
counterflow flames accounting for the effects of changes in inlet tem-
perature, pressure, stretch rate, and equivalence ratio. For this analysis,
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
Fig. 3. SMART analysis of a constant pressure reactor (1 atm) showing (a) a converged solution with a full set of sub-mechanism, (b) the solution after a single iteration, and
(c) a converged solution when only a single sub-mechanism is identified. The overall convergence for (a) and (c) is shown in (d). The reactor was initialized with stoichiometric
methane/air mixture at 1500 K.
both nozzles have the same composition of air and fuel, and the
distance between the nozzles is 4 cm. The inlet temperature is 800 K,
and the pressure is 1 atm. The nominal stretch rate is set to 1000 s−1.
In order to determine the contributions of different sub-mechanisms
(Thermal, N2O, NO2, and Prompt) to the formation of NO, Cho and
Chung abstract the NO production rates directly from the results of the
full-mechanism flame simulation. The Thermal mechanism is defined
as the NO production obtained through the sum of three reactions (-
R178, R179, and R180). The same process is applied for the N2O (R182,
R199, and -R228) and the NO2(-R186, -R187, R188, R189, and R281)
mechanisms. The Prompt mechanism is then defined as the remaining
reactions, i.e. the difference between the summation of the other three
mechanisms from the total NO formation. The corresponding reactions
are listed in the supplementary material. This approach uses a single
numerical simulation containing the full chemistry set, but is unable to
deconflate the coupled reaction sets such as the Thermal and Prompt
sub-mechanisms.
Cho and Chung also provide an extended analysis of the Prompt and
the Thermal mechanism under equivalence ratio variation to demon-
strate the improvements in this approach compared to a previous RPA
technique proposed by Nishioka et al. [15]. In this previous work, the
contributions of the Thermal and Prompt sub-mechanisms were distin-
guished by using two simulations, one containing only the reactions
defined in the Thermal sub-mechanism and another containing the full
chemistry set. Since the differences between those approaches and the
impact on the results are already discussed in detail by Cho und Chung,
this paper focuses solely on a comparison with the further developed
method proposed by Cho and Chung. Additionally, we will discuss these
types of subtractive, multi-simulation approaches in more detail in Case
Studies 2 and 3.
Fig. 4 shows a replication of Cho and Chung’s results for the NO
emission index as a function of the variation in equivalence ratio.
Figs. 4a and b show the variation in contributions from the Thermal and
Prompt sub-mechanisms, while Fig. 4c and d show the contributions
from the N2O and NO2sub-mechanisms defined by Cho and Chung
along with the additional sub-mechanisms defined in the SMART algo-
rithm. The defining initiation reactions for all sub-mechanisms in the
SMART analysis are listed in the supplementary material. The results
show that the approach from Cho and Chung does capture some of
the coarse qualitative trends such as the maximum in Thermal EINO at
stoichiometric conditions, the reduction in Thermal and Prompt EINO
when shifting towards lower equivalence ratios, and the relative magni-
tude of the sub-mechanisms under lean and stoichiometric conditions.
However, differences appear in the quantitative contributions of the
different sub-mechanisms, the relative importance of the Thermal and
Prompt sub-mechanisms under rich conditions, and the exclusion of
NNH, HNO and Reburn sub-mechanisms which show comparable and
often more significant contributions.
While the analyses for the Thermal mechanism show similar be-
haviour for lean and stoichiometric flames, the SMART mechanism
shows a much more rapidly declining relative impact for rich flames
compared to Cho and Chung. This is driven principally by higher
𝑁atom rates of production through reactions R240 and R190 and
reduction in 𝑁atom production through -R178. In the method of Cho
and Chung, the production of NO through R179 and R180 by reaction
with this 𝑁atom is then over-attributed to the Thermal mechanism
despite the fact that the nitrogen was originally fixed in the Prompt
mechanism. As a consequence the relative importance of the Thermal
mechanism is overstated. However, as the Prompt mechanism depends
on the concentration of CH-radicals in the flame zone, an increase in
importance of the Prompt mechanism for richer flames is expected.
In order to support this assumption, Fig. 5 shows the results of the
SMART analysis over the flame for the leanest and richest conditions.
In the lean case, Prompt NO is produced in a narrow region early in
the flame, while the Thermal mechanism gradually increases and then
broadly operates downstream the Prompt region. For the rich flame,
almost the entire NO production is due to the Prompt route. NO is
principally produced by the reactions R179 and R180 and therefore
depends on the availability of atomic nitrogen in the flame. Cho and
Chung allocate both reactions to the Thermal mechanism. However, in
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
Fig. 4. NO Emission index over equivalence ratios calculated by the Cho and Chung Method and the SMART algorithm for the major mechanisms: Thermal and Prompt (a) and
the minor mechanisms (b), also expressed as the percentage of the total EINO in c and d, respectively. The inlet temperature is 300 K, the pressure 1 atm and the strain rate
1000 s−1.
Fig. 5. SMART analysis results showing the NO production rates for all sub-mechanisms over the flame for a lean (a) and a rich (b) premixed methane-air counterflow flame with
an equivalence ratio of 0.5 and 1.4, respectively. The inlet temperature is 800 K, the pressure 1 atm and the strain rate 1000 s−1.
rich flames, nitrogen is also produced by R240, whereas the thermal
nitrogen formation through R178 regresses. The SMART algorithm
traces the nitrogen flow starting from initiation reactions and avoids
fixing NO production rates from R179 and R180 to a single sub-
mechanism. The NO formation fluxes are allocated according to the
origin of the consumed nitrogen with help of the allocation factor
(Eq. (3)). Meaning all atomic nitrogen that was produced by Prompt
initiation R240 and then converted to NO is accounted for as Prompt
NO. The SMART results also reveal that the NO production via the
NNH route is a relevant contributor, especially in lean flames. For
𝜙= 0.5the NNH route is responsible for 40 percent of the NO emission.
Neglecting the NNH formation and defining Prompt NO as a remainder,
can lead to an overestimation of the latter if the former becomes a
relevant contributor. Therefore, we designed the SMART algorithm
to catch all NO fluxes that originate from an unspecified route in
the undefined mechanisms, giving the user a chance to identify and
possibly reconsider neglected formation routes.
3.2. Case study 2: Effects of hydrogen addition on NO formation in ultra-
lean methane-air counterflow premixed flames
The authors, Guo et al. [16], performed numerical simulations to
investigate the effect of hydrogen addition on the extinction limits
and NO𝑥emission characteristics in ultra-lean premixed methane/air
counterflow flames. The study showed that hydrogen enrichment tech-
nology allows stable combustion under ultra-lean conditions, resulting
in significant CO2and NO emission reduction. Analysing NO formation,
the authors conclude that hydrogen addition causes an increased NO
emission if the equivalence ratio remains constant. The elevated NO
production is due to an enhancement in the rate of NNH and N2O
intermediate NO formation routes.
The flame configuration in this study is an axisymmetric laminar
counterflow premixed flame. Guo et al. utilized a revised version of the
code from Kee et al. [32] and an optically thin radiation model [33].
Since the replication was conducted with Cantera, the corresponding
default radiation model by Liu and Rogg [34] was utilized for the
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H. Helbig and M.D. Bohon
simulation. While this may introduce some small variation between the
models, the observed net effect on NO production showed negligible
differences. The inlet temperature and the pressure are 300 K and
1 atm, respectively. A global stretch rate of 30 s−1was used. The
fuel is a variable mixture of methane and hydrogen with the volume
fraction of hydrogen varying between 0 and 0.8. For this case study,
the equivalence ratio of 0.47 was chosen, which is the leanest flame
investigated by Guo et al. in the original work.
Besides NNH and N2O, Guo et al. investigate the contributions of
the less relevant NO formation mechanisms, which in this flame con-
figuration include the Thermal and Prompt sub-mechanisms. In order
to distinguish between the individual contributions of the mechanisms
to the total NO formation, the proposed analysis method needs four
repeated flame simulations, successively removing sets of initiation
reactions from the base kinetic NO formation mechanism. The chosen
initiation reactions for the procedure are listed in the supplementary
material.
The first simulation (SIM1) is the full-kinetic mechanism simulation,
here GRI 3.0. For the second simulation (SIM2), the initiation reactions
of the Prompt routes were removed, followed by the third simulation
(SIM3), where both Prompt and NNH initiations were absent. The last
simulation removed the initiations reactions of the prompt, the NNH
and the N2O mechanisms.
Consequently, the NO production rate from SIM4 is considered to be
the Thermal contribution. The Prompt NO is determined by subtracting
the NO formation of SIM2 from SIM1. The NNH intermediate route
translates to SIM1 minus SIM2. Thus, the difference between SIM2 and
SIM3 is attributed to the N2O mechanism.
Fig. 6 shows the replicated integral results of the NO formation
analysis utilizing the method proposed by Guo et al. and the SMART
algorithm for the NNH and the N2O routes on the left (Fig. 6a) and
the minor mechanisms on the right (Fig. 6b). Guo et al. defined the
emission index of NO based on the consumption of O2rather than fuel
consumption (as is more typically done), since hydrogen is not only
an initial fuel but also an intermediate species in this investigation. At
first glance, the two approaches reach similar overall conclusions for
the contributions of NO emissions. N2O and NNH are identified as the
most significant routes for NO production, with the rising importance
of NNH as the H2volume fraction increases. Prompt and Thermal
are less relevant, with the Thermal EINO almost zero. However, the
SMART analysis incorporates seven sub-mechanisms and only requires
a single flame simulation. While HNO, Reburn and the undefined sub-
mechanisms are negligible in this case, the results identify an NO
depletion by the NO2mechanism for the pure methane/air flame,
which slowly diminishes as the H2volume fraction increases. The
differences in the two approaches and the significance of the NO2
mechanism becomes even more evident when observing the NO rate
of production over the simulated flame, as depicted in Fig. 7. The zero
point was set to the position of the peak production rate of NO. The
nozzle is on the left, and stagnation point is on the right. The SMART
analysis shows that before the major NO production through the N2O
and NNH routes begins, forward-diffused NO is first converted to NO2
and then back to NO. However, in flames with a smaller H2volume
fraction, less NO2is converted back to NO, resulting in an overall
depletion through the NO2route as was shown in 6b. However, when
the rates of NO production are extracted using the method of Guo
et al. the results in Fig. 7a suggest that the conversion of NO to NO2
and back is caused by all analysed mechanisms. This is also visibly
reflected in the emission index in 6, especially for small H2volume
fractions, where N2O, NNH and Prompt show slightly lower values than
the SMART results. Since Guo et al. did not intend to analyse the local
contributions distributed over the flame, extracting this information
from the simulation data comes with limitations, however it does
serve to demonstrate the method by which their approach determines
the individual sub-mechanism contributions. This effect is even more
clearly demonstrated in Fig. 8 which shows the NO production rates
for a configuration with an H2volume fraction of 0.8. The important
take-away from this demonstration, is that while the integral values
capture the general trend in global NO𝑥formation shown in Fig. 6,
the means by which the kinetic mechanism reaches these conclusions
is not representative of the true flux of fixed nitrogen within the
flame. Consequently, the approach of Guo et al. predicts the NO𝑥
formation to be nearly proportionally distributed across the associated
sub-mechanisms throughout the flame as in Fig. 7a or in a wildly
oscillatory distribution as in Fig. 8a. Additionally, the temperature
distributions for each individual simulation is slightly affected by the
removal of the initiation reaction resulting in a shift in the temperature
field. It is likely this temperature shift that causes the strong oscillations
in the sub-mechanisms due to misalignment of the simulations. Because
the SMART algorithm is a post-processing script that operates on a
single simulation, interference with the underlying kinetics is avoided.
For comparison, by tracing the actual flux of fixed nitrogen, the
SMART algorithm is able to determine the distribution of NO formation
within the flame and capture the local features of the NO𝑥chemistry.
Here it can be seen that the earlier, low temperature portion of the
flame is characterized by an initial conversion of forward diffused
NO to NO2and then a reversion to NO. In Fig. 7b, the principal
formation of NO is through the NNH, N2O, and Prompt mechanisms,
with a small amount of Reburn and negligible Thermal NO (due to the
low temperature). These formation mechanisms reasonably correspond
to the variation in the total rate of NO formation predicted by the
simulation. The NO formation in Fig. 8b is similar, however the NNH
comprises a much greater proportion of the NO formation due to the
higher hydrogen content.
3.3. Case study 3: Pressure effects on NO formation in methane-air coun-
terflow partially premixed flames
The authors Naik and Laurendeau [19] conducted quantitative, spa-
tially resolved linear laser-induced fluorescence (LIF) measurements of
nitric oxide concentration in laminar methane/air counterflow partially
premixed and non-premixed flames for elevated pressures. And more
relevant for this work, they also executed a numerical simulation with
a pathway analysis that provides relative contributions of different NO
formation mechanisms at various pressures. The analysis showed that
NO formation is dominated by the Prompt at lower pressure, whereas,
Thermal and N2O pathways become more relevant at higher pressures.
In this case study, the replicated flame configuration is a methane/
air counterflow partially premixed flame with an equivalence ratio
of 1.6 at the left inlet. The inlet temperature is set to 300 K and
the strain rate to 20 s−1. The pressure increases from one up to 15
atmospheres. Naik and Laurendeau conducted the numerical simula-
tions using OPPDIF, an opposed flame code from Sandia [35]. The
effect of gas-phase radiation was accounted for by adding a radiation
source term via a subroutine provided by Gore et al. [36]. Again, small
variations between the replicated and the original simulation might
be introduced here, as the simulation heat loss is determined by the
default Cantera function. However, the observed differences between
the peak temperatures and peak NO concentrations reported in the
Naik and Laurendeau study and the replicated results were found to
be negligible.
The presented method for calculating the contributions of the Ther-
mal, the N2O, the NNH, and the Prompt mechanism to the total NO
production is, like the Guo et al. approach, a multi-simulation sub-
traction method. However, Naik and Laurendeau do not progressively
remove mechanisms but only take out a single sub-mechanism in each
simulation, gaining independent results for each sub-mechanism and
avoiding adding any occurring discrepancies to the remaining sub-
mechanism. Hence, the contribution of each sub-mechanism is the
difference between the full-mechanisms simulation (Sim1) and the
respective incomplete mechanism simulation. The Thermal mechanism
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H. Helbig and M.D. Bohon
Fig. 6. Emission index for ultra-lean premixed counterflow flames over varying volume fractions of H2for major mechanism: NNH and N2O (a) and the minor mechanisms (b).
The equivalence ratio is 0.47, the strain rate 30 s−1, the fresh mixture temperature and pressure are 300 K and 1 atm, respectively.
Fig. 7. NO rate of production for an ultra-lean premixed counterflow flame calculated with the Guo et al. method (a) and with the SMART algorithm (b), also showing the
temperature curve for each flame simulation. The equivalence ratio in this setup is 0.47 and the H2volume fraction 0.4. The fresh mixture temperature, pressure and strain rate
are 300 K, 1 atm, and 30 s−1respectively.
Fig. 8. NO rate of production for an ultra-lean premixed counterflow flame calculated with the Guo et al. method (a) and with the SMART algorithm (b), also showing the
temperature curve for each flame simulation. The equivalence ratio in this setup is 0.47 and the H2volume fraction 0.8. The fresh mixture temperature, pressure and strain rate
are 300 K, 1 atm, and 30 s−1respectively.
was determined with a simulation without the Thermal initiation re-
action (R178) (Sim2). The N2O route was accounted for by removing
R181 to R185 from the mechanism (Sim3). For the fourth simulation
(Sim4), the NNH initiation reaction (R208) was omitted, and for the
determination of the Prompt mechanism, R240 was left out in the last
simulation (Sim5). A table with the relevant reaction equations for this
procedure is provided in the supplementary material.
In their work, the authors express the contributions of the different
sub-mechanisms as relative effects to the total peak concentration
of NO in the flames and support their analysis in more detail with
quantitative reaction path diagrams and sensitivity analysis. For this
work, we extract the emission index and the NO rates of production
throughout the flames from the simulation data. Because the local
analysis was not originally intended by Naik and Laurendeau, the
extraction is constrained, but it nevertheless demonstrates the method
used to determine the individual contributions of the sub-mechanisms.
The SMART analysis captures the same qualitative trend for the
contributions of the sub-mechanisms, confirming the dominance of the
Prompt route for atmospheric pressure and a rapid decrease as the
pressure rises, whereas the Thermal and N2O mechanisms gain impor-
tance with elevated pressure levels. These relative contributions of all
mechanisms to the emission index of NO are displayed in Fig. 9. The
Naik and Laurendeau analysis only captures the emission of the four
defined mechanisms (Thermal, NNH, N2O, and Prompt). For all flame
configurations, the remaining NO emission, that is not caused by one
of these four mechanisms, is below ten percent. However, the source
of the remainder is not conveyed and the authors could not identify
it. The SMART algorithm is designed to capture all fluxes leading to
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H. Helbig and M.D. Bohon
Fig. 9. Emission index of NO for partially premixed methane/air counterflow flames over rising pressure for major mechanism and the total of all mechanism (a) as well as the
minor mechanisms (b). The equivalence ratio is 1.6, the strain rate 20 s−1and the fresh mixture temperature is 300 K.
Fig. 10. NO rate of production for a partially premixed counterflow flame under atmospheric pressure calculated with the Naik and Laurendeau method (a) and with the SMART
algorithm (b), also showing the temperature curve for the full mechanism simulation. The equivalence ratio, the strain rate, and the fresh mixture temperature are 1.6, 20 s−1,
and 300 K, respectively.
NO formation or depletion so that the sum of all contributions equals
the total NO emission. The EINO from both undefined mechanisms
lie between one and three percent. Also, the contributions from the
HNO and the N2O mechanisms are within a negligible range. However,
the relative contributions from Reburn show a significant impact on
mitigating NO emission. In order to gain more comprehension on the
interaction between the individual sub-mechanisms, Figs. 10 and 11
show the NO rates of production over the flame for 1 atm and 6 atm,
respectively. The corresponding sub-figures show the results obtained
by the Naik and Laurendeau method (a) and by the SMART algorithm
(b). For the flame under atmospheric pressure, the NO production rates
bring out the effect of including the Reburn mechanism in the analysis.
In the Naik and Laurendeau results, the initial NO consumption and
the majority of the peak in NO formation around x=0 is attributed
to the Prompt formation. However, the SMART analysis reveals that
NO is consumed by the Reburn, leading to an overall depletion of NO
before peak production. Moreover, the SMART results show that the
peak production rate is not only due to the Prompt mechanism, but that
intermediate species formed through NO depletion react to reform NO.
The Reburn is primarily initiated by R249, R274, and R255. Due to the
spatial overlap with the Prompt pathway and the shared intermediate
species, e.g., HCN is a product of Reburn (R255) and Prompt initiation
(R240), both pathways are strongly coupled. The Naik and Laurendeau
results show that omitting the Reburn route in the analysis leads to
an overall underestimation of the Prompt NO emission by attributing
the depletion fraction to the Prompt pathway. This underestimation
also occurs despite the prediction of an overall higher peak rate of
NO formation by the authors and the omission of a significant amount
of NO production shifting to earlier in the flame. For the flame with
a pressure of 6 atm, the production rates in Fig. 11 show for both
RPA methods two peaks. This trend is already visible for the flame
with atmospheric pressure; however, with the rising importance of the
Thermal and the N2O route, it becomes more evident. The segregation
is linked to the availability of hydrocarbon or hydroxyl radicals and
atomic oxygen, in relation to the proximity to the fuel/air or air nozzle,
respectively. Thus, the right peak (closer to the air inlet) is caused by
Thermal, N2O, and NNH formation routes and the left peak (closer to
the fuel/air mixture side) is due to Prompt NO production. The SMART
analysis, however, reveals that the left peak is also reached through NO
production by the Reburn mechanism, converting intermediate species
back to NO. For the Naik and Laurendeau analysis, shares of the Reburn
route are allocated to the Thermal and the N2O mechanism.
Since the SMART algorithm is a graph-based approach, the adja-
cency matrices of the sub-graphs are transferable to a multi-directional
graph to further the comprehension of the pathways between the inter-
mediate species. This is similar to a quantitative reaction path diagram,
but now includes the distinction between the sub-mechanisms. Fig. 12
shows the fluxes of the Thermal, N2O, NNH, and NO2for pressures of
1 atm, 6 atm, and 15 atm as well as Prompt and Reburn routes for
the same pressures. The allocated molar fluxes between the species
are numerically integrated along the central axis of the flame and
normalized by the maximum value. Normalized fluxes below 0.01 are
omitted for reasons of clarity.
The results show the absolute and relative increase of the Thermal,
N2O, and NNH mechanisms with rising pressure. The NNH and N2O
mechanisms have individual formation routes directly from N2via
NNH or N2O to NO (depending on the mechanism), but, they also
share an NO formation route via the intermediate species of NH and
HNO, which for the N2O mechanism becomes more relevant for higher
pressures (6 atm and 15 atm). At atmospheric pressure, this route
for the N2O mechanism is below the threshold. For higher pressures,
the path analysis also reveals an interconnection of the N2O and the
Thermal route via atomic nitrogen, which is formed in the Thermal
mechanism directly by the initiation reaction. For the N2O mechanism,
however, 𝑁is formed from NH. For both pathways, the nitrogen is
then fixed in NO. The NO2mechanism appears for the flame at 15
Combustion and Flame 262 (2024) 113330
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H. Helbig and M.D. Bohon
Fig. 11. NO rate of production for a partially premixed counterflow flame with a pressure of 6 atm calculated with the Naik and Laurendeau method (a) and with the SMART
algorithm (b), also showing the temperature curve for the full mechanism simulation. The equivalence ratio, the strain rate, and the fresh mixture temperature are 1.6, 20 s−1,
and 300 K, respectively.
Fig. 12. Multi-directional-flux-graph obtained from the SMART analysis describing NO formation routes of selected sub-mechanism for partially premixed methane/air counterflow
flames for rising pressure. The formation routes for the Thermal, the N2O, the NNH and the NO2mechanism are shown for pressures of 1 atm (a), 6 atm (b), and 15 atm (c). The
pathways of the Prompt and Reburn mechanisms are depicted for the same pressures in d, e, and f, respectively.
atm pressure and is secluded from the other mechanisms as NO gets
converted to NO2and back.
In the bottom sub-figures d, e, and f of Fig. 12, the Prompt is
mapped together with the Reburn, since both mechanisms share a
pool of intermediates and their pathways overlap. However, as an
NO formation route, the Prompt begins with unfixing nitrogen from
N2and forming HCN and N. From the latter, the pathway directly
terminates with the production of NO. But, when tracking the paths
via HCN, the flux splits here into the production routes of NCO and
HOCN. From NCO, the paths NH and via HNO to NO, whereas the
route via HOCN leads first to HNCO, then forming NH2, and finally
joining in on the formation of NO via NH followed by HNO. With rising
pressure, the path via NCO becomes less relevant compared to the path
via HOCN. The Reburn mechanism is initiated when NO reacts with
carbon-containing intermediates. For all pressures, HCN, HCNO, and
HNCO are the products of the initiation. At atmospheric pressure, CN
is also produced at a small rate, while for higher pressures, H2CN is
additionally formed. After forming HCN and HNCO in the initiation
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H. Helbig and M.D. Bohon
process (or as a secondary step through HCNO) the Reburn primarily
follows the same reaction path to (re-)produce NO as the Prompt
mechanism. While this depletion of NO followed by the reconversion
back to NO was already detected in the production rates throughout the
flame in Figs. 10 and 11, the network depictions of the fluxes between
the species show that the overall decrease of NO emission, as seen in
the emission index in Fig. 9, is caused by fixation of nitrogen in the
pool of intermediate species of the Reburn mechanism. The nitrogen is
not re-bound in molecular nitrogen. The multi-graph again emphasizes
the connectedness of the individual reaction paths. The production of
all sub-mechanisms, except thermal, proceeds largely via NH-HNO-NO
or for all mechanisms except NNH via 𝑁directly to NO.
4. Conclusions
This work introduced the SMART algorithm as a new method for
reaction path analysis. The SMART algorithm applies a purely post-
processing approach, which creates a ledger of the flow of a target
element (in this case fixed nitrogen) through the chemical network.
In doing so, the contribution of different sub-mechanisms and/or the
influence of individual reactions is directly quantified without inter-
ference in the flame chemistry during simulation. This quantification
allows for both localized distributions indicating when specific sub-
mechanisms are most important as well as integrated values indicating
the contributions of sub-mechanisms across the flame.
The effectiveness of this approach was demonstrated in three case
studies of previous works investigating the pathways of NO formation,
each of which used a different analysis method. For all case studies, the
SMART algorithm accurately reproduced the total NO emissions from
the flames. Moreover, the algorithm traced and allocated the fluxes of
six defined NO formation mechanisms (Thermal, NNH, N2O, NO2, HNO,
and Reburn), along with two undefined mechanisms that catch any re-
maining NO production or consumption. The graph-based approach of
the SMART algorithm enabled monitoring of the individual production
rates throughout the flame and the display of allocated molar fluxes
as quantitative reaction path diagrams. These multifaceted results,
including overall NO emissions, NO production rates, and reaction path
diagrams, provided broad interpretive possibilities and additional in-
sight into the NO formation routes of the replicated flames. Overall, the
analysis with the SMART algorithm revealed several benefits compared
to the replicated methods in the three case studies.
In the first case study, the SMART algorithm successfully separated
the interconnected pathways, specifically the Thermal and Prompt.
These pathways were linked by shared intermediate species (N) and
reactions (R179, R180), which resulted in an underprediction of the
Prompt mechanism in the original results. However, the SMART reso-
lution of the production rates across the richest flame clearly shows
the significance of the Prompt route, consistent with the increase
in the reaction rates of the main initiation reaction and the greater
concentration of CH radicals.
In the second case study, the SMART algorithm was able to convey
transparent results for the individual production rates over the flame.
As a pure post-processing routine, the SMART algorithm avoids inter-
ference with the kinetic chemistry and reliably abstracts production
rates for the NO formation mechanisms throughout the flame, which
is essential for understanding and evaluating the relative contributions
of the analysed sub-mechanisms to the total NO emission.
In the third case study, the SMART algorithm visualizes the results
in a quantitative reaction path diagram separating the fluxes according
to the contributions of the sub-mechanisms, highlighting the intercon-
nection of the formation and destruction pathways. The tracing method
of the SMART algorithm allows for the separation of the pathways of
the Reburn and Prompt mechanisms, although these mechanism have
an almost identical intermediate species set.
We think that due to these advantages, this new method is a valu-
able contribution to reaction path analysis, enhancing the capabilities
of evaluation and comprehension over previous methods.
CRediT authorship contribution statement
Hannah Helbig: Performed research, Wrote code, Analyzed results,
Wrote the paper. Myles D. Bohon: Supervised research, Analyzed
results, Wrote the paper.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgements
This work was supported by the Heinrich Böll Foundation.
Appendix A. Supplementary data
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.combustflame.2024.113330.
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