# Comparative analysis of five $NH_{3}$/air oxidation mechanisms

Shahid Rabbani
Department of Mechanical Engineering
Khalifa University of Science and Technology
Abu Dhabi, UAE 127788
<EMAIL_ADDRESS>
& Dimitris M. Manias
Department of Electrical Engineering and Computer Science
Khalifa University of Science and Technology
Abu Dhabi, UAE 127788
<EMAIL_ADDRESS>
& Dimitrios C. Kyritsis
Research and Innovation Center on CO2 and H2
Department of Mechanical Engineering
Khalifa University of Science and Technology
Abu Dhabi 127788, United Arab Emirates
<EMAIL_ADDRESS>
& Dimitris A. Goussis
Research and Innovation Center on CO2 and H2
Department of Mechanical Engineering
Khalifa University of Science and Technology
Abu Dhabi 127788, United Arab Emirates
<EMAIL_ADDRESS>
Corresponding Author<EMAIL_ADDRESS>
###### Abstract

Five recently developed chemical kinetics mechanisms for ammonia oxidation are
analysed and compared, in the context of homogeneous adiabatic autoignition.
The analysis focuses on the ignition delay and is based on the explosive mode
that is shown to drive the process. Using algorithmic tools based on the
Computational Singular Perturbation algorithm, the reactions responsible for
the generation of the explosive mode are identified, along with the variables
(species mass fractions and temperature) that associate the most to this mode.
Comparison of these sets of reactions and variables, obtained for each
mechanism, allows to correlate the differences in the predictive outcomes from
the mechanisms with specific reactions. The major differences identified,
which lead to different ignition delay times, relate to (i) the relative
duration of chemical and thermal runaways (a sizeable chemical runaway
develops only in some mechanisms) and (ii) the dominant chemistry during the
chemical runaway (chemistry involving species with two nitrogen atoms is
active only in some mechanisms). The major similarities identified refer to
the thermal runaway and in particular to (i) the chemical activity, which is
supported mainly by $OH$-producing reactions and by reactions producing their
reactants and (ii) the thermal activity, which is dominated by strongly
exothermic $OH$-consuming reactions.

_Keywords_ Ammonia $\cdot$ Ignition Control $\cdot$ CSP Diagnostics

##  1 Introduction

A renewed interest in using ammonia in the power generation and mobility
sectors has recently emerged in terms of not only its utility as a hydrogen
carrier, but also as a direct fuel. Ammonia is a carbon-free energy carrier
that can provide high energy density and its production, transportation and
storage can be delivered through well-established infrastructure. The use of
ammonia as a sustainable fuel was discussed comprehensively by Kobayashi et
al. (Kobayashi et al., 2019; Valera-Medina et al., 2021). An economic
comparison of utilization of conventional fuels with the one of ammonia showed
that ammonia was the least expensive fuel in terms of $/GJ (Kobayashi et al.,
2019). Ammonia was also ranked third in terms of GJ/m3, which was in agreement
with the results of Zamfirescu et al. (Zamfirescu and Dincer, 2009) and Cesaro
et al. (Cesaro et al., 2021), who estimated the levelized cost of electricity
from green ammonia to be between $167 and $197 per MWH in 2040. Cardoso et al.
(Cardoso et al., 2021) proposed subsidizing ammonia as a fuel and Valera-
Medina et al. (Valera-Medina et al., 2021) suggested that the use of ammonia
as a fuel will expand as a result of the continuous decrease in cost of power
from renewable sources, something that is supported by the findings of Fasihi
et al. (Fasihi et al., 2021) for ammonia production from hybrid PV-wind power
plants.

Ammonia has been suggested as fuel for solid oxide fuel cells (Siddiqui and
Dincer, 2018; Lan and Tao, 2014), but what is of interest in this paper is its
use as a fuel for combustion (Frigo and Gentili, 2013; Grannell et al., 2008;
Reiter and Kong, 2011), which comes with serious technical challenges. These
relate to low flame speed (less than 7 cm/s for stoichiometric mixture at
standard atmospheric conditions (Frigo and Gentili, 2013)), low heating value
(LHV = 18.8 MJ/kg, as opposed to order of 45 MJ/kg for paraffins (Grannell et
al., 2008)), as well as slow kinetics, as evidenced by long ignition delays
and flame stability problems (Tian et al., 2009; Mathieu and Petersen, 2015;
Glarborg et al., 2018). As a result, currently the main interest in ammonia
relates to marine engines, which rotate at low speeds and utilize large
quantities of fuel (Hansson et al., 2020; Ammonia Energy Association, ;
A.P.Moller—Maersk Press Release, ).

Modelling ammonia oxidation kinetics has developed over the last two decades
and, for obvious reasons, has recently advanced in a substantial manner. Among
others, mechanisms have been proposed by by Tian et al. (Tian et al., 2009),
Mathieu et al. (Mathieu and Petersen, 2015), Glarborg et al. (Glarborg et al.,
2018), Shrestha et al. (Shrestha et al., 2018), Li et al. (Li et al., 2019),
Stagni et al. (Stagni et al., 2020), Zhang et al. (Zhang et al., 2021), Okafor
et al. (Okafor et al., 2019) and Otomo et al. (Otomo et al., 2018). The
relative lack of experimental data has so far limited the validity of the
proposed mechanisms to specific combustion configurations as well as ranges of
initial temperature, pressure, and equivalence ratio. For example, the
mechanism by Mathieu et al. (Mathieu and Petersen, 2015) has been validated in
experiments with $NH_{3}/O_{2}$ mixtures at high temperatures and pressures.
However, this mechanism failed to produce satisfactory results at intermediate
temperature and pressure conditions (Wang et al., 2021). Similarly, the
mechanism by Tian et al. (Tian et al., 2009) predicted laminar flame speeds
with acceptable accuracy, but it lacked accuracy in predicting ignition delay
times. The results of Glarborg et al. (Glarborg et al., 2018) were in good
agreement with ignition delay measurements in a variety of initial conditions;
however, they exceedingly overestimated reactivity in the $NH_{3}/H_{2}$
mixture (Wang et al., 2021). Similarly, the mechanism by Shrestha et al.
(Shrestha et al., 2018) performed well on shock-tube data, such as in (Mathieu
et al., 2013; Burke and Klippenstein, 2017; Mathieu and Petersen, 2015), but
the model underestimated ignition delay time by factor of 2.5 for ignition of
$NH_{3}/H_{2}$ mixtures (Dai et al., 2020).

It is exactly because of these discrepancies that some recent works have
attempted comparisons between the proposed mechanisms, almost exclusively in
the context of agreement with experimental data from ignition experiments in
shock tubes and rapid compression machines. Wang et al. (Wang et al., 2021)
tested the mechanisms of Otomo et al. (Otomo et al., 2018), Glarborg et al.
(Glarborg et al., 2018), Li et al. (Li et al., 2019), Okafor et al. (Okafor et
al., 2019) and Shrestha et al. (Shrestha et al., 2018) against the RCM data of
He et al. (He et al., 2019) and the shock-tube data of Shu et al. (Shu et al.,
2019) and chose to use Otomo’s mechanism for high-pressure autoignition
computations in diesel-combustion conditions (100-200 bar). However, the
comparison presented in the paper, could also justify the choice of other
mechanisms. Otomo’s mechanism was also supported by a recent comparison of no
less than 15 mechanisms that was performed in the context of $NH_{3}$
pyrolysis by Alturaifi et al. (Alturaifi et al., 2022). The authors compared
the predictions of each of the 15 mechanisms on the speciation of the
pyrolysis of 0.5%-molar $NH_{3}$ and of a mixture of 0.42%-molar $NH_{3}$ and
2%-molar $H_{2}$ in $Ar$ in a shock tube, pointed to Otomo’s predictions as
demonstrating best agreement with the experimental results, and proceeded to
present an improved model for the pyrolysis. Then, they incorporated the
improved pyrolysis model in all oxidation mechanisms they considered. This
modification reduced the initial discrepancy in predicted laminar flame speeds
from a factor of three to a factor of 1.5. The scatter of laminar flame speed
results by a factor of two among mechanism predictions and experimental data
was in agreement with the early findings of Xiao et al. (Xiao and Valera-
Medina, 2017). A similar methodology in terms of comparing with experimental
results was followed by da Rocha et al. (da Rocha et al., 2019) who compared
the predictions of ten mechanisms in terms of ignition delay times, flame
speeds, and $NOx$ generation. The flame-speed data were also compared with
experimental measurements. The authors described the state of the art very
accurately by stating that “ _currently available chemical kinetic mechanisms
predict rather scattered ignition delay times, laminar flame speeds, and $NOx$
concentrations in $NH_{3}$ flames, indicating that improvements in sub-
mechanisms of $NH_{3}$ and $NH_{3}/H_{2}$ oxidation are still needed_". The
conclusion is compounded by a recent quantitative assessment of the relative
accuracy with which eight $NH_{3}$ oxidation mechanisms could capture
experimental ignition-delay-time results (Kawka et al., 2020), which showed
that the accuracy of the computational prediction can exceed the standard
deviation of the experimental measurements by more than a factor of ten. This
was probably a conclusion more important than pointing to the mechanisms by
Glarborg et al. (Glarborg et al., 2018) and Shrestha et al. (Shrestha et al.,
2018) as “best” in terms of the particular comparison and, of course, it is
nothing new in combustion research. Rather, it is reminiscent of the decades
of gradual progress that it took in order to converge to widely acceptable
mechanisms for the (chemically much more complicated) oxidation of methane,
and indicative of how much is “ _still needed_ " (da Rocha et al., 2019).

Much as the comparison of the predictions of the proposed mechanisms with
experimental data is valuable, especially at what seems to be an early stage
in the construction of high-fidelity models for the oxidation of $NH_{3}$,
what seems to be missing from the published literature is a comparison of the
proposed mechanisms in terms of the chemical dynamics they imply. In this
paper, we aspire to cover this shortage by examining in detail the explosive
dynamics of the adiabatic, isochoric autoignition of $NH_{3}$/air mixtures as
this is described by the mechanisms of Glarborg et al. (Glarborg et al.,
2018), Shrestha et al. (Shrestha et al., 2018), Li et al. (Li et al., 2019),
Stagni et al. (Stagni et al., 2020), and Zhang et al. (Zhang et al., 2021).
The choice of the particular mechanisms was determined mainly by indications
of “good performance” in the comparisons cited above and much more by
practicalities in terms of availability of the mechanisms to the authors and
size of the analysis. Clearly, the methodology can be applied to other
mechanisms that have appeared or will appear in the future.

In order to analyze the chemical dynamics described by each of the mechanisms
considered here, we used the algorithmic tools of Computational Singular
Perturbation algorithm (Lam and Goussis, 1989, 1994; Hadjinicolaou and
Goussis, 1998). CSP allows for the identification of (i) the mode that drives
the process and (ii) the reactions and variables that relate the most to this
mode (Valorani et al., 2003; Goussis and Najm, 2006; Kazakov et al., 2006; Lu
et al., 2008; Prager et al., 2011; Diamantis et al., 2015a). The driving mode
can be of either explosive or dissipative character; i.e., the reactions
responsible for the emergence of this mode tend to lead the process away or
towards equilibrium (Lam and Goussis, 1989). CSP can handle both types of
modes and provide all relevant diagnostics. This work is following up on our
previous CSP analyses of $NH_{3}$ oxidation that were focused on the
determination of additives for ignition control and $NO$ reduction (Khalil et
al., 2019, 2021), as well as on highlighting differences with methane
oxidation (Manias et al., 2021).

In what follows, we start with a presentation of the physical problem of the
homogeneous, isochoric, adiabatic autoignition and we present ignition-delay
computations with the use of the five mechanisms under consideration for a
wide range of initial temperatures, pressures and equivalence ratios. Then, we
use CSP diagnostics in order to discuss the predictions of the employed
mechanisms in terms of the relative duration of chemical vs. thermal runaway
and we point to a conclusion that seems to be unanimous among the employed
mechanisms, despite their serious differences in terms of dynamic behavior;
i.e. that $NH_{3}$ autoignition has a relatively short chemical runaway and
the longest part of the ignition delay comprises a thermal runaway.
Additionally, we compare the mechanisms in terms of predictions for the
initiation of the autoignition, the chemical activity that supports the
relatively long thermal runaway, and the generation of heat through exothermic
reactions. In this manner, we aspire to demonstrate the chemical features that
each of the mechanisms highlights, which can guide model selection for
specific engineering applications. In order to help the reader focus, we
present the detailed CSP diagnostics for each mechanism in appendices and
reserve the main body of the paper for the results that relate to the
comparison of the ways in which the five mechanisms under consideration
capture the underlying chemical fundamentals.

##  2 Chemical Kinetics Mechanisms

The chemical kinetics mechanisms considered are those developed by Glarborg et
al. (Glarborg et al., 2018), Shrestha et al. (Shrestha et al., 2018), Li et
al. (Li et al., 2019), Stagni et al. (Stagni et al., 2020) and Zhang et al.
(Zhang et al., 2021):

  1. 1.

Glarborg$\\_$2018 (Glarborg et al., 2018): was compiled by extending previous
studies of this group (Dagaut et al., 2008; Mendiara and Glarborg, 2009a, b;
Lopez et al., 2009) and using experimental data. The mechanism was validated
for initial temperatures and pressures and equivalence rations T0 = 285 - 2500
K, p0 = 0.045 - 1.38 atm, $\phi$ = 0.002 - 2.0.

  2. 2.

Shrestha$\\_$2018 (Shrestha et al., 2018): was developed on the basis of
previous ones (Skreiberg et al., 2004; Baulch et al., 2005; Rasmussen et al.,
2008; Mendiara and Glarborg, 2009a; Hong et al., 2010) and of experimental
data. The validity range of this mechanism is T0 = 298 - 2455 K, p0 = 0.045 -
30 atm, $\phi$ = 0.12 - 2.0.

  3. 3.

Li$\\_$2019 (Li et al., 2019): was developed by assembling three different
mechanisms (ara, ; Shrestha et al., 2018; Tian et al., 2009) and then using
Directed Relation Graph with Error Propagation (DRGEP) for its simplification.
This mechanism is valid in the range T0 = 1000 - 2000 K, p0 = 0.98 - 49.34
atm, $\phi$ = 0.5 - 2.0.

  4. 4.

Stagni$\\_$2020 (Stagni et al., 2020): was generated on the basis of
experimental data acquired by the authors and previous theoretical work done
by the group (Faravelli et al., 2003; Ranzi et al., 2012; Metcalfe et al.,
2013; Song et al., 2019; Chen et al., 2019). The range of validity of the
mechanism is T0 = 500 - 2000 K, p0 = 0.8 - 98.7 atm, $\phi$ = 0.5 - 2.0.

  5. 5.

Zhang$\\_$2021 (Zhang et al., 2021): was compiled on the basis of new
experiments and previously constructed mechanisms (Burke and Klippenstein,
2017; Hashemi et al., 2015; Mei et al., 2019). It was validated in the range
T0 = 800 - 1280 K, p0 = 1 - 10 atm, $\phi$ = 0.25 - 1.0.

The number of species and reactions related to nitrogen chemistry in these
chemical kinetics mechanisms are listed in Table 1. In addition, indicative
values are listed of the ignition delay time $t_{ign}$ and of the minimum
value of the time scale that relates to the explosive mode $\tau_{e,min}$,
computed at T0=1100 K, p0=2 atm and $\phi$=1.0. This set of parameters was
selected based on the following considerations. First, this set is within the
range of validity of all mechanisms. Second, the selected values of T0 and
$\phi$ are relevant to the operation of gas turbines and reciprocating
internal combustion engines. Finally, the selected value of p0 is relatively
low, because in Ref. (Glarborg et al., 2018) the Glarborg$\\_$2018 mechanism
was validated at low pressures, although it was later suggested that it is
valid at higher pressures as well (Valera-Medina et al., 2021).

It is shown in Table 1 that there are no significant differences in the number
of species and reactions of the various mechanisms. However, significant
differences are recorded for the computed $t_{ign}$, although the values of
$\tau_{e,min}$ do not vary as much.

Table 1: Number of species and reactions in the five $NH_{3}/air$ chemical kinetics mechanisms, along with $t_{ign}$ and $\tau_{e,min}$ at T0=1100 K, p0=2 atm and $\phi$=1.0. Mechanism |  Species |  Reactions |  $t_{ign}$ (s) |  $\tau_{e,min}$ (s)
---|---|---|---|---
Glarborg$\\_$2018 | 33 | 211 | 0.27 | 1.32$\times 10^{-6}$
Shrestha$\\_$2018 | 34 | 261 | 0.14 | 1.85$\times 10^{-6}$
Li$\\_$2019 | 34 | 252 | 0.13 | 1.88$\times 10^{-6}$
Stagni$\\_$2020 | 31 | 203 | 0.64 | 2.15$\times 10^{-6}$
Zhang$\\_$2021 | 34 | 262 | 0.38 | 2.23$\times 10^{-6}$

All five mechanisms include the species $Ar$, $H$, $H_{2}$, $H_{2}NN$,
$H_{2}NO$, $H_{2}O$, $H_{2}O_{2}$, $He$, $HNO$, $HNO_{2}$, $HNOH$, $HO_{2}$,
$HONO$, $HONO_{2}$, $N$, $N_{2}$, $N_{2}H_{2}$, $N_{2}H_{3}$, $N_{2}H_{4}$,
$N_{2}O$, $NH$, $NH_{2}$, $NH_{2}OH$, $NH_{3}$, $NNH$, $NO$, $NO_{2}$,
$NO_{3}$, $O$, $O_{2}$ and $OH$. In addition, Glarborg$\\_$2018 includes $HON$
and $O_{3}$ and Shrestha$\\_$2018, Li$\\_$2019 and Zhang$\\_$2021 include
$HON$, $OH^{*}$ and $HNO_{3}$. Regarding the reactions in the five chemical
kinetics mechanisms, there are various degrees of overlap. In terms of
nitrogen chemistry, Shrestha$\\_$2018 shares the subsets of $H_{2}NO$ and
$NO_{3}$ of reactions with Glarborg$\\_$2018\. Li$\\_$2019 shares a very large
portion of nitrogen chemistry with Shrestha$\\_$2018\. Stagni$\\_$2020 shares
the the $NH_{2}$ and part of the $NNH$ subsets of rections with
Glarborg$\\_$2018 and the $NH_{2}OH$, $HNOH$ and $NH_{2}OH$ subsets of
reactions with Shrestha$\\_$2018\. Finally, Zhang$\\_$2021 shares a large part
of nitrogen chemistry with Shrestha$\\_$2018\. In terms of hydrogen reactions,
there is no shared chemistry between Shrestha$\\_$2018 and Glarborg$\\_$2018\.
Similarly, there is no shared chemistry between Li$\\_$2019 and both
Shrestha$\\_$2018 and Glarborg$\\_$2018, except the $OH^{*}$ chemistry, which
is shared with Shrestha$\\_$2018\. Stagni$\\_$2020 shares all hydrogen-
chemistry reactions with Li$\\_$2019, except the $OH^{*}$ chemistry.
Zhang$\\_$2021 shares all hydrogen reactions with Glarborg$\\_$2018, with the
exception of a few reactions.

These five mechanisms have been included in a number of investigations that
compared the agreement of computational predictions of various mechanisms with
experimental data in several combustion configurations; i.e., shock-tubes,
laminar flames, stirred reactors (Wang et al., 2019; da Rocha et al., 2019;
Kawka et al., 2020; Wang et al., 2021; Alturaifi et al., 2022; Dagaut, 2022).
Apart from comparing profiles (e.g., various variables, reaction rates,
ignition delays, flame speeds), the agreement among the various mechanisms was
assessed on the basis of sensitivity analysis (SA), rate of production
analysis (RoPA) and element flux analysis (EFA). Such methodologies do
identify some reactions as important. These are reactions that either
participate in the equilibria generated due to the fast dynamics or drive the
process and research should concentrate on them, in order to improve the
validity of the mechanism. SA of global quantities (ignition delay or flame
speed) does not provide information on the progress of the process.
Additionally, SA of both global and local (species mass fraction) quantities,
as well as RoPA and EFA, cannot distinguish between reactions that participate
in established equilibria or drive the process. Moreover, EFA results were
provided at one instance only, so the conclusions reached were of limited
value. Although these methodologies do not provide the time-resolved details
that CSP does, certain conclusions can be drawn. A widely accepted conclusion
refers to the importance of the sequence $NH_{3}\rightarrow NH_{2}\rightarrow
H_{2}NO\rightarrow HNO$ (Wang et al., 2019; da Rocha et al., 2019; Kawka et
al., 2020; Wang et al., 2021; Dagaut, 2022). In general, there is an agreement
on the reactions that realise this path. Several works also mention the
importance of reactions involving species with two atoms of nitrogen in their
molecules (to be hereafter referred to as $N2$ chemistry) (Wang et al., 2019;
da Rocha et al., 2019; Wang et al., 2021; Dagaut, 2022). In this case, there
is no set of reactions that is accepted to realise this path. It is also noted
that the importance of the influence of $OH$ is explicitly pointed out in
Refs. (Wang et al., 2021; Dagaut, 2022) and is implicitly supported by the
reactions identified in Refs. (Wang et al., 2019; da Rocha et al., 2019).

Despite the significant similarities in the mechanisms considered here, Table
1 shows that there are significant differences in the computed values of IDT,
$t_{ign}$, while there is a better agreement on the minimum value of the
explosive time scale $\tau_{e,min}$ that occurs at about $t_{ign}$. A broader
picture of the differences in $t_{ign}$ is provided in Fig. 1, where the
variation of $t_{ign}$ in a wide range of values of T0, p0 and $\phi$ is
displayed. It is shown that $t_{ign}$ computed from the five mechanisms
follows the same trends. Moreover, it is shown that the values of $t_{ign}$
produced by Shrestha$\\_$2018 and Li$\\_$2018 are in general very close. The
profiles of $t_{ign}$ produced by Glarborg$\\_$2018 follow those of
Shrestha$\\_$2018 and Li$\\_$2018, however they over-predict IDT. Finally, the
profiles of $t_{ign}$ produced by Stagni$\\_$2020 and Zhang$\\_$2021 are close
to the profiles generated by the other mechanisms only for certain range of
values of T0, p0 and $\phi$.

Figure 1: The Ignition Delay Time $t_{ign}$ as a function of T0, p0 and
$\phi$.

Figure 1 shows that the largest drop of $t_{ign}$ with increasing values of To
is provided by Zhang$\\_$2021 and the smallest by Glarborg$\\_$2018\. Also,
the largest drop of $t_{ign}$ with increasing values of po is provided by
Stagni$\\_$2020 and Zhang$\\_$2021 for low values of To and by Stagni$\\_$2020
and Glarborg$\\_$2018 for large values of To, while the smallest drop is
provided by Shrestha$\\_$2018 and Li$\\_$2018\. Finally, it is shown that the
largest sensitivity to $\phi$ is exhibited by Stagni$\\_$2020 and the smallest
by Zhang$\\_$2021\. The increase of $t_{ign}$ with $\phi$, indicated by all
mechanisms is in contrast with what happens with hydrocarbons and alcohols
(Tingas et al., 2018a; Rabbani et al., 2022a, b).

Given that all five mechanisms considered involve the same species, either
exactly or with very few differences, the large differences in the prediction
of $t_{ign}$, displayed in Fig. 1, have to be attributed to differences in the
influence of reactions in each mechanism. In the following, the species and
reactions that influence the most $t_{ign}$ will be identified and compared
among the five mechanisms.

##  3 CSP methodology

The process of homogeneous autoignition is a multi-scale problem. The fast
time scales become exhausted quickly and the process is then characterised by
the slower time scales (Lam and Goussis, 1989; Goussis and Lam, 1992; Valorani
and Goussis, 2001; Valorani et al., 2003). The time scale that characterises
the process during the ignition delay time (IDT) for each of the five cases
considered will be identified and analysed with the tools developed in the
context of the Computational Singular Perturbation (CSP) algorithm.

The adiabatic, isochoric and homogenous autoignition is governed by the
equations:

$\displaystyle\dfrac{d\mathbf{y}}{dt}=$
$\displaystyle\dfrac{1}{\rho}\mathbf{W}\cdot\sum_{k=1}^{2K}\mathbf{S}_{k}R^{k}$
(1) $\displaystyle\dfrac{dT}{dt}=$ $\displaystyle\dfrac{1}{\rho
c_{v}}\left(-\mathbf{h}_{c}\cdot\mathbf{W}+\mathrm{R}T\mathbf{U}\right)\cdot\sum_{k=1}^{2K}\mathbf{S}_{k}R^{k}$
(2)

where the vector $\mathbf{y}$ includes the $N$ species mass fractions, $T$ is
the temperature, $\rho$ the mixture density and $c_{v}$ the heat capacity at
constant volume (Law, 2010). Assuming $K$ reactions in the chemical kinetics
mechanism, their forward and backward rates are considered separately in Eqs.
(1) and (2), where $R^{k}$ denotes the rate of the $k$-th unidirectional
reaction and $\mathbf{S}_{k}$ represents the related N-dim. stoichiometric
vector. The N-dim. vector $\mathbf{h}_{c}$ denotes the species absolute
enthalpies, $\mathbf{W}$ is a $N\times N$ diagonal matrix of the species
molecular weights, $\mathrm{R}$ is the universal gas constant and
$\mathbf{U}=[1,1,\cdots,1]$.

CSP considers the governing Eqs. (1) and (2) in the vector form:

$\\\
\dfrac{d\mathbf{z}}{dt}=\sum_{n=1}^{N+1}\mathbf{a}_{n}f^{n}\hfill\qquad\qquad\qquad\hfill
f^{n}=\mathbf{b}^{n}.~{}\mathbf{g}(\mathbf{z})$ (3)

where $\mathbf{z}=[\mathbf{y},T]^{T}$ is a ($N$+1)-dim. vector,
$\mathbf{a}_{n}$ is the ($N$+1)-dim. column CSP basis vector, $\mathbf{b}^{n}$
is the related ($N$+1)-dim. dual row vector
($\mathbf{b}^{i}\cdot\mathbf{a}_{j}=\delta_{j}^{i}$) (Lam and Goussis, 1989,
1994) and $\mathbf{g}(\mathbf{z})$ represents the summation of the $2K$ terms
in Eqs. (1) and (2):

$\mathbf{g}(\mathbf{z})=\mathbf{\hat{S}}_{1}R^{1}+\mathbf{\hat{S}}_{2}R^{2}+\dots+\mathbf{\hat{S}}_{2K}R^{2K}$
(4)

where $\mathbf{\hat{S}}_{k}$ is the generalized stoichiometric vector, the
relation of which to $\mathbf{S}_{k}$ is presented in (Diamantis et al.,
2015a). In Eq. (3) the amplitude $f^{n}$ is set positive, by appropriately
changing the sign of $\mathbf{b}^{n}$ and $\mathbf{a}_{n}$. Each CSP mode
$\mathbf{a}_{n}f^{n}$ in Eq. (3) is characterized by a single time scale, say
$\tau_{n}$; $n=1$ refers to the fastest time scale ($\tau_{1}$) and $n=N$ to
the slowest one ($\tau_{N}$). Assuming the M fastest modes being exhausted,
Eq. (3) simplifies to the reduced model:

$\\\
\dfrac{d\mathbf{z}}{dt}\approx\sum_{n=M+1}^{N+1}\mathbf{a}_{n}f^{n}\hfill\qquad\qquad\qquad
f^{m}\approx 0$ (5)

where $m=1,\dots,M$ (Lam and Goussis, 1989, 1994). The M algebraic equations
in Eq. (5) approximate the equilibria that are generated by the exhausted
modes, within which the solution evolves according the system of differential
equations. The availability of the reduced model in Eq. (5) allows for the
identification of the reactions and variables that control the evolution of
the process. In summary, the following tools are used here:

  * •

_Time scale Participation Index_ (TPI): measures the relative contribution of
each reaction to the development of the time scale $\tau_{n}$, that is
associated with the $n$-th CSP mode $\mathbf{a}_{n}f^{n}$ (Diamantis et al.,
2015b; Goussis and Najm, 2006). A significant positive/negative value of the
TPI suggests that the reaction promotes the explosive/dissipative character of
$\tau_{n}$. An explosive/dissipative time scale is generated by reactions that
tend to drive the system away from/towards equilibrium (Tingas et al., 2017,
2018b; Song et al., 2018).

  * •

_Amplitude Participation Index_ (API): measures the relative contribution of
each reaction to magnitude of the amplitude $f^{n}$ of the $n$-th CSP mode
(Massias et al., 1999; Diamantis et al., 2015a). In the case of an exhausted
modes, large APIs are exhibited by reactions that participate in the
equilibria generated by the fast dynamics, while in the case of active modes
(either explosive or dissipative), a positive/negative value of API is
exhibited by a reaction acting towards strengthening/weakening the impact of
the $n$-th mode (Fotache et al., 1997; Kazakov et al., 2006; Manias et al.,
2019).

  * •

_Pointer_ (Po): identifies the variables (species mass fraction or
temperature) that relate the most to the $n$-th mode (Goussis and Lam, 1992;
Valorani et al., 2003; Manias et al., 2016).

Naturally, the modes that drive the autoignition process during IDT are among
those represented in Eq. (5), the dynamics of which involves the slow time
scales. Since all modes in this equation are active, the time scale of a
driving mode must be among the fastest of these time scales and its amplitude
must be among the largest, so that this mode has a substantial impact. In
order to identify these modes, the time scales and amplitudes of all modes in
Eq. (3) will be examined next.

###  3.1 Time scale of explosive mode

Figure 2 displays the time scales $\tau_{n}$ of all modes for the cases
considered here. The dynamics of all mechanisms exhibit a wide range of time
scales. Among them, there is an explosive time scale $\tau_{e}$, which extends
throughout IDT and disappears at the point where the temperature undergoes a
steep increase at ignition. The minimum value of $\tau_{e}$ is recorded close
to the inflection point of the temperature profile.

Figure 2: The developing dissipative $\tau_{d}$ (grey) and explosive
$\tau_{e}$ (red) time scales during ignition delay, overlaid with the
temperature profile (blue); T0=1100 K, p0=2 atm and $\phi$=1.0.

In the case of Glarborg$\\_$2018, the explosive time scale $\tau_{e}$
initially remains approximately constant, then decelerates and subsequently
starts accelerating at an increasing rate. In the case of Shrestha$\\_$2018
and Li$\\_$2019, there is a gradual deceleration from the start, at the end of
which $\tau_{e}$ accelerates similarly to Glarborg$\\_$2018\. Finally, in the
case of Stagni$\\_$2020 and Zhang$\\_$2021, $\tau_{e}$ decelerates immediately
after the start of the process and then follows the accelerating pattern of
the previous cases. In all cases considered, Fig. 2 shows that $\tau_{e}$ is
among the slowest time scales. In the Stagni$\\_$2020 and Zhang$\\_$2021
cases, $\tau_{e}$ is positioned above a time scale gap throughout IDT, which
is a clear indication that the explosive mode is among those driving the
process, given that the faster modes are all dissipative. In the cases of
Glarborg$\\_$2018, Shrestha$\\_$2018 and Li$\\_$2019 $\tau_{e}$ is positioned
above a time scale bandwidth that is sparse.

Figure 3: The amplitudes $f^{n}$ of all modes during IDT for T0=1100 K, p0=2
atm and $\phi$=1.0; the amplitude of the explosive mode is shown in red.

###  3.2 Amplitude of explosive mode

An additional assessment on whether the explosive time scale is the driving
one during IDT can be obtained by comparing its amplitude $f^{e}$ with that of
all other modes. This comparison is performed through the results shown in
Fig. 3, where the amplitudes $f^{n}$ of all modes are displayed for all cases
considered. It is shown that the amplitude of the explosive mode is the
largest or among the largest throughout the IDT.

Both time scale and amplitude considerations suggest that the explosive mode
is among the ones driving the process during IDT. Therefore, the analysis that
follows will concentrate on the origin of the explosive dynamics. The
explosive mode was first introduced in the field of chemical kinetics by the
first CSP paper (Lam and Goussis, 1989) and was subsequently analyzed with CSP
tools (Fotache et al., 1997; Lee et al., 2005; Kazakov et al., 2006; Goussis
and Najm, 2006; Mittal et al., 2008; Chen et al., 2007; Lu et al., 2008).
Later on, after the introduction of _Chemical Explosive Mode Analysis_ (CEMA)
(Lu et al., 2010), which consists of concepts and tools developed previously
by CSP (Goussis et al., 2021), the explosive mode is analyzed with both CSP
and CEMA. CSP considers the explosive mode as one of the slow ones, so its
dominance is subject to verification (Goussis and Najm, 2006; Tingas et al.,
2017, 2018c; Rabbani et al., 2022a). In contrast, CEMA considers this mode
independently of all other slow modes, thus promoting the misconception that
it is always dominant (Shan and Lu, 2014; Steinberg et al., 2021; Huang et
al., 2022; Zhu et al., 2022; Guo et al., 2022; Desai et al., 2022).

##  4 CSP diagnostics

All CSP diagnostics for the explosive mode are displayed in Appendix A
(Glarborg$\\_$2018), Appendix B (Shrestha$\\_$2018), Appendix C (Li$\\_$2019),
Appendix D (Stagni$\\_$2020) and Appendix E (Zhang$\\_$2021). Parts of the
discussion that follows are based on the data and findings included in these
Appendices.

###  4.1 Chemical vs Thermal runaway

Figure 4: The developing explosive time scale, full (red) and temperature-
truncated (black) Jacobian along the ignition delay time, overlaid with the
temperature profile (blue); T0=1100 K, p0=2 atm and $\phi$=1.0. P1 to P5
denote instances at which CSP diagnostics are computed.

An important characteristic of the oxidation of widely used carbon-containing
fuels, such as hydrocarbons and alcohols, is that they demonstrate a chemical
runaway, during which an extensive radical pool is generated. This extends
over a substantial part of IDT, because it involves the formation of stable
intermediate species, such as aldehydes, $H_{2}O_{2}$ and $CO$. This part of
IDT is followed by the thermal runaway, which is mainly driven by strong
exothermic reactions. It is therefore interesting to explore whether these
features manifest themselves in ammonia autoignition.

Given that the explosive mode characterises the dynamics during IDT, a simple
method was introduced in Ref. (Diamantis et al., 2015a) in order to estimate
the parts occupied by the chemical and thermal runaways. This method consists
in comparing the explosive time scale in the dynamics of the species and
temperature Eqs. 1 and 2 with that in the dynamics of only the species Eq. 1.
In the chemical runaway the two explosive time scales are close to each other,
since temperature has minor influence there, while in the thermal runaway the
two time scales deviate significantly. This methodology was employed
successfully in autoignition studies of of various fuels; e.g., (Tingas et
al., 2018d, b, a; Rabbani et al., 2022b).

For each of the five cases considered here, Fig. 4 displays the explosive time
scale computed on the basis of the full Jacobian of the system in Eqs. 1 and 2
and of the truncated Jacobian of the system in Eq. 1. It is shown that a
relatively large chemical runaway develops in Glarborg$\\_$2018,
Shrestha$\\_$2018 and Li$\\_$2019, while in Stagni$\\_$2020 and Zhang$\\_$2021
it is practically non-existent. This is more clearly demonstrated in Fig. 5,
where it is shown that a chemical runaway is indeed developing in
Stagni$\\_$2020 and Zhang$\\_$2021, but its length is negligible. In all
cases, the chemical runaway extends in the part of the IDT where $\tau_{e}$
decelerates and ends before $\tau_{e}$ starts to accelerate. It is noted that
the two mechanisms that have negligible chemical runaway yield the two longest
IDTs; see Table 1 and Fig. 4.

The differences in the relative extent of chemical and thermal runaway will be
examined in the context of the reactions that relate to the explosive mode,
which was shown to dominate along the two runaways. In the following, the
reactions that contribute to the impact of this mode will by identified and
discussed, along with those that determine the time frame of its action. The
impact of the mode is characterised by its amplitude and the time frame of its
action by the explosive time scale. The API and the TPI indices will identify
the reactions that contribute to the amplitude (impact) and time scale (time
frame), respectively, of the explosive mode.

Figure 5: The developing explosive time scale, full (red) and temperature-
truncated (black) Jacobian along the initiation period; T0=1100 K, p0=2 atm
and $\phi$=1.0.

###  4.2 Initiation of the oxidation process according to the five mechanisms

The reactions producing the largest API indices at P1 ($t=0$) are
$NH_{2}+HO_{2}\leftarrow NH_{3}+O_{2}$ and $NH_{2}+H(+M)\leftarrow NH_{3}(+M)$
for Glarborg$\\_$2018 and only $NH_{2}+HO_{2}\leftarrow NH_{3}+O_{2}$ for
Shrestha$\\_$2018, Li$\\_$2019, Stagni$\\_$2020 and Zhang$\\_$2021\. This
difference is due to the fact that the rate of reaction
$NH_{2}+HO_{2}\leftarrow NH_{3}+O_{2}$ is much smaller in Glarborg$\\_$2018,
while that of $NH_{2}+H(+M)\leftarrow NH_{3}(+M)$ does not vary much, as shown
in Table 2. This finding is in agreement with the results in Ref. (Valera-
Medina et al., 2021). Therefore, the resulting smaller relative contribution
of the first reaction to the amplitude of the explosive mode $f^{e}$ in
Glarborg$\\_$2018, allows the contribution of the second reaction to increase.

Table 2: The Arrhenius constants A [$kmol/m^{3}\cdot s\cdot K^{b}$], b and E [$kJ/kmol$] and the reaction rate $R^{i}$ [$kmol/m^{3}\cdot s$]; the rates computed at $T(0)=1100~{}K$. Numbers in parentheses denote powers of ten. | $NH_{2}+HO_{2}\leftarrow NH_{3}+O_{2}$ | $NH_{2}+H(+M)\leftarrow NH_{3}(+M)$
---|---|---
| $A$ | $b$ | $E$ | $R^{i}$ | $A$ | $b$ | $E$ | $R^{i}$
Glarborg$\\_$2018 | 7.72(5) | 1.50 | 60633 | 3.67(7) | 5.17(17) | -0.55 | 108662 | 7.48(10)
Shrestha$\\_$2018 | 2.24(8) | 1.67 | 58983 | 4.15(10) | 9.00(16) | -0.39 | 110300 | 3.39(10)
Li$\\_$2019 | 1.09(8) | 1.76 | 57676 | 4.54(10) | 9.00(16) | -0.39 | 110300 | 3.39(10)
Stagni$\\_$2020 | 1.42(10) | 1.28 | 55224 | 2.64(11) | 3.50(30) | -5.22 | 111160 | 2.45(9)
Zhang$\\_$2021 | 1.08(9) | 1.71 | 54671 | 4.47(11) | 2.20(16) | 0.00 |  93470 | 8.01(11)

Figure 6: The sequence of reactions that exhibit significant contributions to
the amplitude of the explosive mode (large APIs) at P1 and P2; T0=1100 K, p0=2
atm and $\phi$=1.0. All reactions tend to increase the impact of the explosive
mode (large positive APIs), except for reaction $HO_{2}+NH_{2}\rightarrow
NH_{3}+O_{2}$ in Stagni$\\_$2020 and Zhang$\\_$2021, which tends to decrease
its impact (large negative API). Note that $OH$-producing reactions contribute
to the explosive dynamics sooner in Stagni$\\_$2020 and Zhang$\\_$2021 and
later in Glarborg$\\_$2018\. The colors black, red, blue, cyan and green are
meant to highlight successive phases of the process, as they are described in
the text.

Figure 6 lists the sequence of reactions driving the process at P1 and P2. It
is shown that effectively, in all cases the process starts with the production
of $NH_{2}$ and $HO_{2}$. A radical pool starts forming, consisting initially
of these two species with subsequent formation of $H_{2}NO$ and $H_{2}O_{2}$
in all mechanisms and $OH$ in all mechanisms except in Glarborg$\\_$2018\. In
parallel, $N2$ chemistry (involving species with two atoms of nitrogen in
their molecule) starts developing in Shrestha$\\_$2018, Li$\\_$2019 and
Zhang$\\_$2021\. This chemistry starts with the reaction
$NH_{3}+NH_{2}\rightarrow N_{2}H_{3}+H_{2}$, leading to the formation of
$N_{2}H_{2}$ and $N_{2}H_{4}$, as shown in Fig. 6. In Glarborg$\\_$2018 and
Stagni$\\_$2020 there is no $N2$ chemistry manifesting itself at the start of
the process.

The fate of this radical pool differs drastically between two groups of
mechanisms. In Glarborg$\\_$2018, Shrestha$\\_$2018 and Li$\\_$2019 there is
no significant opposition to the impact of the explosive mode (no reactions
with negative APIs, large in magnitude). In contrast, for Stagni$\\_$2020 and
Zhang$\\_$2021 there is a substantial opposition from reaction
$HO_{2}+NH_{2}\rightarrow NH_{3}+O_{2}$ leading its reactants to an early
steady state, as shown in Fig. 7. The action of this reaction in diminishing
the impact of the explosive mode is pronounced only in Stagni$\\_$2020 and
Zhang$\\_$2021 and is due to the relatively large $HO_{2}$ and $NH_{2}$
concentrations in these two cases, right after the start of the process, as
shown in Fig. 7. These large concentrations are realised because the rate of
$NH_{3}+O_{2}\rightarrow NH_{2}+HO_{2}$ is much larger in Stagni$\\_$2020 and
Zhang$\\_$2021, as shown in Table 2.

Figure 7: Mass fraction profiles of $NH_{2}$ and $HO_{2}$ vs time scaled with
$t_{ign}$; T0=1100 K, p0=2 atm and $\phi$=1.0. It is shown that in
Stagni$\\_$2020 and Zhang$\\_$2021 $NH_{2}$ and $HO_{2}$ attain quasi-steady-
state status immediately after the start of the process.

The reactions contributing the most to the amplitude of the explosive mode
(large APIs), listed in Fig. 6, are not necessarily the same with those that
contribute to its time scale $\tau_{e}$ (large TPIs). For example, although
the $NH_{2}$ and $HO_{2}$-producing reactions $NH_{2}$ \+ $HO_{2}$
$\leftarrow$ $NH_{3}$ \+ $O_{2}$ and $NH_{2}$ \+ $H(+M)$ $\leftarrow$
$NH_{3}(+M)$ contribute the most to the amplitude $f^{e}$ at P1 in
Glarborg$\\_$2018, it is the $NH_{2}$ and $HO_{2}$-consuming reactions
$NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ and $NH_{3}$ \+ $HO_{2}$
$\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$ that contribute the most to the
explosive time scale $\tau_{e}$ there; see Appendix A. However, as the process
progresses, the sets of reactions that contribute the most to $f^{e}$ and
$\tau_{e}$ become similar. For example at P3 in Glarborg$\\_$2018, the
reactions promoting the most $f^{e}$ and $\tau_{e}$ are the $NH_{2}$,
$H_{2}NO$ and $HO_{2}$-consuming reactions $NH_{2}+O_{2}\rightarrow
H_{2}NO+O$, $H_{2}NO+O_{2}$ $\rightarrow HNO+HO_{2}$ and
$NH_{2}+HO_{2}\rightarrow H_{2}NO+OH$, while the one opposing the most is
$NH_{2}+NH_{2}(+M)\rightarrow N_{2}H_{4}(+M)$.

In general, the reactions that contribute to most to $\tau_{e}$ within the
chemical runaway are those whose reactants are identified by the CSP Pointer.
Such reactants are $HO_{2}$, $NH_{2}$ and $H_{2}NO$ for all five mechanisms
and $N_{2}H_{3}$ and $N_{2}H_{4}$ only for Shrestha$\\_$2018, Li$\\_$2019 and
Zhang$\\_$2021, that involve $N2$ chemistry.

  * •

The reaction that involves $HO_{2}$ as a reactant and contributes
significantly to promoting $\tau_{e}$ is $NH_{3}+HO_{2}\rightarrow
NH_{2}+H_{2}O_{2}$. Its influence is mainly manifested at the start of the
process; i.e., in the vicinity of P1.

  * •

For $NH_{2}$, the related reaction is $NH_{2}+O_{2}\rightarrow H_{2}NO+O$ and
its influence is exercised up until the middle of the chemical runaway regime
in all five mechanisms considered. In addition, the $NH_{2}$-consuming
reaction $NH_{3}+NH_{2}\rightarrow N_{2}H_{3}+H_{2}$ is shown to promote
$\tau_{e}$ during the first half of the chemical runaway in the three
mechanisms exhibiting $N2$ chemistry.

  * •

As for $H_{2}NO$, the reaction that contributes the most to $\tau_{e}$, is
$H_{2}NO+O_{2}\rightarrow HNO+HO_{2}$ and its influence lasts throughout the
chemical runaway in Glarborg$\\_$2018 (P1-P3) and Stagni$\\_$2020 (P1-P2) and
in the second half of this period in Shrestha$\\_$2018 (P2-P3), Li$\\_$2019
(P2-P3) and Zhang$\\_$2021 (P2).

  * •

In the case of $N_{2}H_{3}$, there are two reactions that promote $\tau_{e}$:
$N_{2}H_{3}\rightarrow N_{2}H_{2}+H$ and $N_{2}H_{4}+NH_{2}\leftarrow
N_{2}H_{3}+NH_{3}$. Both reactions promote $\tau_{e}$ in Shrestha$\\_$2018 and
Li$\\_$2019, while only the former acts in this manner in Zhang$\\_$2021\. In
all three cases, the influence of these reactions appears at the start of the
chemical runaway regime; P1 and P2 in Shrestha$\\_$201 and P1 in Li$\\_$2019
and Zhang$\\_$2021.

  * •

Finally, the $N_{2}H_{4}$-consuming reaction $NH_{2}+NH_{2}(+M)\leftarrow
N_{2}H_{4}(+M)$ promotes $\tau_{e}$ and is active only in Shrestha$\\_$201 and
Li$\\_$2019, at the start of the chemical runaway regime.

Figure 8: Schematic representation of the two paths that determine the
explosive time scale, which characterises the oxidation process within the
chemical runaway. The path in the left panel is functioning in all five
mechanisms, while that in the right panel is functioning only in
Shrestha$\\_$2018, Li$\\_$2019 and Zhang$\\_$2021.

These findings suggest the existence of two paths that determine the explosive
time scale, which characterises the oxidation process within the chemical
runaway. First is the path $NH_{3}\rightarrow NH_{2}\rightarrow
H_{2}NO\rightarrow HNO$. Then there is the path that can manifest itself as
either $(NH_{3},\ NH_{2})\rightarrow N_{2}H_{3}\rightarrow
N_{2}H_{4}\rightarrow NH_{2}$ or $(NH_{3},\ NH_{2})\rightarrow
N_{2}H_{3}\rightarrow N_{2}H_{2}~{}\text{and}~{}H$. Both paths are depicted in
Fig. 8. The first path is functional in all five mechanisms, while the second
path relates to $N2$ chemistry that is functional only in Shrestha$\\_$2018,
Li$\\_$2019 and Zhang$\\_$2021\. It is noted that rection $H+O_{2}\rightarrow
O+OH$ is shown to promote $\tau_{e}$ early in the chemical runaway in all
three cases involving $N2$ chemistry. Apparently, this is due to reaction
$N_{2}H_{3}\rightarrow N_{2}H_{2}+H$, which is shown to promote $\tau_{e}$.

Significant opposition to $\tau_{e}$ is manifested in Shrestha$\\_$2018,
Li$\\_$2019 and Zhang$\\_$2021 via the $H$-consuming reaction
$NH_{3}+H\rightarrow NH_{2}+H_{2}$ and, more significantly so in
Stagni$\\_$2020 and Zhang$\\_$2021, via reaction $NH_{2}+HO_{2}\rightarrow
NH_{3}+O_{2}$. As discussed earlier, the latter reaction was also shown to
oppose the impact of the explosive mode, by acting towards decreasing its
amplitude $f^{e}$. Finally, in the case of Shrestha$\\_$2018 and Li$\\_$2019,
in which $N2$ chemistry is present, the $N_{2}H_{3}$-consuming reactions
$N_{2}H_{3}+NH_{2}\rightarrow H_{2}NN+NH_{3}$ and $NH_{3}+NH_{2}\leftarrow
N_{2}H_{3}+H_{2}$ provide a minor opposition.

###  4.3 Dynamics of the thermal runaway

The reactions contributing towards the generation of $\tau_{e}$ do not
demonstrate as large variety between mechanisms in the thermal runaway as in
the chemical runaway. These reactions can be classified in the following three
main sets:

  * •

$OH$-producing reactions that support $\tau_{e}$: $NH_{2}+NO\rightarrow
NNH+OH$ and $NH_{2}+HO_{2}\rightarrow H_{2}NO+OH$ that are relevant to all
five mechanisms,

  * •

non-$OH$-producing reactions that support $\tau_{e}$:
$H_{2}NO+O_{2}\rightarrow HNO+HO_{2}$ that is relevant to all five mechanisms,
$NH_{2}+NO_{2}\rightarrow H_{2}NO+NO$ that is relevant to Shrestha$\\_$2018
and Li$\\_$2019 and $NH_{3}+O_{2}\rightarrow HNO+HO_{2}$ that is relevant to
Stagni$\\_$2020 and Zhang$\\_$2021.

  * •

$NH_{2}$-depleting reactions that oppose $\tau_{e}$: $NH_{2}+NO\rightarrow
N_{2}+H_{2}O$ that produces the equilibrium products and applies in all five
cases, $NH_{2}+NO_{2}\rightarrow N_{2}O+H_{2}O$ that applies in
Shrestha$\\_$2018 and Li$\\_$2019 and $NH_{2}+HO_{2}\rightarrow NH_{3}+O_{2}$
that produces the initial reactants and applies in Stagni$\\_$2020 and
Zhang$\\_$2021.

These findings suggest that an ubiquitous action supporting $\tau_{e}$
originates from the $NH_{2}$-consuming reactions in the first set that produce
$OH$. The strongest support originates from $NH_{2}+NO\rightarrow NNH+OH$ in
the cases where a sizeable chemical runaway develops (Glarborg$\\_$2018,
Shrestha$\\_$2018 and Li$\\_$2019) and from $NH_{2}+HO_{2}\rightarrow
H_{2}NO+OH$ in the cases where a chemical runaway practically does not exist
(Stagni$\\_$2020 and Zhang$\\_$2021). The reactions in the second set recycle
species that are reactants in the reactions of the first set; i.e., $H_{2}NO$,
$NO$, $HO_{2}$. The action of the reactions in the third set, that oppose
$\tau_{e}$, is due to (i) the competition for $NH_{2}$ with the promoting ones
in the first and second sets and and (ii) the production of stable species,
such as $N_{2}$, $H_{2}O$ and $N_{2}O$.

The conclusions stated previously apply throughout the thermal runaway, except
the final part where $\tau_{e}$ decreases rapidly towards its minimum value at
$P_{5}$. In this final part all five mechanisms provide similar diagnostics.
In particular, in all five cases considered, the following reactions
contribute in promoting $\tau_{e}$:

  * •

The largest contribution originates from the $OH$-producing reaction
$H+O_{2}\rightarrow O+OH$.

  * •

The two directions of $OH+H_{2}\leftrightarrow H+H_{2}O$ provide the second
largest contributions to $\tau_{e}$, which cancel each other; the
$H$-producing forward direction promoting $\tau_{e}$.

  * •

The third largest contribution originates from the $H$ and $OH$-producing
reaction $O+H_{2}\rightarrow H+OH$.

The next largest contributions originate from reactions that oppose the
explosive character of $\tau_{e}$; i.e., $NH_{3}+H\rightarrow NH_{2}+H_{2}$ in
Glarborg$\\_$2018, $NH+OH\rightarrow NO+H_{2}$, $HO_{2}(+M)\leftarrow OH+O$,
$NH+OH\rightarrow HNO+H$ in Shrestha$\\_$2018, $NH+OH\rightarrow NO+H_{2}$,
$NH_{3}+O\rightarrow NH_{2}+OH$ in Li$\\_$2019, $NH+H\rightarrow N+H_{2}$,
$N_{2}O+H\rightarrow N_{2}+OH$, $HNO+H\rightarrow NO+H_{2}$ in Stagni$\\_$2020
and $NH_{2}+H\rightarrow NH+H_{2}$, $NH_{3}+H\rightarrow NH_{2}+H_{2}$,
$NH+OH\rightarrow HNO+H$ in Zhang$\\_$2021.

In summary, the explosive activity at the end of the thermal runaway is
supported by the $OH$-producing reactions $H+O_{2}\rightarrow O+OH$ and
$O+H_{2}\rightarrow H+OH$. The major opposition to the explosive mode is due
to reactions that consume their reactants $H$ and $O$ and that deplete $OH$.
Clearly, the key feature of the explosive mode at this point is the generation
of $OH$.

###  4.4 Exothermic reactions

Figure 9: The most exothermic reactions at P5; T0=1100 K, p0=2 atm and
$\phi$=1.0.

The role of the $OH$-producing reactions in promoting $\tau_{e}$ during the
thermal runaway regime is revealed when examining the reactions contributing
the most to the rate of heat release. Figure 9 displays the reactions
exhibiting the largest volumetric heat release rate at P5 in all five cases
considered; see Ref. (Rabbani et al., 2022a) for the computation of the heat
release rate of each of the two directions of a reaction. As shown in Fig. 4,
at this point the temperature undergoes a steep increase. It is shown in Fig.
9 that the largest contributions to the heat release rate in all cases
considered originate from the $OH$-consuming and $H_{2}O$-forming reactions
$OH+H_{2}\rightarrow H+H_{2}O$ and $NH_{3}+OH\rightarrow NH_{2}+H_{2}O$. The
largest contribution is provided by the first reaction in Glarborg$\\_$2018
and Shrestha$\\_$2018, by the second reaction in Stagni$\\_$2020, while the
two reactions provide similar contributions in Li$\\_$2019 and Zhang$\\_$2021.

As stated previously, reaction $OH+H_{2}\rightarrow H+H_{2}O$ contributes
significantly to $\tau_{e}$, since it produces the radical for reaction
$H+O_{2}\rightarrow O+OH$, which promotes $\tau_{e}$ the most. However, this
contribution is cancelled by its reverse direction. In contrast, reaction
$NH_{3}+OH\rightarrow NH_{2}+H_{2}O$ provides only negligible contribution to
$\tau_{e}$. These findings suggest that the explosive mode is not linked
directly to the steep temperature rise that is recorded at the end of the
thermal runaway; e.g, through an overall strongly exothermic reaction.
Instead, it is linked indirectly via the production of the $OH$-radicals by
the reactions that support its explosive character, which are consumed by the
most exothermic reactions in order to form the most stable species $H_{2}O$. A
similar relation of the explosive mode with the temperature evolution was also
recorded in the super-adiabatic-temperature phenomenon in the post-ignition
region of light alcohols (Rabbani et al., 2022a; Manias et al., 2022).

Figure 10: Profiles of $OH$ mass fraction and of the rate of change of
temperature vs time scaled with $t_{ign}$; T0=1100 K, p0=2 atm and $\phi$=1.0.

The influence of the generation of $OH$ on the rate of temperature rise is
highlighted in Fig. 10, where the profiles of the mass fraction of this
species and of the rate of change of the temperature are displayed, against
time scaled with $t_{ign}$. The profiles in Fig. 10 are in agreement with the
findings displayed in Fig. 6, where it is shown that $OH$-producing reactions
participating in the explosive dynamics show up the soonest in Stagni$\\_$2020
and Zhang$\\_$2021, in which cases the chemical runaway lasts for a very short
period; see Fig. 4. In contrast, the $OH$-producing reactions show up the
latest in Glarborg$\\_$2018, which exhibits the longest chemical runaway. The
profiles displayed in the left panel of Fig. 10 validate these conclusions,
since it is shown that a sizeable $OH$ pool develops earliest in
Stagni$\\_$2020 and Zhang$\\_$2021 and last in Glarborg$\\_$2018\. These
findings are reflected in the right panel of Fig. 10, where it is shown that
the initial steep temperature rise is manifested first in Stagni$\\_$2020 and
Zhang$\\_$2021 and last in Glarborg$\\_$2018.

Consideration of the $t_{ign}$ values in Table 1, in the light of the $OH$
profiles in the left panel of Fig. 10, reveals that the mass fraction of $OH$
at the the last portion of the thermal runaway correlates strongly with the
duration of IDT. Specifically, Fig. 10 shows that near IDT the five mechanisms
rank as follows in an ascending order of $y_{OH}$: the mechanism producing the
smallest $y_{OH}$ is Stagni$\\_$2020, followed by Zhang$\\_$2021, then by
Glarborg$\\_$2018 and ultimately by Li$\\_$2019 and Shrestha$\\_$2018, the
$y_{OH}$-profiles of which are very close to each other. This is the way the
five mechanisms rank in descending order of IDT, as shown in Table 1. This is
another demonstration of the role of $OH$ in sustaining the thermal activity
that leads to ignition, as shown further by the results in Fig. 11. There,
values of heat release rate $\dot{Q}(t=0.95t_{ign})$ and of ignition delay
time $t_{ign}$ are displayed as a function of $y_{OH}(t=0.95t_{ign})$. The
general trend is that the larger the mass fraction of $OH$ short before
ignition, the higher the heat release and the shorter the ignition delay time.
Figure 11 shows that this conclusion clearly applies to Stagni$\\_$2020,
Zhang$\\_$2021, Glarborg$\\_$2018 and Li$\\_$2019\. It is noted that, although
$y_{OH}(t=0.95t_{ign})$ in Shrestha$\\_$2018 is larger than in Li$\\_$2019,
the related values of $\dot{Q}(t=0.95t_{ign})$ and $t_{ign}$ are very close.
This can be attributed to the difference in the Arrhenius constants of the
most exothermic reaction close to ignition $OH+H_{2}\rightarrow H+H_{2}O$ (see
the panels of Fig. 9 related to these two mechanisms), resulting in a reaction
rate constant in Shrestha$\\_$2018 that is smaller that in Li$\\_$2019.

Figure 11: Values of the heat release rate (HRR) $\dot{Q}$ and of the ignition
delay time (IDT) $t_{ign}$ vs.the values of the $OH$ mass fraction at 95% of
the IDT, $y_{OH,95\%}$; T0=1100 K, p0=2 atm and $\phi$=1.0.

The discussion above leads to the following two major findings:

  * •

The early $OH$-formation in Stagni$\\_$2020 and Zhang$\\_$2021 activates the
most exothermic reactions immediately after the start of the process. This
leads to the early completion of the chemical runaway regime and to the
fastest transition to the thermal one. In contrast, the late $OH$-formation in
Glarborg$\\_$2018 leads to the longest chemical runaway and the slowest
transition to the thermal one.

  * •

The duration of IDT, $t_{ign}$, correlates strongly with the maximum value of
$OH$ mass fraction that manifests itself in the last portion of the thermal
runaway.

##  5 Conclusions

In Ref. (Kawka et al., 2020), Kawka et. al stated that “ _the quantitative
description of homogeneous ammonia combustion is still a challenge_ ". This
challenge was addressed here by analyzing five recently developed chemical
kinetics mechanisms with the algorithmic tools of CSP. The following
conclusions were reached by analysing the explosive mode that was shown to
drive the process:

  * •

The process is initially driven by nitrogen chemistry and transitions
gradually to being driven by hydrogen chemistry towards the end of ignition
delay time. There is a general qualitative agreement, in terms of reactions
and variables related to ignition, among the five mechanisms considered, as it
relates to the period in which hydrogen chemistry dominates. This agreement is
demonstrated by the relatively small differences of $\tau_{e,min}$, displayed
in Table 1, which are recored at the end of IDT where hydrogen chemistry
dominates. In contrast, there are substantial differences in the period in
which nitrogen chemistry dominates. This disagreement is demonstrated by the
very large differences of $t_{ign}$ displayed in the same Table.

  * •

A major such difference relates to the influence of $N2$ chemistry on the
mechanism for ignition during the chemical runaway; such chemistry is active
only in Shrestha$\\_$2018, Li$\\_$2019 and Zhang$\\_$2021 and contributes to
the decrease of $t_{ign}$. This action of $N2$ chemistry is due to the
generation of $H$ and to the fact that three $NH_{2}$ radicals are produced
for each one consumed (see Fig. 8).

  * •

A serious issue relates to the duration of the chemical runaway; in
Glarborg$\\_$2018, Shrestha$\\_$2018, Li$\\_$2019 a sizeable chemical runaway
develops, while in Stagni$\\_$2020 and Zhang$\\_$2021 is extremely short. The
cause of very short chemical runaways is the very fast generation of the
radical pool, which slows down the process at the start of the thermal
runaway, as it is evident in Figs. 4, 5, 7 and 10, through the activation of
the inverse direction of the radical-producing reactions. A typical example is
$NH_{3}+O_{2}\rightarrow HO_{2}+NH_{2}$, which is the initiation reaction in
all five mechanisms considered, the inverse of which exhibits a strong
opposition to the explosive mode only in Stagni$\\_$2020 and Zhang$\\_$2021.

  * •

During the thermal runaway, in all five mechanisms considered the explosive
mode is mainly generated by the $OH$-producing reactions $NH_{2}+NO\rightarrow
NNH+OH$ and $NH_{2}+HO_{2}\rightarrow H_{2}NO+OH$ and by other reactions that
support these two through the production of the required reactants $HO_{2}$
and $NO$. These support reactions differ among the various mechanisms.

  * •

The explosive mode is not linked directly to the steep temperature rise at the
end of the thermal runaway; e.g, through an overall strongly exothermic
reaction. Instead, it is linked indirectly via the production of the
$OH$-radicals by the reactions that support its explosive character, which are
consumed by the most exothermic reactions in order to form the most stable
species $H_{2}O$.

  * •

Although temperature is the variable related the most to the explosive mode at
the end of the thermal runaway, where the steep temperature rise materialises,
the $OH$-producing reactions that mainly support this mode are not the most
exothermic. Instead, in all five cases considered, the two most exothermic
reactions in this period are the $OH$-consuming $OH+H_{2}\rightarrow H+H_{2}O$
and $NH_{3}+OH\rightarrow NH_{2}+H_{2}O$. The former reaction is the most
exothermic in Glarborg$\\_$2018, Shrestha$\\_$2018 and Li$\\_$2019 (where a
sizeable chemical runaway develops) and the latter is the most exothermic one
in Stagni$\\_$2020 and Zhang$\\_$2021 (where the chemical runaway is very
short). The less exothermic reactions differ among the mechanisms.

The purpose of this analysis was to identify thermochemical action that is
either common or different among the five mechanisms considered and exerts
significant influence on ignition delay. A detailed examination was carried
out for each mechanism and the output was synthesised in order to reach
conclusions on major features of the ignition process. These conclusions point
to specific directions for future research, in order to construct a reliable
mechanism. In particular, time-resolved measurements of $OH$ will clarify the
issue of the duration of the chemical runaway. Also, an accurate determination
of the the Arrhenius parameters of the initiation reaction
$NH_{3}+O_{2}\rightarrow NH_{2}+HO_{2}$ will be very important in this regard.
Finally, measurements of species like $N_{2}H_{2}$, $N_{2}H_{3}$ and
$N_{2}H_{4}$ will determine the influence of $N2$ chemistry.

The present analysis considers only one set of values of T0, p0 and $\phi$. It
is reasonable to assume that the features of the underlying chemistry (e.g.,
the role of $N2$ chemistry, the relative length of chemical runaway, etc)
might vary with these parameters. This dependance has to be explored in future
research.

## Acknowledgements

The support from Khalifa University of Science and Technology, via projects
CIRA-2019-033 and and RC2-2019-007, is gratefully acknowledged.

##  Appendix A CSP Diagnostics for the Glarborg$\\_$2018 mechanism

The reactions related to the explosive dynamics during IDT in
Glarborg$\\_$2018 are listed in Table A.1. The reactions exhibiting large APIs
at P1 and P2 are shown in Fig. A.1, while at points P3 to P5 they are the same
with those exhibiting large TPIs. All TPI and Po CSP diagnostics for the
explosive mode, computed at points P1 to P5 shown in Fig. 4, are displayed in
Table A.2.

Table A.1: The reactions contributing the most to the explosive dynamics in Glarborg$\\_$2018\. Bold/regular font represents exothermic/endothermic reactions. 1f | : |  $H$ \+ $O_{2}$ $\rightarrow$ $O$ \+ $OH$ | 648f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$
---|---|---|---|---|---
3f | : |  $O$ \+ $H_{2}$ $\rightarrow$ $OH$ \+ $H$ | 654f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$
4f/b | : |  $OH$ \+ $H_{2}$ $\leftrightarrow$ $H$ \+ $H_{2}O$ | 655f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$
633b | : |  $NH_{2}$ \+ $H(+M)$ $\leftarrow$ $NH_{3}(+M)$ | 699f/b | : |  $H_{2}NO$ \+ $M$ $\leftrightarrow$ $HNOH$ $+M$
634f/b | : |  $NH_{3}$ \+ $H$ $\leftrightarrow$ $NH_{2}$ \+ $H_{2}$ | 705f | : |  $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$
637f | : |  $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$ | 706f | : |  $H_{2}NO$ \+ $NH_{2}$ $\rightarrow$ $HNO$ \+ $NH_{3}$
643b | : |  $NH_{2}$ \+ $HO_{2}$ $\leftarrow$ $NH_{3}$ \+ $O_{2}$ | 771f | : |  $HNOH$ \+ $NH_{2}$ $\rightarrow$ $H_{2}NN$ \+ $H_{2}O$
644f | : |  $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ | 772f | : |  $NH_{2}$ \+ $NH_{2}(+M)$ $\rightarrow$ $N_{2}H_{4}$ (+ $M$)

Figure A.1: Reactions exhibiting large APIs at points P1 and P2 in
Glarborg$\\_$2018; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parentheses
denote API values. The different colors are meant to highlight several phases
of the process as they are described in the text.

Figure A.1 shows the way the oxidation process commences according to
Glarborg$\\_$2018\. It is shown that at P1 the reactions that exhibit large
APIs are 643b (0.59): $NH_{2}$ \+ $HO_{2}$ $\leftarrow$ $NH_{3}$ \+ $O_{2}$
and 633b (0.40): $NH_{2}$ \+ $H(+M)$ $\leftarrow$ $NH_{3}(+M)$, where the
numbers in parentheses denote the API value. At P2 reactions 643b and 633b are
still exhibiting large APIs, albeit smaller ones; i.e., (0.13) and (0.09)
respectively. In addition, the $HO_{2}$ produced by reaction 643b activates
reaction 637f (0.18) $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+
$H_{2}O_{2}$, while the $NH_{2}$ produced by both 643b and 633b activates
reaction 648f (0.45) $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$. In
summary, as shown in Fig. A.1, the initiation of the oxidation process
involves reactions producing initially $NH_{2}$ and $HO_{2}$ at P1 and
additionaly $H_{2}NO$, $H_{2}O_{2}$ and $O$ at P2. At P1 and P2 the Po indices
displayed in Table A.2 show that three of these species ($NH_{2}$, $HO_{2}$
and $H_{2}NO$) are the ones related the most to the explosive mode. These
species are reactants of the reactions 648f (0.70): $NH_{2}$ \+ $O_{2}$
$\rightarrow$ $H_{2}NO$ \+ $O$, 637f (0.23): $NH_{3}$ \+ $HO_{2}$
$\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$ and 705f (0.07) $H_{2}NO$ \+ $O_{2}$
$\rightarrow$ $HNO$ \+ $HO_{2}$, which exhibit the largest TPIs, so they
contribute the most to the explosive time scale $\tau_{e}$; i.e., they are the
reaction that determine the time scale that characterizes the autoignition
process there.

Figure 4 shows that up to P3, $\tau_{e}$ does not change much. This is
reflected in the CSP diagnostics at this point, displayed in Table A.2, which
are not that much different from those in P1 and P2. In particular, it is
shown that, in addition to reactions 648f, 705f and 637f, a significant
contribution to the explosive character of the mode is provided by reaction
644f: $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ and a small
opposition is provided by reaction 772f: $NH_{2}$ \+ $NH_{2}(+M)$
$\rightarrow$ $N_{2}H_{4}$ (+ $M$) that deprives $NH_{2}$ from the promoting
reactions 648f and 644f.

At P4 there is a large number of reactions either favoring or opposing the
explosive character of the mode, as shown in Table A.2. Among those providing
the largest contributions, are the two directions of reaction 699, which
cancel each other. Of the remaining reactions, those that promote the most the
explosive character (positive TPIs) are 655f: $NH_{2}$ \+ $NO$ $\rightarrow$
$NNH$ \+ $OH$, 705f: $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$ and
1f: $H$ \+ $O_{2}$ $\rightarrow$ $O$ \+ $OH$. The common feature of these
reactions is that they produce radicals related to hydrogen chemistry. The
reactions that oppose the most the explosive character (negative TPIs) are
reactions 654f, 706f and 771f, which (i) deprive the promoting reactions 655f
and 705f of the reactants of $NH_{2}$ and $H_{2}NO$ and (ii) lead to the
formation of stable species, such that $N_{2}$, $NH_{3}$ and $H_{2}O$. The
variable associated the most to the explosive mode is temperature. This is a
reasonable finding, since P4 lies in the thermal runaway, as shown in Fig. 4.

Table A.2: The largest TPI and Po values for the explosive mode in Glarborg$\\_$2018; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parenthesis denote powers of ten. Bold/regular font represents exothermic/endothermic reactions. P1 | P2 | P3 | P4 | P5
---|---|---|---|---
t1 =0.00(0)s | t2 =1.11(-2)s | t3 =5.45(-2)s | t4 =1.84(-1)s | t5 =2.65(-1)s
$\tau_{e,f}$=7.71(-3)s |  $\tau_{e,f}$=7.70(-3)s |  $\tau_{e,f}$=6.46(-3)s |  $\tau_{e,f}$=7.32(-2)s |  $\tau_{e,f}$=1.32(-6)s
TPI
648f | +0.60 | 648f | +0.60 | 648f | +0.25 | 655f | +0.14 | 1f | +0.26
637f | +0.23 | 637f | +0.23 | 644f | +0.18 | 654f | -0.12 | 4f | +0.15
705f | +0.07 | 705f | +0.07 | 705f | +0.16 | 705f | +0.10 | 4b |  -0.14
|  |  |  | 637f | +0.08 | 699f |  -0.06 | 3f | +0.05
|  |  |  | 772f |  -0.05 | 706f |  -0.05 | 634f |  -0.04
|  |  |  |  |  | 699b | +0.05 |  |
|  |  |  |  |  | 1f | +0.05 |  |
|  |  |  |  |  | 771f |  -0.05 |  |
|  |  |  |  |  | 634f |  -0.04 |  |
|  |  |  |  |  | 644f | +0.04 |  |
Po
$NH_{2}$ | +0.65 | $NH_{2}$ | +0.65 | $NH_{2}$ | +0.43 | $T$ | +0.81 | $T$ | +0.73
$HO_{2}$ | +0.25 | $HO_{2}$ | +0.25 | $HO_{2}$ | +0.26 | $H_{2}$ | +0.16 | $NH_{3}$ | +0.49
$H_{2}NO$ | +0.07 | $H_{2}NO$ | +0.07 |  $H_{2}N$O | +0.20 |  |  | $H_{2}O$ |  -0.40
|  |  |  |  |  |  |  | $H$ | +0.14

At the point P5, where $\tau_{e}$ reaches minimum value, right before tign,
Table A.2 shows that the major contributions towards the explosive character
of $\tau_{e}$ originate from hydrogen chemistry. In particular, the largest
contribution is produced from reaction 1f: $H+O_{2}\rightarrow O+OH$. The next
contributions are produced from the two directions of reaction 4f:
$OH+H_{2}\leftrightarrow H+H_{2}O$, which cancel each other. The third largest
contribution in promoting $\tau_{e}$ originates from reaction 3f:
$O+H_{2}\rightarrow H+OH$. The major opposition to $\tau_{e}$ originates from
reaction 634f: $NH_{3}$ \+ $H$ $\rightarrow$ $NH_{2}$ \+ $H_{2}$, which
consumes the reactant $H$ of the most promoting reaction 1f. In summary, the
peak of explosive dynamics associates with the production of mainly $OH$ and
secondarily of $H$ and $O$ radicals, which are reactants of the $OH$-producing
reactions 1f and 3f.

##  Appendix B CSP Diagnostics for the Shrestha mechanism

The reactions contributing to the explosive dynamics during IDT in
Shrestha$\\_$2018 are listed in Table B.1. Figure B.1 displays the reactions
exhibiting large APIs at P1 and P2. At P3 to P5 the reactions exhibiting large
APIs are similar to those exhibiting large TPIs. Table B.2 displays the TPI
and Po indices for the explosive mode at P1 to P5, which are indicated in Fig.
4.

Table B.1: The reactions contributing the most to the explosive time scale in Shrestha$\\_$2018\. Bold/regular font represents exothermic/endothermic reactions. 1f | : |  $O_{2}$ \+ $H$ $\rightarrow$ $OH$ \+ $O$ | 701f | : |  $NH_{2}$ \+ $NO_{2}$ $\rightarrow$ $N_{2}O$ \+ $H_{2}O$
---|---|---|---|---|---
3f | : |  $O$ \+ $H_{2}$ $\rightarrow$ $H$ \+ $OH$ | 702f | : |  $NH_{2}$ \+ $NO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $NO$
4f/b | : |  $OH$ \+ $H_{2}$ $\leftrightarrow$ $H$ \+ $H_{2}O$ | 704b | : |  $NH_{2}$ \+ $HONO$ $\leftarrow$ $NH_{3}$ \+ $NO_{2}$
10f | : |  $H$ \+ $O_{2}$ (+$M$) $\rightarrow$ $HO_{2}$ (+$M$) | 718b | : |  $NH_{2}$ \+ $NH_{2}$ (+$M$) $\leftarrow$ $N_{2}H_{4}$ (+$M$)
17b | : |  $O$ \+ $OH$ \+ $M$ $\leftarrow$ $HO_{2}$ \+ $M$ | 724b | : |  $N_{2}H_{4}$ \+ $NH_{2}$ $\leftarrow$ $N_{2}H_{3}$ \+ $NH_{3}$
672f | : |  $NH$ \+ $OH$ $\rightarrow$ $HNO$ \+ $H$ | 729f | : |  $N_{2}H_{3}$ $\rightarrow$ $N_{2}H_{2}$ \+ $H$
673f | : |  $NH$ \+ $OH$ $\rightarrow$ $NO$ \+ $H_{2}$ | 742f | : |  $N_{2}H_{3}$ \+ $NH_{2}$ $\rightarrow$ $H_{2}NN$ \+ $NH_{3}$
691f | : |  $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ | 789f | : |  $NH_{3}$ \+ $H$ $\rightarrow$ $NH_{2}$ \+ $H_{2}$
692b | : |  $NH_{2}$ \+ $HO_{2}$ $\leftarrow$ $NH_{3}$ \+ $O_{2}$ | 792f | : |  $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$
693f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ | 793f | : |  $NH_{3}$ \+ $NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$
698f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$ | 852f | : |  $H_{2}NO$ \+ $NH_{2}$ $\rightarrow$ $HNO$ \+ $NH_{3}$
699f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$ | 854f | : |  $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$

Figure B.1 shows that, according to Shrestha$\\_$2018, the oxidation process
at P1 starts with reaction 692b (1.00): $NH_{2}$ \+ $HO_{2}$ $\leftarrow$
$NH_{3}$ \+ $O_{2}$, (the numbers in parentheses denote the API value). At P2
reaction 692b is still active, exhibiting the largest API. The products of
this reaction, $HO_{2}$ and $NH_{2}$, now activate reactions 693f (0.06):
$NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$, 793f (0.05) $NH_{3}$ \+
$NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$ and 792f (0.06): $NH_{3}$ \+
$HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$. The $N_{2}H_{3}$ generated
activates reactions 724b (0.06): $N_{2}H_{4}$ \+ $NH_{2}$ $\leftarrow$
$N_{2}H_{3}$ \+ $NH_{3}$ and 729f (0.11): $N_{2}H_{3}$ $\rightarrow$
$N_{2}H_{2}$ \+ $H$. In turn, the $H$ generated activates reaction 1f (0.06):
$O_{2}$ \+ $H$ $\rightarrow$ $OH$ \+ $O$. Therefore, the commencement of the
oxidation process, according to Shrestha$\\_$2018, initially involves a
reaction producing $NH_{2}$ and $HO_{2}$ and then reactions producing
$H_{2}NO$, $N_{2}H_{3}$, $N_{2}H_{4}$, $N_{2}H_{2}$, $OH$, $O$ and $H$. Table
B.2 shows that almost all of these products are identified by the Po as
associated the most with the explosive mode at P1 and P2; e.g., $N_{2}H_{3}$
and $NH_{2}$ are the most pointed at P1 and $H_{2}NO$, $N_{2}H_{3}$ and
$NH_{2}$ at P2. The pointed species are reactants of the reactions with the
largest TPIs, listed in Table B.2; e.g., $N_{2}H_{3}$, $NH_{2}$ and $H_{2}NO$
are reactants of reactions 729f, 693f, 742f, 793f, 724b and 854f.

Figure B.1: Reactions exhibiting large APIs at points P1 and P2 in
Shrestha$\\_$2018; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parentheses
denote API values. The different colors are meant to highlight several phases
of the process as they are described in the text.

According to Fig. 4, P3 is positioned towards the end of the chemical runaway
regime. In comparison to P1 and P2, the CSP diagnostics on Table B.2 show that
$N2$ chemistry is not influencing the explosive dynamics any longer, while the
influence of $H_{2}NO$ and of $NO$ increases. This is reflected in both the
TPI and Po indices. In particular, the reactions contributing the most to the
explosive dynamics are 699f: $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$ and
854f: $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$, which involve as
reactants $NO$ and $H_{2}NO$, that exhibit the largest Pos and produce $OH$
and $HO_{2}$. The reactions providing the largest opposition are 698f:
$NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$ and 852f: $H_{2}NO$ \+
$NH_{2}$ $\rightarrow$ $HNO$ \+ $NH_{3}$, which consume $NO$ and $H_{2}NO$
towards mostly stable species.

This general pattern continues at P4, which is located in the thermal runaway
regime, where the $NH_{2}$ and $NO$ consuming reactions 699f and 698f keep
providing the largest contribution and opposition to $\tau_{e}$, respectively.
A number of reactions that consume $H_{2}NO$ and $NH_{2}$ provide minor
contributions. The ones promoting $\tau_{e}$ generate $HO_{2}$ and $OH$ (854f,
691f, 1f), in tandem with the major contributor 699f that produces $OH$. At
this point, Po identifies the temperature as the variable related the most to
the explosive mode. This feature is in agreement with Fig. 4, where it is
shown that at about P4 the temperature starts increasing sensibly.

Table B.2: The largest TPI and Po values for the explosive mode in Shrestha$\\_$2018; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parenthesis denote powers of ten. Bold/regular font represents exothermic/endothermic reactions. P1 | P2 | P3 | P4 | P5
---|---|---|---|---
t1 =0.00(0)s | t2 =5.24(-3)s | t3 =2.91(-2)s | t4 =9.80(-2)s | t5 =1.41(-1)s
$\tau_{e,f}$=7.31(-3)s |  $\tau_{e,f}$=1.05(-2)s |  $\tau_{e,f}$=1.76(-2)s |  $\tau_{e,f}$=3.60(-2)s |  $\tau_{e,f}$=1.85(-6)s
TPI
729f | +0.20 | 729f | +0.12 | 699f | +0.18 | 699f | +0.21 | 1f | +0.20
1f | +0.17 | 742f | -0.11 | 854f | +0.15 | 698f | -0.16 | 4f | +0.13
789f | -0.11 | 854f | +0.10 | 698f | -0.11 | 854f | +0.05 | 4b | -0.13
724b | +0.09 | 693f | +0.08 | 852f | -0.06 | 691f | +0.04 | 673f | -0.04
693f | +0.08 | 724b | +0.06 | 702f | +0.06 | 702f | +0.04 | 17b | +0.03
792f | +0.08 | 1f | +0.06 | 691f | +0.05 | 701f | -0.04 | 3f | +0.03
718b | +0.07 | 793f | +0.06 | 701f | -0.05 | 852f | -0.03 | 672f | -0.03
793f | +0.07 | 704b | +0.05 |  |  | 1f | +0.03 |  |
10f | -0.06 | 789f | -0.05 |  |  |  |  |  |
Po
$N_{2}H_{3}$ | +0.46 | $H_{2}NO$ | +0.24 | $H_{2}NO$ | +0.47 | $T$ | +0.88 | $T$ | +0.81
$NH_{2}$ | +0.22 | $N_{2}H_{3}$ | +0.20 | $NO$ | +0.16 | $H_{2}NO$ | +0.05 | $NH_{3}$ | +0.43
$HO_{2}$ | +0.13 | $NH_{2}$ | +0.18 | $NO_{2}$ | +0.14 | $NO$ | +0.05 | $H_{2}O$ | -0.37
$N_{2}H_{4}$ | +0.12 | $NO_{2}$ | +0.12 | $NH_{2}$ | +0.07 |  |  | $N_{2}$ | -0.17
|  | $N_{2}H_{2}$ | +0.10 | $H_{2}O_{2}$ | +0.06 |  |  | $O_{2}$ | +0.13
|  | $N_{2}H_{4}$ | +0.09 |  |  |  |  | $H$ | +0.07
|  |  |  |  |  |  |  | $OH$ | +0.05
|  |  |  |  |  |  |  | $H_{2}$ | +0.05

At P5, Table B.2 shows that hydrogen chemistry dominates the explosive
dynamics, mainly through the $OH$-producing reaction 1f, while the next
largest contributions originating from the two directions of reaction 4 cancel
each other. Several reactions provide minor contributions. The ones supporting
$\tau_{e}$ produce $OH$ (17b and 3f), while the ones opposing $\tau_{e}$
consume that species (673f and 672f).

##  Appendix C CSP Diagnostics for the Li mechanism

Table C.1 displays the reactions contributing to the explosive dynamics during
IDT in Li$\\_$2019\. The reactions exhibiting large APIs at P1 and P2 are
displayed in Fig. C.1. At P3 to P5 the reactions exhibiting large APIs are
similar to those exhibiting large TPIs. Table C.2 displays the TPI and Po
indices for the explosive mode at P1 to P5, shown in Fig. 4.

Table C.1: The reactions contributing the most to the explosive time scale in Li$\\_$2019\. Bold/regular font represents exothermic/endothermic reactions. 3f | : |  $H_{2}$ \+ $O$ $\rightarrow$ $H$ \+ $OH$ | 539b | : |  $NH_{2}$ \+ $HONO$ $\leftarrow$ $NH_{3}$ \+ $NO_{2}$
---|---|---|---|---|---
4f/b | : |  $H_{2}$ \+ $OH$ $\leftrightarrow$ $H$ \+ $H_{2}O$ | 553b | : |  $NH_{2}$ \+ $NH_{2}(+M)$ $\leftarrow$ $N_{2}H_{4}(+M)$
6f | : |  $O_{2}$ \+ $H$ $\rightarrow$ $O$ \+ $OH$ | 559b | : |  $N_{2}H_{4}$ \+ $NH_{2}$ $\leftarrow$ $N_{2}H_{3}$ \+ $NH_{3}$
35f | : |  $N$ \+ $NO_{2}$ $\rightarrow$ $N_{2}O$ \+ $O$ | 564f | : |  $N_{2}H_{3}$ $\rightarrow$ $N_{2}H_{2}$ \+ $H$
508f | : |  $NH$ \+ $OH$ $\rightarrow$ $NO$ \+ $H_{2}$ | 577f | : |  $N_{2}H_{3}$ \+ $NH_{2}$ $\rightarrow$ $H_{2}NN$ \+ $NH_{3}$
526f | : |  $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ | 605f | : |  $HNOH$ \+ $NH_{2}$ $\rightarrow$ $H_{2}NN$ \+ $H_{2}O$
527b | : |  $NH_{2}$ \+ $HO_{2}$ $\leftarrow$ $NH_{3}$ \+ $O_{2}$ | 624f | : |  $NH_{3}$ \+ $H$ $\rightarrow$ $NH_{2}$ \+ $H_{2}$
528f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ | 627f | : |  $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$
533f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$ | 628f/b | : |  $NH_{3}$ \+ $NH_{2}$ $\leftrightarrow$ $N_{2}H_{3}$ \+ $H_{2}$
534f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$ | 684f | : |  $H_{2}NO$ \+ $NH_{2}$ $\rightarrow$ $HNO$ \+ $NH_{3}$
536f | : |  $NH_{2}$ \+ $NO_{2}$ $\rightarrow$ $N_{2}O$ \+ $H_{2}O$ | 686f | : |  $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$
537f | : |  $NH_{2}$ \+ $NO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $NO$ |  |  |

A comparison of Fig. C.1 and Table C.2 with Fig. B.1 and Table B.2,
respectively, reveals that the most important features related to the
explosive dynamics of Li$\\_$2019 are similar to those of Shrestha$\\_$2018.

In particular, the oxidation process starts at P1 with reaction 527b: $NH_{2}$
\+ $HO_{2}$ $\leftarrow$ $NH_{3}$ \+ $O_{2}$ and proceeds at P2 with the
addition of reactions 528f: $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+
$O$, 628f: $NH_{3}$ \+ $NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$, 627f:
$NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$, 559b:
$N_{2}H_{4}$ \+ $NH_{2}$ $\leftarrow$ $N_{2}H_{3}$ \+ $NH_{3}$, 564f:
$N_{2}H_{3}$ $\rightarrow$ $N_{2}H_{2}$ \+ $H$ and 6f: $O_{2}$ \+ $H$
$\rightarrow$ $OH$ \+ $O$. Similarly to Shrestha$\\_$2018, the products of
these reactions at P1 and P2, notably among them $N_{2}H_{3}$, $H_{2}NO$ and
$NH_{2}$, exhibit large Pos and are reactants of the reactions exhibiting the
largest TPIs, as shown in Table C.2. These reactions are similar to those
exhibiting large TPIs at P1 and P2 in Shrestha$\\_$2018, but re-ordered in
terms of TPI. For example, at P1 the largest TPI in Li$\\_$2019 is exhibited
by reaction 559b: $N_{2}H_{4}$ \+ $NH_{2}$ $\leftarrow$ $N_{2}H_{3}$ \+
$NH_{3}$, which has the fourth largest TPI in Shrestha$\\_$2018, where it is
denoted as reaction 724b. At the same point, the largest TPI in
Shrestha$\\_$2018 is exhibited by reaction 729f: $N_{2}H_{3}$ $\rightarrow$
$N_{2}H_{2}$ \+ $H$, which has the fifth largest TPI in Li$\\_$2019, where it
is denoted as 564f.

Figure C.1: Reactions exhibiting large APIs at points P1 and P2 in
Li$\\_$2019; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parentheses denote
API values. The different colors are meant to highlight several phases of the
process as they are described in the text.

From P3 on, the similarities between Li$\\_$2019 and Shrestha$\\_$2018 keep
increasing, in terms of both TPI and Po. In particular, at P3 and P4, the
major contributions to $\tau_{e}$ originate from reactions 534f: $NH_{2}$ \+
$NO$ $\rightarrow$ $NNH$ \+ $OH$ and 686f: $H_{2}NO$ \+ $O_{2}$ $\rightarrow$
$HNO$ \+ $HO_{2}$ (699f and 854f, respectively in Shrestha$\\_$2018), while
the major opposition originates from reaction 533f: $NH_{2}$ \+ $NO$
$\rightarrow$ $N_{2}$ \+ $H_{2}O$ (698f in Shrestha$\\_$2018). Closely similar
are the pointed variables; e.g. the most pointed $H_{2}NO$ at P3 and $T$ at
P4.

Also, a stronger similarity between Li$\\_$2019 and Shrestha$\\_$2018 is
manifested at P5, where the explosive dynamics are dominated by hydrogen
chemistry and $OH$ production and the most pointed variable is the
temperature.

Table C.2: The largest TPI and Po values for the fast explosive mode in Li$\\_$2019; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parenthesis denote powers of ten. Bold/regular font represents exothermic/endothermic reactions. P1 | P2 | P3 | P4 | P5
---|---|---|---|---
t1 =0.00(0)s | t2 =5.15(-3)s | t3 =2.66(-2)s | t4 =9.00(-2)s | t5 =1.28(-1)s
$\tau_{e,f}$=4.90(-3)s |  $\tau_{e,f}$=8.22(-3)s |  $\tau_{e,f}$=2.38(-2)s |  $\tau_{e,f}$=3.30(-2)s |  $\tau_{e,f}$=1.88(-6)s
TPI
559b | +0.21 | 628f | +0.11 | 534f | +0.20 | 534f | +0.21 | 6f | +0.24
553b | +0.17 | 539b | +0.11 | 533f | -0.15 | 533f | -0.17 | 4f | +0.13
628f | +0.15 | 686f | +0.09 | 686f | +0.11 | 686f | +0.05 | 4b | -0.13
6f | +0.11 | 559b | +0.08 | 684f | -0.05 | 536f | -0.05 | 508f | -0.04
564f | +0.10 | 528f | +0.08 | 536f | -0.05 | 537f | +0.04 | 3f | +0.04
624f | -0.07 | 577f | -0.08 | 526f | +0.04 | 526f | +0.04 | 624f | -0.03
528f | +0.07 | 628b | -0.05 | 605f | -0.04 |  |  |  |
35f | -0.05 |  |  | 537f | +0.04 |  |  |  |
Po
$N_{2}H_{3}$ | +0.41 | $NH_{2}$ | +0.26 | $H_{2}NO$ | +0.42 | $T$ | +0.86 | $T$ | +0.80
$NH_{2}$ | +0.28 | $H_{2}NO$ | +0.25 | $NO$ | +0.24 | $H_{2}NO$ | +0.05 | $NH_{3}$ | +0.46
$N_{2}H_{4}$ | +0.22 | $NO_{2}$ | +0.23 | $NO_{2}$ | +0.14 | $NO$ | +0.05 | $H_{2}O$ | -0.39
$HO_{2}$ | +0.06 | $N_{2}H_{3}$ | +0.12 | $H_{2}$ | -0.11 |  |  | $N_{2}$ | -0.17
|  | $N_{2}H_{4}$ | +0.12 | $H_{2}O_{2}$ | +0.07 |  |  | $O_{2}$ | +0.12
|  | $H_{2}$ | -0.09 | $NH_{2}$ | +0.07 |  |  | $H$ | +0.08
|  | $N_{2}H_{2}$ | +0.05 | $T$ | +0.06 |  |  | $OH$ | +0.05

##  Appendix D CSP Diagnostics for the Stagni mechanism

Table D.1 lists the reactions generating the explosive dynamics in
Stagni$\\_$2020 during IDT. The reactions exhibiting large APIs at P1 and P2
are displayed in Fig. D.1, while at P3 to P5 the reactions exhibiting large
APIs exhibit large TPIs as well. Table D.2 displays the TPI and Po indices for
the explosive mode at P1 to P5, that are indicated in Fig. 4.

Table D.1: The reactions contributing the most to the explosive time scale in Stagni$\\_$2020\. Bold/regular font represents exothermic/endothermic reactions. 3f/b | : |  $H_{2}$ \+ $OH$ $\leftrightarrow$ $H$ \+ $H_{2}O$ | 37f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$
---|---|---|---|---|---
5f | : |  $O_{2}$ \+ $H$ $\rightarrow$ $O$ \+ $OH$ | 38f | : |  $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $OH$ \+ $H_{2}NO$
9f | : |  $H_{2}O_{2}$ (+$M$) $\rightarrow$ $OH$ \+ $OH$ (+$M$) | 74f | : |  $NH_{2}$ \+ $NO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $NO$
26f/b | : |  $NH_{3}$ \+ $H$ $\leftrightarrow$ $H_{2}$ \+ $NH_{2}$ | 76f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$
29f | : |  $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$ | 77f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$
30f/b | : |  $NH_{3}$ \+ $O_{2}$ $\leftrightarrow$ $NH_{2}$ \+ $HO_{2}$ | 171f | : |  $H_{2}NO$ \+ $NH_{2}$ $\rightarrow$ $HNO$ \+ $NH_{3}$
36f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $OH$ | 172f | : |  $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$

Figure D.1: Reactions exhibiting large APIs at points P1 and P2 in
Stagni$\\_$2020; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parentheses
denote API values. The different colors are meant to highlight several phases
of the process as they are described in the text.

The explosive dynamics in Stagni$\\_$2020 resembles to an extent that of
Glarborg$\\_$2018\. In particular, as shown in Fig. D.1, the oxidation process
commences at P1 with reaction 30f (1.00): $NH_{3}$ \+ $O_{2}$ $\rightarrow$
$NH_{2}$ \+ $HO_{2}$, which is one of the two initiation reactions in
Glarborg$\\_$2018; see Fig. A.1. At P2, this reaction retains a dominant
impact, assisted by reactions 38f (0.07): $NH_{2}$ \+ $HO_{2}$ $\rightarrow$
$OH$ \+ $H_{2}NO$, 29f (0.06): $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+
$H_{2}O_{2}$ and 37f (0.06): $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+
$O$; the last two also encountered in Glarborg$\\_$2018\. However, in this
case there is a strong opposition to the impact of the explosive mode from 30b
(-0.12): $NH_{3}$ \+ $O_{2}$ $\leftarrow$ $NH_{2}$ \+ $HO_{2}$. It is due to
the action of this reaction that the chemical runaway is terminated very fast,
as shown in Fig. 4, since the action of the initiation reaction 30f is
suppressed. The products of reactions 30f, 38f, 29f and 37f, that promote the
impact of the explosive mode (large and positive APIs) exhibit large Pos and
are reactants of the reactions exhibiting large TPIs, listed in Table D.2. For
example, at P1, $NH_{2}$ ($Po=0.71$) is the product of reaction 30f
($API=1.00$) and a reactant of reaction 37f: $NH_{2}$ \+ $O_{2}$ $\rightarrow$
$H_{2}NO$ \+ $O$ ($TPI=0.64$). Similarly, at P2, $NH_{2}$ ($Po=0.38$) is a
product of reactions 30f ($API=0.67$) and 29f ($API=0.06$) and a reactant of
30b: $NH_{3}$ \+ $O_{2}$ $\leftarrow$ $NH_{2}$ \+ $HO_{2}$ ($TPI=-0.34$), 37f:
$NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ ($TPI=0.23$) and 38f:
$NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $OH$ \+ $H_{2}NO$ ($TPI=0.1$6).

According to Table D.2, at P1 and P2 the reactions promoting the explosive
character of the mode (positive TPIs) the most are 37f at P1 and 37f and 38f
at P2. What distinguishes this case from all previous ones, is the strong
opposition to the explosive character of the mode (negative TPI) by reaction
30b: $NH_{3}$ \+ $O_{2}$ $\leftarrow$ $NH_{2}$ \+ $HO_{2}$, which exhibits the
largest TPI in magnitude right after the initiation of the process. This is
due to the fact that reactions 37f and 38f compete with reaction 30b for the
same reactants and to the fact that the products of 30b are stable molecules,
while those of 37f and 38f are not. This action of reaction 30b is in contrast
to the action of its forward direction, which is the one promoting the most
the impact of the explosive mode, as evidenced by the fact that it exhibits
the largest API at both P1 and P2; see Fig. D.1. As a result of this action of
reaction 30b, $\tau_{e}$ decelerates abruptly, as shown in Fig. 4.

Table D.2: The largest TPI and Po values for the fast explosive mode in Stagni$\\_$2020; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parenthesis denote powers of ten. Bold/regular font represents exothermic/endothermic reactions. P1 | P2 | P3 | P4 | P5
---|---|---|---|---
t1 =0.00(0)s | t2 =3.40(-4)s | t3 =1.40(-1)s | t4 =4.53(-1)s | t5 =6.44(-1)s
$\tau_{e,f}$=4.65(-3)s |  $\tau_{e,f}$=1.80(-2)s |  $\tau_{e,f}$=4.56(-1)s |  $\tau_{e,f}$=1.81(-1)s |  $\tau_{e,f}$=6.97(-6)s
TPI
37f | +0.64 | 30b | -0.34 | 30b | -0.17 | 30b | -0.17 | 5f | +0.21
172f | +0.14 | 37f | +0.23 | 38f | +0.13 | 38f | +0.12 | 3f | +0.11
29f | +0.09 | 38f | +0.16 | 30f | +0.08 | 30f | +0.09 | 3b | -0.10
36f | +0.05 | 172f | +0.08 | 76f | -0.08 | 76f | -0.08 | 26f | -0.07
9f | +0.05 | 36f | +0.02 | 172f | +0.06 | 77f | +0.06 | 26b | +0.05
|  |  |  | 74f | +0.05 | 172f | +0.05 |  |
Po
$NH_{2}$ | +0.71 | $NH_{2}$ | +0.38 | $T$ | +0.95 | $T$ | +0.98 | $T$ | +0.96
$H_{2}NO$ | +0.13 | $H_{2}NO$ | +0.35 | $H_{2}$ | +0.06 | $H_{2}$ | +0.04 | $O_{2}$ | -0.59
$HO_{2}$ | +0.10 | $NO_{2}$ | +0.11 |  |  |  |  | $NH_{3}$ | +0.38
$H_{2}O_{2}$ | +0.05 | $N_{2}H_{4}$ | +0.07 |  |  |  |  | $H_{2}$ | +0.22
|  |  |  |  |  |  |  | $H_{2}O$ | -0.16
|  |  |  |  |  |  |  | $H$ | +0.07
|  |  |  |  |  |  |  | $OH$ | +0.07

The situation at P3 and P4 is an extension of that at P2. In particular, the
opposing action of 30b is still present, counteracting 38f, which is the
strongest proponent of explosive dynamics at P3 and P4. The influence of
reaction 38f is still assisted by that of reactions 37f and 172f in producing
$OH$ by recycling $HO_{2}$ and $H_{2}NO$. This promoting action is further
supported by reactions 30f (which produces the reactants of the most promoting
reaction 38f), 74f (which produces the reactant of 172f) and 77f (which
produces OH). The opposing action of 30b is supported by reaction 76f, which
consumes $NH_{2}$ that is the reactant of the promoting reactions 38f, 74f and
77f. According to Fig. 4, P3 and P4 belong to the thermal runaway, therefore
the most pointed variable is the temperature, as shown in Table D.2. In fact,
at P4 the combined action of the promoting reactions leads to $OH$ generation.
In particular, reactions 38f and 77f produce $OH$, while reactions 30f and
172f produce $NH_{2}$ and $HO_{2}$ that are reactants of 38f and 77f. At P5
hydrogen chemistry dominates and the $OH$ generation intensifies. This species
is generated by reaction 5f. The $H$-consuming reactions 3b and 2f oppose the
explosive character of the mode, since $H$ is reactant of the most promoting
reaction 5f. The most pointed variable is still temperature.

It is noted that in Stagni$\\_$2020 there is no substantial influence of
$N2$-chemistry.

##  Appendix E CSP Diagnostics for Zhang mechanism

The reactions contributing to the explosive dynamics during IDT in
Zhang$\\_$2021 are listed in Table E.1. The reactions exhibiting large APIs at
P1 and P2 are displayed in Fig. E.1, while those exhibiting large APIs at P3
to P5 are similar to those exhibiting large TPIs. Table E.2 displays the TPI
and Po indices for the explosive mode at P1 to P5, shown in Fig. 4.

Table E.1: The reactions contributing the most to the explosive time scale in Zhang$\\_$2021\. Bold/regular font represents exothermic/endothermic reactions. 1f | : |  $H$ \+ $O_{2}$ $\rightarrow$ $O$ \+ $OH$ | 69f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $N_{2}$ \+ $H_{2}O$
---|---|---|---|---|---
3f | : |  $O$ \+ $H_{2}$ $\rightarrow$ $OH$ \+ $H$ | 89f | : |  $N_{2}H_{2}$ \+ $HO_{2}$ $\rightarrow$ $NNH$ \+ $H_{2}O_{2}$
4f/b | : |  $OH$ \+ $H_{2}$ $\leftrightarrow$ $H$ \+ $H_{2}O$ | 103f | : |  $N_{2}H_{3}(+M)$ $\rightarrow$ $N_{2}H_{2}$ \+ $H(+M)$
58f | : |  $NH_{2}$ \+ $H$ $\rightarrow$ $NH$ \+ $H_{2}$ | 161f | : |  $NH_{3}$ \+ $H$ $\rightarrow$ $NH_{2}$ \+ $H_{2}$
63f | : |  $NH_{2}$ \+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ | 163f | : |  $NH_{3}$ \+ $HO_{2}$ $\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$
64f/b | : |  $NH_{2}$ \+ $HO_{2}$ $\leftrightarrow$ $NH_{3}$ \+ $O_{2}$ | 165f | : |  $NH_{3}$ \+ $NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$
65f | : |  $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ | 221f | : |  $H_{2}NO$ \+ $O_{2}$ $\rightarrow$ $HNO$ \+ $HO_{2}$
68f | : |  $NH_{2}$ \+ $NO$ $\rightarrow$ $NNH$ \+ $OH$ |  |  |

Figure E.1: Reactions exhibiting large APIs at points P1 and P2 in
Zhang$\\_$2021; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parentheses
denote API values. The different colors are meant to highlight several phases
of the process as they are described in the text.

The explosive dynamics in Zhang$\\_$2021 resembles closely that of
Stagni$\\_$2020 and to an extent that of Glarborg$\\_$2018\. As shown in Fig.
E.1, the oxidation process commences at P1 with reaction 64b (1.00): $NH_{3}$
\+ $O_{2}$ $\rightarrow$ $NH_{2}$ \+ $HO_{2}$, which is the same as the
initiation reaction in Stagni$\\_$2020 and one of the two initiation reactions
in Glarborg$\\_$2018; see Figs. D.1 and A.1. With one exception, the reactions
contributing significantly to the impact of the explosive mode at P2 (large
APIs) are the same as in Stagni$\\_$2020\. That is, the dominant impact is
provided by reaction 64b (0.69), assisted by reaction 63f (0.04): $NH_{2}$ \+
$HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$, 163f (0.07): $NH_{3}$ \+ $HO_{2}$
$\rightarrow$ $NH_{2}$ \+ $H_{2}O_{2}$ and 165f (0.06): $NH_{3}$ \+ $NH_{2}$
$\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$. Reactions 64b, 63f and 163f are active
at this point in Stagni$\\_$2020, while reaction 165f is not. It is noted
that, contrary to Stagni$\\_$2020, now there is substantial $N2$ chemistry via
reaction 165f. Similarly to Stagni$\\_$2020, the major opposition to the
impact of the explosive dynamics is provided by reaction 64f (-0.10): $NH_{3}$
\+ $O_{2}$ $\leftarrow$ $NH_{2}$ \+ $HO_{2}$, which again leads to a fast
termination of the chemical runaway, as shown in Fig. 4, since it counteracts
the influence of the most important initiation reaction 64b.

Similarly to all previous cases, the products of reactions 64b, 63f, 163f,
165f, which exhibit large and positive APIs, thus promoting the impact of the
explosive mode, demonstrate large Pos and are reactants of the reactions with
large TPIs, listed in Table E.2. According to Table E.2, $\tau_{e}$ at P1 is
supported by reactions 165f and 65f, which consume $NH_{2}$ that is produced
there by the large-API reaction 64b. Further support to $\tau_{e}$ is provided
by reaction 103f that consumes $N_{2}H_{3}$ produced by reaction 165f.
Reaction 103f also produces $H$, which feeds into reaction 1f that also
supports $\tau_{e}$. The main opposition to the explosive character of the
mode at P1 is provided by reaction 161f, which consumes $NH_{3}$ and $H$ that
are reactants of the main supporting reactions.

At P2 the reactions supporting $\tau_{e}$ the most are 165f: $NH_{3}$ \+
$NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$ and 1f: $H$ \+ $O_{2}$
$\rightarrow$ $O$ \+ $OH$ that survive from P1, in addition to 63f: $NH_{2}$
\+ $HO_{2}$ $\rightarrow$ $H_{2}NO$ \+ $OH$ and 221f: $H_{2}NO$ \+ $O_{2}$
$\rightarrow$ $HNO$ \+ $HO_{2}$. Similarly to Stagni$\\_$2020 and in contrast
to all other cases considered, there is a strong opposition to the explosive
character realised by reaction 64f: $NH_{2}$ \+ $HO_{2}$ $\leftrightarrow$
$NH_{3}$ \+ $O_{2}$, the reverse direction of which provides the largest
contribution to the impact of the explosive mode (largest API). Like
Stagni$\\_$2020, this action of reaction 64f decelerates the process.

Table E.2: The largest TPI and Po values for the fast explosive mode in Zhang$\\_$2021; T0=1100 K, p0=2 atm and $\phi$=1.0. Numbers in parenthesis denote powers of ten. Bold/regular font represents exothermic/endothermic reactions. P1 | P2 | P3 | P4 | P5
---|---|---|---|---
t1 =0.00(0)s | t1 =3.24(-4)s | t3 =7.64(-2)s | t4 =2.65(-1)s | t5 =3.78(-1)s
$\tau_{e,f}$=2.52(-3)s |  $\tau_{e,f}$=9.45(-3)s |  $\tau_{e,f}$=2.81(-1)s |  $\tau_{e,f}$=1.03(-1)s |  $\tau_{e,f}$=2.23(-6)s
TPI
161f | -0.19 | 64f | -0.24 | 63f | +0.19 | 64f | -0.18 | 1f | +0.23
165f | +0.18 | 165f | +0.16 | 64f | -0.18 | 63f | +0.17 | 4f | +0.14
103f | +0.18 | 63f | +0.09 | 64b | +0.10 | 64b | +0.10 | 4b | -0.13
1f | +0.17 | 161f | -0.08 | 69f | -0.09 | 69f | -0.09 | 58f | -0.04
65f | +0.11 | 1f | +0.07 | 221f | +0.06 | 221f | +0.06 | 3f | +0.04
163f | +0.06 | 221f | +0.07 | 68f | +0.04 | 68f | +0.05 | 161f | -0.04
|  | 89f | +0.04 | 219f | -0.04 |  |  |  |
Po
$NH_{2}$ | +0.49 | $N_{2}H_{2}$ | +0.38 | $T$ | +1.01 | $T$ | +0.99 | $T$ | +0.82
$N_{2}H_{3}$ | +0.29 | $H_{2}NO$ | +0.33 |  |  |  |  | $NH_{3}$ | +0.53
$HO_{2}$ | +0.10 | $NH_{2}$ | +0.20 |  |  |  |  | $H_{2}O$ | -0.46
$H_{2}O_{2}$ | +0.05 | $H_{2}$ | -0.19 |  |  |  |  | $N_{2}$ | -0.18
|  | $N_{2}H_{3}$ | +0.17 |  |  |  |  |  |

A comparison of the CSP diagnostics in Stagni$\\_$2020 and Zhang$\\_$2021,
displayed in Tables D.2 and E.2, reveals that columns P2 to P4 display similar
results, with the exception of the most promoting reaction in P2, which is
37f: $NH_{2}$ \+ $O_{2}$ $\rightarrow$ $H_{2}NO$ \+ $O$ in Stagni$\\_$2020 and
165f: $NH_{3}$ \+ $NH_{2}$ $\rightarrow$ $N_{2}H_{3}$ \+ $H_{2}$ in
Zhang$\\_$2021\. In particular, at P3 and P4 the opposing influence of
reaction 64f persists, assisted by reaction 69f, both consuming $NH_{2}$. The
major proponent of $\tau_{e}$ are the $NH_{2}$ and $HO_{2}$-consuming reaction
63f, the $NH_{3}$ and $O_{2}$ consuming reaction 64b, the $H_{2}NO$-consuming
reaction 221f and the $NH_{2}$-consuming reaction 68f. As in Stagni$\\_$2020,
the combined action of the promoting reactions 63f, 64b, 221f and 68f leads to
$OH$-generation. This is achieved directly by reactions 63f and 68f and
indirectly by 64b and 221, which produce $NH_{2}$ and $HO_{2}$ that are
reactants of 63f and 68f. P3 and P4 belong to the thermal runaway, as shown in
Fig. 4. Therefore, the variable associated the most to the explosive mode is
the temperature, as shown in Table E.2.

At P5, where $\tau_{e}$ records minimum value, the reactions supporting
$\tau_{e}$ the most relate to hydrogen chemistry, similarly to all other cases
considered. In particular, the largest contribution originates from reaction
$H$ \+ $O_{2}$ $\rightarrow$ $O$ \+ $OH$, assisted by reaction 3f: $O$ \+
$H_{2}$ $\rightarrow$ $OH$ \+ $H$ , while the contributions of the two
directions of reaction 4: $OH$ \+ $H_{2}$ $\leftrightarrow$ $H$ \+ $H_{2}O$
effectively cancel. The major opposition originate from reactions 58f and
161f, which relate to nitrogen chemistry and consume $H$, which is reactant
the major proponent to $\tau_{e}$, reaction 1f. The temperature remains the
most pointed variable.

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