# The recombination region in lean steady premixed H2 flames.

José Graña-Otero<EMAIL_ADDRESS>Dept. of Mechanical Engineering,
University of Kentucky.
279 RGAN Building, Lexington, KY. USA. Phone: (859) 218-0645

###### Abstract

H2 premixed flames are well known for a long, trailing region where the
unburnt H2 and the super-equilibrium concentrations of radicals left past the
fuel consumption layer gradually decay to thermodynamic equilibrium. This
recombination region, it’s argued here, is a second order effect induced by
the premature quenching of the shuffle reactions, which inhibits the decay to
equilibrium in the first approximation.

Its structure and kinetics are studied in detail to capture the small but
finite reaction rates accounting for its characteristic length scale, which is
large enough to render the diffusive transport negligible, hence deactivating
the upstream feedback link with the main flame structure. It is isothermal and
can be described, in moderately lean flames, by just the distribution of H2 as
the sole degree of freedom, a drastic reduction consequence of the strongly
constrained evolution imposed by the almost exactly quenched shuffle
reactions.

The rate of decay to equilibrium of H2 and radicals is thus dictated by the
balance between the convective transport and chemical sinks controlled by the
HO2 kinetics, which becomes dominant after the shuffle reactions quench.
Essentially, it is a two-step mechanism with the first, rate-limiting step
converting O2 into HO2, whereas the second one, comprising multiple fast,
parallel pathways, depletes radicals as HO2 is converted into O2 and H2O.

Generalizations to very lean and nearly stoichiometric flames, when the one
degree of freedom model is not applicable, as well as the effect of heat
losses are also briefly discussed.

###### keywords:

Hydrogen premixed flames , Recombination region , Reduced hydrogen kinetics ,
Shuffle reactions , Hydroperoxyl kinetics.

††journal: Combustion and Flame

Nomenclature $\dot{\mathcal{C}}$ total molar consumption rate
($kmole/(m^{3}s)$). $\mathcal{D}_{\ce{H2}}$ $\ce{H2}$ mixture mass-averaged
diffusivity ($m^{2}/s$). $\mathcal{D}$ operator
$X_{\ce{H2}}^{-1}\partial()/\partial\xi$ defined in (7). $\mathcal{E}$
relative deviation from exact partial equilibrium. $[i]$ molar concentration
of species $i$ ($kmole/m^{3}$). $k_{i}$ Arrhenius reaction rate constant in
units $kmole$, $m^{3}$ and $s$. $\mathcal{K}_{i}$ equilibrium constant of the
reaction $i$.. $\mathcal{L}$ local length scale of the H2 distribution.
$\mathcal{L}^{o}$ reference length scale (6). $\dot{m}^{\prime\prime}_{L}$
laminar flame mass consumption flux ($kg/(m^{2}s)$). $\mathcal{M}_{i}$
molecular weight of species $i$. $\dot{\mathcal{N}}$ net molar production rate
$\dot{\mathcal{P}}-\dot{\mathcal{C}}$. $\dot{\mathcal{P}}$ total molar
production rate ($kmole/(m^{3}s)$). $Pe_{r}$ Peclet number in the
recombination region. $\dot{\mathscr{S}}^{e}$ molar production rate from all
out-of-equilibrium reactions ($kmole/(m^{3}s)$). $u$ flow velocity relative to
the flame ($m/s$). $x$ coordinate along the flame ($mm$). $X$ mole fraction.
$Y$ mass fraction. $\acute{X}$ mole fraction derivative with respect to
$X_{\ce{H2}}$. $z$ reduced H2 concentration
$[\ce{\ce{H2}}]^{b}/[\ce{\ce{H2}}]$. Greek: $\alpha$ branching ratio
$\dot{\omega}_{17}/\dot{\omega}_{13}$. $\delta_{L}$ flame thickness.
$\gamma_{m}$ scaled rate $\dot{\omega}^{\prime e}_{m,f}/\dot{\omega}^{\prime
e}_{13,r}$ of the HO2 reaction $m$ (B). $\tilde{\Gamma}_{4}$ regular part
$\tilde{\omega}_{4}/(1-z^{2})$ of the rate of reaction 4 (14). $\phi$
equivalence ratio. $\psi_{i}$ scaled mass fraction $Y_{i}/Y^{e}_{i}$.
$\tilde{\Omega}_{i}$ scaled reaction rates of the shuffle reactions
$\dot{\omega}^{e}_{i,f}/\dot{\omega}^{e}_{13,f}$. $\dot{\omega}_{i}$ molar
reaction rate of reaction $i$ ($kg/m^{3}$). $\rho$ density ($kg/m^{3}$).
$\tau$ characteristic time ($s$). $\xi$ coordinate $x$ scaled with
$\mathcal{L}^{o}$. subscripts: f, r forward, reverse. superscripts: b burnt
thermodynamic equilibrium. e shuffle reactions partial equilibrium
concentrations (3). u fresh mixture. $0$ beginning of the recombination
region. accents: $\tilde{}$ rates scaled with $\dot{\omega}_{13f}$. $\dot{}$
magnitude per unit time .

##  1 Introduction.

The structure of flames has often been investigated using an overall
irreversible reaction between the fuel and the oxidizer, with a rate
proportional to the mass fraction of the limiting reactant and an Arrhenius
dependence on the temperature (see for instance the classic references by Ya.
B. Zel’dovich [1], F. A. Williams [2], A. Liñán [3] and Sivashinsky [4]). The
chemical reaction is in this case essentially confined to the so-called
reaction or fuel consumption layer. Outside it, the chemical reaction is
either frozen, with too small a rate to be significant when the temperature is
too low; or in equilibrium, with zero net rate because one of the reactants
has been depleted.

In these models, chemical equilibrium is quickly reached right downstream of
the reaction layer, once the deficient reactant has been exhausted. Without
source terms, the temperature and species remain constant downstream of steady
flames, so zero-gradients are the appropriate downstream boundary conditions.

Flames with more complex and realistic kinetic mechanisms can be as well
understood using the same basic framework [5]; namely a thin reaction layer,
where the chemical activity is confined, flanked by transport regions where
reactions are either frozen or in equilibrium.

However the presence of many competing chemical reactions introduce additional
structure inside the fuel consumption layer, with nested and overlapping sub-
layers, that complicate extraordinarily the analysis, even in relatively
simple cases such as hydrogen [6, 7, 8] or methane [9, 10, 11].

These difficulties are due to the presence of highly reactive intermediate
radicals such as O, OH and H and many others present in complex fuels.
Although present in much smaller amounts than the main species, radicals are
nevertheless produced in the reaction layer in quantities largely exceeding
the values corresponding to equilibrium at the local temperature.

Most of these intermediate species are typically short-lived and their
concentrations decay rapidly outside the fuel consumption layer.

However, intriguingly, despite their high reactivity in both H2 and
hydrocarbon fuels, O, OH and H typically decay at much slower rates than most
of other radicals and can be found at relatively large distances downstream of
the fuel consumption layer in large amounts compared with the final
equilibrium values.

All the same, chemical equilibrium, or equivalently, zero-gradients of
temperature and species are still used as boundary conditions right downstream
of the fuel consumption layer, somehow ignoring the tail of slow decay of
these radicals.

To understand these aspects, this paper analyzes the structure of the high-
temperature region – hereafter referred to as the recombination region –,
where H2 and radicals leaking downstream of the fuel consumption layer decay
to the burnt thermodynamic equilibrium state.

Figure 1: Distributions, in a $\phi=0.6$ H2-air flame, of the molar fractions
$X$ of the main species and radicals in the fuel consumption layer (top left)
and in the recombination region (top right) where the burnt thermodynamic
equilibrium is finally attained. The values corresponding to this state –
denoted in ?he text with the $b$ superscript – are represented by the small
dash-dot segments (indistinguishable from the solid lines in the case of O2
and H2O). The origin of $x$ coincides with the HO2 peak production rate, found
in the fuel consumption layer for all values of the equivalence ratio $\phi$.
The bottom plot shows the variation with $\phi$ of the final burnt equilibrium
values $b$. The dash-dot lines represent the corresponding values $X^{0}_{i}$
at the beginning of the recombination region; these are evaluated as the
maxima of O, OH and H, or the values of $X^{0}_{i}$ at the location of the
maximum of $X_{\ce{H}}$ in the case of H2 and O2. $X_{\ce{O2}}$ is plotted
separately for clarity.

This investigation is relevant to the structure of the OH* – excited OH –
chemiluminescence radiation, studied by Graña and Mahmoudi in [12] where it is
shown that a significant fraction of the total radiated intensity can be
originated and supported by the underlying H2 flame recombination region.
Similarly, the recombination region plays an important role in the
$\ce{NO_{x}}$ distribution structure in H2 flames. More specifically, as will
be shown in coming work, the persistent concentrations of NO and N2O left
behind the fuel consumption layer are actually supported by the underlying
recombination region studied in this paper.

The rest of the paper starts with Section 2. The recombination region, giving
an overview of the main properties of the recombination region. Section 3.
Kinetics in the recombination region, analyzes in detail its kinetic structure
using a reduced version of the H2-air mechanism which is simple but still
contains the essential elements. Section 4. Formulation, studies in detail the
problem of the structure of the recombination region, showing how it can be
simplified and solved. Finally, Section 5. Discussion and conclusions,
discusses the limitations and applicability of the findings. In particular,
the effect of heat losses on the structure of the recombination region is
addressed in some detail to determine when, and to some extent how, the
analysis should be modified in their presence.

##  2 The recombination region.

Chemical equilibrium, or equivalently zero gradients of the temperature and
species distributions, are the appropriate boundary conditions downstream of a
planar, steady premixed flame. These are to be applied past the flame, at a
distance large enough to ensure that thermodynamic equilibrium has been
attained. However, in analytical studies, they are often applied immediately
downstream of the fuel consumption layer, as if equilibrium were reached right
after it. The predictions so obtained are in good agreement with the numerical
or experimental results. Nevertheless, when solving numerically (exactly) this
problem, it is found that these conditions need to be enforced at a much
larger distance downstream of the fuel consumption layer in order to avoid
unphysical discontinuities in the gradients.

Figure 1 shows an example of such numerical solutions in the case of a H2-air
flame with an equivalence ratio $\phi=0.6$. These results and those of the
rest of the paper have been obtained with a customized version of the open
source code Cantera [13], together with the Glarborg et al. [14] kinetic
mechanism with inert N2. The results obtained with the full (including the N2
oxidation) mechanism are indistinguishable from the results presented below.
Similar results, with inessential differences in the numerical results, are
obtained as well with the San Diego [15] or the more recent Konnov [16]
mechanisms.

As can be seen, the temperature, O2 and H2O rapidly reach their final
equilibrium concentrations, so the zero-gradient boundary conditions seem
accurate for them right after the fuel consumption layer. However, the
concentrations of H2 and radicals slowly decay over a relatively large length
until they finally reach the zero-gradient condition at the final burnt
equilibrium state.

Understanding this behavior and in particular the length scale associated with
the radicals decay requires the analysis of the interplay between transport
and kinetics, which are briefly overviewed in the rest of this section, also
anticipating the main results analyzed in detail in the rest of the paper.

Figure 2: The left plot represents the H2 mass fraction distributions for
several representative values of the equivalence ratio. The dots mark the
location where $Y_{\ce{H2}}$ has decayed to $Y^{b}_{\ce{H2}}/Y_{\ce{H2}}=0.9$
with $Y^{b}_{\ce{H2}}$ the final burnt equilibrium mass fraction. The right
plot represents the corresponding distributions of the Péclet number
$Pe_{r}=u\mathcal{L}/\mathcal{D}_{\ce{H2}}$, based on $\mathcal{L}$ and on the
H2 mass diffusivity. This plot shows that the recombination region is
essentially a convective-reactive region, with $Pe_{r}\gg 1$, for lean flames
with not to small values of $\phi$.

###  2.1 Transport.

The reduced but finite reaction rates prevalent in the recombination region
make its length scale larger than, but comparable to, that of the preheat
region, as suggested by Figure 1. As a result of this length scale disparity,
the gradients in the recombination region are much smaller than those in the
fuel consumption layer; they can therefore be set to zero – chemical
equilibrium in a first approximation – when the solution is seen with the
scale of the fuel consumption layer. Accordingly, radicals concentrations
remain constant downstream of the fuel consumption layer at distances
comparable to its length scale, as in the top plot of Figure 1. However, when
the proper scale for the much longer recombination region is used, as in the
middle plot of this Figure, concentrations are seen to slowly decay and
finally level off to zero slopes – chemical equilibrium to all orders.

This characteristic length – larger even than that of the preheat region where
convective and diffusive transport are of the same order of magnitude – render
the transport of species and enthalpy dominated by convection in the first
approximation. A quantitative measure of the relative magnitude of each
transport mode, shown in Figure 2, is the Péclet number
$Pe_{r}=u\mathcal{L}/\mathcal{D}_{\ce{H2}}$, based on the local characteristic
length $\mathcal{L}=Y_{\ce{H2}}/(\partial Y_{\ce{H2}}/\partial x)$. $Pe_{r}$
is also of the order of the ratio $\mathcal{L}/\delta_{L}$ of $\mathcal{L}$ to
the flame thickness (without the recombination region)
$\delta_{L}\sim\mathcal{D}_{\ce{H2}}/u$, based on the H2 mass diffusivity and
on the local flow velocity $u$ (in a frame of reference attached to the
flame), of the order of the burnt products flame speed111The preheat region
length $\delta_{L}$ should be estimated using the values of
$\mathcal{D}_{\ce{H2}}$ and $u$ upstream the fuel consumption layer (at the
fresh mixture temperature for instance), whereas those entering in the
definition of $Pe_{r}$ are evaluated in the recombination region, around the
burned equilibrium temperature. However, even in the hottest flames near
stoichiometry, the length $\delta_{L}$ remains essentially of the same order
of magnitude on both sides of the flame (see Figure 3)..

The recombination region is thus characterized by moderately larger than unity
Péclet (or Reynolds) numbers. Therefore, the transport problem for the
distribution of species and temperature transitions from the diffusion-
controlled regime of the fuel consumption layer to the convection-controlled
regime of the recombination region. Or in mathematical terms, from the
elliptic character of the transport problem in the fuel consumption layer to a
parabolic character in the recombination layer. Once in the recombination
region, information is only transported downstream, away from the fuel
consumption layer.

Figure 3: Distributions in terms of the temperature of relevant physico-
chemical magnitudes, showing in particular that despite the moderately large
variations of the H2 diffusion coefficient $\mathcal{D}_{\ce{H2}}$, the length
scale $\mathcal{D}_{\ce{H2}}/u$ changes at most by a factor of 5 between the
cold and hot sections of the flame.

The recombination region structure is thus slave of the conditions left past
the fuel consumption layer, which are in turn independent of the evolution in
the recombination region. This explains why the condition of chemical
equilibrium applied on the hot edge of the fuel consumption layer works well.
The recombination region can thus be ignored, in the first approximation, when
the interest is in calculating the flame velocity.

Figure 4: Distributions of the net $\dot{\mathcal{N}}$ consumption/production
(solid/dash-dot lines) rates of H2, O, H and OH in the fuel consumption layer
(top) and downstream of it in the recombination region (bottom). The
transition between these two regions can be identified by the abrupt drop of
two orders of magnitude of the H2 consumption rate and also by the transition
from net production to net consumption of the three radicals O, H and OH.

###  2.2 Kinetics.

The distributions of the net consumption rates of radicals and H2 of Figure 4
show that, although at reduced rates compared with those in the fuel
consumption layer, the flame is still reacting downstream of the fuel
consumption layer. These plots also illustrate the change of sign of the net
production rates of radicals which transition from net production in the fuel
consumption layer to net consumption in the recombination region. They clearly
show thus, as anticipated, the role of this region depleting, down to the
final thermodynamic equilibrium state, the unburned H2 and the excess of
radicals left past the fuel consumption layer.

Label |  | Reaction |
---|---|---|---
shuffle reactions |  |
1 |  O2 + H | $\leftrightarrow$ |  O + OH
4 |  H2 + OH | $\leftrightarrow$ |  H2O + H
5 |  2 OH | $\leftrightarrow$ |  H2O + O
HO2 reactions |  |
13 |  H + O2 + M | $\leftrightarrow$ |  HO2 + M
17 |  HO2 + O | $\leftrightarrow$ |  O2 + OH
19 |  HO2 + OH | $\leftrightarrow$ |  H2O + O2
Table 1: Reduced H2-O2 kinetic mechanism in the recombination region. This
mechanism reproduce the exact (that is, the numerical results with the full
mechanism) net production rates of all the species with an accuracy of at
least 10% in the recombination region of lean flames. The first three, the
shuffle reactions, are out of equilibrium in the fuel consumption layer but
remain in nearly exact partial equilibrium throughout the recombination
region. The last three reactions represent the basic recombination engine
recycling radicals and H2 to products. Although these three reactions are
early in the recombination region out of equilibrium (see Figure 4), this
subset is considered as reversible in order to capture the final thermodynamic
equilibrium, where all reactions eventually reach partial equilibrium. The
reaction numbers refer to Glarborg et al.’s mechanism [14], excerpted in D .

This reactivity is ultimately associated with the irreversibility of the fuel
consumption layer, where the fast, out-of-equilibrium conversion of H2 to
products involves radicals in amounts well in excess over the values
corresponding to chemical equilibrium. However, the reactions responsible for
the H2 fast consumption – the shuffle reactions in Table 1 – abruptly quench
at the hot edge of the fuel consumption layer, leaving H2 and the radicals O,
OH and H frozen in the first approximation, unable to decay to equilibrium.
This transition to partial equilibrium (quench) of the shuffle reactions,
illustrated in Figure 5 in the specific case of the reaction 4, marks the end
of the fuel consumption layer and the beginning of the recombination region.

Figure 5: The left plot illustrates, for several representative lean flames,
the transition between the fuel consumption layer (upstream) and the
recombination region (downstream) using as indicator the net reaction rate of
the shuffle reaction 4, H2 + OH $\leftrightarrow$ H2O + H. The transition can
be identified by the drop, sharper and deeper as stoichiometry is approached,
to values much smaller than unity of $\dot{\omega}_{4}/\dot{\omega}_{4,f}$. As
shown in the right plot, this drop – quench of reaction 4 – leaves past the
fuel consumption layer values of the mass fraction $Y_{H_{2}}$ that are still
orders of magnitude larger than the final equilibrium values $Y^{b}_{H_{2}}$.
The temperature $T/T^{b}$, scaled with its final value $T^{b}$, is also shown
for reference. Notice that all the magnitudes displayed in the right plot have
unity final equilibrium values, showing that, especially in low-$\phi$ flames,
the recombination region is characterized by a many orders of magnitude decay
of the scaled mass fraction $Y_{H_{2}}/Y^{b}_{H_{2}}$. Note also that close to
stoichiometry, a significant amount of the heat release occur downstream of
the fuel consumption layer, making inaccurate the assumption of an isothermal
recombination region. The vertical line marking the beginning of the
recombination region is suggestive, not a sharp transition.

The almost exact partial equilibrium of the three shuffle reactions,
maintained throughout the entire recombination region, gives three expressions
(see (3) below) yielding the radicals O, OH and H concentrations in terms of
that of H2. These expressions replace the corresponding transport equations,
hence reducing the number of degrees of freedom by three. If, in addition, the
temperature and O2 and H2O concentrations are assumed to be constants, as
suggested by Figure 1, the full flame problem can be reduced to just solving
the H2 distribution, given by the ordinary differential equation derived below
in (12).

The source term in this equation is due to the HO2 kinetics, which drives the
chemical activity in the recombination region. These reactions had in the fuel
consumption layer much smaller rates than those of the shuffle reactions. In
the recombination region however, they remain out of equilibrium, i.e. with
widely different forward and reverse rates, and become dominant, setting in
particular the scale for the deviations from partial equilibrium of the
shuffle reactions.

In summary, the realistic description of H2 flames with the detailed kinetic
mechanism can be reduced, in the recombination region, to a one-degree-of-
freedom, thick, isothermal, diffusion-free structure trailing the flame,
similar to that found by Liñán [17] as the slow recombination regime in his
analysis of H2-air flames, based on a simplified, two-step model due to
Zel’dovich [18].

##  3 Kinetics in the recombination region.

The basic recombination working principle can be exemplified with the reduced
mechanism of Table 1, which reproduces the net production and consumption
rates with errors of the order of 10%. The characteristic scales and the
fundamental mathematical structure found with it remain the same when the full
mechanism is used.

Figure 6:  Kinetics in the fuel consumption layer, showing that the shuffle
reactions (1, 4 and 5) are out of equilibrium throughout it but converge to
partial equilibrium on its hot side. The kinetics in the recombination region
is instead characterized by the almost exact partial equilibrium (balanced
forward (solid) and reverse (dash-dot) rates) of the shuffle reactions; in
contrast, the reactions of the HO2 kinetics are clearly out of equilibrium,
with much faster forward than reverse rates. Note as well that the deviations
from partial equilibrium of the shuffle reactions (for clarity only the net
rate (dash line) of reaction 4 is plotted) are of the order of the forward
rates of the HO2 reactions. The bottom plot shows the distribution of the
branching ratio $\alpha=\dot{\omega}_{17}/\dot{\omega}_{13}$ and
$1-\alpha=\dot{\omega}_{19}/\dot{\omega}_{13}$ together with their combined
net rate (yellow line). Deviations of this line from unity, notably early in
the recombination region, indicate inaccuracies of the reduced mechanism. The
vertical line suggesting the origin $x\approx 0.5$ mm of the recombination is
guided by the plot on the left.

This mechanism comprises the three shuffle reactions and the three most
relevant reactions from the full HO2 kinetics, excerpted in D . This last
subset represents the engine driving the conversion of the radicals and H2
excess into products. As shown in Figure 6, these reactions are left past the
fuel consumption layer out of equilibrium, with much larger forward than
reverse rates, only reaching partial equilibrium asymptotically, as the final
burnt equilibrium state is approached.

In contrast, the shuffle reactions are, in the first approximation, in partial
equilibrium, with balanced forward and reverse rates throughout the entire
recombination layer. However, the deviations of the shuffle reactions from the
exact partial equilibrium still need to be considered. This can be easily seen
from the kinetics of H2, whose consumption rate $\dot{\mathcal{N}}_{\ce{H2}}$
is simply given by the net rate of shuffle reaction 4 (see also Figure 8
below), which is non-zero only in the approximation beyond partial
equilibrium. These deviations from exact partial equilibrium are in turn, as
will be shown, of the order of the rates of the HO2 kinetics
$\dot{\mathcal{N}}_{\ce{H2}}\sim\dot{\omega}_{4}\sim\dot{\omega}_{13}$.

###  3.1 The recombination mechanism.

The distributions of the rates of the HO2 reactions are represented in Figure
6, which shows that, early in the recombination region, the forward rates of
reactions 13, 17 and 19 are several orders of magnitude faster than the
reverse ones. The kinetics of HO2 can thus be seen as a two-step mechanism.
First, HO2 is produced through reaction 13f: H + O2 + M $\rightarrow$ HO2 + M.
This is the rate limiting step in the HO2 kinetics and therefore sets the
characteristic chemical time in this region. Balancing its production rate,
the chain-breaking reactions 17 and 19 consume HO2 and radicals converting
them into O2 and H2O. This basic scheme, namely reaction 13 producing HO2 and
the rest of the reactions consuming and maintaining it in steady state,
remains unmodified when the full HO2 mechanism is considered.

The steady state of HO2 can be written as
$\dot{\omega}_{13}\approx\dot{\omega}_{17}+\dot{\omega}_{19}$, which shows
that the mass of HO2 is conserved. HO2 serves thus just as the intermediate in
the engine pumping radicals to products.

The overall stoichiometry of the HO2 subset can be made more explicit using
the branching ratio $\alpha=\dot{\omega}_{17}/\dot{\omega}_{13}$, so
$1-\alpha\approx\dot{\omega}_{19}/\dot{\omega}_{13}$, both plotted in Figure
6. Using it to eliminate HO2 among the three reactions gives

$\displaystyle\ce{H}+\alpha\ce{O}+(1-2\alpha)\ce{OH}\rightarrow(1-\alpha)\ce{H2O},$
(1)

which clearly shows that the overall effect is the conversion of radicals to
H2O.

Note also the absence of O2 from this equation, showing that its mass is
conserved, just as that of HO2. In short, HO2 is produced from O2 through
reaction 13 and recycled back to O2 through the recombination reactions 17 and
19.

The overall reaction (1) shows that the HO2 submechanism represents net
consumption rates – sinks – of radicals and the production rates – sources –
of H2O, which in turn induce corresponding net rates on the shuffle reactions.
These are thus removed from partial equilibrium and forced to sustain net
rates of the order of those of the HO2 reactions, i.e.
$\dot{\omega}_{1,4,5}\sim\dot{\omega}_{13}$. Intuitively, – or using Le
Chatelier’s principle – it can be anticipated that the deviations are such as
to promote the relaxation back to partial equilibrium; that is, they are
faster in the chain-breaking direction, which converts the excess of radicals
and H2 into O2 and H2O. For instance, reaction 4 is slightly faster in the
forward than in the reverse direction, so it produces one molecule of H2O per
each consumed molecule of H2.

More quantitatively, Figure 6 shows that the net rates of the shuffle
reactions are two orders of magnitude smaller that their corresponding forward
or reverse rates; that is, the deviations (the net rates) from exact partial
equilibrium are small:
$\dot{\omega}_{i,f}-\dot{\omega}_{i,r}\sim\dot{\omega}_{13}\sim
10^{-2}\dot{\omega}_{i,f}\sim 10^{-2}\dot{\omega}_{i,r}$ for $i=1,4,5$.
Ultimately, the factor of two orders of magnitude separating these rates is
determined by the ratios $k_{13,f}/k_{1,4,5;f}\sim 10^{-2}$ of the reaction
constants of reaction 13 to those of the shuffle reactions, as shown in Figure
7. In the multiple-scales and perturbation methods terminology,
$\epsilon_{1,4,5}=k_{13,f}/k_{1,4,5;f}$ are the small parameters causing the
phenomena described in this paper.

Figure 7: Distributions, in a $\phi=0.6$ H2-air flame, of the shuffle
reactions equilibrium constants $\mathcal{K}_{1,4,5}$, which reach their final
high-temperature asymptotic values quickly downstream of the hot edge of the
fuel consumption layer, so they are nearly constants throughout the
recombination region. Also shown in dash lines the ratios
$k_{13,f}/k_{1,4,5;f}\sim 10^{-2}$. The short dash-dot segments represent the
burnt equilibrium values.

###  3.2 Partial equilibrium of the shuffle reactions.

Although well-known, the partial equilibrium of the shuffle reactions is
briefly discussed here to introduce some nomenclature that will be used
hereafter. Balancing their forward and reverse rates yields

$\displaystyle[\ce{O}]\approx\mathcal{K}_{1}\mathcal{K}_{4}\frac{[\ce{O2}]}{[\ce{H2O}]}\,[\ce{H2}],$
(2a)
$\displaystyle[\ce{OH}]\approx\sqrt{\frac{\mathcal{K}_{1}\mathcal{K}_{4}}{\mathcal{K}_{5}}\,[\ce{O2}][\ce{H2}]},$
(2b)
$\displaystyle[\ce{H}]\approx\sqrt{\frac{\mathcal{K}_{1}\mathcal{K}_{4}^{3}}{\mathcal{K}_{5}}\frac{[\ce{O2}]}{{[\ce{H2O}]}^{2}}\,[\ce{H2}]^{3}},$
(2c)

with the equilibrium constants $\mathcal{K}_{i}=k_{i,f}/k_{i,r}$, functions of
only the temperature. The relative errors
$|[\ce{S}]_{exact}-[\ce{S}]_{approx}|/[\ce{S}]_{exact}$ (S = O, OH and H) of
these approximations are of the order of the relative deviations
$\mathcal{E}_{i}=(\dot{\omega}_{i,f}-\dot{\omega}_{i,r})/\dot{\omega}_{i,f}$
from partial equilibrium of the shuffle reactions (see Graña and Mahmoudi
[12]), which, for not too lean flames, are typically of order
$\mathcal{E}_{i}\sim 10^{-2}$ (see Section 5. Discussion and conclusions. and
Figure 15 for more details).

Guided by Figures 1 and 14, which show that the temperature soon reaches its
final equilibrium value, the equilibrium constants $\mathcal{K}_{i}$ can be
assumed, throughout the recombination region, to be constants equal to their
final burnt equilibrium values at $T^{b}$. Similarly, Figure 1 suggests that
the concentrations of O2 and H2O can as well be replaced in these expressions
by the corresponding burnt equilibrium values.

With these approximations, the equations (2) can be more compactly written
using the burnt equilibrium concentrations $[\ce{O}]^{b}$, $[\ce{H}]^{b}$ and
$[\ce{OH}]^{b}$ as

$\displaystyle\frac{[\ce{\ce{O}}]^{e}}{[\ce{\ce{O}}]^{b}}=\left(\frac{[\ce{\ce{OH}}]^{e}}{[\ce{\ce{OH}}]^{b}}\right)^{2}=\left(\frac{[\ce{\ce{H}}]^{e}}{[\ce{\ce{H}}]^{b}}\right)^{2/3}=\frac{[\ce{\ce{H2}}]}{[\ce{\ce{H2}}]^{b}}=\frac{1}{z},$
(3)

introducing the reduced H2 concentration
$z=[\ce{\ce{H2}}]^{b}/[\ce{\ce{H2}}]\leq 1$. The superscript $e$ remarks the
use of the burnt equilibrium values of the equilibrium constants and O_2 and
H2O concentrations.

Despite being excellent approximations, these expressions are not yet enough
because the concentration of H2 is still undetermined. They anticipate however
that, in lean flames, the H2 concentration is the only degree of freedom in
the recombination region.

Figure 8: Variation with the equivalence ratio $\phi$ of the net H2
consumption rate $\dot{\mathcal{N}}_{\ce{H2}}$ and of the most contributing
reactions $\dot{\omega}_{i}$, showing that, almost exactly,
$\dot{\mathcal{N}}_{\ce{H2}}=\dot{\omega}_{4}$. All the rates are scaled with
the net rate $\dot{\omega}_{13}$ of reaction 13, showing that this is the
proper scaling for the kinetics in this region. The rates are evaluated at
$x=20$ mm – with $x=0$ mm the location where HO2 attains its maximum rate;
$x=0$ is thus representative of the fuel consumption layer location, and also
of the beginning of the recombination region given the length contrast between
both regions. This plot thus shows the asymptotic ordering, which is also
representative of the ordering closer to the fuel consumption layer.

##  4 Formulation

The structure of the recombination region is, in moderately lean flames,
essentially determined by the dynamics of H2. The characteristic length scale
is thus given by the balance between the transport rate of H2 and its
consumption rate through reaction 4. This balance is first examined below in
orders of magnitude to estimate the characteristic scales which will then be
used to formulate and solve the problem.

These order of magnitude estimations are carried out here because they are
usually even more insightful than actually solving the resulting problem,
whose results can be easily guessed once the physics of the problem is
understood. They are also helpful cleaning the problem out from small
unimportant effects, and help making sure that what remains is relevant. In
addition, the estimations are often enough to solve engineering problems, when
trends are more important than precise numerical results.

###  4.1 Scales in the recombination region.

The H2 consumption rate $\dot{\mathcal{N}}_{\ce{H2}}$ is almost exactly given
by $\dot{\mathcal{N}}_{\ce{H2}}=\dot{\omega}_{4}$ through the entire range of
$\phi$ in lean flames as illustrated in Figure 8. This Figure shows also that
$\dot{\omega}_{13}$ is the appropriate scaling. Thus, in the first
approximation

$\displaystyle\dot{\mathcal{N}}_{\ce{H2}}\approx\dot{\omega}_{4}\sim\dot{\omega}^{e}_{13,f}\approx[\ce{H}]^{e}/\tau^{b}_{13,f},$
(4)

where $[\ce{H}]$ approximated with $[\ce{H}]^{e}\propto[\ce{H2}]^{3/2}$, given
by (3), which is based on the final equilibrium values of the temperature and
O2 and H2O concentrations, also used to evaluate the burnt equilibrium value
of the reaction time $\tau^{b}_{13,f}=1/(k_{13,f}[\ce{O_{2}}])|^{b}$.

Figure 9: Comparison between $\mathcal{L}=Y_{\ce{H2}}/\partial_{x}Y_{\ce{H2}}$
(solid lines) of Figure 2 and $\mathcal{L}^{o}$ (dash-dot lines) defined in
(6). $\mathcal{L}^{o}$ represents well the order of magnitude of $\mathcal{L}$
through the recombination region except, as expected, close to the final
equilibrium, $z\rightarrow 1$, because only the forward rate of 13 is
accounted for in $\mathcal{L}^{o}$. The dotted lines, included for reference,
correspond to the fuel consumption layer, taken to roughly occur for $x<0.6$
mm.

On the other hand, the previous discussion suggests that convection can be
used to estimate the species transport rate, so the H2 mass conservation
equation reads, in orders of magnitude, as

$\displaystyle\dot{m}^{\prime\prime}_{L}\frac{\partial Y_{\ce{H2}}}{\partial
x}\sim\mathcal{M}_{\ce{H2}}\dot{\omega}_{4}\sim\mathcal{M}_{\ce{H2}}\dot{\omega}^{e}_{13,f},$
(5)

with $\dot{m}^{\prime\prime}_{L}=\rho u$ the mass flow rate of mixture
consumed by the flame per unit surface and time.

The characteristic length $\partial x\sim\mathcal{L}^{o}$ along which the H2
mass fraction undergoes variations of the order of itself, $\partial
Y_{\ce{H2}}\sim Y_{\ce{H2}}$, is thus, in terms of the mole fractions $X_{i}$
to absorb for simplicity the molar masses,

$\displaystyle\mathcal{L}^{o}=\frac{\dot{m}^{\prime\prime}_{L}Y_{\ce{H2}}}{\mathcal{M}_{\ce{H2}}\dot{\omega}^{e}_{13,f}}=\mathcal{L}^{o,b}\frac{X_{\ce{H2}}}{X^{e}_{\ce{H}}}=\mathcal{L}^{o,b}\sqrt{z},$
(6)

with $\mathcal{L}^{o,b}=u^{b}\tau^{b}_{13,f}$ the final equilibrium value of
$\mathcal{L}^{o}$, written in terms of the flame’s products side velocity
$u^{b}=\dot{m}^{\prime\prime}_{L}/\rho^{b}$, quickly reached past the fuel
consumption layer.

$\mathcal{L}^{o}$ is thus the length required to see variations of the H2
concentration of the order of $[\ce{H2}]$ moles per unit volume with a
consumption rate of order $[\ce{H}]^{e}/\tau^{b}_{13,f}\propto[\ce{H2}]^{3/2}$
moles per unit volume and time, so, going down the recombination region, it
takes longer to reduce $[\ce{H2}]$ because its consumption rate decreases
faster than linearly with $[\ce{H2}]$.

Although considered below, the reverse rate of reaction 13 is not taken into
account in (4) for simplicity. Figure 9 shows however that $\mathcal{L}^{o}$
compares well with $\mathcal{L}=Y_{\ce{H2}}/\partial_{x}Y_{\ce{H2}}$, except,
naturally, close to the final equilibrium ($z\rightarrow 1$).

###  4.2 Dimensionless formulation.

The transport equations for the steady flame propagation [19, 2, 1] can thus
be written in dimensionless form using the scaled coordinate along the flame
$\partial\xi=\partial x/\mathcal{L}^{o}\sim 1$ and the scaled rates
$\tilde{\omega}_{i}=\dot{\omega}_{i}/\dot{\omega}_{13,f}\sim 1$. Dropping in
addition the diffusion terms leads to the following form of the equations for
H2 and radicals

$\displaystyle\frac{\partial\ln
Y_{\ce{H2}}}{\partial\xi}=-\tilde{\omega}_{4},$ (7a)
$\displaystyle\mathcal{D}X_{\ce{O}}=\tilde{\omega}_{5}+\tilde{\omega}_{1}+\tilde{\mathscr{S}}^{e}_{\ce{O}},$
(7b)
$\displaystyle\mathcal{D}X_{\ce{OH}}=\tilde{\omega}_{1}-\tilde{\omega}_{4}-2\tilde{\omega}_{5}+\tilde{\mathscr{S}}^{e}_{\ce{OH}},$
(7c)
$\displaystyle\mathcal{D}X_{\ce{H}}=\tilde{\omega}_{4}-\tilde{\omega}_{1}+\tilde{\mathscr{S}}^{e}_{\ce{H}},$
(7d)
$\displaystyle\mathcal{D}X_{\ce{HO2}}=\tilde{\mathscr{S}}^{e}_{\ce{HO2}},$
(7e)

with the operator
$\mathcal{D}\varphi=X_{\ce{H2}}^{-1}\partial\varphi/\partial\xi$. This system
must be complemented with expressions for the rates, discussed in B and C, and
with appropriate initial conditions.

The equations for the temperature, O2 and H2O are left out of (7) because, as
discussed in A, they show that the corresponding variations, of order
$\partial(T-T^{b})/(T^{b}-T^{u})\sim\partial X_{\ce{O2}}\sim\partial
X_{\ce{H2O}}\sim X_{\ce{H2}}$, are (not too close to stoichiometry), as shown
in Figure 1 and in Figure 14 below, much smaller than the respective values
$(T-T^{b})/(T^{b}-T^{u})$, $X_{\ce{O2}}$ and $X_{\ce{H2O}}$, thus validating
approximating them as constants, as done for instance deriving (3).

The rates in (7) have been split into those collected in
$\tilde{\mathscr{S}}^{e}_{i}$, which include sources of species $i$ due to
reactions that are out of equilibrium, i.e. with widely different forward and
reverse rates. These can therefore be evaluated in the first approximation
using (3) and expressed as functions of only $z$, as done in B for the HO2
reactions.

$\tilde{\mathscr{S}}^{e}_{i}$ comprises mainly the HO2 reactions, so it scales
with $\dot{\omega}_{13}$. For instance, in the approximation of the reduced
mechanism of Table 1, $\tilde{\mathscr{S}}^{e}_{\ce{O}}=-\tilde{\omega}_{17}$,
$\tilde{\mathscr{S}}^{e}_{\ce{OH}}=\tilde{\omega}_{19}-\tilde{\omega}_{17}$
and $\tilde{\mathscr{S}}^{e}_{\ce{H}}=-\tilde{\omega}_{13}$. However, other
out-of-equilibrium reactions not involving HO2, such as reaction 9, H + O + M
$\leftrightarrow$ OH + M, (see Figure 10) can be included in
$\tilde{\mathscr{S}}^{e}_{i}$.

Figure 10: Reaction rates, scaled with $\dot{\omega}_{4}$, of the most
relevant reactions in the H2-O2 mechanism (see D). Solid and dash lines
(coincident due to the almost exact partial equilibrium of all but reaction 9)
represent forward and reverse reaction rates respectively. Dash-dot lines
represent the corresponding net rates. The symbol $\sim$ links reactions with
proportional reaction rates; these are reactions with the same chemical
formula, but different temperature dependence rate, which therefore become
proportional in the recombination region where the temperature is constant.

At variance, the shuffle reactions rates $\tilde{\omega}_{1,4,5}$ are left out
of $\tilde{\mathscr{S}}^{e}_{i}$ because they are in almost exact partial
equilibrium, with $\tilde{\omega}_{1,4,5}\approx 0$ – used to derive (3) –,
and need to be evaluated in the next order of approximation.

To do so, the scaled mass fractions $\psi_{i}=Y_{i}/Y^{e}_{i}$ of $i=$ O, OH
and H are introduced as independent variables instead of $Y_{i}$.
$\tilde{\omega}_{1,4,5}$ can then be written, as shown in Appendix C, as
algebraic functions of $\psi_{i}$

$\displaystyle\tilde{\omega}_{1}=\tilde{\Omega}^{e}_{1}\left(\psi_{\ce{H}}-\psi_{\ce{O}}\psi_{\ce{OH}}\right),$
(8a)
$\displaystyle\tilde{\omega}_{4}=\tilde{\Omega}^{e}_{4}\left(\psi_{\ce{OH}}-\psi_{\ce{H}}\right),$
(8b)
$\displaystyle\tilde{\omega}_{5}=\tilde{\Omega}^{e}_{5}\left(\psi_{\ce{OH}}^{2}-\psi_{\ce{O}}\right),$
(8c)

with the coefficients
$\tilde{\Omega}^{e}_{i}=\dot{\omega}^{e}_{i,f}/\dot{\omega}^{e}_{13,f}$,
functions of only $z$.

It follows from these expressions and from $\tilde{\Omega}^{e}_{i}\gg 1$ (see
C) that the new variables $\psi_{i}$ are such that $|\psi_{i}-1|\ll 1$, which
confirms that the equilibrium concentrations (3) are excellent approximations.
They can be used in particular to evaluate the derivatives on the left hand
sides of (7) as

$\displaystyle\partial X_{k}/\partial\xi\approx\acute{X}^{e}_{k}\cdot\partial
X_{\ce{H2}}/\partial\xi,$ (9)

with $\acute{X}^{e}_{k}=\partial X^{e}_{k}/\partial X_{\ce{H2}}$ the
derivatives with respect to $X_{\ce{H2}}$, functions in the first
approximation of only $z$.

The system (7) can thus be written as

$\displaystyle\frac{\partial\ln
Y_{\ce{H2}}}{\partial\xi}=-\tilde{\omega}_{4},$ (10a)
$\displaystyle-\tilde{\omega}_{4}\acute{X}^{e}_{\ce{O}}=\tilde{\omega}_{5}+\tilde{\omega}_{1}+\tilde{\mathscr{S}}^{e}_{\ce{O}},$
(10b)
$\displaystyle-\tilde{\omega}_{4}\acute{X}^{e}_{\ce{OH}}=\tilde{\omega}_{1}-\tilde{\omega}_{4}-2\tilde{\omega}_{5}+\tilde{\mathscr{S}}^{e}_{\ce{OH}},$
(10c)
$\displaystyle-\tilde{\omega}_{4}\acute{X}^{e}_{\ce{H}}=\tilde{\omega}_{4}-\tilde{\omega}_{1}+\tilde{\mathscr{S}}^{e}_{\ce{H}},$
(10d) $\displaystyle\tilde{\mathscr{S}}^{e}_{\ce{HO2}}=0,$ (10e)

where the equations for O, OH and H have been divided by (7a) to eliminate
$\partial X_{\ce{H2}}/\partial\xi$. Additionally, the last equation (10e) is
written taking into account that HO2 is in steady state, as shown in B, so
$\partial X_{\ce{HO2}}/\partial\xi\approx 0$.

This formulation (10), in terms of $Y_{\ce{H2}}$ for H2 and of $\psi_{i}$ for
the radicals O, OH and H as dependent variables, is a differential-algebraic
system involving explicitly the spatial derivative of $Y_{\ce{H2}}$. In
contrast, $\psi_{i}$ enter only implicitly through the rates
$\tilde{\omega}_{1,4,5}$. These rates can thus be used temporarily as
independent variables in place of $\psi_{i}$. Once the rates
$\tilde{\omega}_{1,4,5}$ are determined, as done next, $\psi_{i}$ follow from
(8).

Thus, the equations for O, OH and H represent three linear algebraic relations
from which the rates $\tilde{\omega}_{1,4,5}$ can be solved explicitly in
terms of $\tilde{\mathscr{S}}^{e}_{i}$ and $\acute{X}^{e}_{i}$ as done in C,
giving in particular

$\displaystyle\tilde{\omega}_{4}\approx-\frac{\tilde{\mathscr{S}}^{e}_{\ce{OH}}+2\tilde{\mathscr{S}}^{e}_{\ce{O}}+3\tilde{\mathscr{S}}^{e}_{\ce{H}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}}.$
(11)

This expression for $\tilde{\omega}_{1}$ and also $\tilde{\omega}_{4,5}$ are
proportional to $\tilde{\mathscr{S}}^{e}_{\ce{i}}$, and therefore to the HO2
reactions rates, i.e. $\tilde{\omega}_{1,4,5}\propto\tilde{\omega}^{e}_{13}$,
thus proving the statement, anticipated above without proof, that the HO2
kinetics governs the deviations from exact partial equilibrium of the shuffle
reactions.

The approximation (11) is general regarding the HO2 kinetics used to evaluate
$\tilde{\mathscr{S}}^{e}_{i}$. Since all these rates depend ultimately on the
concentrations of O, OH and H, which are obtained from (3), including more
reactions, or even of the H2O2 kinetics, is thus straightforward save for the
extended required calculations.

Figure 11: Distribution of the normalized $\dot{\omega}_{4}$ for several
representative values of $\phi$, showing that $\dot{\omega}_{4}$ scales with
$\dot{\omega}_{13}\sim\dot{\omega}_{13,f}(1-z^{2})$, with the factor $1-z^{2}$
accounting for the zero net rate at $z=1$; $\dot{\omega}_{13,f}$ carries the
order of magnitude of the rate and accounts for the effect of $\phi$. The
dash-dot lines represent the approximation (11) evaluated with the full
(exact) chemistry, showing that it is excellent around $\phi=0.6$. The dotted
lines represent, for reference, the fuel consumption layer taken to roughly
occur for $x<0.5$ mm.

Using $\tilde{\omega}_{4}$ back in (10a) leads finally to the differential
equation governing the distribution of H2

$\displaystyle\frac{\partial\ln
z}{\partial\xi}=\frac{\tilde{\mathscr{S}}^{e}_{\ce{OH}}+2\tilde{\mathscr{S}}^{e}_{\ce{O}}+3\tilde{\mathscr{S}}^{e}_{\ce{H}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}}.$
(12)

or in terms of the physical variables

$\displaystyle\frac{\partial\ln Y_{\ce{H2}}}{\partial
x}=-\frac{\mathcal{M}_{\ce{H2}}}{\dot{m}_{L}}\frac{\dot{\mathscr{S}}^{e}_{\ce{OH}}+2\dot{\mathscr{S}}^{e}_{\ce{O}}+3\dot{\mathscr{S}}^{e}_{\ce{H}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}},$
(13)

The source term of (12), function of only $z$ (of $Y_{\ce{H2}}$ in (13)) in
the first approximation – so these are simply quadratures –, has the form (see
B)

$\displaystyle\tilde{\omega}_{4}=\dot{\omega}_{4}/\dot{\omega}_{13,f}=\tilde{\Gamma}_{4}(z)(1-z^{2}),$
(14)

with $\tilde{\Gamma}_{4}(z)$ a regular, finite function of order unity,
plotted for representative values of $\phi$ in Figure 11.

Equation (12) together with (14) give the asymptotic form of the decay to
equilibrium as $1-z\propto e^{-\xi/\Lambda}$ for $\xi\rightarrow\infty$, with
a dimensionless length scale $\Lambda=(2\tilde{\Gamma}_{4}^{b})^{-1}$,
monotonously decreasing with $\phi$ as follows from the behavior of
$\tilde{\Gamma}_{4}^{b}=\tilde{\Gamma}_{4}(z=1)$ displayed in Figure 11. The
$\phi$-dependence of $\Lambda$ thus confirms the larger thickness of the
recombination region of leaner flames as anticipated in Figure 2.

The equation (12) (and (13)) requires an initial condition giving the starting
value of H2. This initial condition is the matching condition with the
solution upstream of the recombination region and is obtained as the constant,
in the first approximation, value of H2 amount left downstream of the fuel
consumption layer, after the shuffle reactions reach partial equilibrium, with
the rates of the HO2 reactions set to zero to prevent its slow decay. Instead,
to explore the errors associated with the approximations discussed previously
and to remove as much as possible errors associated with matching, the initial
condition is taken as the value of the exact solution at some specific
location downstream of the fuel consumption layer. As shown below, the errors
associated with the approximate solution decrease as the initial condition is
chosen at larger distances downstream of the fuel consumption layer. This is
so because the difference between the exact and the approximate value of the
slope $\partial\ln z/\partial\xi$ decreases approaching the burnt equilibrium
where the difference is zero. Thus, an upper bound of the errors of the
simplified model can be obtained by choosing the initial close to the fuel
consumption layer as done below.

Figure 12: The left plot compares the numerical solution of the full problem,
labelled as Exact (solid lines), with the solution of (10a) using i) the exact
rate $\dot{\omega}_{4}$ (dash lines) and ii) evaluating exactly the
approximation (11) of $\dot{\omega}_{4}$ (dash-dot lines). The right plot
represents the corresponding relative errors. The initial condition to
integrate (12) has been (arbitrarily) chosen as the value of
$Y_{\ce{H2}}(x_{0})$ from the exact solution, with $x_{0}$ = 0.5 mm, which is
roughly where the recombination region starts. As discussed in Figure 13,
choosing to start the solution further downstream decreases the relative
error.

###  4.3 Results.

In order to illustrate the validity of (10a) and of (12), Figure 12 compares
the exact – numerical solution of the full flame problem – distribution of
$Y_{\ce{H2}}$, with solutions of (10a) and (12) integrated using the exact
numerical solution to evaluate their right hand sides. The corresponding
relative errors measure in the first case the effect of neglecting the
diffusive transport in the full problem; whereas in the second case the errors
also include the additional effect of approximating $\tilde{\omega}_{4}$ with
(11).

Figure 13: On the left, the solution, for $\phi=0.6$, of (12) using the
reduced HO2 mechanism of Table 1 and the expressions for the rates, based on
(3), given in B. To understand the effect of the initial conditions
$Y^{0}_{H_{2}}(x_{0})$, the solutions for $x_{0}=0.5$ mm (green line),
$x_{0}=1$ mm (yellow line) and $x_{0}=10$ mm (blue), with $Y^{0}_{H_{2}}$
evaluated from the exact solution, are included. The inset on the lower right
corner represents the associated errors; for reference, also shown (purple
lines) the errors for $\phi=0.6$ from Figure 12. On the right, the solution
and corresponding error for $\phi=0.4$ using the same mechanism and
$x_{0}=1.0$.

These results show that neglecting the diffusive transport is a good
approximation in not too lean flames but becomes inaccurate, as Figure 2
anticipated, especially early in the recombination region, in slower flames.
On the contrary, the approximation (11), based on constant values of the
temperature and O2 concentration provides an excellent approximation not too
close to stoichiometry, as expected from Figure 1.

On the other hand, Figure 13 shows the solution of (12) for $\phi=0.4$ and 0.6
using the reduced mechanism of Table 1, i.e.
$\tilde{\mathscr{S}}^{e}_{\ce{O}}=-\tilde{\omega}_{17}$,
$\tilde{\mathscr{S}}^{e}_{\ce{OH}}=\tilde{\omega}_{19}-\tilde{\omega}_{17}$
and $\tilde{\mathscr{S}}^{e}_{\ce{H}}=-\tilde{\omega}_{13}$, with the rates
$\tilde{\omega}_{13,17,19}$ evaluated as functions of $z$ using the
expressions given in B. The corresponding errors, with a peak of around 20%,
include the additional effect of using the reduced kinetics instead of the
full mechanism (purple lines), and also of using (3).

##  5 Discussion and conclusions.

The structure and kinetics of the recombination region has been studied in
detail in lean H2 flames, with the main result that this region can be
described using the H2 concentration as only degree of freedom. Its
distribution is given by the solution of the first-order ordinary differential
equation (12), or (13) in physical variables, whereas the O, OH and H
distributions follow from that of H2 using the equilibrium relation (3).

Figure 14: Distributions of the mass fractions $Y_{i}$ of O, H, OH and O2,
normalized with the corresponding final burnt equilibrium values $Y^{b}_{i}$.
The thin dash lines are the approximations (3), based on constant values
$\theta,Y_{\ce{H2O}}/Y^{b}_{\ce{H2O}},Y_{\ce{O2}}/Y^{b}_{\ce{O2}}\approx 1$
throughout this region. These approximations are excellent in sufficiently
lean flames (left plot). However, as stoichiometry is approached (right plot),
the temperature and the amount of O2 significantly deviate – notice the
different vertical right scale range on each plot –, early in the
recombination region, from their final equilibrium value.

This significant simplification is based on the simultaneous partial
equilibrium of the three shuffle reactions, which is accurate in not too lean
flames as shown in Figures 12 and 15. It is based also on the assumption of
constant values of the temperature and O2 and H2O concentrations, which reach
their asymptotic values quickly downstream of the fuel consumption layer in
flames not too close to stoichiometry flames as illustrated in Figure 14.
Thus, this simplified model with just one degree of freedom works well in
moderately lean flames, around $\phi\approx 0.6$, but requires to include more
degrees of freedom in very lean flames below $\phi<0.4$, and close to
stoichiometry.

In the limit of very lean flames, the shuffle reaction 1, and reaction 4 to a
lesser degree, cease to be in partial equilibrium as illustrated in Figure 15.
The partial equilibrium of reaction 5 however still holds, furnishing a
relationship between OH and H, which can be used to reduce by one the number
of degrees of freedom. Additionally, H tends to be in steady state in the
recombination region in these ultra lean flames hence eliminating the
corresponding transport equation in favor of an explicit algebraic
relationship providing H in terms of the rest of variables. Thus, the
recombination region requires in this limit the concentrations of H2 and of O
as degrees of freedom. On the other hand, despite the declining reactivity in
this limit, diffusive transport needs to be accounted for, as Figures 2 and 12
suggest, clearly as a result of the also decreasing flame consumption rate of
fresh mixture $\dot{m}^{\prime\prime}_{L}$, and therefore of the convective
transport rate.

Figure 15: Deviations from exact partial equilibrium of the shuffle reactions.
Solid ($\dot{\omega}_{i,f}>\dot{\omega}_{i,r}$) and dash-dot
($\dot{\omega}_{i,f}<\dot{\omega}_{i,r}$) lines represent the recombination
region, whereas thin dash lines represent the region upstream of it – fuel
consumption layer and preheat region – and are removed from the bottom plot
for clarity. The green and red shadowing mark the regions with errors are
smaller than 10% and 1% respectively. The side color bars are guides to the
lines color-$\phi$ mapping.

Close to stoichiometry, on the other hand, the transport equations for the
temperature and for the O2 must be solved together with that for H2. This is
so because a non-negligible fraction of the total heat release occurs in the
recombination region, so the temperature can not be assumed to be constant any
more. In addition, O2 is found in these flames in amounts comparable to those
of H2 and can not be assumed to be constant as seen in Figure 14.

###  5.1 Heat losses

The long length scale associated with the recombination region suggests that
heat losses may become significant and change the structure of the
recombination region (if this is not disrupted, for instance by a turbulent
flow). The paper’s results, however, still apply unchanged in the presence of
moderate heat losses, when the induced temperature variations $\partial
T_{HL}$ are small compared with the local temperature in the recombination
region $\partial T_{HL}/T\approx\partial T_{HL}/T^{b}\ll 1$ (in similarity
with, as shown in A, the small temperature changes induced by the combustion
of the unburned H2 left past the fuel consumption layer). For instance, H2O in
the products induce heat radiation losses of the order of $\partial T/\partial
x\sim X_{\ce{H2O}}\kappa_{\ce{H2O}}\sigma
T^{4}/(\dot{m}^{\prime\prime}_{L}c_{p})\sim 10^{2}$K/m, with
$\kappa_{\ce{H2O}}\sim 1m^{-1}$ the H2O mean absorption coefficient[20, 21]
and $\sigma$ the Stefan-Boltzmann constant. This heat losses rate induce in
turn, over the thickness of the recombination region (see Figure 2),
temperature variations such that $\partial T_{HL}/T\sim 10^{-2}$, which can
therefore be neglected.

However, more intense heat losses can change the structure. For instance,
Joulin and Clavin [22] showed that unstretched flames are led to extinction
when the heat loses are increased to produce gradients on the products side of
order $\partial T/\partial x\sim\beta^{-1}(T^{b}-T^{u})/\delta_{L}$, with
$\delta_{L}$ the flame thickness (without the recombination region) and
$\beta\sim 10$, the Zeldovich number222$\beta=E(T^{b}-T^{u})/R{T^{b}}^{2}$ is
the Zeldovich number, which is not well defined in non-Arrhenius kinetics; it
can however be estimated as $\beta\sim d\ln U_{L}/d\ln T^{b}\sim 10$ not too
close to the flammability limits (where it diverges).. Heat losses of this
order induce, over the thickness of the recombination region, temperature
variations such that $\partial
T_{HL}/T\sim\beta^{-1}(T^{b}-T^{u})/T^{b}(\mathcal{L}/\delta_{L})$, which is
of order unity, or even larger in slow flames.

Thus, before extinction, heat loses such that $\partial T_{HL}/T\sim 1$ can
occur, inducing changes in the recombination region structure. In this case
however, the equation for the temperature would be decoupled from the species
distributions (ignoring the small effect on minor species on properties such
as the mixture-averaged heat conductivity or specific heat). The problem for
the H2, O, OH and H distributions would not be autonomous anymore because the
imposed temperature distribution, function of the coordinate along the
recombination region, would enter for instance in the equilibrium constants of
the shuffle reactions in the conditions (2). In addition, low temperatures
could bring the shuffle out of equilibrium, inducing effects similar to those
previously discussed for very lean flames. Thus, even in this case of strong
heat losses, when $\partial T_{HL}/T\sim 1$, these can be accounted for with
modifications similar to those considered above to extend the validity to
extreme values of $\phi$, so the main results still apply.

##  6 Appendixes.

##  Appendix A Temperature and products.

The equations for $\theta$, H2O and O2, are

$\displaystyle\frac{1}{X_{\ce{H2}}}\frac{\partial\theta}{\partial\xi}=\sum_{i}{q_{i}\tilde{\omega}_{i}},$
(15a) $\displaystyle\frac{1}{X_{\ce{H2}}}\frac{\partial
X_{\ce{O2}}}{\partial\xi}=\tilde{\omega}_{1}+\tilde{\mathscr{S}}^{e}_{\ce{O2}},$
(15b) $\displaystyle\frac{1}{X_{\ce{H2}}}\frac{\partial
X_{\ce{H2O}}}{\partial\xi}=\tilde{\omega}_{4}+\tilde{\omega}_{5}+\tilde{\mathscr{S}}^{e}_{\ce{H2O}},$
(15c)

with $\theta=(T-T^{u})/(T^{b}-T^{u})$, the temperature increment over the
fresh mixture temperature $T^{u}$, and $q_{i}$ the molar heat release from
each reaction $i$, scaled with $\mathcal{M}_{m}c_{p,m}(T^{b}-T^{u})$ with
$\mathcal{M}_{m}$ and $c_{p,m}$ the mixture molar mass and heat capacity per
unit mass respectively.

Both sides of these equations are of order unity and therefore the variations
of these variables are proportional to the local amount $X_{\ce{H2}}$ of H2
being burned, $\partial(\theta,X_{\ce{O2}},X_{\ce{H2O}})\sim
X_{\ce{H2}}\ll\theta,X_{\ce{O2}},X_{\ce{H2O}}$, which holds not too to
stoichiometry as shown for instance in Figure 1.

##  Appendix B HO2 reaction rates.

The HO2 reactions are out of equilibrium in the recombination region and can
therefore be evaluated in the first approximation, using the equilibrium
concentrations $[\ce{O}]^{e}$, $[\ce{OH}]^{e}$, $[\ce{H}]^{e}$ from (3). The
resulting approximate rates carry errors due to the non-zero deviations from
partial equilibrium of the shuffle reactions (see Figure 15) and to the
assumption of constant values of T, $[\ce{O2}]$ and $[\ce{H2O}]$.

The forward rate of reaction 13, the rates scale, can be written, with
$\dot{\omega}^{b}_{13,f}=(k_{13,f}[\ce{O2}][\ce{H}])|^{b}$, as

$\displaystyle\dot{\omega}_{13,f}=k_{13,f}[\ce{O2}][\ce{H}]\approx\dot{\omega}^{b}_{13,f}/z^{3/2}.$
(16)

On the other hand, for any HO2 reaction,

$\displaystyle\tilde{\omega}_{m}=\dot{\omega}_{m}/\dot{\omega}_{13,f}\approx\tilde{\Gamma}_{m}(z)(1-z^{2}),$
(17)

with $\tilde{\Gamma}_{m}(z)$ finite, well-behaved, order unity functions of
$z$ plotted in Figure 17 for representative m.

This form of the rates is based on the steady state of HO2 (see Figure 16),
which has the form

$\displaystyle[\ce{\ce{HO2}}]^{e}=\frac{\dot{\omega}^{e}_{13,f}+\sum_{q}{\dot{\omega}^{e}_{q,r}}}{\dot{\omega}^{\prime
e}_{13,r}+\sum_{q}{\dot{\omega}^{\prime e}_{q,f}}},$ (18)

obtained from the balance, $\dot{\omega}^{e}_{13}=\sum_{q\neq
13}{\dot{\omega}^{e}_{q}}$, between the HO2 net production and consumption
rates. The sums include all the HO2 reactions $q$ except reaction 13, and the
primed rates are evaluated without $[\ce{HO2}]$.

Figure 16: Scaled rates of the relevant reactions of the HO2 kinetics (the
numbers refer to the mechanism in Appendix D). The net consumption rate of HO2
is of the order of $\dot{\mathcal{N}}_{\ce{HO2}}/\dot{\omega}_{13}\sim
10^{-4}$, meaning that HO2 is well in steady state. The rates are evaluated at
$x=20$ mm, as in Figure 8 where the choice is explained in detail.

To derive (17) for $m\neq 13$ first write
$\dot{\omega}_{m}\approx\dot{\omega}^{\prime
e}_{m,f}[\ce{HO2}]^{e}-\dot{\omega}^{e}_{m,r}$, which with (18) gives

$\displaystyle\dot{\omega}_{m}/(\dot{\omega}^{\prime
e}_{13,r}\mathcal{D})\approx\dot{\omega}^{\prime
e}_{m,f}\dot{\omega}^{e}_{13,f}-\dot{\omega}^{e}_{m,r}\dot{\omega}^{\prime
e}_{13,r}+\sum_{q}{(\dot{\omega}^{\prime
e}_{m,f}\dot{\omega}^{e}_{q,r}-\dot{\omega}^{e}_{m,r}\dot{\omega}^{\prime
e}_{q,f}}),$ (19)

with $\mathcal{D}$ defined below in (20).

Using the equilibrium concentrations (3) yields
$\dot{\omega}^{e}_{m,r}\dot{\omega}^{\prime e}_{13,r}/(\dot{\omega}^{\prime
e}_{m,f}\dot{\omega}^{e}_{13,f})\approx z^{2}$, so the numerator of the first
term is $\dot{\omega}^{\prime e}_{m,f}\dot{\omega}^{e}_{13,f}(1-z^{2})$; the
rest of the terms vanish since
$\dot{\omega}^{e}_{p,f}/\dot{\omega}^{e}_{p,r}\propto z^{-1/2}$ for any $p\neq
13$, leading to (17) with $\tilde{\Gamma}_{m}(z)$ given, for $m\neq 13$, by

$\displaystyle\tilde{\Gamma}_{m}=\frac{\gamma_{m}}{\mathcal{SD}}\leq 1,$ (20a)
$\displaystyle\gamma_{m}=\dot{\omega}^{\prime e}_{m,f}/\dot{\omega}^{\prime
e}_{13,r}\approx\gamma^{b}_{m}/z^{\alpha_{m}/2},$ (20b)
$\displaystyle\mathcal{SD}=1+\sum_{m}{\gamma_{m}},$ (20c) and with
$\alpha_{19}=1$, $\alpha_{17}=2$, $\alpha_{14,15}=3$ for the reactions
represented in Figure 17, which also depicts these approximations.

The HO2 steady state gives $\dot{\omega}_{13}$ in turn with the same form (17)
with $\tilde{\Gamma}_{13}=\sum_{m\neq
13}{\tilde{\Gamma}_{m}}=(\mathcal{SD}-1)/\mathcal{SD}$.

Figure 17: Distributions of the functions $\tilde{\Gamma}_{m}$ (17) for the
more relevant reactions of the HO2 mechanism for $\phi=0.6$. The dash lines
are the approximations (20a). $\tilde{\Gamma}_{13}=\sum_{m\neq
13}{\tilde{\Gamma}_{m}}$, which shows that the reactions 17 and 19 are
dominant in most of the recombination region. Reactions 14 and 15 become
relevant at the interface with the fuel consumption layer, because they are
dominant inside it, as shown by Fernández-Galisteo et al. [7, 8], who use
reactions 14, 15 and 19 (their 5, 6, 7 respectively) to model HO2 in the fuel
consumption layer; note however that they consider these reactions as
irreversible.

##  Appendix C Shuffle reaction rates.

Evaluation of $\tilde{\Gamma}_{4}$. The three middle equations of (10) give

$\displaystyle\tilde{\omega}_{1}\approx\frac{\tilde{\mathscr{S}}^{e}_{\ce{OH}}\acute{X}^{e}_{\ce{H}}-(\tilde{\mathscr{S}}^{e}_{\ce{O}}+\tilde{\mathscr{S}}^{e}_{\ce{H}})\acute{X}^{e}_{\ce{OH}}+(\tilde{\mathscr{S}}^{e}_{\ce{OH}}+\tilde{\mathscr{S}}^{e}_{\ce{H}})\acute{X}^{e}_{\ce{O}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}},$
$\displaystyle\tilde{\omega}_{4}\approx\frac{\tilde{\mathscr{S}}^{e}_{\ce{OH}}+2\tilde{\mathscr{S}}^{e}_{\ce{O}}+3\tilde{\mathscr{S}}^{e}_{\ce{H}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}},$
$\displaystyle\tilde{\omega}_{5}\approx\frac{\tilde{\mathscr{S}}^{e}_{\ce{OH}}\acute{X}^{e}_{\ce{H}}-(\tilde{\mathscr{S}}^{e}_{\ce{O}}+\tilde{\mathscr{S}}^{e}_{\ce{H}})\acute{X}^{e}_{\ce{OH}}+(\tilde{\mathscr{S}}^{e}_{\ce{OH}}+\tilde{\mathscr{S}}^{e}_{\ce{H}})\acute{X}^{e}_{\ce{O}}}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}},$

with the rates $\tilde{\mathscr{S}}^{e}_{i}$ proportional to the HO2 reaction
rates. For instance, with the reduced mechanism of Table 1,
$\tilde{\mathscr{S}}^{e}_{\ce{O}}=-\tilde{\omega}_{17}$,
$\tilde{\mathscr{S}}^{e}_{\ce{OH}}=\tilde{\omega}_{19}-\tilde{\omega}_{17}$
and $\tilde{\mathscr{S}}^{e}_{\ce{H}}=-\tilde{\omega}_{13}$, which, with (20),
give $\tilde{\omega}_{4}\approx\tilde{\Gamma}_{4}(z)(1-z^{2})$, where

$\displaystyle\tilde{\Gamma}_{4}(z)\approx\frac{\tilde{\Gamma}_{19}-\tilde{\Gamma}_{17}-2\tilde{\Gamma}_{17}+3(\tilde{\Gamma}_{17}+\tilde{\Gamma}_{19})}{\acute{X}^{e}_{\ce{OH}}+2(\acute{X}^{e}_{\ce{O}}+1)+3\acute{X}^{e}_{\ce{H}}}.$
(22)

Figure 18: On the left, the distribution of the reaction rates ratios
$\tilde{\Omega}^{e}_{i}$ for a flame with $\phi=0.6$, showing that they soon
reach their asymptotic values $\tilde{\Omega}^{b}_{1}$ and
$\tilde{\Omega}^{b}_{4}$, plotted in the right graph as functions of $\phi$.
$\tilde{\Omega}^{e}_{5}$ has instead the form
$\tilde{\Omega}^{e}_{5}=\tilde{\Omega}^{b}_{5}\sqrt{z}$ (thin green dash
line), with the asymptotic value $\tilde{\Omega}^{b}_{5}$ also represented in
the right plot. Note that $\tilde{\Omega}^{e}_{5}\gg 1$, so reaction 5 is in
partial equilibrium for all $\phi<1$. Reactions 1 and 4 however become out of
partial equilibrium at the low-$\phi$ limit (see Figure 15).

Deviations from zero net rates. The non-vanishing rates of the shuffle
reaction are obtained in the approximation beyond partial equilibrium. The
approach is exemplified first with reaction 4, with a rate
$\dot{\omega}_{4}=k_{4,f}[\ce{H2}][\ce{OH}]-k_{4,r}[\ce{H2O}][\ce{H}]$, which
gives at equilibrium
$\dot{\omega}^{e}_{4,f}=k_{4,f}^{b}[\ce{H2}][\ce{OH}]^{e}=(k_{4,r}[\ce{H2O}])|^{b}[\ce{H}]^{e}=\dot{\omega}^{e}_{4,r}$.
Taking $\dot{\omega}^{e}_{4,f}=\dot{\omega}^{e}_{4,r}$ as common factor leads
to

$\displaystyle\dot{\omega}_{4}/\dot{\omega}^{e}_{4,f}=\frac{k_{4,f}}{k_{4,f}^{b}}\psi_{\ce{OH}}-\frac{k_{4,r}[\ce{H2O}]}{(k_{4,r}[\ce{H2O}])|^{b}}\psi_{\ce{H}},$
(23)

in terms of the reduced concentrations $\psi_{s}=[\ce{s}]/[\ce{s}]^{e}$.

Finally, approximating $[\ce{H2O}]/[\ce{H2O}]^{b}$ and the ratios
$k_{4,(f,r)}/k_{4,(f,r)}^{b}$ by unity and dividing by
$\dot{\omega}^{e}_{13,f}=(k_{13,f}[\ce{O_{2}}])|^{b}[\ce{H}]^{e}$ gives

$\displaystyle\tilde{\omega}_{4}=\tilde{\Omega}^{e}_{4}\left(\psi_{\ce{OH}}-\psi_{\ce{H}}\right),$
(24a) and similarly for reactions 1 and 5
$\displaystyle\tilde{\omega}_{-1}=\tilde{\Omega}^{e}_{-1}\left(\psi_{\ce{O}}\psi_{\ce{OH}}-\psi_{\ce{H}}\right),$
(24b)
$\displaystyle\tilde{\omega}_{5}=\tilde{\Omega}^{e}_{5}\left(\psi_{\ce{OH}}^{2}-\psi_{\ce{O}}\right),$
(24c)

with $\tilde{\Omega}^{e}_{i}=\dot{\omega}^{e}_{i,f}/\dot{\omega}^{e}_{13,f}$
functions of $z$ plotted in Figure 18.

Using (3) shows that $\tilde{\Omega}^{e}_{1}$ and $\tilde{\Omega}^{e}_{4}$
are, in the first approximation, constants; whereas,
$\tilde{\Omega}^{e}_{5}\approx\tilde{\Omega}^{b}_{5}\sqrt{z}$, as Figure 18
confirms in both cases.

##  Appendix D Glarborg et al.’s mechanism.

The following two tables list the subsets of the Glarborg et al.’s H2-O2
kinetic mechanism[14] used in this work. The reactions involving N have been
removed. This introduces only negligible, non-essential numerical differences
compared with the exact calculations with the full mechanism. The numbers used
in the text are as shown here.

| Label |  | Reaction |
---|---|---|---
H2-O2 subset |  |
1 |  H + O2 | $\leftrightarrow$ |  O + OH
2, 3 |  O + H2 | $\leftrightarrow$ |  OH + H
4 |  OH + H2 | $\leftrightarrow$ |  H2O + H
5, 6 |  OH + OH | $\leftrightarrow$ |  H2O + O
6 |  H2O + M | $\leftrightarrow$ |  H + OH + M’
7 |  H2 + M | $\leftrightarrow$ |  H + H + M
8 |  H2 + Ar | $\leftrightarrow$ |  H + H + Ar
9 |  H + O + M | $\leftrightarrow$ |  OH + M
10 |  O + O + M | $\leftrightarrow$ |  O2 + M’
11 |  H2O + M | $\leftrightarrow$ |  H + OH + M’
12 |  H2O + H2O | $\leftrightarrow$ |  H + OH + H2O
|  | Label |  | Reaction |
---|---|---|---
HO_2-H2O2 subset |  |
13 |  H + O2 + M | $\leftrightarrow$ |  HO2 + M
14 |  HO2 + H | $\leftrightarrow$ |  H2 + O2
15 |  HO2 + H | $\leftrightarrow$ |  OH + OH
16 |  HO2 + H | $\leftrightarrow$ |  H2O + O
17 |  HO2 + O | $\leftrightarrow$ |  OH + O2
18 |  HO2 + OH | $\leftrightarrow$ |  H2O + O2
19 |  HO2 + OH | $\leftrightarrow$ |  H2O + O2
20 |  HO2 + HO2 | $\leftrightarrow$ |  H2O2 + O2
21 |  HO2 + HO2 | $\leftrightarrow$ |  H2O2 + O2
22 |  H2O2 + M | $\leftrightarrow$ |  OH + OH
23 |  H2O2 + H | $\leftrightarrow$ |  H2O + OH
24 |  H2O2 + H | $\leftrightarrow$ |  HO2 + H2
25 |  H2O2 + O | $\leftrightarrow$ |  HO2 + OH
26 |  H2O2 + OH | $\leftrightarrow$ |  HO2 + H2O
27 |  H2O2 + OH | $\leftrightarrow$ |  HO2 + H2O
Table 2: Glarborg et al.’s H2-O2 kinetic mechanism[14].

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