# Effects of three-dimensional slit geometry on flashback of premixed hydrogen
flames in perforated burners

Filippo Fruzza  Hongchao Chu  Rachele Lamioni<EMAIL_ADDRESS>Temistocle Grenga  Chiara Galletti  Heinz Pitsch  Dipartimento di Ingegneria
Civile e Industriale, Università di Pisa, 56122 Pisa, Italy  Institute for
Combustion Technology, RWTH Aachen University, Aachen 52056, Germany  Faculty
of Engineering and Physical Sciences, University of Southampton, Southampton
SO17 1BJ, UK

###### Abstract

Addressing flashback represents a pivotal challenge in the advancement of
innovative perforated burners intended to substitute natural gas with hydrogen
in household appliances. Current numerical models, employing 2D configurations
to estimate flashback velocities in the slits of such burners, offer valuable
insights with reasonable computational costs. However, the inherent complexity
of the phenomena suggests that a 2D model may inadequately capture flashback
dynamics, resulting in inaccurate estimations of flashback limits. In this
study, 3D simulations are employed for the first time to explore the impact of
the three-dimensional shape of slits on the flashback limits of hydrogen-
premixed flames. Steady-state simulations are conducted to compute flashback
limits for three equivalence ratios ($\phi=0.6$, $0.8$, and $1.0$),
investigating slits with fixed width and varying lengths up to
$8\text{\,}\mathrm{mm}$. Additionally, transient simulations are performed to
investigate the flashback dynamics. The results are compared with those from
2D configurations to assess the reliability of the infinite slit
approximation. Notably, 2D simulations significantly underestimate flashback
limits as the critical initiation region is consistently located at the slit
extremities, which are neglected in 2D configurations. For the same reason,
the flashback velocity exhibits weak dependence on slit length, since the
flashback is consistently initiated at the slit far ends regardless of length.
The physical mechanisms driving the initiation of flashback in that zone are
identified as preferential diffusion effects, which cause the enrichment of
the mixture at the slit extremities, and enhanced heat transfer promoted by
the enclosed geometry, which increases the pre-heating of the fresh gases in
that regions. These findings underscore the necessity of taking into account
three-dimensional effects in numerical simulations for accurate estimations of
the flashback limits in domestic burners.

###### keywords:

Hydrogen , Flashback , Premixed flame , Perforated burner

††journal: …

##  1 Introduction

As Europe strives to meet the ambitious targets set by the EU Green Deal, the
role of green hydrogen has garnered significant attention [1, 2]. One of its
promising applications is in the decarbonization of heating for residential
and commercial buildings, especially when the transition to electrification,
is financially burdensome and technically complex, like in older buildings [3,
4]. These applications often rely on domestic end-user devices like condensing
boilers equipped with perforated burners, which inject premixed fuel-air
mixtures into combustion chambers, generating short-length flames that fit
within the compact space between the burner and heat exchanger coils [5, 6, 7,
8, 9]. The introduction of hydrogen, either in its pure form or as a blend
with natural gas, presents an opportunity for substantial emissions reduction
[10, 11, 12, 13].

However, the use of hydrogen as a fuel presents unique challenges due to its
distinct physical properties compared to natural gas, which is currently the
dominant fuel source for these devices. Its laminar flame speed is
approximately six times higher than that of natural gas under stoichiometric
conditions, and its flame thickness is much smaller. Furthermore, hydrogen
exhibits a broader flammability range [14]. Hydrogen-air mixtures have a lower
effective Lewis number, altering combustion characteristics and complicating
the stabilization of premixed hydrogen flames, introducing thermo-diffusive
instabilities [15]. Given that domestic burners typically operate in premixed
conditions, understanding these effects is crucial in the design process to
prevent safety issues and performance problems, such as elevated burner
temperatures and flashback.

As a first approximation, the stabilization of premixed flames is primarily
determined by the ratio of flame speed to bulk flow velocity. However,
numerous distinct factors significantly contribute to flame stabilization,
encompassing a range of phenomena such as flame-wall conjugate heat transfer,
curvature, stretch rate, Soret diffusion, preferential diffusion effects, and
interactions between the flame and the surrounding walls [16, 17, 18, 19, 20,
21]. Notably, the influence of wall heat losses and wall temperatures has been
emphasized in several studies [22, 23, 24, 25]. Furthermore, the significance
of flow-flame interactions in the dynamics of boundary-layer flashback,
especially in hydrogen-enriched swirled flames, has been demonstrated in
various investigations [26, 27, 28].

Experimental examinations of flashback limits for H2-enriched mixtures in
domestic boilers have been undertaken in previous studies [29, 30, 31]. In
parallel, numerical assessments conducted by Vance et al. [32, 19, 33] have
explored the influence of parameters such as heat losses, stretch, Lewis
number, and Soret diffusion on the stabilization of H2-air and H2-CH4-air
flames within 2D configurations. Similarly, 2D configurations have been
employed in research by Fruzza et al. [34] and Flores-Montoya et al. [35],
both of which investigated the flashback limits of H2-enriched mixtures in
multi-slit burners. These studies have collectively revealed two distinct
flashback regimes contingent upon the hydrogen content. Recently, Fruzza et
al. [36] have further explored this configuration, employing stochastic
sensitivity analysis methods to investigate the relative impact of H2 content,
equivalence ratio, and geometrical parameters on the flashback limits of
highly H2-enriched lean flames.

However, the range of applicability of the infinite slit approximation,
commonly assumed in 2D configurations, has not been verified in the
literature. Moreover, the physical mechanisms governing the flashback of
premixed hydrogen flames are likely influenced by three-dimensional effects.
For instance, preferential diffusion is driven by curvature, a three-
dimensional quantity. Therefore, 2D models may not accurately capture the
actual flashback dynamics in a real burner, leading to inaccurate estimations
of flashback velocities. In this study, we conduct, for the first time,
numerical simulations in 3D configurations to explore the effect of the three-
dimensional shape of the slit on the flashback limits of H2-air flames. The
numerical model incorporates conjugate heat transfer between the flame and the
burner plate. We adopt a dual approach, using both steady-state simulations to
investigate the effect of slit length over a wide range at different
equivalence ratios and transient simulations to capture flashback dynamics.
The results are then compared with those obtained in 2D configurations to
assess the reliability of the infinite slit approximation.

##  2 Configuration and numerical methods

In this study, we simulate a segment of the perforated plate of a real burner
plate typically used in domestic condensing boilers. 3D configurations
representing arrays of holes or slits of different shapes and sizes are
considered. The 3D configuration is shown in Figure 1 along with the
computational domain.

Figure 1: Left panel: computational domain (fluid zone in ligth gray, solid
zone in dark grey). Bottom panel: slit geometry. Boundary conditions are
indicated in red.

Due to the symmetries of the problem, the computational domain can be reduced
to a quarter of the entire slit, with symmetry boundary conditions on the
symmetry planes. The fluid domain extends enough both downstream
(${H_{\mathrm{out}}=$8\text{\,}\mathrm{mm}$}$) and upstream
(${H_{\mathrm{in}}=$4\text{\,}\mathrm{mm}$}$) of the solid. The slit width is
denoted by $W$, while the distance between two adjacent slits is denoted by
$D$. The distance between the centers of the round ends of the slit is
represented by $L$, so that $L=$0\text{\,}\mathrm{mm}$$ corresponds to a
circular hole of diameter $W$. The burner plate thickness is
${T=$0.6\text{\,}\mathrm{mm}$}$ for all cases, while the slit width remains
fixed at ${W=$0.5\text{\,}\mathrm{mm}$}$. The slit length is varied within the
range $L\in\left[$0\text{\,}\mathrm{mm}$-$8\text{\,}\mathrm{mm}$\right]$,
which covers a practical investigation range. For a given combination of the
parameters $W$ and $L$, the distance between slits $D$ can be adjusted to fix
the porosity of the burner, defined as $A_{\mathrm{slit}}/A_{\mathrm{tot}}$,
where $A_{\mathrm{slit}}$ is the perforated area and $A_{\mathrm{tot}}$ is the
sum of the plate and the perforated areas. We use H2-air mixtures at different
equivalence ratios $\phi$. Uniform velocity and uniform temperature of
${T_{u}=$300\text{\,}\mathrm{K}$}$ are set at the inlet, and a pressure outlet
with ${p=$1\text{\,}\mathrm{a}\mathrm{t}\mathrm{m}$}$ is imposed at the
outlet. At the fluid-solid interface, a no-slip boundary condition is given
for the velocity, and zero-mass flux is imposed for the species equations. No
thermal boundary conditions are needed at the fluid-solid interface since the
heat fluxes are computed directly as described below.

The transport equations of mass, momentum, energy, and mass fractions of
chemical species are given as:

$\displaystyle\frac{\partial\rho}{\partial
t}+\mathbf{\nabla}\cdot\left(\rho\mathbf{v}\right)=0$ (1)
$\displaystyle\frac{\partial}{\partial
t}\left(\rho\mathbf{v}\right)+\mathbf{\nabla}\cdot\left(\rho\mathbf{v}\mathbf{v}\right)=-\mathbf{\nabla}p+\mathbf{\nabla}\cdot\left(\mathbf{\bar{\tau}}\right)$
(2) $\displaystyle\begin{split}&\frac{\partial}{\partial t}\left(\rho
E\right)+\mathbf{\nabla}\cdot\left(\mathbf{v}\left(\rho E+p\right)\right)=\\\
&=\mathbf{\nabla}\cdot\left(k\mathbf{\nabla}T+\sum_{j=1}^{N}h_{j}\left(\sum_{k=1}^{N-1}\rho
D_{m,jk}\mathbf{\nabla}Y_{k}+D_{T,j}\frac{\mathbf{\nabla}T}{T}\right)\right)-\sum_{j=1}^{N}h_{j}\omega_{j}+S_{rad}\end{split}$
(3) $\displaystyle\frac{\partial}{\partial t}\left(\rho
Y_{i}\right)+\mathbf{\nabla}\cdot\left(\rho\mathbf{v}Y_{i}\right)=\mathbf{\nabla}\cdot\left(\sum_{j=1}^{N-1}\rho
D_{m,ij}\mathbf{\nabla}Y_{j}+D_{T,i}\frac{\mathbf{\nabla}T}{T}\right)+\omega_{i},$
(4)

where $\rho$ is the density, $\mathbf{v}$ is the velocity vector, $p$ is the
pressure and $\mathbf{\bar{\tau}}$ is the stress tensor. The gas phase is
modeled as an ideal gas. $T$ is the temperature, $h_{i}$, $Y_{i}$ and
$\omega_{i}$ are the enthalpy, the mass fraction, and the net rate of
production of the $i$th species, respectively, and
$E=\sum_{i=1}^{N}h_{i}Y_{i}-p/\rho+\absolutevalue{\vec{v}}^{2}/2$. $k$ is the
mass-weighted thermal conductivity of the mixture, $D_{m,ij}$ are the
generalized Fick’s law diffusion coefficients of the species $i$ in species
$j$, and $D_{T,i}$ are the thermal diffusion coefficients of the $i$th
species. Finally, $S_{rad}$ is the energy source associated with radiation.
The burner plate is modeled as a solid with the properties of the stainless
steel typically used for this kind of burner, with density
${\rho_{s}=$7719\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$}$, specific
heat
${c_{p,s}=$461.3\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$}$,
and thermal conductivity
${k_{s}=$22.54\text{\,}\mathrm{W}\text{\,}{\mathrm{m}}^{-1}\text{\,}{\mathrm{K}}^{-1}$}$.
Inside the solid domain, we solve the energy equation:

$\frac{\partial}{\partial
t}\left(\rho_{s}h_{s}\right)=\nabla\cdot\left(k_{s}\nabla T\right)$ (5)

where $h_{s}=\int_{T_{0}}^{T}c_{p,s}dT$ is the sensible enthalpy of the solid
material.

The equations are solved on a structured grid, with characteristic cell size
in the reaction region of ${\Delta x=$25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$\simeq\delta_{F}/13}$, where $\delta_{F}$ is the 1D unstretched flame
thickness. The mesh is slightly stretched in both the $x$ and $y$ directions
towards the domain boundaries and in the $z$ direction towards the inlet and
outlet, far from the reaction zone. We employ detailed chemistry, utilizing a
reduced mechanism comprising 9 chemical species and 22 reactions. This reduced
mechanism is derived from the Kee-58 skeletal mechanism [37]. This choice
ensures consistency with prior research involving CH4/H2 mixtures up to 100%
H2, where the same mechanism was utilized [34, 36]. Full multicomponent
diffusion is modeled through generalized Fick’s law coefficients derived by
the Maxwell-Stefan equations [38, 39, 40]. Soret diffusion is modeled using
the following empirically-based composition-dependent expression provided by
Kuo [41]:

$D_{T,i}=-2.59\times
10^{-7}T^{0.659}\left[\frac{M_{i}^{0.511}X_{i}}{\sum_{j=1}^{N}M_{j}^{0.511}X_{j}}-Y_{i}\right]\cdot\left[\frac{\sum_{j=1}^{N}M_{j}^{0.511}X_{j}}{\sum_{j=1}^{N}M_{j}^{0.489}X_{j}}\right],$
(6)

where $M_{i}$, $X_{i}$, and $Y_{i}$ are the molar mass, molar fraction, and
mass fraction, respectively, of the species $i$. Radiation is modeled by means
of the gray Discrete Ordinates (DO) method [42], assuming the emissivity of
the fluid-solid interface to be 0.85. The conjugate heat transfer (CHT)
between the fluid and the solid zones is modeled to consider the interaction
between the flame and the burner plate. The CHT is modeled using Fourier’s Law
to compute the heat flux through the fluid-solid interface [38].

##  3 Solution methodology

We perform steady-state simulations using a pressure-based coupled algorithm
[38] and a second-order scheme for spatial discretization. Given the large
number of simulations required for the parametric variations investigated in
this study, a steady-state approach is needed for reasons of computational
cost. We start from a relatively high inlet velocity, ensuring a stable flame
solution. The inlet velocity is then systematically reduced until the steady-
state solver no longer converges to a stable flame solution, signifying the
attainment of the critical inlet velocity for flashback. To accurately
estimate this flashback inlet velocity, the minimum decrement of the inlet
velocity is set at ${\Delta
V_{\mathrm{in}}=$0.01\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$}$, where
$V_{\mathrm{in}}$ is the uniform inlet velocity. Neglecting the density
variations of the mixture due to the high burner plate temperatures, the cold-
flow bulk velocity at the slit entry is defined as

$V_{\mathrm{S}}=\frac{A_{\mathrm{tot}}}{A_{\mathrm{slit}}}V_{\mathrm{in}}.$
(7)

Following [33], the flashback velocity $V_{\mathrm{FB}}$ is defined as

$V_{\mathrm{FB}}=\left.V_{\mathrm{S}}\right|_{FB}=\frac{A_{\mathrm{tot}}}{A_{\mathrm{slit}}}\left.V_{\mathrm{in}}\right|_{FB}.$
(8)

To validate the reliability of the steady-state approach, we perform transient
simulations on a domain that encompasses the entire slit geometry, rather than
just a quarter. This is undertaken to capture potential asymmetries in
flashback dynamics. The PISO implicit algorithm is employed [38], with a
second-order scheme for time discretization. The solution methodology remains
the same as described above, with a minimum velocity decrement of ${\Delta
V_{\mathrm{in}}=$0.01\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$}$. In the fluid
domain, a time step of ${\Delta t=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$
is employed. However, the characteristic time scales of heat conduction within
the solid domain are considerably higher. Consequently, implementing a uniform
time step across the entire computational domain would require excessively
long simulations for the burner temperature to stabilize, rendering it
infeasible due to computational costs. To address this, a larger time step for
the solid zone, $\Delta t_{s}=10^{3}\Delta t$, is applied. Similar
methodologies have been previously applied in flashback simulations [34, 35].
It is worth noting that further reducing the solid time step does not change
the estimated flashback velocities, but does impact the flashback dynamics:
while effective for steady-state solutions, this approach has limitations
during the flashback occurrence, as the artificially accelerated heating of
the solid introduces unphysical flame oscillations. Therefore, once the
flashback velocity is determined using this method, the final stage of the
simulation, corresponding to the flashback occurrence, is repeated with a
uniform time step of ${\Delta t=$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$}$ in
both phases. This ensures an accurate capture of the flashback dynamics
without being influenced by the unphysical heating of the solid.

For transient simulations, two test cases were considered: a circular hole
with a diameter of ${W=$0.5\text{\,}\mathrm{mm}$}$ and a slit with a width of
${W=$0.5\text{\,}\mathrm{mm}$}$ and length of ${L=$2\text{\,}\mathrm{mm}$}$,
at an equivalence ratio of $\phi=0.6$. The transient approach exhibited no
significant discrepancies compared to the steady-state method. Therefore, we
opted for the faster steady-state approach to estimate flashback velocities
across a broad range of geometrical and operational parameters. A more
detailed comparison between the two approaches can be found in the
Supplementary Material.

##  4 Results and discussion

###  4.1 Preliminary analysis

All previous numerical studies on flashback in multi-slit burners have been
conducted using 2D configurations, corresponding to the infinite slit
approximation where the far ends of the slit are neglected. When performing a
2D simulation of a slit array, the domain represents a cross-sectional profile
of the actual slit, as shown in Figure 2.

Figure 2: Visualization of the 2D configuration.

Its geometry fully determined by the width $W$, the spacing between adjacent
slits $D$, and the plate thickness $T$. In two dimensions, the porosity is
defined as $W/(D+W)$. The question we aim to address is: What length does the
actual 3D slit need to have to be accurately represented by a 2D
configuration?

To answer this question, a preliminary study is carried out by comparing the
results obtained from a 2D simulation with those of 3D simulations of slits
with increasing length. The burner temperature is chosen as the test quantity
for this analysis due to its significant impact on flashback [34]. Both 2D and
3D simulations are performed at $\phi=0.6$ with a fixed inlet velocity
corresponding to
${V_{\mathrm{S}}=$4\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$}$. In all cases,
the plate thickness is ${T=$0.6\text{\,}\mathrm{mm}$}$, the slit width is
${W=$0.5\text{\,}\mathrm{mm}$}$, and the distance between two adjacent slits
is ${D=$1\text{\,}\mathrm{mm}$}$. For the 3D simulations, the length $L$ is
varied in the range $1\text{\,}$-$50\text{\,}\mathrm{mm}$. The burner
temperature as a function of the slit length, along with the results of the 2D
case, is shown in Figure 3.

Figure 3: Burner plate temperature as a function of the slit length for
${W=$0.5\text{\,}\mathrm{mm}$}$ and ${D=$1\text{\,}\mathrm{mm}$}$. The red
dashed line represents the 2D result.

The 2D simulation predicts a burner temperature of
${T_{B}=$812\text{\,}\mathrm{K}$}$. In contrast, 3D simulations predict a
higher burner temperature of ${T_{B}=$911\text{\,}\mathrm{K}$}$ for
${L=$1\text{\,}\mathrm{mm}$}$, which decreases to
${T_{B}=$826\text{\,}\mathrm{K}$}$ when ${L=$10\text{\,}\mathrm{mm}$}$,
ultimately converging with the 2D result as $L$ approaches
$50\text{\,}\mathrm{mm}$. Higher burner temperatures for shorter slits can be
readily explained by an increasing impact of the slit extremities, where
additional heat is transferred from the flame to the burner plate. As the
length is increased, the relative impact of the extremity regions decreases,
until the 2D result, for which these regions are completely neglected, is
recovered.

The findings from this preliminary analysis highlight the inadequacy of the
infinite slit approximation for slits shorter than $50\text{\,}\mathrm{mm}$.
Considering that slits in practical devices are usually much shorter than
$50\text{\,}\mathrm{mm}$, 2D configurations prove to be inadequate
representations of the actual geometry. This underscores the necessity of
utilizing 3D simulations to accurately capture the behavior of practical
devices, accounting for the complete geometric complexity of the slit
configuration, and motivates a thorough analysis of the influence of the third
dimension on flashback limits.

###  4.2 Effect of the slit length

To analyze the effect of the third dimension on the flashback limits, we
compute the flashback velocities at three equivalence ratios, $\phi=0.6$,
$0.8$, and $1.0$, for slits with different lengths. The considered values of
$L$ are ${L=$0\text{\,}\mathrm{mm}$}$ (circular hole),
$0.5\text{\,}\mathrm{mm}$, $1\text{\,}\mathrm{mm}$, $2\text{\,}\mathrm{mm}$,
$4\text{\,}\mathrm{mm}$, and $8\text{\,}\mathrm{mm}$. Longer slits are not
investigated because they are not practically relevant for real burners. For
all cases, a fixed porosity of $A_{\mathrm{slit}}/A_{\mathrm{tot}}=0.2$ is
imposed by adjusting $D$. In Figure 4 we plot $V_{\mathrm{FB}}$ versus $L$ for
various equivalence ratios. Both unnormalized and normalized values are
presented in panels (a) and (b), respectively. The flashback velocity and the
slit length are scaled with the 1D unstretched laminar flame speed $s_{L}$ and
thermal flame thickness $\delta_{T}$, respectively. Additionally, we include
the flashback velocity computed for a 2D configuration representing an
infinite slit with the same width and porosity imposed for 3D cases,
corresponding to $W=$0.5\text{\,}\mathrm{mm}$$ and
$D=$2\text{\,}\mathrm{mm}$$.

Figure 4: Flashback velocity plotted against slit length for various
equivalence ratios: (a) non-normalized values and (b) normalized values with
respect to $s_{L}$ and $\delta_{T}$. 2D results are indicated by dashed
horizontal lines.

As expected, $V_{\mathrm{FB}}$ increases with an increase in the equivalence
ratio due to a higher laminar flame speed. Conversely, the normalized
flashback velocity $V_{\mathrm{FB}}/s_{L}$, which is a more reliable indicator
of the flashback propensity as it removes the trivial dependence on the
laminar flame speed, exhibits the opposite trend. As flashback of hydrogen
flames is strongly influenced by preferential diffusion effects, a higher
flashback propensity for leaner mixtures is expected [33, 36]. The flashback
velocity increases rapidly between $L=$0\text{\,}\mathrm{mm}$$ and
$L=$1\text{\,}\mathrm{mm}$$. However, for $L>$1\text{\,}\mathrm{mm}$$, the
dependence of the flashback velocity on the slit length diminishes
significantly. This observation suggests a distinct behavior of circular holes
and very short slits compared to longer slits, which exhibit a consistent
behavior irrespective of their length. The 2D results exhibit significant
deviations from the 3D results, with the latter being approximately 1.5 times
higher. Notably, unlike the observed behavior of the burner plate temperature
in Figure 3, the 3D results for the flashback velocity do not converge to the
2D values for large slit lengths.

To understand the distinct behaviors of the circular hole configuration and
the slit configuration, we investigate the differences in the flame structure
at the flashback limits for the circular hole and a slit. In Figures 5 and 6,
we show temperature, local equivalence ratio, and normalized H2 consumption
rate profiles for the circular hole configuration and the slit with length
${L=$2\text{\,}\mathrm{mm}$}$ (transverse and longitudinal section) at the
flashback limit, for the case ${\phi=0.6}$. The local equivalence ratio
$\varphi$ is defined using the Bilger formula [37] and normalized with the
mixture equivalence ratio $\phi_{\mathrm{in}}=0.6$. The molecular H2
consumption rate $\omega_{\ch{H2}}$ is normalized as
$\bar{\omega}_{\ch{H2}}=\omega_{\ch{H2}}/\textnormal{max}(\omega_{\ch{H2},\textnormal{1D}})$,
where $\textnormal{max}(\omega_{\ch{H2},\textnormal{1D}})$ is the maximum H2
consumption rate obtained for the corresponding unstretched 1D flame. To
visualize the flame front, iso-contours of progress variable are plotted. The
progress variable is defined as $C=1-Y_{H_{2}}/Y_{H_{2},u}$, where $Y_{H_{2}}$
denotes the mass fraction of H2, and $Y_{H_{2},u}$ represents its value in the
unburnt mixture.

Figure 5: (a) Temperature, (b) normalized equivalence ratio, and (c)
normalized H2 consumption rate profiles for the circular hole at the flashback
limit for the case ${\phi=0.6}$. Progress variable iso-contours are plotted in
white. The solid zone representing the burner plate is delimited by the dashed
line.

Figure 6: (a) Temperature, (b) normalized equivalence ratio, and (c)
normalized H2 consumption rate profiles for the slit with length
${L=$2\text{\,}\mathrm{mm}$}$ at the flashback limit for the case
${\phi=0.6}$. Progress variable iso-contours are plotted in white. The solid
zone representing the burner plate is delimited by a dashed line.

For the circular hole, we observe the flame front to be very close to the hole
exit. The flame thickness is comparable with the radius of the hole, and the
flame is attached both at the top and the inner sides of the burner plate. The
enclosed geometry leads to high pre-heating of the fresh gases, caused by the
heating of the flow when passing through the hot hole: the heat lost by the
flame to the burner at the top of the plate is given back to the flow on the
bottom and the inner sides of the plate, causing the gases to be preheated
when reaching the flame front. Due to the small width of the channel, the
mixture temperature is approximately uniform when reaching the flame zone,
with $T\simeq$800\text{\,}\mathrm{K}$$ at $C=0.2$. In Figure 5(b) we observe
variations in the local equivalence ratio caused by preferential diffusion.
Two sources of preferential diffusion are observed: one induced by the flame
front curvature and the other by Soret diffusion. For mixtures with effective
Lewis numbers smaller than one, preferential diffusion effects cause $\varphi$
to be reduced at the tip of the flame due to negative curvature, leading to
the enrichment of the flame base region [43]. Additionally, Soret diffusion
causes light species like H2 to diffuse toward hotter regions in the presence
of temperature gradients, leading to the enrichment of the mixture close to
the hot burner plate [19]. However, for the circular hole, no significant
enrichment is observed in the flame region. The shape of the flame is
approximately hemispherical, suppressing curvature-induced preferential
diffusion effects. Moreover, no significant Soret-induced enrichment is
observed at the flame base. This flame configuration suggests that the main
mechanism driving the flashback is the pre-heating of the unburnt gases:
despite the flow velocity increasing with its temperature due to a decrease in
density, the flame speed increases faster than linearly with the unburnt
mixture temperature [41]. When the inlet velocity is decreased, the flame
finds a new stable position, closer to the burner plate, further increasing
the solid temperature and the pre-heating of the mixture. When a critical
inlet velocity and a critical temperature are reached, the flame speed
surpasses the bulk velocity, causing the flame to flashback into the hole.

Conversely, in the case of the slit, a significantly more pronounced influence
of preferential diffusion is observed. The elongated shape of the slit induces
non-uniformities in the curvature of the flame front, resulting in the
enrichment of the mixture in the flame base regions along both the sides of
the slit and its extremities. Additionally, the presence of temperature
gradients pushes the fuel toward the hot burner plate surface due to the Soret
effect, causing further enrichment in the flame base region. As a consequence
of these two effects, a peak of $\bar{\omega}_{\ch{H2}}\simeq 1.5$ is observed
at the flame base. Mixture pre-heating is more prominent near the far end,
where the enclosed geometry promotes heat transfer. Consequently, due to an
enhanced flame speed, the flame stabilizes further upstream in comparison to
the slit side, attaching to the inner wall of the burner plate. This
positioning designates this region as the likely initiation zone for
flashback: if the heightened flame speed, resulting from both increased H2
consumption rates and mixture pre-heating, exceeds the flow velocity in the
critical region near the wall, where the flow velocity is low, flashback is
initiated.

The crucial role of the slit extremities in initiating the flashback also
accounts for the disparities between 2D and 3D results illustrated in Figure
4. The flashback limits computed for 3D configurations do not converge to the
2D values for long slits, indicating intrinsically different flashback
mechanisms for the two configurations. To gain deeper insights into this
observation, the analysis of the flashback dynamics in 3D configurations is
particularly valuable.

###  4.3 Flashback dynamics

In this section, we analyze the flashback dynamics occurring in a three-
dimensional slit. To do this, we make use of the transient simulation for a
slit with ${L=$2\text{\,}\mathrm{mm}$}$ at $\phi=0.6$ described in Section 3.
Transient simulations provide the opportunity to observe the flashback
dynamics, which is not possible using the steady-state approach.

Figure 7 depicts a sequence of three snapshots captured during the last stage
of the simulation. The snapshots illustrate the evolution of the flame front
after the reduction of the inlet velocity from the last stable velocity to the
flashback velocity.

Figure 7: Snapshots of the transient simulation for the slit with
${L=$2\text{\,}\mathrm{mm}$}$, corresponding to four instants during the
occurrence of flashback. Iso-surfaces of progress variable at $C=0.3$ are
shown to visualize the flame front. The burner plate geometry is included as a
reference. $t=0$ marks the moment of the reduction of the inlet velocity from
the last stable velocity to $V_{\mathrm{FB}}$.

An iso-surface of progress variable, corresponding to $C=0.3$, is plotted to
identify the flame front. The initial instant, denoted as $t=0$, marks the
moment of the reduction of the inlet velocity. At $t=0$, the flame front is in
its initial stable position (Figure 7 (a)). During the first
$40\text{\,}\mathrm{m}\mathrm{s}$, we observe minimal alterations in the flame
front position. By ${t=$40\text{\,}\mathrm{m}\mathrm{s}$}$, flashback
initiates rapidly within the slit (Figure 7 (b)). The initiation is asymmetric
and originates at the ends of the slit. Here, the flame front moves backward
towards the entrance of the slit, traversing it entirely on that side (Figure
7 (c)), ultimately driving the entire flashback process (Figure 7 (d)). For a
better visualization of the important physical mechanisms involved, a colored
3D visualization of the last stable flame, corresponding to Figure 7 (a), is
shown in Figure 8. The iso-contour of progress variable corresponding to
$C=0.3$ is colored by temperature (a), normalized local equivalence ratio (b),
and normalized molecular H2 consumption rate (c).

Figure 8: Iso-contour of progress variable $C=0.3$ for a slit with
${L=$2\text{\,}\mathrm{mm}$}$ at the flashback limit for the case ${\phi=0.6}$
(top view). The iso-contours are colored by: (a) temperature, (b) normalized
local equivalence ratio, and (c) normalized H2 consumption rate.

A temperature peak is observed at the slit far ends, where the enclosed
geometry promotes heat transfer, leading to higher pre-heating (Figure 8(a)).
Preferential diffusion effects, driven by curvature and Soret effect, induce
the enrichment of the mixture in the same regions. Both of these phenomena
contribute to an increase in the H2 consumption rate (Figure 8 (c)), and a
subsequent increase in the flame speed. Consequently, this region emerges as
the flashback initiation zone, with the local flame speed exceeding the flow
velocity and triggering flashback.

The observation of this distinctive dynamics explains the weak dependence of
the flashback velocity on the length of the slit above a certain value (Figure
4), as we anticipate the critical region for flashback initiation to be
consistently located at the ends of the slit, irrespective of the specific
value of $L$. Furthermore, the crucial role played by the slit extremities
makes flashback an intrinsically three-dimensional phenomenon, with enhanced
pre-heating in these regions causing the initiation of flashback at higher
inlet velocities compared to 2D configurations. In 2D, flashback is initiated
when the imbalance between the flow velocity and the flame speed is reached at
the sides of the slit [33, 34, 36, 35], and this occurs at lower inlet
velocities. Consequently, the 3D results in Figure 4 do not converge to those
obtained in 2D configurations, where the slit extremities are consistently
neglected.

##  5 Conclusions

In this study, we investigated the influence of the three-dimensional geometry
of the slits on the flashback limits of hydrogen-premixed flames in perforated
burners. We conducted steady-state and transient simulations to compute the
flashback limits for slits with varying lengths and compared the results with
those obtained in 2D configurations. Furthermore, we performed transient
simulations to investigate the three-dimensional flashback dynamics.

A preliminary analysis was conducted to estimate the range of applicability of
the infinite slit approximation, commonly employed in 2D simulations used in
the literature for flashback velocity estimations. We computed the burner
plate temperature, chosen as the test quantity, for a given inlet velocity in
three-dimensional slits with increasing length and in a 2D configuration. Our
results highlighted the inadequacy of the infinite slit approximation, as the
3D results converged to the 2D ones only for slit lengths beyond practical
interest.

Flashback limits were computed for slits with fixed width and increasing
length, ranging from ${L=0}$ (corresponding to a circular hole) to
${L=$8\text{\,}\mathrm{mm}$}$. The circular hole was found to be more
resistant to flashback, due to weaker preferential diffusion effects compared
to the slits. However, for ${L>0}$, the flashback velocity was found to
exhibit a weak dependence on the slit length. The results were compared with
the flashback velocity obtained in a 2D configuration. Notably, in this case,
the 3D results do not converge to those obtained in 2D configurations for
large $L$, and the 2D results were found to significantly underestimate the
flashback velocity.

The flashback dynamics in a slit was investigated in detail using transient
simulations. The flashback was found to be initiated asymmetrically at one
extremity of the slit, where the imbalance between the flame speed and the
flow velocity causes the flame front to move upstream, ultimately driving the
entire flashback process. The location of the initiation of the flashback is
due to two combined mechanisms. Firstly, mixture enrichment is observed at the
flame base in the slit extremities region due to curvature-induced
preferential diffusion and the Soret effect. Furthermore, the enclosed
geometry of the slit far ends promotes the heat transfer from the hot burner
plate to the fresh mixture, increasing its pre-heating. Both these effects
increase the flame speed in that critical zone, triggering the initiation of
flashback. This observation explains the weak dependence of the flashback
velocity on the length of the slit, as we anticipate the critical region for
flashback initiation to be consistently located at the ends of the slit,
irrespective of the specific value of $L$. Furthermore, the crucial role
played by the slit extremities explains why the 3D results do not converge to
those obtained in 2D configurations, where the slit extremities are always
neglected.

## Author contributions

## Acknowledgements

This research is funded by the Ministry of University and Research (MUR) and
Immergas S.p.A., Brescello, RE (Italy), as part of the PON 2014-2020 “Research
and Innovation” resources - Green/Innovation Action - DM MUR 1061/2021 and DM
MUR 1062/2021.

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