# Analysis of Gaseous and Gaseous-Dusty, Premixed Flame Propagation in Obstructed Passages with Tightly Placed Obstacles

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Formulation

#### 2.1. Two-Dimensional (2D) Geometry

#### 2.2. Cylindrical Geometry

## 3. Results and Discussion

#### 3.1. Gaseous Combustion

_{4})-air burning, as well as that of propane (C

_{3}H

_{8})-air, for comparison, with various blockages $\alpha =0,1/3,1/2,2/3$ employed. The horizontal dotted lines in Figure 2b show the speeds of sound for the methane-air, $354\mathrm{m}/\mathrm{s}$, and propane-air, $340\mathrm{m}/\mathrm{s}$, mixtures (Table 1). The case of no obstacles, $\alpha =0$, reproduces, completely, the situation of “finger + DL” flame acceleration [3]. It is noted that this acceleration is limited in time such that the flame would start decelerating when its skirt contacts a sidewall at $t~0.089\mathrm{s}$ and $0.072\mathrm{s}$ for methane-air and propane-air burning, respectively. It is also noted that by these times, the propane-air flame overcomes the sound threshold $340\mathrm{m}/\mathrm{s}$, whereas the methane-air flame stops at ${U}_{tip}~288\mathrm{m}/\mathrm{s}$. In contrast, in an obstructed channel, $\alpha >0$, acceleration is unlimited in time until the flame approaches the speed of sound and can eventually trigger a detonation. We should recall, at this point, that approaching the near-sonic values by the flame front will eventually break the incompressible approach, adopted in Equation (6), and the entire present formulation. Indeed, to describe the DDT stage accurately, we have to incorporate the impacts of gas compressibility into the present analysis. This can be done by considering the formulation of Section 2 as the zeroth-order approach in $Ma\to 0$, and then extending it to account for the finite $Ma$ according to the methodology employed earlier for unobstructed [27] and obstructed [28] geometries. However, such a rigorous extension of the present formulation to account for the compressibility effects requires a separate work and is presented elsewhere [29]. It is also noticed that the analytical incompressible formulation of Section 2 does not involve pressure as a parameter (except for the fact that pressure comes indirectly through the thermal-chemical parameters such as ${U}_{f}$ or $\mathsf{\Theta}$, which are taken for ambient, atmospheric pressure indeed in the present section). However, if the present formulation is extended to account for a finite $Ma$, as discussed above, then pressure, its variations and gradient, are directly involved in a revised formulation, with the ambient, atmospheric pressure imposed as the initial conditions and a boundary condition in the open side of the passage.

_{4})-air and propane (C

_{3}H

_{8})-air flames of various equivalence ratios. Specifically, Figure 10 presents ${Z}_{rud}$ versus $\varphi $ for various blockage ratios, including the case of no obstacles, $\alpha =0$, in the 2D (Figure 10a) and cylindrical (Figure 10b) geometries. Overall, Figure 10 agrees with our analysis above in that the shortest run-up distances are observed for a slightly fuel-rich methane-air flame of $\varphi ~1.1$. In the 2D case, Figure 10a, we have ${Z}_{rud}~8.19\mathrm{m}$, $7\mathrm{m}$, and $5.46\mathrm{m}$ for $\alpha =1/3,1/2,\mathrm{and}2/3$, respectively. The case of $\alpha =0$ in a 2D geometry is not relevant because a flame skirt contacts a sidewall and stops accelerating before the DDT event for all $\varphi $ considered, which is in line with the findings of [3]. For the lean or rich methane-air flames, the run-up distances are much higher, namely, up to $80\mathrm{m}$ for $\varphi =0.6$ and up to $35\mathrm{m}$ for $\varphi =1.4$ (still in the 2D geometry). In the cylindrical-axisymmetric configuration, for the fastest methane-air flames with $\varphi =1.1$ we found the run-up distances as small as ${Z}_{rud}~5.31\mathrm{m}$, $4.11\mathrm{m}$, $3.45\mathrm{m}$, and $2.64\mathrm{m}$ for $\alpha =1/3,1/2,\mathrm{and}2/3$, respectively. For lean or rich methane-air burning, the run-up distances are much higher in the cylindrical geometry, i.e., up to $40\mathrm{m}$ for $\varphi =0.6$ and up to $18\mathrm{m}$ for $\varphi =1.4$. It is noted that, unlike the 2D configuration, in the cylindrical-axisymmetric case, the detonation is predicted for methane-air burning even in the case of $\alpha =0$, when the equivalence ratio is in the range $0.8\le \varphi \le 1.3$, which generally agrees with [3]. Overall, for the same geometry, $\alpha $ and $\varphi $, the run-up distances are dramatically shorter for the C

_{3}H

_{8}-air flames as compared with the CH

_{4}-air flames; this would overcome the sound threshold for additional equivalence ratios in the case of $\alpha =0$, being within the ranges $1\le \varphi \le 1.2$ and $0.7\le \varphi \le 1.4$ for the 2D and cylindrical-axisymmetric geometries, respectively.

#### 3.2. Validation of Gaseous Formulation

#### 3.3. Extension to Gaseous-Dusty Environment

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${R}_{f}$ | radius of a global spherical expanding flame front |

$C$ | constant defined in Equation (1) |

$t$ | time |

$n$ | Darrieus–Landau instability exponent |

${k}_{DL}$ | Darrieus–Landau cutoff wavenumber |

${U}_{f}$ | laminar flame velocity |

${\rho}_{fuel}$ | density of fuel mixture |

${\rho}_{burnt}$ | density of burnt gas |

${L}_{f}$ | flame thickness |

${D}_{th}$ | thermal diffusivity coefficient |

${U}_{DL}$ | instantaneous radial flame velocity |

$H$ | half-width of a two-dimensional (2D) passage |

$R$ | radius of a cylindrical passage |

$\mathsf{\Delta}z$ | obstacle spacing |

$x$, r | radial direction |

$z$ | axial direction |

${R}_{f}\left(t\right)$ | flame “skirt” |

$\mathsf{\Delta}z$ | obstacle spacing |

${u}_{x}$, ${u}_{r}$ | radial velocity |

${u}_{z}$ | axial velocity |

${Z}_{tip}$ | flame tip position |

${U}_{tip}$ | flame tip velocity |

${t}_{obs}$ | the time flame skirt touches an obstacle |

${R}_{f}\left({t}_{obs}\right)$ | flame “skirt” radius at ${t}_{obs}$ |

${Z}_{tip}\left({t}_{obs}\right)$ | flame tip position at ${t}_{obs}$ |

${U}_{DL}\left({t}_{obs}\right)$ | global flame velocity at ${t}_{obs}$ |

${t}_{rud}$ | flame run-up time |

${c}_{0}$ | speed of sound |

${Z}_{rud}$ | flame run-up distance |

${R}_{f,o}$ | flame “skirt” in [15] |

${t}_{f}\left(z\right)$ | the time instant at which the fresh gas between obstacles at the position $z$ starts burning |

${U}_{d,f}$ | laminar flame velocity in an “effective” gaseous-dusty environment |

${C}_{p}$ | specific heat of gaseous air-fuel mixture |

${C}_{T}$ | entire specific heat |

${T}_{f}$ | flame temperature with particles |

${T}_{b}$ | adiabatic flame temperature based on purely methane-air equivalence ratio |

${T}_{u}$ | unburnt gas temperature |

$E$ | activation energy |

${R}_{u}$ | gas constant |

$M$ | molar masses |

$m$ | original masses |

${C}_{s}$ | specific heat of dust particles |

${n}_{s}$ | number of particles per unit volume |

${V}_{s}$ | volume of a single particle |

${\rho}_{s}$ | density of dust |

$\rho $ | density of gaseous-dusty fuel-air mixture |

${\rho}_{u}$ | density of gas |

${c}_{s}$ | concentration of the particles |

${r}_{s}$ | radius of a particle |

$Q$ | volumetric heat release |

${L}_{v}$ | heat of gasification per unit volume |

${n}_{air}$ | number of moles of air per unit volume |

${V}_{C{H}_{4}}$ | volume of methane |

${V}_{air}$ | volume of air |

${n}_{product}$ | number of moles of the burning products |

$\alpha $ | blockage ratio |

$\varphi $ | equivalence ratio |

${\lambda}_{DL}$ | Darrieus–Landau critical wavelength |

$\mathsf{\Theta}\equiv {\rho}_{u}/{\rho}_{b}$ | thermal expansion ratio |

$\beta $ | defined as $\sqrt{\mathsf{\Theta}\left(\mathsf{\Theta}-1\right)}$ |

${\varphi}_{s}$ | modified equivalence ratio in the gaseous-dusty air mixture |

$f$ | flame |

$DL$ | Darrieus–Landau |

$fuel$ | fuel mixture |

$burnt$ | burnt gas |

$th$ | thermal |

$tip$ | tip |

$obs$ | obstacle |

$rud$ | run-up distance |

$0$ | initial |

$f,o$ | flame in obstructed passage |

$d$ | dust |

$s$ | particle |

$T$ | total |

$b$ | burnt |

$u$ | unburnt |

$act$ | actual |

$st$ | stoichiometric |

$product$ | product |

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**Figure 1.**Illustration of flame acceleration inside an obstructed passage (

**a**) and delayed burning between obstacles (

**b**).

**Figure 2.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a 2D geometry for stoichiometric ($\varphi =1$) methane (CH

_{4})-air and propane (C

_{3}H

_{8})-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 3.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a cylindrical-axisymmetric geometry for stoichiometric CH

_{4}-air and C

_{3}H

_{8}-air burning with various blockage ratios, $\alpha =0,1/3,1/2,2/3$.

**Figure 4.**Comparison of the 2D and cylindrical-axisymmetric geometries. Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) for stoichiometric CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 5.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**,

**c**) and velocity ${U}_{tip}$ (

**b**,

**d**) in a 2D (

**a**,

**b**) and cylindrical-axisymmetric (

**c**,

**d**) geometry for stoichiometric CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$ and various power factors $n=1.33,1.4,1.5$.

**Figure 6.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a 2D geometry for lean ($\varphi =0.8$), stoichiometric ($\varphi =1$), and rich ($\varphi =1.2$) CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 7.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a 2D geometry for highly-lean ($\varphi =0.6$) and highly-rich ($\varphi =1.4$) CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 8.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a cylindrical-axisymmetric geometry for lean ($\varphi =0.8$), stoichiometric ($\varphi =1$), and rich ($\varphi =1.2$) CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 9.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**) and velocity ${U}_{tip}$ (

**b**) in a cylindrical-axisymmetric geometry for highly-lean ($\varphi =0.6$) and highly-rich ($\varphi =1.4$) CH

_{4}-air burning with various blockage ratios $\alpha =0,1/3,1/2,2/3$.

**Figure 10.**The flame run-up distance versus the equivalence ratio $\varphi $ for CH

_{4}-air and C

_{3}H

_{8}-air burning at various blockage ratios: $\alpha =0,1/3,1/2,2/3$ in a 2D (

**a**) and cylindrical-axisymmetric (

**b**) geometries.

**Figure 13.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**,

**c**) and velocity ${U}_{tip}$ (

**b**,

**d**) in a 2D (

**a**,

**b**) and cylindrical-axisymmetric (

**c**,

**d**) geometries for lean CH

_{4}-air burning of $\varphi =0.7$, without and with dust particles (combustible, inert, and combined) of particle radius ${r}_{s}=75\mathsf{\mu}\mathrm{m}$ and concentration ${c}_{s}=50\mathrm{g}/{\mathrm{m}}^{3}$, for various blockage ratios, $\alpha =0,1/3,2/3$.

**Figure 14.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**,

**c**) and velocity ${U}_{tip}$ (

**b**,

**d**) in a 2D (

**a**,

**b**) and cylindrical-axisymmetric (

**c**,

**d**) geometries for lean CH

_{4}-air burning of $\varphi =0.7$, without and with dust particles (combustible, inert, and combined) of particle radius ${r}_{s}=75\mathsf{\mu}\mathrm{m}$ and concentration ${c}_{s}=120\mathrm{g}/{\mathrm{m}}^{3}$, for various blockage ratios, $\alpha =0,1/3,2/3$.

**Figure 15.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**,

**c**) and velocity ${U}_{tip}$ (

**b**,

**d**) in a 2D (

**a**,

**b**) and cylindrical-axisymmetric (

**c**,

**d**) geometries for lean CH

_{4}-air burning of $\varphi =0.7$, without and with dust particles (combustible, inert, and combined) of particle radius ${r}_{s}=75\mathsf{\mu}\mathrm{m}$ and concentration ${c}_{s}=250\mathrm{g}/{\mathrm{m}}^{3}$, for various blockage ratios, $\alpha =0,1/3,2/3$.

**Figure 16.**Time evolution of the flame tip position ${Z}_{tip}$ (

**a**,

**b**) and velocity ${U}_{tip}$ (

**c**,

**d**) in a 2D (

**a**,

**b**) and cylindrical-axisymmetric (

**c**,

**d**) geometries for lean CH

_{4}-air burning of $\varphi =0.7$ with and without dust particles (combustible, inert, and combined) of particle radius ${r}_{s}=10\mathsf{\mu}\mathrm{m}$ and concentration ${c}_{s}=120\mathrm{g}/{\mathrm{m}}^{3}$, for various blockage ratios, $\alpha =0,1/3,2/3$.

**Table 1.**The parameters for methane-air and propane-air combustion: the thermal expansion ratio, $\mathsf{\Theta}$, the laminar burning velocity, ${U}_{f}$, and the sound speed in the fuel mixture, ${c}_{0}$, versus the equivalence ratio, $\varphi $.

Methane-Air Fuel Mixtures | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\varphi $ | 0.6 | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3 | 1.4 |

$\mathsf{\Theta}\equiv {\rho}_{u}/{\rho}_{b}$ | 5.54 | 6.11 | 6.65 | 7.12 | 7.48 | 7.55 | 7.43 | 7.28 | 7.09 |

${U}_{f}$($\mathrm{m}/\mathrm{s}$) | 0.089 | 0.169 | 0.254 | 0.325 | 0.371 | 0.383 | 0.345 | 0.250 | 0.137 |

${c}_{0}$($\mathrm{m}/\mathrm{s}$) | 351.5 | 352.1 | 352.7 | 353.3 | 353.9 | 354.5 | 355.1 | 355.6 | 356.2 |

Propane-Air Fuel Mixtures | |||||||||

$\varphi $ | 0.63 | 0.7 | 0.8 | 0.9 | 1 | 1.1 | 1.2 | 1.3 | 1.4 |

$\mathsf{\Theta}\equiv {\rho}_{u}/{\rho}_{b}$ | 6.04 | 6.56 | 7.15 | 7.66 | 8.02 | 8.08 | 8 | 7.88 | 7.74 |

${U}_{f}$($\mathrm{m}/\mathrm{s}$) | 0.147 | 0.217 | 0.303 | 0.374 | 0.418 | 0.429 | 0.399 | 0.322 | 0.226 |

${c}_{0}$($\mathrm{m}/\mathrm{s}$) | 343 | 342.3 | 341.6 | 340.8 | 340.1 | 339.5 | 338.8 | 338.1 | 337.5 |

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## Share and Cite

**MDPI and ACS Style**

Kodakoglu, F.; Demir, S.; Valiev, D.; Akkerman, V.
Analysis of Gaseous and Gaseous-Dusty, Premixed Flame Propagation in Obstructed Passages with Tightly Placed Obstacles. *Fluids* **2020**, *5*, 115.
https://doi.org/10.3390/fluids5030115

**AMA Style**

Kodakoglu F, Demir S, Valiev D, Akkerman V.
Analysis of Gaseous and Gaseous-Dusty, Premixed Flame Propagation in Obstructed Passages with Tightly Placed Obstacles. *Fluids*. 2020; 5(3):115.
https://doi.org/10.3390/fluids5030115

**Chicago/Turabian Style**

Kodakoglu, Furkan, Sinan Demir, Damir Valiev, and V’yacheslav Akkerman.
2020. "Analysis of Gaseous and Gaseous-Dusty, Premixed Flame Propagation in Obstructed Passages with Tightly Placed Obstacles" *Fluids* 5, no. 3: 115.
https://doi.org/10.3390/fluids5030115