# Spherical Diffusion Flames of Ethylene in Microgravity: Multidimensional Effects

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Statement of the Problem

_{2}O and CO

_{2}) and diatomic molecules (N

_{2}and O

_{2}), as well as soot, which can form during combustion.

- In microgravity conditions, the acceleration of gravity ${g}_{i}$ is negligible;
- The evolution of the SDF is symmetrical with respect to the axis of the supply tube;
- The porous medium in the flow region can be modeled by flow resistance according to the Darcy law and heat exchange with the fluid according to the Newton law, i.e., the porous medium can be represented by added momentum and heat sources, ${\left(\frac{\partial P}{\partial {x}_{i}}\right)}_{s}$ and ${\Psi}_{s}$ respectively, in the governing equations. In addition, since the porous medium reduces the volume accessible for fluid, the local flow velocity, ${U}_{i}$ and superficial velocity inside the porous medium, ${u}_{i}$ are coupled by the undirected porosity value $\phi $: ${U}_{i}=\phi {u}_{i}$;
- The thermophysical and structural parameters of the PB material are constant;
- Thermal radiation of PB is negligible; PB absorbs thermal radiation of soot, H
_{2}O, CO_{2}, and (for the sake of generality) N_{2}and O_{2}; - Gas-phase and catalytic reactions in the PB are absent;
- The gas flow is laminar;
- A multicomponent gas mixture obeys the thermal and caloric equations of state of an ideal gas and possesses variable thermophysical properties;
- The effect of thermodiffusion is negligible;
- Soot is an equivalent gas with the molecular mass of atomic carbon, when simulating soot reactions;
- Soot particles are the clusters of 20–25 carbon atoms, have the corresponding constant size, and do not coagulate;
- The radiation heat flux is caused solely by soot, H
_{2}O, CO_{2}, N_{2}, and O_{2}emittance; - The outer wall of the computational domain is impermeable, noncatalytic, and isothermal.

_{2}H

_{2}as a precursor of soot C. Except for soot C, all other substances involved in the reactions of soot formation and oxidation are included in the detailed reaction mechanism (DRM) of ethylene oxidation [29], containing 48 species and 209 reversible elementary reactions. The kinetic parameters of reactions, namely, the pre-exponential factor, ${A}_{k}$, activation energy ${E}_{k}$, and the temperature exponent in the expression for the rate of the $k$th reaction, were determined using the thoroughly tested DRM of soot formation [30] from the condition of the best agreement between the results of calculations for the soot yield obtained on the basis of DRM and on the basis of the macrokinetic mechanism. For estimating the effect of soot radiation, it is conditionally assumed (assumption 11) that soot particles possess the specific (per unit mass) emitting surface, ${S}_{soot}=6/\left({d}_{soot}{\rho}_{soot}\right)$ (here ${d}_{soot}$ is the conditional soot particle size, ${\rho}_{soot}$ is the soot density), which is directly connected to the soot mass fraction ${Y}_{soot}$. Thus, the simplest radiation model is used without radiation reabsorption and scattering by soot particles.

_{2}O, CO

_{2}, N

_{2}, and O

_{2}. Assumption 13 is conventional.

_{2}O and CO

_{2}are taken from the polynomials in [32]; for N

_{2}and O

_{2}, ${a}_{l}$ is independent of the gas temperature and is assumed to be equal to 0.1. The transport coefficients of the gas, as well as the effective diffusion coefficients of species in the gas mixture and specific heats, are calculated by the formulae presented in [33].

_{2}H

_{4}+ 3O

_{2}= 2CO

_{2}+ 2H

_{2}O, whereas ${W}_{O}$ = 32 kg/kmol and ${W}_{F}$ = 28 kg/kmol are the molecular masses of oxygen and ethylene. It is generally believed that flames with low ${Z}_{st}$ are more prone to soot formation than flames with large ${Z}_{st}$ [17,18].

#### 2.2. Numerical Solution

^{2}, ${R}_{s}$ = 0.0032 m, ${r}_{\mathrm{\infty}}$ = 0.288 m, $\phi $ = 0.5, $\kappa ={10}^{-13}$ m

^{2}; $d={10}^{-5}$ m; ${\epsilon}_{s}$ = 0.8, ${\rho}_{s}$ = 4000 kg/m

^{3}, ${c}_{s}$ = 650 J/(kg·K), ${\lambda}_{s}$ = 5 W/(m·K), ${d}_{soot}$ = 2 nm, ${\rho}_{soot}$ = 2000 kg/m

^{3}, $N$ = 49, $L$ = 213.

## 3. Results and Discussion

#### 3.1. Experiments

#### 3.2. Calculations

#### 3.2.1. 1D Calculations

#### 3.2.2. 2D Calculations

#### Cold Flow

#### Reactive Flow

## 4. Conclusions

- (1)
- There exist the unambiguous dependences of the ratio of flame radius to fluid mass flow rate through the PB, ${R}_{\mathrm{f}}/{G}_{\mathrm{i}\mathrm{n}}$, on the stoichiometric mixture fraction ${Z}_{st}$ for normal and inverse flames;
- (2)
- The growth rate of normal flames decreases with ${Z}_{st}$ and they become more prone to be stationary. At ${Z}_{st}=$ 0.36–0.411, the normal flames become almost stationary, i.e., their growth rates become very low;
- (3)
- Contrary to normal flames, the growth rate of inverse flames increases with ${Z}_{st}$ and they become less prone to be stationary. At ${Z}_{st}=$ 0.203–0.218, the inverse flames become almost stationary, i.e., their growth rates become very low;
- (4)
- The maximum growth rate of inverse flames is considerably less than that of normal flames. This means that the lifetime of inverse flames must be longer than that of the normal flames at the identical value of ${R}_{\mathrm{f}}/{G}_{\mathrm{i}\mathrm{n}}$;
- (5)
- At the same extinction radius, the extinction of the inverse flame occurs later than the extinction of the normal flame due to the lower growth rate;
- (6)
- The flame radius at extinction is approximately constant in the wide range of ${G}_{\mathrm{i}\mathrm{n}}$. A simple scaling law from the literature underestimates the flame radius at extinction;
- (7)
- The 2D cold-flow calculations show that the supply tube blocks a part of the surface of the porous burner in the vicinity of the south pole. Due to this blockage, the normalized accumulated mass flow rate of fluid through the southern hemisphere is about 15% less than through the northern hemisphere. The local mass flow rate of fluid through the porous burner is nonuniform with the maximum flow rate attained in the angular interval [−20°, −40°] in the southern hemisphere;
- (8)
- The cold isothermal gas supply tube looks more corresponding to the experimental observations than the adiabatic supply tube, as it exhibits the characteristic asymmetry in both flame shape and temperature distribution (luminosity) observed in the experiments. It could be thus assumed that flame quenching near the gas supply tube observed in the experiments is caused by the cold gas supply tube;
- (9)
- The 2D calculations reveal the oscillatory evolution of ethylene diffusion flames with slow alterations in flame shape and temperature caused by the incepience of torroidal vortices in the surrounding gas;
- (10)
- The introduction of directional microgravity on the level of 0.01$g$ allows complete suppression of flame oscillations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{l}$ | Emissivity of the $l$th emitting gas |

${A}_{k}$ | Pre-exponential factor, ($k$ = 1, …, $L$) |

${c}_{p,l}$ | Specific heat at constant pressure, ($l$ = 1, …, $N$) |

${c}_{s}$ | Solid skeleton heat capacity |

${(C/O)}_{c}$ | Threshold local C/O atomic ratio |

$d$ | Characteristic size of the solid skeleton |

${d}_{soot}$ | Conditional soot particle size |

${D}_{l}$ | Effective diffusion coefficient of the $l$th species, ($l$ = 1, …, $N$) |

${E}_{k}$ | Activation energy, ($k$ = 1, …, $L$) |

${\left(\frac{\partial p}{\partial {x}_{i}}\right)}_{s}$ | added momentum source in a porous medium |

${G}_{in}$ | Inlet mass flow rate |

${g}_{i}$ | $i$th component of the vector of acceleration of gravity $g$; |

$H$ | Mean gas static enthalpy |

${H}_{l}^{0}$ | Standard enthalpy of formation of the $l$th species, ($l$ = 1, …, $N$) |

$I$ | Mean gas total enthalpy |

${j}_{l}$ | Molecular mass flux of the $l$th species, ($l$ = 1, …, $N$) |

${j}_{lj}^{t}$ | Turbulent mass flux of the $l$th species, ($l$ = 1, …, $N$), (j = 1, 2, 3) |

$L$ | Total number of chemical reactions in the gas |

${n}_{k}$ | Temperature exponent, ($k$ = 1, …, $L$) |

$N$ | Number of gas species |

$P$ | Mean gas pressure |

${P}_{0}$ | Initial pressure |

${q}_{j}$ | Molecular heat flux, (j = 1, 2, 3) |

${q}_{j}^{t}$ | Turbulent heat flux, (j = 1, 2, 3) |

$\dot{Q}$ | Mean source of energy due to chemical transformations |

${R}_{0}$ | Length of the buffer channel |

${R}_{f}$ | Flame radius |

${R}_{s}$ | Radius of porous sphere |

${r}_{\mathrm{\infty}}$ | Radius of the outer wall of the chamber |

$R$ | Universal gas constant |

${S}_{in}$ | Passage area of gas supply tube |

${S}_{PB}$ | Specific surface area of the porous burner |

${S}_{soot}$ | Specific emitting surface area |

$t$ | Time |

${t}_{ign}$ | Time of ignition |

$T$ | Temperature |

${T}_{0}$ | Initial temperature |

${T}^{0}$ | Standard temperature |

${T}_{c}$ | Threshold local temperature of soot formation |

${T}_{ign}$ | Ignition temperature |

${T}_{s}$ | Temperature of porous sphere |

${u}_{i}$ | Superficial velocity inside the porous medium, (j = 1, 2, 3) |

${U}_{i}$ | The $i$th component of the mean gas velocity vector, (j = 1, 2, 3) |

$V$ | Chamber volume |

${\dot{w}}_{l}$ | Mean source of mass due to chemical transformations, ($l$ = 1, …, $N$) |

$W$ | Molecular mass |

${W}_{O}$ | Molecular mass of oxidizer |

${W}_{F}$ | Molecular mass of fuel |

${x}_{j}$ | Cartesian coordinate, (j = 1, 2, 3) |

${X}_{l}$ | Volume fraction of the $l$th emitting gas |

${Y}_{i0}$ | Initial species mass fractions, ($i$ = 1, …, $N$) |

${Y}_{i,in}$ | Inlet species mass fractions, ($i$ = 1, …, $N$) |

${Y}_{l}$ | Mean mass fraction of the $l$th species, ($l$ = 1, …, $N$) |

${Y}_{soot}$ | Soot mass fraction |

${Z}_{st}$ | Stoichiometric mixture fraction |

${\alpha}_{s}$ | Heat transfer coefficient between gas and porous sphere |

${\delta}_{s}$ | Delta function |

${\epsilon}_{s}$ | Coefficient of radiation absorption by the porous sphere material |

$\kappa $ | Permeability |

$\lambda $ | Thermal conductivity of the $l$th species |

${\lambda}_{s}$ | Solid skeleton thermal conductivity |

$\mu $ | Dynamic viscosity of gas |

${\nu}_{O}$ | Stoichiometric coefficient of oxidizer in the overall reaction equation |

${\nu}_{F}$ | Stoichiometric coefficient of fuel in the overall reaction equation |

${\upsilon}_{l,k}^{\prime}$ | Stoichiometric coefficients of the $l$th species in the reactants of the $k$th reaction, ($l$ = 1, …, $N$), ($k$ = 1, …, $L$) |

${\upsilon}_{l,k}^{\u2033}$ | Stoichiometric coefficients of the $l$th species in the products of the $k$th reaction, ($l$ = 1, …, $N$), ($k$ = 1, …, $L$) |

$\rho $ | Mean gas density |

${\rho}_{s}$ | Solid skeleton density |

${\rho}_{soot}$ | Soot density |

$\sigma $ | Stefan–Boltzmann constant |

${\tau}_{ij}$ | Tensor of viscous stresses |

${\tau}_{ij}^{t}$ | Tensor of turbulent stresses |

$\phi $ | Porosity |

${\Psi}_{s}$ | Added heat source in porous medium |

$\Omega $ | Heat source/sink other than that of chemical nature |

${\Omega}_{s}$ | Heat source/sink for porous sphere |

${\Omega}_{sg}$ | Radiation absorption |

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**Figure 2.**Photograph of the porous burner mounted on a gas supply tube (

**a**) and the schematics of the 2D (

**b**) and 1D (

**c**) computational domain.

**Figure 3.**Some dimensions of the 2D (

**a**) and 1D (

**b**) computational domains, as well as initial and boundary conditions.

**Figure 4.**Comparison of predicted (curve) and measured (black circles) flame radius (

**a**) and temperature (

**b**) for the normal flame No. 4 (19189K1).

**Figure 5.**Comparison of predicted (curve) and measured (red circles) flame radius (

**a**) and temperature (

**b**) for the inverse flame No. 29 (22018G3). Temperature measurements are not available.

**Figure 6.**Calculated dependences of ${R}_{\mathrm{f}}/{G}_{\mathrm{i}\mathrm{n}}$ vs. ${Z}_{st}$ for normal (black circles) and inverse (red circles) flames at five instants of time: (

**a**) 10 s, (

**b**) 20 s, (

**c**) 30 s, (

**d**) 40 s, and (

**e**) 50 s after ignition. Numbers correspond to the flame numbers in Table 1 and Table 2. Curves N and I correspond to the best fits of calculated data for normal and inverse flames at 10 s after ignition.

**Figure 7.**Calculated dependences of ${R}_{\mathrm{f}\mathrm{e}}/{G}_{\mathrm{i}\mathrm{n}}$ vs. ${Z}_{st}$ for normal (black circles) and inverse (red circles) flames at extinction. Numbers correspond to the flame numbers in Table 1 and Table 2. Curves N and I correspond to the best fits of the calculated data for normal and inverse flames at 10 s after ignition (see Figure 6a).

**Figure 9.**Cold flow streamlines from the porous burner for the conditions of normal flame No. 1 (19115B1).

**Figure 10.**The calculated cold-flow dependences of the normalized local mass flow rate (

**a**) and normalized accumulated mass flow rate of fuel (

**b**) through the porous burner on the angle in the polar section of the sphere: −90° corresponds to the south pole, 0° corresponds to the equator, and +90° corresponds to the north pole of the sphere (normal flame No. 1 (19115B1)).

**Figure 11.**The calculated evolution of normal flame No. 1 (19115B1) with the isothermal supply tube.

**Figure 14.**The calculated evolution of inverse flame No. 25 (21328D1) with an isothermal supply tube and a microgravity of 0.01g.

**Figure 15.**Video frames of the evolution of a highly sooty inverse flame. Arrows show the motion of some characteristic points at the soot cloud edge.

No. | Flame | Combustion Chamber | Porous Burner | p, atm | ${\mathit{Z}}_{\mathit{s}\mathit{t}}$ | ${\mathit{R}}_{\mathbf{f}\mathbf{e},\mathbf{e}\mathbf{x}\mathbf{p}}$, mm | ${\mathit{R}}_{\mathbf{f}\mathbf{e},\mathbf{c}\mathbf{a}\mathbf{l}\mathbf{c}}$, mm | ${\mathit{T}}_{\mathbf{f}\mathbf{e},\mathbf{c}\mathbf{a}\mathbf{l}\mathbf{c}}$, K | |||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{X}}_{\mathit{O}2}$ | ${\mathit{X}}_{\mathit{N}2}$ | ${\mathit{X}}_{\mathit{C}2\mathit{H}4}$ | ${\mathit{X}}_{\mathit{N}2}$ | ${\mathit{G}}_{\mathbf{i}\mathbf{n}}$, mg/s | |||||||

1 | 19115B1 | 0.203 | 0.797 | 1.000 | 0.000 | 0.660 | 1.020 | 0.062 | - | - | - |

2 | 19206L6 | 0.203 | 0.797 | 0.291 | 0.709 | 1.980 | 1.010 | 0.184 | - | - | - |

3 | 19171D4 | 0.376 | 0.624 | 1.000 | 0.000 | 2.520 | 1.010 | 0.106 | 32 | 28.0 | 1200 |

4 | 19189K1 | 0.374 | 0.626 | 0.502 | 0.498 | 5.010 | 1.310 | 0.191 | 31 | 27.5 | 1200 |

5 | F10 | 0.380 | 0.620 | 1.000 | 0.000 | 1.224 | 1.239 | 0.107 | - | - | - |

6 | F02 | 0.391 | 0.609 | 0.288 | 0.712 | 1.800 | 1.190 | 0.300 | - | - | - |

7 | F08 | 0.386 | 0.614 | 0.288 | 0.712 | 3.603 | 1.263 | 0.297 | - | - | - |

8 | F05 | 0.400 | 0.600 | 0.288 | 0.712 | 4.514 | 1.250 | 0.304 | - | - | - |

9 | 19156C2 | 0.366 | 0.634 | 1.000 | 0.000 | 2.529 | 1.040 | 0.104 | 32 | 28.0 | 1200 |

10 | 19142J3 | 0.356 | 0.644 | 1.000 | 0.000 | 2.529 | 0.990 | 0.101 | 31 | 28.0 | 1200 |

11 | 19150N1 | 0.296 | 0.704 | 0.168 | 0.832 | 4.885 | 1.010 | 0.360 | - | - | - |

12 | 19150G3 | 0.338 | 0.662 | 0.288 | 0.712 | 8.779 | 1.050 | 0.272 | 35.5 | 30.5 | 1200 |

13 | 19175A3 | 0.391 | 0.609 | 1.000 | 0.000 | 1.960 | 1.270 | 0.110 | - | - | - |

14 | 19206A5 | 0.207 | 0.793 | 0.288 | 0.712 | 8.779 | 1.010 | 0.189 | 31 | 34 | 1200 |

15 | 19206G1 | 0.205 | 0.795 | 1.000 | 0.000 | 2.529 | 1.010 | 0.062 | 32 | 32 | 1200 |

16 | 19206G4 | 0.201 | 0.799 | 1.000 | 0.000 | 0.822 | 1.010 | 0.061 | - | - | - |

17 | 19206L4 | 0.195 | 0.805 | 0.288 | 0.712 | 2.835 | 1.010 | 0.180 | 19 | 19 | 1200 |

18 | 19123F1 | 0.206 | 0.794 | 0.490 | 0.510 | 2.640 | 1.010 | 0.120 | 23 | 24 | 1200 |

19 | 19123L1 | 0.202 | 0.798 | 1.000 | 0.000 | 2.510 | 1.010 | 0.061 | 33 | 32 | 1200 |

20 | 19189J3 | 0.378 | 0.622 | 0.502 | 0.498 | 5.010 | 1.300 | 0.192 | 31 | 28 | 1200 |

21 | 19200H3 | 0.285 | 0.715 | 0.131 | 0.869 | 4.430 | 1.020 | 0.411 | - | - | - |

22 | 19115F1 | 0.204 | 0.796 | 0.292 | 0.708 | 2.180 | 1.040 | 0.184 | - | - | - |

23 | 19123A2 | 0.209 | 0.791 | 1.000 | 0.000 | 1.620 | 1.000 | 0.063 | 25 | 26 | 1200 |

24 | 19123C1 | 0.207 | 0.793 | 0.290 | 0.710 | 4.460 | 1.000 | 0.188 | 24 | 25 | 1200 |

No. | Flame | Combustion Chamber | Porous Sphere | $\mathit{p}$, atm | ${\mathit{Z}}_{\mathit{s}\mathit{t}}$ | ${\mathit{R}}_{\mathbf{f}\mathbf{e},\mathbf{e}\mathbf{x}\mathbf{p}}$, mm | ${\mathit{R}}_{\mathbf{f}\mathbf{e},\mathbf{c}\mathbf{a}\mathbf{l}\mathbf{c}}$, mm | ${\mathit{T}}_{\mathbf{f}\mathbf{e},\mathbf{c}\mathbf{a}\mathbf{l}\mathbf{c}}$, K | |||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{X}}_{\mathit{C}2\mathit{H}4}$ | ${\mathit{X}}_{\mathit{N}2}$ | ${\mathit{X}}_{\mathit{O}2}$ | ${\mathit{X}}_{\mathit{N}2}$ | ${\mathit{G}}_{\mathbf{i}\mathbf{n}}$, mg/s | |||||||

25 | 21328D1 | 0.257 | 0.743 | 0.212 | 0.788 | 10.05 | 1.03 | 0.211 | - | - | - |

26 | 21349M3 | 0.270 | 0.730 | 0.212 | 0.788 | 9.11 | 1.00 | 0.203 | - | - | - |

27 | 22018H2 | 0.097 | 0.903 | 0.497 | 0.503 | 6.37 | 1.01 | 0.614 | 24 | 20 | 1200 |

28 | 22018J1 | 0.096 | 0.904 | 0.318 | 0.682 | 9.73 | 1.01 | 0.514 | 25 | 21 | 1200 |

29 | 22018G3 | 0.098 | 0.902 | 0.850 | 0.150 | 7.89 | 1.01 | 0.720 | 31 | 28 | 1200 |

30 | 22018G2 | 0.099 | 0.901 | 0.850 | 0.150 | 5.90 | 1.01 | 0.718 | 28 | 24 | 1200 |

31 | 22018G1 | 0.099 | 0.901 | 0.850 | 0.150 | 3.90 | 1 | 0.718 | - | - | - |

32 | 21328N5 | 0.080 | 0.920 | 0.850 | 0.150 | 2.27 | 0.96 | 0.759 | - | - | - |

33 | 22035J2 | 0.096 | 0.904 | 0.850 | 0.150 | 5.90 | 0.51 | 0.725 | - | - | - |

34 | 21340M1 | 0.121 | 0.879 | 0.850 | 0.150 | 9.22 | 1.01 | 0.676 | - | 29 | 1200 |

35 | 21349N3 | 0.251 | 0.749 | 0.212 | 0.788 | 9.10 | 0.52 | 0.215 | - | - | - |

36 | 21349N4 | 0.246 | 0.754 | 0.212 | 0.788 | 10.03 | 0.52 | 0.218 | - | - | - |

37 | 22018B1 | 0.168 | 0.832 | 0.850 | 0.150 | 4.7 | 1 | 0.601 | - | - | - |

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## Share and Cite

**MDPI and ACS Style**

Frolov, S.M.; Ivanov, V.S.; Frolov, F.S.; Semenov, I.V.
Spherical Diffusion Flames of Ethylene in Microgravity: Multidimensional Effects. *Fire* **2023**, *6*, 285.
https://doi.org/10.3390/fire6080285

**AMA Style**

Frolov SM, Ivanov VS, Frolov FS, Semenov IV.
Spherical Diffusion Flames of Ethylene in Microgravity: Multidimensional Effects. *Fire*. 2023; 6(8):285.
https://doi.org/10.3390/fire6080285

**Chicago/Turabian Style**

Frolov, Sergey M., Vladislav S. Ivanov, Fedor S. Frolov, and Ilya V. Semenov.
2023. "Spherical Diffusion Flames of Ethylene in Microgravity: Multidimensional Effects" *Fire* 6, no. 8: 285.
https://doi.org/10.3390/fire6080285