# Soot Formation in Spherical Diffusion Flames

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## 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.

- 1.
- The gas supply tube does not affect the evolution of the SDF.
- 2.
- All processes are spherically symmetric.
- 3.
- 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 $: ${U}_{i}=\phi {u}_{i}$.
- 4.
- The structural and thermophysical parameters of the PS material are constant.
- 5.
- PS absorbs thermal radiation of soot, H
_{2}O, CO_{2}, N_{2}and O_{2}; thermal radiation of PS is negligible. - 6.
- Catalytic and gas-phase reactions in the PS are absent.
- 7.
- The gas flow is laminar.
- 8.
- The gas mixture obeys the ideal-gas thermal and caloric equations of state; gas thermophysical properties are variable.
- 9.
- The effect of thermodiffusion is negligible.
- 10.
- Soot is an equivalent gas with the molecular mass of atomic carbon, when simulating soot reactions.
- 11.
- Soot particles are the clusters of 20–25 carbon atoms, have the corresponding constant size, and do not coagulate.
- 12.
- The radiation heat flux is caused solely by soot, H
_{2}O, CO_{2}, N_{2}and O_{2}emittance. - 13.
- The outer wall of the chamber is impermeable, isothermal, and non-catalytic.

_{2}H

_{2}+ C

_{2}H

_{2}= C + C + C

_{2}H

_{4}

_{2}= CO + CO

_{2}O = H

_{2}+ CO

_{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 DRM of ethylene oxidation [36] containing 48 species and 209 reversible elementary reactions. The kinetic parameters of reactions (I)–(IV), 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, are presented in Table 1.

_{2}O, CO

_{2}, N

_{2}and O

_{2}. Assumption 13 is conventional.

_{2}O and CO

_{2}are taken from the polynomials in [38], for N

_{2}and O

_{2}${a}_{l}$ is independent of the gas temperature and is assumed to be equal to 0.1. The dynamic viscosity $\mu $ and thermal conductivity $\lambda $ of the gas, as well as the effective diffusion coefficients of species in the gas mixture, ${D}_{l}$, and specific heats ${c}_{p,l}$ are calculated by the formulae presented in [39].

_{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. Large values of ${Z}_{st}$ correspond to flames with oxygen excess, and small values correspond to flames with fuel excess. Normal flames in an atmosphere of oxygen diluted with nitrogen, with the supply of undiluted ethylene to the PS, correspond to small values of ${Z}_{st}$. Inverse flames in an atmosphere of ethylene diluted with nitrogen, with the supply of undiluted oxygen to the PS, correspond to large values of ${Z}_{st}$. It is generally believed that flames with low ${Z}_{st}$ values are more prone to soot formation than flames with large ${Z}_{st}$ values [40,41].

#### 2.2. Numerical Solution

^{2}, ${r}_{s}$ = 0.0032 m, ${r}_{\infty}$ = 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}, and $N$ = 49, $L$ = 213.

## 3. Results of Experiments and Calculations

^{4}in Figure 7a and 10

^{6}in Figure 7b) in order to consider the details of the soot spatial distribution. It can be seen that at the point at which the temperature reaches its maximum value, the concentrations of ethylene and oxygen are almost zero, and the concentrations of intermediate and final reaction products reach values close to their maxima. In both flames, at some distance from the flame, the mass fractions of all intermediate and final reaction products drop to zero, whereas the mass fractions of the initial substances and the gas temperature are restored to their initial values. Despite the similarity of the structures of the normal and inverse flames, they fundamentally differ in the spatial distributions of soot. If in a normal flame, soot is present only inside the flame (between the PS and the temperature maximum), then in an inverse flame, soot is present only outside the flame (outside the temperature maximum).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

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

${A}_{k}$ | Pre-exponential factor |

${c}_{p,l}$ | Specific heat at constant pressure |

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

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

$d$ | Characteristic size of solid skeleton |

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

${D}_{l}$ | Effective diffusion coefficient of the $l$th species |

${E}_{k}$ | Activation energy |

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

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

$H$ | Mean gas static enthalpy |

${H}_{l}^{0}$ | Standard enthalpy of formation of the $l$th species |

$I$ | Mean gas total enthalpy |

${j}_{l}$ | Molecular mass flux of the $l$th species |

${j}_{lj}^{t}$ | Turbulent mass flux of the $l$th species |

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

${m}_{soot,\mathsf{\Sigma}}$ | Total soot mass |

$\dot{{m}_{soot,\mathsf{\Sigma}}}$ | Cumulative rate of soot formation |

${n}_{k}$ | Temperature exponent |

$N$ | Number of gas species |

$P$ | Mean gas pressure |

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

${q}_{j}$ | Molecular heat flux |

${q}_{j}^{t}$ | Turbulent heat flux |

$\dot{Q}$ | 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 |

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

$R$ | Universal gas constant |

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

${S}_{PS}$ | Specific surface area of the porous sphere |

${S}_{soot}$ | 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 |

${U}_{i}$ | The $i$th component of the mean gas velocity vector |

$V$ | Chamber volume |

${\dot{w}}_{l}$ | Mean source of mass due to chemical transformations |

$W$ | Molecular mass |

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

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

${x}_{j}$ | Cartesian coordinate |

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

${Y}_{i0}$ | Initial species mass fractions |

${Y}_{i,in}$ | Inlet species mass fractions |

${Y}_{l}$ | Mean mass fraction of the $l$th species |

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

${Y}_{soot,\mathsf{\Sigma}}$ | Integral 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 |

${\upsilon}_{l,k}^{\u2033}$ | Stoichiometric coefficients of the $l$th species in the products of the $k$th reaction |

$\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 |

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

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

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

## References

- Haynes, B.S.; Wagner, H.G. Soot formation. Prog. Energy Combust. Sci.
**1981**, 7, 229–273. [Google Scholar] [CrossRef] - Kennedy, I.M. Models of soot formation and oxidation. Prog. Energy Combust. Sci.
**1997**, 23, 95–132. [Google Scholar] [CrossRef] - Richter, H.; Howard, J.B. Formation of polycyclic aromatic hydrocarbons and their growth to soot—A review of chemical reaction pathways. Prog. Energy Combust. Sci.
**2000**, 26, 565–608. [Google Scholar] [CrossRef] - Karatas, A.E.; Gülder, Ö.L. Soot formation in high pressure laminar diffusion flames. Prog. Energy Combust. Sci.
**2012**, 38, 818–845. [Google Scholar] [CrossRef] - Wang, Y.; Chung, S.H. Soot formation in laminar counterflow flames. Prog. Energy Combust. Sci.
**2019**, 74, 152–238. [Google Scholar] [CrossRef] - Lapuerta, M.; Rodríguez–Fernández, J.; Sánchez-Valdepeñas, J. Soot reactivity analysis and implications on diesel filter regeneration. Prog. Energy Combust. Sci.
**2020**, 78, 100833. [Google Scholar] [CrossRef] - Michelsen, H.A.; Colket, M.B.; Bengtsson, P.-E.; D’Anna, A.; Desgroux, P.; Haynes, B.S.; Miller, J.H.; Nathan, G.J.; Pitsch, H.; Wang, H. A review of terminology used to describe soot formation and evolution under combustion and pyrolytic conditions. ACS Nano
**2020**, 14, 12470–12490. [Google Scholar] [CrossRef] - Xi, J.; Yang, G.; Cai, J.; Gu, Z. A review of recent research results on soot: The formation of a kind of carbon-based material in flames. Front. Mater.
**2021**, 8, 695485. [Google Scholar] [CrossRef] - Gleason, K.; Carbone, F.; Sumner, A.J.; Drollette, B.D.; Plata, D.L.; Gomez, A. Small aromatic hydrocarbons control the onset of soot nucleation. Combust. Flame
**2021**, 223, 398–406. [Google Scholar] [CrossRef] - Martin, J.W.; Salamanca, M.; Kraft, M. Soot inception: Carbonaceous nanoparticle formation in flames. Prog. Energy Combust. Sci.
**2022**, 88, 100956. [Google Scholar] [CrossRef] - He, J.; Ying, Y.; Chen, M.; Liu, D. Soot formation characteristics in hybrid pyrolysis of zero-carbon fuel ammonia and ethylene mixtures. Front. Energy Res.
**2022**, 10, 996813. [Google Scholar] [CrossRef] - Santoro, R.J.; Yeh, T.T.; Horvath, J.J.; Semerjian, H.G. The Transport and Growth of Soot Particles in Laminar Diffusion Flames. Combust. Sci. Technol.
**1987**, 53, 89–115. [Google Scholar] [CrossRef] - Glassman, I. Soot formation in combustion processes. Proc. Combust. Inst.
**1989**, 22, 295–311. [Google Scholar] [CrossRef] - Glassman, I.; Nishida, O.; Sidebotham, G. Critical temperatures of soot formation. In Soot Formation in Combustion; Springer: Berlin/Heidelberg, Germany, 1994; pp. 316–324. [Google Scholar] [CrossRef]
- Sunderland, P.B.; Faeth, G.M. Soot formation in hydrocarbon/air laminar jet diffusion flames. Combust. Flame
**1996**, 105, 132–146. [Google Scholar] [CrossRef] [Green Version] - Glassman, I. Sooting laminar diffusion flames: Effect of dilution, additives, pressure, and microgravity. Proc. Symp. Combust.
**1998**, 27, 1589–1596. [Google Scholar] [CrossRef] - Atreya, A.; Agrawal, S.; Sacksteder, K.; Baum, H. Observations of methane and ethylene diffusion flames stabilized around a blowing porous sphere under microgravity conditions. In Proceedings of the 32nd Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 1994. [Google Scholar]
- Tse, S.D.; Zhu, D.; Sung, C.-J.; Ju, Y.; Law, C.K. Microgravity burner-generated spherical diffusion flames: Experiment and computation, Combust. Flame
**2001**, 125, 1265–1278. [Google Scholar] [CrossRef] - Sunderland, P.B.; Axelbaum, R.L.; Urban, D.L.; Chao, B.H.; Liu, S. Effects of structure and hydrodynamics on the sooting behavior of spherical microgravity diffusion flames. Combust. Flame
**2003**, 132, 25–33. [Google Scholar] [CrossRef] - Santa, K.J.; Chao, B.H.; Sunderland, P.B.; Urban, D.L.; Stocker, D.P.; Axelbaum, R.L. Radiative extinction of gaseous spherical diffusion flames in microgravity, Combust. Flame
**2007**, 151, 665–675. [Google Scholar] [CrossRef] [Green Version] - Chernovsky, M.K.; Atreya, A.; Im, H.G. Effect of CO
_{2}diluent on fuel versus oxidizer side of spherical diffusion flames in microgravity. Proc. Combust. Inst.**2007**, 31, 1005–1013. [Google Scholar] [CrossRef] - Christiansen, E.W.; Tse, S.D.; Law, C.K. A computational study of oscillatory extinction of spherical diffusion flames. Combust. Flame
**2003**, 134, 327–337. [Google Scholar] [CrossRef] - Liu, S.; Chao, B.H.; Axelbaum, R.L. A theoretical study on soot inception in spherical burner-stabilized diffusion flames. Combust. Flame
**2005**, 140, 1–23. [Google Scholar] [CrossRef] - Lecoustre, V.R.; Sunderland, P.B.; Chao, B.H.; Axelbaum, R.L. Numerical investigation of spherical diffusion flames at their sooting limits. Combust. Flame
**2012**, 159, 194–199. [Google Scholar] [CrossRef] - Frenklach, M.; Mebel, A.M. On the mechanism of soot nucleation. Phys. Chem. Chem. Phys.
**2020**, 22, 5314–5331. [Google Scholar] [CrossRef] [PubMed] - Frenklach, M.; Clary, D.W.; Gardiner, W.C., Jr.; Stein, S.E. Detailed kinetic modeling of soot formation in shock-tube pyrolysis of acetylene. Proc. Combust. Inst.
**1985**, 20, 887–901. [Google Scholar] [CrossRef] - Commodo, M.; Kaiser, K.; De Falco, G.; Minutolo, P.; Schulz, F.; D’Anna, A.; Gross, L. On the early stages of soot formation: Molecular structure elucidation by high-resolution atomic force microscopy. Combust. Flame
**2019**, 205, 154–164. [Google Scholar] [CrossRef] - Irimiea, C.; Faccinetto, A.; Mercier, X.; Ortega, I.-K.; Nuns, N.; Therssen, E.; Desgroux, P.; Focsa, C. Unveiling trends in soot nucleation and growth: When secondary ion mass spectrometry meets statistical analysis. Carbon
**2019**, 144, 815–830. [Google Scholar] [CrossRef] - Mitra, T.; Zhang, T.; Sediako, A.D.; Thomson, M.J. Understanding the formation and growth of polycyclic aromatic hydrocarbons (PAHs) and young soot from n-dodecane in a sooting laminar coflow diffusion flame. Combust. Flame
**2019**, 202, 33–42. [Google Scholar] [CrossRef] - Agafonov, G.L.; Bilera, I.V.; Vlasov, P.A.; Kolbanovskii, Y.A.; Smirnov, V.N.; Teresa, A.M. Soot formation at pyrolysis and oxidation of acetylene and ethylene in shock tubes. Kinet. Catal.
**2015**, 56, 15. [Google Scholar] [CrossRef] - Irace, P.H.; Lee, H.J.; Waddell, K.; Tan, L.; Stocker, D.P.; Sunderland, P.B.; Axelbaum, R.L. Observations of long duration microgravity spherical diffusion flames aboard the International Space Station. Combust. Flame
**2021**, 229, 111373. [Google Scholar] [CrossRef] - Frolov, S.M.; Medvedev, S.N.; Frolov, F.S. Spherical diffusion flame of ethylene in the spaceflight experiment “Adamant”. Combust. Explos.
**2021**, 14, 9–21. [Google Scholar] [CrossRef] - Williams, F.A. Combustion Theory; The Benjamin/Cummings Publishing Company, Inc.: Menlo Park, CA, USA, 1985; pp. 636–637. [Google Scholar]
- Hou, D.; Lindberg, C.S.; Manuputty, M.Y.; You, X.; Kraft, M. Modeling soot formation in a benchmark ethylene stagnation flame with a new detailed population balance model, Combust. Flame
**2019**, 203, 56–71. [Google Scholar] [CrossRef] - Basevich, V.Y.; Medvedev, S.N.; Frolov, S.M.; Frolov, F.S.; Basara, B.; Prieching, P. Macrokinetic model for calculating soot emission in Diesel engine. Combust. Explos.
**2016**, 9, 36–46. [Google Scholar] - Basevich, V.Y.; Belyaev, A.A.; Posvyanskii, V.S.; Frolov, S.M. Mechanisms of the oxidation and combustion of normal paraffin hydrocarbons: Transition from C1–C10 to C11–C16. Rus. J. Phys. Chem. B
**2013**, 7, 161–169. [Google Scholar] [CrossRef] - Forchheimer, P. Wasserbewegung durch boden, Zeitschrift des Vereines deutscher Ingenieure. Sci. Res.
**1901**, 45, 1781–1788. [Google Scholar] - Available online: https://tnfworkshop.org/radiation/ (accessed on 20 November 2022).
- Reid, R.C.; Prausnitz, J.M.; Sherwood, T.K. The Properties of Gases and Liquids; McGrawHill: New York, NY, USA, 1977. [Google Scholar]
- Skeen, S.A.; Yablonsky, G.; Axelbaum, R.L. Characteristics of non-premixed oxygen-enhanced combustion: II. Flame structure effects on soot precursor kinetics resulting in soot-free flames. Combust. Flame
**2010**, 157, 1745–1752. [Google Scholar] [CrossRef] - Kumfer, B.; Skeen, S.A.; Axelbaum, R.L. Soot inception limits in laminar diffusion flames with application to oxy-fuel combustion. Combust. Flame
**2008**, 154, 546–556. [Google Scholar] [CrossRef] - Available online: https://www.flickr.com/photos/space-flames (accessed on 20 November 2022).
- Available online: https://www.facebook.com/space.flames (accessed on 20 November 2022).
- Irace, P.H.; Waddell, K.; Constales, D.; Yablonsky, G.; Kim, M.; Sunderland, P.B.; Axelbaum, R.L. On the existence of steady state gaseous microgravity spherical diffusion flames in the presence of radiation heat loss. To appear. Proc. Combust. Inst.
**2023**. [Google Scholar] [CrossRef] - Xia, F.; Axelbaum, R.L. Simplifying the complexity of diffusion flames through interpretation in C/O ratio space. Comput. Math. Appl.
**2013**, 10, 1625–1632. [Google Scholar] [CrossRef] - Kumfer, B.; Skeen, S.A.; Chen, R.; Axelbaum, R.L. Measurement and analysis of soot inception limits of oxygen-enriched coflow flames. Combust. Flame
**2006**, 147, 233–242. [Google Scholar] [CrossRef] - Minutolo, P.; Gambi, G.; D’Alessio, A. Properties of carbonaceous nanoparticles in flat premixed C
_{2}H_{4}/air flames with C/O ranging from 0.4 to soot appearance limit. Proc. Symp. Combust.**1998**, 27, 1461–1469. [Google Scholar] [CrossRef] - Hura, H.S.; Glassman, I. Soot formation in diffusion flames of fuel/oxygen mixtures. Proc. Symp. Combust.
**1989**, 22, 371–378. [Google Scholar] [CrossRef] - Valencia-López, A.M.; Bustamante, F.; Loukou, A.; Stelzner, B.; Trimis, D.; Frenklach, M.; Slavinskaya, N.A. Effect of benzene doping on soot precursors formation in non-premixed flames of producer gas (PG). Combust. Flame
**2019**, 207, 265–280. [Google Scholar] [CrossRef] - Gleason, K.; Gomez, A. Detailed study of the formation of soot precursors and soot in highly controlled ethylene(/toluene) counterflow diffusion flames. J. Phys. Chem. A
**2022**. [Google Scholar] [CrossRef] [PubMed] - Frolov, S.M.; Avdeev, K.A.; Ivanov, V.S.; Vlasov, P.A.; Frolov, F.S.; Semenov, I.V.; Belotserkovskaya, M.S. Evolution of the soot-particle size distribution function in the cylinder and exhaust system of piston engines: Simulation. Atmosphere
**2023**, 14, 13. [Google Scholar] [CrossRef]

**Figure 1.**Photograph of the porous sphere mounted on a gas supply tube (

**a**) and the schematic of the computational domain (

**b**).

**Figure 3.**Examples of the time histories of the flame radius: (

**a**) normal flame; (

**b**) inverse flame. Symbols correspond to the experiment, and the curves correspond to the calculation.

**Figure 4.**Examples of the time histories of the flame temperature: (

**a**) normal flame; (

**b**) inverse flame. Symbols correspond to the experiment, and the curves correspond to the calculation.

**Figure 5.**Examples of time histories of the size (

**a**) and temperature (

**b**) of a quasi-stationary inverse flame. Symbols correspond to the experiment, and the curves correspond to the calculation.

**Figure 6.**Calculated time histories of the temperature of normal and inverse flames undergoing radiative extinction.

**Figure 7.**Typical calculated structures of normal (

**a**) and inverse (

**b**) diffusion flames in 20 s after ignition. Soot mass fractions are increased by a factor of 10

^{4}(

**a**) and 10

^{6}(

**b**).

**Figure 8.**Dynamics of change in the calculated total soot mass ${m}_{soot,\mathsf{\Sigma}}$ versus time for flames with different values of ${Z}_{st}$. Normal flames are represented by blue dots, while inverse flames are represented by red dots.

**Figure 10.**Calculated dependences of the cumulative rate of soot formation $\dot{{m}_{soot,\mathsf{\Sigma}}\left(t\right)}$ on the C/O atomic ratio and temperature at the point of maximum soot concentration in the flame structure at different times after ignition: (

**a**) 2 s; (

**b**) 5 s; (

**c**) 10 s; red dots: $\dot{{m}_{soot,\mathsf{\Sigma}}\left(t\right)}>$ 0, and blue dots: $\dot{{m}_{soot,\mathsf{\Sigma}}\left(t\right)}<$ 0.

**Figure 11.**Calculated dependences of (

**a**) temperature on the local atomic ratio C/O and (

**b**) temperature ${T}_{0.53}$ and atomic ratio ${\left(C/O\right)}_{1305}$ on the stoichiometric mixture fraction ${Z}_{st}$ in normal and inverse flames in 2 s after ignition [24].

**Figure 12.**Calculated dependences of temperature on the local C/O atomic ratio in normal (blue curves) and inverse (red curves) flames in 2 (

**a**), 10 (

**b**), 20 (

**c**), and 30 s (

**d**) after ignition. The regions colored with red fill correspond to the parametric domains of soot formation.

Reaction | ${\mathit{A}}_{\mathit{k}},[\mathbf{L},\mathbf{mol},\mathbf{s}]$ | ${\mathit{E}}_{\mathit{k}}/\mathit{R},\left[\mathbf{K}\right]$ | ${\mathit{n}}_{\mathit{k}}$ |
---|---|---|---|

I | $2\times $ 10^{16} | 40,000 | 0 |

II | $1\times $ 10^{15} | 40,000 | 0 |

III | $1\times $ 10^{15} | 40,000 | 0 |

IV | $1\times $ 10^{12} | 0 | 0 |

Flame | Combustion Chamber | Porous Sphere | p, atm | |||
---|---|---|---|---|---|---|

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

19115B1 | 0.203 | 0.797 | 1.000 | 0.000 | 0.660 | 1.020 |

19206L6 | 0.194 | 0.806 | 0.288 | 0.712 | 0.660 | 1.010 |

19171D4 | 0.363 | 0.637 | 1.000 | 0.000 | 1.372 | 1.010 |

19189K1 | 0.363 | 0.637 | 0.476 | 0.524 | 1.372 | 1.310 |

F10 | 0.380 | 0.620 | 1.000 | 0.000 | 1.224 | 1.239 |

F02 | 0.391 | 0.609 | 0.288 | 0.712 | 1.800 | 1.190 |

F08 | 0.386 | 0.614 | 0.288 | 0.712 | 3.603 | 1.263 |

F05 | 0.400 | 0.600 | 0.288 | 0.712 | 4.514 | 1.250 |

19156C2 | 0.366 | 0.634 | 1.000 | 0.000 | 2.529 | 1.040 |

19142J3 | 0.356 | 0.644 | 1.000 | 0.000 | 2.529 | 0.990 |

19150N1 | 0.296 | 0.704 | 0.168 | 0.832 | 4.885 | 1.010 |

19150G3 | 0.338 | 0.662 | 0.288 | 0.712 | 8.779 | 1.050 |

19175A3 | 0.391 | 0.609 | 1.000 | 0.000 | 1.960 | 1.270 |

19206A5 | 0.207 | 0.793 | 0.288 | 0.712 | 8.779 | 1.010 |

19206G1 | 0.205 | 0.795 | 1.000 | 0.000 | 2.529 | 1.010 |

19206G4 | 0.201 | 0.799 | 1.000 | 0.000 | 0.822 | 1.010 |

19206L4 | 0.195 | 0.805 | 0.288 | 0.712 | 2.835 | 1.010 |

19115M4 | 0.193 | 0.807 | 0.476 | 0.524 | 1.380 | 1.020 |

19123F1 | 0.206 | 0.794 | 0.490 | 0.510 | 2.640 | 1.010 |

19123F2 | 0.206 | 0.794 | 0.489 | 0.511 | 2.640 | 1.010 |

19123F3 | 0.205 | 0.795 | 0.490 | 0.510 | 2.640 | 1.010 |

19123L1 | 0.202 | 0.798 | 1.000 | 0.000 | 2.510 | 1.010 |

19123L2 | 0.201 | 0.799 | 1.000 | 0.000 | 2.510 | 1.010 |

19150N1 | 0.351 | 0.649 | 0.168 | 0.832 | 4.820 | 1.040 |

19189J3 | 0.378 | 0.622 | 0.502 | 0.498 | 5.010 | 1.300 |

19200H3 | 0.285 | 0.715 | 0.131 | 0.869 | 4.430 | 1.020 |

19115F1 | 0.204 | 0.796 | 0.292 | 0.708 | 2.180 | 1.040 |

19123A2 | 0.209 | 0.791 | 1.000 | 0.000 | 1.620 | 1.000 |

19123A3 | 0.208 | 0.792 | 1.000 | 0.000 | 1.620 | 1.000 |

19123A4 | 0.208 | 0.792 | 1.000 | 0.000 | 1.620 | 1.000 |

19123C1 | 0.207 | 0.793 | 0.290 | 0.710 | 4.460 | 1.000 |

Flame | Combustion Chamber | Porous Sphere | $\mathit{p}$, atm | |||
---|---|---|---|---|---|---|

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

21328D1 | 0.257 | 0.743 | 0.212 | 0.788 | 10.05 | 1.03 |

21349M3 | 0.270 | 0.730 | 0.212 | 0.788 | 9.11 | 1 |

22018H2 | 0.097 | 0.903 | 0.497 | 0.503 | 6.37 | 1.01 |

22018J1 | 0.096 | 0.904 | 0.318 | 0.682 | 9.73 | 1.01 |

22018G3 | 0.098 | 0.902 | 0.850 | 0.150 | 7.89 | 1.01 |

22018G2 | 0.099 | 0.901 | 0.850 | 0.150 | 5.90 | 1.01 |

22018G1 | 0.099 | 0.901 | 0.850 | 0.150 | 3.90 | 1 |

21328N5 | 0.080 | 0.920 | 0.850 | 0.150 | 2.27 | 0.96 |

22035J2 | 0.096 | 0.904 | 0.850 | 0.150 | 5.90 | 0.51 |

21340M1 | 0.121 | 0.879 | 0.850 | 0.150 | 9.22 | 1.01 |

21340M2 | 0.274 | 0.726 | 0.262 | 0.738 | 8.8 | 1.01 |

21349N3 | 0.251 | 0.749 | 0.212 | 0.788 | 9.10 | 0.52 |

21349N4 | 0.246 | 0.754 | 0.212 | 0.788 | 10.03 | 0.52 |

22018B1 | 0.168 | 0.832 | 0.850 | 0.150 | 4.7 | 1 |

22024F1 | 0.187 | 0.813 | 0.412 | 0.588 | 9.16 | 1.01 |

22024B1 | 0.189 | 0.811 | 0.850 | 0.150 | 5.9 | 1.01 |

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## Share and Cite

**MDPI and ACS Style**

Frolov, S.M.; Ivanov, V.S.; Frolov, F.S.; Vlasov, P.A.; Axelbaum, R.; Irace, P.H.; Yablonsky, G.; Waddell, K.
Soot Formation in Spherical Diffusion Flames. *Mathematics* **2023**, *11*, 261.
https://doi.org/10.3390/math11020261

**AMA Style**

Frolov SM, Ivanov VS, Frolov FS, Vlasov PA, Axelbaum R, Irace PH, Yablonsky G, Waddell K.
Soot Formation in Spherical Diffusion Flames. *Mathematics*. 2023; 11(2):261.
https://doi.org/10.3390/math11020261

**Chicago/Turabian Style**

Frolov, Sergey M., Vladislav S. Ivanov, Fedor S. Frolov, Pavel A. Vlasov, Richard Axelbaum, Phillip H. Irace, Grigoriy Yablonsky, and Kendyl Waddell.
2023. "Soot Formation in Spherical Diffusion Flames" *Mathematics* 11, no. 2: 261.
https://doi.org/10.3390/math11020261