# Mathematical Modeling of Non-Premixed Laminar Flow Flames Fed with Biofuel in Counter-Flow Arrangement Considering Porosity and Thermophoresis Effects: An Asymptotic Approach

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Modeling

#### 2.1. Lycopodium Characteristics

#### 2.2. Porosity of Lycopodium Particles

#### 2.3. Thermophoretic Force

#### 2.4. The Flame Structure

^{3}. Therefore, the initial position of the flame sheet was presumed to be on the left-hand side of the stagnation plane. It should be noted that the flame sheet position was the axial distance from the stagnation plane at which the flame front was detected.

#### 2.5. Mathematical Modeling of the Flame

- For simplicity, it was assumed that values of density and specific heat were constant and the momentum of the fuel and oxidizer streams were the same.
- The vaporization process occurred in a very thin zone (asymptotic limit).
- Thermal radiation and heat losses were disregarded.
- Lycopodium particles were uniformly dispersed. In other words, the size and shape of the particles were assumed to be the same.
- A large Zeldowich number was presumed. Thus, the thickness of the flame zone would be too small.
- The ambient temperature $({T}_{\pm \infty})$ was assumed to be 300 K.
- In order to analytically solve the coupled complex conservation equations of mass and energy in the considered zones, it was assumed that a gaseous fuel with a certain chemical composition evolved from the asymptotic vaporization of the lycopodium particles. Therefore, pyrolysis was disregarded as clearly considered in References [11,20,43].
- No chemical interaction occurred between the solid particles before the vaporization front.

#### 2.5.1. Dimensionalized Governing Equations

#### 2.5.2. Normalization of the Governing Equations

#### 2.5.3. Boundary and Jump Conditions

Preheat zone: | ${R}_{1}:\{x|-\infty x\le {x}_{v}\}$ |

Post vaporization zone: | ${R}_{2}:\{x|{x}_{v}\le x\le {x}_{f}\}$ |

Oxidizer zone: | ${R}_{3}:\{x|{x}_{f}\le x\infty \}$ |

#### 2.5.4. Solution of the Governing Equations

_{1}: $-\infty \le x\le {x}_{\upsilon}$

_{2}: ${x}_{\upsilon}\le x\le {x}_{f}$

_{3}: ${x}_{f}\le x\le +\infty $

#### 2.6. Flame Zone Analysis

#### 2.7. Calculation of the Thermophoretic Force

_{1}: $-\infty \le x\le {x}_{\upsilon}$

## 3. Results and Discussion

^{3}. Increasing the Lewis number of the fuel decreased the amount of fuel mass approaching the reaction zone, which shifted the flame sheet toward the fuel nozzle. It is notable that the mass fraction of the fuel for the reaction process increased with an increase in the mass particle concentration. Therefore, further fuel mass diffusivity pushed the flame sheet toward the oxidizer zone.

## 4. Conclusions

- The flame temperature increased by decreasing the volume porosity, fuel and oxidizer Lewis numbers.
- The flame sheet position moved toward the fuel nozzle with an increasing volume porosity and fuel Lewis number.
- The flame sheet position shifted toward the fuel nozzle by decreasing the oxidizer Lewis number.
- The thermophoretic force increased by decreasing the volume porosity, fuel and oxidizer Lewis numbers.
- The critical strain rate increased by decreasing the volume porosity, fuel and oxidizer Lewis numbers.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$a$ | Strain rate |

$\overline{C}$ | Mean thermal velocity of gas molecular |

${C}_{a}$ | gaseous phase specific heat $\left(\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}\right)$ |

${C}_{k}$ | temperature jump coefficient |

${C}_{p}$ | Solid particle specific heat $\left(\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}\right)$ |

${C}_{t}$ | Temperature creep coefficient |

${C}_{m}$ | Velocity jump coefficient |

${C}_{s}$ | Gas velocity discontinuities coefficient |

${D}_{C}$ | Damkohler number |

${D}_{F}$ | Mass diffusivity coefficient of gaseous fuel (m^{2}/s) |

${D}_{O}$ | Mass diffusivity coefficient of oxidizer (m^{2}/s) |

${D}_{0E}$ | Critical Damkohler number |

${D}_{T}$ | Thermal diffusivity coefficient (m^{2}/s) |

$E$ | activation energy (kj) |

$erfi\left(x\right)$ | Error function |

${f}_{e}$ | Porosity factor |

$H$ | Heavi side function |

${k}_{g}$ | Gas thermal conductivity $\left(\frac{\mathrm{kJ}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{K}}\right)$ |

${k}_{p}$ | Lycopodium thermal conductivity $\left(\frac{\mathrm{kJ}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{K}}\right)$ |

${k}_{T}$ | Constant Defined in Equation (4) |

$L$ | Mean free path |

$Le$ | Lewis number |

$m$ | Mixture molecular mass ($\frac{\mathrm{kg}}{\mathrm{mol}}$) |

${m}_{f}$ | Fuel molecular mass ($\frac{\mathrm{kg}}{\mathrm{mol}}$) |

${m}_{O}$ | Oxygen molecular mass ($\frac{\mathrm{kg}}{\mathrm{mol}}$) |

${n}_{p}$ | Number of particle per volume unit |

$Q$ | Reaction heat per unit of fuel mass $\left(\frac{\mathrm{kJ}}{\mathrm{kg}}\right)$ |

$q$ | Dimensionless heat |

$R$ | Universal constant of gases $\left(\frac{{\mathrm{m}}^{3}\mathrm{Pa}}{\mathrm{mol}\xb7\mathrm{K}}\right)$ |

${r}_{p}$ | Particle radius |

$T$ | Fuel temperature $\left(\mathrm{K}\right)$ |

${T}_{a}$ | activation temperature (K) |

${T}_{f}$ | Flame temperature (K) |

${T}_{v}$ | Particle Start temperature of vaporization $\left(\mathrm{K}\right)$ |

${W}_{F}$ | molecular weight of fuel |

$x$ | Dimension length |

${x}_{f}$ | Flame position |

${x}_{v}$ | Vaporization front position |

${Y}_{F}$ | Gaseous fuel mass fraction |

${Y}_{O}$ | Oxidizer mass fraction |

${Y}_{s}$ | Particle mass fraction |

${y}_{F}$ | Dimensionless fuel mass fraction |

${y}_{O}$ | Dimensionless oxidizer mass fraction |

${y}_{s}$ | Dimensionless mass fraction of solid particles |

$Z$ | Secondary coordinate axis |

$Ze$ | Zeldovich number |

Greek symbols | |

$\alpha $ | Initial mass fraction of oxidizer |

$\epsilon $ | Volume porosity |

Θ | Dimensionless Temperature |

$\lambda $ | Thermal conductivity of fuel or oxidizer $\left(\frac{\mathrm{kJ}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{K}}\right)$ |

$\mu $ | Dynamic viscosity |

${\upsilon}_{F}$ | Fuel stoichiometric coefficient |

${\upsilon}_{O}$ | oxidizer stoichiometric coefficient |

${\upsilon}_{product}$ | Product stoichiometric coefficient |

$\rho $ | Density $\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ |

${\rho}_{a}$ | Gaseous phase density $\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ |

${\rho}_{p}$ | Density of Solid particle $\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ |

${\tau}_{vap}$ | constant time characteristic of vaporization |

${\omega}_{v}$ | Particle vaporization rate $\left(\frac{\mathrm{kg}}{\mathrm{m}\xb7{\mathrm{s}}^{2}}\right)$ |

${\omega}_{F}$ | Rate of chemical reaction $\left(\frac{\mathrm{kg}}{\mathrm{m}\xb7{\mathrm{s}}^{2}}\right)$ |

$\varnothing $ | Constant and equal to 0.941 |

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**Figure 1.**Scanning electron microscopy of lycopodium biofuel particles [22].

**Figure 2.**Schematic representation of the porous biofuel particle considered in the current analysis.

**Figure 8.**Flame temperature against the fuel Lewis number for several values of mass particle concentration.

**Figure 9.**Flame sheet position against the fuel Lewis number for different mass particle concentrations.

**Figure 10.**Flame temperature against the effective equivalence ratio (${\varphi}_{u}$) considering several lycopodium radii.

**Figure 12.**Oxidizer and gaseous fuel mass fractions against the position for different mass particle concentrations.

**Figure 13.**Critical flow strain rate against the oxidizer Lewis number for several porosity factors.

Parameter | Value |
---|---|

${k}_{p}$ | 1.446538 × ${10}^{-4}$ $\frac{\mathrm{kj}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{K}}$ |

${k}_{g}$ | 0.3468 × ${10}^{-4}$ $\frac{\mathrm{kj}}{\mathrm{m}\xb7\mathrm{s}\xb7\mathrm{K}}$ |

${C}_{k}$ | 1.147 |

${C}_{t}$ | 2.2 |

${C}_{m}$ | 1.146 |

Property | Value | Property | Value |
---|---|---|---|

${\rho}_{p}$ | 1000 $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | ${C}_{p}$ | 5.67 $\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${\rho}_{a}$ | 1.2 $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | ${C}_{a}$ | 1.001 $\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ |

$Q$ | 64,895.4 $\frac{\mathrm{kJ}}{\mathrm{kg}}$ | $q$ | 0.4 |

$r$ | 12 $\mathsf{\mu}\mathrm{m}$ | $n$ | 12 $\mathrm{Giga}$ |

${T}_{in}$ | 300 K | ${T}_{vap}$ | 650 K |

$v$ | 2 | ${y}_{O,+\infty}$ | 0.13 |

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**MDPI and ACS Style**

Bidabadi, M.; Ghashghaei Nejad, P.; Rasam, H.; Sadeghi, S.; Shabani, B.
Mathematical Modeling of Non-Premixed Laminar Flow Flames Fed with Biofuel in Counter-Flow Arrangement Considering Porosity and Thermophoresis Effects: An Asymptotic Approach. *Energies* **2018**, *11*, 2945.
https://doi.org/10.3390/en11112945

**AMA Style**

Bidabadi M, Ghashghaei Nejad P, Rasam H, Sadeghi S, Shabani B.
Mathematical Modeling of Non-Premixed Laminar Flow Flames Fed with Biofuel in Counter-Flow Arrangement Considering Porosity and Thermophoresis Effects: An Asymptotic Approach. *Energies*. 2018; 11(11):2945.
https://doi.org/10.3390/en11112945

**Chicago/Turabian Style**

Bidabadi, Mehdi, Peyman Ghashghaei Nejad, Hamed Rasam, Sadegh Sadeghi, and Bahman Shabani.
2018. "Mathematical Modeling of Non-Premixed Laminar Flow Flames Fed with Biofuel in Counter-Flow Arrangement Considering Porosity and Thermophoresis Effects: An Asymptotic Approach" *Energies* 11, no. 11: 2945.
https://doi.org/10.3390/en11112945