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The quasi-steady model of the combustion of a fuel droplet has been modified. The approach involved the modification of the quasi-steady model to reflect the difference in constant properties across the flame front. New methods for accurately estimating gas constants and for estimating Lewis number are presented. The proposed theoretical model provides results that correlate favorably with published experimental results. The proposed theoretical model also eliminates the need for unguided adjustment of thermal constants or the complex analysis of the variation of thermal properties with temperature and can serve as a basis for analysis of other combustion conditions like droplets cloud and convective and high-pressure conditions.

Fuel droplet models are used to describe the influence of droplet size and ambient conditions on fuel combustion in devices such as diesel engines, rocket engines, gas turbines, oil fired boilers, and furnaces [

In this paper, the simple quasi-steady fuel droplet combustion model is modified for higher accuracy by assuming discontinuity in the heat transfer and transport coefficients across the flame sheet and nonunity Lewis number for the inner and outer region. A method for estimating property constants for the two regions is recommended. Note that while the discontinuity in Imaoka and Sirignano [

In the derivation of the classical droplet combustion model, the following assumptions are made [

Burning droplet is spherical and surrounded by a spherically symmetric flame in a quiescent infinite medium.

Burning process is quasi-steady.

Fuel is a single component and pressure is uniform and constant.

Gaseous species are of 3 types: fuel vapor, oxidizer, and combustion products.

Stoichiometric proportions of fuel-oxidizer are at flame.

Unity Lewis number is assumed.

Radiation heat transfer is negligible.

No soot or liquid water is present.

Uniform species thermal constants:

These assumptions are good, but the following assumptions changes will be made in order to improve the accuracy of the model.

Unity Lewis is assumed only at the source of diffusing species, and nonunity Lewis number is assumed in the outer and inner regions. This assumption is made because, at the sources of diffusing specie, the generation of the diffusing specie causes the thermal diffusivity to balance the mass diffusivity, while, away from the source of diffusing species, the thermal diffusivity and mass diffusivity have different values depending on the species concentration, species properties, and temperature profile.

The property of the inner region is different from the property of the outer region. This assumption is made because the average temperature in the outer region is different from the average temperature in the inner region and the species composition in the outer region is different from the species composition in the inner region.

The new assumption that the property of the inner region is different from the property of the outer region is shown in Figure

Separation of inner region and outer region species properties.

In order to relate the outer constants to the inner constants, let

In the outer region, Fick’s law in terms of the constant fuel mass flow rate can be presented in the form [

The species properties for the inner region are estimated as follows:

For the estimation of Lewis number in the two regions, unity Lewis number is assumed at the source of diffusing specie. Therefore, in the inner region where fuel diffuses from the droplet surface, unity Lewis number is assumed at the droplet surface
_{x}_{y}_{2} dominates the product and outer region, and the property of air can be used to estimate

As an example, calculation of combustion variables for the case of _{7}H_{16}) droplet combustion in air was done. Both the simple quasi-steady model and the proposed new model (modified quasi-steady model) were used. Ambient conditions were used, that is,

The calculation results using the simple quasi-steady model are given in Table

Simple quasi-steady model

Iteration | ^{3} cm^{2}/s] | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2100 | 1235.8 | 0.079415 | 0.107941 | 0.090825 | 4.1634 | 8.46 | 968.9 | 35.00 | 5.73 | 1131.1 |

2 | 968.9 | 670.2 | 0.053133 | 0.053444 | 0.053258 | 3.0872 | 8.71 | 1201.2 | 35.40 | 4.59 | 232.3 |

3 | 1201.2 | 786.4 | 0.056942 | 0.068886 | 0.061720 | 3.3730 | 8.64 | 1125.1 | 35.30 | 4.85 | 76.1 |

4 | 1125.1 | 748.3 | 0.055603 | 0.063637 | 0.058817 | 3.3013 | 8.66 | 1142.9 | 35.32 | 4.73 | 17.8 |

5 | 1142.9 | 757.2 | 0.055909 | 0.064849 | 0.059485 | 3.3030 | 8.66 | 1142.5 | 35.32 | 4.78 | 0.4 |

Fixed variables:

Proposed model

Iteration | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2100 | 1235.7 | 0.090857 | 3.5920 | 1199.0 | 0.077116 | 1.1739 | 1.73 | 0.82 | 0.18 | 0.996 | 58.9 | 2474.8 | 11.62 | 7.00 | 374.8 |

2 | 2474.8 | 1423.1 | 0.101818 | 3.7666 | 1386.4 | 0.089698 | 1.2083 | 1.72 | 0.75 | 0.16 | 1.017 | 67.1 | 2542.8 | 10.37 | 7.75 | 68 |

3 | 2542.8 | 1457.1 | 0.103923 | 3.7945 | 1420.4 | 0.092212 | 1.2145 | 1.72 | 0.73 | 0.15 | 1.022 | 68.9 | 2556.2 | 10.14 | 7.89 | 13.4 |

4 | 2556.2 | 1463.8 | 0.104341 | 3.7999 | 1427.1 | 0.092714 | 1.2157 | 1.73 | 0.73 | 0.15 | 1.023 | 69.3 | 2558.7 | 10.09 | 7.92 | 2.5 |

5 | 2558.7 | 1465.1 | 0.104422 | 3.8009 | 1428.4 | 0.092811 | 1.2160 | 1.73 | 0.73 | 0.15 | 1.023 | 69.3 | 2559.2 | 10.09 | 7.93 | 0.5 |

Fixed variables:

Proposed model

Iteration | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2100 | 1221.0 | 0.106872 | 2.6318 | 1199.0 | 0.077116 | 1.1739 | 2.80 | 0.82 | 0.18 | 0.95 | 63.43 | 2349.1 | 11.79 | 7.32 | 249.1 |

2 | 2349.1 | 1345.6 | 0.114822 | 3.0807 | 1323.6 | 0.085242 | 1.1968 | 2.45 | 0.77 | 0.17 | 0.96 | 68.46 | 2387.7 | 10.95 | 7.83 | 38.6 |

3 | 2387.7 | 1364.9 | 0.116093 | 3.1512 | 1342.9 | 0.086586 | 1.2003 | 2.41 | 0.76 | 0.16 | 0.96 | 69.38 | 2394.6 | 10.81 | 7.91 | 6.9 |

4 | 2394.6 | 1368.3 | 0.116320 | 3.1638 | 1346.3 | 0.086827 | 1.2010 | 2.40 | 0.76 | 0.16 | 0.96 | 69.54 | 2395.8 | 10.79 | 7.92 | 1.2 |

Fixed variables:

The proposed new model calculation results are compared favorably with published experimental results on the combustion of

Proposed model

Experimental results [ | Proposed model | 1999 model [ | 1991 model [ | 1950s–1970s model [ | |
---|---|---|---|---|---|

2305 | 2559.2 | 2631 | 2631 | 1142.5 | |

^{3} cm^{2}/s) | 7.2–11.4 | 7.93 | 14.4 | 7.9 | 4.74 |

3–10 | 10.09 | 33.2 | 9 | 35.2 |

The proximity between calculated and measured values has been greatly improved by the new model proposed in this paper. The proposed new model accurately predicts

The estimated flame temperature seems to have the greatest error, and the most probable source of error is in the estimation of

Most of the available tabulated fuel vapor thermal conductivities ranges up to 500 K, and this points out the need for having thermal conductivities tables or curve fits that ranges up to 1500 K or higher in order to use and achieve results with less error.

The simple quasi-steady model of a fuel droplet was modified to reflect the difference in constant properties across the flame sheet. Two average temperatures were used: one for the inner region and the other for the outer region. The two average temperatures were used to evaluate the assumed constant specific heat and thermal conductivities for the two regions. Nonunity Lewis number was assumed for the two regions while unity Lewis number was assumed at the source of diffusing species, which implies that unity Lewis number was assumed at the flame sheet for the outer region and at the liquid-vapor interface for the inner region. The Lewis numbers obtain in the sample calculation falls within the range that has been observed experimentally [

Smaller transfer number (~1)

Transfer or Spalding number

Specific heat constant of gas [J/kg

Constant

Droplet diameter [m]

Mass diffusivity [m^{2}/s]

Latent heat of vaporization [J/kg]

Evaporation rate constant [m^{2}/s]

Thermal conductivity of gas [W/m

Molecular weight [kg/kmol]

Fuel mass flow rate [kg/s]

Lewis number

Constant: Z ratio

Pressure [atm]

Interface to liquid heat transfer per unit mass (droplet heating) [J/kg]

Gas constant [J/Kg

Radius [m]

Temperature [atm]

Time [s]

Droplet life time [s]

Number of carbon atoms in fuel molecule

Mass fraction [kg/kg]

Number of hydrogen atoms in fuel molecule

Enthalpy of combustion [J/kg]

Thermal diffusivity [m^{2}/s]

Oxidizer-to-fuel stoichiometric mass ratio [kg/kg]

Density [kg/m^{3}]

Mole fraction [kmol/kmol].

Outer region

Inner region.

Initial condition

Free stream—far from surface

Boiling point

Droplet

Flame

Fuel

Gas

Guess

Interface

Liquid

Oxidizer

Droplet surface

Universal.