## Spectral Radiative Properties of Diesel Fuel Droplets

* Following from: *The Mie solution for spherical particles, Radiative properties of semi-transparent spherical particles

The importance of absorption and scattering of thermal radiation by fuel droplets in diesel engines is widely discussed in the literature (Mengüç et al., 1985; Viskanta and Mengüç, 1987; Lage and Rangel, 1992, 1993a,b; Chang and Shieh, 1995; Lage et al., 1995; Dombrovsky and Zaichik, 2001; Dombrovsky et al., 2001; Sazhin et al., 2001; Dombrovsky, 2002; Abramzon and Sazhin, 2005, 2006; Tseng and Viskanta, 2005, 2006; Viskanta, 2005; Sazhin, 2006; Goldfarb et al., 2007). Absorption of thermal radiation by droplets gives a contribution to their heating and evaporation, and affects the autoignition of fuel vapor. It should also be noted that absorption and scattering of thermal radiation by fuel droplets considerably change the radiation field in the combustion chamber of a diesel engine.

The absorption of radiation by fuel droplets was considered first by Friedman and Churchill (1965). They calculated the JP-4 fuel droplet absorption coefficient from the transmission data using the Bouger-Beer law. Tuntomo et al. (1992) presented the results of measurements of indices of absorption and refraction of heptane and decane in the wavelength range from 2.6 to 15 μm. Kelly et al. (1989) reported the results of measurements of absorbance of *n*-heptane, benzene, isooctane, and a number of samples of unleaded petrol in the range of 0.66-1.215 μm. The thermophysical and chemical properties of some of these fuels (heptane, decane) are close to those of diesel fuels. At the same time, comparison of the absorption spectrum of a realistic diesel fuel measured by Dombrovsky et al. (2001, 2003) and that of heptane and decane reported by Tuntomo et al. (1992) shows that the former can differ substantially from the later two, presumably due to the contribution of various additives. The material in the present article is based on the more realistic spectral properties of diesel fuels from research done by Dombrovsky et al. (2003, 2004).

It was shown by Dombrovsky (2002) that the simplest model of gray medium used by Mengüç et al. (1985) may lead to significant errors in both the radiation absorption by fuel droplets and the integral radiation flux to the cylinder head. This means that numerical simulation of the combined heat transfer in diesel engines should include the calculation of the main spectral characteristics of the fuel droplets. The natural limitations of the computational time by calculation of the combined processes in the combustion chamber lead to the situation when only very simple models for radiative properties of combustion products and fuel droplets can be employed. It was shown by Dombrovsky et al. (2003) that one can use approximate equations (3) from the article Radiative properties of semi-transparent spherical particles to calculate the efficiency factor of absorption and the transport efficiency factor of scattering for the fuel droplets. At the same time, it is inconvenient to use the tabulated experimental data for absorption spectra of various fuels in the computer code. Following the article Radiative properties of semi-transparent spherical particles (Dombrovsky et al., 2004), we will also use the analytical approximations of spectral dependences *n*(λ) and κ(λ), which are convenient for engineering calculations.

The dependences κ(λ) for typical diesel fuels from the article Radiative properties of semi-transparent spherical particles (Dombrovsky et al., 2003) in the wavelength range from 2 to 6 μm are given in Fig. 1. The data for shorter wavelengths are not included because the measurements in the range 1.1 < λ < 2 μm were not reliable. The number of fuels in Fig. 1 should be treated as follows: 1, the diesel fuel used in cars (yellow fuel); and 2, the diesel fuel used in off-road equipment in which dye has been added for legislative purposes (pink fuel).

**Figure 1. Spectral index of absorption of diesel fuels according to Dombrovsky et al. (2003, 2004): solid lines, measurements; dashed lines, analytical approximations (1a)-(1g) and (2).**

One can see that the absorption peaks for the two fuels are practically the same and that the absorption in the ranges of semi-transparency is considerably different. Dependences κ(λ) can be approximated as follows:

(1a) |

(1b) |

(1c) |

(1d) |

(1e) |

(1f) |

(1g) |

where λ is expressed in microns and coefficients *b*_{j} correspond to the minimum absorption in the following ranges of fuel semi-transparency: *b*_{1} = 4, *b*_{2} = 3.4, and *b*_{3} = 3.8 for fuel 1; and *b*_{1} = 3.4, *b*_{2} = 3, and *b*_{3} = 2.7 for fuel 2. The latter values of coefficients were determined at wavelengths λ_{1} = 2.2 μm, λ_{2} = 3.3 μm, and λ_{3} = 4.9 μm as follows:

(2) |

One can see in Fig. 1 that the fine structure of the absorption spectra is not described by Eqs. (1a)-(1g) and (2), but this is not important for radiation heat transfer in diesel engines.

The index of refraction *n*(λ) can be calculated by using the following Kramers-Krönig relation (Nussenzveig, 1972; Bohren and Huffman, 1983; Lucarini et al., 2005):

(3) |

For practical calculations it is more convenient to use the so-called subtractive Kramers-Krönig analysis suggested by Ahrenkiel (1971). This analysis is based on the calculation of the difference between *n*(λ) and the measured value of the refractive index at a certain wavelength λ_{1}:

(4) |

Equation (4) is less sensitive to the limitation of the range of integration when compared with Eq. (3). For numerical integration in a limited spectral range, Eq. (4) has been rearranged as follows:

(5) |

where

and δ is a small parameter. The choice of λ_{min} and λ_{max} is determined by the location of the main absorption bands in the spectrum. The calculation by Dombrovsky et al. (2003) based on Eq. (5) with λ_{1} = 0.4358 μm, *n*(λ_{1}) = 1.46, λ_{min} = 0.2 μm, and λ_{max} = 10 μm results in dependence *n*(λ), in which short-wave part is shown in Fig. 2. Because of the same absorption peak, the calculated index of refraction is practically the same for both fuels considered. To avoid complicated calculations for every diesel fuel, one can use the following analytical approximation of the resulting curve:

(6) |

with parameters *n*_{0} = 1.46 and λ_{m} = 3.4 μm. As usual, the wavelength in Eq. (6) is expressed in microns. One can see in Fig. 2 that the error of this approximation is insignificant.

**Figure 2. Index of refraction of a diesel fuel: solid line, calculation using the Kramers-Krönig relation; dashed line, approximation (6).**

The calculated efficiency factor of absorption *Q*_{a} and the transport efficiency factor of extinction *Q*_{tr} for diesel fuel droplets are presented in Fig. 3. The term “exact calculations” refers to the Mie calculations based on the detailed spectral dependences *n*(λ) and κ(λ). The approximate calculations are based on approximations (1a)-(1g), (2), and (6) for spectral optical constants and approximations (3a) and (3b) from the article Radiative properties of semi-transparent spherical particles for the efficiency factors.

**Figure 3. Efficiency factor of absorption and transport efficiency factor of extinction for diesel fuel droplets: solid lines, exact calculations; dashed lines, approximate calculations.**

One can see that a combination of the suggested analytical approximations enables us to obtain a good estimate of the main radiative characteristics of diesel fuel droplets. Obviously, a similar approach can be used in a more wide spectral range by taking into account the new data for optical constants of various fuels reported recently by Sazhin et al. (2004, 2007). According to these studies, the absorption index weakly increases with the wavelength in the range of 1 < λ < 2 μm for diesel fuel and in more wide range of 0.8 < λ < 2 μm for gasoline fuel. This means that approximation (1b) can also be used at shorter wavelengths. More detailed material on the radiative properties of diesel fuel droplets can be found in recent study by Dombrovsky and Baillis (2010).

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