Near-Infrared Properties of Droplets of Aluminum Oxide Melt
Following from: The Mie solution for spherical particles, Radiative properties of semi-transparent spherical particles
Leading to: Radiative properties of polydisperse systems of independent particles, Radiation heat transfer in a solid-propellant rocket engine, Thermal radiation from non-isothermal particles in combined heat transfer problems
The interest in the radiative properties of aluminum oxide at high temperatures was initiated by investigations of the thermal radiation of combustion products in solid-propellant rocket engines (SPREs). Some rocket propellants contain aluminum powder, which increases the combustion temperature. By combustion of such a propellant, many aluminum oxide particles of the so-called condensed phase (c-phase) are produced. The c-phase mass concentration can reach about 40% with respect to the total mass of the combustion products (Shishkov et al., 1988; Davenas, 1993).
The particle sizes in the combustion chamber, supersonic nozzle, and exhaust jet of a rocket engine are very different and depend on a number of parameters. The optimal nozzle design and the engine performance strongly depend on the c-phase disperse composition. The data for c-phase particles are also important for correct calculation of radiative and combined heat transfer in the combustion chamber and supersonic nozzle (Dombrovsky, 1996a,b; Duval et al., 2004). One should also remember the decisive contribution of c-phase particles in the exhaust plume signature (Laredo and Netzer, 1993; Dombrovsky, 1996a). An additional material on this subject can be found in the studies done by Dombrovsky and Baillis (2010). The above motivation has lead to the detailed study of the size and morphology of alumina particles in solid rockets and exhaust jets over the past years (Bartlett et al., 1963; Povinelli and Rosenstein, 1964; Cheung and Cohen, 1965; Crowe and Willoughby, 1967; Jenkins and Hoglund, 1969; Dobbins and Strand, 1970; Eisel et al., 1975; Kraeutle, 1977; Gossé et al., 2003, 2006; Gritsenko et al., 1980; Bakhir et al., 1980b; Hermsen, 1981; Strand et al., 1981; Mueller and Kessler, 1985; Cofer et al., 1987, 1989; Akiba et al., 1990; Reed and Calia, 1993; Laredo et al., 1994; Hespel et al., 2003; Stabroth et al., 2006). Theoretical modeling of the size distributions is very difficult and, in engineering practice, sometimes empirical expressions for the mean particle diameter in engine performance prediction are employed (Hermsen, 1981). The usual range of particle diameters in the nozzle is from 1 to 10 μm. The size distributions of particles are often described by the two-parameter family of gamma distribution. At the same time, very large particles can be found near the combustion front, and the sampling of the particles at a small distance from the propellant surface gives the distributions with two maxima (Bakhir et al., 1980b; Gritsenko et al., 1980). It is generally accepted that the small particle concentration maximum is due to homogeneous condensation of oxide vapors, with the second maximum formed by burning of the metal droplets. Relatively small particles are observed near the nozzle throat, but there is particle growth in the supersonic nozzle (Jenkins and Hoglund, 1969). In the studies done by Strand et al. (1981), Akiba et al. (1990), and Stabroth et al. (2006), one can find the data for aluminum oxide particles in the atmosphere after the flight of a rocket or spacecraft with an aluminized solid propellant rocket engine. These data are important for some ecological problems.
Because of the decisive role of the condensed phase particles in the radiation heat transfer in rocket engines, considerable attention was given to the determination of the aluminum oxide optical properties. The reader is referred to the studies by Malitson (1962), Oppenheim and Even (1962), Bauer and Carlson (1964), Carlson (1965), Gryvnak and Burch (1965), Adams (1967), Mularz and Yuen (1972), Bakhir et al. (1977, 1980a), Lingart et al. (1982a,b), Konopka et al. (1983), Rubtsov et al. (1984), Anfimov et al. (1993), and Plastinin et al. (1998) on this subject. Some of these works dealt with pure aluminum oxide, but the majority refer to the condensed phase of solid propellant combustion products. The data obtained by different authors vary considerably from each other, particularly regarding the absorption index. This may be connected not only to the experimental error, but with real differences due to various admixtures (Dobbins and Strand, 1970; Rieger, 1979; Pluchino and Masturzo, 1981).
The temperature dependence of the index of refraction of aluminum oxide is not strong and an approximation based on the following three-term Sellmeier dispersion equation for synthetic sapphire at room temperature suggested by Malitson (1962) is used:
where λ is expressed in microns. Following the work of Dombrovsky (1982), we approximate the index of refraction using the linear temperature dependence:
The calculated dependences n(λ) for several values of temperature from Tmin = 1800 K to Tmax = 3500 K are given in Fig. 1. The value of Tmin corresponds to the maximum overcooling of the molten aluminum oxide found by Vazhinsky (1980), and Tmax corresponds approximately to the adiabatic temperature in the combustion chamber of the SPRE.
Figure 1. Index of refraction of aluminum oxide at high temperatures.
The spectral index of absorption of aluminum oxide at high temperatures is not well known. As mentioned above, there is no good agreement between the data obtained by different authors. The latter statement is illustrated in Fig. 2, where the approximation of data by Bakhir et al. (1977) at T = 2950 K suggested by Dombrovsky (1982)
Figure 2. Index of absorption of aluminum oxide: (a) at 1800 < T < 2600 K and (b) at T ≥ 2600 K; 1, Mularz and Yuen (1972); 2, Rubtsov et al. (1984); 3, Adams (1967); 4, Carlson (1965); 5, Bakhir et al. (1977); and 6, approximation (3).
is also shown. Here, as earlier, the wavelength is expressed in microns.
According to Dombrovsky (1982), we use the following approximation for the temperature dependence of the index of absorption in the temperature range 2500 < T < 3500 K:
The resulting dependences κ(λ) are given in Fig. 3. One can see that the aluminum oxide melt is semi-transparent in the shortwave range and the temperature dependence of the absorption index is significant.
Figure 3. Index of absorption of molten aluminum oxide at high temperatures.
For analysis of the solidification of molten aluminum oxide droplets, the data for the index of absorption near the equilibrium melting temperature Tm = 2320 K is needed. First of all, one should note that the absorption index of mono-crystal aluminum oxide (synthetic sapphire) at temperatures near the melting point is not the same as the absorption index of poly-crystal oxide after melting and subsequent solidification. In the first case, the absorption index increases sharply by melting. However, there is no such effect for poly-crystal aluminum oxide. For this reason, one cannot expect a sharp decrease in the index of absorption by solidification of the melt. The following average temperature coefficient was given by Rubtsov et al. (1984):
By ignoring the weak spectral dependence in the wavelength range from 0.4 to 1.2 μm, one can use the following approximation of the absorption index temperature dependence in this range:
The same expression can be used to describe the experimental data by Mularz and Yuen (1972) but with coefficient κ(Tm) = 3 × 10-4. The longer wavelengths up to λ = 4μm are also important for thermal radiation at temperature T ≈ Tm when the maximum of thermal radiation takes place at λm ≈ 1.5 μm. Keeping in mind the expectable increase of the absorption index in the range 1.2 < λ < 4 μm, the same absorption coefficient of aluminum oxide α(T) = 4πκ(T) / λm over the wavelength range in analysis of radiative cooling and solidification of the melt droplets can be used (for further details, see Dombrovsky, 2007; Dombrovsky and Dinh, 2008; article Thermal radiation from non-isothermal particles in combined heat transfer problems).
Some results of the Mie theory calculations for a monodisperse system of aluminum oxide particles at parameters typical of combustion products in SPREs are presented in Fig. 4. The above introduced specific absorption coefficient Ea and the specific transport extinction coefficient Etr = Ea + Estr were calculated by using the following approximation for aluminum oxide density:
Figure 4. Typical spectral dependences of specific absorption and transport extinction coefficients of monodisperse alumina particles in SPRE combustion products.
Linear function (7) corresponds to the experimental temperature dependences found by Kirshenbaum and Cahill (1960) and Mitin and Nagibin (1970). One can see in Fig. 4 that scattering is much greater than absorption and the extinction curves Etr(λ) strongly depend on particle radius a. It is also important that there is a significant temperature dependence of specific absorption coefficient Ea, which follows the corresponding dependence of the index of absorption. The particles considered are in the Mie scattering region. Therefore, we observe the main maximum of transport extinction, as discussed in the article Radiative properties of semi-transparent spherical particles. The corresponding dependences for polydisperse systems of molten aluminum oxide particles typical of the SPRE combustion chamber are considered in the article Radiative properties of polydisperse systems of independent particles.
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