## Radiative Properties of Polydisperse Systems of Independent Particles

* Following from: *The Mie solution for spherical particles; The scattering problem for cylindrical particles

In practice, we have usually to deal with disperse systems consisting of particles of various sizes. This means that absorption and scattering characteristics of a small (“differential”) volume element of the polydisperse medium should be determined from the corresponding characteristics of single particles of different sizes. This procedure is quite clear and simple in the case of negligible dependent scattering effects when both near-field interaction between the neighboring particles and far-field interference of the radiation scattered by these particles can be ignored. Mishchenko et al. (2004) have recently reported a detailed and consistent analysis of scattering by a small volume element starting from the first principles. Particularly, they considered a difference between the approach considering a group of particles as a complex single particle and the concept of a single scattering of radiation by a small volume element, which is a basis of the phenomenological radiation transfer theory. The conditions of applicability of the usual additive procedure in calculations of the RTE (radiative transfer equation) coefficients for sparsely and randomly positioned particles were discussed, including the interference of radiation scattered by various particles in the vicinity of the exact forward direction. The latter effect seems to be not important for radiation heat transfer problems considered in this book. Therefore, we will use the traditional procedure to determine the main characteristics of radiation absorption and scattering of disperse systems.

### General Relations

It is assumed that small volume elements contain a large number of particles, so that a representative local size distribution of the particles can be introduced. In the case of spherical particles, the number of particles with a radius in the range from *a* to *a* + *da* in a unit volume is expressed as *N*_{p}*F*(*a*)*da*, where *N*_{p} is the total number of particles in this volume, and *F*(*a*) is the size distribution function. The above-introduced function is normalized as follows:

(1) |

In some applications, the integral volume distribution function is experimentally determined as

(2) |

Obviously, one should use the following relation to calculate the ordinary normalized size distribution function:

(3) |

The expressions for absorption, scattering, and extinction coefficients and scattering (phase) function of a polydisperse system of spherical particles through the corresponding parameters of single particles are as follows (hereafter we omit subscript λ for brevity):

(4a) |

(4b) |

Note that we consider here the scattering function for randomly polarized radiation. Obviously, the expressions for transport scattering and extinction coefficients of a polydisperse system of spherical particles are similar to Eq. (4a),

(5) |

It is convenient to introduce the volume fraction of particles

(6) |

and rewrite Eqs. (4a) and (5) in the form

(7) |

Following the common practice, we use here the ordinary notation for the integral parameters of the particle size distribution (Dombrovsky, 1996),

(8) |

The description of disperse systems of cylindrical particles or fibers is more complex because of the dependence of their optical properties on orientation with respect to the direction of incident radiation (even for randomly polarized radiation). One often comes across disperse systems of particles randomly oriented in space (isotropic system) or in parallel planes (transversely isotropic system). In these cases, it is convenient to introduce the efficiency factors *Q*_{a} and *Q*_{tr} averaging over orientations according to equations from the article The scattering problem for cylindrical particles. In the case of a transversely isotropic system, the values of *Q*_{a} and *Q*_{tr} depend on the angle of illumination θ. As a result, the corresponding coefficients α, β_{tr} depend also on the radiation incidence.

The additional difficulty of calculations for polydisperse systems of cylindrical particles is connected with the possible difference of the particle length. For a polydisperse medium containing cylindrical particles of the same length, the expressions for absorption and transport extinction coefficients are similar to those for spherical particles,

(9) |

Size distributions of particles and fibers may be very complex because they are formed by quite different processes of the disperse system generation or production. Sometimes, there are several extrema of the distribution function. But in many cases one can consider the following two-parameter gamma distribution widely employed for disperse composition of natural and industrial aerosols (Levin, 1961; Cadle, 1966; Tien and Lee, 1982: Blokh, 1988; Dombrovsky, 1996):

(10) |

This distribution has a maximum at *a* = *a*_{m} = *B*/*A*, and the average particle radii can be easily calculated as

(11) |

The typical curves *F*(*a*) are shown in Fig. 1.

**Figure 1. Gamma distribution of particle sizes: 1, A = 3 μm^{-1}; 2, A = 2 μm^{-1}.**

### Monodisperse Approximation

The calculations for polydisperse systems of particles are usually very time consuming. Therefore, it is of interest to analyze a possibility of the so-called monodisperse approximation with an appropriate average radius of the particles. Obviously, the monodisperse approximation is expected to be accurate when efficiency factors are almost independent of the particle radius. In this case, Eq. (7) for spherical particles is radically simplified as

(12) |

This equation looks like that for a monodisperse system consisting of particles of radius equal to *a*_{32}. This means that one can use the so-called Sauter radius *a*_{32} as an equivalent average radius of the polydisperse spherical particles. This limiting case is often realized for disperse systems containing large particles. One can refer to results for particles of various substances obtained by Dombrovsky (1976, 1979, 2007), Buckius and Hwang (1980), Mengüç and Viskanta (1985), Goodwin and Mitchner (1989), Im and Ahluwalia (1993), Liu and Swithenbank (1993), Manickavasagam and Mengüç (1993), Caldas and Semião (1999a,b), and Sharma and Jones (2002). It was recently confirmed by Maruyama et al. (2008) that the Sauter radius of water droplets can be used in radiation heat transfer in a fog layer.

A more radical simplification is observed for very small particles. In the Rayleigh region, scattering is negligible and the absorption efficiency factor *Q*_{a} is directly proportional to the diffraction parameter *x*. As a result, the coefficient of absorption α is simply proportional to the volume fraction of particles, and independent of particle size distribution (see article Rayleigh scattering).

It is interesting to consider the case of arbitrary-size particles of a weakly absorbing material when exponential approximation (3a) from the article Radiative properties of semitransparent spherical particles can be employed for the efficiency factor of absorption. Obviously, the monotonic dependence of *Q*_{a} on particle radius *a* is favorable for the use of a monodisperse approximation with the particle radius *a* = *a*_{32}. According to Eq. (3b) from the article Radiative properties of semitransparent spherical particles, the transport scattering efficiency factor *Q*_{s}^{tr} has a maximum at ρ = 2*x*(*n* - 1) = 5. Therefore, the use of a monodisperse approximation in the region of the maximum may lead to a considerable overestimation of scattering. The above-discussion is confirmed by numerical data for typical problem parameters presented in Fig. 2. The values of the specific absorption coefficient and transport extinction coefficient, defined as

(13) |

are plotted in this figure as functions of wavelength at constant indices of refraction and absorption. The computational results for two different particle size distributions with the same Sauter radius *a*_{32} = 2 μm are very close to each other. The monodisperse approximation gives excellent results for the absorption coefficient. As was expected, the deviation of the approximate calculations from the exact numerical data for the transport extinction coefficient in the Mie resonance region is considerable. Of course, this deviation is not so important in some applied heat transfer problems [see, for instance, papers by Dombrovsky (1976) and Dombrovsky et al. (2003)], but one can remember the real case when a monodisperse approximation gives too crude results even for integral (over the spectrum) radiation flux from an isothermal particle cloud (Dombrovsky, 1976, 1996).

**Figure 2. Specific absorption coefficient and transport scattering coefficient of polydisperse spherical particles at m = 1.5 - 0.01i: 1, gamma distribution with A = 2 μm^{-1}, B = 1 μm^{-1}; 2. gamma distribution with A = 3 μm^{-1}, B = 3 μm^{-1}; 3, monodisperse approximation with a = a_{32} = 2 μm.**

It goes without saying that a monodisperse approximation is totally inapplicable for thermal radiation of nonequilibrium (multitemperature) polydisperse systems when particles of different size have considerably different temperatures. The known examples are the thermal radiation of particles in plasma spraying (Dombrovsky and Ignatiev, 2003; Fauchais, 2004), the radiation from two-phase combustion products in exhaust plumes of aluminized-propellant rocket engines (Laredo and Netzer, 1993; Cai et al., 2007), and the radiative cooling of core melt droplets in nuclear fuel-coolant interaction (Dombrovsky, 2007; Dombrovsky et al., 2009).

It is obvious that all the above comments concerning the monodisperse approximation are qualitatively correct for polydisperse particles of different shapes. But the engineering problems for fibrous materials are usually close to thermal equilibrium when there is no considerable difference between the temperatures of fibers of different sizes in a small volume of the material. At the same time, it is not obvious that a monodisperse approximation is a good choice for estimating the specific spectral properties of fibrous materials.

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