Most sprays or ensembles of drops contain drops of different sizes. Measurements will provide information on the fractions of drops of different sizes which can be presented in terms of number or volume (mass) as illustrated in Figure 1. A number of equations of varying complexity (two to four adjustable constants) have been suggested to describe such distributions.

**Figure 1. Example of number and volume based drop size distributions showing the position of the Sauter mean diameter and other important mean diameters.**

However, for many purposes a single number characterizing the drop size is required. Sometimes, the median diameter is employed: 50% of the drops are larger (in number or volume terms) than the median and 50% are smaller. These are usually identified as D_{NM} and D_{VM}. In some cases, an arithmetic average diameter will suffice to describe the distribution, but because the drop surface area and volume are proportional to the square and cube of the diameter, respectively, a more complex description is required.

A general mean diameter can be defined by

or in terms of a finite number of discrete size classes,

The mean diameter with the same ratio of volume to surface area as the entire ensemble is known as the Sauter mean diameter. It corresponds to values of p = 3 and q = 2 in the above equations. This particular diameter is named after the German scientist who first employed it. Dr Ing J. Sauter worked at the Laboratorium für Technische Physik of the Technische Hochschule, Müchen on aspects of internal combustion engines. In particular, he studied atomization in carburettors and as part of his work devised a technique to determine the size of drops produced based on the absorption/scattering of light. The technique depends on the fact that absorption/scattering is proportional to the surface area of the drops. The value per unit volume of liquid contains a term equal to the right hand side of Eqs. (1) or (2) with p = 3 and q = 2, i.e., the Sauter mean diameter.

For most drop size distributions, the Sauter mean diameter, D_{32}, is larger than the arithmetic, D_{10}, surface, D_{20}, and volume, D_{30}, mean diameters. The relative position of the different diameters is shown in Figure 1 Williams (1990) quotes an example where the Sauter mean diameter, D_{32}, is 18 mm and the other diameters are D_{10} = 5.5 μm; D_{20} = 7.5 μm; D_{30} = 10 μm. The number and volume (or mass) median diameter were 4.2 and 24 μm, respectively.

The different mean diameters are appropriate for different purposes as illustrated in Table 1.

The Sauter mean diameter is probably the most commonly used mean as it characterizes a number of important processes. Chin and Lefebvre (1985) suggest that it is the best measure of the fineness of sprays.

#### REFERENCES

Chin, J. S. and Lefebvre, A. H. (1985) Some comments on the characterization of drop-size distribution in sprays, Proceedings of ICLASS-85, IV/A/1/1-12.

Lefebvre, A. H. (1989) *Atomization and Spray*, Hemisphere, New York.

Mugele, R. A. and Evans, H. D. (1951) Droplet distribution in sprays, *Ind. Eng. Chem.*, 43, 1317-1324.

Sauter, J. (1926) Grössenbestimmung von Brennstoffteilchen, *Forschungsarbeiten auf dem Gebiete des Ingenieurwesens*, Heft 279.

Sauter, J. (1928) Untersuchung der von Spritzvergasern gelieten Zerstäbung, *Forschungsarbeiten auf dem Gebiete des Ingenieurwesens*, Heft 312.

Williams, A. (1990) *Combustion of Liquid Fuel Sprays*, Butterworths, London.