Fractals and aerosol characterization

Abstract

B. B. Mandelbrot introduced the term FRACTALS with his 1977 book: Fractals, form, chance and dimension (Freeman, San Francisco, 1977); stressing the general set theoretical concepts underlaying the description of irregular and fragmented spatial patterns, defying simple geometric description. Very often such patterns can be judged by ascribing them a dimension, which may be any rational number. This concept, somewhat modified and extended, seems to be suitable to describe in a simple way such features of aerosols as the irregular and i nhomogeneous spatial distribution of the particles, which are difficult to describe otherwise. The concept of relative fractal dimension, relative with respect to the system considered, could be used as a simple measure of its inhomogeniety. If we think of dimension, we usually have the topological dimension in mind. However, there are certain, so far unused mathematical models available, e.g. a curve with infinite length almost filling continuously a two-dimensional figure, which may be characterized ascribing it a fractional dimension, i.e. a dimension between 1 and 2. Especially turbulence might be described in this way but since the size of the volume considered influences the result the size or size range in which the model is applicable has to be defined too. Hence, since size (and/or size range) and dimension has to be given to describe the situation properly the fractal assigned to the system together with the size, the reference size, should be called a relative fractal. An aerosol dispersed from a point source into a flow field (e.g. smoke stack) has to be assigned at least dimension one if the flow is strictly laminar and the aerosol not dispersing at all. If the flow is homogeneously turbulent the dimension assigned to any volume sufficiently far downstream will be three. The real situation will always be between these two limiting cases. Hence any aerosol cloud can be ascribed a relative dimension between one and three, for the characteristic length defining the reference volume. The relative fractal dimension of an aerosol (parameter) will be particularly important for comparing the results of non-linear reactions which are only comparable if the fractal dimensions for the systems considered are identical. This concept of describing complex systems might mark the beginning of a new era of aerosol research converting arts like aerosol sampling and aerosol diagnosis into a true science.