Extraction of the 3-D capacity dimension from 2-D projections of fractal flocs

Extraction of the 3-D capacity dimension from 2-D projections of fractal flocs

Maggi, F.

Civil Engineering and Geosciences

Hydraulic Engineering

2003-05-07

The optical measurements of flocs foreseen in the settling column are 2-D projections of 3-D fractal structures formed in a turbulent environment by means of aggregation and break-up processes. Based on numerical analysis, the resulting structures appear non-homogenous fractals with a full spectrum of fractal dimensionalities, MAGGI(2002). The dimensionality that possesses a high value in our investigation is the capacity dimension. It relates to the floc mass and, therefore, it is directly usable in modelling the settling velocity, WINTERWERP (1999), and many other geometrical properties. On the other hand, we have noticed that the spectrum of fractal dimensionalities can give indications on the rate of growth of the flocs and on the dynamics of the processes involved in flocculation, MAGGI (2002). Despite the easy computation of all dimensionalities within the space of projection, the evaluation of the 3-D capacity dimension of the flocs from 2-D projections is still an open question. Complications arise because there is not yet a full theory which covers the problem of n-dimensional projections of fractals embedded in an m-dimensional domain. Little knowledge exists about how a project ion of a fractal affects the capacity dimension and, moreover, the full spectrum of fractal dimensionalities. This investigation is focused on the numerical characterisation of the fractal structure of unknown objects from their projections. The main reason to approach this problem is the considerable consequence of a direct extraction of 3-D information from 2-D measurements in the modelling of cohesive aggregates. Even if the problem could be reducible to an empirical estimation of the relation (and correlation) between the dimensions of two sets (the 3-D original set and the 2-D projected set), we still must explore theoretical research in literature. In the light of this, we discuss and investigate the problem of projections and cross-sections of 3-D fractals into a 2-D Euclidian space. In particular, we show the uncertainty when computing analytically the capacity dimension of 3-D fractals from 2-D projections. Subsequently, we perform two series of numerical experiments in order to show that the theory can be rigorously applied to a specific class of fractals (homogeneously distributed within the domain) and that, for fractal aggregates such as mud flocs (non-homogeneously distributed) the theory yields distorted results with respect to the numerical ones. We then establish whether we can apply it to our measurements or that we have to follow a different strategy. Next, we consider an empirical relation between a 2-D fractal (perimeter-based) dimension of the projection and the 3-D capacity dimension of the aggregates, that are representative of mud flocs. We have learned that the theory is applicable to homogeneously distributed sets. For non-homogeneously distributed sets the numerical results diverge from the theory significantly. We have performed a numerical experiment to relate the 3-D capacity dimension and the perimeter-based dimension of 2-D projections of artificially generated fractal flocs. We have observed that a hyperbolic-like correlation is well representative of the transformation of 3-D information into 2-D information. We have considered that this way to achieve information is independent from the length scales considered because the perimeter-based dimension does not relate to any of the lengths, but rather to the resolution. The (theoretical and numerical) results herein discussed enable us to apply the knowledge here developed to the recordings of the flocs produced in the settling column.

flocs
capacity dimension
projection
experiment

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TU Delft, Department of Hydraulic Engineering

Report no. 1-03

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(c) 2003 TU Delft, Department of Hydraulic Engineering