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Fractal Analysis
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Most branches of science and engineering are now using fractal analysis for characterizing natural or synthetic particles, complex physical or chemical processes, and complex signatures of instruments. Peleg (1993) and Barrett and Peleg (1995) reviewed the applications of fractal analysis in food science. These included particulates characterization, non-linear kinetics, agglomeration and crystallization, mixing of viscous liquids, diffusion in non-uniform media, and characterizing jagged signatures. Fractal dimensions have been successfully used to describe the ruggedness and geometric complexities of both natural and synthetic particles (Peleg and Normand 1985; Yano and Nagai 1989; Nagai and Yano 1990; Graf 1991; Barletta and Barbosa-Canovas 1993; Peleg 1993; Rahman 1997). Peleg (1993) also applied fractal geometry in the study of the shape of broccoli. Similarly, fractal analysis has also been applied to characterize native and physically or chemically transformed food particles. Fractal analysis can predict the efficiency of the transformation process and food particle properties, such as adsorption capacity, solubility, puffing ability, chemical reactivity, and emulsifying ability to optimize food ingredient selection for product development and process design (Rahman 1997). Applications have also been made in studying textural properties of foods (Barrett et al. 1992; Rohde, Normand, and Peleg 1993; Barrett and Peleg 1995). Examples are: acoustic signature analysis of crunchy food (Peleg 1993), image analysis (Barrett et al. 1992; Peleg 1993), analysis of the cell size distribution of puffed corn extrudates (Barrett and Peleg 1995), fractal reaction kinetics (Kopelman 1988), diffusion in fractal surfaces (Nyikos and Pajkossy 1988); gel strength by rheology and fractal analysis (Bremer, van Vliet, and Walstra 1989); pore size distribution for porosimetry data (Ehrburger-Dolle, Lavanchy, and Stoeckle 1994); and moisture sorption isotherms (Suarez-Fernandez and Aguerre 2000). The fractal dimensions of solid surfaces are known not to be constant but to range from 2 (flat) to 3 (volume-filling). They depend not only on the composition of the material but also on how it was produced. This fractal property has a strong influence on the efficiency of chemical reactions since most chemical reactions take place on the surface and the higher the dimension, the greater the efficiency.
The fractal behaviour of gravity field elements: case study in Egypt
Published in Journal of Spatial Science, 2023
The immediate result of a fractal analysis is the fractal dimension , which represents a measure of the roughness, irregularity and complexity of the spatial phenomena under consideration. Fractal planar curves, such as contours and profiles, have a fractal dimension range . On the other hand, fractal surfaces possess a fractal dimension domain . Thus, the fractal dimension is a real number which is essentially greater than the topological dimension and less than the Euclidian dimension of the investigated feature (McClean 1990). According to the computational algorithm followed, there exist many kinds of fractal dimensions. Examples are the Power Spectral Density (PSD), Box-counting, Regularization and Variogram fractal dimensions (Barnsley 1993, Jiang 1998).
Simulation study on the electro-osmotic characteristic of a dehumidification fin
Published in Science and Technology for the Built Environment, 2022
Shanshan Cai, Xuan Sun, Xu Li, Song Li, Xue Xue
Two reconstruction methods are presented to derive the internal structure of the solid desiccant. The first method is the Sierpinski carpet model (Yu et al. 2020), which generates a fractal structure. Fractal analysis is a useful tool to describe structures with irregular component sizes and phase arrangement medium. The basic Sierpinski geometry is shown in Figure 3. If the black square represents a solid particle, then the model is considered a pore mass fractal model; if the black square represents a pore, then the model should be treated as a solid mass fractal model. The solid desiccant can be considered as either a solid or pore mass fractal. However, the pore dimensions can hardly be determined; thus, the black square is treated as a pore mass fractal in the model. The specifications of the Sierpinski geometry can be computed from Equations (1) and (2). The basic inputs, such as the fractal dimension (D), voidage (ε), and diameter of the particle (Zp), were determined from preliminary experiments. where and are the total side length and particle side length of the Sierpinski carpet, respectively; is the order of the fractal, and is the dimension of the Euclidean space.