Fractal dimension operations

Here, D< )s is the fractional diffusivity defined as (4-2d,) (2d,-4) )(3-dP) j js a constant related to the fractal dimension and R0 is the side length of a square electrode), and dv dtv means the Riemann-Liouville mathematical operator of fractional derivative ... 

The above operation is iterated at various segment sizes. Finally, the self-similar fractal dimension of the profile embedded by the two-dimensional space is given by... 

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

We are drawn to the conclusion that log-log fractal plots are useful for the correlation of adsorption data - especially on well-defined porous or finely divided materials. A derived fractal dimension can also serve as a characteristic empirical parameter, provided that the system and operational conditions are clearly recorded. In some cases, the fractal self-similarity (or self-affine) interpretation appears to be straightforward, but this is not so with many adsorption systems which are probably too complex to be amenable to fractal analysis. 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

In classical, continuum theories of diffusion-reaction processes based on a Fickian parabolic partial differential equation of the form, Eq. (4.1), specification of the Laplacian operator is required. Although this specification is immediate for spaces of integral dimension, it is less straightforward for spaces of intermediate or fractal dimension [47,55,56]. As examples of problems in chemical kinetics where the relevance of an approach based on Eq. (4.1) is open to question, one can cite the avalanche of work reported over the past two decades on diffusion-reaction processes in microheterogeneous media, as exemplified by the compartmentalized systems such as zeolites, clays and organized molecular assemblies such as micelles and vesicles (see below). In these systems, the (local) dimension of the diffusion space is often not clearly defined. 

The early fractal studies of humic materials alluded to the application of sample fractal dimensions for the description of aggregation mechanisms operant in these systems. In the first paper reporting a fractal nature for humic materials in solution, Osterberg noted that the observed fractal dimensions were indicative of an RLA process [26]. 

The principle of the box counting method mainly involves an iteration operation to an initial square, whose area is supposed to be 1 and which covers the entire graph. The initial square is divided into four sub-squares and so on. After the n times operations, the number of sub-squares, which contain the discrete points of the profile graph are counted and the length L of the profile is approximately obtained. Then the fractal dimension is calculated as D=l+log L/(n.log2). 

CNC milling operations are carried ont on mild steel work-pieces to get machined snrfaces for different combinations of spindle speed, feed rate and depth of cnt. The generated snrfaces are measnred nsing the Talysnrf instrnment and fnrther processed to get fractal dimension (D). The experimental resnlts are nsed to develop the prediction model nsing artificial nenral networks to model fractal dimension. For ANN, fnll factorial design of experiments is considered and the experimental resnlts are presented in Table 5.4 (Sahoo et ah, 2008). 

In the insert the dependence p on D is shown, from which p decrease at microgel fractal dimension growth follows. This assumes Levy flights probability decrease at the system viscosity increase. Hence, the ability to control active time gives the possibility of reaction conrse operation. 

In the early DLA simulation [11], that is, with a fractal pattern of dimension approximately 2.5, the Witten-Saners model (WS) was obtained. This model repeats the operation in which multiple small balls that move freely around a small ball fixed at the origin make contact with each other, adhere and form clusters (the particle-cluster aggregation process). If we assume all balls move freely, collide and form clusters irreversibly (cluster-cluster aggregation [12, 13]), more general cases can be handled. In this model, when a model experiment was done under the same conditions as the WS model, the fractal dimension was 1.75. Figure 5 shows the experimental results of a model experiment of cluster-cluster aggregation. As the number of small balls increases, the process of network growth can be seen better. 

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