Surface-perimeter method

The algorithms most fi equently used for calculation of fractal coefficients from the AFM results are [58] Fourier spectrum integral method, surface-perimeter method, structural function method and variable method. To determine the surface dimension by the Fourier spectrum integral method it is necessary to obtain the picture of the surface 2D FFT generating amplitude and time of the matrix (more detail s are given in paper [58]. Assuming the surface function as f(x,y), the Fourier transform in two-dimensional space can be expressed as [58] ... 

Particle shapes influence properties such as surface area, bulk density, flow, and so on. A number of methods are available for describing shape from simpler qualitative descriptions, through property ratios, to techniques that employ fast Fourier transformations to describe the projected perimeter of the particle. The measurement of the shape and the relevance of the data obtained are generally the two difficulties associated with particle shape. Fortunately, in the processing of materials physically unlike those in chemical processing, shape is perhaps is less significant and is more often than not inherently accounted for in the nominal diameter. 

Values of effectiveness factors in washcoat layers with non-uniform thickness around the channel perimeter have been studied by Hayes et al. (2005). However, the applicability of (even the generalized) effectiveness factor approach is quite limited in complex systems with competing reactions, surface deposition of reaction components, non-linear rate laws and under transient operating conditions (e.g. periodically operated NSRC). Typically, the effectiveness factor method can be used for more accurate prediction of CO, H2 and HC oxidation light-off and conversions in DOC. 

This method also requires that 0=0. A variation on the method eliminates the detachment step. Here one holds the plate at the point where the bottom of the plate is in the plane of the liquid surface. At this point the surface tension holds up the weight of the meniscus, using the perimeter of the plate (p) ... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

For the Wilhelmy plate method, a thin plate with a perimeter of about 4 cm is lowered to the surface of a liquid and the downward force directed on the plate is measured. Surface tension is the force divided by the perimeter of the plate. For this method to be valid, the liquid should completely wet the plate before the measurement, which means that the contact angle between the plate and the liquid is zero. Furthermore, the position of the plate should be correct, which means that the lower end of the plate is exactly on the same level as the surface of the liquid. Otherwise the buoyancy effect must be calculated separately. 

The Wilhelmy method. In the top picture a plate of the solid surface is lowered into a submerging liquid. The liquid pushes up on the solid sample with force due to the buoyancy and the surface tension, and these forces are measured by instruments attached to the arm above the sample and depend on the length d, surface tension n, and wetted length / (the perimeter of the sample along the line of contact of the air, liquid, and solid). In the bottom picture the sample is being raised and the liquid exerts a downward force. 

Surface tension is defined as the energy per unit area of surface or the force per unit length. In principle, by the detachment method, the liquid adhering to the bottom of the body is raised up till the column of the liquid hanging from the body is broken. If the perimeter of the basis of the body is known, then the force necessary to detach the body... 

Size can refer to volume, area, or length, and therefore pore-size distribution may be defined in terms of any one of these properties. In practice, the definition of size adopted is highly dependent upon the method of measurement. For example, the area size distribution of pores is often measured by image analysis of soil thin sections, while water retention data are usually interpreted in terms of the distribution of pore diameters (Bullock Thomasson, 1979). For consistency with the definition of the Peclet number, we have chosen to define size in terms of length, L. Dullien (1991) has proposed the following interrelationships between the different definitions of size L = VIS in three-dimensions or L=AJP in two-dimensions, where V is volume, S is surface area, A is cross-sectional area and P is perimeter. These relations can be used to compare pore-size distributions measured using different methods. 

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