Spherical-material model

The Spherical-Material Model. The finite-element model used for the spherical-material model is a single rubber sphere surrounded by an annulus of epoxy resin. The complete cell can be represented by axisymmetric elements in an analogous manner to the cylindrical model, as shown in Figure 2. The same number and types of elements were used for this model. Analyses were also undertaken assuming a hole instead of the rubber particle the grid then consisted of the epoxy annulus alone. 

Figure 2. Development of the finite-element mesh for the spherical-material model, showing the shape of the deformed grid for application of unidirectional load (---). The interface on the grid is indicated by the arrows.

The predictions for the values of v for epoxy resin filled with glass beads have been repeated using the spherical-material model. The predictions from the two models are compared with experimental values in Figure 5. The values predicted using the spherical-material model are far closer to the experimental values than the values predicted using the cylindrical model. The better fit of the spherical-material model is particularly marked in the higher range of volume fraction. 

Figure 6 compares experimental results with finite-element predictions obtained using the spherical-material model. The predictions obtained using the lower value of v for the rubbery phase are lower than those obtained using the higher value. However, both finite-element predictions underestimate the experimentally measured reduction of modulus with volume fraction, although the difference is only about 0.2 GPa at about a 25% volume fraction. Thus, there is reasonable agreement between the predicted and the experimental results. 

On average, the correct shape of the Voronoi cell is spherical. Thus, the correct overall material model for the assumption of a random distribution is a collection of spherical cells of different sizes, each containing a single sphere. Strictly speaking, any overall property of the material should be obtained by summing the contributions from the different cell sizes. This summing may be carried out by application of a dispersion factor to the property value found for the cell describing the overall volume fraction (12). The results presented here were obtained for the cells that describe the overall volume fraction. The application of the spatial statistical model to take into account the effect of the variable cell size is the subject of current work. 

HIAC device.An HIAC/ROYCO model size 3000, 6 channels (Pacific Scientific Co., USA - Silver Spring, MD.) fitted with an HR60H sensor, with standard size range of 1-120 im, flow rate 10 ml min and supplied by the manifacturer already calibrated with standard spherical materials was employed. 

Often, Hertz s work [27] is presented in a very simple form as the solution to the problem of a compliant spherical indentor against a rigid planar substrate. The assumption of the modeling make it clear that this solution is the same as the model of a rigid sphere pressed against a compliant planar substrate. In these cases, the contact radius a is related to the radius of the indentor R, the modulus E, and the Poisson s ratio v of the non-rigid material, and the compressive load P by... 

Bartle et al. [286] described a simple model for diffusion-limited extractions from spherical particles (the so-called hot-ball model). The model was extended to cover polymer films and a nonuniform distribution of the extractant [287]. Also the effect of solubility on extraction was incorporated [288] and the effects of pressure and flow-rate on extraction have been rationalised [289]. In this idealised scheme the matrix is supposed to contain small quantities of extractable materials, such that the extraction is not solubility limited. The model is that of diffusion out of a homogeneous spherical particle into a medium in which the extracted species is infinitely dilute. The ratio of mass remaining (m ) in the particle of radius r at time t to the initial amount (mo) is given by ... 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Gas diffusion in the nano-porous hydrophobic material under partial pressure gradient and at constant total pressure is theoretically and experimentally investigated. The dusty-gas model is used in which the porous media is presented as a system of hard spherical particles, uniformly distributed in the space. These particles are accepted as gas molecules with infinitely big mass. In the case of gas transport of two-component gas mixture (i = 1,2) the effective diffusion coefficient (Dj)eff of each of the... 

PALS is based on the injection of positrons into investigated sample and measurement of their lifetimes before annihilation with the electrons in the sample. After entering the sample, positron thermalizes in very short time, approx. 10"12 s, and in process of diffusion it can either directly annihilate with an electron in the sample or form positronium (para-positronium, p-Ps or orto-positronium, o-Ps, with vacuum lifetimes of 125 ps and 142 ns, respectively) if available space permits. In the porous materials, such as zeolites or their gel precursors, ort/zo-positronium can be localized in the pore and have interactions with the electrons on the pore surface leading to annihilation in two gamma rays in pick-off process, with the lifetime which depends on the pore size. In the simple quantum mechanical model of spherical holes, developed by Tao and Eldrup [18,19], these pick-off lifetimes, up to approx. 10 ns, can be connected with the hole size by the relation ... 

It has long been recognized that the validity of the BKW EOS is questionable.12 This is particularly important when designing new materials that may have unusual elemental compositions. Efforts to develop better EOSs have been based largely on the concept of model potentials. With model potentials, molecules interact via idealized spherical pair potentials. Statistical mechanics is then employed to calculate the EOS of the interacting mixture of effective spherical particles. Most often, the exponential-6 (exp-6) potential is used for the pair interactions ... 

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