Arbitrary shape

Gibbons R M 1969 Scaled particle theory for particles of arbitrary shape Mol. Phys. 17 81... 

The nearest thing to a complete justification of equations (11.3)i for pellets of arbitrary shape, is an argument given by W. E. Stewart [74], which does not depend on any particular choice of flux relations. In Chapter 10 it was pointed out that all isothermal flux models must have the general form... 

Now consider a catalyst pellet of arbitrary shape occupying a region V... 

The only other situation in which equations (11.3) are firmly established for a pellet of arbitrary shape is the case of a mixture at the limit of bulk diffusion control, with diffusion describable in terms of the dusty... 

Though the case of constant matrix elements and the example investigated by Hite are the only situations for which Che stoichiometric relations have been fully established in pellets of arbitrary shape, it is worth mentioning situations in which these relations are known not to hold. When the composition and pressure at the surface of the pellet may vary in an arbitrary way from point to point it seems unlikely on intuitive grounds that equations (11.3) will be satisfied, and Hite and Jackson [77] confirmed by direct computation that there are, indeed, simple situations in which they are violated. Less obviously, direct computation [75] has also shown them to be violated even when the pressure and composition of the environment are the same everywhere, in the case where finite resistances to mass transfer exist at the surface of Che pellet. 

Bashin A A and K Namboodiri 1987. A Simple Method for the Calculation of Hydration Enthalpies c Polar Molecules with Arbitrary Shapes. Journal of Physical Chemistry 91 6003-6012. 

Foams that ate relatively stable on experimentally accessible time scales can be considered a form of matter but defy classification as either soHd, Hquid, or vapor. They are sol id-1 ike in being able to support shear elastically they are Hquid-like in being able to flow and deform into arbitrary shapes and they are vapor-like in being highly compressible. The theology of foams is thus both complex and unique, and makes possible a variety of important appHcations. Many features of foam theology can be understood in terms of its microscopic stmcture and its response to macroscopically imposed forces. 

Fig. 3. The effect of crack growth on potential energy in a loaded body where (a) is a cracked body of arbitrary shape with a load P appHed, and (b) is the change in potential energy in the body owing to incremental crack growth, Sa. Other terms are defined in text.

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

This means that there is a cross-over temperature defined by (1.7) at which tunneling switches off , because the quasiclassical trajectories that give the extremum to the integrand in (2.1) cease to exist. This change in the character of the semiclassical motion is universal for barriers of arbitrary shape. 

Use a mechanics of materials approach to determine the apparent Young s modulus for a composite material with an inclusion of arbitrary shape in a cubic element of equal unit-length sides as In the representative volume element (RVE) of Figure 3-17. Fill in the details to show that the modulus is... 

Schadler, G. H., 1992, Solution of Poisson s equation for arbitrary shaped overlapping or nonoverlapping charge densities in terms of multipole moments, PAy. Rev. 545 11314. 

The proof of convergence of scheme (19) reduces to the estimation of a solution of problem (21) in terms of the approximation error. In the sequel we obtain such estimates using the maximum principle for domains of arbitrary shape and dimension. In an attempt to fill that gap, a non-equidistant grid... 

Because of this, there is a real need for designing the general method, by means of which economical schemes can be created for equations with variable and even discontinuous coefhcients as well as for quasilinear non-stationary equations in complex domains of arbitrary shape and dimension. As a matter of experience, the universal tool in such obstacles is the method of summarized approximation, the framework of which will be explained a little later on the basis of the heat conduction equation in an arbitrary domain G of the dimension p with the boundary F... 

In the preceding sections this trend of research was due to serious developments of the Russian and western scientists. Specifically, the method for solving difference equations approximating an elliptic equation with variable coefficients in complex domains G of arbitrary shape and configuration is available in Section 8 with placing special emphasis on real advantages of MATM in the numerical solution of the difference Dirichlet problem for Poisson s equation in Section 9. 

Suppose that masses are located inside a volume of an arbitrary shape and imagine that the potential of the field due to these masses is constant on some surface S, surrounding the masses. Fig. 4.3. Also assume that the potential tends to zero at infinity. Then, as was shown in Chapter 1, the potential at any point p outside the surface S is equal to... 

Rashin, A.A., M.A. Bukatin, J. Andzelm, and T. Hagler. 1994. Incorporation of reaction field effects into density functional calculations for molecules of arbitrary shape. Biophys. Chem. 51, 375. 

Truong, T. N. and E. V. Stefanovich. 1995. A new method for incorporating solvent effects into classical, ab initio molecular-orbital and density functional theory frameworks for arbitrary shape cavity. Chem. Phys. Lett. 240, 253. 

FIGURE 11.22 Arbitrary shaped spot with a seed in the middle. 

For a non-interacting spin-1/2 system subjected to a PIP of an arbitrary shape, the Hamiltonian in the rotating frame with a RF carrier frequency /rf... 

An input pulse of fairly arbitrary shape is put into the process. This pulse starts and ends at the same value and is often Just a square pulse (i.e., a step up at time zero and a step back to the original value at a later lime t ). See Fig. 14.3. The response of the output is recorded. It typically returns eventually to its original steadystate value, If C(,j and m, are perturbations from steadystate, they start and end at zero. The situation where the output does not return to zero will be discussed in Sec. 14.3.4,... 

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

Umashankar, K., Taflove, A., and Rao, S.M., 1986, Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects, IEEE Trans. Antennas. Propagat. 34(6) 758-766. 

The various probabilities on the rhs of Eq. (9.5.3) depend on the shape of the adsorbent molecule in each of the occupancy states. It is very difficult to compute these probabilities for arbitrary shapes. It is, however, intuitively clear that the whole term y ( 1,1) becomes nearly unity when the two ligands are small compared with the size of the adsorbent molecule, and when the separation between the sites is large compared with the diameter of the solvent molecules. The whole term y (l, 1) will be unity when the ligands are buried within the adsorbent molecule, in which case there is no excluded volume change in the reaction... 

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