One-Dimensional Geometries

For more general applications involving electrolyte ions of various charges and sizes or for mixed boundary conditions in which the surface 

As with the analytical solutions presented above, we consider in detail the PB cell model and take the bulk model limit at the end. We begin the derivation of the finite-element PB algorithm by writing the mobile charge density of Eq. [4] as an explicit functional of the potential  

For further reference below, we have also included any nonelectrostatic contributions to the potential of mean force in the form of an activity coefficient Ay(2 ) = y r) — j R). Using Eq. [378] with Eq. [3] gives 

Each lattice point or finite element in the system must be assigned to represent either part of the polyelectrolyte or its ionic environment. To account for a finite distance of closest approach to the polyelectrolyte surface by ions of varying size, a rolling-sphere algorithm may be 

Differential equation [379] can be converted into an integral representation by integrating over the th volume element (with volume vj) to get 

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The one-dimensional geometry of a radially expanding ring is perhaps the simplest for considering fundamental aspects of the fracture and fragmentation process. In a ductile metal ring, fracture proceeds through the multiple... 

Recent work on spatial stabilization has been directed towards the production of one-dimensional flames (Fll, P10). These may be either flat, cylindrical, or spherical. The primary purpose of such flames has been to measure velocities accurately and to provide a flame that can be described by a one-dimensional theory. The measurement of temperature and composition profiles is meaningful, of course, only in a flame in which the geometry is known. One-dimensional geometry greatly reduces the labor required to analyze such profiles in order to study kinetics. 

Among the technical methods proper to the one dimensional geometry, one may cite the Bethe ansatz [19], the bosonization techniques [18], and, more recently, the Density-Matrix Renormalization Group (DMRG) method (20, 21] and a closely related scheme which is directy considered in this note, the Recurrent Variational Approach (RVA) [22, 21], The two first methods are analytical and the third one is numerical the RVA method is in between. 

Using the information discussed so far, we can now return to the gedanken flame experiment with the idea of considering modified numerical methods in order to reduce the computational cost. The goal is to calculate the propagation of a flame front across a one-meter tube using a one-dimensional geometry with a fixed detailed chemical reaction rate scheme. 

Based on the above discussion, the physics of the ion acceleration process can be theoretically modeled under the following assumptions, leading to the formulation of a relatively simple system of equations which can be investigated analytically and numerically. First of all, let us restrict our analysis to a one-dimensional geometry. The electron population can be described as a two-temperature Boltzmann distribution, where the subscripts c and h refer to the cold and hot electron components, respectively,... 

That is, the solution for Ihe two-dimensional short cylinder of height a and radiu.s r is equal to the product of the noiidimcusionalized solutions for the oue-dimensional plane wall of thickness a and the long cylinder of radius r , which are the two geouieiiies whose intersection is the short cylinder, as shown in Fig. 4—35. We generalize this as follows the solution for a multidimensional geometry is the product of the solutions of the one-dimensional geometries whose intersection is the multidimensional body. 

A modified form of the product solution can also be used to determine the total transient heat transfer to or from a multidintensional geometry by using the one-dimensional values, as shown by L. S. Langston in 1982. The transient heat transfer for a two-dimensional geometry formed by the intersection of two one-dimensional geometries 1 and 2 is... 

The major advantage of potential probes is that they directly measure the local potential in the electrolyte and allow a straightforward and quantitative interpretation of the data. Besides, they are easy to operate and inexpensive. Their obvious disadvantage is that they readily hinder the transport of the reacting species and/or shield the electric field. This has restricted their use to one-dimensional geometries. 24,28,30,149,160,161,166,167 In many applications, spatial resolution was... 

A theoretical model to relate the Wiener spectrum to the toner deposit parameters is difficult to construct because the mathematical difficulties of dealing with projections of transforms of probability distributions quickly "hide" any simple relationships. Models have been constructed however for a crowded monolayer photographic emulsion (11), and for multilayers of emulsion (12). Although the analysis was done for one-dimensional geometry, extension to two dimensions was outlined. A different approach will be used here, which relies on the linearity property of the Fourier transform, and assumes that the location of the toner particles is independent of neighbors. 

Traditionally oscillations have been observed and presented in a one dimensional geometry, a line. In mathematical terms this is x = x(t) as shown in Fig. IV. 1. 

In this chapter we shall consider steady conduction in one-dimensional geometry. Although the main objective is conduction, convection described in terms of an assumed heat transfer coefficient will be included whenever it is pertinent. This may be the case when the heat transfer is desired in terms of ambient temperatures (Section 2.2) or when heat loss normal to the direction of conduction is essential, as in the case of extended surfaces (Section 2.4). Here we continue to employ the five-step formulation but somewhat less explicitly than the way we used it in Chapter 1. Each reader should tailor the degree of elaboration of this formulation to his or her particular needs. 

One dimensional geometry Because the scale of phase separation in the junction is small compared to the device size, an effective medium approach is often applied, and one-dimensional symmetry assumed. This means neglecting the effect of local electric fields at the interface, and differences in behavior between electrons or ions that are near or further from the interface or grain boundaries. The effect of electrolyte on electron transport can be included through an effective diffusion coefficient. 

Here djdn denotes the derivative in the direction normal to the interface.) To the best of our knowledge these interface conditions have not been derived from the fundamental interface condition (2.3) and the hypotheses already made in deriving the multigroup diffusion equations (except in the very important special case of one-dimensional geometry). To illustrate the difficulties we substitute the condition (3.1) into equation (2.3) and find that... 

To provide some data in one-dimensional geometry, the results from homogenized DTF-IV (Ref. 7) and KENO IV calculations were compared for a finite and infinite system geometry. Based on these analyses, the critical slab thickness for a slab of rods 60 lattice unite long is 244 mm (1219 x 914.4 x 244 mm). An Infinite Slab of the same rods would have a 213-mm critical thickness, as predicted by DTF-IV. 

Fife refers to a set of partial differential equations like eqs. (6.28) as a propagator—controller system, for reasons that will become clear later. We recall that the small parameter e 1 makes u change much more rapidly than v. It also makes the time scale of the chemical change much shorter than that of the diffusion that is, the diffusion terms are small compared with the chemical rates of change. Note that all the chemistry is in the functions / and g, which are constructed so that neither concentration can grow to infinity. We will want to consider either a one-dimensional geometry (spatial variable x) or a radially symmetric two-dimensional system, like a Petri dish. 

The difficulty in solving the problem lies in the complexity of the equation system which describes the evolution of the material. This point has already been underlined in an earlier attempt to rationalize the selection of the equations. The main sets of equations of this system are (for simplicity we will consider a one-dimensional geometry)... 

Confine the process to a geometrically simple structure - a one-dimensional geometry, state the authors. They point out that ID flow in the absence of lateral nonuniformities is one prerequisite for uniform product quality - a major benefit of PI. 

Furthermore, the behavior of adsorbate molecules in nano-eonfinement is fundamentally different than in the bulk phase, whieh eould lead to the design of new sorbents [78]. Finally, their one-dimensional geometry could allow for aligrunent in desirable orientations [79] for given separation devices to optimize the mass transport. Despite possessing sueh at-... 

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