Kissing number problem

To maximize the coordination number C, what s the best we can do The question is an old one, known as the Gregory-Newton or kissing number problem, and it has a well-known answer. But let s carry out a crude geometrical analysis just to see that the answer makes sense. 

We now turn our attention to steady states of reaction-transport systems. We focus first on steady states that arise in RD models on finite domains. Such models are important from an ecological point of view, since they describe population dynamics in island habitats. The main problem consists in determining the critical patch size, i.e., the smallest patch that can minimally sustain a population. As expected intuitively, the critical patch size depends on a number of factors, such as the population dynamics in the patch, on the nature of the boundaries, the patch geometry, and the reproduction kinetics of the population. The first critical patch model was studied by Kierstead and Slobodkin [228] and Skellam [414] and is now called the KISS problem. A significant amount of work has focused on systems with partially hostile boundaries, where individuals can cross the boundary at some times but not at others, or systems where individuals readily cross the boundary but the region outside the patch is partially hostile, or a combination of the above. In this chapter we deal with completely hostile boundaries and calculate the critical patch size for different geometries, reproduction processes, and dynamics. 

Since Luyben identified the snowball effect (Luyben, 1994), the sensitivity of reactor-separator-recycle processes to external disturbances has been the subject of several studies (e.g., Wu and Yn, 1996 Skogestad, 2002). Recent work by Bildea and co-workers (Bildea et al., 2000 and Kiss et aL, 2002) has shown that a critical reaction rate can be defined for each reactor-separator-recycle process using the Damkohler number. Da (dimensionless rate of reaction, proportional to the reaction rate constant and the reactor hold-up). When the Damkohler number is below a critical value, Bildea et al. show that the conventional unit-by-unit approach in Figure 20.15 leads to the loss of control. Furthermore, they show that controllability problems associated with exothermic CSTRs and PFRs are resolved often by controlling the total flow rate of the reactor feed stream. 

This represents the solution for the kissing problem in three dimensions and is valid for icosahedral and cuboctahedral morphologies. For other shapes, the reader may refer to a paper from the group of Martin [29]. Particles possessing the above number of atoms are said to be in a closed-shell configuration. The number of atoms required to fill up coordination shell completely, nx, of a particular shell, is given by... 

Let us consider some of the most common problems in an adhesively bonded joint. Bonding problems in a three-layer step-lap joint, for example, are illustrated in Fig. 1. Ultrasonic techniques to find voids, weak cohesive properties, and delaminations are available and are quite reliable. The weak interface or kissing bond detection situation, on the other hand, has a limited number of solutions useful only in special situations. No generalized technique is yet available to depict a weak interface or kissing bond reliably, although a number of promising techniques have emeiged. 

---