Collections of processing elements

The angular brackets denote an average over all initial times. Because most materials are composed of a collection of different elements, the neutron scattering process represents the contribution of each atomic nucleus to the scattering process, weighted by the scattering length, bj, of the individual... 

The finite element method is a systematic procedure of approximating continuous functions as discrete models. This discretization involves finite number of points and subdomains in the problem s domain. The values of the given function are held at the points, so-called nodes. The non-overlapping subdomains, so-called finite elements, are connected together at nodes on their boundaries and hold piecewise and local approximations of the function, which are uniquely defined in terms of values held at their nodes. The collection of discretized elements and nodes is called the mesh and the process of its construction is called meshing. A typical finite element partition of a two-dimensional domain with triangular finite elements is given in Fig. 1. 

As was mentioned, the integrations in the first and last terms in (8.10) are over the whole collection of finite elements. Therefore, when it comes to obtaining a nonlinear equation at a specific degree of freedom of a specific node, only the integrals from the elements adjacent to this node contribute to the equation. The process of accounting for contributions from adjacent elements is called assembly and is one of the basic procedures in the finite element method. The same is true for the second term in (8.10) with the only difference that the element integrals are computed over the boundary, but the process is the same. The above-mentioned treatment can be found in a number of references, for example [3-5]. 

A number of special processes have been developed for difficult separations, such as the separation of the stable isotopes of uranium and those of other elements (see Nuclear reactors Uraniumand uranium compounds). Two of these processes, gaseous diffusion and gas centrifugation, are used by several nations on a multibillion doUar scale to separate partially the uranium isotopes and to produce a much more valuable fuel for nuclear power reactors. Because separation in these special processes depends upon the different rates of diffusion of the components, the processes are often referred to collectively as diffusion separation methods. There is also a thermal diffusion process used on a modest scale for the separation of heflum-group gases (qv) and on a laboratory scale for the separation of various other materials. Thermal diffusion is not discussed herein. 

Additional sources of the elements are tin slag and scrap. For instance, cassiterite deposits, in Australia, Brazil, Thailand and some other countries, contain a significant amount of tantalum. The bulk of this tantalum is collected in the slag and processed separately. Recycling of various tantalum-bearing scrap is also a veiy important source for tantalum production. These scrap materials include powder surplus from sintering operations, scrap from mill products, rejected and used capacitors, scrapped cutting tools and furnace hardware. 

Automatization of all stages of the analytical process is a trend that can be discerned in the development of modern analytical methods for chemical manufacture, to various extents depending on reliability and cost-benefit considerations. Among the elements of reliability one counts conformity of the accuracy and precision of the method to the specifications of the manufacturing process, stability of the analytical system and closeness to real-time analysis. The latter is a requirement for feedback into automatic process-control systems. Since the investment in equipment for automatic online analysis may be high, this is frequently replaced by monitoring a property that is easy and inexpensive to measure and correlating that property with the analyte of interest. Such compromise is usually accompanied by a collection of samples that are sent to the analytical laboratory for determination, possibly at a lower cost. 

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