Attainable region dimension

As it has been shown, in AR approach the construction of the boundary is essential. Feinberg and Hidebrandt (1997) demonstrated that the boundary of attainable region could be assembled only by means of combinations of PFRs, CSTRs, and DSRs (differential side-stream reactors). The calculation of these elementary reactors is simple. The construction of an AR is not complicated, if the analysis is restricted at two dimensions. Moreover, 3-D representation could be visualised relatively easy with an appropriate computer tool. In the case of systems of higher dimensions the method relies rather on numerical techniques, and is not easy to apply. 

Thus far, the attainable region has been shown for the analysis of systems with two key compositions to be tracked. In the following, the principle of reaction invariants is used to reduce the composition space in systems of larger dimension. 

The dimension of i is equal to the number of species minus the number of elements (atoms) in the species. When this dimension is two or less, the principle of reaction invariants permits the application of the attainable region to complex reaction systems. This is illustrated in the following example, introduced by Omtveit et al. (1994). 

By evoking the principle of reaction invariants, the number of species that need to be tracked for this system is reduced to two so that the attainable region can be shown in two dimensions. Accordingly,... 

The material covered in this book is organized into two sections. It may be helpful to refer to Figure P.l for an overview of the organization of chapters. Section I (Chapters 1-5) focuses on the basics of attainable region (AR) theory. Importantly, this section introduces a different way of viewing chemical reactors and reactor networks. The examples discussed in Section I are of a simpler nature, with an emphasis on describing all problems in two dimensions only. Section I is best read in a sequential fashion. 

Like many other reservoirs in the temperate regions, Mequinensa is monomictic. The thermal stratification begins in spring, intensifies and attains its maximum in summer. In the autumn the water column mixes and water temperature is uniform in the vertical dimension in winter. The summer stratification is more intense close to the dam [39]. In this area, during the stratification period the surface temperatures can attain 24-27°C, while at the bottom remain around 14—16°C [36-39]. 

Williams (49), Ward (79), and Jancar et al. (89) proposed an approximate model of mixed mode of fracture to account for the effect of finite specimen dimensions for Kc and G, respectively. The basic idea in both theories is a substitution of the actual distribution of fracture toughness across the cross-section by a simple bimodal distribution, assuming plane strain value in the center and plane stress value at the surface area of the specimen. Size of the plastic zone IR relative to the specimen width B gives the contribution of plane stress regions and is a measure of the displacement of the state of stress at the crack tip from the plane strain conditions. Note that this approach can be used only if the mode of failure does not change with the test conditions or material composition (i.e., it attains its brittle character). 

Conditioning. After the hardboard has been hot-pressed or has been heat-treated or oil-tempered, the moisture content is well below what will be attained at equilibrium with the atmosphere in normal use. Such very dry boards will change dimensions upon picking up moisture and may warp. It is important, therefore, to humidify the boards under controlled conditions before packaging. The desired equilibrium moisture content (EMC) reached will vary from 5 to 12 percent, depending upon the nature of the board and the general humidity conditions in the region of use. 

---