Superstructure Methods

AR construction using superstructure formulations are generally targeted toward mathematically sophisticated readers, which require the use of mathematical optimization techniques. Brief overviews of two superstructure methods are described later, although readers interested in this field of work should consult Burri et al. (2002), Davis et al. (2008), Kokossis and Floudas (1990), Manousiouthakis et al. (2004), Posada and Manousiouthakis (2008), Rooney and Biegler (2000), and Zhou and Manousiouthakis (2006, 2007, 2009) for further details. 

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From this perspective, AR theory should not be viewed as a competing method to optimization or superstructure methods, but rather as complementary tool used to benchmark current designs. Superstructures can be used in conjunction with AR theory to both set reactor performance targets and design the reactors needed to achieve these targets. 

A candidate AR construction method that utilizes hyperplanes to carve away unachievable space shall be discussed in Section 8.5.2. Linear constraints, such as non-negativity constraints on component concentrations and flow rates, may also be expressed in the form of a hyperplane equation. Hyperplanes therefore also arise in establishing bounds in state space. In Section 8.6, superstructure methods shall be described for the computation of candidate ARs. These methods, at their core, rely on the solution of a large... 

Superstructure methods. The superstructure approach was briefly mentioned in Chapter 1. A reactor superstructure is designed to approximate the performance characteristics for various scenarios. Combining the superstructure in different arrangements produces a range of outputs that is, in turn, an approximation to the AR. Superstructure methods result in a system of algebraic equations and must be solved using a non-LP method. 

V SIDE NOTE Similarities between shrink-I Tj wrap method and other superstructure methods... 

In later sections, we discuss superstructure methods (LP formulations and the IDEAS framework) that are similar in nature to the shrink-wrap method. A major difference between the methods is that the shrink-wrap method is geometric in nature. 

Since computation of the AR is carried out via many small CSTRs in the total connectivity model, interpretation of the associated optimal reactor structure is not clear using the method. This is a trait observed for many superstructure methods. Nevertheless, candidate ARs for systems with highly nonlinear kinetics may be found with this approach since the method does not solve for exit concentrations, but rather volumes and flow rates. 

The IDEAS approach is a reactor superstructure method that represents all reactor networks as the combination of two generalized blocks. When the system is viewed in this manner, the resulting equations describing the problem can be made linear. An advantage of this is that traditionally nonlinear reactor network problems may then be solved via an LP technique, such as that described by the LP formulations in Section 8.6.1. And as a result, the solution to the linear system is guaranteed to be globally optimal. 

Alternate methods for finding the AR, via supersumc-ture methods, were also briefly discussed. The IDEAS framework, in particular, is a generalized AR construction scheme that may also be utilized to solve both AR and non-AR-related problems. Superstructure methods rely on mathematical programming techniques and are hence not always easy to program and implement without prior knowledge and training. 

For more complex network designs, especially those involving many constraints, mixed equipment specifications and so on, design methods based on the optimization of a superstructure can be used. 

As with the case for water minimization, the graphical methods used for effluent treatment and regeneration have some severe limitations. As before, multiple contaminants are difficult to handle, constraints, piping and sewer costs, multiple treatment processes and retrofit are all difficult to handle. To include all of these complications requires an approach based on the optimization of the superstructure. 

In the Equivalent Static Design Method, foundations are typically designed for the peak reactions obtained from the superstructure dynamic analysis. These reactions are treated as static loads, disregarding any time phase relationship. The basis for equivalent static design is discussed in 7M 5-856. 

The structural peculiarities of the whole class of isomorphous compounds PbsAOs (A = S, Se, Cr) are up to now investigated only by single crystal and powder X-ray methods. Thus two t)q)es of superstructures obtained are 4 3 2 from single crystal dififractometry and 2 3 2 from powder dififractometry. This contradiction between the two methods can be solved by using the SAED method as it is done here with PbsMoOs. SAED investigations undoubtedly confirmed that 4 3 2 superstructure of deformed a-PbO is more probable to this class of compounds [18]. 

The crystal chemistry of BajRC C has been systematically studied by single-crystal and powder diffraction methods with R = La, Pr,... Yb, in addition to the conventional yttrium compound [(52)(53) (54) and references therein]. With the exception of La, Pr, and Tb, the substitution of Y with rare-earth metals has little or no effect on the superconductivity, with the values of Tc ranging from 87 to 95K. Also, a relatively small change is observed in the cell constants of these compounds. The La, Pr, and Tb-substituted materials are not superconductors. A detailed structural analysis of the Pr case (52) did not show any evidence of a superstructure or the presence of other differences with the atomic configuration of the yttrium prototype. 

Figure 5 presents the results of tensile tests for the HPC/OSL blends prepared by solvent-casting and extrusion. All of the fabrication methods result in a tremendous increase in modulus up to a lignin content of ca. 15 wt.%. This can be attributed to the Tg elevation of the amorphous HPC/OSL phase leading to increasingly glassy response. Of particular interest is the tensile strength of these materials. As is shown, there is essentially no improvement in this parameter for the solvent cast blends, but a tremendous increase is observed for the injection molded blend. Qualitatively, this behavior is best modeled by the presence of oriented chains, or mesophase superstructure, dispersed in an amorphous matrix comprised of the compatible HPC/OSL component. The presence of this fibrous structure in the injection molded samples is confirmed by SEM analysis of the freeze-fracture surface (Figure 6). This structure is not present in the solvent cast blends, although evidence of globular domains remain in both of these blends appearing somewhat more coalesced in the pyridine cast material.

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