Two-dimensional constructions

The van de Vusse system will be studied in detail in Chapter 5 (two-dimensional constructions) and also in Ch ter 7 (three-dimensional constructions). 

The manner by which final PFR trajectories are approached on the AR boundary is highly specialized. We have already noticed how CSTRs may be used as connecting structures to PFRs for two-dimensional constructions in Chapters 4 and 5. Similar kinds of reactor arrangements fulfill the same duty in -dimensional space as well. These structures are aptly termed connectors in AR theory, which are solution trajectories that satisfy the CSTR and DSR equations. Connectors are, in fact, either ... 

Although DSRs do not serve any important function in two-dimensional constructions, theorem 2 shows that they are a critical feature of the AR boundary in higher dimensions—critical DSRs serve as connectors to final PFR extreme points. Theorem 2 also describes the role that mixing and feed points play in relation to connectors on the boundary. Notice that if a connector satisfying theorem 2 is available, then we can conclude that mixing and feed points to connectors (critical CSTRs or DSRs) must take compositions from mixing lines (lineations) that also reside on the AR boundary. 

Many aspects of the Van de Vusse system have already been discussed in previous chapters— particularly with respect to two-dimensional constructions in Chapter 5 and critical reactors in Chapter 6— but we have not yet given a full description of how the AR for the three-dimensional system can be generated. We now wish to describe the AR constmction for this system for two noteworthy reasons ... 

Flexibility. Higher dimensional systems can be broken down into a number of smaller and simpler two-dimensional problems. For systems involving exactly two independent reactions, the resulting AR can still be computed with the method. Two-dimensional constructions provide a minimum working dimension from which candidate regions may be determined. 

We can assemble higher dimensional polytopes by taking the convex hull of points generated from many two-dimensional constructions. This is done using iso-compositional reactors that operate in a specific two-dimensional shce within IR". Many planes, orientated in various positions in space, are required in order to fully capture the form of the higher dimensional polytope. 

Growth After the initial candidate AR has been determined, the region is grown by successive application of two-dimensional constructions as follows ... 

A number of simple, isothermal, two-dimensional constructions were described in Chapter 5. In particular, the well-known Van de Vusse and isola systems were studied. The isola problem allowed us to demonstrate the influence that degenerate kinetics have on the construction of the AR. Calculating the CSTR locus is not always easy, particularly when complicated kinetics is present. We also investigated how an AR involving residence time, r, may be constructed, which are represented as unbounded regions in state space. Residence time constructions are feasible since r... 

Figure 2.5 Principle setup of two-dimensional, constructal network (a) Unidirectional network with fork-like structure and four levels (b) bidirectional network with antenna-like structure and six levels.

Recall that user innovation theoiy defines lead userness as a two-dimensional construct (Luthje and Herstatt 2004 von Hippel 2005), the two dimensions being "ahead of the trend and "high expected benefit. Still, all lead user studies but one (Franke et al. 2006) measure lead userness on one continuous scale including items for both components (Franke et al. 2006 Jeppesen and Frederiksen 2006 Kratzer and Lett 2009 Kratzer and Lettl 2008 Morrison et al. 2004 Morrison et al. 2000 Schreier et al. 2007 Schreier and... 

The microstructure of Pb-Sn solders can be described by the binary alloy phase diagram. The phase diagram is a two-dimensional construct that predicts phase development as a function of the temperature and material composition. An important caveat is that the phase diagram represents the material at equilibrium. Strictly speaking, equilibrium is achieved when the material has been cooled at an infinitely slow rate such that no further changes occur to the microstructure at the target temperature. 

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