Model superstructure

FIG. 8 Phase diagram of a Langmuir monolayer in a model of grafted stiff Lennard-Jones chains. LE denotes a disordered expanded phase, LC-U a condensed phase with untilted chains, LC-NN and LC-NNN condensed phases with collective tilt towards nearest neighbors and next-nearest neighbors, respectively, and LC-mod a phase which has a superstructure and an intermediate direction of tilt. (From Stadler and Schmid [151].)... 

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

Figure 9.22. Scanning force microscopy images of polyethylene films formed on a model planar chromium polymerization catalyst. The small white stripes are lamellar crystals. These form the well-known spherulite superstructure upon crystallization from the...

If a model is available for the reaction chemistry and kinetics, then a temporal superstructure can be developed to represent a batch reactor in the time dimension with a series of reactor compartments that connect to each other sequentially in the time dimension3. This temporal superstructure network, representing a batch reactor, is created... 

Thus, the design of a batch reactor can be based on the optimization of a temporal superstructure. Given a simulation model with a mathematical formulation, the next step is to determine the optimal values for the control variables of a batch reaction system. 

One significant advantage of the simplified superstructure in Figure 18.29 is that each exchanger can be modeled by a linear equation if the supply and target temperatures of the streams are fixed. The area for each heat exchanger is... 

Figure 26.53 presents a superstructure for the design of an effluent treatment system involving three effluent streams and three treatment processes17. The superstructure allows for all possibilities. Any stream can go to any effluent process and potential bypassing options have been included. Also, the connections toward the bottom of the superstructure allow for the sequence of the treatment processes to be changed. To optimize such a superstructure requires a mathematical model to be developed for the various material balances for the system and costing correlations included. Such a model then allows... 

Figures 4.1 and 4.2 depict the superstructures on which the mathematical model is based. Figure 4.1 represents a situation where reusable water storage does not exist. In this situation, water used in each water using operation j can be supplied from the fresh water header, the recycle/reuse water header or a combination of both headers. Water from each operation j can be recycled to the same operation, reused in downstream processes and/or dispensed with as effluent. On the other...

The mathematical model presented in this section is based on the superstructures given in Figs. 4.1 and 4.2 and is made up of the following sets, variables and parameters. 

The mathematical formulation comprises of a number of mass balances and scheduling constraints. Due to the nature of the processes involved, the time aspect is prevalent in all the constraints in some form or another. A superstructure is used in the derivation of the mathematical model, as discussed in the following section. A description of the sets, variables and parameters can be found in the nomenclature list. 

Fig. 9.1 Superstructure for the inherent storage model with a central storage vessel (Gouws and...

The mathematical model is based on the superstructure shown in Fig. 11.2. The heat transfer fluid in heat storage remains in the storage vessel during heat transfer with only the process fluid pumped around. The superstructure also shows that each unit is capable of receiving external heating or cooling in addition to direct and indirect heat integration. 

The individual objects that compose the model are screws which simulate the molecules of the helical polymer their individual chirality is responsible for the spontaneous formation of the overall helical structure. In the model, the screws are left-handed and have a rather compact thread, that is, the pitch-to-diameter ratio of the individual screws is small In this case, left-handed screws generate a left-handed overall helical structure. If instead the thread is loose, left-handed screws may generate a right-handed superstructure (see discussion in Section 3). 

Flexible frame type structures are normally independent of the foundation dynamics and the associated mass is not included. For shear wall type structures, the effect of the superstructure is more pronounced and should be included in the analysis. In general, the foundation model should include all structural elements which tend to move rigidly with the foundation. Refer to TM 5-856 (volume 7), Section 9.06 and TR 4921 for further details. 

Figure 10. Illustration of influence of domain formation in block copolymers according to the models of Helfand and coworkers. The free energy is shown as a function of the size and separation of the domains of varying composition. (The spontaneous separations may be analogous to the way superstructure is formed in natural polymers of plants and animals.)...

In the mathematical optimization based approaches first a superstructure is created which has embedded a large number of alternative designs. Then mathematical techniques like MINLP are used to find the optimum process within the specified superstructure. For the products considered here there are two big hurdles preventing the large scale use of these techniques (Hill, 2004). Firstly a lot of the physico-chemical phenomena occurring are not completely understood. This makes rigorous modeling difficult. Secondly there is a lack of relevant property models for structured products. 

Choice of lattice linear dimensions Lx, Ly (usually Lx = Ly = L, apart from physically anisotropic situations, such as the ANWII model ). Only finite lattices can be simulated. Usually boundary effects are diminished by the choice of periodic boundary conditions, but occasionally studies with free boundaries are made. Note that Lx, Ly must be chosen such that there is no distortion of the expected orderings in the system e.g. for the model of where due to a third nearest neighbor interaction superstructures with unit cells as large as 4 x 4 did occur, L must be a multiple of 4. 

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