Problem Superstructure

4 Wastewater Minimisation in Multiproduct Batch Plants Single Contaminants 

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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. 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

Consider now ways in which the best arrangement of a distillation sequence can be determined more systematically. Given the possibilities for changing the sequence of simple columns or the introduction of prefractionators, side-strippers, side-rectifiers and fully thermally coupled arrangements, the problem is complex with many structural options. The problem can be addressed using the optimization of a superstructure. As discussed in Chapter 1, this approach starts by setting up a grand flowsheet in which all structural features for an optimal solution are embedded. 

If the rate of feed addition, rate of product takeoff, temperature and pressure are known in each time interval, a simulation of the reactor can be carried out in that time interval. The problem is that the conditions will change from one time interval to subsequent time intervals. The profile of the dynamic variables (feed addition, product takeoff, temperature and pressure) need to be known through time. In the approach described in Chapter 3 for profile optimization5, a shape can be imposed for a given variable through time and the dynamic variables optimized in conjunction with the temporal superstructure. One profile for each dynamic variable is assigned to the... 

One of the ways to avoid this problem is to simplify the superstructure to remove some of the structural options in Figure 18.28a9. This is done in Figure 18.29. This structure is created by splitting each hot stream into a number of branches equal to the number of cold streams and splitting each cold stream into a number of branches equal to the number of hot streams. In this way, a structure is created that allows each hot stream to be matched with each cold stream9. 

Note that the A Tmin for the problem must be fixed in order to remain an MILP problem. Fixing A Tmin fixes the composite curves and the temperatures across each enthalpy interval or block. Unfortunately, this would not necessarily lead to the best network, as the initial superstructure was already simplified with many structural options missing. But this can be allowed for by first carrying out the... 

Thus, the introduction of constraints and complex columns demands a simultaneous solution of the sequencing and heat recovery problems. This can be carried out on the basis of the optimization of a superstructure. 

To include all of these complexities requires a different approach from the one described so far. The design approach based on the optimization of a superstructure can be used to solve such problems14. Figure 26.36 shows the superstructure for a problem involving two operations and a single source of fresh water14. The superstructure allows for reuse from Operation 1 into Operation 2, reuse from Operation 2 to Operation 1, local recycles around both operations, fresh water supply to both operations and... 

Maximum water reuse can be identified from limiting water profiles. These identify the most contaminated water that is acceptable in an operation. A composite curve of the limiting water profiles can be used to target the minimum water flowrate. While this approach is adequate for simple problems, it has some severe limitations. A more mathematical approach using the optimization of a superstructure allows all of the complexities of multiple contaminants, constraints, enforced matches, capital and operating costs to be included. A review of this area has been given by Mann and Liu21. 

In spite of these progresses, and perhaps because of them, some early basic problems have not received all the attention they deserved, even if their solutions are likely to fill crucial gaps in our understanding of chromatin function. Among them, the so-called linking number paradox [6,7], previously reviewed by one of the authors [8], has raised heated debate in the past [9-11]. As will be shown below, this paradox is at the heart of nucleosome conformational dynamics within the context of chromatin superstructure. 

This chapter explains the general representation of a petrochemical planning model which selects the optimal network from the overall petrochemical superstructure. The system is modeled as a mixed-integer linear programming (MILP) problem and illustrated via a numerical example. 

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