Uneven time discretization

Recent approaches tend to adopt the uneven discretization of the time horizon of interest wherein each time point along the time horizon coincides with either the start or the end of a task (Schilling and Pantelides, 1996). In addition to accurate representation of time this approach results in much smaller number of time points, hence fewer binary variables, as shown in Fig. 1.8b. 

A mathematical formulation based on uneven discretization of the time horizon for the reduction of freshwater utilization and wastewater production in batch processes has been developed. The formulation, which is founded on the exploitation of water reuse and recycle opportunities within one or more processes with a common single contaminant, is applicable to both multipurpose and multiproduct batch facilities. The main advantages of the formulation are its ability to capture the essence of time with relative exactness, adaptability to various performance indices (objective functions) and its structure that renders it solvable within a reasonable CPU time. Capturing the essence of time sets this formulation apart from most published methods in the field of batch process integration. The latter are based on the assumption that scheduling of the entire process is known a priori, thereby specifying the start and/or end times for the operations of interest. This assumption is not necessary in the model presented in this chapter, since water reuse/recycle opportunities can be explored within a broader scheduling framework. In this instance, only duration rather start/end time is necessary. Moreover, the removal of this assumption allows problem analysis to be performed over an unlimited time horizon. The specification of start and end times invariably sets limitations on the time horizon over which water reuse/recycle opportunities can be explored. In the four scenarios explored in... 

The model is derived to take into consideration the possibility of multiple storage vessels which are dedicated to the storage of certain wastewater. The formulation shares some of the characteristics of the multiple contaminant model presented in the previous chapter. This is due to the fact that both formulations have roots in the scheduling methodology derived by Majozi and Zhu (2001). Furthermore, the uneven discretization of the time horizon is used as the time representation. 

The presented mathematical formulation is an extension of the scheduling model proposed by Majozi and Zhu (2001), which uses a state sequence network (SSN) representation. This formulation is based on an uneven discretization of time framework (see Chapter 2, Fig. 2.4) as shown in Fig. 10.1. A time point corresponds to the beginning of a particular task and is not necessarily equidistant from the preceding and the succeeding time points, as it encountered in discrete-time formulations. The... 

More significantly, a suboptimal objective value of 2944.1 rcu was reported as an optimal solution. Using the uneven discretization of time formulation proposed in this chapter, a globally optimal value of 3081.8 rcu was obtained in 24.5 CPU s. Only 72 binary variables were necessary and the model solution was based on... 

The case study was solved using the uneven discretization of time formulation presented in this chapter. The mathematical model for the scenario without heat integration (standalone mode) involved 88 binary variables and gave an objective value of 1060 rcu. This value corresponds to the production of 14 t of product and external utility consumption of 12 energy units of steam and 20 energy units... 

A uneven discretization of time mathematical formulation for direct heat integration of multipurpose batch plants has been presented. The formulation results in smaller problems compared to the discrete-time formulation, which renders it applicable to large-scale problems. Application of the formulation to an industrial case study showed an 18.5% improvement in objective function for the heat-integrated scenario relative to the standalone scenario. 

As aforementioned, the mathematical model proposed in this chapter is an extension of the mathematical formulation presented in Chapter 10. It is based on the uneven discretization of the time horizon as shown in Fig. 11.1 and entails the following sets, variables and parameters. 

---