Unit scheduling constraints

The direct recycle/reuse scheduling constraints are given in constraints (9.29), (9.30), (9.31), (9.32) and (9.33). Constraint (9.29) states that water can only be recycled/reused to a processing unit provided the unit is operating at that time point. However, the constraint also states that the recycle/reuse of water is not a prerequisite for the operation of a processing unit. Constraints (9.30) and (9.31) ensure that the time at which water is recycled/reused corresponds to the time at which the water is produced. Constraints (9.32) and (9.33) ensure that the time at which water is recycled/reused corresponds to the starting time of the task using the water. 

In published mathematical formulations scheduling constraints for wastewater storage in processing units are not required, since this option is ignored. However, in this formulation the option to store wastewater in idle processing units is included, hence scheduling constraints that govern this occurrence are required. 

The final scheduling constraints are the feasibility constraints and time horizon constraints. Constraint (9.68) ensures that a processing unit can only process one task... 

In this example, there are two reactors in which reactions 1, 2 and 3 can be performed. Equal mean reaction times for the different reactions in each of the reactors imply similar performances for the reactors. The overall process consists of four units, i.e. heater, reactor 1, reactor 2 and separator. In order to handle the usage of feed C in two distinct reactions, i.e. reactions 1 and 3, different states were assigned to each of the streams of feed C, i.e. states, v3 and s4, respectively. In this example, scheduling is performed over an 8-h time horizon. It should be noted that reactors 1 and 2 are suitable for performing reactions 1, 2 and 3, which implies that constraint (2.13) is crucial. Constraints that exhibit similar structure to those presented in example 1 are not repeated. 

In this chapter, state sequence network (SSN) representation has been presented. Based on this representation, a continuous-time formulation for scheduling of multipurpose batch processes is developed. This representation involves states only, which are characteristic of the units and tasks present in the process. Due to the elimination of tasks and units which are encountered in formulations based on the state task network (STN), the SSN based formulation leads to a much smaller number of binary variables and fewer constraints. This eventually leads to much shorter CPU times as substantiated by both the examples presented in this chapter. This advantage becomes more apparent as the problem size increases. In the second literature example, which involved a multipurpose plant producing two products, this formulation required 40 binary variables and gave a performance index of 1513.35, whilst other continuous-time formulations required between 48 (Ierapetritou and Floudas, 1998) and 147 binary variables (Zhang, 1995). 

The first constraints that are dealt with in the zero effluent scheduling formulation are the unit mass balance constraints. These constraints account for the movement of mass between the various units and storage vessels. 

Due to the nature of the process constraints have to be derived that capture the essence of time. The first of these considered deal with the scheduling of the tasks in a unit. 

The first minor change to the mass balance constraints from the scheduling formulation is found in constraint (8.2), which defines the size of a batch. In the synthesis formulation, the batch size is determined by the optimal size of the processing unit. Due to this being a variable, constraint (8.2) is reformulated to reflect this and is given in constraint (8.59). The nonlinearity present in constraint (8.59) is linearised exactly using Glover transformation (1975) as presented in Chapter 4. 

The above constraints deal with the mass flows between the various units in a batch plant. They do not consider the timing of the streams, tasks and such. Therefore, further constraints have to be derived to ensure the correct sequencing and scheduling of the processes, streams and tasks. 

Enhanced continuous-time unit-specific event-based formulation for short-term scheduling of multipurpose batch processes Resource constraints and mixed storage policies. Ind. Eng. Chem. Res., 43, 2516-2533. 

Reactor systems that can be described by a yield matrix are potential candidates for the application of linear programming. In these situations, each reactant is known to produce a certain distribution of products. When multiple reactants are employed, it is desirable to optimize the amounts of each reactant so that the products satisfy flow and demand constraints. Linear programming has become widely adopted in scheduling production in olefin units and catalytic crackers. In this example, we illustrate the use of linear programming to optimize the operation of a thermal cracker sketched in Figure E 14.1. 

In model constraints given next, Q is a wrap-around operator (Shat et al., 1993), r, holds the duration of tasks in number of time intervals (5=8 h) and set K, gives the tasks belonging to chemical z. The objective function minimizes the total cost of the schedule in relative money units (r.m.u.). Eq 2 ensures that the volume handled by the task does not exceed the capacity of the vessel Vm. Eq 3 ensures that material production only occurs if the corresponding task is executed. The periodic schedule features exactly one batch of each chemical (eq 4). Eqs 5-6 are the excess resource balances. Eq 7 ensures that the start-up procedure does not require more units than those available. 

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