Scheduling constraints

The constraints that comprise the scheduling module of the model are divided into four groups, namely, task scheduling, direct recycle/reuse scheduling, storage scheduling and time horizon constraints. 

The task scheduling constraints used in this model are similar to those that have been presented in full in Chapter 2. However, the reader should note that the time variables in these constraints relate to water using operations only. These constraints will therefore not be discussed here, since the full description of the constraints can be found in the preceding chapters, particularly Chapter 2. However, these constraints are presented below. 

The constraints used for the scheduling of the recycle/reuse in the model are similar to those used in the previous multiple contaminant wastewater minimisation methodology, (see constraints (6.31), (6.32), (6.33), (6.34) and (6.35)). Again, the reader is alerted to the fact that the following constraints apply to water using operations, i.e. they are not as generalised as in Chapter 6. This is due to the fact that 

Scheduling constraints have to be derived to account for the timing of multiple streams leaving a storage vessel. Constraint (7.36) ensures that water leaving a storage vessel at a later time point does so at a later absolute time in the time horizon. Constraints (7.37) and (7.38) ensure that the time at which two streams leave a storage vessel at a time point corresponds to the same time for each. 

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The constraints considered in the mathematical formulation are divided into two modules. The first deals with the mass balance constraints and the second with the sequencing and scheduling constraints. The mass balance constraints for the case where there is no central storage are slightly different to those for the case where there is. The mass balances for each are described in the mass balance module below. The sequencing and scheduling module will be described, for both cases, in a subsequent section. The nomenclature for all the sets, variables and parameters can be found in the nomenclature list. 

The production scheduling model has been presented in detail in Chapter 2 of this textbook, but is briefly presented in this section of the chapter for purposes of continuity and facilitation of understanding. For a detailed explanation of each of the production scheduling constraints, the reader is referred to Chapter 2. 

Sequencing and Scheduling Constraints Associated with Storage... 

The final group of sequencing and scheduling constraints comprise of feasibility constraints and time horizon constraints. 

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. 

The scheduling constraints presented above hold for each storage vessel in the process. It must be noted that there is no interaction between the various storage vessels, since each storage vessel is independent of the other. 

The constraints that comprise the mathematical formulation are presented in two modules. The first deals with the mass balance constraints and the second the scheduling constraints. 

The first scheduling constraints considered are binary variable constraints governing wastewater reuse. These constraints ensure the correct reuse of water, since water is only reused in distinct amounts at certain points in the time horizon. 

Scheduling constraints also have to be derived to ensure the time at which wastewater reuse occurs is correct within the time horizon. 

The final scheduling constraints considered are the time horizon constraints. These constraints ensure that each event occurs within the time horizon of interest and are given in constraints (8.45), (8.46), (8.47), (8.48), (8.49), (8.50) and (8.51). 

The synthesis model is based on the zero effluent scheduling formulation with the addition of design specific constraints. The scheduling constraints used in the synthesis formulation are identical to those used in the scheduling formulation and the mass balance constraints are virtually identical except for a few minor changes in the limiting capacities, amount of water and amount of product. These changes will be discussed below. 

It is important to note the impact of the reactor on the resulting model. Since the reactor does not use water at all, water mass balances around the reactor are not required. The reactor can also be excluded from the reuse scheduling as the operation of the reactor does not directly affect the reuse of water. However, task scheduling constraints are still required for the reactor as are raw material and product mass balances. 

Following the task scheduling constraints are the constraints dealing with the scheduling of the direct recycle/reuse of wastewater. 

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. 

Apart from the binary variable constraints given above, scheduling constraints are also required to ensure the transfer of water occurs at the correct time in the time... 

In the case where there is a central storage vessel, scheduling constraints also have to be derived to account for the timing of water entering and exiting the vessel relative to the overall operation. 

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... 

Constraints (10.1), (10.2), (10.8)-(10.14), in conjunction with the overall plant scheduling constraints, constitute a complete MILP formulation for direct heat integration in batch processes in a situation where the batch size is allowed to vary at different instances along the time horizon of interest. 

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