Water reuse/recycle constraints

Scenario 4 Formulation for fixed water quantity with reusable water storage Constraints (4.18), (4.19), (4.3), (4.20), (4.16), (4.17), (4.21), (4.22), (4.23), (4.24), (4.25) and (4.26) together constitute a complete water reuse/recycle model for a situation in which the quantity of water in each water using operation is fixed. This is also a nonconvex MINLP for which exact linearization is not possible. 

The following constraints address the time dimension for water reuse/recycle in batch processes in the absence of central reusable water storage. 

The results for this scenario were obtained using GAMS 2.5/CPLEX. The overall mathematical formulation entails 385 constraints, 175 continuous variables and 36 binary/discrete variables. Only 4 nodes were explored in the branch and bound algorithm leading to an optimal value of 215 t (fresh- and waste-water) in 0.17 CPU seconds. Figure 4.5 shows the water reuse/recycle network corresponding to fixed outlet concentration and variable water quantity for the literature example. It is worth noting that the quantity of water to processes 1 and 3 has been reduced by 5 and 12.5 t, respectively, from the specified quantity in order to maintain the outlet concentration at the maximum level. The overall water requirement has been reduced by almost 35% from the initial amount of 165 t. 

The overall model for scenario 1, which is MILP, entails 1320 constraints, 546 continuous and 120 discrete/binary variables. 52 nodes were explored in the branch and bound algorithm and the optimal freshwater requirement of 1767.84 kg was reached in 1.61 CPU seconds. Figure 4.9 shows the corresponding water reuse/recycle network. 

The corresponding mathematical formulation entails 5534 constraints, 1217 continuous and 280 binary variables. An average of 4000 nodes were explored in the branch and bound search tree. The solution required three major iterations and took 309.41 CPU seconds to obtain the optimal solution of 1285.50 kg. This corresponds to 45.53% reduction in freshwater demand. A water reuse/recycle network that corresponds to this solution is shown in Fig. 4.11. 

Table 4.4 is the summary of the mathematical model and the results obtained for the case study. The model for scenario 1 involves 637 constraints, 245 continuous and 42 binary variables. Seventy nodes were explored in the branch and bound algorithm. The model was solved in 1.61 CPU seconds, yielding an objective value (profit) of 1.61 million over the time horizon of interest, i.e. 6 h. This objective is concomitant with the production of 850 t of product and utilization of 210 t of freshwater. Ignoring any possibility for water reuse/recycle, whilst targeting the same product quantity would result in 390 t of freshwater utilization. Therefore, exploitation of water reuse/recycle opportunities results in more than 46% savings in freshwater utilization, in the absence of central reusable water storage. The water network to achieve the target is shown in Fig. 4.14. 

This mathematical model is made up of two sets of constraints that are built within the same framework. One set of constraints focuses on the exploration of water reuse/recycle opportunities and the other on proper sequencing to capture the time dimension. Although this model has been presented in detail in Chapter 4, it is presented here in sufficient detail to facilitate understanding. 

Constraints (4.1) states that the inlet stream into any operation j comprises of the reuse/recycle streams from all water using operations / as well as the fresh water stream. As mentioned early on in this chapter, reuse refers to the use of an outlet water stream from operation j in another operation /, whereas recycle refers to the use of an outlet water stream from operation j in the same operation j. Constraint... 

Constraints (4.27) states that if water is recycled from operation j to operation / at a given time point p, then operation / should commence at time point p. However, the fact that operation / commences at time point p does not necessarily mean that there is a corresponding recycle/reuse stream at time point p. This is due to the fact that operation / could be using freshwater instead of recycle/reuse stream. Constraints (4.28) and (4.29) together ensure that water recycle/reuse from operation j to operation / coincides with the completion of operation j at time point p. Similarly, constraints (4.30) and (4.31) ensure that water recycle/reuse from operation jto operation/ coincides with the start of operation/ at time point p. Constraints (4.32) states that any operation j will start after the previous task in the same operation j is complete at time point p. Constraints (4.33) and (4.34) respectively state that if an operation j starts or ends at two distinct time points, then the later time... 

Constraints (4.35) and (4.36) stipulate that when the water stream is transferred from operation j to reusable water storage, then the time of transfer should coincide with the completion of operation j. however, operation j will only be completed and able to transfer water to storage at time point p if it started at time point p- 1. Also, the fact that operation j commenced at time point p- 1 does not necessarily mean that it will transfer water to storage at time point p, since this water could be immediately reused/recycled and/or dispensed with as effluent. This is captured by constraints (4.37). The following constraints (4.38), (4.39) and (4.40) are similar to (4.35), (4.36) and (4.37), but apply to the outlet stream of reusable water storage. 

Figure 4.9 shows that 1767.84 kg of freshwater is required over the 7.5 h time horizon. This corresponds to 25% reduction in freshwater requirement compared to the situation without water recycle/reuse. Although water from process A is at a relatively lower concentration of 0.1 kg salt/kg water, the time constraints in the absence... 

Constraints (5.1) states that the inlet stream into any operation j is made up of recycle/reuse stream, fresh water stream and a stream from reusable water storage. On the other hand, the outlet stream from operation j can be removed as effluent, reused in other processes, recycled to the same operation and/or sent to reusable water storage as shown in constraints (5.2). Constraints (5.3) is the mass balance around unit j. It states that the contaminant mass-load difference between outlet and inlet streams for the same unit j is the contaminant mass-load picked up in unit j. The inlet concentration into operation j is the ratio of the contaminant amount in the inlet stream and the quantity of the inlet stream as stated in constraints (5.4). The amount of contaminant in the inlet stream to operation j consists of the contaminant in the recycle/reuse stream and the contaminant in the reusable water storage stream. Constraints (5.5) states that the outlet concentration from any unit j is fixed at a maximum predefined concentration corresponding to the same unit. It should be noted that streams are expressed in quantities instead of flowrates, which is indicative of any batch operation. The total quantity of water used at any point in time must be within bounds of the equipment unit involved as stated in constraints (5.6). Following are the storage-specific constraints. 

Due to the discontinuous availability of wastewater for reuse, constraints have to be formulated that ensure the correct timing of water reuse when it occurs. Constraints (8.24) and (8.25) ensure that the time at which wastewater is produced and the time at which the wastewater is recycle/reused correspond to the same time. 

If the assumption that the contaminant mass in the wastewater is relaxed, then the additional raw material in the form of the contaminant mass has to be accounted for. The wastewater in this case not only supplements the water in the raw material, but also any other raw materials used in product formulation. The raw material balance given in constraint (8.1) is reformulated to account for the additional raw material source. Constraint (8.1) is split into a water balance and a raw material balance for the other components required in product formulation. The water balance is given in constraint (8.52). The balance for the other components used in the product formulation is given in constraint (8.53). Due to the fixed ratio of water and other components in product formulation and the fixed batch size, the amount of water and the amount of other components are fixed. Therefore, in constraints (8.52) and (8.53) the amount of water and amount of other raw material is fixed. The water balance, in constraint (8.52), states that the amount of water used in product is comprised of freshwater, water from storage and directly recycle/reused water. Constraint (8.53), the mass balance for the other components, states that the mass of other components used for product is the mass from bulk storage, the mass in directly recycled/reused water and the mass in water from storage. 

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. 

Constraints (4.38) and (4.39) state that when water stream is transferred from storage to any operation j for reuse, then the time of transfer must coincide with the start of operation j. Constraints (4.40) ensures that whenever a water stream is transferred from storage to operation j at time point p, then operation j must commence at time point p. However, operation j can start at time point p even if there is no reusable water stream transferred from storage, since water could be received from recycle/reuse and fresh water streams. 

The formulation for this scenario entails 1411 constraints, 511 continuous and 120 binary variables. The reduction in continuous variables compared to scenario 1 is due to the absence of linearization variables, since no attempt was made to linearize the scenario 2 model as explained in Section 4.3. An average of 1100 nodes were explored in the branch and bound search tree during the three major iterations between the MILP master problem and the NLP subproblem. The problem was solved in 6.54 CPU seconds resulting in an optimal objective of 2052.31 kg, which corresponds to 13% reduction in freshwater requirement. The corresponding water recycle/reuse network is shown in Fig. 4.10. 

The overall model for this scenario involves 5614 constraints, 1132 continuous 280 binary variables. Three major iterations with an average of 1200 nodes in the branch and bound search tree were required in the solution. The objective value of 1560 kg, which corresponds to 33.89% reduction in freshwater requirement, was obtained in 60.24 CPU seconds. An equivalent of this scenario, without reusable water storage, i.e. scenario 2, resulted in 13% reduction in fresh water. Figure 4.12 shows the water recycle/reuse network corresponding to this solution. 

Mass balance constraints (6.1), (6.3) and (6.5) need to be reformulated to account for the water from storage. The water into a unit in this case is not only comprised of freshwater and directly recycled/reused water, but also water from storage. This... 

Sequencing Constraints for Recycle/Reuse in the Absence of Reusable Water Storage... 

Sequencing Constraints that Associate Production Scheduling and Water Recycle/Reuse... 

The mass balances that are first considered are those that deal with the mass flow around a unit. The first of these is an inlet water balance, given in constraint (7.1). The water into a unit is the sum of the directly recycled/reused water, freshwater and water from the various storage vessels. The outlet water balance is presented in constraint (7.2). This constraint states that water leaving a processing unit is either directly recycled/reused, discarded as effluent or sent to one or more storage vessels... 

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... 

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