Water reuse/recycle constraints requirement

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. 

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... 

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 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. 

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