The proposed ratio (Z*k,sk/C*k,SR_max), where Z*k,SKj = Zk,SKj - Ck,FW and C*k,SR_max = Ck,SR_max - Ck,FW. helps to identify the source prioritisation as this ratio constitutes the constant term in expression (3). If the ratio is smallest for contaminant ’k’, then the constant term:\(F_{SK1}\left[1-\left(\frac{Z_{k,SK1}-C_{k,FW}}{C_{k,SR\_\max\ }-C_{k,FW}}\right)\right]\)has the largest value which means the source allocation is more likely to follow the prioritisation sequence for contaminant ’k’. The optimal source allocation should be done to minimise this term. However, this does not mean that the sink should follow exactly the source prioritisation sequence for contaminant ’k’, because different sources are traded-off by other contaminants. The proposed concentration ratio shows the limiting contaminant for a specific sink, and also show which source prioritisation order is likely to be followed.
Classification of sinks to Below or Above Pinch Regions
Based on the definition of Material Pinch Point, the material streams located below the Pinch Region require a fresh resource, while the one located above the Pinch Region does not require fresh resource and the unused sources are discharged. In other words, the Pinch Point separates the material sinks into two parts [27]:
For a multi-contaminant case, it is imperative to determine the prior classification of the sinks to Below or Above the Pinch Regions in each contaminant cascade. This is because the sequence of sink is needed to be determined so that the allocation of source is performed. The existence of multiple contaminants makes the identification difficult. To address this problem, the individual concentration ratio of the source to the sink (CSR/ZSK) plays an important role. The following heuristic for classification of sinks to above or below Pinch Region is used. For detailed explanation please refer to Chin et al. [27].
  1. Determine the ratio of shifted sink concentration (Z*k,SKj = Zk,SKj - Ck,FW) to the shifted source concentration (C*k,SRi = Ck,SRi - Ck,FW), i.e. Z*k,sk/C*k,SRi for each sink-source pair and for each contaminant ’k’, Where and
  2. For each sink ’j’,
  1. Identify the smallest value of the concentration ratio of sink ’j’ with all the sources.
  2. Identify which pairs have ratio \(\geq\)1 and which have <1.
  3. Calculate the number of sources (NSR+) that contribute to the ratio \(\geq 1\) and the number of sources (NSR-) that contribute to ratio <1.
  4. If the total flowrate of NSR+\(\geq\) total flowrate of NSR-, then sink ’j’ is classified as a sink that does not require fresh resource. Else, sink ’j’ is classified as a sink that requires fresh resource.
  5. Source allocation pattern for multi-contaminants

In the domain of single contaminant problems, the water sources are prioritised with the ascending order of concentration. This strategy is well-proven in the paper of El-Halwagi et al. [9], which means cleaner (higher quality) sources should be fully used up first. However, the source prioritisation becomes complicated when multiple contaminants are involved. The following sub-sections explain the source allocation characteristics for a multi-contaminant problem, and to infer the source allocation steps to be followed, to achieve the minimum freshwater target.
Consider the two-contaminant example presented in Table 1 below, and Example 1a features a problem with conflicting sources. If the only contaminant ’A’ is considered, SR1 is cleaner than SR2, while for contaminant ’B’, SR2 is cleaner than SR1. The prioritisation sequence of the sources is not obvious in this case. Example 1b features a problem with non-conflicting sources. It is apparent that SR1 and SR2 are not conflicting in this case, like CA,SR1> CA,SR2, and CB,SR1> CB,SR2, which means SR2 is preferred over SR1.
Table 1: A two-contaminant problem with both examples of conflicting and non-conflicting sources