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]:
- Above the Pinch Point, the sinks do not require freshwater, i.e. the
total freshwater target, FFWT=0.
- Below the Pinch Point, the sinks require freshwater, i.e. the total
freshwater target, FFWT>0.
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].
- 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
- For each sink ’j’,
- Identify the smallest value of the concentration ratio of sink ’j’
with all the sources.
- Identify which pairs have ratio \(\geq\)1 and which have <1.
- Calculate the number of sources (NSR+) that contribute
to the ratio \(\geq 1\) and the number of sources
(NSR-) that contribute to ratio <1.
- 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.
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