Overall graphical design
procedure
For a more general ’k’ contaminants case and many sources available. The
ideal case of source allocation depends on the number of contaminants.
For 2 contaminants, it should be paired with 2 sources (3 equations:
Sink mass balance and both impurity constraints with 3 unknowns:
FFw, FSR1, FSR2). For
’k’ contaminants, then pair with ’k’ sources (’k’ equations with ’k’
unknowns). The following Source Allocation Step is presented to identify
the optimal source flowrates for a specific sink:
- For a specific sink, check whether the Pinch Point occurs at the CCs
that correspond to the main limiting contaminant. The main limiting
contaminant is the one producing the smallest ratio of sink’s
concentration to the maximum source concentration (see Section 2.2.1).
If the Pinch is at the limiting contaminant, go to step (vi), else
continue to step (ii).
- For a sink, if there are ’k’ contaminants, then pair with ’k’ sources
that are arranged in the order based on the contaminant cascade that
the sink is assigned to.
For example, if SK1 is assigned to contaminant cascade ’A’, then pair
with the first ’k’ sources that are arranged with ascending order of
concentration ’A’. This is explained in the next point.
Set \(F_{Fw,SK1}\) for contaminant ’k1’ = \(F_{Fw,SK1}\) for
contaminant ’k2’ for each binary contaminant pairs minus 1
(kC2 – 1). Determine the flowrates
of the first ’k’ sources using Eq(3) by using one of the sources as a
reference source.
- For example, if there are 3 contaminants, i.e. k={A, B, C}, then set
a condition \(F_{Fw,SK1}=F_{Fw,SK1}\) for each binary pairs, i.e.
A-B, B-C, A-C. By setting \(F_{Fw,SK1}\) for A = \(F_{Fw,SK1}\) for B
and \(F_{Fw,SK1}\ \)for B = \(F_{Fw,SK1}\) for C, this means the
freshwater requirement for A is set to equal to for C as well. Setting
the freshwater conditions is sufficient for 2 of the pairs
(3C2 – 1).
- For 3 contaminants case, solve the system of equations with Eq(5) for
both A-B and B-C pairs. If SR1, SR2 and SR3 are the first 3 sources in
the cascade, determine the flowrate of SR1 and SR2 by setting SR3 as a
reference source. This is to achieve the limits for 3 contaminants. As
there are 3 equations (1 sink mass balance equation and 2 Eqs(5) for
two of the contaminant pairs) for 3 contaminants, to ensure zero
degrees of freedom, 3 sources are needed.
- Repeat this by setting each source as the reference source until all
the flowrates have been identified.
- If one of the source flowrates is determined as negative, this means
the scenario where all contaminants have Pinch Points does not exist.
If this is the case, repeat the procedures (i) to (iv) for ’k-1’
contaminants. If not, go to step (vi)
- Check if the sources flowrates are enough to cover the sink. If one of
the sources has not enough flowrates, fully use up the source. Repeat
the procedures (i) to (iv) with the next ’k’ sources.
For example, for the 3 contaminants example, if the optimal flowrates
of SR1 exceed the available SR1, fully use up SR1 for the sink. Repeat
the similar procedures using the next three sources, i.e. SR2, SR3 and
SR4.
Repeat procedures (i) to (v) until all sinks are fulfilled
Based on these observations, the following overall design procedures
using graphical methodology with Pinch concept are proposed:
- Assign the sink to proper contaminant cascade. This can be done by
determining the ratio of sink’s concentration to the maximum source
concentration (see Section 2.1)
- For each sink, determine whether it should be Below or Above the
Pinch.
- Start the design procedures with the contaminant cascade that contains
the sink that has the highest flowrate.
- Arrange the sink in the contaminant cascade with ascending order of
the concentration of contaminant based on the cascade it has been
assigned to.
- Perform the Source Allocation Steps for the sinks that are Below the
Pinch.
- Repeat the procedures (iii) to (v) until the sinks (Below the Pinch)
for the next contaminant cascade.
- Allocate the remaining sources to the sinks that are Above the Pinch.