consumer i = n, we have no constraints remaining that can be directed in the proper
sense to bound the ∆Pcv,n term in the objective. Note that because the assignment of
the numerical value for the consumer index i is arbitrary, its assignment will have
no impact on what we have shown here and that this result would hold for any set
of consumer indices.
Since we have n variables ∆Pcv,i in the objective, one for each of the n consumers,
and the objective is monotonically increasing in each of these variables, we must
have n constraints bounding the ∆Pcv,i from below. Above we have shown that at
most (n 1) of these constraints could result from eq 4-2. The only remaining
constraints on the ∆Pcv,i are the set formed by eq 4-5. Let's assume for the moment
that all consumers have the same minimum pressure differential requirement for
their control valves, i.e., that ∆Pcvm,i is the same for all i. Now we see that it must be
the consumer with the minimum value of ∆Pcv,i whose constraint from eq 4-5 is
active. This is true since a consumer with any greater value would cause at least one
of the other constraints from the set of eq 4-5 to be violated, i.e., if consumer i has the
minimum control value pressure loss
for all k ≠ i.
∆Pcv,k > ∆Pcv,i
And if for all i and k the minimum allowable control valve pressure losses are equal
∆Pcvm,i = ∆Pcvm,k .
Now, if the constraint for consumer i is active
∆Pcv,i > ∆Pcvm,i
then the constraint for consumer k can not be active
∆Pcv,k > ∆Pcvm,k .
We have already shown that (n 1) of the constraints from the set of eq 4-2
are active and can thus be treated as equalities. These (n 1) constraints
force the pressure loss summations to be equal for all n values of the consumer's
index i
∑ (
∆Ps + ∆Pr )j + ∆Pcv + ∆Phe =
∑ (
∆Ps + ∆Pr )j + ∆Pcv + ∆Phe
for all i ≠ k.
j
j
i
k
This means that we identify the consumer whose value of ∆Pcv,i is at its mini-
mum allowed ∆Pcvm,i by finding the consumer with the maximum value for the
quantity
∑ (
∆Ps + ∆Pr )j + ∆Pcvm,i + ∆Phe .
j
i
Once the pipe sizes are known, this quantity is easily calculated. This consumer who
has the minimum value of ∆Pcv,i is our so called "critical" consumer.
It follows then that there are only two cases where more than one of the
constraints from the set of eq 4-5 may be active:
1. In the case where the ∆Pcvm,i values are identical for all consumers, if the
maximum value of
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