hydrodynamic losses in the piping always being lower for the lower flow rates and
thus the total pressure increase necessary at the plant is reduced.
A simplified objective function
If all the constraints are shown to be satisfied, then we have found the optimal
solution to the multiple-consumermultiple-pipe problem and we need not do any
further calculations. If, however, we find that constraints have been violated, we
need to refine our design. The first reaction of the designer when faced with this
result should be to closely examine the nature of the constraint that has been
violated. Often these constraints are "soft" and may be changed if the optimum
design indicates so. For example, in this problem one such constraint would be the
maximum allowed pressure. The designer has the option of using a higher pressure
class of piping if he or she feels it is warranted. This, of course, will most likely add
to the cost and, if this additional cost is significant, the designer may choose to
evaluate the design subject to the original constraint and the revised constraint, as
well to determine which one yields the lowest cost when the additional cost of the
higher pressure class piping is included.
Now that we have shown activity for some of the constraints, let's consider the
problem again with a reduced objective and determine if solution is possible. Thus,
in our reduced objective, we are only interested in the terms in the objective function
that relate to variable piping costs, since we have shown that constraint activity
determines the values of the other decision variables. Thus, our problem can be
restated as
(
)j
min. Cpt = ∑ Chl + Cpev + Cpv
′
(5-10)
j
′
where Cpt is the total diameter variable pipe costs for the system ($).
The constraints to which this solution is subject are that the absolute pressure
levels not be exceeded. The activity of the constraints of eq 4-2 and 4-5 fixes the
pressure increase at the heating plant and therefore the pumping power for the
system. Thus, at this point we no longer need to include the pumping power
consumed in the piping in our reduced objective function, eq 5-10. If we remove the
pumping power from eq 5-10, it becomes a monotonically increasing function of
pipe diameter. Thus, for the problem to be bounded, we must have monotonically
decreasing constraints on the pipe diameters. The sum of eq 5-8 and 5-9 forms one
such equality constraint for each pipe diameter that can be directed to bound it
properly. This constraint is
(
)d,s + (ρ1 c )d,r md+c L d (5+b+c) j - (∆Ps + ∆Pr )j ≡ < 0
b
2+c 1 c
a ε (4 / π)
˙2
ρ
(5-11)
(
)
with the monotonicities being dj- , ∆Ps-j , ∆Pr-j .
,
,
Notice that the constraints in the set of eq 5-5 will bound ∆Ps,j and ∆Pr,j in the
opposite sense to this constraint, so these nonobjective variables are bounded above
and below as required by MP2.
Thus, our problem is now to use these active constraints to solve for the diameters.
We will have one constraint from eq 5-11 for each pipe segment in the system. In
addition, we have already shown that we have (n 1) active constraints from eq 4-
2, where n is the total number of consumers. We have one additional active
constraint from eq 4-5 for the critical consumer. However, at this point we are unsure
as to which consumer is the critical consumer; thus, we must include all n of the
constraints from the set of eq 4-2. In addition, we would still need to include all of
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