practice, since pumping energy, inherently more expensive than heat energy, is
being converted into frictional heating of the fluid. As noted at the close of the last
chapter, there is an alternative to these high control valve pressure losses: reduce the
pipe sizes further such that the pressure differential at the consumer's control valve
is reduced. Such a practice was proposed by DFF (1985), where they suggest
reducing the size of the "service pipes," those ultimately connecting the consumer
to the network. It may also be possible to reduce some of the pipes sizes within the
network as well. For our example problem, we see that we have limited options.
Consumer 1 is our critical consumer, so we cannot reduce any of the pipe sizes
servicing this customer; this rules out the pipe segments (6,1), (5,6) and (8,5). The
remaining pipe segments are (7,3), (6,7), (5,4) and (7,2). Thus, we investigate the
possibility of reducing the size of these pipes.
First, let's look at pipe segment (5,4). This is the only pipe segment serving
consumer 4; thus, this is the only option for reducing the pressure loss in this
consumer's control valve. What we would like to do is find the minimum pipe size
that will not violate the constraint of eq 4-2 for consumer 4. Effectively, what has
happened here is that we have removed the pumping power term from the objective
function so it now becomes monotonically increasing in d(5,4). We need to find the
constraint that will bound d(5,4) below. While not immediately obvious, eq 4-2 forms
such a constraint on d(5,4) when directed as follows
(∆P (
).
h1 = ∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe - ∆Php ≡ < 0
+
+
s, 5,4) , ∆Pr,(5,4)
j
From eq 4-4, we see that ∆Ps,(5,4) and ∆Pr,(5,4) are related to d(5,4) by
h2 = {[(ρ1 c)d,s + (ρ1 c)d,r]
[(a/2) εb (4/π)2+c md+c L d(5+b+c)](5,4)} (∆Ps)(5,4) (∆Pr)(5,4) ≡< 0
˙2
(
)
with the monotonicities being h2 = d(5,4) , ∆Ps(5,4) , ∆Pr(5,4) .
,
,
So, we see that d(5,4) is bounded below by h2 and that the non-objective variables
∆Ps,(5,4) and ∆Pr,(5,4) are bounded below by this constraint and above by h1,, as
required by the second monotonicity principle (see Papalambros and Wilde 1988).
Now we can use constraints h1 and h2 to find the optimal value of d(5,4). To do so we
treat h1 as a strict equality and solve for (Ps)(5,4) + (Pr)(5,4). We then substitute the
result into h2, again treating it as a strict equality, and solve for d(5,4). The result is
d(5,4) = 0.0614 (m) .
The discrete diameters that bracket this value are 0.0545 and 0.0703 m. The lower
bracketing discrete diameter will cause constraint h1 to be violated since a decrease
in d(5,4) will increase ∑ (∆Ps + ∆Pr ) . The optimal discrete diameter determined previ-
j
j
ously was 0.0703 m; thus, we are unable to improve on this result.
Let's look at the remaining pipe segments (7,3), (6,7) and (7,2). These pipe
segments serve both consumers 2 and 3. Consumer 2 is served by pipe segments (6,7)
and (7,2) and consumer 3 is served by pipe segments (7,3) and (6,7). Both consumers
are served by pipe segment (6,7); thus, any decisions we make about this pipe
segment must be checked to ensure that both consumer constraints (eq 4-2) are
obeyed. Also, notice that if we decrease one of the pipe sizes and this violates
constraint h1, we may be able to increase the other pipe size in the pair serving that
consumer such that the total costs for the pipes and heat losses are reduced but
constraint h1 is still satisfied. It is also possible that a pipe size could be reduced or
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