practice, since pumping energy, inherently more expensive than heat energy, is

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 *(

).

+

+

s, 5,4) , ∆*P*r,(5,4)

From eq 4-4, we see that ∆*P*s,(5,4) and ∆*P*r,(5,4) are related to *d*(5,4) by

[(*a*/2) εb (4/π)2+*c *md+*c *L d(5+*b+c*)](5,4)} (∆*P*s)(5,4) (∆*P*r)(5,4) ≡< 0

˙2

(

)

with the monotonicities being h2 = *d*(5,4) , ∆*P*s(5,4) , ∆*P*r(5,4) .

,

,

So, we see that *d*(5,4) is bounded below by *h*2 and that the non-objective variables

∆*P*s,(5,4) and ∆*P*r,(5,4) are bounded below by this constraint and above by *h*1,, as

required by the second monotonicity principle (see Papalambros and Wilde 1988).

Now we can use constraints *h*1 and *h*2 to find the optimal value of *d*(5,4). To do so we

treat *h*1 as a strict equality and solve for (*P*s)(5,4) + (*P*r)(5,4). We then substitute the

result into *h*2, again treating it as a strict equality, and solve for *d*(5,4). The result is

The discrete diameters that bracket this value are 0.0545 and 0.0703 m. The lower

bracketing discrete diameter will cause constraint *h*1 to be violated since a decrease

in *d*(5,4) will increase ∑ (∆*P*s + ∆*P*r ) . The optimal discrete diameter determined previ-

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 *h*1, 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 *h*1 is still satisfied. It is also possible that a pipe size could be reduced or

67