1. Proceed by first finding the set of servicing pipe segments unique to the critical consumer; this will

be his final service pipe only. Increase the diameter of that pipe segment until it reduces his pressure

loss to the same level as the consumer with the next highest pressure losses.

2. Now identify the pipe segments that these two consumers alone share and increase those pipe

diameters enough to reduce their pressure losses to the level of the next highest consumer. Note that

it may be that there are no shared pipe segments for these two consumers alone. In that event proceed

to the next step directly.

3. Again look for pipe segments shared by the three consumers with the highest pressure losses and

increase the diameters of those pipe segments enough to bring the pressure losses of these three

consumers to the level of the consumer with the fourth highest pressure loss. Once again, in the event

that no shared pipe segments exist, proceed directly to the next step.

4. Repeat this procedure until no consumers remain with pressure losses exceeding the constraints.

lowest cost up to that point in the process.

To address the instances where more than one alternative is available at a

particular step in either of the processes outlined in Figures 9 and 10, we would like

a strategy that minimizes cost. Let's investigate the effect of pipe diameter to see if

it would be to our advantage to choose smaller or large pipes as candidates for the

diameter increase.

First, we note that the capital cost *C*pv,j is a linearly increasing function of pipe

diameter. Thus, an incremental increase in pipe diameter would have the same effect

regardless of the absolute value of the pipe diameter.

The cost of heat loss *C*hl,j is a somewhat complicated function of the pipe diameter.

It also includes an approximation introduced in Chapter 2. Within the range of

validity of the approximation (0.025 m ≤ *d *≤ 1.0 m), we can see how the heat loss cost

behaves by examining its slope as shown in Figure 11.

The slope of the heat loss cost as plotted below in Figure 11 is essentially the first

term of eq 2-24 with the values of the parameters taken from the example of Chapter

3. From Figure 11 we see that the slope of the heat loss curve is always positive within

our range of interest. This tells us that whenever we increase the pipe diameter we

will increase heat losses, as we would expect. We also see that the slope is a

decreasing function of the diameter, except for pipe diameters over about 0.75 m,

where it becomes a slightly increasing function. For the portion of the range where

the slope is decreasing, we know that an incremental change in pipe diameter will

result in less increase in heat loss for larger diameters than for smaller ones.

The pressure loss as a function of pipe diameter is given by the sum of eq 5-8 and

5-9, which is our former eq 4-4

∆*P*s&r = (*a *εb (4/π)2+*c *A6 m d2+*c *L d (5+*b+c*))j.

˙

(4-4)

If we take the partial derivative of this pressure loss with respect to diameter, we

have

∆*P*s&r/ *d *= (5 + *b + c*) (*a *εb (4/π)2+*c *A6 m d2+*c *L d (6+*b+c*))j.

˙

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