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.

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*. C*pt = ∑ *C*hl + *C*pev + *C*pv

′

(5-10)

′

where *C*pt 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 *m*d+*c *L d (5+*b*+*c*) j - (∆*P*s + ∆*P*r )j ≡ < 0

b

2+*c * 1 *c*

*a *ε (4 / π)

˙2

ρ

(5-11)

(

)

with the monotonicities being dj- , ∆*P*s-j , ∆*P*r-j .

,

,

Notice that the constraints in the set of eq 5-5 will bound ∆*P*s,j and ∆*P*r,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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