′

(2-19)

′

where Ct = *C*t [(1+ *PVF*m&rAm&r)(*A*1np+ *A*3L)]

′

Minimizing Ct with respect to *d *is, of course, equivalent to minimizing the

′

original total cost function *C*t. Therefore, we can work with Ct for convenience. If

we neglect the first term, which represents the cost of heat losses, we have a

geometric programming problem (Papalambros and Wilde 1988) with zero degrees

of difficulty. Without specifying the parameter values, we see from inspection that

the weights of the two remaining terms will be

With heat losses neglected, at the optimum pipe diameter the variable costs

associated with pumping are 1/(6 + *b + c*) of the total variable costs. The variable

costs attributable to pipe capital and maintenance costs are the remaining portion.

Here, the variable costs represent that portion which is a function of our decision

variable, the pipe diameter. Also note that the pumping costs include the variable

portion of the capital cost of the pumps and the maintenance associated with that

portion, as well as the pumping energy costs.

Considering a more specific case, if the values of parameters *b *and *c *found in the

example given for eq 2-11 (*b *= 0.152, *c *= 0.0568) are used, we find the following

values for the weights

These results vary very little over the range of values found for *b *and *c *in

Appendix A. Thus, when heat losses are neglected, we find this very simple solution

is applicable in most cases.

Once values for the remaining parameters are known, the pipe diameter is found

by using the equations given above and the two terms of the objective function

remaining. The resulting expression is

(2-20)

It should be noted that this solution obtained using geometric programming theory

also could have been easily obtained using classical differential methods, as used

later. The advantage of the geometric programming method is that it ensures that a

a local extremum and require the evaluation of second order terms to determine the

nature of the extremum, i.e., maximum or minimum.

To arrive at this simple expression for the pipe diameter, we have neglected the

heat losses. Because the cost of heat losses will always be greater than zero, we have

constructed a lower bounding problem for our original problem by neglecting this

cost, i.e.

The cost of any design that includes heat losses can never be less than the same

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