∆*x*i = 0.050 m

ε = 5 105 m

For the application, we consider a main portion of a distribution system that

including pipeline heat losses, of 25 MW. At the temperature difference specified

˙

above, this would require a "design" or maximum flow of md = 100 kg/s. We also

assume that the length of the pipeline is 1000 m. Note, however, that the length used

will not affect the diameter determined, as the length could be factored out of each

of the variable terms in the objective function eq 2-23, and the calculation would then

be done on a unit length basis. To arrive at a realistic total cost, which includes the

cost fixed with respect to *d*, the calculations here are for the system length specified

above. We also assume that only one pump is associated with the system.

For the problem described above, we arrive at the following values for the

parameters in the objective function:

γ = 0.0231 (dimensionless)

The calculation of the above parameters is straightforward with the exception of

program adapted from Ferziger (1981), which uses Romberg integration. The

program is included in Appendix B.

Before solving eq 2-24 to determine the optimum diameter, we first find an

approximate solution using eq 2-20, which neglects the heat losses. From eq 2-20 we

solve for the diameter directly, obtaining *d *= 0.216 m. Using this value of *d *as an

initial estimate, we can proceed to solve eq 2-24. We know that the solution to eq 2-

24, which includes heat losses, will be a smaller diameter than the solution to eq 2-

20, which does not include heat losses, since heat losses are an increasing function

of the diameter. Various "root finder" methods can be used to find the solution to

eq 2-24. Guided by the value obtained above, a simple trial-and-error method was

used here, which yielded a solution to three significant digits with several function

evaluations. The optimal diameter *d *was found to be 0.208 m. The total cost for this

design is *C*t =
||content||
.11 106. In the following section, this result will be compared to one

obtained using a common design rule of thumb.

Ideally, an analysis similar to the one above would be used to size all major district

heating pipes. In reality, however, most systems are designed on the basis of rules

of thumb that have evolved from practice. Although such rules of thumb may prove

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