ks = 1.3 W/m C
Hp = 1.0 m
∆xi = 0.050 m
ε = 5 105 m
a = 0.119 (dimensionless)
b = 0.152 (dimensionless)
c = 0.0568 (dimensionless)
Ts = 120C
Tr = 60C.
Application
For the application, we consider a main portion of a distribution system that
would serve a large number of consumers. We assume a maximum heating load,
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.
Solution
For the problem described above, we arrive at the following values for the
parameters in the objective function:
γ = 0.0231 (dimensionless)
A9 = .58 106/m
I1 = .56 104
I3 = 44.1 $ m5.095
A1 = 60/pump
A3 = .18 105.
The calculation of the above parameters is straightforward with the exception of
I3. The integral in the I3 parameter was evaluated numerically by a FORTRAN
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 Ct =
||content||
.11 106. In the following section, this result will be compared to one
obtained using a common design rule of thumb.
Comparison with a design based on a 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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