unnecessary. With the varying return temperature produced by our consumer
model, however, we will need to carry out this integration. Assuming that the cost
of heat Ch is constant, our new equation for I1 becomes
2/n1
(Ts + Ta )
2
]
(Tmg )0 8760
(
∫ A14 + A15 cos(2πt / 8760) / A13
I1 = A16
- Tm At +
dt
2(Ts - Ta ) 0
2
(3-26)
where A16 = PVFh L Ch 4πki ($/[C hr]).
Because the cosine function is raised to a non-integer power, it is not possible to
carry out the integration in eq 3-26 analytically. Once again we have used the
Romberg method of numerical integration to evaluate the integral. The calculation
of I1 was done using the FORTRAN program I1EQ3-26, which is included in
Appendix B. Using the parameter values assumed earlier in this section, we obtain
I1 = .33 104. Thus, we find that including the consumer model reduces the value
of the I1 parameter by 14.4% from 8.56 104 to 7.33 104.
Now that we have new values for the parameters that are affected by the
consumer model, we can recompute the optimal diameter for the sample application
given in Chapter 2. We proceed as before, i.e., 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 = 0.210 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. Guided by the value
obtained above, a simple trial-and-error method was once again used here. This
method yielded a solution to three significant digits with only four function
evaluations. The optimal diameter d was found to be 0.203 m. The total cost for this
design is found to be Ct =
||content||
.064 106 using eq 2-19. By coincidence, the optimal
diameter we have found also is one of the standard discrete diameter pipes
available; thus, it is not necessary for us to compute total costs for other discrete
diameters as before.
The addition of the consumer model has changed the optimal diameter from
0.208 to 0.203 m, a decrease of only 2.5%. The optimal discrete diameter remains
unchanged. While in this particular case, the inclusion of the consumer model had
no net effect on the choice of optimal discrete diameter, this obviously will not
always be the case.
The cost predicted for any pipe diameter is also changed slightly by the addition
of the consumer. The total cost with and without the consumer model is given in
Table 4. Pressure drops and costs for discrete pipe sizes under maximum flow
conditions with and without the consumer model (pipe data from Marks 1978).
Nominal
Inside diameter
Ct , w/o
Ct, with
∆Pd
pipe size
schedule 40
consumer model
consumer model
($ 106)
($ 106)
(in.)
(in.) (m)
(Pa/m)
--
8.187
0.208
340
1.111
1.065
8
7.981
0.203
384
1.112
1.064
10
10.020
0.255
120
1.178
1.140
12
11.938
0.303
50
1.305
1.267
30
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