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 *C*h is constant, our new equation for *I*1 becomes

2/*n*1

(Ts + *T*a )

2

]

(*T*mg )0 8760

(

∫ *A*14 + *A*15 cos(2π*t */ 8760) / *A*13

- *T*m *A*t +

d*t*

2(*T*s - *T*a ) 0

2

(3-26)

where *A*16 = *PVF*h L Ch 4π*k*i ($/[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 *I*1 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

of the *I*1 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

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 *C*t =
||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

Ct , w/o

Ct, with

∆Pd

--

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