expanded objective would be if we also wanted to determine the optimal size for the

consumer's heat exchanger equipment, which would require including these costs

in the objective as well. Here, however, we will not address these additional issues.

With these limitations in scope our objective function becomes

(

)

+ *C*pvc +

min. *C*st = *C*fixed + ∑ *C*hl + *C*pev + *C*pv

(5-1)

(

)

∑ ∫ ∆*P*cv,i + ∆*P*he,i (mi /ρr ) d*t*

˙

ηp ηpm * i yr*

where *C*st = total system cost ($)

this portion of their costs ($)

that portion of pipe cost ($)

to piping pressure losses, and the maintenance and repair on that

portion of pump costs ($)

sumer ($).

Notice that the density used in the last term of this equation is the taken at the

return condition. This is done because the pumps are usually located on the return

side of the system at the heating plant. The cost of pumps, which was previously

lumped with the piping cost, has been broken out as a separate cost since the number

of pumps will be discrete for the system.

˙

In general, the mass flow rate for any consumer mi and the pressure losses at the

consumer (∆*P*cv + ∆*P*he)i will be functions of time. Previously, we assumed that the

mass flow rate over the yearly cycle was given by eq 3-25. Since the pressure loss in

the consumer's heat exchanger ∆*P*he,1 is a function of mass flow rate, as given by eq

4-6, it will also be a function of time. As we will show later, the pressure loss in the

consumer's control valve ∆*P*cv,i will be used to "balance" the network. Hence, it will

become a function of time in most all cases as well. We will have some choices as to

the best way to balance the network using the consumer's control valve, as will also

be shown later.

In eq 5-1 we have separated the cost of the pumps into the fixed costs, that portion

which does not depend on pump capacity, and the variable costs, which are

attributable to either pressure losses at the consumer or in the piping network. We

have also separated the fixed portion of the pipe cost as well from that portion that

depends on pipe diameter. Effectively, this does not change our objective function

as far as terms that contain the pipe diameters are concerned, since the fixed costs

of the pipes and pumps are not considered in determining the optimum pipe

diameter, as can be seen from eq 2-19. For a multiple pipe system, these fixed costs

are

(5-2)

The variable cost of pumps attributable to pressure losses at the consumer *C*pvc

will be determined by the pressure losses and flow rate at the design condition. It is

this condition for which the pressure difference between the supply and return at the

heating plant, as well as the mass flow rate, are greatest. Thus, the pumps must be

sized for this condition. This portion of the pump cost will be given by

[

]

(5-3)

˙

d

43

Integrated Publishing, Inc. |