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
(
)
+ Cpvc +
min. Cst = Cfixed + ∑ Chl + Cpev + Cpv
j
(5-1)
(
)
∑ ∫ ∆Pcv,i + ∆Phe,i (mi /ρr ) dt
Ce
˙
ηp ηpm i yr
where Cst = total system cost ($)
Cfixed = fixed cost of pipes and pumps, and the maintenance and repair on
this portion of their costs ($)
Cpv = diameter variable cost of pipes and the maintenance and repair on
that portion of pipe cost ($)
Cpev = diameter variable cost of pumps and pumping energy attributable
to piping pressure losses, and the maintenance and repair on that
portion of pump costs ($)
Cpvc = variable cost of pumps attributable to the pressure losses at the con-
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 (∆Pcv + ∆Phe)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 ∆Phe,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 ∆Pcv,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
Cfixed = (1 + PVFm&r Am&r ) (A1np + A3 ∑ Lj ) .
(5-2)
j
The variable cost of pumps attributable to pressure losses at the consumer Cpvc
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
[
]
Cpvc = A2 (1+PVFm&rAm&r) ∑ (∆Pcv,i + ∆Phe,i )(mi /ρr ) .
(5-3)
˙
i
d
43
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