PPf = frictional pumping power, exclusive of pump and pump driver
inefficiencies (W).
The first integral term represents the total cost of electrical energy input to drive
the pumps. The second integral term is the value of heat recovered in frictional
heating of the fluid. The actual pumping power and the fluid frictional portion are
related as follows
PPa = PPf /ηpηpm
(27)
where ηp = pump efficiency (dimensionless)
ηpm = efficiency of the motor driving the pump (dimensionless).
The pumping power of a closed system with return lines will not be affected by
elevation differences within a network and therefore they need not be considered
here. Elevation differences will, however, become a factor in determining the
absolute pressure level within a network. A constraint will arise owing to absolute
pressure limitations of the piping. This will be addressed later.
Now we assume that the product of the pump and motor efficiency can be
expressed as a function of the fraction of maximum volumetric flow. A similar
approach was used by Phetteplace (1981) based on data from Gartman (1970). This
gives an expression of the form
ηpηpm = Aη ( m /ρ) (ρd/ md )
˙
˙
(28)
where Aη =
empirical coefficient (dimensionless)
˙
m=
mass flow rate (kg/s)
˙
md =
maximum (design) mass flow rate (kg/s)
ρ=
fluid density (kg/m3)
ρd =
fluid density at design conditions (kg/m3).
The frictional pumping in the supply or return line is given by
PPf = 2 (2/π)2 f L ρ2 m3 d5
˙
(29)
where f is a friction factor (dimensionless).
Using the above expression for both the supply and return pipes, we substitute
the results, along with our earlier result for PPa, and our expression for PPf, into our
original expression for the pumping energy cost and simplify to obtain
Ceρ md
˙
PVFh
∫
Cpe = d5A11
Ch A8m3 fdt

(210)
˙
A mρ
˙ d
η
PVFe
yr
where A11 = (4/π)2 PVFe L (m)
(
)


A8 = ρs 2 + ρr 2 / 2 (s and r subscripts denote supply and return conditions
respectively) (m6/kg2).
Now we would like to find a simple function to approximate the friction factor f
over a range of interest. A simple power function relationship would be desirable to
keep the number of terms to a minimum and thus not complicate the above
expression further. The form of such a function is suggested by the dimensionless
groups of the Moody diagram (Jeppson 1976). A method of finding an approximat
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