COST OF PUMPING-continue

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

(2-7)

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 )

˙

(2-8)

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

˙

(2-9)

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 = d-5A11

Ch  A8m3 fdt

-

(2-10)

˙

 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-

7