ing function by converting to logarithmic variables and using a least-squares curve
fit was developed (Appendix A) to fit an equation of the form
f = a (ε/d)b Rec
where a, b and c = coefficients determined by curve fitting (dimensionless)
ε = absolute roughness of the piping (m)
Re = Reynolds number for the pipe flow (dimensionless).
As an example, the following coefficients are obtained over the parameter range
a = 0.119
b = 0.152
c = 0.0568
50 ≤ T ≤ 130
0.5 ≤ v ≤ 4.5
0.050 ≤ d ≤ 0.770
where T = water temperature (C)
v = flow velocity (m/s).
When compared to the Colebrook and White equation (Jeppson 1976), the
maximum error of this approximation is 6.9%, with the average error over the range
given being only 1.1%. If more accuracy is required, a much better result could be
obtained by narrowing the parameter ranges. Some examples of results for other
parameter sets are given in Appendix A. The coefficients will be carried for the
general case in the derivations following to allow for values obtained with other
By expressing the Reynolds number as a function of the quantities previously
used in the formulation, our equation for the friction factor becomes
f = a (4/π)c εb d(b+c) mc
where is dynamic viscosity (Pa s).
Now if we substitute this result into our expression for the cost of pumping
energy and simplify, we obtain
Cpe = I2d (5+b+c)
Ch A7 m3+c dt $ m 5+b+c
I2 = A12
η ˙ d PVFe
A7 = [(ρ 2 c)s + (ρ2c)r]/2 (m6+c sc/kg2+c)
A12 = a (4/π)2+c εb PVFe L (mb+1).