For incompressible flow in circular conduits (pipes), the head losses can be

calculated using the Darcy-Weisbach equation

(A-1)

where *f *= friction factor (dimensionless)

Numerous other expressions have been proposed for calculating frictional head

losses. The Darcy-Weisbach equation is, however, the most fundamentally ap-

pealing as it can be derived analytically while the other relationships are empirical

in nature (Jeppson 1976). For laminar flow it is possible to show that the friction

factor *f *is a function of the Reynolds number alone. Unfortunately, the flow in heat

distribution piping is seldom laminar. For turbulent flow the friction factor has been

determined empirically to be a function of the Reynolds number and the relative

roughness of the pipe. A number of correlations have been proposed for the friction

factor. Those correlations that give the best agreement with the experimental data

are implicit in the friction factor. This renders them impractical for analyses such as

this one. For this analysis, and other applications, it would be desirable to have a

simple expression that would provide sufficient accuracy over limited ranges of

interest. To keep the expression as simple as possible, while allowing it to be an

accurate approximation, a method is developed here that yields a one-term power

function.

To approximate friction factor information in the form of implicit equations or

empirical data, we can develop our approximation using the least-squares method.

First, we assume a desired form for our expression for the friction factor

^

(A-2)

^

where *f *= predicted friction factor (dimensionless)

ε = absolute roughness of the piping (m)

If we assume that for any set of values for *RR *and *Re *we have an observed friction

factor *f*, we would like to minimize the sum of the squares between the *f *'s predicted

by our equation and all the observed *f *'s within the range of interest

()

^2

min ∑ *f *- *f *.

(A-3)

The summation is taken over all the observations available within the range of

interest for the parameters *RR *and *Re*. In the event that we are trying to approximate

an implicit empirical expression, we would choose incremental values of *RR *and *Re*

over the range of interest and use these to calculate a corresponding *f *value. This

approach will be illustrated later in this appendix. To accomplish the minimization,

we first convert to a linear form by making the following substitutions

(A-4)

79