APPENDIX A: APPROXIMATION OF THE FRICTION FACTOR
For incompressible flow in circular conduits (pipes), the head losses can be
calculated using the Darcy-Weisbach equation
hf = f L v2/2 g d
(A-1)
where f = friction factor (dimensionless)
L = pipe length (m)
v = flow velocity (m/s)
d = inside diameter of the pipe (m).
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
^
f = a RRb Rec
(A-2)
^
where f = predicted friction factor (dimensionless)
a, b, and c = coefficients determined by the least-squares method (dimensionless)
RR = ε/d = relative roughness of the pipe (dimensionless)
ε = absolute roughness of the piping (m)
Re = Reynolds number for the pipe flow (dimensionless).
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
Y = ln f
(A-4)
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