as this one. The Newton-Raphson method has been used in SUBROUTINE CWFF.
This method uses knowledge of the first derivative of the function to find the
solution. A description of the method can be found in nearly any reference on
numerical methods, such as Forsythe et al. (1977).
To use the Newton-Raphson method to solve the implicit Colebrook-White
equation, an initial estimate of the f value is needed. An explicit equation for the
friction factor f can be used for this. The explicit equation does not need to be
extremely accurate to yield a suitable first estimate. The equation given by Wood
(1966) is a good explicit relationship for turbulent flow and can be used. Wood's
equation is
(
)
0.44
1.62RR0.134
0.225
f = 0.094 RR
+ 0.53 RR + 88 RR
.
(A-18)
/ Re
To calculate the friction factor using either eq A17 or A18 requires that we know
the Reynolds number Re and the relative roughness RR. To calculate these param-
eters, we need to specify the fluid density and dynamic viscosity as well as the pipe
diameter and absolute roughness and the flow velocity. The fluid properties are a
function of the temperature of the fluid and to a lesser extent its pressure as well.
Here, we will assume that the fluid is at its saturation pressure for the temperature
specified. Two FORTRAN subroutines were written to determine the fluid proper-
ties. The first, SUBROUTINE SATLN, calculates the saturation pressure for water
given the temperature. The second, SUBROUTINE WTRTBL, calculates the density
and dynamic viscosity given the temperature and pressure. The main program
FFCONST and each of the subroutines mentioned above are included in Appendix
B.
Using the program FFCONST, a number of the constants a, b and c were
determined for several sets of parameters. Table A1 summarizes the results. In each
of the examples of Table A1 the absolute roughness of the pipe was taken as 4.6
105 m.
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