Equation 3-9a, expressed in U.S.
customary units, is When R is very large (greater than about 105), the
flow is fully turbulent and /depends only on rough-
., - (IWOO®' V" . ( »;2e)" (3-9b) S ness. In the transition zone between turbulent and
laminar flow, both roughness and R affect/, which can
be calculated from a semianalytical expression devel-
where hf is feet per 1000 ft, Q is gallons per minute, D oped by Colebrook [4]:
is pipe diameter in inches, and C, again, is given in
Table B-5 (Appendix B).
210 3 7 ( }
See Subsection "Friction Coefficients" for a dis-
cussion of the H-W C factor.
1
jf- - H - v/J
Oi (zlV , 2.5l} n 1T.
where e is the absolute roughness in millimeters (or
inches or feet) and D is the inside diameter in millime-
Darcy-Weisbach Equation ters (or inches or feet), so that z/D is dimensionless.
The Moody diagram, Figure B-I (Appendix B), was
The equation for circular pipes is developed from Equation 3-12 [5]. Note that the
curves are asymptotic to the smooth-pipe curve (at the
* = /Z^ (3-10) left). To the right, curves calculated from the Cole-
brook (also called Colebrook-White) equation are
where h is the friction headloss in meters (feet), /is a indistinguishable from the horizontal lines for fully
coefficient of friction (dimensionless), L is the length developed turbulent flow given in Prandtl [6]. The
of pipe in meters (feet), D is the inside pipe diameter probable variation of / for commercial pipe is about
in meters (feet), v is the velocity in meters per second ±10%, but this variation is masked by the uncertainty
(feet per second), and g is the acceleration of gravity, of quantifying the surface roughness.
9.81 m/s2 (32.2 ft/s2). The advantages of the Darcy- An explicit, empirical equation for /was developed
Weisbach equation are as follows: by Swamee and Jain [7]:
• It is based on fundamentals. / = — (3-13)
2
• It is dimensionally consistent.
• It is useful for any fluid (oil, gas, brine, and sludges). , (e/D 5.14]
10gl
• It can be derived analytically in the laminar flow {37 + ^J
region.
• It is useful in the transition region between laminar The value of /calculated from Equation 3-13 differs
flow and fully developed turbulent flow. from /calculated from the Colebrook equation by less
• The friction factor variation is well documented. than 1%.
The disadvantage of the equation is that the coeffi- Friction headloss can be determined from the
cient / depends not only on roughness but also on Darcy-Weisbach equation in a number of ways:
Reynolds number, a variable that is expressed as • Use one of the Appendix tables (B-I to B -4) to find
the appropriate pipe size. Compute R, find /from the
R = v-» (3-11) Moody diagram, and compute an accurate value of h
V from the Darcy-Weisbach equation. Because /
where R is Reynolds number (dimensionless), v is changes only a little for large changes of R, no second
velocity in meters per second (feet per second), D is trial is needed. Compare the h so obtained with the
the pipe ID in meters (feet), and V is kinematic viscos- value in Tables B-I to B -4 for an independent check.
ity in square meters per second (square feet per sec- • Program Colebrook's Equation 3-12 to find /as an
ond) as given in Appendix A, Tables A-8 and A-9. iterative subroutine for solving Equation 3-10 with
a computer. Once programmed (a simple task even
Determination of f for a hand-held, card-programmable calculator),
any pipe problem can be solved in a few seconds.
In the laminar flow region where R is less than 2000,/ • Use the Swamee-Jain expression for / in the
equals 64/ R and is independent of roughness. Darcy-Weisbach equation. Equation 3-13 could
Between Reynolds numbers of 2000 and about 4000, even be used as a first approximation for iteration
flow is unstable and may fluctuate between laminar of Equation 3-12.
and turbulent flow, so / is somewhat indeterminate. • Refer to the extensive tables of flow by Ackers [8].
