Water Hydraulics - Mathematics and Statistics for the Environment - page 599

where
h f = Head loss (ft).
f = Coefficient of friction.
L = Length of pipe (ft).
V = Mean velocity (ft/sec).
D = Diameter of pipe (ft).
g = Acceleration due to gravity (32.2 ft/sec 2 ).
Q = Flow rate, (ft 3 /sec).

The Darcy-Weisbach formula was meant to apply to the flow of any fluid, and into this friction
factor was incorporated the degree of roughness and an element known as the Reynolds number ,
which is based on the viscosity of the fluid and the degree of turbulence of flow. The Darcy-Weisbach
formula is used primarily for head loss calculations in pipes. For open channels, the Manning equa-
tion was developed during the later part of the 19th century. Later, this equation was used for both
open channels and closed conduits.
In the early 1900s, a more practical equation, the Hazen-Williams equation, was developed for
use in making calculations related to water pipes and wastewater force mains:

Q = 0.435 × CD 2.63 × S 0.54 (22.21)

where
Q = Flow rate (ft 3 /sec).
C = Coefficient of roughness ( C decreases with roughness).
D = Hydraulic radius R (ft).
S = Slope of energy grade line (ft/ft).

22.5.1 C F aCtor
As mentioned in Chapter 14, the C factor , as used in the Hazen-Williams formula, designates the
coefficient of roughness. C does not vary appreciably with velocity, and by comparing pipe types
and ages it includes only the concept of roughness, ignoring fluid viscosity and Reynolds number.
Based on experience (experimentation), accepted tables of C factors have been established for pipe
and are given in engineering tables. Generally, the C factor decreases by one with each year of pipe
age. Flow for a newly designed system is often calculated with a C factor of 100, based on averaging
it over the life of the pipe system.
Note: A high C factor means a smooth pipe; a low C factor means a rough pipe.

22.6 CHARACTERISTICS OF OPEN-CHANNEL FLOW
Basic hydraulic principles apply in open-channel flow (with water depth constant) although there is
no pressure to act as the driving force (McGhee, 1991). Velocity head is the only natural energy this
water possesses, and at normal water velocities it is a small value ( V 2 /2 g ). Several parameters can be
(and often are) used to describe open-channel flow; however, we begin our discussion by addressing
several characteristics of open-channel flow, including whether it is laminar or turbulent, uniform
or varied, or subcritical, critical, or supercritical.

22.6.1 l laminar and t urbulent F loW
Laminar and turbulent flows in open channels are analogous to those in closed pressurized conduits
(e.g., pipes). It is important to point out, however, that flow in open channels is usually turbulent. In
addition, laminar flow essentially never occurs in open channels in either water or wastewater unit
processes or structures.