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164 Chapter 5 Water Hydraulics, Transmission, and Appurtenances
In the preceding expressions, f, n, and m are the friction or roughness factors determined
by hydraulic experiments using water only. Of the above hydraulic equations, Robert
Manning’s expression is commonly used for both open channels and closed conduits.
Combining Eqs. 5.18 and 5.22 will give the following Manning formula using U.S.
customary units:

v (1.486>n)(r) 0.67 (s) 0.5 (U.S. Customary Units) (5.25)

where v velocity, ft/s; n coefficient of roughness, dimensionless; r hydraulic radius,
ft; and s slope of energy grade line, ft/ft.
Equation 5.26 is the equivalent Manning formula using SI units:
v (1>n)(r) 0.67 (s) 0.5 (SI Units) (5.26)

where v velocity, m/s; n coefficient of roughness, dimensionless; r hydraulic ra-
dius, m; and s slope of energy grade line, m/m.
For a pipe flowing full, the hydraulic radius, Eq. (5.19), becomes
2
r [( >4) D ]>( D) D>4 (5.27)
where r hydraulic radius, ft or m; and D pipe diameter, ft or m.
Substituting for r into Eqs. (5.25) and (5.26), the following Manning equations are ob-
tained for practical engineering designs for circular pipes flowing full:

v (0.59>n)(D) 0.67 (s) 0.5 (U.S. Customary Units) (5.28)
Q (0.46>n)(D) 2.67 (s) 0.5 (U.S. Customary Units) (5.29)
3
where Q flow rate, ft /s; v velocity, ft/s; D pipe diameter, ft; s slope of energy
grade line, ft/ft; and n roughness coefficient, dimensionless. For SI measurements:
v (0.40>n)(D) 0.67 (s) 0.5 (SI Units) (5.30)
Q (0.31>n)(D) 2.67 (s) 0.5 (SI Units) (5.31)

3
where Q flow rate, m /s; v velocity, m/s; D pipe diameter, m; s slope of energy
grade line, m/m; and n roughness coefficient, dimensionless.
The Hazen-Williams formula is most widely used in the United States to express flow
relations in pressure conduits or conduits flowing full, the Manning formula in free-flow
conduits or conduits not flowing full. The Hazen-Williams formula, which was proposed
in 1905, will be discussed in this chapter; Manning’s formula is taken up in Chapter 13 on
hydraulics of sewer systems.
The following notation is used for the U.S. customary units: Q rate of discharge,
3
gpm, gpd, MGD, or ft /s as needed; d diameter of small circular conduits, in.; D
2
2
diameter of large circular conduits, ft; v mean velocity, ft/s; a D >4 d >576
2
cross-sectional area of conduit, ft ; r a/wetted perimeter D>4 d>48 hydraulic
radius, ft; h loss of head, ft; L conduit length, ft; and s h >L hydraulic gradient,
f
f
dimensionless.
3
There is another set of notation to be used for the SI units: Q rate of discharge, m /s;
d diameter of small circular conduits, mm; D diameter of large circular conduits, m;
2
v mean velocity, m/s; a cross-sectional area of conduit, m ; r hydraulic radius, m;
h loss of head, m; L conduit length, m; and s h >L hydraulic gradient, dimen-
f
f
sionless.