Encyclopedia of Environmental Science and Engineering, Volume I and II

WATER FLOW 1277

Equation (12) states that beyond a certain Reynolds number,

when the flow is fully turbulent, the friction factor is influ-

enced only by the relative roughness,D and independent of

the Reynolds number.

Transition Flow Most commercial pipe flows do not

follow either the smooth pipe or rough pipe equations.

Colebrook and White proposed a transitional flow equation

which would be asymptotic to both:

1

2

37

251

10

f

eD

Rfe

log 

.

.

.

⎛ /

⎝

⎜

⎞

⎠

⎟ (13)

Equation (13) approaches the smooth pipe equation for low

and the rough pipe equation for high values of the Reynolds

number respectively. Unlike Nikuradse’s , which represents

the actual height of the sand grains, the  of Colebrook—

White’s equation is not an actual roughness dimension but a

representative height describing the roughness projections. It is

referred to as the equivalent sand-grain diameter since the fric-

tion loss it represents is the same as the equivalent sand-grain

diameter; Table 2 gives experimentally observed values:

TABLE 2

Equivalent sand-grain diameter

Pipe material  (mm)

Riveted steel 9.14

Rough concrete 3.05

Smooth concrete 0.31

Steel 0.05

Moody Diagram (Moody, 1944.) The Moody Diagram

(Figure 2) summarises and solves graphically the four fric-

tion factor equations Eqs. (7), (11), (12), (13) as well as

delineating the zones of the various flow regimes. The line

separating transitional and fully turbulent flow is given by

Rouse’s equation:

1

f^200

R

D

 e . (14)^

Mannning ’ s Equation The Manning equation, although

originally developed for open channel flow, has often been

extended for use in pressure conduits. The equation is usu-

ally favored for rough textured material (rough concrete,

unlike rock tunnels) and cross-sections that are not circular

(rectangular, horseshoe). It is most commonly given in the

form:

v

N

 RS

(^1) 23 12//
, (15)
in which N  roughness coefficient. Equation (15) can also
be transformed to:
hN
L
R
V
f19 6 2
2
43
2
.,/
g
(16)
S
N
AR

Q^22
243 /.
(17)
FIGURE 2 Pipe friction factors.
Friction factor f
Critical
Smooth pipes
0.01
0.02
0.03
0.04
0.05
0.06
103 104 105 106 107
Relative roughness
/D
Fully turbulent
0.0001
0.004
0.001
0.002
0.004
/D = 0.010
Reynolds number VD/ν
C023_003_r03.indd 1277C023_003_r03.indd 1277 11/18/2005 11:12:12 AM11/18/2005 11:12:12 AM