Encyclopedia of Environmental Science and Engineering, Volume I and II

1276 WATER FLOW

where r  D /4 for flow at full bore. The use of Darcy’s

equation in the form given by Eq. (5) is sometimes extended

to open channel flow.

The determination of the friction factor, f, depends on

the flow regime, that is, whether the flow is laminar, critical,

transitional, smooth, turbulent or rough fully turbulent.

Laminar Flow Consider the mean pipe velocity, V, as

given by Hagen-Poiseuille’s equation for laminar flow:

V

g

m

SD

32

(6)

in which S  energy slope.

Combining Eq. (4) with Eq. (6) and noting that S  Hf/L,

g  rg, and nmr, the friction factor is given by:

f

e



64

R

, (7)^

where Re  nDg is the Reynolds number. Eq. (7) can be

used for all pipe roughness as the friction factor in lami-

nar flow is independent of the wall protuberances and is

inversely proportional to the Reynolds number. The energy

loss varies directly as the mean pipe velocity in laminar flow

which persists up to a Reynolds number of about 2000.

The velocity profile, which has a parabolic distribu-

tion, can be obtained from Hagen-Poiseuille’s equation. The

velocity, u, at any radius, r, of the pipe of diameter, D, is

given by:

u

SD



g

44 m

2

⎛ r 2

⎝⎜

⎞

⎠⎟

. (8)

At the centre line, the velocity is a maximum:

u

SD

max

g

m

2

16

(9)

The mean velocity is:

uu

SD

mean()max/^232.

g^2

m

(10)

Critical Flow From a Reynolds number of about 2000 and

extending to 4000 lies a critical zone where the flow may be

either laminar or turbulent. The flow regime is unstable and

no equation adequately describes it.

Smooth Turbulent For pipes fabricated from hydraulically

smooth materials such as copper, plexiglass and glass, the

flow is smooth turbulent for a Reynolds number exceeding

4000. The von Karman-Nikuradse smooth pipe equation is:

1

(^208010)
f
log (Rfe ).. (11)
Equation (11) indicates that the friction factor depends on
the fluid properties and deceases with increasing Reynolds
number.
Rough Fully Turbulent Nikuradse experimented with
pipes artificially roughened with uniform sand grains. The
results were fitted to the theory of Prandtl–Karman to give
the well known rough-pipe equation:
1
(^211410)
f
log (D/).. (12)
TABLE 1
Fluid properties
Fluid
Temperature

C
Mass density
kg/m^3
Specific weight
kN/m^3
Dynamic viscosity
N-s/m^3
Kinematic viscosity
m^2 /s
Surface tension
N/m
Water 0 1000 9.81 1.75  10 ^3 1.75  10 ^6 0.0756
— 5 1000 9.81 1.52  10 ^3 1.52  10 ^6 0.0754
— 10 1000 9.81 1.30  10 ^3 1.30  10 ^3 0.0742
Mercury 20 13,570 133.1 1.56  10 ^3 1.15  10 ^7 0.514
Sea water 20 1028 10.1 1.07  10 ^3 1.04 10 ^6 0.073
Moving
plate
Applied
force
V
dy
u+dy
u
du
u = 0 Stationary
plate
y
FIGURE 1 Fluid shear.
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