Page 89 - Mechanical Engineers' Handbook (Volume 4)
P. 89
78 Fluid Mechanics
11.2 Fully Developed Laminar Flow in Ducts
The velocity profile in circular tubes is that of a parabola, and the centerline velocity is
2
p
R
u max
L 4
and the velocity profile is
1 2
u r
u max R
where r is the radial location in a pipe of radius R. The average velocity is one-half the
maximum velocity, V u max /2.
The pressure gradient is
p 128 Q
L D 4
which indicates a linear increase with increasing velocity or flow rate. The friction factor
for circular ducts is ƒ 64/Re or ƒ 16/Re and applies to both smooth as well as
D
D
rough pipes, for Reynolds numbers up to about 2000.
For noncircular ducts the value of the friction factor is ƒ C/Re and depends on the
duct geometry. Values of ƒ Re C are listed in Table 8.
11.3 Fully Developed Turbulent Flow in Ducts
Knowledge of turbulent flow in ducts is based on physical models and experiments. Physical
models describe lateral transport of fluid as a result of mixing due to eddies. Prandtl and
´
von Karman both derived expressions for shear stresses in turbulent flow based on the Reyn-
´
olds stress ( u v ) and obtained velocity defect equations for pipe flow. Prandtl’s
equation is
u u u u R
max max
2.5 ln
/ v y
0
where u max is the centerline velocity and u is the velocity a distance y from the pipe wall.
´
´
von Karman’s equation is
u max u u max u
/ v
0
1 1
y
y
1
ln 1 R R
In both, is an experimentally determined constant equal to 0.4 (some experiments show
better agreement when 0.36). Similar expressions apply to external boundary layer flow
when the pipe radius R is replaced by the boundary layer thickness . Friction factors for
smooth pipes have been developed from these results. One is the Blasius equation for Re D
5
10 and is ƒ 0.316/Re 1/4 obtained by using a power-law velocity profile u/u max
D
1/7
(y/R) . The value 7 here increases to 10 at higher Reynolds numbers. The use of a log-
arithmic form of velocity profile gives the Prandtl law of pipe friction for smooth pipes:
