2.7 Laminar and Turbulent Flow in Ducts
9
8
7
6
B = 5.5+2.5In(u k M
s * I
B = 8.5
transition
completely rough
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
u k
loglo --T
Fig. 2.35: Factor B (adapted from Schlichting, 1960)
75
3. fully rough regime when u.ks/v > 70. All the roughness elements reach
outside the laminar sublayer and the predominant part of the resistance
to flow is due to drag acting on them. Further increase of the Reynolds
number brings no change of flow pattern, and consequently factor B
remains constant, B = 8.5 (Fig. 2.35).
Finally, we return for a moment to an average velocity, ii, for laminar flow
given by Eq. (2.104). After some rearrangements, this equation becomes:
b.p
64/lJ ii 2
Pwg
iiPw D2 2g'
(2.115)
This equation expresses a relationship between the piezometric head and velocity head (see Sect. 2.3.5) and is valid only for laminar flow in smooth pipes.
However, for real fluid flow in non-smooth pipes, some pressure loss b.p or head
loss b.H can be expected due to drag at the pipe wall and shear in the turbulent motion itself. This is usually expressed by the Darcy-Weisbach equation
as:
b.p
I ii 2
b . H = - = j - -
Pwg
D 2g'
(2.116)
in which j is the friction factor. The head loss in Eq. (2.116) is proportional to
the length of pipe and to the quadratic flow velocity, as is expected. Therefore,