Page 88 - Mechanical Engineers' Handbook (Volume 4)

11 Viscous Fluid Flow in Ducts 77

velocity profile also changes and becomes adjusted to a fixed shape. When these have reached
constant conditions, the flow is called fully developed flow.
The momentum equation for a pipe of diameter D gives the pressure gradient as
dp 4 d dV
V 2 V
0
dx D dx dx
which shows that a pressure gradient overcomes wall shear and increases momentum of the
fluid either as a result of changing the shape of the velocity profile (d /dx) or by changing
the mean velocity along the pipe (dV/dx is not zero for gas flows).
For fully developed incompressible flow
dp
p 4 0

dx L D
and a pressure drop simply overcomes wall shear.
For developing flow in the entrance, 1 initially and increases to a constant value
downstream. Thus, the pressure gradient overcomes wall shear and also increases the flow
momentum according to

dp 4 d
0 V 2
dx D dx
4
For fully developed flow, ⁄3 for laminar flow and
1.03 for turbulent flow in round
pipes.
For compressible gas flow beyond the entrance, the velocity profile becomes essentially
fixed in shape, but the velocity changes because of thermodynamic effects that change the
density. Thus, the pressure gradient is
dp 4 0 dV
V
dx D dx
Here is essentially constant but dV/dx may be significant.

11.1 Fully Developed Incompressible Flow
2
The pressure drop is
p (ƒL/D)( V /2) Pa, where ƒ is the Darcy friction factor. The
2
Fanning friction factor ƒ ƒ/4 and then
p (4ƒ /D)( V /2), and the head loss from
pipe friction is

p ƒ V 2 (4ƒ ) V 2
L
L
h m
ƒ
D 2g D 2g
The shear stress varies linearly with radial position, (
p/L)(r/2), so that the wall
2
2
shear is (
p/L)(D/4), which may then be written ƒ V /8 ƒ V /2.
0 0
A shear velocity is defined as v / V ƒ/8 V ƒ /2 and is used as a
0
normalizing parameter.
For noncircular ducts the diameter D is replaced by the hydraulic or equivalent diameter
D 4.4/P, where A is the flow cross section and P is the wetted perimeter. Thus, an
h
annulus between pipes of diameter D and D , D being larger, the hydraulic diameter is D 2
2
1
1
D .
1