Environmental Engineering Reference
In-Depth Information
The mean velocity is given by:
R
2
8
D
p
D
x
v
mean
¼
Regarding the corresponding head
h
of the fluid as a measure for pressure,
it is also possible to write:
R
2
8
@
h
@x
v
mean
¼
n
Taking into account that the total flux through the pipe is known, one
obtains the classical result of Hagen
4
and Poiseuille
5
ð
R
prdr ¼
pR
4
8
D
p
D
x
Q ¼
v
x
ðrÞ
2
0
The formula was first experimentally developed by Hagen and by
Poiseuille independently. According to the Hagen-Poiseuille formulae there
is a linear relation between the flux
Q
and the pressure gradient
D
x,
and
between the characterising velocities and the pressure gradient. Such
a relationship is typical for situations in which the friction on solid walls is
a dominant process. A linear relation between velocity and pressure, or flux
and hydraulic head, holds not only in systems of pipes of small diameter but
for porous media flow in general. Darcy's Law for porous media states
exactly such a relation (see Chap. 11.3).
The validity of the given formulae is limited by the dimensionless
Reynolds-number of
Re ¼
D
p
/
2300, where the diameter is taken as characteristic
length and the mean velocity as characteristic velocity.
, the focus will be on analytical solutions, which can be
applied for special cases only. In the following subchapter cases will be considered
in which internal friction can be neglected. Thereafter special systems are in the
focus which are dominated by friction.
Using MATLAB
®
4
Gotthilf Heinrich Ludwig Hagen (1797-1884), German engineer.
5
Jean Louis Marie Poiseuille (1799-1869), French physician and physicist.
