Mechanics of Fluids

value in eqn 12.7 gives
that is,
where
�
��
�
c = � 1
ϱ � 1/K + d/(tE) �
�
c =
�
K ′
Q
1 1 d
= +
K ′ K tE
(12.8)
Here K ′ may be regarded as the effective bulk modulus of the liquid when
in the pipe. (As mentioned earlier, the transmission of elastic waves also
occurs through the material of the pipe walls, but the effect of these waves
is normally negligible.)
In cases where the longitudinal movement of the pipe is restrained, as
for example in the fuel-injection system of a diesel engine (where pressure
transients have particular significance), the effective bulk modulus of the
liquid is slightly different. It may be calculated by again determining the
appropriate value of δA/A. However, even if all longitudinal movement of
the pipe is prevented, the resulting wave velocity is little affected, and, for a
thin-walled pipe, eqn 12.8 is sufficiently accurate for most purposes.
A few figures will show the orders of magnitude. A representative value
of K for water is 2.05 GPa and so the wave velocity in a rigid pipe is
�
� �
K
=
ϱ
�
� �
2.05 × 109 N · m−2 =
10 3 kg · m −3
�
(2.05 × 10 6 m 2 · s −2 ) = 1432 m · s −1
Other liquids give values of the same order.
For a steel pipe (E = 200 GPa) of, say, 75 mm diameter and 6 mm
thickness
1
=
K ′
1
2.05 × 10 9 Pa +
75 mm
6mm× 200 × 10 9 Pa
whence K ′ = 1.817 × 109 Pa and so
�
�1.817 �
× 109 N · m−2 c =
= 1348 m · s −1
10 3 kg · m −3
Although c by its dependence of K varies slightly with pressure and temperature,
sufficient accuracy is usually obtained by regarding it as constant for a
particular pipe-line. (Gas bubbles in the liquid, however, appreciably reduce
the value of K and therefore of c.)
It is important to realize that the velocity of wave propagation is not a
velocity with which particles of matter are moving. For this reason the word
Pressure transients 563