Mechanics of Fluids

530 Compressible flow of gases
✷
=
=
M = 1.715
Across the shock
�
1 + 1
3 × 1.82� 4
1.8 2
� 1 + 0.2M 2 � 6
M 2
p2
=
p1
2γ M2 1
� �4 3
= 1.828 for helium
4
1
for air, whence (by trial)
1.26 − (γ − 1)
(from eqns 11.24 and 11.29)
γ + 1
= 2.8 × 1.7152 − 0.4
2.4
= 3.265
∴ p1 = 30 kPa/3.265 = 9.19 kPa
From eqn 11.32,
p0,1
p1
=
=
�
γ − 1
1 +
∴ p0,1 = 46.4 kPa
2 M2 1
� γ /(γ −1)
�
1 + 0.2 × 1.715 2� 3.5
= 5.05
11.10 COMPRESSIBLE FLOW IN PIPES OF CONSTANT
CROSS-SECTION
We have so far considered flow of a gas where friction may, at least to
a first approximation, be neglected. Friction must be accounted for, however,
when flow takes place in pipe-lines and similar conduits. Here we
shall discuss steady flow in a pipe of constant cross-section and (except in
Section 11.10.3) we shall assume the velocity to be sufficiently uniform over
the section for the flow to be adequately described in one-dimensional terms.
Since the change of pressure resulting from friction gives rise to a change of
density, and thus of velocity, matters are more complicated than for an
incompressible fluid. The properties of the fluid are also affected by heat
transferred through the walls of the pipe. If the pipe is well insulated the
heat transfer may be negligible and the changes therefore adiabatic (but not,
of course, isentropic). In short pipes where no specific provision is made
for heat transfer the conditions may approximate to adiabatic. On the other
hand, for flow at low velocities in long, uninsulated pipes an appreciable
amount of heat may be transferred through the pipe walls, and if the temperatures
inside and outside the pipe are similar the flow may be approximately
isothermal (i.e. at constant temperature). This is so, for example, in long
compressed-air pipe-lines and in low-velocity flows generally.