Mechanics of Fluids

From the Fanno Tables at pc/pB = 0.278, MB = 0.302 and
(fl max P/A)B = 5.21. For a circular pipe P/A = 4/d, so
(flmaxP/A)A − (flmaxP/A)B =
Compressible flow in pipes of constant cross-section 539
f × 12 m × 4
0.02 m
= 21.37 − 5.21
yielding f = 0.00673
(b) At exit, pc/p = 77.8/100 = 0.778. From the Fanno Tables,
(fl max P/A) = 0.07. Hence
L m 21.37 − 0.07
=
12 m 21.37 − 5.21
yielding L = 15.82 m
(c) Rewriting eqn 11.79, and evaluating it at B:
m = ApM
� γ � 1 + (γ − 1)M 2 /2 �
RT0
� 1/2
= π
4 (0.02 m)2 × (2.8 × 10 5 ) Pa × 0.302
� � �
1.4 × 1 + 0.4 × (0.302) 2 /2
×
287 J · kg −1 · K −1 × 294 K
= 0.109 kg · s −1
� 1/2
11.10.2 Isothermal flow in a pipe
In Section 11.10.1 we assumed that there was no heat transfer across the
walls of the pipe. In general, of course, any quantity of heat may be transferred
to or from the fluid. This general case is too complicated to be
considered in this book, but a particular example of practical interest is
that in which the heat transfer is such as to keep the temperature of the
fluid constant, that is, in which the flow is isothermal. For gases such flow
is usually achieved at low velocities in long pipes not thermally insulated
because there is then opportunity for sufficient heat transfer through the
pipe walls to maintain the gas at (or near) the temperature of the surroundings.
Although Mach numbers are usually low the assumption of constant
density is untenable because of the significant changes of pressure.
We again seek the integral of the momentum equation (11.67):
dp
ρ
fu2
+ udu + dl = 0
2A/P
The heat transfer through the walls invalidates the energy equation (11.9)
but we introduce the condition T = constant. For a perfect gas the pressure
is then proportional to the density, so from the continuity relation
✷