Chapter IX: Flow of Compressible Fluids in Pipes

§ 89. _Flow of Air in Long Pipes._--When air flows through a long
pipe, by far the greater part of the work expended is used in
overcoming frictional resistances due to the surface of the pipe. The
work expended in friction generates heat, which for the most part must
be developed in and given back to the air. Some heat may be
transmitted through the sides of the pipe to surrounding materials,
but in experiments hitherto made the amount so conducted away appears
to be very small, and if no heat is transmitted the air in the tube
must remain sensibly at the same temperature during expansion. In
other words, the expansion may be regarded as isothermal expansion,
the heat generated by friction exactly neutralizing the cooling due to
the work done. Experiments on the pneumatic tubes used for the
transmission of messages, by R. S. Culley and R. Sabine (_Proc. Inst.
Civ. Eng._ xliii.), show that the change of temperature of the air
flowing along the tube is much less than it would be in adiabatic
expansion.

§ 90. _Differential Equation of the Steady Motion of Air Flowing in a
Long Pipe of Uniform Section._--When air expands at a constant
absolute temperature [tau], the relation between the pressure p in
pounds per square foot and the density or weight per cubic foot G is
given by the equation

p/G = c[tau], (1)

where c = 53.15. Taking [tau] = 521, corresponding to a temperature of
60° Fahr.,

c[tau] = 27690 foot-pounds. (2)

The equation of continuity, which expresses the condition that in
steady motion the same weight of fluid, W, must pass through each
cross section of the stream in the unit of time, is

G[Omega]u = W = constant, (3)

where [Omega] is the section of the pipe and u the velocity of the
air. Combining (1) and (3),

[Omega]up/W = c[tau] = constant. (3a)

Since the work done by gravity on the air during its flow through a
pipe due to variations of its level is generally small compared with
the work done by changes of pressure, the former may in many cases be
neglected.

Consider a short length dl of the pipe limited by sections A0, A1 at a
distance dl (fig. 99). Let p, u be the pressure and velocity at A0, p
+ dp and u + du those at A1. Further, suppose that in a very short
time dt the mass of air between A0A1 comes to A´0A´1 so that A0A´0 =
udt and A1A´1 = (u + du)dt1. Let [Omega] be the section, and m the
hydraulic mean radius of the pipe, and W the weight of air flowing
through the pipe per second.

From the steadiness of the motion the weight of air between the
sections A0A´0, and A1A´1 is the same. That is,

W dt = G[Omega]u dt = G[Omega](u + du) dt.

By analogy with liquids the head lost in friction is, for the length
dl (see § 72, eq. 3), [zeta](u²/2g)(dl/m). Let H = u²/2g. Then the
head lost is [zeta](H/m)dl; and, since Wdt lb. of air flow through the
pipe in the time considered, the work expended in friction is
-[zeta](H/m)Wdl dt. The change of kinetic energy in dt seconds is the
difference of the kinetic energy of A0A´0 and A1A´1, that is,

(W/g) dt {(u + du)² - u²}/2 = (W/g)u du dt = W dH dt.

The work of expansion when [Omega]udt cub. ft. of air at a pressure p
expand to [Omega](u + du)dt cub. ft. is [Omega]p du dt. But from (3a)
u = c[tau]W/[Omega]p, and therefore

du/dp = -c[tau]W/[Omega]p².

And the work done by expansion is -(c[tau]W/p)dpdt.

The work done by gravity on the mass between A0 and A1 is zero if the
pipe is horizontal, and may in other cases be neglected without great
error. The work of the pressures at the sections A0A1 is

p[Omega]u dt - (p + dp)[Omega](u + du) dt
= -(pdu + udp)[Omega] dt

But from (3a)

pu = constant,

p du + u dp = 0,

and the work of the pressures is zero. Adding together the quantities
of work, and equating them to the change of kinetic energy,

WDH dt = -(c[tau]W/p) dp dt - [zeta](H/m)W dl dt

dH + (c[tau]/p) dp + [zeta](H/m) dl = 0,

dH/H + (c[tau]/Hp) dp + [zeta]dl/m) = 0 (4)

But

u = c[tau]W/[Omega]p,

and

H = u²/2g = c²[tau]²W²/2g[Omega]²p²,

.: dH/H + (2g[Omega]²p/c[tau]W²) dp + [zeta] dl/m = 0. (4a)

For tubes of uniform section m is constant; for steady motion W is
constant; and for isothermal expansion [tau] is constant. Integrating,

log H + g[Omega]²p²/W²c[tau] + [zeta]l/m = constant; (5)

for

l = 0, let H = H0, and p = p0;

and for

l = l, let H = H1, and p = p1.

log (H1/H0) + (g[Omega]²}/W²c[tau]) (p1² - p0²) + [zeta]l/m = 0. (5a)
where p0 is the greater pressure and p1 the less, and the flow is from
A0 towards A1.

By replacing W and H,

log (p0/p1) + (gc[tau]/u0²p0²)(p1² - p0² + [zeta]l/m = 0 (6)

Hence the initial velocity in the pipe is

u0 = [root][{gc[tau](p0² - p1²)} / {p0²([zeta]l/m + log (p0/p1)}]. (7)

When l is great, log p0/p1 is comparatively small, and then

u0 = [root][(gc[tau]m/[zeta]l) {(p0² - p1²)/p0²}], (7a)

a very simple and easily used expression. For pipes of circular
section m = d/4, where d is the diameter:--

u0 = [root][(gc[tau]d/4[zeta]l) {(p0² - p1²)/p0²}]; (7b)

or approximately

u0 = (1.1319 - 0.7264 p1/p0) [root](gc[tau]d/4[zeta]l). (7c)

§ 91. _Coefficient of Friction for Air._--A discussion by Professor
Unwin of the experiments by Culley and Sabine on the rate of
transmission of light carriers through pneumatic tubes, in which there
is steady flow of air not sensibly affected by any resistances other
than surface friction, furnished the value [zeta] = .007. The pipes
were lead pipes, slightly moist, 2¼ in. (0.187 ft.) in diameter, and
in lengths of 2000 to nearly 6000 ft.

In some experiments on the flow of air through cast-iron pipes A.
Arson found the coefficient of friction to vary with the velocity and
diameter of the pipe. Putting

[zeta] = [alpha]/v + [beta], (8)

he obtained the following values--

+------------------+--------+-------+--------------------+
| Diameter of Pipe | | | [zeta] for 100 ft. |
| in feet | [alpha]| [beta]| per second. |
+------------------+--------+-------+--------------------+
| 1.64 | .00129 | .00483| .00484 |
| 1.07 | .00972 | .00640| .00650 |
| .83 | .01525 | .00704| .00719 |
| .338 | .03604 | .00941| .00977 |
| .266 | .03790 | .00959| .00997 |
| .164 | .04518 | .01167| .01212 |
+------------------+--------+-------+--------------------+

It is worth while to try if these numbers can be expressed in the form
proposed by Darcy for water. For a velocity of 100 ft. per second, and
without much error for higher velocities, these numbers agree fairly
with the formula

[zeta] = 0.005(1 + (3/10)d), (9)

which only differs from Darcy's value for water in that the second
term, which is always small except for very small pipes, is larger.

Some later experiments on a very large scale, by E. Stockalper at the
St Gotthard Tunnel, agree better with the value

[zeta] = 0.0028(1 + (3/10)d).

These pipes were probably less rough than Arson's.

When the variation of pressure is very small, it is no longer safe to
neglect the variation of level of the pipe. For that case we may
neglect the work done by expansion, and then

z0 - z1 - p0/G0 - p1/G1 - [zeta](v²/2g)(l/m) = 0, (10)

precisely equivalent to the equation for the flow of water, z0 and z1
being the elevations of the two ends of the pipe above any datum, p0
and p1 the pressures, G0 and G1 the densities, and v the mean velocity
in the pipe. This equation may be used for the flow of coal gas.

§ 92. _Distribution of Pressure in a Pipe in which Air is
Flowing._--From equation (7a) it results that the pressure p, at l ft.
from that end of the pipe where the pressure is p0, is

p = p0 [root](1 - [zeta]lu0²/mgc[tau]); (11)

which is of the form

p = [root](al + b)

for any given pipe with given end pressures. The curve of free surface
level for the pipe is, therefore, a parabola with horizontal axis.
Fig. 100 shows calculated curves of pressure for two of Sabine's
experiments, in one of which the pressure was greater than atmospheric
pressure, and in the other less than atmospheric pressure. The
observed pressures are given in brackets and the calculated pressures
without brackets. The pipe was the pneumatic tube between Fenchurch
Street and the Central Station, 2818 yds. in length. The pressures are
given in inches of mercury.

_Variation of Velocity in the Pipe._--Let p0, u0 be the pressure and
velocity at a given section of the pipe; p, u, the pressure and
velocity at any other section. From equation (3a)

up = c[tau]W/[Omega] = constant;

so that, for any given uniform pipe,

up = u0p0,
u = u0p0/p; (12)

which gives the velocity at any section in terms of the pressure,
which has already been determined. Fig. 101 gives the velocity curves
for the two experiments of Culley and Sabine, for which the pressure
curves have already been drawn. It will be seen that the velocity
increases considerably towards that end of the pipe where the pressure
is least.

§ 93. _Weight of Air Flowing per Second._--The weight of air
discharged per second is (equation 3a)--

W = [Omega]u0p0/c[tau].

From equation (7b), for a pipe of circular section and diameter d,

W = ¼[pi] [root](gd^5(p0² - p1²)/[zeta]lc[tau]),
= .611[root](d^5(p0² - p1²)/[zeta]l[tau]). (13)

Approximately

W = (.6916 p0 - .4438 p1)(d^5/[zeta]l[tau])^½. (13a)

§ 94. _Application to the Case of Pneumatic Tubes for the Transmission
of Messages._--In Paris, Berlin, London, and other towns, it has been
found cheaper to transmit messages in pneumatic tubes than to
telegraph by electricity. The tubes are laid underground with easy
curves; the messages are made into a roll and placed in a light felt
carrier, the resistance of which in the tubes in London is only ¾ oz.
A current of air forced into the tube or drawn through it propels the
carrier. In most systems the current of air is steady and continuous,
and the carriers are introduced or removed without materially altering
the flow of air.

_Time of Transit through the Tube._--Putting t for the time of transit
from 0 to l,
_
/l
t = | dl/u,
_/0

From (4a) neglecting dH/H, and putting m = d/4,

dl = g d[Omega]²p dp/2[zeta]W²cr.

From (1) and (3)

u = Wc[tau]/p[Omega];

dl/u = g d[Omega]³p² dp/2[zeta]W³c²[tau]²;
_
/p0
t = | g d[Omega]³p² dp/2[zeta]W³c²[tau]²,
_/p1

= gd[Omega]³(p0³ - p1³)/6[zeta]W³c²[tau]². (14)

But

W = p0u0[Omega]/c[tau];

.: t = gdc[tau](p0³ - p1³)/6[zeta]p0³u0³,

= [zeta]^(½)l^(3/2)(p0³ - p1³)/6(gc[tau]d)^(½)(p0² - p1²)^(3/2); (15)

If [tau] = 521°, corresponding to 60° F.,

t = .001412 [zeta]^(½)l^(3/2)(p0³ - p1³)/d^(½)(p0² - p1²)^(3/2); (15a)

which gives the time of transmission in terms of the initial and final
pressures and the dimensions of the tube.

_Mean Velocity of Transmission._--The mean velocity is l/t; or, for
[tau] = 521°,

u_mean = 0.708 [root]{d(p0² - p1²)^(3/2)/[zeta]l(p0³ - p1³)}. (16)

The following table gives some results:--

+-----------+-----------------+----------------------------------+
| | Absolute | |
| | Pressures in | Mean Velocities for Tubes |
| | lb. per sq. in. | of a length in feet. |
+-----------+--------+--------+------+------+------+------+------+
| | p0 | p1 | 1000 | 2000 | 3000 | 4000 | 5000 |
+-----------+--------+--------+------+------+------+------+------+
| Vacuum | 15 | 5 | 99.4 | 70.3 | 57.4 | 49.7 | 44.5 |
| Working | 15 | 10 | 67.2 | 47.5 | 38.8 | 34.4 | 30.1 |
| | | | | | | | |
| Pressure | 20 | 15 | 57.2 | 40.5 | 33.0 | 28.6 | 25.6 |
| Working | 25 | 15 | 74.6 | 52.7 | 43.1 | 37.3 | 33.3 |
| | 30 | 15 | 84.7 | 60.0 | 49.0 | 42.4 | 37.9 |
+-----------+-----------------+------+------+------+------+------+

_Limiting Velocity in the Pipe when the Pressure at one End is
diminished indefinitely._--If in the last equation there be put p1 =
0, then

u´_mean = 0.708 [root](d/[zeta]l);

where the velocity is independent of the pressure p0 at the other end,
a result which apparently must be absurd. Probably for long pipes, as
for orifices, there is a limit to the ratio of the initial and
terminal pressures for which the formula is applicable.