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.
[Illustration: FIG. 100.]
_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.
[Illustration: FIG. 101.]
§ 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)