Page:EB1911 - Volume 14.djvu/80

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68⁠

HYDRAULICS

[FLOW IN RIVERS

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

ζ＝0.0028 (1 + 3/10d).

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 − ζ (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 √{1 − ζlu₀² / mgcτ};

(11)

which is of the form

p＝√ (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.

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＝cτW / Ω＝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.

Fig. 101.

§ 93. Weight of Air Flowing per Second.—The weight of air discharged per second is (equation 3a)—

W＝Ωu₀p₀ / cτ.

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

W＝14π √ (gd ⁵ (p₀² − p₁²) / ζ lcτ),
⁠＝.611 √ (d ⁵ (p₀² − p₁²) / ζ lτ).

(13)

Approximately

W＝(.6916p₀ − .4438p₁) (d ⁵ / ζ lτ)^1/2.

(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 3/4 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,

$t=\int _{0}^{l}dl/u,$

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

dl＝gdΩ²p dp / 2ζW²cr.

From (1) and (3)

u＝Wcτ / pΩ;
dl/u＝gdΩ³p² dp / 2ζW³c²τ²;

t＝ $\int _{p_{1}}^{p_{0}};$ gdΩ³p² dp / 2ζW³c²τ²,

＝gdΩ³ (p₀³ − p₁³) / 6ζW³c²τ².

(14)

But

W = p₀u₀Ω / cτ;

∴ t＝gdcτ (p₀³ − p₁³) / 6ζp₀³u₀³,
＝ζ^1/2 l^3/2 (p₀³ − p₁³) / 6(gcτd)^1/2 (p₀² − p₁²)^3/2;

(15)

If τ = 521°, corresponding to 60° F.,

t＝.001412 ζ^1/2 l^3/2 (p₀³ − p₁³) / d ^1/2 (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 τ = 521°,

u_mean＝0.708 √ {d (p₀² − p₁²)^3/2 / ζ l (p₀³ − p₁³)}.

(16)

The following table gives some results:—

		Absolute Pressures in ℔ per sq. in.		Mean Velocities for Tubes of a length in feet.
		p₀	p₁	1000	2000	3000	4000	5000
Vacuum Working	$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	15	5	99.4	70.3	57.4	49.7	44.5
Vacuum Working		15	10	67.2	47.5	38.8	34.4	30.1
Pressure Working	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	20	15	57.2	40.5	33.0	28.6	25.6
		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 √ (d / ζl);

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.

X. FLOW IN RIVERS AND CANALS

§ 95. Flow of Water in Open Canals and Rivers.—When water flows in a pipe the section at any point is determined by the form of the boundary. When it flows in an open channel with free upper surface, the section depends on the velocity due to the dynamical conditions.

Suppose water admitted to an unfilled canal. The channel will gradually fill, the section and velocity at each point gradually changing. But if the inflow to the canal at its head is constant, the increase of cross section and diminution of velocity at each point attain after a time a limit. Thenceforward the section and velocity at each point are constant, and the motion is steady, or permanent regime is established.

If when the motion is steady the sections of the stream are all equal, the motion is uniform. By hypothesis, the inflow Ωv is constant for all sections, and Ω is constant; therefore v must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the bed has a uniform slope. In that case the motion is uniform, the depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section to section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections.

Fig. 102.

In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal