Page:EB1911 - Volume 14.djvu/79

COMPRESSIBLE FLUIDS IN PIPES] The true value of h must lie between h′ and hb. Choose a new value of h, and recalculate Q1, Q2, Q3. Then if

Q1 > Q2 + Q3 in case I.,

or

Q1 + Q2 > Q3 in case III.,

the value chosen for h is too small, and a new value must be chosen.

If

Q1 < Q2 + Q3 in case I.,

or

Q1 + Q2 < Q3 in case III.,

the value of h is too great.

Since the limits between which h can vary are in practical cases not very distant, it is easy to approximate to values sufficiently accurate.

§ 88. Water Hammer.—If in a pipe through which water is flowing a sluice is suddenly closed so as to arrest the forward movement of the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one and gradually dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J. Joukowsky at Moscow in 1898 (Stoss in Wasserleitungen, St Petersburg, 1900), and the results are generally confirmed by experiments made by E. B. Weston and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in. diameter, from 1000 to 2500 ft. in length. The sluice closed in 0.03 second, and the fluctuations of pressure were automatically registered. The maximum excess pressure due to water-hammer action was as follows:—

In some cases, in fixing the thickness of water mains, 100 ℔ per sq. in. excess pressure is allowed to cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance (see also Carpenter, Am. Soc. Mech. Eng., 1893).

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, 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, (1)

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

c = 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

Gu = W = constant, (3)

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

up/W = c = 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 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,

Wdt = Gudt = G (u+du)dt.

By analogy with liquids the head lost in friction is, for the length dl (see § 72, eq. 3), (u2/2g) (dl/m). Let H = u2/2g. Then the head lost is (H/m)dl; and, since Wdt ℔ of air flow through the pipe in the time considered, the work expended in friction is − (H/m)W dl 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 = WdHdt.

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

du / dp = −cW / p2.

And the work done by expansion is −(cW/p) dp dt.

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

pudt−(p+dp) (u + du) dt = −(pdu + udp) dt

But from (3a)

pu = constant, pdu+udp = 0,

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

WdHdt = −(cW/p) dp dt − (H/m) W dl dt dH + (c/p) dp + (H/m) dl = 0, dH/H + (c/Hp) dp + dl / m = 0 (4)

But

u = cW / p,

and

H = u2/2g = c22W2 / 2g2p2,

∴ dH/H + (2g2p / cW2) dp + dl / m = 0. (4a)

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

log H + g2p2 / W2c + 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) + (g2 / W2c) (p12 − p02) + 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 / u02p02) (p12 − p02 + l/m = 0  (6)

Hence the initial velocity in the pipe is

u0 = √ [{gc (p02 − p12)} / {p02 ( l/m + log (p0 / p1) }]. (7)

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

u0 = √ [ (gcm/ 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 = √ [ (gcd / 4 l) {(p02 − p12) / p02} ]; (7b)

or approximately

u0 = (1.1319 − 0.7264 p1/p0) √ (gcd / 4l). (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 = .007. The pipes were lead pipes, slightly moist, 2 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

= /v + , (8)

he obtained the following values—

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

= 0.005 (1 + 3/10d), (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.