HYDRAULICS.] HYDROMECHANICS 491 For tubes of uniform section m is constant ; for steady motion W is constant ; and for isothermal expansion T is constant. Integrat ing, log H + + (* = constant . (5); for and for Z = 0, let H = H , and p =p ; 1 = 1, let H = H lf and p =p l . where p is the greater pressure and p : the less, and the flow is from A towards Aj. By replacing W and H, Hence the initial velocity in the pipe is 2 C p 1 . . . . (7). When Z is great, log ^- is comparatively small, and then Pi germ pp 2 - ~ , 7 )
a very simple and easily used expression. For pipes of circular section m = , where d is the diameter : or approximately = V i gcrd p<?-pi* (W; C* 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 feet per second, and without much error for higher velocities, these numbers agree fairly with the formula / n .... (9), (o 1 + ^ which only differs from Dairy s value for water in that the second term, which is always small except for very small pipes, is larger. Some more recent experiments on a very large scale, by M. Stoekalper at the St Gotthard Tunnel, agree better with the value These pipes were probably less rough than M. 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, ami then - z - PS- "l n --- G 7i - 2r/ m (10), precisely equivalent to the equation for the flow of water, r and 2, being the elevations of the two ends of the pipe above any datum, Poandp, the pressures, G and G, the densities, and v the mean velocity in the pipe. This equation may be used for the flow of coal gas. 84. Distribution of Pressure in a Pipe in which Air is Flowing. From equation (la) it results that the pressure f>, at I feet from that end of the pipe where the pressure is^ , is
- -P Jl- 2
/ V myc-r which is of the form p= /al + b for any given pipe with given end pressures. The curve of free sur face level for the pipe is, therefore, a parabola with horizontal axis. Fig. 100 shows calculated curves of pressure for two of Mr Sabine s 83. Coefficient of Friction for .. Jnwin of the experiments by Me sf transmission of light carriers tl
- here is steady flow of air not sen
other than surface friction, furnis were lead pipes, slightly moist, and in lengths of 2000 to nearly ( Some experiments on the flow c jeen made by M. Arson. He foun with the velocity and diameter of H he obtained the following values- 4ir. A discussion by Professor ssrs Culley & Sabine on the rate irough pneumatic tubes, in which sibly affected by any resistances ied the value = 007. The pipes !j inches (0 187 ft.) in diameter, 000 feet. f air through cast-iron pipes have d the coefficient of friction to vary the pipe. Putting 1 +0. (8), !7 Diameter of Pipe in Feet. a
f for 100 feet per second. 1-64 1-07 83 338 266 164 00129 00972 01525 03604 03790 04518 00483 00640 00704 00941 00959 01167 00484 00650 00719 00977 00997 01212 1575 8454. Fi. experiments, in one of which the pressure was greater than atmo< spheric 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 be tween Fenchurch Street and the Central Station, 2818 yards in length. The pressures are given in inches of mercury. Variation of Velocity in the Pipe. Let p , u 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-. erW , = constant so that, for any given uniform pipe, (12); which gives the velocity at any section in terms of the pressure, which has already been determined. Fig. 101 gives the velocity 4-2 27ft. Fie. 101. 84.S-t.Fl. curves for the two experiments of Messrs Culley & 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. 85. Weight of Air Flowing per Second. The weight of air dis charged per second is (equation 3a) From equation (7b), for a pipe of circular section and diameter d, 611 Approximately (B 2 - V, J /I t Q r (18). ( -6916ft,- -4438^) 86. Application to tlit Case of Pneumatic Tubes for the Trans mission 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 iu.
