490 HYDROMECHANICS [HYDRAULICS, where Q is the common discharge of the two portions of the pipe. Hence _ h -h c l. A d^ from which h is easily obtained. If then h is greater than hb, opening the sluice between X and B will allow flow towards B, and the case in hand is case I. If h is less than hb, opening the sluice will allow flow from B, and the case is case III. If h ==hb, the case is case II., and is already completely solved. The true value of h must lie between h and 7;&. Choose a new value of h, and recalculate Q 1( Q 2 , Q 3 . Then if 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 A , A x at a dis tance dl (fig. 99). Let p, u be the pressure and velocity at A c , p + dp and
ill !
i i - " O u + du those at Fur _. ther, suppose that in a very short time dt the mass of air between A Aj conies to AjAj so that A A. =udt and AjA( = (u + du)dt. Let fi 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 A Ao, and AjAJ is the same. That is, = Gnudt = Gn(u + du)dt . By analogy with liquids the head lost in friction is, for the length dl (see 69, eq. 3), dl
- t(- Lit
2g m Let H= 2g head lost is Then the and, since "Wdt pounds of air flow through the pipe in the time considered, the work expended in friction is the value chosen for h is too small, and a new value must be chosen ! If Qj-cQtj + Qu in case I., or Qi + Qj-^Qa 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. IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES. 81. Flow of Air in Long Pipes. 1 When air flows through a long pipe, by far the greater part of the work expended is r,S5(l in over coming Motional 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 con ducted 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 Messrs Culley & Sabine, show that the change of temperature of the air flowing along the tube is much less than it would be in adiabatic expansion. 82. Differential Equation of the Steady Motion of Air Flowing in a Long Pipe of Uniform, Section. When air expands at a constant absolute temperature T, 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 where c= 53 15. Taking r = 521, corresponding to a temperature of 60 Fahr., CT = 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 Gflit = W = constant ...... (3), where fl is the section of the pipe and u the velocity of the air. Combining (1) and ^3), W = CT = constant 1 This investigation was first given in the Proc Jnst. of Civil Engineers, vol, zlili. The change of kinetic energy in dl seconds is the difference of the kinetic energy of A Ao and AjAj, that is, g W = 9 The work of expansion when fiudt cubic feet of air at a pressure p expand to Q.(u + du)dt cubic feet is H pdudt . But from (3a) crW u*=-- , cip du crW dp fip 2 And the work done by expansion is _crW, P The work done by gravity on the mass between A and A x is zero if the pipe is horizontal, and may in other cases be neglected with out great error. The work of the pressures at the sections AoAj i? pnudt- (p + dp)n(u + du}dt = - (pdu + udp)ndt . But from (So,) pu = constant, pdu + udp = , and the work of the pressures is zero. Adding together the quan tities of work, and equating them to the change of kinetic energy, -dpdt . f-rW - dpdt- H But and , M + c CrW _ ~ 2g
H
