492 HYDROMECHANICS [HYDRAULICS. a li^ht felt earner, the resistance of which in the tubes in London is only f 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 to I, t.f l *L t Jo U From da) neglecting , and putting m = , From (1) and
dl But 1 gdcr I > o o C/V ^n 6 (flfcrd)* If T = 521, corresponding to 60 F. , <= -001412^ P*-P*-. 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 ; or, for = 521, / 7 / O ON 3 . . . (16). u =0-708 mean / V The following table gives some results: Absolute Pres sures in tb per sq. inch. Mean Velocities for Tubes of a length in ft. Po Pi 1000 2000 3000 4000 5000 Vacuum Working. 15 5 99-4 70-3 57-4 49-7 44-5 15 10 67 2 47-5 38-8 34-4 30-1 Pressure Working. 20 15 57-2 40-5 33-0 28-6 25-6 25 15 74-6 527 43-1 37-3 33-3 30 15 84-7 60 49-0 42-4 37-9 Limiting Velocity in the Pipe, when the Pressure at one End diminished indefinitely. If in the last equation there be pul
- 1 = 0, then
0-708
- /?
V a where the velocity is independent of the pressure p a 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 anc terminal pressures for which the formula is applicable. X. FLOW IN RIVERS AND CANALS. 87. Flow of Water in Open Canals and Rivers. When wate flows in a pipe the section at any point is determined by the forn of the boundary. When it flows in an open channel with free uppe surface, the section depends on the velocity due to the dynamica conditions. Suppose water admitted to an unfilled canal. The channel wil 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 eacl point attain after a time a limit. Thenceforward the section anc velocity at each point are constant, and the motion is steady, or per manent regime is established. If when the motion is steady the sections of the stream are all qual, the motion is uniform. By hypothesis, the inflow Civ is con- tant for all sections, and n is constant ; therefore v must be con- tant also from section to section. The case is then one of uniform teady motion. In most artificial channels the form of section is onstant, and the bed has a uniform slope. In that case the motion s uniform, the depth is constant, and the stream surface is parallel o the bed. If when steady motion is established the sections are nequal, the motion is steady motion with varying velocity from ection to section. Ordinary rivers are in this condition, especially vhere the flow is modified by weirs or obstructions. Short unob- tructed lengths of a river may be treated as of uniform section vithout great error, the mean section in the length being put for he actual sections. In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than ,hose near the bottom and sides. The ordinary formulae for the low of streams rest on an 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 lypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to a a b b without deformation. The omponent of the weight parallel
- o the channel bed balances the
friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the tream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean velocity of the stream, and is not in any simple relation with it, for channels of different forms. The theory is therefore obviously based on an imperfect hypothesis. However, by taking variable values for the coefficient of friction, the errors of the ordinary formulae are to a great extent neutralized, and they may be used without leading to practical errors. Formulae have been obtained based on less restricted hy potheses, but at present they are not practically so reliable, and are more complicated than the formulae obtained in the manner de scribed above. .88. Steady Flow of Water with Uniform Velocity in Channels of Constant Section. Let aa , lib (fig. 103) be two cross sections normal to Fig. 103. the direction of motion at a distance dl. Since the mass aa bb moves uniformly, the external forces acting on it are in equilibrium. Let n be the area of the cross sections, x the wetted perimeter, pq + qr + rs, of a section. Then the quantity m = is termed the hydraulic mean depth of the section. Let v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the be slope or fall of the stream in feet, per foot, being the ratio . The external forces acting on aa bb parallel to the direction of motion are three : (a) The pressures on aa and bb , which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same. (b) The component of the weight W of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aa bb is Gndl, and the component of the weight in the direction of motion is Gndl x the cosine of the angle between Wy and ab, that is, Gndl cos abc = Gndl = Gnidl. ab (c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area ydl ^ rubbing surface arid to a function of the velocity which may be written f(v) ; f(v) being the friction per square foot at a velocity v. Hence the friction is --^dlf(v). Equating the sum of the forces to 261*0 f(v) -7^- But it has been already shown ( 63) that./()-G , (D-
(2).
