CHAPTER XI. HYDRAULICS. 361. The word Hydraulics means primarily the science of the Motion of Water in Pipes; but it is now extended to cover the elementary parts of the practical science of the Motion of Fluids. This includes the Discharge from Orifices, the Theory of Hydraulic Machinery, such as Water Wheels, Turbines, Paddle Wheels and Screw Propellers, Injectors, etc., which can be treated by the aid of Torricelli's and Bernoulli's Theorems; and the Motion in Pipes, Canals, and Rivers, taking into account the effect of Fluid Friction so far as it can be treated in an elementary manner. 362. Torricelli's Theorem. The velocity of discharge of water from a small orifice a depth h below the free surface was given by Torricelli (1643) as the velocity v acquired in falling from the level of the free surface, so that v2=gh, or v=√(2gh); and v is then called the velocity due to the head h. This is argued by asserting that the hydrostatic energy of the water, Dh ft-lb per ft3, or h ft-lb per lb, becomes converted on opening the orifice into the kinetic energy Dv2/g ft-lb/ft3, or v2/g ft-lb/lb. 462 TORRICELLI'S THEOREM ON Thus the jet of water, if directed nearly vertically upwards, would nearly reach the level of the free surface; and if directed in any other direction will form a parabolic jet, of which the directrix lies in the free surface of the still liquid. The cross section of the jet OVR, while continuous and not shattered into drops, will be inversely as the velocity; and the horizontal component of the velocity being constant, equidistant vertical planes will intercept equal quantities of water, so that G the C.G. of the water will coincide with the C.G. of the parabolic area cut off by the chord; and the height of the C.G. of the jet cut off by a horizontal chord OR will be two-thirds of the height of the vertex (fig. 104, p. 469). If the jet could be instantaneously reduced to rest and frozen, it could stand as an arch, without shearing stress across normal sections. If the vessel is in motion, the velocity of efflux v is still taken as due to the head of the pressure p; in this way the efflux from an orifice in a rotating vessel (Barker's Mill or a Turbine) is given (§ 345) by v=(2gz+y2w2), or from an orifice in an ascending or descending bucket; balanced by a counterpoise at the end of a rope over a pulley by v={2(g±a)z} (§322); the student may work out the motion of the buckets completely as an exercise. 363. The velocity of efflux v must be reckoned not exactly at the orifice, but a little in front at the point where the jet is seen to contract to its smallest cross section; this part is called the vena contracta, and the ratio of the cross section of the vena contracta to that of the orifice is called the coefficient of contraction, and denoted by c1. FLOW AND JETS OF WATER. 463 Practically, in consequence of friction, the velocity v at the vena contracta is a little less than (2gh), and the ratio of v to (2gh) is called the coefficient of velocity, and denoted by c,. Now if the area of the vena contracta is A ft2, and of the orifice is B ft2, A=c1B; and the flow of water is Av=c1Вv=c1c2В/(2gh) ft3/sec; and the product c12 is denoted by c, and called the coefficient of discharge. The flow of water through a standard vertical orifice one in2 in section, under a head of 6 ins, is called the miner's inch; since B-1÷144 ft2, h=0·54 ft, and we may put on the average c=0.62, this flow is about 1.5 ft3/minute. 364. Torricelli's Theorem is still employed when the head varies, as in filling or emptying a reservoir or lock, in sinking a ship by a hole under water, or in pouring out water from a vessel through a spout; and now, if X denotes the area of the surface of the water at a height x above the orifice B, giving the time t of filling or emptying the vessel between the levels x and h; this may be worked out for vessels of various form, as the cylinder, cone, sphere, etc. Thus, for example, if an orifice of one ft2 be opened in the bottom of a sheet iron tank, 30 ft long, 20 ft broad, and 9 ft deep, drawing 4 ft of water, the tank will sink in about 26 minutes, taking c=0.6. 464 FLOW WITH VARIABLE HEAD. If the orifice in a vertical wall is large, and the variations of head over its area is taken into account, and if y denotes the breadth of the orifice at a depth x below the surface, the efflux in ft3/sec is, with c=1, Q=/y√(2gx)dx and t=/Xdx/Q. Thus if h, h' denote the depth of the top and bottom of a rectangular orifice of breadth b, Q=b./(29) / "x3dx=3b_/(2g)(ha —h ́1); h' so that the average velocity of efflux is due to the head and this is 4h, if h'= 0. 4/h-h For example, the time of draining to a depth of 3 inches the ditch of a fortress, one mile long, 30 ft broad, and 9 ft deep, by a vertical cut 2 ft broad, is 13 hours; and to lower the depth to one inch will take 12 hours more. 365. The flow of water is DAv = DA (2gh) lb/sec, possessing momentum DAv2/g= 2DAh second-lb/sec; this will therefore be the thrust in lbs of the jet against a fixed plane perpendicular to its direction. This thrust is double the hydrostatic thrust due to the head h; thus, for instance, the water of Niagara, falling 162 ft, can balance a column of water 324 ft high in a J-shaped tube, with its lower mouth under the fall. The energy of the jet is DAv. v2/g= DAvh ft-lb/sec; and therefore the H.P. (horse-power) is DAv3/550g=DAvh/550. With a metre and kilogramme as units, D=1000, g=981; and 75 kg-m/sec is the cheval-vapeur. MOMENTUM AND ENERGY OF A JET. 465 Thus a jet of water 10 ins in diameter, issuing under a head of 600 ft has 7300 H.P.; these large jets are used for hydraulic mining in California, the nozzle being controlled by an apparatus called a hydraulic giant. 366. Denoting by p the hydrostatic pressure Dh lb/ft2 due to the head h, then v = (2gp/D); and the jet discharges A(2gp/D) fts/sec, or A/(2gpD) lb/sec, possessing momentum 2Ap sec-lb, and energy and H.P. Thus the velocity of efflux from the Hydraulic Power main (§ 14) would be 333 f/s, and the flow through a hole inch diameter, with c = 0.65, would be 40,000 gallons in 24 hours. Again the pressure in the air vessel in fig. 78, for a steady flow of V ft3/sec of water through a delivery pipe A ft2 in section, is DV2/gA2 lb/ft2. 367. Suppose that two fluids, water and steam for instance, are issuing by two nearly equal nozzles, of cross sections A and a ft2, from a vessel in which the (gauge) pressure is p; denoting the density of the steam by 8, then, according to Torricelli's theorem, G.H. α D the delivery in lb of the steam jet = α the momentum of the steam jet the energy or H.P. of the steam jet = α the energy or H.P. of the water jet AV 8 20 |