HYDRAULICS.] HYDROMECHANICS 475 Supposing the discharge into the air, so that Pi where the first terra on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of the coefficient of contraction at EF. Let c^O 64, the value for simple orifices, then the coefficient of velocity is c,= - L= = = 0-87 .... (2). The actual value of c c found by experiment is 82, which does not differ more from the theoretical value than might be expected if the friction of the mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH, l-00 82, It is easy to see rom e eq less than atmospheric pressure. Q = 82nV2^ from the equations that the pressure p at EF is eric pressure. Eliminating v lt we get -~ 3 h nearly . . . G o = pa - | Gh Ib per sq. ft. (3) If a pipe connected with a reservoir on a lower level is introduced into the mouthpiece at the part where the contraction is formed (fig. 58), the water will rise in this pipe to a height p nearly. G If the distance X is less than this, the water from the lower reservoir will be forced continuously into the jet by the atmospheric pressure, and discharged with it. This is the crudest form of a kind of pump known as the jet pump. 45. Convergent Mouthpieces. With convergent mouthpieces there is a contraction within the mouthpiece causing a loss of head, and a diminution of the velocity of discharge, as with cylindrical mouthpieces. There is also a second contraction of the stream out side the mouthpiece. Hence the discharge is given by an equation of the form where n is the area of the external end of the mouthpiece, and the section of the contracted jet beyond the mouthpiece. Convergent Mouthpieces (CasteVs Experiments}. Smallest diameter of orifice = 05085 feet. Length of mouthpiece = 2 6 diameters. An sic of Convergence. Coefficient of Contraction, Cc Coefficient of Velocity, Cr Coefficient of Discharge, C
999 830 829 1 36 1 -000 866 866 3 10 1-001 894 895 4 10 1-002 910 912 5 26 1-004 920 924 7 52 998 931 929 8 58 992 942 934 10 20 987 950 938 1-2 4 986 955 942 13 24 983 962 946 14 28 979 966 941 16 36 969 971 938 19 28 953 970 924 21 945 971 918 23 937 974 913 29" 58 919 975 896 40 D 20 887 980 869 48 50 861 984 847 The maximum coefficient of discharge is that, for a mouthpiece with a convergence of ly 1 24 . The values of c, and c c must here be determined by experiment. The above table gives values sufficient for practical purposes. Since the contraction be yond the mouthpiece increases with the convergence, or, what is the same thing, c c diminishes, and on the other hand the loss of energy di minishes, so that c, increases with the convergence, there is an angle for which the product c c c v , and consequently the discharge, is a maxi mum. 46. Divergent Con- oidal Mouthpiece. Suppose a mouth piece so designed that there is no abrupt change in the section or velocity fig. 58. of the stream passing through it. It may have a form at the inner end approximately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 59. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let o>, v, p be the section, velocity, and pressure at CD, and n, v lt p 1 the same quantities at EF, p a being as usual the atmospheric pressure, or pressure on the free surface AB. Then, since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece, If the jet discharges into the air, p l =^ a and 7- 2 5--*; or, if a coefficient is introduced to allow for friction, where c e is about 97 if the mouthpiece is smooth and well- formed. Q = fl/i 1 ! = c,n/ 2yfi . Hence the discharge depends on the area of the stream at EF, and not at all on that at CD, and the latter may be made as small as we please without affecting the amount of water discharged. There is, however, a limit to this. As the velocity at CD is greater than at EF the pressure is less, and therefore less than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p = 0, the continuity of flow is impossible. In fact the stream disengages itself from the mouth piece for some value of p greater than (fig. 60). From the equations, P.,P-( V 1^* G G v 20 ;
Let = wi. Then
