470 HYDKOMECHANICS [HYDRAULICS. A a V. THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES. 33. Minimum Coefficient of Contraction. Re-entrant Mouth piece of Borda. In one special case the coefficient of contraction can be determined theoretically, and, as it is the case where the convergence of the streams- approaching the orifice takes place through the greatest possible angle, the coefficient thus determined is the minimum coefficient. Let fig. 42 represent a vessel with vertical sides, 00 being the free water surface, at which the pressure is p a - Suppose the liquid issues by a hori zontal mouth- J2 O_ piece, which is re-entrant and of the greatest length which permits the jet to spring clear from the inner end of the orifice, without adhering to its sides. With such an orifice the velocity near the points CD is neg ligible, and the pressure at those points may be -^ taken equal to the hydrostatic pres sure due to the depth from the free surface. Let n be the area of the mouthpiece AB, o> that of the con tracted jet aa. Suppose that in a short time t, the Fi S- 42 - mass OOaa comes to the position O O a a ; the impulse of the horizontal external forces acting on the mass during that time is equal to the horizontal change of momentum. The pressure on the side OC of the mass will be balanced by the pressure on the opposite side OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (p a + GA)fi. On the surface an 1 section AaaB of the jet, the horizontal re sultant of the pressure is equal to the atmospheric pressure p a acting on the vertical projection AB of the jet ; that is, the resultant pressure -is- p a ft. Hence the resultant horizontal force for the whole mass OOaa is (p a + Gh)n-p a a = Ghn. Its impulse in the time t is GkCtt. Since the motion is steady there is no change of momentum between O O and aa. The change of horizontal momen tum is, therefore, the difference of the horizontal momentum lost in the space OOO O and gained in the space aaa a . In the former space there is no horizontal momentum. The volume of the space aaa a is uvt ; the mass of liquid in that Equating impulse to (I G space is uvt ; its momentum is 9 g momentum gained, wv* 9 gh n v^ = 2gh, and But a result confirmed by experiment with mouthpieces of this kind. A similar theoretical investigation is not possible for orifices in plane surfaces, because the velocity along the sides of the vessel in the neighbourhood of the orifice is not so small that it can be neg lected. The resultant horizontal pressure is therefore greater than Ghn, and the contraction is less. The experimental values of the coefficient of discharge for a re-entrant mouthpiece are 5149 (Borda), G 5547 (Bidone), 5324 (Weisbach), values which differ little from the theoretical value, "5, given above. fl 4 - ocit y f filaments issuing in a Jet. A jet is composed fluid filaments or elementary streams, which start into motion at some point in the interior of the vessel from which the fluid is dis charged and gradually acquire the velocity of the jet. Let Mm. ng 4d, be such a filament, the point M being taken where the velo city is insensibly small, and m at the most contracted section of the jet, where the filaments have become parallel and exercise uniform mutual pressure. Take the free surface AB for datum line, and let A B p lt v 1} h v be the pressure, velocity, and depth below datum at M ; p, v, h, the corresponding quantities at m. Then 26, eq. (3), -i + ii = _p + |__A (1). But at M, since the velocity is insensible, the pressure is the hydro static pressure due to the depth ; that is, ^ = 0, p^pa + GJi^ At m, p=p a , the atmospheric pressure round the jet. Hence, insert ing these values, = /2gh = 8 025 V/i (2); (2a). That is, neglecting the viscosity of the fluid, the velocity of fila ments at the contracted section of the jet is simply the velocity due to the difference of level of the free surface in the rese rvoir and the orifice. If the ori fice is small in dimen sions compared with h, the filaments will all have nearly the same velocity, and if h is measured to the centre of the orifice, the equa tion above gi^es the mean velocity of the jet. Case of a Submerged Orifice. Let the orifice discharge below the level of the tail water. Then usin 26, Fig. 44. the notation shown in fig. 44, we have at M, ^ = 0,
- at m, z> Gh 3 +p a . Inserting these values in (3),
Pa -- h. 2 (3), where h is the difference of level of the head and tail water, and may be termed the effective head producing flow. Case ivhcre the Pressures are different on the Free Surface and at the Orifice. Let the fluid flow from a vessel in which the pressure is p into a vessel in which the pressure is p, fig. 45. The pressure p will produce the same effect as a layer of fluid of thickness -^2 added to the head water ; and the pressure^ will produce the same effect as a layer of thickness G added to the tail water. Hence the effective difference of level, or effec tive head producing flow, will be Fig. 45. Po
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