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52
[DISCHARGE OF ORIFICES
HYDRAULICS


The values of cf and cs must here be determined by experiment. The above table gives values sufficient for practical purposes. Since I the contraction beyond the mouthpiece increases -E 1.:;;: . - Tgl with the ionvergenicr, what is the same t ing, N! Zi 66 diminishes, and on the 5 other hand the loss of fi I energy diminishes, so Q Q that Uv increases with Q J the convergence, there “ Q 2;:fjj;;g is an angle for which the § -- ', . product c¢ cv, and con/ N sequently the discharge, I -is a maximum. |= K 5 § 5I. Divergent Corz)f oidal Mouthpiice.-Sup- pose a mout piece so I designed that there is - Q no abrupt change in the 1. L, , l[ section or velocity of "" the stream passing through it. It may A 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. 60. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let av, t', p be the section, velocity and pressure at CD, and Q, 1/1, pl the same quantities at EF, pn being as usual the atmospheric pressure, or pressure on the free surface AB. Then, FIG. 59. DISCHARGE WITH VARYING HEAD A B


since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece, h-I-p.1/G=v2/2g-l-p/G —|— =v12/2g-l-pl/G. Q Ili; the jet discharges into I t eair, p1=p.1;an 1 1/12/2g=h; Zi vi = / (2gh); 5 or, if a coefficient is introduced to allow for friction, ic E “vi =cv/ (2gh): 2 | where Cv is about o~97 if I the mouthpiece is smooth 5' ' 5 and well formed. » ~'—*- '*-"- i'-“ "'-'*"" ' Q = S2211 =6vQ/ (2gh). Xf' Hence the discharge depends on the area of the stream at EF, and not at ID F all on that at CD, and the latter may be made as small as we please without FIG 60 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 le ss than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p=o, the continuity of flow is impossible. In fact the stream disengages itself from the ° mouthpiece for some value of p greatir than o (fig. 61). From the equations, / P/f' =1>f»/G * (vi -11 2)/22° l WV Let il/w =m. Thenl U = 7] m; s P/G=pa/Giv.2<m2-1>/2g

=Pu/G-(Wiz-1)}l;

" ' whence we find that p/G $

will become zero or negative

~ Q/~ § ~{1<h+1>./G>/hz = /11 +Pa/Ghf; FIG 5I or, putting pa/G=34 ft., if S2/w§ ~/{(h-l-34)/h}, In practice there will be an interruption of the full bore How with a less ratio of Q/cv, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is Q =w~' f2g(h+Pa/GN: that is, the discharge is the same as for a well-bell mouthed mouthpiece of area w, and without the expanding part, discharging into a vacuum. § 52. Jet Pump.-A divergent mouthpiece may be arr angled to act as a pump, as shown in hg. 62. The water which supplies the energy required for pumping enters at A. The water to be pumped enters DD where the velocity is greatest DD the stream enlarges in section, at B. The streams combine at and the pressure least. Beyond . 6" ~. // f mM/ 'af

/ / g 3 EQTQ
 f

% // f /' /0 / 'S s)§ / //, § gg£f M 5 .,

.L gg; - f p s § T * '< <1s. 駠;?§ ?;

E§ " , . H el; =, / / f-, sge gas. as ~< -§ s- " = / /, /. /V t s =g Mk, M //, I / i s D i' / f / .», '/, ,¢ 'mls / / r I if %>s~ Y/ / nhl / 7/sg N/ l;' Z // / J li* I //§ fi /M o / /, <>=®3 >.§ T;§ fj';?-Jlfgill* at i/ if g/ $54 » il [1 li ;Z” hi, s i ii g

fi. lilli . fi ll I U

A r and its pressure increases, till it is sufficient to balance the h ead due to the height of the lift, and the water Hows away by the discharge pipe C. FIG. 63 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pumping; "" ' "' "" """"'"""' ' " 7 5§ :§ ?'§§ 71 ia;-':vi'E=.; if I """!“"" I I a if i) 1? - fK - .. - - -~ .; -;-. - -1——~ .: a J. Q QE 1 D D f”2 l? I B v .i» FIG. 63. B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped. § 53. Flow from a Vessel when the Effective Head 'varies with the Time.—Various useful problems arise relating to the time of emptying and filling vessels, reservoirs, lock chambers, &c., where the How is dependent on a head which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section. Time of Emptyirzg or Filling a Verlical-sided Lock Chamber.-Suppose the lock chamber, which has a water surface of Sl square ft., is emptied through a sluice in the tail gates, of area ai, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 64) be the initial difference of level, h the difference =;E?-E15-;§ “1§ —. - -C»—-H dh A 1, , x . I It 1 i Tuul water lamb 3»;§ a~;T~§ e§§ >- -ésezf

F IG. 64. of level after l seconds. Let -dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount -Sldh, and the outfiow from the sluice in the same time is cw/ (2gh)dt. Hence the differential equation connecting h and t is

caul (2gh)dt-i-Qh =(J.