1849."I 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



263 



But rf. 



When in the equation (/) the altitude of pressure H' = is 

 assumed, we find for the quantity of discharge through a semi- 

 elliptical orifice of which the upper horizontal edge coincides with 

 the surface of the water, — 



w{.(aw)}=.<^.|.M5-i.i5-i.i.i::-..}. 



And when the integral is taken between limits j? ^ and j; = o, it 

 follows that 



a 



J^ </j;V{'<a=--^-)} =2a?{i-|.(i)-a,(j.i)-3^a.i.t)--- } 



= a2 . 0-4793 ; 

 and, therefore, — 



0. Q — k/>y{iga) . ba . 0-9586. 



By comparing equations (o and d), we find for the proportion of 

 the quantities of dischai-ge through two orifices which are equal 

 semi-ellipses, but in inverted positions, 



Q; : Qjj = 1-1504. : 0-9586; or, Q^ = 1-200 Q^,.* 



Ordinarily, the quantity of discharge through these orifices is 

 computed by the formula 



p. Q = 0-617 v'(+i'H')ai = c\/{igll') . a; 



where a is the area of the orifice, and H' the altitude of pressure 

 above the centre. In order to exhibit the deviation of this formula 

 from the strict results of the Torricellian law, a small table is 

 given below of the co-efficients of the quantities of discharge 

 through circular orifices for difl^erent altitudes of pressure. For 

 this purpose the equation (j) is employed, and in order to adapt 

 it to the object in view, a is put := b ^ r, and for the altitude 

 of pressure above the top of the orifice, H =; )nr ; which gives 

 H' = ii + r=z {m + I) r. 



This value being substituted in the equation mentioned, gives 

 ?. Q=aT)^V(4i,H')r2^.2{4-a.i)(i.i)(».+ l)-2 



-(i-i-H)(^-H)('«+ir* - (i•i•M-Tff■A)(M•l-l)('»-^l)-'' -••-•}. 



When H = 0, the top of the orifice lies in the surface of the 

 water, and m = and H' := r ; whence for this value of H', when 

 the last equation is compared with the preceding, — 



Q = 0-5924, \/{igr)r-w, c — 0-5924. 



Put successively for ?« the numbers i, i, |, 1 , and there will 



be found for the consequent values of the co-efficients c the fol- 

 lowing : — 



Values ofm J j J 1 2 3 4 



Altitude H' r ^r jr Jr 2r 3r 4r 5r ..(m + loir 

 Co-effi,;iente-5924 -6031 -6076 -6102 -6118 -6147 '6156 -6163 ... (iir)2... 



The mere inspection of this short table shows that the co- 

 efficient c in the equation (/)) can be considered constant, and 

 = ii")- only when the altitude of pressure exceeds lOt: For cal- 

 culating the quantity of water for altitudes less than 10;-, the 

 general equation (7) must be employed. For altitudes above I Or 

 we may, however, put Q =: (|Tr)2^(4^H').j-^ir. In the equation (q) 

 the co-efficient for great and small altitudes remains = (jt)-'^ and 

 thence it follows that the contraction is independent of the velo- 

 city. 



It must here be observed that the above values of c in the praxis 

 are capable of direct application only when the sinking of the level 

 of the reservoir is very small. In the preceding investigations, 

 the level in the reservoir was considered as constant ; without this 

 condition, the integration of equation (H § 9) would be much more 

 difficult. The experiments, on the contrary, show, with small alti- 

 tudes of pressure, a sinking of the level above the orifice — which 

 indeed is only small, but in strictness does not agree with the 

 theoretical suppositions. On these grounds, the formulie, when 

 applied for small altitudes measured immediately above the orifice, 

 require a correction depending on the sinking of the level. On 

 the other hand, for the altitude of pressure for constant water- 



* The area of the seml-elllpse Is aiir, and a4=J{a4w) . -. Substitute this value of ai 

 Id equation (o), anti omit the area and the co-efflcient; and then the velocity becomes 

 V(, = ^(4ga) . —— . By a similar method are found the velocities for all other orifices. 

 When in all the formula, a=4 for the radius of the orilice, the quantities of discharge 

 and velocities furciicuUi uniices aiedetermiuec. 



levels — that is, for altitudes in large reservoirs measuring 1 to li 

 yards above the orifice — the correction is inconsiderable. With 

 circular orifices, the sinking is probably much smaller than with 

 rectangular orifices ; for the upper edge of the orifice, when it 

 coincides with the surface of the water, has for the first form only 

 one point, — for the second form has its whole breadth, without any 

 pressure. 



The older experiments give, it is true, in contradiction to the 

 above table, an increase of the co-efficient c for small altitudes ; 

 only it is very probable that the altitudes were measured imme- 

 diately above the orifices, and therefore were found too small from 

 the sinking of the water-level; and that, for the same reason, the 

 increase of the co-efficient is a single inference from a measure- 

 ment of the altitude too small from the sinking of the level. 



The older experiments are in general but little adapted for in- 

 vestigation from theoretical inferences, as they partly were con- 

 ducted within too narrow limits, partly were not capable of being 

 compared together on account of the great diversity of the me- 

 thods of experiment, of the apparatus, and generally of all the 

 details. 



It is, therefore, very possible, that notwithstanding such expe- 

 riments, which for small decreasing altitudes give increasing co- 

 efficients, the CO- efficients of circular orifices, however, diminish 

 with the altitude of pressure, and follow an analogous law with 

 that which we shall hereafter find for square orifices. These 

 doubts can only be removed by as accurate and general experi- 

 ments as Poncelet and Lesbros conducted for rectangular orifices. 



11. 



Discharge through Quadrilateral Orifices. 



Let efcd be an orifice of the form of a trapezium, of which the 

 parallel sides are horizontal. Leta6 = H, ef—m, cd — l, bn=^b, 

 bo = .i, st=-y, bg — z. Then 



J ef ■ og 



St : e/=: og : bg ; and st = — - — ; 



or, 



y = 



m{z — J?) 



f\ — 



E7 



S^ — M 



Vg 



Whenx=-bii = b,i/ or st = cd:=l; and 



- , '"(^ — *) 



therefore I =: . 



z 



Obtain from this equation the value 



of z, and put it in the above expression 



for 3/, and it follows that 



tub — inj; -t- Ix 



y= b ■ 



Put this value of y in the equation 

 (H§9), and designate by k' the co- 

 efficient for orifices of quadrilateral and 

 like forms, and we have generally for 

 the quantity of discharge 



pmb — mjc + l.v , ,,,, . ^ ,^ 

 a. Q = // V( W 1 ''■^' VC H + ,r).* 



Put j/ (H -I- a') = t(, BXiAludx = «. Then/yudjr = u-y — I dy. 



/» /• -1 l~~nt 



Bat /udx = dx j/ (H -h or) = i(H -|- xY ; and dy = -^ dx. 



Therefore, Jydx ^/{n -\- x) - y.| (H +x)^ - §-^(ir(H +,.)*■ 



J^x{H + xy^ = liH + x)l 



But 



Hence, finally, substituting for y its value 

 Jydx{U+x) = 3 J ^^ (H+,r)T- 



2 2 l~m 

 '3 ■ 5 b 



(H-f-.,)*. 



Take this integral between limits x =: and x^=b, and the 

 equation (a) gives for the quantity of discharge 



b. Q = fr' ^{ig) . UliH+b)^-mm +§(/-m)5^^"-^t^%- 



* The numerical vulue of k' is constant only for the square and at eqnal altituHes <if 

 jjresBiire. It v.'ries, as we shall see herealier for other rectangular forms of the orifice, 

 aL-cordiiig tu a proportion depending ou the heights or vertical sides of these oribcta. 



