292 



THE CIVIL ENGINEER AND ARCHITECrs JOURNAL. 



[OCTOBEB, 



11 -\- b or II + / or II + m = PI. The equ.ition (r), for example, 

 gives in this case — 



Q = k\'{ig)^{ H'^ - 2(ir- im)i + (11- m)?} . 



13. 



e 



/: 



Co-efficient of Contraction of Square Orifices. 



In order to estimate the quantity of discharge from a square 

 orifice, the value of co-efficient k' must still be found. As the 

 magnitude of the contraction by § 7 is dependent on the mag-iii- 

 tude of the diagonal or central width of the orifice, the co-efficient 

 for different forms of the orifice must vary in proportion to these 

 widths or secant lines. The co-efficient for a square orifice must, 

 therefore, be considerably different from that for a circular orifice; 

 it may however be estimated as soon as the mean value of all the 

 secant lines of a square is known. 



First, then, the co-efficients for a square and a circular orifice 

 under otherwise similar circumstances, must be in the inverse pro- 

 portion of their contracting forces. 

 e. T /T Ti Now these forces, by § 7, are pro- 



portional to the radii or secant- 

 lines, and conseqently are as the 

 radius of the circle to the mean se- 

 cant of the square. By the mean 

 secant is here meant the mean value 

 of all the secants, as ci, which can 

 lie between cd and ce. 



In order to find these values, the 

 method in § 2 for estimating y will 

 be adopted. 



Let di::=x, ic=y, cd^=h!—ieb; then 



y = v{(iSO'-' + -r'}; and, /^(;u?=/rf^V{(iO'+.^°} 



Take this integral between the limits x = and x=: ^l ; then, 

 ' "ydj! = iil)- { i V2 + i log (1 -}- v^2) } . 



Divide now hy/dj; = J/, and we find lor the mean secant, 



a. »' = i/{V2 + log(l+V2)}. 



The sqjiare, however, of which the side =/ has an area equal to 

 that of tlie circle of which the radius =r; whence r-ir = P, and 

 / =:: r^ir. Substitute this value in the above expression, and we 

 find 



6. J/' = irv/7r{v2 + log(l-hV2)}- 



Call now the contracting force in the circular orifice /„, and in 

 the square orifice /,; then, fo '■ f'= r : y; and 



/, _ r _ 4 



/, y VT{V2 + log(I + v/2)}* 



We have seen in § 10, respecting circular orifices, that the co- 

 efficient of contraction, by adopting the true velocities, is constant 

 for all altitudes of pressure and ^ (i "■)*. By comparison of square 

 orifices with circular this, however, is no longer the case, while the 

 velocities in botli orifices at the same depth below the surface are 

 neither equal nor for different altitudes related by a constant 

 proportion. Hence it follows that the relation of these co-effici- 

 ents to one another, depends not merely on the contracting forces 

 but also on the velocities. As these velocities, however, for dif- 

 ferent positions of the orifices give results differing from one 

 another, the comparison ultimately consists in finding points which 

 with respect to the velocities are to one another in constant relation. 

 When such points ai-e found, the velocities which tal<e place at 

 them, will serve for a nearer computation of the co-efficient. Of 

 all points which can here come into consideration, those may be 

 considered most proper in which the pressures sustained by the 

 water towards the orifice are in equilibrium, or the centres of pres- 

 sure. 



The next step is to determine the relation of the co-efficients of 

 square and circular orifices to their velocities. Let the expression 

 Q = /cFV be taken, in which the quantity Q is the product of the 

 co-efficient Ar, the area of the orifice F, and the velocity V: then, 

 wh en Q remains invariable, the product k\ must be constant; and i 



k must increase in the same proportion in which V decreases, and 

 conversely. Hence it follows that the co-efficients are in inverse 

 proportion to the velocities; they are also in inverse proportion to 

 the contracting forces. Put, therefore, V^ and V, for the veloci- 

 ties at the points of comparison or centres of pressure, and k, k', 

 the co-efficients of contraction of circular and square orifices: 

 then k : Ic — /,V, :/uV„; and therefore for the co-efficients of 

 square orifices. 



Designate, for circular and square orifices, the distance of the 

 centre of pressure below the surface of the water by H^ and H, ; 

 the altitudes at the upper edges of the orifices by Ag and /», ; the 

 radius of the circle by r; and the side of the square by /. Then 

 we know that 



«° = '^o + '- + i(Af^'- "^ = '^' + i'+I5(£H0- 



Until now nothing is determined respecting the relative positions 

 of the orifices; they must however be assumed so that the mutual 

 relation of the velocities to one another may not be neglected. 

 This object will be attained when the under edges are at the same 

 depth, and then it follows that 



A„ -1- 2r = A, -I- ;; or, A^ -1- 2r — / = A,. 



Substituting this value in the above expression for H^, 



H, = Ao + 2r - y + '" 



12(Ao 4- 2;- - iO' 

 When both orifices are of equal magnitude, r'vz^P^ or rz= 



Substitute this value of r, and put A^ ^ ml, we find 

 . H„^[..+ ;L+|4..(„.+ -L)|--], 



^'"+f,-4+{'^-('"+J,-0}"'> 

 Vq designate the velocities at the centres of 

 and as ^ =; (ii^)'* we have by equation (c). 



V'^{2-t-log(l-Hv^2)} 



= 0-6064. 



Substitute this value in equation (rf), and we find for the general 

 expression for the co-efficients for square orifices, 



g. i'=-60';4-[ni + -5M190+ 



4ir(m+-5641S0; 



}*{ 



m + •628379 +r 



1J(»1+ 620379) J 



For m = 0, the altitude of pressure = 0, and we find k' = -5836. 

 For m — a very large value, the expressions in the brackets are 

 together nearly equal to unity, and i'= -eoei. For m = 1000, the 

 altitude of pressure H= 1000 1, and k' =; •60638. In the following 

 table the co-efficients for square orifices for different values of m, 

 from to 1000 are collected. 



Table I. 



The sinking of the level (§ 10) vertically above the orifice of 



IT 



the reservoir is observable for values of to = -y = 5, and has con- 

 siderable influence only when m is less than 5. We will therefore 

 call those altitudes of pressure for which m is greater than 5, for 

 which also the sinking of the level may be neglected without sen- 

 sible error, greater altitudes; and those for wkich m is less than 5, 



