Apr. 33,1917 
Flow through Submerged Rectangular Orifices 
109 
DERIVATION OF FORMULA FOR MODIFIED RECTANGULAR ORIFICES 
Several unsuccessful attempts were fnade to derive a simple and accu¬ 
rate expression of the variation of the exponent and coefficient values in 
the individual equations No. 13 to 23, inclusive. As has been previously 
mentioned, the exponents and coefficients plot as a series of disconnected 
curves with the depths and lengths of orifices as the governing factors, 
apparently. The difficulty was experienced in determining just what fac¬ 
tors should be used in an expression of the law of variation; and, though 
the following form is sufficiently accurate for practical purposes, it does 
not faithfully represent the variation. A more exact expression could 
have been obtained by using the same factors and expressing the varia¬ 
tions as curves instead of straight lines, but the resulting formula would 
have been so complicated as to make it of doubtful practical value. It 
will be observed that the discharge formula appears to be more compli¬ 
cated than it really is, and the influence of velocity of approach has been 
expressed in terms of the wetted area of the channel of approach, which 
may be determined more easily, because there is a standard size of box 
for each length of orifice. 
The experimental discharge data for the standard conditions of orifices 
and orifice boxes were plotted logarithmically against the areas of the 
orifices. The resulting series of curves were for each constant difference 
in head. From these curves the smoothed or balanced discharge values 
were taken and plotted logarithmically against the difference in heads. 
The equations of the average straight-line curves drawn through these 
points are given in Table II. The exponents in these equations were 
plotted against the ratio of the area of the orifice in square feet to the 
area of the wetted cross section of the channel of approach, as shown 
in figure 12. As previously noted, the exponents are in groups for the 
several areas of orifices with the same depth of orifice, and each group 
forms a detached curve which is probably parabolic in shape. To avoid 
a very complicated expression of their law of variation, they were 
assumed to be represented by straight lines which were parallel and had 
equal intercepts on the Y axis. The equation of each individual line is 
given in figure 12. The constants in these equations were plotted 
against the depths of the orifices (fig. 13), and the equation of the 
resulting curve was obtained. The substitution of this value in the 
equations given in figure 12 and with an average value for the coefficient 
of the ratio gave n =0.4945 + 0.05 jd — — ^ — as the general expression 
of the exponent of the head. 
