NOTCHES AND WEIRS 



141 



ART. 540. EXPERIMENTAL RESULTS AND FORMULAE. 



Adopting the base formula Q = K b H^ cub. ft. per sec., where K = 



- x/ 2 g C, many experiments have been carried out to determine the 

 o 



value of K. Of these, perhaps those of Francis, Fteley and Stearns, 

 Smith, Bazin, and Williams are the most reliable. 



Francis, 1 in a preliminary series of experiments carried out in 1848 

 and 1851, assumed the formula Q = K b H x , and found x to vary slightly 

 with the head, having a mean value 

 1-47. 



In his later experiments, however, 

 carried out in 1852, he adopted, for 

 convenience, the constant value 1-5, 

 and determined the corresponding 

 values of K. 



From these experiments, carried 

 out on weirs from 8 to 10 feet long, 

 and with heads ranging from *62 

 to 1'56 feet, he deduced the empirical 

 formula 



Q = 3-33 ( b H } H* cub. ft. per sec. 



for the flow over a sharp-edged rectangular notch, where n = 2, 1, 

 or 0, according as the stream has 2, 1, or end contractions, the bottom 

 contraction being free in every case. 



Thus for two end contractions Q = 3 '33 (b '2 H) H$ c.f.s. 

 one contraction Q = 3'33 (b 'I H} H% c.f.s. 



no Q = 3-33 b H% c.f.s. 



That this formula, though empirical, has a rational basis may be seen 

 if we consider the state of affairs in a long and shallow notch having two 

 end contractions (Fig. 77). 



If the notch is sufficiently broad, a certain section in the middle will 

 have stream lines which, as seen from above, are parallel, while the two 

 end sections have stream lines which suffer an amount of curvature 

 depending on their distance from the ends. The distance over which this 

 curvature extends is found to be approximately m H, where m = 1'5, 



FIG. 77. 



Lowell Hydraulic Experiments (New York, 1883), 



