500 



HYDRODYNAMICS. 



Discharge 

 of Water 

 from large 

 Openings. 



Table cal- 

 culated for 

 weirs, by 

 DrKobison. 



It appeared indeed that AF depended on the form of 

 the wasteboard, as might have been expected. When 

 the board was very thin and had a considerable depth, 

 AF was much greater than when the board was thick 

 or narrow, and placed on the top of a broad damhead, 

 as in Fig. 8. 



Du Buat's general formula, viz. 



D = 



G (l (^\ H* may be accommo- 



dated to any ratio between AF and AL, in place of 

 the ratio of | adopted in the formula. Thus, if AF = 

 in x AL, m being a fractional co-efficient less than 1, 

 the formula becomes 



Dr Robison has calculated the following Table from 

 Du Buat's formula, which is suited to English inches. 



TABLE VI. Containing tie quantity of Water dischar- 

 ged over a Weir. 



We have added to this Table a third column contain- 

 ing the quantities of water discharged, as inferred from 

 experiments made in this country, and examined by Dr 

 Robison, who found that they in general gave a dis- 

 charge v 3 greater than that which is deduced from Du 

 Buat's formula. We would recommend it therefore to 

 the engineer to employ the third column in hia prac- 

 tice. 



The preceding Tables and formula suppose that the 

 water from which the discharge is made is perfectly 

 stagnant ; but if it should happen to reach the opening 

 with any velocity, we have only to multiply the area 

 of the section by the velocity of the stream. 



When the quantity of water discharged over a weir is 

 known, the depth of the edge of the wasteboard, or H, 

 may be found from the following formula. 



11.41 72 l 



over a weir when the height H is one inch, and the Discharge 

 real discharge to the theoretical discharge as 9536 to of Wate ' 

 1000. These numbers, however, suppose the length ^ ( 

 of the weir to be infinite, or to be so great that the con- .__ -_- 

 traction at its two ends produces no perceptible effect 

 in diminishing the discharge. The formula, therefore, 

 of Michelotti includes only the contractions produced by 

 the upper edge of the wasteboard. 



In order to calculate the discharge of rectangular Experi- 

 orifices reaching to the surface, M. Eytelwein repre- mentsofS 

 sents the velocity, which varies as the square root of E >' telwtan> 

 the height, by the ordinates of a parabola and the quan- 

 tity of water discharged by the area of a parabola T of 

 that of the circumscribing rectangle. Hence the quan- 

 tity of water discharged may be found by taking y of 

 the velocity due to the mean height, and allowing for 

 the contraction of the vein. This mode of calculation 

 M. Eytelwein has found to agree wonderfully with the 

 experiments of Du Bual already given, as well as with 

 several accurate experiments of his own. 



M. Eytelwein takes the case of a lake, in which a 

 rectangular opening is made without any lateral walls, 

 three feet wide, and reaching two feet below the surface 

 of the water. In this case, as appears from the follow- 

 ing Table of co-efficients given by that engineer, the 

 co-efficient for finding the velocity as corrected for 

 contraction, is 5.1. Hence H being the height, we 

 have |\/H X 5.1 ; and since H = 2 feet in the present 

 instance, we have the corrected mean velocity =: 4.8 

 feet ; and as the area is 3x2 = 6, the quantity of 

 water discharged in a second is 28. 8 cubic feet. Put- 

 ting C for the co-efficient corrected for contraction, W 

 the width of the aperture, and H its depth below the 

 surface, we have the general formula, 



for the quantity of water in cubic feet according to 

 Eytelwein. 



As the same co-efficient answers for a weir of con- 

 siderable extent, we may deduce from the preceding 

 formula the depth or breadth necessary for the dis- 

 charge of a given quantity of water. Thus let it be 

 required in a lake with a weir three feet broad, and in 

 which the water stands five feet above the weir, to know 

 how much the weir must be widened in order that the 

 water may stand a foot lower, we have the velocity 

 = T v' 5 X 5.1, and the quantity of water = T -v/ 5 X 

 5.1 X 3 X 5; but as it is required that the height 

 H shall be reduced one foot, or from 5 to 4, we have 

 the velocity suited to this rr T ^/4 X 5.1, and conse- 

 quently the section will be 



TV /5X5.1X3X5 



7.5 

 4 



Experi- 

 ments of 

 JHchelotti. 



The experiments of Michelotti give 0.2703 v'H for 

 the number of cubical inches discharged in a second 



|^/4X5.1 v^4 



and the height is 4, the breadth must be 



4.19 feet. 



If the surface of water always stands at the same 

 height AB in the vessel ABCD, Fig. 9. and if the la- 

 teral orifice, of considerable magnitude, is m n op, then 

 we have only to determine by the preceding methods 

 the quantities of water discharged by the open orifices 

 rpos, rmns, and the difference between these quan- 

 tities will give the discharge for the orifice mnop. The 

 same result may be obtained with nearly the same ac- 

 curacy, by taking the velocity due to the centre of gra- 

 vity of the orifice below AD, and correcting it by its 

 proper co-efficient. 



On the dis- 

 charge from 

 vertical ori- 

 fices of con- 

 siderable 

 magnitude 

 with a con- 

 stant head 

 of water. 



PI.ATJE 

 CCCXVIII. 

 fig. 9. 



