Discharge 

 of Water 

 from Ori- 

 fices. 



498 



In the following Table, given by the Abbe Bos- 

 sut, he has compared the theoretical with the real dis- 

 charges from an orifice one inch in diameter, and the 

 different altitudes of the fluid in the reservoir. The real 

 discharges in column 3d were not determined by direct 

 experiment, but were ascertained with the precaution 

 indicated in the three preceding rules, and may be 

 considered to be as accurate as if they had been ob- 

 tained from direct experiment. The fourth column was 

 computed by M. Prony.* 



TABLE II. Comparison of the Theoretic tvith the Real 

 discharges from an Orifice one inch in diameter. 



Comparison 

 of the theo- 

 retic with 

 the real dis- 

 charges. 



HYDRODYNAMICS. 



as deduced from the preceding experiments of Bossut, t 

 Q=0.6l938AT,v/2gH, 



A being the area of the orifice iu square feet, H the al- 

 titude of the fluid in feet, T the time, g the force of gra- 

 vity at the end of a second, and Q the quantity of water 

 in cubic feet. As \/2 g is a constant quantity, and is 

 equal to 7.77125, we have 



Q = 4.818 ATv/H~for orifices of any form. 

 If the orifices are circular, and if d represents their dia- 

 meter, then 



Q= 3.7842 d 2 Tv/H. 

 From the second of these equations we obtain 



Q 



Discharge 

 ofWater 

 from Ori- 

 fices. 



Art 



4.818 



It appears from this Table, that the real as well as 

 the theoretical discharges are nearly proportional to 

 the square roots of the heights of the fluid in the reser- 

 voir. Thus for the heights 1 and 4, whose square roots 

 are as 1 to 2 feet, the real discharges are 2722 and 

 5436, which are to one another as 1 to 1.997, very 

 nearly as 1 to 2. 



By means of the formula in the preceding page, we 

 may easily apply the above Table to the determina- 

 tion of the quantities discharged under different alti- 

 tudes of water in the reservoir, and from orifices of 

 different sizes. Let it be required, for instance, to 

 determine the quantity of water discharged from an 

 orifice of 3 inches in diameter, under an altitude of 30 

 feet. Then, since the real quantities discharged are 

 in the compound ratio of the orifices, and the square 

 roots of the altitudes of the water, and since the theore- 

 tical discharge by an orifice 1 inch in diameter, under 

 an altitude of 15 feet is 16968 cubical inches in a mi- 

 nute, we have 1 v/ 15 : 9^ 30 = 16968 : 215961, the 

 theoretical discharge. But since the theoretical is to the 

 real discharge as 1 to .62, the above value being dimi- 

 nished in that ratio, gives 133309 cubic inches for the 

 real quantity of water discharged by the orifice. 



The following formula have been given by M. Prony, 



T= 



Q 



4.818 A y'H 



_ Q_ 



~ (4.818AT)' 



These formulae will be found to give very accurate re- 

 sults ; but if we wish to obtain a still higher degree of 

 accuracy, we must not Use the mean co-efficient 0.6194, 

 but the one in the Table which comes nearest to the 

 circumstances of the case. Thus if the head of water 

 happens to be small, such as 1 foot, then we must take 

 the co-efficient 0.62 133, and if it happens to be great, we 

 must use the least co-efficient 0.61716. 



In order to determine the velocity with which the 

 fluid is discharged, we must first obtain the theoretical 

 velocity, whichisV= v /32.174.v'H=8.0l6 ^/H in Eng- 

 lish inches. That is, the velocity acquired by falling 

 through any height H, is found by multiplying the 

 square root of the height by 8.016. But as the real ve- 

 locity of the issuing fluid is to its theoretical velocity as 

 0.6194 to 10, we have 4.965 V^H as the measure of the 

 real velocity, or in round numbers 5V/H; that is the ve- 

 locity in a second of time in English feet is five times 

 the square root of the height of the fluid in the reser. 

 voir ; or, if we prefer expressing these values in inches, 

 then since 32.2 feet = 772 inches, and v/772 = 27.78, 

 we have V = 27.78 v/H for the theoretical velocity, 

 and V = 17.206 y'H for the velocity at a simple orifice. 



In order, however, to obtain the velocity more accu- 

 rately, we should deduce the co- efficient of \/H, not 

 from the medium co> efficient in the preceding Table, 

 but from the co-efficient in the Table which approaches 

 nearest to the circumstances of the experiment. 



The following Table contains a series of experiments 

 by M. Michelotti, which were made on a most magni- 

 ficent scale, and with the utmost accuracy. As they 

 extend to apertures of three inches both square and 

 circular, and to altitudes twice as great as those em- 

 ployed by Bossut, they form an excellent supplement to 

 his experiments. We consider them indeed as much 

 more valuable than those of Bossut, as the quantities of 

 water discharged in each experiment were prodigiously 

 greater than his. The reservoir employed was 20 feet 

 high, and three feet square within, and had openings at 

 different distances from the top. The water flowed into 

 a cistern whose area was 289 square feet, and whose 

 figure was uniform, and the quantity of it was ascertain- 

 ed in French feet, by measuring its height in the cistern. 



To deter- 

 mine the 

 velocity of 

 the issuing 

 fluid. 



iMichelot- 

 ti's experi- 

 ments. 



See his Architecture Hydrauliyue, torn. i. p. 369. 



The meusures are in French feet, which are to English feet as 1066 is to 1000. 



