514 



HYDRODYNAMICS. 



Motion of The fourth column in the preceding Table is calcula- 



Water in A/ (D* d*) 



Pipes and ted from the formula q = Q X yr, ' which is 



Canals. 



v -Y- ' thus obtained. We have already seen in Chap. I. of 

 Part II. that the pressure of the fluid on the pipe is 



measured by h =rj-. Then, if Q is the quantity of 



water which would have been discharged in a given 

 time under the head or pressure h, the quantity of wa- 

 ter q discharged in the same time under the head 



or pressure h =rj- , will be thus found Q : q = */ h : 



?=Qx 



D 2 



The agreement of the formula with the experiments 

 is very striking. From this method of considering the 

 subject, M. Bossut deduces a very simple method of 

 determining the discharge from a long tube subject to 

 friction from the expenditure of an orifice perforated in 

 its sides. Let x denote the ratio of the expenditure of 

 the proposed pipe having regard to friction, to the 

 expenditure upon the supposition that there is no fric- 

 tion ; or, which is the same thing, let x = . By sub- 

 stituting x in place of =75 in the preceding formula, 



we have q = Q -/(I *'), and * = H. Let 



us now suppose that the tube has 2 inches diameter, 

 that the Q head of water is 3 feet, that the lateral 

 orifice is 6 lines, and that it discharges at the orifice 

 1000 cubic inches in a minute. This orifice, as appears 

 from former experiments, would give 1 178 cubic inches 

 in a minute, if the extremity of the pipe were stopped, 

 that is, Q= 1178 cubic inches, whilst q is only 1000 

 cubic inches. By putting these values in the equa- 



2\ 



we have x 0.5289. But by 



tion x = 



. 

 Q 



Table II. p. 498, this additional tube would give 24504 

 cubic inches in a minute, abstracting the effects of fric- 

 tion ; hence the effects of friction being included, it will 

 discharge 0.5289x24504= 12952 cubic inches in a mi- 

 nute. The preceding observations are also applicable 

 to inclined tubes, whether straight or curved. 



In the formation of pipes, it is necessary to give them 

 a much greater thickness than that which is necessary 

 to resist the pressure indicated by the preceding Table, 

 for the pipes are exposed to several forces which are not 



taken into consideration. The following Table con- Motion of 

 tains the thickness of leaden and iron pipes, which were Water in 

 used in France in the time of Bossut. 



The thickness of pipes ought to increase with the 

 head of water, and the strain should always be calcula- 

 ted from the whole height of the reservoir, and upon 

 the supposition that the pipe is stopped at one end. 



Bossut' 's Experiments on the Motion of Water in Canals. 



The experiments of Bossut on this subject were made Bossut's . 

 upon an open canal, the bottom of which was on a level periments 

 with the bottom of the reservoir from which the water on the mo- 

 flowed. The orifice by which the water issued into the tion . of * 

 canal from the reservoir had constantly a horizontal 

 width of 5 inches, but the height of the orifice was made 

 to vary by raising or depressing a.slider, so as to obtain 

 a rectangular opening of various heights. In order to 

 measure the velocity of the water in the canal, Bossut 

 tried various ways ; but he ultimately preferred the me. 

 thod of finding it by observing the time which elapsed 

 between the opening of the orifice, and the arrival of the 

 water at different parts of the canal. The velocity thus 

 found is obviously less than the velocity of the water when 

 the current is perfectly established. But there is a con- 

 stant ratio between these two velocities, in consequence 

 of which the one may be safely inferred from the other. 

 The canal was 105 feet long, and was divided into five 

 equal parts, and also into three equal parts ; so that 

 each of the fifth parts was 2 1 feet, and each of the third 

 parts 35 feet long. In order to ascertain the arrival of 

 the water at these different parts of the canal, small 

 wheels like those used by children were placed at each 

 point of division; and the commencement of their mo- 

 tion, which indicated the arrival of the water at that 

 point, was instantly perceived by the person who count* 

 ed the oscillations of the pendulum. When the canal 

 was horizontal, the following were the results. 



TABLE VII. Containing the Velocity of Water in different parts of a Rectangular Horizontal Canal 105 feet long, 



under different Altitudes of Fluid in the Reservoir. 



