Motion of 



Water in 



1'ipes and 



Canals. 



528 



-14930 hence we shall obtain 1.7 136 for the velocity 

 to feet per second, or 20.5632 for the velocity ,n .nche., 

 which, by the rule already explained, gives 1 

 Scotch pints per minute. 

 Quantity of water delivered 

 by the pipe, 



HYDRODYNAMICS. 



V = 0. 



189.4, Scotch pints per 

 minute. 



In applying this formula to Example II. in p. 526 % re- 

 lative to the velocity in a canal as measured by Mr W att, 

 we have d =29.126 and/=8 inches. 



iloiion <rf 



Water in 



Pipes and 



Canals. 



uy me pf**, - - - , 



Ditto determined by Eytel- 189.77 



wein's formula, .... 

 Ditto by Du Dual's formula, ,~~.-~ 

 From which it appears, that in this case Eytelwein s 

 formula is the most correct of the two die error being 

 only 0.37, while in Du Buat's it is 1 27 S ot h 



Kytelwein's 

 rale for ca- 

 nals and ri- 

 vers. 



tv ot water HI CHUB*" iiv * j ^ . 



aiders the friction as nearly proportional to the , square 

 of the velocity, not because a number of part cles pro- 

 portional to the velocity is torn asunder in a time pro- 

 portionally short, but because, when a body is moving 

 in lines of a given curvature, the deflecting forces are 

 as the squares of the velocities; for it is obvious, , to 

 the particles of water which touch the sides and bottom 

 of the canal must be deflected, in consequence oft 

 vations and depressions on the surface upon which they 

 slide nearly into the same curvihneal path, whatever 

 be the velocity with which they move. We may there- 

 fore consider the friction as nearly proportional to the 

 square of the velocity, and as nearly the ^ " 

 denths It will, however, vary according to the surface 

 of e flu d which is in contact with the so id, m propor- 

 tion to the whole quantity of fluid j that is the friction 



or a dven quantity of water will be directly as the sur- 

 face of the bTttom and sides of a canal, oras the penme- 



eoflhe section in contact with the water ; or supposing 

 the whole quantity of water to be spread <*" 



Stttt^tt 



J ^Hfe?to=Ra 



"eight, or the fore, that urge, t he p.rtj cle, .long the 

 inclined plane, will vary as the height of t 



," teC.th i, given, or a, the fall in any given d,,. 

 Si Hence it S>W that the fnct.on, which i. 



SsfettiSfc^iSijSa 



&&&ss&? ^3r= 

 afe^r^s=2B 



^*F^!rt^l:ffiSHft 



Hence =0.91^8x29-126 = 0.91 X 15.264 = 13.890, 

 a result which agrees very nearly with the mean ve- 

 locity as ascertained by Mr Watt. 



The preceding fonnula is applicable only to a canal, 

 or to a straight river flowing through an equable chan- 

 nel M Eytelwein has shewn that the velocity is in 

 general a little greater, when the bottom is horizontal 

 than when it is parallel to the surface, and that the 

 velocity in curved channels is always greater on t 

 convex than on the concave side. It is not easy to 

 give a rule for the decrease of the velocity from the 

 surface to the bottom of a stream of water, as it 

 sometimes found to be a maximum below the surface. 



The following are the velocities in the Arno and 

 the Rhine : 



ARXO. 



Depth in Velocity in inches 

 feet. per second. 



2 39^ 



4 38 



8 37 



16 33- 



17 31 



M Eytelwein considers that an approximate value 

 of the mean velocity may be obtained by deducting 

 ^ for every foot of the whole depth. 



- . 



and 





^ the velocity of Water 

 open Canals. 



IN our history of Hydrodynamics, vre have already 

 uiven a general view of the labours of Chezy, Girard, 

 and Prony, in the composition of formulae I 

 mining the velocity of water in conduit pipes and open 

 Canals As the formula obtained by these eminent en- 

 Sneers have all the same character, both from their ex- 

 treme simplicity, and from their containing only alge- 

 brakal quantities, we have thought it proper to give an 

 account of them in the same Section. In doing this, 

 we shall adopt the notation of M. Prony, and reta 

 coefficients as he has given them in French metres. 



Investiga 

 tions of 

 Chezy, G 

 rard, aotl 



Tronr. 





to preserve ts equay 



be proportionally increased or diminished by 

 the Square of the velocity, in the ratio of the 

 mean depth, or the velocity in the ratio of its 

 TSS U expect, therefore, that the vdo- 

 cities will be conjointly as the square root ot the 

 dSrS depth, and of the fall in a given distance 

 or as amean proportional between these two lines. If we 

 Uke tToTng ish miles fora given length, we must find 

 Snean proportional between the hydraulic mean depth, 

 and theVin two English miles ; and having ascertain- 

 ed the relation which this bears to *e velocity ma pa, 

 ticular case, we may easily determine it in all other case 

 According to M. Eytelwein's formula, this mean pro- 

 portionaHs ^ of the velocity, or 0. 9 1 times the velocity 



hydraulic mean deth m 



The following are the symbols which he employs : 

 x the leneth of the pipe or canal. 

 I = Se 2rence of level between the two extremi- 



ties of the pipe. 



, - the area of the section of the pipe or canal. 

 = the perimeter of the section in contact with the 



s - the accelerating force of gravity, or 32.174. 



I) = the diameter of the tube. 



R _.l = the mean radius, or the hydraulic mean 



I = the d sFne'of the inclination of the pipe or ca. 



nal. 

 U=the mean velocity in the section a. 



being the mean pro 



ve have 





