161 



THE CIVIL EXGINEEU AND ARCHITECT'S JOURNAL. 



[M^ 



That we may f,nve an exact idea of the resistance they offer to 

 the motion, let us suppose a conduit in which, perpendicular to its 

 axis, we have phiced a diaphragm ov thin partition pierced with an 

 orifice. The stream, on arriving at tliis point, will contract and 

 reduce itself to the size of the a|perture, taking a greater velocity 

 in proportion as the section is smaller; and this velocity will always 

 lie greater than it would have heen in this part of the conduit 

 without the jiartition. The force necessary to produce the extra 

 velocity, the direction of the motion remaining the same, will evi- 

 dently be due to the resistance offered hy the contraction. 



Let B be the diameter of the orifice, yn its coefficient for contrac- 

 tion. The velocity through this point requiring to he greater than 

 in the conduit, and following the inverse ratio of the sections, will 



D' 



be then expressed by ■0155r- — „v- ihe excesses of force, or loss 

 III- Iv 



of head arising from the contraction, will therefore he 



■01J5r(-:rT7:— l) = ■!,..„, ^, 



^ III- li^ I \in-li' 



In terms of the discharge, this resistance will be expressed by 

 •02519 Q=('_L__L). 



M. Navier, considering that the stream, on passinz out of the contraction, 

 ifuinedialely resumes the velocity proper to the conduit, instead of ttie dif- 



1 1 



ference between the squares of tlie two terms, 



\|lr\^' D'l 



of their difTerence, 



ImU- D-l 



- and — . takes the square 



But as this opinion is contrary to fact, as 



the experiments given in the next section will show, we must be careful in 

 adopiinjr a result founded on false premises. 



It is hut seldom, however, that we shall have to make use of the 

 above f<U'niula, for in a conduit l)i])e there ought not be any sensi- 

 ble contraction: should one accidentally be found, this formula 

 will serve to give us the value of its resistance It will generally 

 be slight; in some experiments made with sluice valves fixed in 

 the conduits of Toulouse, I found, after diminishing the section of 

 one of them by Jl^, that the discharge was only reduced -j-rij- 



21. If, in the same conduit, below the first contraction there he 

 a second, a third, &e., the resistance from each may be determined 

 by the above formula, and their sum taken. 



But, in order that these resistances may be thus added, they must be in- 

 dependent of each other; that is to say, the fluid, after passing through the 

 first contraction, must have recovered the general velocity of tlie conduit 

 liefore reaching the second. If it were not so, the fluid stream, after leaving 

 the first contraction, would preserve entirely, or in part, the excess of velocity 

 which it had acquired in order to pass tlirough; and a less effort would he 

 necessary for the second, and less in proportion as the distance between the 

 contractions was smaller. 



Eytelwein has made many experiments 

 which fully demonstrate this fact. He took 

 tubes 103 inch in diameter, at either end of 

 which was a cop|)er plate pierced with an 

 orifice -51 inch diameter; their length, or 

 distance between the orifices, being given in 

 the first C(dumn of the accompanying table. 

 They were fitted horizontally to a reservoir, 

 and the discharge made by each ascertained; 

 this discharge, as compared with the theoretic 

 discharge, which is represented by unity, is 

 contained in the second column: it goes on 

 gradually diminishing, and consecpiently indi- 

 cating the resistance increasing, in proportion 



::s the distance between the two orifices is greater. Eytelwein 

 ngain fixed in a tube 1-03 inch diameter, four thin plates, each 

 [lierced with an orifice of '256 inch diameter, and at a distance of 

 ■256 inch from each other; the discharge was then '022. When, 

 however, the i>lates were placed at the distance of ro3 feet from 

 each other, the discharge was not more than -0331. 



22. The observations we have made resj)ecting contractions 

 caused by thin plates pierced with orifices, apply equally to those 

 which would be produced hy very short tubes of a diameter smaller 

 than that of the conduit. I cite the 24th Experiment of Venturi. 

 This eminent philosopher, with great judgment, arranged his appa- 

 ratus to consist of two sorts of tubes alternately; the one B, B, 

 were -HS feet long, and ^-inch diameter; the other C, (J, were 

 I '9 inches diameter, and their length sometimes '289 feet, and 



sometimes ^564 feet. He at first made use of a single tube C; 

 then of two, of three, of four, and lastly of five: he successively 

 ap])lied these various combinations to a reservoir, using a constant 



Fig. 4. 



Iiead of 2^89 feet, and the following are some of the discharges 



obtained: — 



With a single tube, B -0444 cub. feet. 



With a tube, C, added -0329 „ 



With three tubes, C ■0252 „ 



With five tubes, C ^0202 „ 



I have attempted to compare these results with those by the 



methods of calculation I have given: the differences have been 



sometimes great, sometimes inconsiderable; thus, for the last case 



I have had ■0185 cubic feet. 



23. Notwithstanding the great irregularities which these results 

 present, they are well worthy attention, and principally on account 

 of the very striking manner in which they show the effect produced 

 hy enlargements in a pipe; an effect, carried above a certain limit, 

 altogether as prejudicial as that of contractions. 



Venturi's entire apparatus, which was 3^2 feet long, may be con- 

 sidered as a pipe ^inch diameter, having the five enlargements C. 

 It furnished, as we have seen, a discharge of ^0202 cub. feet. He 

 afterwards, with a tube of the same length, but of the uniform 

 diameter of 4-inch, obtained ^0327 cub. feet. The enlargements 

 thus diminishing the discharge in the ratio of 100 to 62. 



24. There is yet one other contraction that ought to be con 

 S'dered — that experienced by the fluid stream on its entry into a 

 pipe of less diameter than that which immediately precedes it. 

 The resistance arising from this contraction will evidently be the 

 same as if, at the entry of the pi]ie, we had placed a plate pierced 

 with an orifice of which the section should be to that of the pipe 

 as 111 to 1 {in being the coefficient belonging to the contraction); 

 and its expression will then be 



•02519 ^-7 _L-l); 



DA m-2 J' 



This is a special case of the general formula (20), where B = D. 



The value of in can only be approximatively. For a very short 

 pipe, as for cylindrical adjutages, it will be '82. But in pijies, 

 jiroperly so called, it approaches nearer to 1; and more so in pro- 

 jiortion to the length of the pipe, and even, according to M. Prony, 

 as the diameter is greater; so that in large conduits, the effect of 

 this contraction is very small. It is still further reduced by con- 

 necting pipes of two diameters by a conical length, gradually dimi- 

 nishing from one to the other. 



Lastly, as we have remarked (5), the effect of the contraction at 

 the head of a pipe is itnplicitly comprised in the values of the 

 coefficients of the fundamental equation; and its effect at the 

 entry of a pipe which branches from a larger conduit, will he com- 

 prised in the determination of the head of such branch, so that we 

 need not in any case make calculation of it. 



Observations on the Practical a)ii>licutitm of the Formutce. 



25. The coetBcienls of the formulae which we have given, especially those 

 concerning the principal resistance — that due to the friction against the inte- 

 rior of the pipe — have been determined by experiments made chiefly on 

 pipes of small diameter and of no gieat length (5); they have been gene- 

 rally well-bored pipes, well joined, and free from incrustations. But can 

 such formula; be safely applied, without modification, to conduits of a differ- 

 ent description — namely, to those used in large distributions of water? This 

 is a question which we must now examine. 



The pipes of which conduits are formed are almost always more or less 

 imperfect, from the effect of the mould, or in casting; their section is no 

 longer exactly circular, and consequently, cateris paribus, it is smaller than 

 it ought to he. Their interior surface presents inequalities which retard the 

 motion. When joineil, the axis of the whole is not always a line without 

 rehatement; the interior is not a perfectly cylindrical surface; the edges of 

 some of the pipes project, and the currents reaching these points, are 

 arrested, divided, and sometimes reflected back again: thus arise eddies in 

 the movement, loss of motive force, and consequently a diminution of the 

 discbarge. Even when the pipes are well cast, so that the channel is very 



