1S.50."| 



THE CIVIL ENGINEKR AND ARCHITECT'S JOURNAL. 



131 



Art. 1.— of SIMPLE CONDUITS. 



In hydraulics, and particularly in connection with water-works, 

 the name of conduit is jriven to a long scries of pipes, joined 

 exactly one to another. The conduit is called simple (in opposition 

 to a -ii/stein of coudnit.i) when it consists of only a single line of 

 pil>es, conducting to its extreme end all the water it receives at 

 its origin. 



1. Straiout Co.nduits of Uniform Diameter. 



Manner nf expressing the Resistance. 



2. For greater simplicity, let us unite in one the two forces 

 which tend tti produce the velocity of flow — the pressure AC at 

 the head of the conduit, and that of FD which arises from the 

 slope: for this purpose, let us imagine that the given conduit CD 

 is placed horizontally at HI, at the bottom of a reservoir, of which 

 the depth AII = AC'+ FD = ED. Nothing will be changed in the 

 data of the problem: we shall have the same force and the same 

 resistance, the latter being independent of the position of the 

 conduit. 



The force of pressure by reason of which the water tends to run 

 out, or more immediately, the vertical height ED, which is the 

 difference of level between the orifice of discharge and the surface 

 of the fluid in the reservoir, is called the head {charge de la con- 

 duite). We shall designate it by H. 



If the conduit offered no resistance to the motion, setting aside 

 the effect of contraction at the entry, the water would run out 

 with a velocity due to this total height, as we have just seen. But 

 it is not so: the resistance of the sides opposing an obstacle, 

 diminishes this velocity; it consequently absorbs a portion of the 

 motive head H. The flow takes place only by virtue of the 

 remaining portion; which portion is simply the height due to the 

 velocity of discharge, or indeed to the velocity at any point of 

 the conduit, since the motion therein is uniform, and the section 

 everywhere the same. 



Let V be this velocity, _ - will be the height due to the velocity 



V' 



or the effective portion of the head; H will then be the por- 

 tion absorbed by the resistance. 



3. We have thus expressed, by the height H, the effort or the 

 force of pressure which drives the water in the conduit; by the 



height -, the force which produces the discharge; and by another 



I'- 

 lineal quantity II — — , the resistance or negative force: although 



it is a principle in mechanics that forces of pressure, or efforts 

 equivalent to weights, ought also to be expressed by weights. I 

 will explain myself on this subject. 



We have, in a former chapter, seen that the absolute pressure on 

 a fluid horizontal plane, or portion of that plane, designated by .?, 

 waspiH'bs-, p being the specific gravity of the pressing liquid. 

 Since, according to the laws of hydrostatics, the pressure is equal 

 on every part of this plane, it will be sufficient, and at the same 

 time convenient, to consider but one part only; this will be an 

 infinitely small one, which we may suppose always of the same 

 area; then s being constant, the pressure will vary only with the 

 specific gravity or the nature of the liquid, and the height of its 

 column: it is in this sense that we say that the height of the 

 column of mercury in the barometer expresses the pressure of the 

 atmosphere. If the pressing liquid remain the same (as will be 

 always the case with water in this chapter), we may also pass 

 over its weight p, which is constant; and the pressure will be 

 expressed simply by H, and will be exclusively proportional to it. 



If we were rigorously to adhere to the principle, we should 

 regard H as the weight of the fluid filament which presses and 

 drives on in the conduit the molecule which is immediately 

 beneath it; and we should represent it by a line, as in elemen- 

 tary statics we represent by lines, forces which are also weights. 



Amount of the Resistanee — Fundamental Equation. 

 i. Since the resistance arises from the effect of the sides, it 

 will be proportional to their superficies— that is to say, to the 

 length of the conduit, and to the circumference of its section, 

 which IS here the wet perimeter; for we are supposing that the 

 flow is made in a full pipe, otherwise we should have the case of a 

 simple canal. In other words, the more the section is enlarged. 



the more also will the resistance of the sides be distributed among 

 a greater number of mcdecules; consequently, it will less affect 

 each of them and the total mass; it will be in inverse ratio to 

 their number, and consequently to the magnitude of the section. 

 In short, here, as in canals, it will be proportional to the square of 

 the velocity plus a fraction of the simple velocity. 



Then, if L be the length of the conduit, S its section, C the wet 

 contour or perimeter, a and 6 two constant coefficients, the expres- 

 sion of the resistance will be 



and we shall have 



CL 



H- - = a^~{v' + bv) (1.) 



5. It remains to determine the coefficients a and b. M. Prony, 

 who was tlie first to undertake this task in an adequate manner, 

 makes use, for the purpose, of fifty-one experiments made by our 

 most able hydraulicians, and which Du Buat had before employed 

 in the establishment of his formula?. He has deduced therefrom, 



a = -OOOS+SS; b = -0498; 



or, in tlie value of English feet, a =: -0001062; b = •'i6339. 



Of these fifty-one experiments, eighteen had been made by Du Buat him- 

 self, upon a tin pipe, of 1063 inches diameter and 65 6 feet long; twenty, 

 six had been made by Uossut, on tubes also of tin, 1-06 inches, 1-42 inches, 

 and 213 inches diameter, and whose lengths varied from 31'95 feet to 192 

 feet ; lastly, seven had been made on the large conduits in the park at Ver- 

 sailles, one was 53 inches diameter and 7478 feet long, and another 19'3 

 inclies diameter and 3834 feet long. 



Twelve years afterwards, Eytelwein treated anew the question 

 of the motion of running waters; he has thought it right to take 

 into consideration the contraction of the vein at the entrance of 

 the conduit, and m being the coefficient for this contraction, he 

 determined (the measures being in metres). 



Or in English feet, 

 H 



V- CL , 



H — , = -0002803 -=- (v- + 



Qgxm- S ^ 



•081 v)" 



CL 



.(II.) 



^gXni- 



— -0000854 -— («f +-275G v) 



But VI, whose effect, besides being imperceptible in large con- 

 duits, is included in the value of «, given by the experiments. 

 Consequently, and paying regard only to the most exact observa- 

 tions, and especially to those of Couplet, I shall adopt the equa- 

 tion. 



[In metres] H 



II- 



[In Eng. feet] H = 



For canals, the equation is, 



-0003425 - - {v- + 



CL , , 

 -0001044 -K- (u- -f-18045 



5 1') 3 



..(III.) 



[In metres] 



H - 



25 



-066 )') 



CL , „ 



-0003655 — - {v- + 



CL 



•OOOllU — {v~ -\--21654^ V 



>i 



.(IV.) 



[In Eng. feetj H - ^^ = 



These two equations are similar and very nearly the same, as should be 

 the case. The slight differences in the numerical coefficients probably arise 

 from errors in the observations. If this be so, as the observations are capa- 

 ble of being made with much greater exactness upon conduits than upon 

 canals or rivers, it may he presumed that the coefficients of the equations for 

 conduits are the more correct. 



6. The section of pipes being a circle, if D represent the dia- 

 meter, we shall have S = n'D', and C — irD; and by putting for 

 IT, w', and g, their numerical values,* the fundamental equation 

 for the motion of water in conduit pipes will become. 



[In metres] II 

 [In feet] H - 



-051 V- = -00137 Yj («^ + 



•055 v) 



.0155 11= =-0004176 — {v' -f--18045 v) ' 



.(V.) 



The velocity is very rarely among the quantities given or 

 required in the problems to be resolved; the discharge is the 



* Tr = 3-1416j ii'=7SS4. 

 •J- =-061 (in me res). — =-0165 {in English feet). 



