18S0.T 



THE CIVIL ENGINEER AND ARCHITECrS JOURNAL. 



875 



diminutions as in the preceding numher, and by reason of the 

 same causes; their summits will only reach the line E(i, which will 

 be their limit (they would be limited by EB, if the conduit were 

 quite open); consequently, the pressure upon any one jioint of 

 which H„ is the depression below the reservoir, will be expressed 

 by H„ — r — -0155 «'-. 



Fig. 6. 



The expression will be the same for an undulating conduit as 

 AH' r K' B; only tlie summits of the columns will no longer be in 

 a right line; the resistances, being proportional to the length of 

 the pipes, will strictly fullovv tlie ratios of AH', AT; but not those 

 of E c, E e' ; a condition necessary in order that the points A, a, a', 

 may be in a right line. 



Total Head; and Effective Head. 



30. We have called the head of a conduit, and designated by H, 

 the difference of level between the surface of tlie fluid in the reser- 

 voir, and the orifice of discharge; it would be the height due to 

 the velocity of discharge, if tlie pipes offered no resistance to the 

 motion. But the resistance diminishes this entire head, so that the 

 effective head o{ the conduit, or the height by virtue of which in 

 reality the fluid runs out, will be less according to the resistance 

 which it will have met with, from the beginning to the extreme 

 end of the pipes; R being that resistance, the effective head will 

 be II — R. 



By analogy, for every point of the conduit, its total head will be 

 the height of the reservoir H^ above it; and its effective head will 

 be this same height diminished by the resistance experienced by 

 fluid from the origin of the conduit to it, or H„ — r. 



Difference between the Head and the Pressure. 



31. Since the pressure upon this same point is Kg — r — '0155 vr, 

 •0155r^ ivill be the difference between it and the head. In general, 

 the height due to the velocity of the water at any point of a con- 

 duit, is the difference which exists there between the effective head, 

 or the head properly so called, and the pressure upon this point. It 

 is not correct to take one for the other; but in large conduits, 

 where the height due to the velocity is very small, the error is 

 seldom or never of consequence. 



Of the PietomHre and its Indications. 



32. The guage tubes which we have supposed to he placed on 

 the conduits, (and which, by the height to which the fluid is raised 

 in them, measure the pressure which exists at the points to which 

 they are fitted,) have been named in France, Piezometres {piesis, 

 pie-seos, pressure; and metron, a measure). 



They serve to give us as it were a physical representation of what is un- 

 derstood by resistance and loss of head. Let us suppose that we have fitted 

 one upon a given point of a conduit situated at H^ below the level of tlie 

 reservoir. According to what has just been said, if the water were in repose 

 in the conduit, it would rise in the tuhe to the height H^ ; when the water 

 is running it will sink, and remain at the height H„ —r — />, h hfing the ele- 

 vation due to the velocity v. The depression, or tlie ditference hetween the 

 two heights will he then, H„ — H„ + r + A ; and in designating it by x, we 

 shall have ,r = r + A ; or, r = x — h ; that is to say, the resistance experienced 

 by the water, from the oriijin of the conduit, to any point of its length, will 

 be represented by the difference of level between ttie surface of the reservoir 

 and the summit of the fnid column in a piezometer fitted upon this point 

 {niinus the height due to the velocity in the conduit, a quantity always very 

 small), if we augment or diQiini:sh the volume of water running in the 

 conduit, and consequently its velocity, hy enlarging or contracting the orifice 

 of discharge, the fluid in the piezometer will hedmie lower or higher, in a 

 slight degree, very nearly proportional to the square of this volume or 

 velocity. 



J The depression ought to be Q" j 



00068 



L 



•0000961. 1 



1)^ J4 / Ua J 



and we shall have to compare the results of theory with those of experi- 

 mcut. 



For a second point of the conduit, taken, for example, lower down stream 

 than the first, we should have in like manner r' = jr' — A, since the velocity 

 of the height due to it. A, remain the same throughout the conduit. Cut- 

 ting off from this equation the first r = x — h, we have r' — r = x' — x. Now, 

 r' — r, the difl'erence between the two resistances, is evidently the resistance 

 met with from the first point to the second; and x' — x, the difference he- 

 tween the depression of the two piezometrical columns helow the reservoir, 

 will he the difference of level hetween the summits of the two columns ; 

 thus, the resistance uhieh the water meets with from one point of a conduit 

 to another, or the loss of head from the first to the second, is indicated by 

 the difference of level between the summits of the fluid columns of two 

 piezometers, fixed one on each of the two points. If the diameter of the 

 pipe on which the second piezometer is fixed were different to the first, 

 then the height h' due to its velocity would no longer be equal to h, and ne 

 should have — 



r' — r' = x' — h' — {x — h) = {x' — x) — {h' — h); 

 that is, the resistance from one point to another would be measured by the 

 difference of level between the two piezometrical summits, minus or plus 

 the difference between the two heights due to the respective velocities, ac- 

 cording as the velocity at the point down stream should be greater or less 

 than the other. 



We see, by these examples, how the piezometer renders perfectly clear 

 the resistance in pipes, and the variations which take place in them; and, 

 consequently, how useful its indications may be. I have such an instrument, 

 made of glass, fixed on one of the conduits of Toulouse, and carried into 

 the office of the engineer of these works; and it indicates to him constantly 

 tlie state of the water, and the disturbances it meets with. 



Thickness required for Conduit Pipes. 



33. [Under this head, D'Aubuisson discusses the theory of 

 the force of pressure tending to burst a conduit pipe; and then 

 from the results of experiment on the cohesive strength of cast- 

 iron, deduces the thickness of pipe necessary for various diameters 

 to withstand this pressure. Adding to this theoretical value, a 

 margin to allow for the sudden shocks to which conduit pipes are 

 liable, and for the imperfections usual in castings, the formulie 

 which he finally submits and has adopted for practice is, calling 

 thickness of pipe e, 



e in inches =^ "39+ in. -|- •015 dia. in inches. 



For pipes of less than 4^ inches diameter, he considers there is no 

 necessity to add the second term of the equation, but makes them 

 all about |-inch in thickness. 



This formulae is for pipes proved to ten atmospheres, or about 

 300 feet head.] 



Art. IV. — Of Systems of Conhuits. 



It is rare in practice that we have to deal with a simple conduit, 

 conveying to its extreme end all the water it receives at its origin. 

 A portion of this is generally carried off, at vai-ious points, by 

 secondary conduits; from these again branch pipes of a third 

 order, so that a large distribution of water in a city or town, pre- 

 sents as it v/ei-e a trunk branched and sub-branched in every 

 direction. 



3+. To determine the circumstances of the motion of water in 

 the different parts of such a system, and that by the knowledge 

 alone of the dimensions and respective position of the several 

 parts, is a complicated problem, of which a solution has not yet 

 been given; and yet the calculations which an engineer has to 

 make relate generally to a system, and not to an isolated conduit. 



To form an idea of the basis on which 1 have established the 

 solution that I am about to give, and which is applicable to at least 

 some cases, let us suppose a system already existing, adapted to a 

 reservoir maintained constantly full, and discharging water througli 

 mouths at the end of various branches. Let the question to be 

 determined be, for instance, the quantity of water flowing out of 

 each mouth (although that is not the object we have now in view), 

 it is evident that we could immediately calculate this quantity if 

 we knew the effective head of water at the end of the branch, that 

 head of water being the height due to the velocity of discharge 

 (30). But after what has been said (30—32) the effective head is 

 the entire head, minus the loss of head or resistance that the fluid 

 has experienced in its passage through the system from the reser- 

 voir to this mouth; so that this problem is reduced to the 

 determination of the amount of the several losses of head. 



Of the Several Losses of Head. 



35. These arise, 1st, and almost solely, from the frictional resist- 

 ance of the sides of the pipe. 2ndly. From the resistance due to 

 the bends. 3rdly. From the change of direction in the movement 



