1850 ] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



103 



The Rexlslance of Brnds and Angles. 



16. Every moving boily, which after having followed a certain 

 direction suddenly changes therefrom, loses a portion of its velo- 

 ritv, represented by the versed sine of the angle formed by the 

 two directions. If it moves in a curved line, it is continually 

 changing its direction; Iiut tlie loss of velocity at each change is 

 onlv an infinitely small one of tlie second order; and consequently, 

 although the number of losses be infinite, the total loss will be 

 only an infinitely small one of the first order, or as nothing: in 

 other words, every moving body which arrives tangentially at a 

 curve and follows it for some length, possesses on quitting it the 

 same velocity it had on its arrival. It follows, that if a bend in a 

 conduit be well formed, and the fluid therein should exactly 

 follow the curve, it would suffer no resistance or loss of velocity. 



But this is not the case: the molecules composing the fluid cur- 

 rent being independent of each other, while those which are in 

 contact with the sides would follow the curvature, the others, 

 beino' directed against the sides, will be reflected by them, or by 

 the intervening particles, at an angle which is sometimes very 

 considerable. For example, the central fillet aC has a tendency to 



strike the side ACB at C, 

 and from thence to be re- 

 flected in the direction Ci. 

 Tlie mutual action of the 

 particles on one another, 

 will produce, in the whole, 

 a loss of velocity; it will 

 be, generally, less than 

 that of the central stream taken alone, but always greater than 

 that of the current bordering on the sides. 



This diminution of velocity, and consequently of discharge, 

 although certain, will yet be very slight. Thus, Bossut, with a 

 pipe of I'OU inch diameter, and oiij feet long, laid horizontally in 

 H straight line, and with a head of 1'07 feet, ol)taiiied a discharge 

 of •7360 cubic feet per minute: then having bent it in a serpentine 

 form, so as to have six well rounded curves, all else remaining tlie 

 same, he obtained '7205 cubic feet per minute. 



We may, however, by multiplying and increasing the acuteness 

 of the bends, render the diminution of discharge very consider- 

 able. Rennie, with a lead pipe, 15 feet long, and j-inch diame- 

 ter, fitted horizontally to a reservoir under a head of 4 feet, 

 obtained a discharge of "419 cubic feet per minute; then having 

 given the same Jiipe fifteen semicircular bends, of 3^ in. radius, and 

 again fitted it to ttie reservoir, the discharge was not moi-e tlian 

 •370 cubic feet: so that the fifteen bends reduced the discharge in 

 the ratio of 100 to 89; with a quadruple head, the reduction was 

 in the ratio of 100 to 88. 



17. 'With regard to the laws regulating the resistance of bends, 

 and to its amount, we are indebted to Dubuat for the first well- 

 observed facts. He has taken various pipes, at first rectilinear, 

 and measured the head necessary to obtain from them a certain 

 volume of water in a certain time: he has then bent them in dif- 

 ferent forms, and in such a manner that the central current had a 

 tendency to be reflected at angles of determined number and 

 acuteness, and again examined the head with which they dis- 

 charged an equal volume of water in an equal time. The differ- 

 ence between the two heads, for the same pipe, at one time recti- 

 linear and at another bent, was evidently the head due to the 

 bends, and consequently the measure of their resistance. The 

 principal of twenty-five experiments, which he has thus made, are 

 given in the following table: — 



Dimeter. length. Nun,be''rt'v 



Inches. 



1-07 



ro/ 



1-07 

 107 

 1^07 

 1-07 

 107 

 107 

 2-13 

 213 



213 



Feet. 



10-39 

 10-39 

 10-39 

 10-39 

 10-39 

 1240 

 12-40 

 6546 

 22-66 

 22-66 



22-66 



1 of 36° 



2 of 36 



3 of 36 



4 of 24-57 

 10 of 36 



4 of 36 



4 of 36 



4 of 36 



4 of 36 



4 of 36 

 f 6 of 24-57] 

 { 5 of 36 



1 of 56-23 



Velocity Resistance 



per due to 



Second, the Bends. 



Feet. 

 7^55 

 7-55 

 7-55 

 7-55 

 6-36 

 5-16 

 2-60 

 2-54 

 7-66 

 5-22 



7-66 



Feet. 

 •067 

 •133 

 •221 

 •133 

 •524 

 •146 

 •036 

 ■035 

 •257 

 •118 



•767 



Coefficient 

 Ueduted. 



•0034 

 •0034 

 •0037 

 •0034 

 •0037 

 •0039 

 ■0039 

 •0039 

 ■0030 

 ■0031 



•0038 



Dubuat concludes from his experiments, that the resistance arising 

 from bends is proportional to the square of the velocity of the 

 fluid, to the number of angles of reflection, and to the square of 

 their sines. 



In this hypothesis the coefficient varies but little, and its mean 

 value is -0037. So that if v be the velocity; n, n, &c. the number 

 of angles of reflection of equal measure; ?, i', &c. the respective 

 measures of the angles, the resistance will be 



\ -0037 ?!-(« sin- j + n siri' i' + ); 



or in function of Q, and taking s for the sum of the squares of all 



Q-' 



the sines, •0061 ^, X s-. 



18. In the application of this formula to any given conduit, we must 

 determine the number and value of tlie angles of 

 reflection for each bend. Now, a simple diagram 

 shows, 1st, that in a pipe bent to an arc of a circle 

 (and no other curves need he admitted in practice), 

 half the diameter of the pipe divided by the radius 

 of the are, will give the versed sine of the angle of 

 reflection, and we may consequently cet its cosine 

 and value in degrees; 2ndly, that the number of 

 degrees in the arc (i. e. the supplement of the angle 

 of the curve), divided hy twice the angle of reflec- 

 tion, will indicate the numher of angles. 



Let us, for ex.imple, take a conduit pipe, -82 feet 

 diameter, conveying T/O cuhic feet of water per 

 second, and which presents a bend of 95°, the radius 

 of the curve being 6-89 feet: what will be the re- 

 sistance occasioned by this bend .* 



According to the rule laid down, the versed sine 

 of the angle of reflection will be ■0595 {= i^y'^), and its cosine ^9405 

 ( = 1 — -0595), the cosine of an angle of 19° 52'; this is the angle of reflec- 

 tion. The arc of curvature 85° ( = 180°-95°) divided by 39 73° (twice the 

 angle of reflection), will give their number; this we shall take as 3, the quo- 

 tient being 2-14. The sine of 19° 52' is •3398, and its square ^1155 : the 

 resistance sought will therefore be 



•00608 — — ^x3x^ll55 = ^0144feet; 



(-82)^ 



a quantity extremely small, although the curve was tolerably acute and the 

 velocity consideiable. For the pipe with fifteen bends, in Rennie's experi- 

 ment, the above method of calculation would give a resistance of 633 feet : 

 the experiment itself, as we shall shortly see, gave MGfeet, which would 

 raise the coefficient of Dubuat from -00608 to ■01113. But such a case as 

 this seldom occurs in practice; nor does the value of the resistance, even if 

 we double the coefficient, often amount to an inch loss of head. 



We may neglect account of the value of this resistance in curves of great 

 radius; tlie angles of reflection, it is true, will he greater, hut not so strong; 

 and the sum of the squares of the sines, and consequently the resistance, 

 will be less. 



19. If the effect of well curved bends is imperceptible, it is not so with 

 angles, properly so called. An experiment of Venturi shows their influence: 

 this savan had three tubes made, r25 feet in length, and 13 inch diameter; 

 one was rectilinear, the second had a bend of 9U° well curved, and the third 

 had an acute angle, also of 90°: under a head of 2-88 feet, they filled a 

 vessel containing 4-84 cuhic feet, respectively in 45", 50", and 70". The bad 

 effect of aniiles is shown still more plainly in the experiments of Rennie : 

 with his pipe 15 feet lung, J-inch diameter, and with a bead of 4 feet, he 

 obtained, per minute, a discharge 



With the rectilinear pipe ^4 19 cubic feet. 



With the fifteen semicircular bends .. ^370 „ 



With 1 right angle ^333 „ 



With 24 right angles ^152 „ 



so that one angle of 90° reduced the discharge more than 15 considerable 

 bends. This fact alone shows with what care all angles should be avoided in 

 the establishment of conduit pipes. 



In seeking the heads which made the three pipes with bends or angles 

 give a discharge (•419 cubic feet) equal to that which was obtained when 

 there was neither angle nor bend, we find them respectively 5-15, 6-33, and 

 30-52 feet. Deducting 4 feet, there remains for the resistance arising from 

 the bends and angles (17) 1.15, 2-33, and 26-54 feet. From which we con- 

 clude that the resistance from a single angle of 90°, was more than double 

 that of fifteen bends; and that of twenty-four angles was only 11-4 times 

 greater than that of a single one. This last result also shows that the 

 resistance of angles and bends is not proportional to their number, as Dubuat 

 had remarked. 1 had also observed a like want of proportion in m-j Experi- 

 ments an the motion of Air in Conduit Pipes ('Annales des Mines,' 1828, 

 p. 453). 



Resistance arising from Contractions. 



20. Contractions, of which we are about to treat, are occasioned 

 by a diminution of the section of the couduit for a very short 

 length. 



23* 



