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64
[STEADY FLOW IN PIPES
HYDRAULICS


organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see “ Deposits in Pipes, ” by Professor ]. C. Campbell Brown, Proc. Inst. Civ. Eng., 1903-1904).

§ 81. Flow of Water through Fire Hose.-The hose pipes used for fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by J. R. Freeman' (Arn. Soc. Civ. Eng. xxi., 1889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. In reducing the results the true diameter has been taken. Let v=mean velocity in ft. per sec.; r=hydraulic mean radius or one-fourth the diameter in feet; hydraulic gradient. Then t'=n/ (ri).

relative velocity of the two parts Diangleter @31g§ ;1§ 1

tates 1 1) 71.

Inches. per min 4

°

Solid rubber 5 2-65 215 -1863 12-50 123-3 hose .» 344 -4714 20°OO' 124-0

Woven cotton, 2-47 200 2464 I3-40' 119-1 rubber lined i, , 299 5269 2O°O0 121-5 Woven cotton, 2'49 200 2427 13-20 117-7 rubber lined i, , 319 5708 21-oo 122-1 Knit cotton, g 2-68 132 0809 7-50 111-6 rubber lined 2, , 299 3931 I7-O0 114-8 |' Knit cotton, 5 2-69 204 2357 11-50 100-1 3 rubber lined l, , 319 5165 I8~OO l05~8 Woven cotton, < 2-12 154 3448 I 4°0O 113-4 rubber lined 2, , 240 7673 21-81 118-4 Woven cotton, 2-53 54-8 0261 3-50 94-3 rubber lined i, , 298 8264 I9-OO 9I~O Unlined linen { 2-60 57-9 0414 3-50 73-9 hose, , 4 331 1624 '20°O0 79-6 § 82. Reduction of a Long Pipe of Varying Diameter to an Equivalent Pipe of Uniform Diameter. Dupuit's Equation.-Water mains for the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calculations may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main. - -:

f<- —» -[1 —~ ->i~= —»»-~ lg ~—~ ~—1-l<~ —- lg- -~—~ as u . I

- .

A 2:11 tl; I:ALI 'Y;

1 ¢

<-~~- - —~ Z- -.- . -»—e-91

B if .1

FIG. 86.

In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let l, , 12... be the lengths,

di, dz... the diameters,

vi, vz... the velocities,

il, iQ... the slopes,

for the successive portions, and let l, d, v and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is

It = 1111 . . .

= I(U12'4l1/2£d1) +s“(f12”'4l2/23112) + ~ -and in the uniform main

il = tt-2-41/21-1>.

If the mains are equivalent, as defined above, s“(v”°4l/2gd) = i“(vF~4J1/21111)+s“(v2”~4li/2gd2)+ . .-But, since the discharge is the same for all portions, § 1rd'2v = %1rd127J1 = 341rd2'27J2 = . . . 171 = Udg/diz; v2=vd'/df . . .

Ziifo suppose that of may be treated as constant for all the pipes. ell

I/fi = (df/dif) (ll/dl) 'l' Q14/1124) (12/112) +—l- <d°/li1'°)l1+<d5/(l2°)l2+...

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge. § 83. Other Losses of Head in Pipesf-Most of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted. Sudden Enlargement of Section.-Suppose a pipe enlarges in section from an area wo to an area wi (fig. 87); then A

111/1/o==wo/wi? E

or, if the section is circular,

W1/'Un ='= (do/'diy:

The head lost at the abrupt change tof velocity has already been

shown to be the head due to the

  • v d

tl,

of the stream. Hence head lost FIG' 87 fit = (Uv -1102/22 = (w1/ wo- IW?/23 = {(fl1/do)”- 1}2”12/28 of f)¢=i'e7]12/Zgr (I)

if fe is put for the expression in brackets. w1/w0- 1.1 1.2 1.5 1 7 1.8 1.0 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0 lil/d0= 1.05 1.10 1.22 1.30 1.34 1.38 1.41 1.58 1.73 1.87 2.00 2.24 2.45 2.65 2.83 g-e= .01 .04 .25 .49 .64 .81 1.00 2.25 4.00 6.25 9.oo16 0025.:>o 36.o49.o Abrupt Contraction of Section.~When water passes from a larger to a smaller section, as in hgs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. 4 §

§ l.;f e;

L

FIG. 88. FIG. 89.

Let w be the section and 11 the velocity of the stream at bb. At aa the section will be ctw, and the velocity (w/c, w)v=v/cl, where eu is the coefficient of contraction. Then the head lost is fy... = (ff/C. -102/2g= (I/c.- 02112/2g; and, if cc is taken 0-64,

b, ,, =0-316112/2g. (2)

The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by friction. For water entering a cylindrical, not bell-mouthed, pipe from a reservoir of indehnitely large size, experiment gives D-=0~505 11”/21- (3)

If there is a diaphragm at the mouth of the pipe as in fig. 89, let wi be the area of this orifice. Then the area of the contracted stream is can, and the head lost is

be =l(w/¢¢w1)" I l”f/2/28

=s“.1»'/2g (4)

if § ', is put for {(w/c¢w1)-IP. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice:- 6¢= .616 .614 .612 .610 .617 .605 .603 .GOI .SQ8 .596 = 231.7 50.99 19.78 9.612 5.256 3.077 1.876 1169 0.734 0.480 Vi.. ' H 4 .4 .., 4 4 4 4 4

When a diaphragm was placed in a tube of uniform section (fig. 90) FIG. 90.

the following values were obtained, wi being the area of the orifice and w that of the pipe:-

C0160-

/ - 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9 1 0 C, = .624 .632 .643 .659 .681 .712 .755 .813 .892 I.O0

£c= 225.9 47.77 30.83 7.801 1.753 1.796 .797 .290 .060 .000