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 J. 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 (Am. 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; i = hydraulic gradient. Then v = n √(ri).
| Diameter in Inches. | Gallons (United States) per min. | i | v | n | |
| Solid rubber hose | 2.65 | 215 | .1863 | 12.50 | 123.3 |
| ” | 344 | .4714 | 20.00 | 124.0 | |
| Woven cotton, rubber lined | 2.47 | 200 | .2464 | 13.40 | 119.1 |
| ” | 299 | .5269 | 20.00 | 121.5 | |
| Woven cotton, rubber lined | 2.49 | 200 | .2427 | 13.20 | 117.7 |
| ” | 319 | .5708 | 21.00 | 122.1 | |
| Knit cotton, rubber lined | 2.68 | 132 | .0809 | 7.50 | 111.6 |
| ” | 299 | .3931 | 17.00 | 114.8 | |
| Knit cotton, rubber lined | 2.69 | 204 | .2357 | 11.50 | 100.1 |
| ” | 319 | .5165 | 18.00 | 105.8 | |
| Woven cotton, rubber lined | 2.12 | 154 | .3448 | 14.00 | 113.4 |
| ” | 240 | .7673 | 21.81 | 118.4 | |
| Woven cotton, rubber lined | 2.53 | 54.8 | .0261 | 3.50 | 94.3 |
| ” | 298 | .8264 | 19.00 | 91.0 | |
| Unlined linen hose | 2.60 | 57.9 | .0414 | 3.50 | 73.9 |
| ” | 331 | 1.1624 | 20.00 | 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.
 |
| 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
| l1, l2 | be the lengths, |
| d1, d2 | the diameters, |
| v1, v2 | the velocities, |
| i1, i2 | 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
| h = i1l1 + i2l2 + . . . |
and in the uniform main
If the mains are equivalent, as defined above,
But, since the discharge is the same for all portions,
| 14πd 2v = 14πd12v1 = 14πd22v2 = . . . |
Also suppose that ζ may be treated as constant for all the pipes. Then
| l/d = (d 4/d14) (l1/d1) + (d 4/d24) (l2/d2) + . . . |
which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge.
 |
| Fig. 87. |
§ 83. Other Losses of Head in Pipes.—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 ω0 to an area ω1 (fig. 87); then
or, if the section is circular,
The head lost at the abrupt change of velocity has already been shown to be the head due to the relative velocity of the two parts of the stream. Hence head lost
if ζe is put for the expression in brackets.
| ω1/ω0 = | 1.1 | 1.2 | 1.5 | 1.7 | 1.8 | 1.9 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | 5.0 | 6.0 | 7.0 | 8.0 |
| d1/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 |
| ζe = | .01 | .04 | .25 | .49 | .64 | .81 | 1.00 | 2.25 | 4.00 | 6.25 | 9.00 | 16.00 | 25.00 | 36.0 | 49.0 |
 |
 |
| Fig. 88. | Fig. 89. |
Abrupt Contraction of Section.—When water passes from a larger to a smaller section, as in figs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. Let ω be the section and v the velocity of the stream at bb. At aa the section will be ccω, and the velocity (ω/ccω) v = v/c1, where cc is the coefficient of contraction. Then the head lost is
and, if cc is taken 0.64,
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 indefinitely large size, experiment gives
If there is a diaphragm at the mouth of the pipe as in fig. 89, let ω1 be the area of this orifice. Then the area of the contracted stream is ccω1, and the head lost is
| ɧc = {(ω/ccω1) − 1}2 v2/2g |
if ζ, is put for {(ω/ccω1) − 1}2. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice:—
| ω1/ω = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
| cc = | .616 | .614 | .612 | .610 | .617 | .605 | .603 | .601 | .598 | .596 |
| ζc = | 231.7 | 50.99 | 19.78 | 9.612 | 5.256 | 3.077 | 1.876 | 1.169 | 0.734 | 0.480 |
 |
| Fig. 90. |
When a diaphragm was placed in a tube of uniform section (fig. 90) the following values were obtained, ω1 being the area of the orifice and ω that of the pipe:—
| ω1/ω = | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
| ce = | .624 | .632 | .643 | .659 | .681 | .712 | .755 | .813 | .892 | 1.00 |
| ξc = | 225.9 | 47.77 | 30.83 | 7.801 | 1.753 | 1.796 | .797 | .290 | .060 | .000 |
