General Mean Values of Constants.
The general formula (Hagen’s)—h/l = mvn/dx·2g—can therefore be taken to fit the results with convenient closeness, if the following mean values of the coefficients are taken, the unit being a metre:—
| Kind of Pipe. | m | x | n |
| Tin plate | .0169 | 1.10 | 1.72 |
| Wrought iron | .0131 | 1.21 | 1.75 |
| Asphalted iron | .0183 | 1.127 | 1.85 |
| Riveted wrought iron | .0140 | 1.390 | 1.87 |
| New cast iron | .0166 | 1.168 | 1.95 |
| Cleaned cast iron | .0199 | 1.168 | 2.0 |
| Incrusted cast iron | .0364 | 1.160 | 2.0 |
The variation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known.
It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:—
| Kind of Pipe. | m | x | n |
| Tin plate | .0265 | 1.10 | 1.72 |
| Wrought iron | .0226 | 1.21 | 1.75 |
| Asphalted iron | .0254 | 1.127 | 1.85 |
| Riveted wrought iron | .0260 | 1.390 | 1.87 |
| New cast iron | .0215 | 1.168 | 1.95 |
| Cleaned cast iron | .0243 | 1.168 | 2.0 |
| Incrusted cast iron | .0440 | 1.160 | 2.0 |
§ 78. Distribution of Velocity in the Cross Section of a Pipe.—Darcy made experiments with a Pitot tube in 1850 on the velocity at different points in the cross section of a pipe. He deduced the relation
where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (Mém. de l’Académie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b = Ri/U2, where U is the mean velocity in the pipe; then V/U = 1 + 9.03 √b. A very useful result for practical purposes is that at 0.74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin.
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| Fig. 84. |
§ 79. Influence of Temperature on the Flow through Pipes.—Very careful experiments on the flow through a pipe 0.1236 ft. in diameter and 25 ft. long, with water at different temperatures, have been made by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 112 in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v1.795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h = ζ (4L/D) (v2/2g), then for heads varying from 1 ft. to nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft. per second, the values of ζ were as follows:—
| Temp. F. | ζ | Temp. F. | ζ |
| 57 | .0044 to .0052 | 100 | .0039 to .0042 |
| 70 | .0042 to .0045 | 110 | .0037 to .0041 |
| 80 | .0041 to .0045 | 120 | .0037 to .0041 |
| 90 | .0040 to .0045 | 130 | .0035 to .0039 |
| 160 | .0035 to .0038 |
This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynolds’s equation is assumed h = mLVn/d3−n, and n is taken 1.795, then values of m at each temperature are practically constant—
| Temp. F. | m. | Temp. F. | m. |
| 57 | 0.000276 | 100 | 0.000244 |
| 70 | 0.000263 | 110 | 0.000235 |
| 80 | 0.000257 | 120 | 0.000229 |
| 90 | 0.000250 | 130 | 0.000225 |
| 160 | 0.000206 |
where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different temperatures Professor W. C. Unwin found that the resistance was proportional to constant × (1 − 0.0021t) and the values of m given above are expressed almost exactly by the relation
In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3% of resistance for 10° F. increase of temperature.
§ 80. Influence of Deposits in Pipes on the Discharge. Scraping Water Mains.—The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. In pipes which convey certain kinds of water, oxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in. and 8-in. pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to 51% of the original discharge. J. G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see “Incrustation of Iron Pipes,” by W. Ingham, Proc. Inst. Mech. Eng., 1899). It was found that by scraping the interior of the pipe the discharge was increased 56%. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly to 10% and afterwards more slowly, the diminution in a year being about 25%.
Fig. 85 shows a scraper for water mains, similar to Appold’s but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or withdrawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the main. At Lancaster after twice scraping the discharge was increased 5612%, at Oswestry 5412%. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation. The scraper can be easily followed when the mains are about 3 ft. deep by the noise it makes. The average speed of the scraper at Torquay is 213 m. per hour. At Torquay 49% of the deposit is iron rust, the rest being silica, lime and organic matter.
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| Fig. 85. Scale 125. |
In the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another character which has led to trouble in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to be an
